Livro Composites

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COMPOSITE MATERIALS By: Dr. Mark V. Bower, P. E.  Copyright 1992-2000 The University of Alabama in Huntsville Huntsville, Alabama Composite Materials INTRODUCTION...........................................................................................................1 Introduction ................................................................................................................1 Definition of Terms.....................................................................................................1 CONSTITUENTS AND FABRICATION TECHNIQUES............................................8 Constituents................................................................................................................8 Reinforcement Forms ...............................................................................................17 Fabrication Techniques............................................................................................21 Hazards .....................................................................................................................42 LAMINA MECHANICS ...............................................................................................48 Preliminaries ............................................................................................................48 Constitutive Relations..............................................................................................53 Engineering Properties for Orthotropic Materials..................................................57 Plane Stress Orthotropic Constitutive Relation .....................................................58 Off-Axis properties of orthotropic lamina................................................................60 STRENGTH OF LAMINA ...........................................................................................65 Tensor Polynomial Failure Criterion.......................................................................65 Quadratic Failure Criterion.....................................................................................66 R-Factor analysis......................................................................................................73 CLASSICAL LAMINATION THEORY .......................................................................75 History ......................................................................................................................75 Preliminaries ............................................................................................................76 Force -- Moment Resultants.....................................................................................77 Equilibrium of a Plate Element ...............................................................................79 Displacement Field Model........................................................................................83 ii Orthotropic Constitutive Relation ...........................................................................86 The Laminated Plate Equations ..............................................................................91 LAMINATES ................................................................................................................97 Introduction ..............................................................................................................97 Alternate Expressions for Laminate Stiffnesses.....................................................97 Simplifying Assumptions on Laminate Structure ..................................................99 Stress Distribution in a Laminate .........................................................................112 Laminate Failure Theories ....................................................................................113 iii Preface This document is a work in progress, as are most books. Every effort has been made to ensure the accuracy of the information contained in it. That does not, unfortunately, guarantee that every equation is without error. Having said this, the reader is encouraged to consult other texts on composite materials. A few are listed on this page. Further, neither the author nor The University of Alabama in Huntsville is responsible for the application of the information contained in this document. Good engineering practice requires the application of sound engineering judgment. The author acknowledges the support of The University of Alabama in Huntsville, the Microsoft Academic Support Program, and the Dell Corporation Academic Support Program. I acknowledge the support and assistance of my wife, Peggy, and children, Renae, Amber, Elizabeth, and Matthew. Further, I acknowledge the support, inspiration, and anointing of Jesus Christ. Without His help, I could not have come this far. References: 1. Mechanics of Composite Materials, R. M. Jones, McGraw-Hill Book Company, Washington, D. C., 1975. 2. Primer on Composite Material: Analysis, J. E. Ashton, J. C. Halpin, and P. H. Petit, Technomic Publishing Co., Inc., Westport, CT, 1969. 3. Introduction to Composite Materials, S. W. Tsai and H. T. Hahn, Technomic Publishing Co., Inc., Westport, CT, 1980. 4. Fundamentals of Composites Manufacturing: Materials, Methods and Applications, A. B. Strong, Society of Manufacturing Engineers Dearborn, MI, 1989. iv INTRODUCTION INTRODUCTION A composite material is defined as a material composed of two or more constituents combined on a macroscopic scale by mechanical and chemical bonds. Typical composite materials are composed of inclusions suspended in a matrix. The constituents retain their identities in the composite. Normally the components can be physically identified and there is an interface between them. Composite materials are classified based on the shape and relative dimensions of the inclusion and the structures. Composite materials are classified as: • Particulate • Filamentary • Laminated In a particulate composite, the major dimension of the inclusion is small compared to the structural dimensions. Particulate composites may be made with small particles, such as glass beads, or with chopped fibers. In filamentary composites, one dimension of the inclusion is of the same order of magnitude as the structural dimensions. Filamentary composite materials may be made from uni-directional tape or cloth. In laminated composite materials, two of the major dimensions of the inclusions are of the same order of magnitude as the structural directions. Sandwich sections are examples of a laminated composite material. Two additional distinctions are made in the classification of composite materials: advanced composite materials are those composites which are made with inclusions that have a modulus greater than that of steel (30 Mpsi, 207 GPa) and volume fraction of inclusions greater than fifty percent, and hybrid composite materials are those composites which are made with two or more different inclusion materials. DEFINITION OF TERMS ANGLEPLY LAMINATE Containing plies alternately oriented at plus and minus a fixed angle other than 90 degrees to the reference direction. Revised: 10 February, 2000 Page 1 COMPOSITE MATERIALS INTRODUCTION 2 ANISOTROPIC Not isotropic; exhibiting different properties when tested along axes in different directions. AUTOCLAVE A pressurized heated chamber used to cure composite materials. autoclave is pressurized with gas, typically air or nitrogen. An BALANCED LAMINATE A composite laminate whose lay-up is symmetrical with relation to the midplane of the laminate. BLEEDER CLOTH A nonstructural layer of material used in manufacture of composite parts to allow the escape of excess gas and resin during cure. B-STAGE An intermediate stage in the polymerization reaction of certain thermosetting resins; the state in which most prepregs are stored and shipped. CAUL PLATE A smooth metal plate used in contact with the lay-up during cure to transmit normal pressure and to provide a smooth surface to the finished laminate. COLLIMATED Rendered parallel, applies to filaments. COUPLING AGENT That part of a sizing or finish, which is designed to provide a bonding link between the reinforcement and the laminating resin. CRAZING Fine resin cracks at or under the surface of a plastic. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS INTRODUCTION 3 CROSSPLY LAMINATE Containing lamina alternately oriented at 0 degrees and 90 degrees. CURE To irreversibly change the properties of a thermosetting resin by chemical reaction, i.e., condensation, ring, closure, or addition. Cure may be accomplished by addition of curing (cross-linking) agents, with or without heat. DELAMINATION The separation of the layers of material in a laminate. DRAPE The ability of broadgoods to conform to an irregular shape. ELONGATION The amount of deformation of the fiber caused by the breaking tensile force, expressed as the percentage of the original length. FIBER PLACEMENT An automated fabrication process in which the machine places fiber bundles along predetermined paths to build up the structure. FILAMENT A long, continuous length of fiber, measured in yards. FILAMENT WINDING An automated fabrication process typically used to produce cylindrical or spherical shape. The machine winds fiber bundles onto a mandrel that is removed after the cure process. FILL Yarn running from selvage to selvage at right angles to the warp in a woven fabric. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS INTRODUCTION 4 FILLER A relatively inert material added to a plastic to modify its strength, permanence, working properties, or other qualities, or to lower costs. FINISH A mixture of materials for treating glass fibers. It contains a coupling agent to improve the bond of resin to glass; and usually includes a lubricant to prevent abrasion and a binder to promote strand integrity. With graphite or other filaments, it may perform either or all of the above functions. FLASH Excess plastic material which forms at the parting line of a mold or which is extruded from a closed mold. GEL COAT A quick-setting resin used in molding processes to provide an improved surface for composites; it is the first resin applied to the mold after the moldrelease agent. HAND LAY-UP The process of placing and working successive plies of the reinforcing material or resin impregnated reinforcement in position on a mold by hand. HYBRID COMPOSITE A composite structure composed of more than two different materials, for example, a laminate with outer laminae of glass/epoxy and inner laminae of graphite/epoxy. HYDROCLAVE Similar to an autoclave except that the chamber is pressurized using heated water or other liquid. INTERLAMINAR SHEAR The shear strength at rupture in which the plane of fracture is located between the layers of reinforcement of laminate. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS INTRODUCTION 5 ISOTROPIC Having uniform properties in all directions. The measured properties of an isotropic material are independent of the axis of testing. LAMINA A single layer or ply of material. laminate. The fundamental building block of a LAY-UP A laminate that has been assembled, but not cured; or a description of the component materials and geometry of a laminate. NON-WOVEN FABRIC A fabric, usually resin-impregnated, in which the reinforcements are continuous and unidirectional; layers may be crossplied. ORTHOTROPIC Having three mutually perpendicular planes of elastic symmetry. PARALLEL LAMINATE A laminate of woven fabric in which the plies are aligned in the same position as originally aligned in the fabric roll. PLASTICIZER For epoxy, a lower molecular weight material added to reduce stiffness and brittleness; it results in a lower glass- transition temperature for the polymer. PULTRUSION A fabrication process used to produce a highly collimated composite shape (rod, bar, etc.). POSTCURE Additional elevated temperature cure, usually without pressure, to improve final properties and/or complete the cure. In certain resins, complete cure and ultimate mechanical properties are attained only by exposure of the cured resin to higher temperatures than those of curing. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS INTRODUCTION 6 POT LIFE The length of time that a resin system retains viscosity low enough to be used in processing. PREPREG; PREIMPREGNATED A combination of mat, fabric, non-woven material, or roving with resin, usually in the B-stage, ready for molding. QUASI-ISOTROPIC LAMINATE A laminate approximating isotropy by orienting plies in several directions. RESIN TRANSFER MOLDING (RTM) A manufacturing process used to produce large composite structures. In this process, a dry lay-up is infused with resin in a molding process. May be found in various forms such as vacuum assisted resin transfer molding (VARTM) or Seeman’ Composite Resin Infusion Molding Process (SCRIMP). Not in wide spread use for advanced composites. ROVING A multiplicity of single ends of continuous filament with no applied twist drawn together as parallel strands. STACKING SEQUENCE The sequence of angles and possibly materials that describes the orientation of the individual lamina in a laminate from top to bottom, e. g., +45/-45/+45/-45, or 0/90/90/0, or 0/+60/-60/0/+60/-60/-60/+60/0/-60/+60/0. Most laminates are composed of a large number of laminae, frequently in repeated patterns, which leads to the use of shorthand notation. Using shorthand notation the first example is written: 2[±45]. The second and third sequences are symmetric about the mid-plane, and thus can be written: [0/90]S and 2[0/±60]S, where the subscript S indicates symmetry. SIZING On glass fibers, the compounds which, when applied to filaments at forming, provide a loose bond between the filaments, and provide various desired handling and processing properties. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS INTRODUCTION 7 SYMMETRIC LAMINATE A laminate that is symmetric in both geometry and material properties about the mid-plane. TACK With prepreg materials, the degree of stickiness of the resin. TAPE LAYING An automated process in which the machine lays a composite tape, either wet or prepreg, on predetermined paths. TOW A loose, untwisted bundle of filaments. TRANSVERSELY ISOTROPIC Having uniform properties in one plane. The measured properties of a transversely isotropic material are independent of the axis of testing within the plane. UNI-DIRECTIONAL LAMINATE A laminate with non-woven reinforcements that are all laid up in the same direction. WARP The yarn running lengthwise in a woven fabric. WET LAY-UP A reinforced plastic which has liquid resin applied as the reinforcement is being laid up. Revised: 10 February, 2000 M. V. Bower CONSTITUENTS AND FABRICATION TECHNIQUES For the purpose of this discussion, composite materials are defined as a marriage of two or more constituent materials on a macroscopic scale. To understand the fabrication techniques associated with composite materials it is important to discuss the types of constituent materials and the fabrication techniques used to produce composite structures. CONSTITUENTS MATRIX MATERIALS Polyester Resins Polyester is a thermoset polymer that is formed from a condensation polymerization. Polyester has been widely used in commercial applications with fiberglass. Applications include: • Boat hulls, • Shower stalls, • Bath tubs, • Car bodies, • Building and roof panels, • Molded furniture, and • Pipes. Advantages for the use of polyester resin include: • Low cost (generally lowest found in composite materials) and • A wide assortment of diacids and diols can be used to give physical and chemical properties. Disadvantages for the use of polyester resin include: • Poor temperature capabilities, • Poor weather resistance, • Shelf life may be limited, and • Poor mechanical properties (stiffness and strength) as compared to advanced composites. Revised: 10 February, 2000 Page 8 COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 9 Epoxy Resins Epoxy is a thermoset polymer that forms a strong rigidly crosslinked network of polymer chains. Epoxy has been widely used in commercial applications with fiberglass, graphite, and aromatic fibers. Applications include: • Aircraft components, • Pressure vessels, • Rocket motor cases, and • Car bodies. Advantages for the use of epoxy resin include: • Excellent adhesion • Excellent mechanical properties (strength and stiffness), • Excellent chemical resistance, • Excellent weather resistance, • Low shrinkage, • Good fatigue strength, • Good corrosion protection, and • Versatility in processing. Disadvantages for the use of epoxy resin include: • Poor high temperature capabilities, • Uncured resin is toxic, • Poor handling properties (uncured), and • Relatively expensive. Epoxies are available in multi-component and single component systems. The cure of epoxy may be through the application of hardeners, a catalytic agent that activates or facilitates crosslinking between the polymer chains, (a two-part system), or through the application of heat or ultra-violet light (a one-part system). Epoxies may be stored at freezer temperatures, which prompts long storage/shelf life. Wide ranges of cure cycles are available. Polyimide and Polybenzimidaole Resins Polyimide and polybenzimidaole (PBI) are thermoplastic polymers with excellent high temperature (600° to 700°) properties. Polyimide and PBI has been used in commercial applications with graphite, and aromatic fibers. Applications include: • Aircraft components. Advantages for the use of polyimide and PBI resin include: • Excellent mechanical properties (strength and stiffness), • Excellent thermal properties, and • Good processability on conventional molding equipment. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 10 Disadvantages for the use of polyimide and PBI resin include: • Variable resistance to solvents depending on specific compound, • Difficult synthesis process, • Difficult fabrication methods, • Resin is toxic and should be handled with great care, • Expensive raw materials, and • Very expensive (more than epoxies). Use of Polyimide and PBI compounds is growing as the knowledge base increases. Phenolic Resin Phenolic is a thermoset polymer with good high temperature properties. Phenolic has had a long history of commercial applications as a general unreinforced plastic and is now being used as a composite resin with graphite, and aromatic fibers. Applications include: • Aircraft components, • Rocket nose cones and nozzles, and • Automotive applications. Advantages for the use of phenolic resin include: • Good mechanical properties (strength and stiffness), • Good thermal properties with an ablative nature, and • Good processability. Disadvantages for the use of phenolic resin include: • Absorbs moisture easily, • Brittle behavior, and • Relatively expensive (more than epoxies). Carbon Matrices Carbon matrices are produced from polymeric resins that are carefully charred in a processed called pyrolysis. Carbon matrices may also be produced by vapor deposition, but the process is limited to structures less that 3/16” thick. Applications include: • Aircraft components, • Rocket nose cones and nozzles, and • Automotive applications, especially brake components. Advantages for the use of carbon matrices include: • Very high specific heat capacity (highest known), • Good mechanical properties (stiffness and strength), • Good toughness, • Good resistance to shock, Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES • • 11 Excellent thermal properties, and Excellent thermal stability. Disadvantages for the use of carbon matrices resin include: • Absorbs moisture easily, • Poor wear resistance, and • Very expensive (more than five times the cost of a phenolic ablative composite). Thermoplastics Matrices Thermoplastic polymers have long chain molecules that are loosely interconnected by weak chemical bonds and mechanical tangling. Because of the structure of thermoplastic polymers they do not require reactive cure cycles or have a distinct melting temperature, displaying fluid like (viscoelastic) behavior at even room temperature. Consequently, these materials lend themselves to molding processes. Thermoplastics include: • Polyethylene, • Nylon, • Polystyrene, • Polyester, • Polycarbonate, • Polyvinylchloride (PVC), • Acrylonitrile butadiene styrene (ABS), • Acrylic, • Polyethylene terephthalate (PET), • Polyetheretherketone (PEEK), • Polyphenylene oxide, et cetera. Advantages for the use of thermoplastics resin include: • Large number of processing methods, • Lower fabrication times, (compared to thermosetting polymers), • Good compression strength after impact, • Good hot/wet compression strength, • Resistant to moisture absorption, and • Easy dyed or given special properties (e.g. flame retardant). Disadvantages for the use of thermoplastic resins include: • High viscosity impairs wet-out of reinforcement, • High consolidation pressures are required, and • Mechanical, chemical, thermal, and electrical properties depend on specific selected. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 12 Ceramic Matrices Organic resins are characterized by the prevasive presence of covalent bonds. Ceramic matrices, by contrast, are characterized by the predominance of ionic bonds, however silicon-carbide (SiC) has covalent bonding. Ceramic solids may be crystalline, vitreous (glass-like), or mixed. Advantages for the use of ceramic matrix include: • Dimensionally stable at high temperatures, • High chemical stability, • High thermal stability, • Excellent mechanical properties (strength and stiffness), • Resistant to moisture absorption, and • Applicable to extreme temperatures (2000° to 4000°). Disadvantages for the use of ceramic matrix include: • Very brittle, • Very high consolidation pressures are required, and • Very expensive to produce and maintain. Phenolic and carbon matrice are sometimes classified as ceramic matrices. Metal Matrices In comparison to organic resins and ceramic matrices, metal matrix composites (MMCs) are characterized by the predominance of metallic bonds. In MMCs discontinous or continuous metal fibers are suspended in a matrix of a differing metal (e.g. aluminum, titanium, magnesium, copper, et cetera). Advantages for the use of metal matrix include: • Outstanding mechanical properties (stiffness and strength) for continuous fiber MMCs, • Good wear resistance, • Thermally conductive, • Good fracture toughness for continuous fiber MMCs, • Good fatigue strength for continuous fiber MMCs, • Resistant to moisture absorption, and • Applicable to extreme temperatures (2000° to 4000°). Disadvantages for the use of metal matrix include: • Significant difficulties associated with the inherent non-wetability of fibers, • Very high consolidation pressures are required, and • Very expensive to produce. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 13 REINFORCING MATERIALS While there is no restriction as to the material used as the reinforcing element for modern composites there are generally three materials that are commonly used: glass, graphite, and organic. These materials are discussed in this section. Glass Fibers Glass fibers have long been used as reinforcing elements. Owens-Illinois and Corning Glass developed a fiberglass manufacturing facility in 1937. Glass is produced from silica sand, limestone, boric acid, and other elements. Types of glass include: • E-glass, • S-glass (and the variation S2-glass), • C-glass, and • Quartz. These are the four primary types of glass used in composite materials. The type of glass is defined by the chemical composition Advantages for the use of glass fibers include: • Applicable to wide range of geometries and sizes, • Seamless construction, • Good strength and durability, • Lower tooling costs, • Increased design flexibility, • Minimal maintenance, and • Corrosion resistant. Disadvantages for the use of glass fibers include: • Mechanical properties are not as good as metals or other reinforcing fibers. The fiber glass production processes are shown in the following figure. Note that the process may begin from stock (marbles) or directly from melt. The use of stock has had better control over the properties. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 14 Fiber glass production processes, (a) marble process and (b) direct-melt process. Carbon/Graphite Fibers Carbon or graphite fibers for structural applications began production in significant quantities in the 1950s. Graphite fibers are among the highest stiffness and highest strength material known today. Types of graphite fibers include: • Polyacrylonitrile (PAN)-Based Fibers • Pitch-Based Fibers • Rayon-Based Fibers Advantages for the use of graphite fibers include: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES • • • • 15 Excellent strength, Excellent stiffness, Excellent specific strength and stiffness, and Corrosion resistant. Disadvantages for the use of graphite fibers include: • Significantly more expensive than glass fibers, and • Brittle behavior. Two graphite fiber production processes (PAN-based and pitch-based) are shown in the following figure. Note that both processes use a two step carbonization/graphitization process to convert the raw fiber into graphite. Characteristic properties for graphite fibers from the three processes is listed in the following table. Graphite fiber production processes, (a) PAN-based process and (b) Pitch-based process. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 16 Carbon Fiber Mechanical Properties4 PAN-based fibers Low Modulus High Modulus Tensile Modulus (Mpsi) 33 56 Tensile Strength (kpsi) 480 350 Elongation (%) 1.4 0.6 Density (g/cc) 1.8 1.9 Carbon Assay (%) 92-97 100 Pitch-based fibers Low Modulus High Modulus Tensile Modulus (Mpsi) 23 55 Tensile Strength (kpsi) 200 350 Elongation (%) 0.9 0.4 Density (g/cc) 1.9 2.0 Carbon Assay (%) 97 99 Rayon-based fibers Tensile Modulus (Mpsi) 5.9 Tensile Strength (kpsi) 150 Elongation (%) 2.5 Density (g/cc) 1.6 Carbon Assay (%) 99 Organic Fibers Organic fibers for structural applications were introduced for commercial applications in 1971. Graphite fibers are among the highest stiffness and highest strength material known today. Types of organic fibers include: • Kevlar Fibers • Nomex Fibers, and • Spectra (ultra highly oriented polyethylene) Fibers. Advantages for the use of organic fibers include: • Very high strength, • Very high stiffness, • Very high specific strength and stiffness, • Excellent impact resistance, • High toughness, and • Corrosion resistant. Disadvantages for the use of organic fibers include: • Significantly more expensive than glass fibers, and • Properties may be affected by environmental factors (e.g. ultra violet radiation). Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 17 Specialty Reinforcements Specialty reinforcements include: • Boron, • Silicon Carbide, and • Other. These reinforcements were originally developed in the 1960s. Advances in graphite and organic reinforcing materials coupled with lower costs associated with them has impacted the growth in applications for specialty reniforcements. Advantages for the use of boron and silicon carbide fibers include: • Very high strength, and • Very high stiffness. Disadvantages for the use of boron and silicon carbide fibers include: • Extremely expensive. REINFORCEMENT FORMS The form of the reinforcements used in composite materials spans a wide range and has a direct impact on the mechanical properties of the structural component. The form of the reinforcing elements also impacts the fabrication techniques that can be used. As discussed in the first chapter, composites are classified based on the geometry of the reinforcing element. Fiber Terminology Fiberous reinforcements have several specific terms used to describe the make-up and geometry. These terms include: • Filament Single fiber produced from a single port in the spinning process. Diameters for common filaments (glass and graphite) range from 0.000015 inches to 0.0005 inches. • Fibers A general term commonly used to refer to a collection of filaments. • Strand Commonly a bundle or group of untwisted, collimated filaments. Used interchangeably with fiber and filament. • Tow A bundle or group of untwisted, collimated filaments usually with a specific count. • Yarn A twisted bundle of continuous filaments. • Roving Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES • • • • 18 A number of tows or yarns collected into a parallel bundle without twisting. Tape A collection of collimated (parallel) filaments usually made from tows held together by a binder, which is typically a B-stage resin. Woven Fabric A two-dimensional material made by interlacing yarns or tows in various patterns. Braiding A three-dimensional material made by interlacing yarns or tows in various patterns. Mat A two-dimensional material made of randomly oriented chopped fibers or swirled continuous fibers that may be held together loosely by a binder. Weave types The textile industry has developed a number of different weaves that are commonly used in applications from clothing and upholstery to composite materials. The specific weave used in a structure may impact the drape in the fabrication process and the mechanical properties of the structure. Typical weaves used in composite materials include: • Plain weave • Basket weave • Crowfoot satin weave • Long-shaft satin or harness weave • Leno weave The typical weaves are shown in the following figure with the machine directions as indicated in the figure. The plain weave is the simplest weave that has uniform strength in two directions when the yarn size and count are similar in the warp and fill directions. Plain weave fabrics are commonly used for: • flat laminates, • printed circuit boards, • narrow fabrics, and • tooling. The basket weave is similar to plain weave except that warp yarns are woven as one over and under tow fill yarns. The weave is less stable than plain weave. Consequently the weave is more pliable and drape is better. Basket weave fabrics are stronger than an equivalent weight/count plane weave fabric. Applications for basket weave fabrics are similar to those for plain weave. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 19 Crowfoot satin weave has improved unidirectional quality with more strength in the fiber directions than plain weave fabrics. The crowfoot satin weave is more pliable than plain weave fabrics and can comply to complex contours and spherical shapes. Applications include: • Fishing rods, • Diving Boards, • Skis, • Aircraft ducts, • Channel, and • Conduit. Long-shaft satin or harness weave has a high degree of drape and stretch in all directions. The weave is less stable than in plain weave fabrics. Applications include: • Aircraft housings, • Radomes, • Ducts, and • Other contoured surfaces. Leno weave produces heavy fabrics for rapid build-up of plies. Leno weave fabrics are used: • As inner cores of thin coatings, • Tooling, and • Repairs. The choice of weave for a particular application will generally be a compromise between structural and fabrication requirements. Unidrectional tape will produce higher strength plies but are more difficult to fabricate. Drape of the cloth can be a major consideration in structures with complex contours. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 20 Common weaves used in composite materials. Cloth Directions and Name Conventions Plain Weave Basket Weave Crowfoot Satin Long-shaft Satin Leno Weave Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 21 FABRICATION TECHNIQUES For the purposes of this presentation the fabrication techniques for composite materials are arbitrarily divided into three catagories they are: manual processes, machine processes, and mass production processes. The type of technique used in a particular application will depend on among other things: • the number of parts to be produced, • the facilities available, • the repeatability of the parts to be produced, • the mechanical properties required in the finished parts, • the materials (resin and reinforecment) to be used, and • the size of the part. Unlike metal manufacturing processes, composite fabrication processes can have significant impact on part quality. Composite materials have received undo criticism in some arenas due in part to inadequate quality control in the fabrication processes. Consequently, it is important for the designer and analyst to understand the composite fabrication processes and to develop an appreciation for the impact of the fabrication processes on system behavior. CURE PROCESSES The majority of composite materials in production today are made with thermoset polymeric resins. Consequently, the structure requires some kind of a cure process to produce the final part. A generic cure cycle is shown in the following figure. 300 150 200 100 T (°F) p (psi) Three hour hold at 250°F with one hour at 150psi. 100 50 2°/min Heating 5°/min Heating Pressure 0 0 0 50 100 150 200 250 300 t (minutes) Generic cure cycle with temperature and pressure required. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 22 Vacuum Bag Processes Whether or not a cure cycle requires additional pressure to achieve consolidation of part the vast majority of all composites produced require use of a vacuum bag. In the vacuum bag process the part is covered by a release ply, next a barrier film, then the appropriate number of bleeder ply to absorb the excess resin from the lay-up, a breather ply to provide a flow path for trapped gasses and volitols released during the cure cycle, and finally a bag. The vacuum bag may be a molded bag or sheet of polymer fitted to the part. Molded bags are more expensive but require less manual labor to install. They are commonly used in high production applications. A generic lay-up with the various vacuum bag components is shown in the following figure. Typical Vacuum Bag Components. Bagging is an important part of processing thermoset composite parts. It has a direct impact on part quality. It is possible for a part, carefully laid-up, to be scrapped due to poor bagging. It is essential that a bag be tightly sealed and leak free and be in perfect contact with the workpiece. A leak free bag is necessary to achieve consolidation of the lay-up and to provide the necessary path to exhaust evolved gasses that may be trapped in the lay-up during the fabrication process or those that are produced by the chemical reactions in the cure cycle. Producing a leak free bag can be challenging, but is not in general impossible. There may be times that a bag loses its seal during the cure cycle. When this happens it is Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 23 typically unrepairable, but the part may not be lost, depending on when the leak develops and what type of cure cycle is in use. Perfect contact with the part must be established in the bagging processes. Failure to establish perfect contact will produce a flawed part that may or may not have to be scrapped due to the failure. Perfect contact appears in two points in the bagging process, bridges and darts. These problems are shown in the following figures. Darts are used in bag construction when flat bag material is used instead of a molded bag. The seams produced on the finished parts due to use of darts is a resin rich point that is not likely to adversly affect the mechanical properties of the part. Bridges on the other hand are more than unsightly belmishes. It is quite common to have internal voids and delaminations in the finished workpiece in the vicinity of bridges. Rubbing tools are used to compact the bag and remove wrinkles. Vacuum Bag Dart Lay-up Gap due to bridging Gap due to dart Vacuum Bag Lay-up Mold Mold Bridging in a vacuum bag process. Dart used in vacuum bag process. Auto- and Hydroclave processes Autoclave and hydroclave processes use additional pressure to consolidate the laminate. Vacuum bags are used with both of these processes. The use of a hydroclave may produce a superior cure cycle due to the improved heat transfer from the liquid medium to the workpiece as compared to the gas used in an autoclave. Typically the added pressures used in these processes range from 50 psi to 200 psi. Research on the application of pressure and duration of the vacuum held on the part for phenolic composites has indicated that part quality can be significantly impacted by these steps. MANUAL TECHNIQUES Manual fabrication techniques for composite materials include manual lay-up and manual spray-up. Of these processes, the manual techniques are dominated by the manual lay-up process. Advantages and disadvantages for these processes are listed in the following table. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 24 Advantages and Disadvantages of Manual Fabrication Techniques Advantages* Disadvantages* Design flexibility Labor-intensive process Large and complex parts can be Only one good (molded) surface is produced obtained Production rate requirements are low Low-volume production process Minimum equipment investment is Quality is related to the skill of the necessary operator Tooling cost is low Longer cure times required Any material that will hold its shape can Product uniformity is difficult to be used as a mold form maintain with in a single part and from part to part Start-up lead time and cost are minimal Waste factor is high Design changes are easily effected Hazards associated with handling the materials are higher Molded-in inserts and structural reinforcements are possible Sandwich constructions are possible Prototyping and pre-production method for high volume molding processes Semi-skilled workers are needed and are easily trained * Note: the horizontal alignment in the table is not intended to imply a relationship between the points. Design considerations for manual techniques include the following: • Minimum inside radius: 0.1875 to 0.25 inches. Tighter radii are possible but not desirable. • Minimum draft recommended: 2°. Split molds can have 0°. • Undercuts: should be avioded but can be made by using split or rubber molds. • Molded in holes: lareg diameter only. • Minium practical thickness: 0.03 inches for manual lay-up, 0.06 inches for manual spray-up. • Maximum practical thickness: unlimited total, 0.25 inches per cure. • Normal thickness variation: +0.03/-0.015 inches for manual lay-up. ±0.025 inches for manual spray-up. • Special construction possible: built-in cores, metal inserts, metal or other edge stiffeners • Bosses: must be tapered. • Fins: special handling required. • Limiting size factor: none, other than mold size, oven size (if required) and handling considerations. • Shape limitations: none. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES • • • 25 Finished surfaces: one (two with special tooling). Gel-coat surface: only one smooth surface, reverse can be coated after molding Molded in labels: Manual Lay-up In a manual lay-up process each ply of a laminate is placed by hand. The advantage to these processes is that little in the way of equipment is required. The disadvantages to these processes include: high degrees of variablitiy between parts, even those produced by a single technician and generally inconsistent quality even when performed by a highly qualified technician. As always the advantages must be weighed against the disadvantages in a given application. Wet Processes In wet lay-up processes the lamina or ply must be saturated with resin before being laid-up in the tooling. Following saturation, excess resin must be removed to avoid having a part that is unacceptably resin rich. The following are the basic steps in a wet lay-up process: 1. Prepare patterns for each ply of the laminate locating all darts and folds required to accurately follow the mold. Minimize the number of overlaps and never superimpose overlaps. If overlapping plies are required, keep the overlap width to 0.75 inches, +0.25 inches/-0.0 inches. A pattern may be used for multiple ply when the pattern is repeated in the stacking sequence. 2. Optional, prepare a lamination kit, cutting all plies to the required pattern, marking their order in the stacking sequence with Teflon tape. 3. Coat tooling with release film or place release ply on tooling. If Gel-coat is desired it should be applied at this time. 4. Prepare the resin pot. Mix the resin components as required and place in a container that is sufficiently large to lay the individual plies for the laminate. 5. Saturate the first ply of material with resin. 6. Strip excess resin from ply. 7. Place first ply on tooling with the orientation specified in the stacking sequence, film side-up. Using a non-stick (Teflon or steel) tool press the ply onto the tooling. Work out all air bubbles and wrinkles. Ensure that the ply is in total contact with the tooling, working out all bridges that occur at fillets and gaps at corners that occur at rounds. Draw excess resin from ply. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 26 8. Remove Teflon tape marking lamina sequence. (This is very important.) 9. Saturate the next ply of material with resin. 10. Strip excess resin from the ply. 11. Apply the next ply with the orientation specified in the stacking sequence, film side-up. Again, using a non-stick (Teflon or steel) tool press the ply onto the lay-up. Work out all air bubbles and wrinkles. Ensure that the ply is in total contact with the previous layer, working out all bridges that occur at fillets and gaps at corners that occur at rounds. Draw excess resin from ply. 12. Repeat steps 8-11 until the specified stacking sequence is completed. 13. Depending on the resin system used, if the fabrication process cannot be completed in a single operation the lay-up should be bagged with a vacuum bag and held under a controlled environment (especially low humidity) until the operation can resume. 14. Prepare the part for the cure process following the bagging procedure. Manual scissors, power shears, and semi-automatic and automatic machines may be used to cut the plies for the lay-up. When computer numerically controlled automatic cutting machines are used, patterns are not necessary. Prepreg Processes 1. Prepare patterns for each ply of the laminate locating all darts and folds required to accurately follow the mold. Minimize the number of overlaps and never superimpose overlaps. If overlapping plies are required, keep the overlap width to 0.75 inches, +0.25 inches/-0.0 inches. A pattern may be used for multiple ply when the pattern is repeated in the stacking sequence. As with the wet process, if automatic cutting machines are used patterns are not required. 2. Coat tooling with release film or place release ply on tooling. 3. Optional, prepare a lamination kit, cutting all plies to the required pattern, marking their order in the stacking sequence on the backing film. Return the kit to storage as quickly as possible to maintain quality. 4. Bring the prepreg to room temperature for the lay-up process. 5. Place first ply on tooling with the orientation specified in the stacking sequence, film side-up. Using a non-stick (Teflon or steel) tool press the ply onto the tooling. Work out all air bubbles and wrinkles. Ensure that the ply is in total contact with the tooling, working out all bridges that occur at Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 27 fillets and gaps at corners that occur at rounds. A hot air gun can be used to increase ply flexibility and tack to place the ply. 6. Remove the backing film from the prepreg. (This is very important.) 7. Apply the next ply with the orientation specified in the stacking sequence, film side-up. Again, using a non-stick (Teflon or steel) tool press the ply onto the lay-up. Work out all air bubbles and wrinkles. Ensure that the ply is in total contact with the previous layer, working out all bridges that occur at fillets and gaps at corners that occur at rounds. 8. Repeat step 7 until the specified stacking sequence is completed. 9. If the fabrication process cannot be completed in a single operation the lay-up should be bagged with a vacuum bag and placed in cold storage under vacuum until the operation can resume or held under a controlled environment (especially low humidity). When lay-up resumes, if the part was placed in cold storage, the part must be brought to room temperature before fabrication can continue. 10. Prepare the part for the cure process following the bagging procedure. Spray-up In the typical spray-up process chopped fibers, usually glass, and resin are simultaneously sprayed onto or into an open mold. Fiber roving is fed through a chopper and injected into a resin stream that is manually directed at the mold. The resin system may be pre-mixed or mixed in the spray-up nozzle. After the composite is sprayed into the mold it is hand rolled to remove air, compact the fibers, and smooth the interior surface. Because of the nature of the process the fibers are randomly oriented within the laminate and the behavior is transversely isotropic. Depending on the resin system used the workpiece will be bagged and cures in the same manner as lay-up parts. MACHINE PROCESSES Machine processes are typically superior to manual processes in quality, quantity, and production time. However, they are also significantly more expensive than manual processes to implement. Simple winding machines are in the tens of thousands of dollars, while the most advanced, sophisticated fiber placement machines are millions of dollars. With capital equipment costs of this magnitude the decision to use machine processes is not a caviler decision. At this writing there are four principal types of machines used for composite fabrication. They are: filament winding, tape placement or tape laying, fiber placement, and pultrusion. Of these four, only the pultrusion process incorporates Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 28 a cure cycle in the process. Parts produced by filament winding, tape laying, and fiber placement must be bagged and cured with the specified cure cycle. Filament Winding Filament winding is the oldest of the machine processes. The process is related to the turning processes used in machining operations. Filament winding is used to produce axisymmetric structures. Parts produced by filament winding include: • Tubes or pipes, • Cylindrical pressure vessels (Rocket motor cases), and • Spherical pressure vessels. Fundamentally the process involves winding roving or tow around a mandral. The winding angle ranges from nearly axial, or longitudinal (0°, axial, can be obtained in special winding operations) to hoop, or circumferential (90°). In the winding operation dry roving is pulled through a resin bath where the roving is saturated with resin. The excess is stripped from the roving and the roving is drawn through a generation ring. The winding head with its generating ring traverses the longitudinal direction of the workpiece riding on a carriage and laying the roving on the mandral. The roving follows a helical path around the mandral, see the figures below. As the carriage reaches the end of the workpiece it reverses direction and lays down another layer in the opposite direction. The process continues until the mandral is completely covered and then the machine moves to the next ply. Because the roving does not completely cover the workpiece in a single pass (except in the hoop direction) the roving is laid down in stripes that alternate in direction (±α). The result is something approaching a woven cloth, similar to a plain weave fabric, except that the fiber (roving) directions are not perpendicular. A typical filament winding machine is shown in the figure. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 29 Fundamental components in a filament winding process. Hoop or circumferential winding. Typical polar winding. Multi-circuit helical winding. Note the overlap of windings. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 30 Typical filament winding machine. One of the most important components in successful filament winding is the determination of the relative speed between the mandral and winding head. These motions determine the wrapping angle and overlap of the roving. The winding angle may be approximated from:  Rω  α = arctan  ,  v  where α is the winding angle, R is the radius of the workpiece, ω is the rotation rate of the workpiece, and v is the longitudinal speed of the winding head. From this equation you can observe that if the workpiece has a change in diameter along the length the rotation rate of the workpiece and/or the speed of the carriage must be adjusted to hold the winding angle constant. The complexity of the problem is further complicated by polar winding at the dome of pressure vessels. Another important component in successful filament winding is tension control. Tension affects resin content, void content, and structural properties. Roving tension ranges from 0.25 lbs. to 1 lb. per bundle or tow. Tension is provided by guide eyes in line, center rotating guide eye, rotating scissor bars, drum-type brakes (which may be electromagnetically controlled), and/or drag through the resin bath. The first three of these tensioners are shown in a figure below. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES Guide eyes in line. Center rotating guide eye 31 Rotating Scissor bars. Advances in filament winding technology include spherical machines and multi-head winders. Spherical winding machines can vessels that are cylindrical or spherical with a single opening. Multi-head machines can produce vessels with a quasi-braided structure. A spherical machine is shown in the following figure. winding produce winding winding Advantages and Disadvantages for Filament Winding Processes. Advantages Disadvantages or Limitations Applicable to parts of widely varying Resin viscosity and pot life must be carefully size. chosen and monitored. Parts with strength in several Programming of the winding can be difficult. directions can be easily made. Excellent material usage. Not all shapes can reasonably be made by filament winding. Operational control of several key Forming after winding and other parameters is important. techniques allow noncylindrical shapes to be made. Flexible mandrels can be retained in Ability to analyze (design) impaired due to the structure to serve as liners for invalidation of key assumptions in tanks. lamination theory. Panels and fittings for reinforcement or attachment can be easily included during the winding process. Parts with high pressure ratings can be made. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 32 A spherical winder. Tape Lay-up Semi-automatic and automatic tape laying machines have been developed to reduce production times, improve consistency within parts and between parts, and improve part quality. Tape lay-up machines use prepreg tape, unidirectional and cloth, and are computer numerically controlled. Through appropriate programming it is possible to eliminate the patterns use in the manual lay-up processes. Semi-automatic and automatic tape laying machines are used to produce flat and contoured laminates. Tape laying machines are limited in their capabilities to surfaces with large radii of curvature. Tape laying machines cannot produce highly geometrically complex parts. Typical automatic tape laying machines are described by the number of degrees of freedom (DOF) or axes of the tape laying head, for example, a head may have three translational degrees of freedom, two rotational degrees of freedom, the ability to start a tape, and the ability to cut the tape. This machine is described as a seven-DOF machine. Figures below show a tape laying head, a flat automatic tape laying machine, a coutoured laminate in a tape laying machine, and a multi-axis tape laying machine. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 33 Grumman tape laying head. Flat automatic lay-up machine. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 34 A contoured laminate on an automatic lay-up machine. Carriage Tape Laying A ten DOF automatic lay-up machine. Fiber Placement Automatic fiber placement machines combine and extend the capability of filament winding and tape laying machines. Fiber placement machines place individual fiber bundles onto a mold. This is reminescent of the filament winding process and in contrast to the laying of a tape with a tape laying machine. However, the fiber placement machine places the fiber bundles in parallel throughout a layer, without the overlapping in a helical winding process. Because of the ability to place fiber bundles the fiber placement process can produce highly geometrically complex shapes with small radii of curvature. Internal radii are limited by the size of the placement head, approximately 6 inches on a Viper Placement Machine and external radii are limited to the minimum bending radius of the fiber bundles, approximately 0.1875 to 0.25 inches. A fiber placement machine is shown in the figure below. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 35 NASA Viper fiber placement machine. Fiber placement is a comparably rapid process. Fiber placement machines can place up to 700 inches of fiber per minute. Further, the complexity of the parts produced by fiber placement is extrordinary. For example, the intake duct for the F-16 is a geometrically complex part with multiple compound curves is produced by fiber placement with relative ease. Pultrusion Pultrusion is an adaptation of the drawing process to composites fabrication. This process produces long relatively narrow cross section with highly ordered and compacted reinforcements. Cross sections produced by pultrusion range from circular to “L”-channel to hat sections. The reinforcing fibers, which may be glass, graphite, or arimid, are generally all oriented along the major direction of the pultrusion. In general the process begins with dry roving that is drawn through a resin bath and into a compaction die. From the compaction die the material is drawn into a curing die where the excess resin is striped and the part is cured with significant pressures. It is possible to draw the workpiece over a bench before the final cure to produce curved (instead of straight) sections. Automotive composite leaf springs are an example of curved pultruded parts, a pultrusion forming operation and automobile springs are shown in the figures. A microwave curing prepreg based pultrusion system is shown in the figure below. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 36 Schematic of a Pultrusion forming system. Automobile leaf springs produced by Pultrusion forming. Schematic of a Pultrusion system using microwave energy to cure the resin system. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES Advantages High material usage compared with lay-up High throughput rate Can give high resin contents Close to fiber tow properties 37 Pultrusion Disadvantages or Limitations Part cross-sections must generally be uniform. Problems can arise when resin or fibers accumulate and build up at the die opening When dies run resin rich to account for fiber anomalies, strength is sacrificed. Voids can result if dies are run with too much opening for the fiber volume. When quick curing systems are used, mechanical properties are often sacrificed. MASS PRODUCTION TECHNIQUES Manual production techniques by their very nature are limited in their production rates. The machine processes are typically more expensive and do not lend themselves to high production volumes. Production of composite parts for applications such as those found in the automotive industry necessitate production processes that have very high production rates. Molding processes are able to meet these demands. Molding Molding of composite materials has its foundation in the metals casting and forming processes. These processes include: sheet molding, bulk molding, thick molding, and liquid molding (resin transfer molding). The advantages of these processes include lower per part tooling costs and higher production rates. Sheet Molding Sheet molding processes were developed in response to a request from the automotive industry. Their desire was for a composite material process that allowed them to use the metal bending and stamping equipment and techniques with which they were familiar. The material used in sheet molding is called sheet molding compound (SMC). In the sheet molding process chopped roving, usually glass fibers, is mixed with resin and deposited between plastic films, usually polyethylene. The material is then used in a stamping like process. The schematic below shows a typical SMC machine. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 38 Schematic of the SMC fabrication process. Note that the flow is from left to right. Thick and Bulk Molding Thick and bulk molding operations are related to the closed die or matched-die casting and forming processes. In these processes the bulk molding compound (BMC) or the thick molding compound (TMC) is prepared similar to the SMC compound, however it is mixed in buld rather than sheet. Advantages and disadvantages of the process are listed below. The TMC process is shown in the figure. Matched-die Molding Advantages Disadvantages or Limitations Both interior and exterior surfaces More equipment is needed than for lay-up. are finished. Complex shapes including ribs and Molds and tooling are costly compared to thin details are possible. lay-up molds. High production rates are possible. Transparent products are not possible with SMC and BMC. Labor costs are low. Molding problems (trapped water, etc.) may cause surface imperfections such as pitting or waviness. Minimum trimming of parts in SMC and BMC have limited shelf-lives. needed. Products have good mechanical properties and close part tolerances. Good consolidation of parts. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 39 Schematic of the TMC fabrication process. Resin Transfer Molding Resin transfer molding is used in a number of variations with a number of different names to identify the variations. Among these processes are: Resin Transfer Molding (RTM), Structural Reaction Injection Molding (SRIM) Resin Injection Molding (RIM), Vacuum-Assisted Resin Injection (VARI), Thermal Expansion Resin Transfer Molding (TERTM), Vacuum Assisted Resin Transfer Molding (VARTM), and Seaman’s Composites Resin Injection Molding Process (SCRIMP) to name a few. All of these processes involvethe same basic steps. The basic steps in RTM are: • Place preform in mold. • Close mold. • Infuse/inject liquid resin into mold. • Cure part in mold. • Open mold. • Remove part from mold. • Clean up part. One of the significant problems in the use of these liquid molding processes is adequate setting of the structural preform by the liquid resin. To produce a high quality composite part it is essential that the reinforcing fiber structures be throughly impregnated with the resin. The resin or matrix in a composite transfers the load from one fiber to the next. If there is no resin present, the loads do not get transferred from fiber to fiber, which results in an inadequate structure. Proper infusion or impregnation of the fiber preform requires a low viscosity resin and an Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 40 extended pot life (two hours or more depending on the size of the part). In addition it is very important to provide a leak path for all gasses that are native to the preform as the resin is injected. Advantages and disadvantages for the RTM process are listed below. Resin Transfer Molding Advantages Disadvantages or Limitations Very large and complex shapes can The mold design is critical and requires be made efficiently and great skill inexpensively. Production times are much shorter Properties are equivalent to matched-die than lay-up. molding (assuming proper fiber wet-out) but are not generally as good as with vacuum bagging, filament winding, or Pultrusion. Clamping pressure is low compared Control of resin uniformity is difficult. to matched-die molding. Radii and edges tend to be resin rich. Surface definition is superior to Reinforcement movement during resin lay-up. injection is sometimes a problem. Inserts and special reinforcements can be added easily. The sill level required for the operator is low. Many mold materials can be used. Parts can be made with better reproducibility that with lay-up. Workers are not exposed to chemicals and vapors as with lay-up. Schematics for the resin transfer molding process and the resin infusion process are shown in the following figures. Note in the RTM process that a pump is used to force the resin into the mold and a press is used to hold the mold closed. It may be advantageous to use pumps to force the resin into the mold. However, high resin flowrates may cause the preform to be dislocated from the desired position. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 41 Schematic of the Resin Transfer Molding Process. Schematic of the Resin Infusion Process. Automated Spray-up The automated spray-up processes are simply an automation of the manual spray-up processes. In some ways the process is related to the automated painting processes used in industries such as the automotive industry. In this application a robotic arm is programmed to spray chopped reinforcement and resin into a mold. A schematic of the automated spray-up process is shown in the following figure. The various letters in the figure designate components of the spray-up machine. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 42 Schematic of an automated spray-up machine. HAZARDS The hazards associated with the production and fabrication of composite materials is higher than in the production of parts from conventional materials. Some of the more exotic conventional materials, such as titanium, magnesium, and berelium, have significant healt hazards associated with their production and/or fabrication processes. Composite materials in general are produced from organic chemical compounds that in their uncured state may pose significant health hazards unless handled with great care. HEALTH INFORMATION TERMINOLOGY To begin the discussion we must again define some terms in the field. First of all we must define the difference between toxicity and hazard. Toxicity is an inherent harmful effect of a chemical. It is a physical property of the chemical. Hazard is controlled by exposure. Exposure to a toxic chemical required for a hazard to exist. A chemical with Acute Toxicity has a harmful effect after single and/or short term exposure. Toxicity is measured in lethal doses and lethal concentrations. The Mean Lethal Dose – LD50 is expressed as a ratio in mg of chemical to kg of body weight. It is the amount of chemical administered by a specific route that is expected to kill 50% of a group of experimental animals. The Mean Lethal Concentration – LC50 is expressed in mg/m3 or parts per million (ppm) in air. It is the concentration of chemical in air that is expected to kill 50% of a group of experimental animals. There are for some chemicals levels below which there is no observable effect. This level is defined as the No Observable Effect Level, NOEL. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 43 The toxicity of a chemical is further characterized by the local effects it causes. These effects include: • Irritation Localized reaction resulting from either single or multiple exposures to a physical or chemical entity at the same site. • Corrosion Tissue destruction in such a way that normal healing is not possible. • Sensitization Allergic reaction to a substance that develops upon repeated exposure. • Chronic Toxicity Characterized by adverse health effects in an animal or person, which has been caused by exposure to a substance of over a significant portion of that animal’s or person’s life, or by long-term effects resulting from a single or a few doses. Two other health terms that are used to describe the toxicity of chemicals are used commonly today but have specific definitions in the health field. The first is carcinogenicity. This is the ability of a substance to cause tumors. Long term testing required to determine if a substance is carcinogenic. The results of these tests are conclusive. The second is mutagenicity. This is the ability of substance to cause changes in the genetic materials of cells. Short term testing can be used to determine if a substance is mutagenic. The results of these short term tests are speculative (non-conclusive). To minimize the hazards associated with working with toxic substances exposure limits are defined. Exposure limits that are defined in terms of Threshold Limit Values (TLV). TLVs assume that the exposed population is composed of normal, healthy adults, and does not address aggravation of pre-existing conditions of illnesses. These limits are not fine lines between safe and dangerous concentrations and should not be used by anyone untrained in the discipline of industrial hygiene. Four important TLVs are: • Threshold Limit Value – Time Weighted Average (TLV-TWA) The time weighted average for a normal 8 hour workday and 40 hour work week, to which nearly all workers may be exposed, day after day without adverse effect. • Threshold Limit Value – Short Term Exposure Limit (TLV-STEL) The concentration to which workers can be exposed continuously for a short period of time (15 minutes) without suffering from (1) irritation, (2) chronic or irreversible tissue damage, or (3) narcosis of sufficient degree to increase the likelihood of accidental injury, impair self-rescue, or materially reduce work efficiency, and provided that the daily TLV-TWA is not exceeded. • Threshold Limit Value – Ceiling (TLV-C) The concentration that should not be exceeded during any part of the workday. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES • 44 Permissible Exposure Limits – (PEL) PELs are legal binding airborne exposure limits, which are issued by the Occupational Safety and Health Administration (OSHA). Proper industrial hygiene is essential to minimize the hazards of working with toxic substances and to ensure that the TLVs are not exceeded. Proper industrial hygiene requires controlling the routes of exposure to the toxic substances. Routes of contact include: • Skin and Eye Contact At risk through skin and eye contact are: Hands, Lower Arms, and Face. Contact with liquids, gases, vapors, or particulates should be minimized to reduce the risk of contact. • Inhalation Inhalation can be a significant route of exposure to toxic substances in composite fabrication. Solvents and other volatiles may be released from the resin systems during the manufacturing and curing of composites. Further, dusts may be generated in the machining of cured composite materials. • Ingestion Ingestion is not typically not a major problem in the fabrication of composite materials provided that there is sufficient control. Proper industrial hygiene requires control of the processes in five areas: Administrative, Engineering, Operations/Process, Safety, and personal. Administrative Controls include proper: handling of materials, training, isolation of operations, personal protective equipment, personal hygiene, warnings and labels, housekeeping, dispensing and storage of chemicals, and emergency instructions. The Engineering Controls include proper: plant layout, design and use of equipment, and exhaust ventilation. Operations/Process Controls include proper: mixing of resins (personal protective equipment as appropriate, and specific mixing instructions -- available and followed), curing operations (use product specific cure cycle), and handling of cured resin systems (as appropriate). Safety Controls are as appropriate. Personal Controls include proper training of all personnel and a commitment by all personnel to maintain a safe, hazard free workplace. This includes a commitment by employers to effectively instruct the employees on site hazards, warning labels, and material safety data sheets. Further, management and employee are responsible for knowing about hazards and taking measures for minimizing exposure. TOXICOLOGICAL PROPERTIES OF COMPONENTS The following section lists some of the toxicological properties of components of composite materials. The information presented below is generic and further specific information should be obtained regarding the specific compounds with which you are dealing. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES • • • • • 45 Epoxy Resins Epoxy resins are always used with curing agents and commonly with a series of other additives. Generally, more demanding handling procedures and controls are recommended for the curing agent or other additives. Epoxies are primary skin and mucous membrane irritants. Some epoxies have sensitizing effects. Hardening and Curing Agents • Aromatic Amine Hardeners These hardeners have slight irritating effect on skin and mucous membranes. They have been shown to cause damage to liver and may decrease ability of blood to transport oxygen to tissues. Exposure should be minimized or avoided. • Aliphatic and Cycloaliphatic Amine Hardeners These hardeners are strong bases. It is a sever irritant and is corrosive. Exposure should be avoided. • Polyaminoamide Hardeners This hardener produces mild irritation of skin and mucous membranes. It may cause sensitization. Exposure should be minimized or avoided. • Amide Hardeners This hardener has a slight irritant effect. Avoid inhaling dust. • Anhydride Curing Agents This hardener is a sever eye irritant and a strong skin irritant. Exposure should be minimized or avoided. Polyurethane Resins • Isocyanates Most commercial isocyanates are highly toxic due to skin and respiratory sensitization, or skin absorption and systemic toxicity. They produce strong irritation of skin and mucous membranes of eyes and respiratory tract. Extreme care is necessary! Good ventilation is required! • Toluene diisocyanate (TDI) Toluene is a mutagen. TLVs for toluene are: TLV-TWA of 0.005 ppm and TLV-STEL of 0.02 ppm. Toluene has no odor below TLV levels. At this time there is no carcinogenic data. It is, however, classified as potentially carcinogenic. Polyols These are cure agents. At this time no particular health hazard is indicated. Phenolic and Amino Resins • Phenol-Formaldehyde Resins These resins have low hazard levels. Phenol and formaldehyde may be absorbed through skin. Good ventilation is recommended and skin sensitization is possible. • Urea- and Melamine-Formaldehyde Resins Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES • • • • • 46 These resins have acute toxicity similar to the phenol-formaldehyde resins. Skin sensitization is possible. Bismaleimides No extensive studies have been performed on bismaleimides at this time. They may cause skin irritation or sensitization. Dust or vapors may irritate eyes, nose, and throat. Thermoplastics Generally thermoplastics are not considered harmful to workers’ health. Skin irritation is not observed and no toxic effects are known to be associated with inhalation of dusts. Burns may present sever hazard with thermoplastics. Styrene Monomer Styrene vapors can cause eye irritation. The liquid will cause eye, skin, and mucous membrane irritation. Styrene has systemic effects on central nervous system, liver, and kidneys have been observed. It is possibly carcinogenic to humans. Reinforcing Materials Most reinforcing materials in and of themselves are non-toxic. However, inhalation of filler may be detrimental to health. Inhalation may produce effects similar to asbestosis. • Carbon and Graphite Fibers Threshold limits have been established for carbon and graphite fibers. The limits are: (TLV-TWA) 10mg/m3 (OSHA) and 3 fibers/cm3 (U. S. Navy). • Aramid Fibers The exposure limit (TLV-TWA) is set by manufacturers at 5 fibers/cm3. No apparent effects from inhalation are observed. • Fiber Glass The exposure limit (TLV-TWA) for fibrous glass is 10mg/m3. NIOSH recommends – 3 fibers/ cm3. Exposure may cause mechanical irritation of eyes, nose, and throat. It is classified as possible human carcinogen. Solvents Contact with most organic solvents causes drying and defatting of skin and dermatitis. Some solvents are directly absorbed through intact skin; absorption is enhanced if skin abraded or irritated. An additional concern is the ability of a solvent to carry other substances though skin with it. • Acetone Acetone is a common laboratory solvent. It was placed on the hazardous list. However, it has been more recently removed. The threshold limits are: a TLV-TWA of 750ppm and a TLVSTEL of 1000ppm. • Methyl ethyl ketone (MEK) Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CONSTITUENTS AND FABRICATION TECHNIQUES 47 In addition to being a solvent, MEK is also used as an accelerator for Gel-coat. It causes eye, nose, and throat irritation. The threshold limits are: a TLV-TWA of 200 ppm and a TLV-STEL = 300 ppm. Revised: 10 February, 2000 M. V. Bower LAMINA MECHANICS PRELIMINARIES A lamina is a flat or nearly flat thin layer of material. In this application the material is a composite material either tape or cloth. In practical engineering applications the lamina is the fundamental building block of the structure. To understand the mechanics of laminated structures it is necessary to understand the mechanics of the individual lamina. NOTATION Recall from solid mechanics that there are six unique stresses at each point in a body. The drawing in Figure 1 shows the nine stresses acting on the faces of an infinitesimal cube at the point in question. The stresses, σ, shown are given x2 σ22 σ21 σ23 σ12 σ32 σ11 σ31 x1 σ13 σ33 x3 Figure 1. Stresses at a point. with two indices, the first index indicates the direction of the normal to the surface on which the stress acts and the second index indicates the direction of the stress component. This is indicial or tensorial notation, σij , where the indices, i and j range form 1 to 3. The nine stresses shown in Figure 1 reduce to six unique stresses as a result of the application of the principle of conservation of angular momentum and assumptions regarding the ability of the material to support an internal couple. The consequence is that the shear stress appear in pairs, σ12 = σ 21 , σ13 = σ 31 , and σ 23 = σ32 , or σij = σ ji . Revised: 10 February, 2000 Page 48 COMPOSITE MATERIALS LAMINA MECHANICS 49 Compact or contracted notation is an alternate method for identifying the six unique stresses at a point. In compact notation a single index is used to identify the individual stress. The relationship between the stress terms in tensorial notation and compact notation is shown in Table 1. Table 1. Relationship Between Stress Terms Expressed In Tensorial Notation And Compact Notation. Tensorial Notation σ11 σ 22 σ33 Compact Notation σ 23 σ4 σ13 σ5 σ12 σ6 1 σ2 σ3 COORDINATE SYSTEMS In analysis of the mechanics of a composite lamina there are four different coordinate systems that may need to be considered. They are: • Principal Material Directions • Structural Directions • Principal Stress Directions • Principal Strain Directions The principal material directions are defined from symmetries of the material. For a filamentary material or cloth the first and second principal material directions are defined by the primary fiber direction. The third direction is taken perpendicular to the lamina. The structural directions are those that are defined by the particular application of the composite material. The principal stress and principal strain directions are calculated from the loading and the associated material behavior. To distinguish between these terms the following convention will be used: • Principal material direction stresses and strains are indicated by numerical subscripts, i. e., 1, 2, ..., and 6, as noted above. • Structural direction stresses and strains are indicated by letter subscripts, i. e., x, y, z, s. This notation scheme is adapted from Jones, and Tsai and Hahn. In this application the subscripts, x, y, and z, indicate normal stresses or strains in the corresponding direction and s indicates Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS 50 the in-plane, xy, shear stress or strain. The out-of-plane stresses are zero as a result of the plane stress assumption. • Principal stress direction stresses and strains are indicated by upper-case Roman numeral subscripts, i. e., I, II, and III. • Principal strain direction stresses and strains are indicated by a superscript, i. e., ε, and upper-case Roman numeral subscripts, i. e., I, II, and III. This system is relatively common within the industry. However, you should pay careful attention to the convention used by any software package in use or when dealing with other organizations. TRANSFORMATION OF STRESSES FROM ONE COORDINATE SYSTEM TO ANOTHER In analysis of the mechanics of a composite lamina transform the stresses from one coordinate system to another. important to define the sense of the coordinate rotation. A rotation angle is defined as positive in the positive z-direction. it is necessary to In this process it is positive coordinate Figure 2 shows the y x2 x1 +θ x3 = z x Figure 2. Relationship between the structural coordinate system and the principal material directions. relationship between the structural axes and the principal material directions of a lamina. In this figure, the positive angle, θ, is shown as a counterclockwise rotation about the z-axis. It is easily shown and well documented in numerous solid mechanics texts that the stresses in the structural directions are transformed to the principal material directions by: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS 51 2 n2 2mn  σ x   σ1   m      2 2 m − 2mn  σ y  , σ 2  =  n σ   − mn mn m 2 − n 2  σ   6   s  1 where m = cosθ and n = sin θ . STRAIN For the analysis of most common engineering structures the deformation is measured in terms of the change in length to the original length and is termed strain. The tensorial definition of strain, εij , is: 1  ∂u ∂u  εij =  i + j  , 2  ∂x j ∂xi  2 where ui are the displacements in the ith coordinate direction. Note that there are only six unique strains since the strain tensor is symmetric, i. e., εij = ε ji . Engineering strain is an alternative measure of the deformation. The definition of engineering strain, eij , is:  ∂ui  ∂x  j eij =   ∂ui + ∂u j  ∂x j ∂xi  i= j . 3 i≠ j The shear strains, i. e., eij i ≠ j , are also commonly presented as γ ij . As with stress, compact or contracted notation is an alternate method for identifying the six unique strains. In compact notation a single index is used to identify the individual stress. The relationship between tensorial strain, engineering strain, and engineering strain expressed in compact notation is shown in Table 2. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS 52 Table 2. Relationship Between Tensorial Strain, Engineering Strain, And Engineering Strain Expressed In Compact Notation. Tensorial Strain ε11 ε 22 ε33 Engineering Strain e11 e22 e33 Engineering Strain Expressed in Compact Notation ε1 ε2 ε3 1 ε 23 2 1 ε13 2 1 ε12 2 e23 = γ 23 ε4 e13 = γ13 ε5 e12 = γ12 ε6 TRANSFORMATION OF STRAINS FROM ONE COORDINATE SYSTEM TO ANOTHER The differences in the definition of tensorial and engineering strains, though seemingly small, are significant. The normal strains in both definitions are equal. However, the shear strains differ by a factor of 1 2 . The consequence of this difference is that engineering strains are not directly transformable from one coordinate system to another; they must first be converted to tensorial strains before the coordinate transformation and then converted back to engineering strain. The end result is that the transformation of the strains measured along the structural directions into the principal material directions is 2 n2 mn  ε x   ε1   m      2 − mn  ε y  , m2 ε 2  =  n ε  − 2mn 2mn m 2 − n 2  ε   6   s  4 where m = cosθ and n = sin θ , as defined previously. PLANE STRESS ASSUMPTION There are two assumptions that are commonly made in the analysis of solids and structures to simplify the governing system of equations. They are: • Plane Stress, and Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS • 53 Plane Strain. The assumption of plane stresses is that the out-of-plane stresses are much much less than the in-plane stresses. In this application assume that the out-of-plane direction is the z- or x3-direction, then σ33 = σ13 = σ 23 = 0 . The plane stress assumption is commonly applied to structures that are very thin in one direction, such as a composite lamina. The assumption of plane strain is that the out-of-plane strains are zero, or much much less than the in-plane stresses. If the out-of-plane direction is the z- or x3-direction, then ε33 = ε13 = ε 23 = 0 . The plane strain assumption is commonly applied to structures that are very thick in the out-ofplane direction. These assumptions have similar impact on the governing system of equations, but they are very different assumptions and they have distinctly different meanings. The plane stress assumption is used in the analysis of composite lamina. CONSTITUTIVE RELATIONS Hooke originally postulated a linear relationship between stress and strain, i. e., σ = Eε , where E is the constant of proportionality. This relationship has been generalized to the full three dimensional stress state at a point. The generalized form of Hooke’s Law in terms of tensorial stresses and strains is: 3 3 σij = ∑∑ Cijkl ε kl , k =1 l =1 where Cijkl is the stiffness tensor. The stiffness tensor is a fourth order tensor which represents, 34 or 81 independent constants. When the symmetries of the stress and strain tensors, and assumptions about the internal strain energy are applied these 81 independent constants are reduced to 21 and the Generalized Hooke’s Law can be expressed as a matrix expression. It is: {σ} = [C ]{ε} , where {σ} is the 1× 6 stress vector, {ε} is the 1× 6 strain vector, and [C ] is the symmetric 6× 6 stiffness matrix. If all of the terms are shown and compact notation is used, the general constitutive relation for any material is: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS  σ1  C11 σ  C  2   12 σ3  C13  = σ 4  C14 σ 5  C15    σ 6  C16 C12 C22 C23 C24 C25 C26 C13 C23 C33 C34 C35 C36 54 C14 C 24 C34 C 44 C45 C 46 C15 C25 C35 C45 C55 C56 C16   ε1  C26  ε 2  C36  ε3    . C46  ε 4  C56  ε 5    C66  ε 6  5 Alternately, the constitutive relation may be written in the compliance form where strains are expressed in terms of stresses. This form is: {ε} = [S ]{σ} , where [S ] is the symmetric 6× 6 compliance matrix, which is the inverse of the stiffness matrix. If all of the terms are shown and compact notation is used, the general compliance form of the constitutive relation for any material is: ε1   S11 ε   S  2   12 ε3   S13  = ε 4   S14 ε 5   S15    ε 6   S16 S12 S 22 S 23 S 24 S 25 S 26 S13 S 23 S33 S 34 S35 S 36 S14 S 24 S34 S 44 S 45 S 46 S15 S 25 S 35 S 45 S 55 S56 S16   σ1  S 26  σ 2  S36  σ3    . S 46  σ 4  S56  σ5    S66  σ 6  6 When observations of or assumptions about material behavior are made conclusions may be made regarding physical planes of symmetry in the material and consequently reduce the number of independent terms in the stiffness matrix. ANISOTROPIC MATERIALS Materials which display no axes of symmetry in response to mechanical loads are defined as anisotropic, literally “not isotropic.” These materials have 21 independent material constants as shown in the equations above. For the analyst this is the worst possible situation. The implications of a full stiffness matrix is that a normal strain will produce normal and shear stresses and, conversely, that shear strains will produce shear and normal stresses. ORTHOTROPIC MATERIALS Materials which display three perpendicular axes of symmetry in response to mechanical loads are defined as orthotropic. These materials have 9 independent material constants and the stiffness form of the constitutive relation is: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS  σ1  C11 σ  C  2   12 σ3  C13  = σ 4   0 σ 5   0    σ 6   0 C12 C22 C23 0 0 0 C13 C23 C33 0 0 0 55 0 0 0 C44 0 0 0 0 0 0 C55 0 0   ε1  0  ε 2  0  ε 3    0  ε 4  0  ε 5    C66  ε6  7 and the compliance form of the constitutive relation is:  ε1   S11 ε   S  2   12 ε3   S13  = ε 4   0 ε 5   0    ε 6   0 S12 S 22 S 23 0 0 0 S13 S 23 S33 0 0 0 0 0 0 S 44 0 0 0 0 0 0 S55 0 0   σ1  0  σ 2  0  σ3    . 0  σ 4  0  σ 5    S 66  σ 6  8 For an orthotropic material it is easily shown through simple linear algebra that:  S 22 S33 − S 23 2  S  S S  13 23 − S12 S33  S S S − S S  12 23 13 22 S [C ] =   0    0   0   S13 S 23 − S12 S33 S 2 S11 S33 − S13 S S12 S13 − S 23 S11 S S12 S 23 − S13 S 22 S S12 S13 − S 23 S11 S 2 S11S 22 − S12 S 0 0 0 0 0 0 0 0 1 S 44 0 0 0 0 1 S55 0 0 0 0  0   0    0  , 0   0   1   S 66  9 where S = S11S 22 S 33 − S11S 23 − S 22 S13 − S33 S12 + 2 S12 S 23 S13 . 2 2 2 This is the most common symmetry found in composite lamina, particularly filamentary lamina. These expressions are also referred to as the on-axis properties. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS 56 TRANSVERSELY ISOTROPIC MATERIALS Materials which display an infinite number of axes of symmetry in one plane in response to mechanical loads are defined as transversely isotropic. These materials have 5 independent material constants and the constitutive relation is:  σ1  C11 σ  C  2   12 σ3  C13  = σ 4   0 σ 5   0    σ 6   0 C12 C11 C13 0 0 0 C13 C13 C33 0 0 0 0 0 0 C44 0 0 0   ε1   ε  0  2   ε3  0   0  ε 4   ε 5  0   1  ε 6  2 (C11 − C12 ) 0 0 0 0 C44 0 10 and the compliance form of the constitutive relation is:  ε1   S11 ε   S  2   12 ε3   S13  = ε 4   0 ε 5   0    ε 6   0 S12 S11 S13 0 0 0 S13 S13 S33 0 0 0 0 0 0 S 44 0 0 0 0 0 0 S 44 0 0   σ1   σ  0  2   σ3  0   . 0  σ 4   σ 5  0   2(S11 − S12 ) σ 6  11 ISOTROPIC MATERIALS Materials which display an infinite number of axes of symmetry in response to mechanical loads are defined as isotropic. These materials have 2 independent material constants and the constitutive relation is:  σ1  C11 σ  C  2   12 σ3  C12  = σ 4   0 σ 5   0    σ 6   0 C12 C11 C12 0 0 0 C12 C12 C11 0 0 0 0 0 0 1 2 (C11 − C12 ) 0 0 0 0 0 0 1 2 (C11 − C12 ) 0 0   ε1   ε  0  2   ε3  0   0  ε 4   ε 5  0   1  ε 6  2 (C11 − C12 ) 12 and the compliance form of the constitutive relation is: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS  ε1   S11 ε   S  2   12 ε3   S12  = ε 4   0 ε 5   0    ε 6   0 S12 S11 S12 0 0 0 57 0 0 0   σ1   σ  0 0 0  2   σ 3  0 0 0   . 2(S11 − S12 ) 0 0  σ 4   σ 5  0 2(S11 − S12 ) 0   0 0 2(S11 − S12 ) σ6  S12 S12 S11 0 0 0 13 ENGINEERING PROPERTIES FOR ORTHOTROPIC MATERIALS Engineering properties are more commonly used to describe the mechanical behavior of materials than stiffnesses or compliances. The engineering properties that you are familiar with are Young’s modulus (also known as modulus of elasticity, or elastic modulus), Shear modulus (also known as modulus of rigidity), and Poisson’s ratio. For an orthotropic material there are three principal directions, associated with the three directions of material symmetry. Associated with each of these directions are a Young’s modulus, Ei , a shear modulus, Gij and a Poisson’s ratio, ν ij . The properties are easily shown to be related to the compliances. The result is:  1  E  1  − ν12  ε1   ε   E1  2   − ν13 ε3   E1  = ε 4   0 ε 5      ε 6   0   0  − ν 21 E2 1 E2 − ν 23 E2 − ν 31 E3 − ν 32 E3 1 E3 0 0 0 0 0 0 0 0 1 G23 0 0 0 1 G13 0 0 0 0 0  0   0   σ1     σ2 0    σ3  .   0  σ 4   σ 5    0  σ 6   1   G12  14 Note in this expression that there are three Young’s moduli, one in each of the principal material directions, three shear moduli, one for each principal material direction pairs, and six Poisson’s ratios. Because the compliance matrix must be symmetric, as discussed earlier, there are only three unique Poisson’s ratios. The Poisson’s rations are interrelated by the expression: ν ij = Ei ν ji . Ej Revised: 10 February, 2000 15 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS 58 Through linear algebraic manipulations the stiffness matrix can be expressed in terms of the nine engineering properties. This relationship is:  E2 − E3ν 23 2  2 E2 E3C   E2ν12 + E3ν13ν 23  E1 E2 E3C  [C ] =  ν13 + ν12 ν 23  E1E2C  0   0  0  E2ν12 + E3ν13ν 23 E1E2 E3C 2 E1 − E3ν13 2 E1 E3C E1ν 23 + ν12 ν13 2 E1 E2C 0 0 0 ν13 + ν12 ν 23 E1E2C E1ν 23 + ν12 ν13 2 E1 E2C 2 E1 − E2ν12 2 E1 E2C 0 0 0 0 0 0 0 0 0 G23 0 G13 0 0 0  0    0   , 0   0  0   G12  16 where E E E − E1 E3ν 23 − E2 E3ν13 − E2 ν12 − 2 − E1 E2 ν12 ν 23 ν13 C= 1 2 3 . 2 2 E1 E2 E3 2 2 2 2 PLANE STRESS ORTHOTROPIC CONSTITUTIVE RELATION The plane stress assumption can be applied to analysis of orthotropic lamina. When the plane stress assumption is applied the constitutive relation is simplified from a 6 × 6 matrix expression to a 3× 3 expression. For the compliance expression a simple static condensation is performed to reduce the system to a system of three equations. The plane stress compliance equation is:  ε1   S11    ε 2  =  S12 ε   0  6  S12 S 22 0 0   σ1    0  σ 2  . S66  σ6  17 Note in this expression that there is no change to the compliances, that is reduced compliance terms are the same as those in the three dimensional compliance expression. Reduction of the stiffness expression is not as simple as the compliance equation. To simplify the stiffness expression one must solve for the out-of-plane normal strain in terms of the in-plane stresses and stiffnesses and substitute the results back into the stiffness expression, i. e., Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS {σ} = [C ]{ε} ⇒ 59 ε3 = − (C13 ε1 + C23ε 2 ) C33 ε4 = 0 , ε5 = 0  σ1  C11 σ  C  2   12  0  C13  = 0  0 0  0    σ 6   0 C12 C22 C23 0 0 0 C13 C23 C33 0 0 0 0 0 0 C44 0 0 0 0 0 0 C55 0 ε1  0    ε2 0    − (C13 ε1 + C23 ε 2 )  0   C33 ,  0   0    0 0   C66    ε  6 which gives: C13   C11 − C 33 σ1   C23    σ 2  = C12 − C33 σ    6  0   C23 C33 C C22 − 13 C33 0 C12 −  0    ε1    0  ε 2    C66  ε 6    This expression is also written as:  σ1  Q11 Q12    σ 2  = Q12 Q22 σ   0 0  6  0   ε1    0  ε 2  , Q66  ε 6  18 where [Q ] is the so-called reduced stiffness matrix. REDUCED STIFFNESSES IN TERMS OF ENGINEERING PROPERTIES The reduced stiffnesses can also be written in terms of the engineering properties. This relationship is: 2  E1  2  E1 − E2 ν12 [Q] =  ν12 E1E2 2 E − E2 ν12  1 0   ν12 E1 E2 2 E1 − E2 ν12 E1E2 2 E1 − E2 ν12 0 Revised: 10 February, 2000  0   0  .  G12   19 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS 60 The reduced compliances are the same as the compliance terms. OFF-AXIS PROPERTIES OF ORTHOTROPIC LAMINA As discussed earlier there are four coordinate systems that are used in the analysis of orthotropic lamina. The stress-strain relationship in the principal material directions is referred to as on-axis. The behavior in the structural directions is referred to as off-axis behavior. Shown earlier were the transformations of stress and strain from the principal material directions to the structural directions. Writing: {σ}12 = [T ]{σ}xy where {σ}12 are the stresses in the principal material directions, {σ}xy are the stresses in the structural directions, and [T ] is the stress transformation matrix, then we can write the inverse relation, {σ}xy = [T ]−1{σ}12 where [T ] is the inverse of the stress transformation matrix. Noting that −1 {ε}12 = [T ′]{ε}xy , where {ε}12 are the strains in the principal material directions, {ε}xy are the strains in the structural directions, and [T ′] is the strain transformation matrix, and {σ}12 = [Q]{ε}12 , then we can write {σ}xy = [T ]−1{σ}12 = [T ]−1[Q]{ε}12 = [T ]−1[Q][T ′]{ε}xy = [Q ]{ε}xy , or σ x  Qxx    σ y  = Qxy σ  Q  s   xs Qxy Qyy Qys Qxs  ε x    Qys  ε y  , Qss   ε s  20 where [Q ] is the off-axis reduced stiffness matrix. When the matrix multiplications are carried out the off-axis reduced stiffnesses are found to be: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS Qxx   m 4    4 Qyy   n Qxy   m 2 n 2  = 2 2 Qss   m n Qxs   m 3n    3 Q ys   mn n4 m4 m 2n2 m 2n2 61 2m 2 n 2 2m 2 n 2 m4 + n4 − 2m 2 n 2 ( − mn3 − mn m 2 − n 2 − m 3n mn m 2 − n 2 ( ) )    Q11   Q22    ,  Q12  2 2  − 2mn m − n Q66   2mn m 2 − n 2  4m 2n 2 4m 2n 2 − 4m 2 n 2 2 m2 − n2 ( ( ) ( ) 21 ) again, m and n are defined as before. Through a similar process the off-axis compliance relation can be shown to be: ε x   S xx    ε y  =  S xy ε   S  s   xs S xy S yy S ys S xs  σ x    S ys  σ y  , S ss  σ s  22 When the matrix where [S ] is the off-axis reduced compliance matrix. multiplications are carried out the off-axis reduced compliances are found to be: S xx   m 4 S   4  yy   n S xy   m 2 n 2  = 2 2  S ss   4m n  S xs   2m3 n    S ys   2mn3 n4 2m 2 n 2 m4 2m 2 n 2 m2 n 2 m4 + n4 − 8m 2 n 2 4m 2 n 2 − 2mn 3 − 2mn m 2 − n 2 − 2m 3 n 2mn m 2 − n 2 ( ( ) ) m2 n2 m2 n2 − m2 n2 2 m 2 − n2 − mn m 2 − n 2 mn m 2 − n 2 ( ( ( ) )     S11   S 22    ,   S12   S   66  23 ) again, m and n are defined as before. INVARIANT PROPERTIES OF ORTHOTROPIC LAMINA When the powers of the trigonometric functions in the transformation equation for the stiffnesses above are converted to the multiple angle form a set of combinations of the lamina properties are discovered. These combinations are defined as invariant stiffness properties of the lamina. The invariants, U iQ , are: 1 [3Q11 + 3Q22 + 2Q12 + 4Q66 ] 8 1 U 2Q = [Q11 − Q22 ] 2 1 U 3Q = [Q11 + Q22 − 2Q12 − 4Q66 ] 8 U1Q = Revised: 10 February, 2000 24 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS 62 1 [Q11 + Q22 + 6Q12 − 4Q66 ] 8 1 U 5Q = [Q11 + Q22 − 2Q12 + 4Q66 ] 2 U 4Q = and the off-axis stiffness properties are related to the invariant properties by: Qxx  U1Q    Q Qyy  U1 Qxy  U 4Q  = Q Qss  U 5 Qxs   0    Qys   0 cos2θ cos4θ   - cos2θ cos4θ  1  0 - cos4θ  Q   U 2  . 0 - cos4θ  Q  U 3  1 sin 2θ sin 4θ  2  1 θ θ sin 2 sin 4  2 25 Similarly, invariant compliance properties, U iS , are found to be: 1 [3S11 + 3S22 + 2S12 + S66 ] 8 1 = [S11 − S 22 ] 2 1 = [S11 + S 22 − 2 S12 − S66 ] 8 1 = [S11 + S 22 + 6 S12 − S 66 ] 8 1 = [S11 + S 22 − 2 S12 + S66 ] 2 U1S = U 2S U 3S U 4S U 5S 26 and the off-axis compliances are related to the invariant properties by:  S xx  U1S S   S  yy  U1  S xy  U 4S  = S  S ss  U 5  S xs   0     S ys   0 cos2θ -cos2θ 0 0 sin2θ sin2θ cos4θ   cos4θ   1  -cos4θ   S   U 2  . -4cos4θ  S  U 3  2sin4θ   -2sin4θ  27 OFF-AXIS ENGINEERING PROPERTIES OF ORTHOTROPIC LAMINA Just as with the on-axis properties, the off-axis properties can be expressed in terms off-axis engineering properties. Associated with the structural directions is a Young’s modulus, E x , E y , a shear modulus, Gs a Poisson’s ratio, ν xy , and Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS 63 coefficients of mutual influence, η x,s , η y ,s , ηs , x , ηs , y . The properties are easily shown to be related to the compliances. The result is:  1  ε x   E x    − ν xy ε y  =  ε   E x  s  η s ,x  E  x − ν xy Ex 1 Ey ηs , y Ey η x ,s   Gs   σ  x η y ,s     σ y  . Gs    σ 1  s  Gs  28 The coefficients of mutual influence, as defined by Lekhnitski, characterize the stretching due to shear, first kind, and the shearing due to stretching, second kind. Specifically, the coefficients of mutual influence of the first kind are defined as: ηi ,s = εi , i = x, y εs and the coefficients of mutual influence of the second kind are defined as: ηs ,i = εs , i = x, y . εi Symmetry of the compliance matrix applies, therefore, η x ,s ηs , x . = Gs Ex and η y ,s ηs , y = . Gs Ey The off-axis engineering properties can be written in terms of the on-axis engineering properties. These relations are:  1 2ν12  2 2 1 4 1 1 m n + = m 4 +  − n , E x E1 E1  E2  G12  1 2ν12  2 2 1 4 1 1 m n + = n 4 +  − m , E y E1 E1  E2  G12 Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINA MECHANICS 64  2 1 2 4ν12 1  2 2 1 m n + m4 + n4 , = 2 + + − Gs E E E G G 2 1 12  12  1 ( ) ν 1 1 1  2 2 m n  , ν xy = Ex  12 m 4 + n 4 −  + −  E1 E2 G12   E1  ( )  2 2ν 1  2  2 2ν12 1  2 n  ,  m −  + − ηs , x = Ex mn  + 12 − E1 G12  E1 G12    E2  E1  2 2ν 1  2  2 2ν12 1  2 m  , n −  + − ηs , y = E y mn  + 12 − E1 G12  E1 G12    E2  E1 where m and n are as defined before. Revised: 10 February, 2000 M. V. Bower STRENGTH OF LAMINA TENSOR POLYNOMIAL FAILURE CRITERION The principle that underlies a failure criterion is that a function of material properties and current load values can be defined that separates the domain of safe response from the failed domain. Figure 3 shows a typical failure surface in the principal material direction stress space. Note that the plot in the figure is a two-dimensional slice out of the three-dimensional failure surface. Also note that the horizontal and vertical scales are not equal. The region inside the curve is the “safe” region while the region outside the surface is failed. Therefore failure is defined to occur on the surface. The function that defines the surface is defined as the failure criterion function. The function may be derived phenomenologically or empirically. For metals the yield criteria proposed by Tresca and Henke and Von Mises have theoretical basis in the mechanics of materials and are supported by experimental evidence. The application of a failure criterion to composite materials is much more problematic. There is minimal theoretical basis that justifies the use of a failure criterion. However, there is ample experimental evidence that the application of a failure criterion to composite materials. f = 0.835 R = 10 1.170 Current State 0 -600 -500 -400 -300 -200 -100 0 100 200 300 σ2 kpsi -10 -20 -30 -40 -50 σ1 kpsi Figure 3. Failure Envelop in the Stress Domain. Revised: 10 February, 2000 Page 65 COMPOSITE MATERIALS STRENGTH OF LAMINA 66 The tensor polynomial in its most general form was presented first by Goldenblat' and Knoppov as an expansion involving the current stress state and various strength tensors. At failure this tensor polynomial is expressed as 1 = f (σ) = Fij (σij ) + Fijkl (σij σ kl ) + Fijklmn (σij σ kl σ mn ) + Fijklmnop (σij σ kl σ mnσ op ) + L , α β γ δ where σij is the stress tensor expressed as a rank two tensor with indices from 1 to 3, and Fij , Fijkl , Fijklmn , and Fijklmnop are the strength tensors. Following the standard notation for tensor mathematics, there is an implicit summation from 1 to 3 for each repeated index in the expression. From this expression, when failure occurs the polynomial expansion equals one, by definition. Tsai and Wu presented the tensor polynomial in the form in which all exponents are one. They showed that there is no loss in generality from the form of Goldenblat' and Knoppov. Thus, the Tsai-Wu form of the tensor polynomial is 1 ≥ f (σ ) = Fij σij + Fijkl σij σ kl + Fijklmn σ ij σ kl σ mn + Fijklmnop σij σ kl σ mn σop + L . Alternately, using contracted or compact notation the tensor polynomial is 1 ≥ f (σ ) = Fi σi + Fij σi σ j + Fijk σi σ j σ k + Fijkl σi σ j σ k σl + L , where σ i is the stress tensor expressed as a rank one tensor with indices from 1 to 6, and Fi , Fij , Fijk , and Fijkl are the strength tensors. Note in this expression the convention used in tensor mathematics is used with the exception that the implicit summation over repeated indices is from 1 to 6. The stresses used in this expression may be in the principal material directions, the principal stress directions, or some structural direction. If stresses other that those from the principal material directions are used, the strength tensors must be transformed appropriately. When the tensor polynomial, f (σ ) , equals one failure occurs. When the function is less than one the lamina is not failed and conversely when the function is greater than one the lamina is failed. QUADRATIC FAILURE CRITERION In the original development of Tsai and Wu, the general tensor polynomial failure criterion was simplified to a quadratic polynomial. If one considers the onaxis, or principal material direction, strength behavior of the material loaded in plane stress, the quadratic form of the tensor polynomial is f (σ ) = F1σ1 + F2σ 2 + F6σ6 + F11σ1 + F22 σ 2 + F66σ 6 + 2 2 2 , F12σ1σ 2 + F21σ1σ 2 + F16 σ1σ 6 + F61σ1σ 6 + F26 σ 2σ6 + F62 σ 2σ6 Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS STRENGTH OF LAMINA 67 or f (σ ) = F1σ1 + F2σ 2 + F6σ6 + F11σ1 + F22 σ 2 + F66σ 6 + 2 2 2 (F12 + F21 )σ1σ2 + (F16 + F61 )σ1σ6 + (F26 + F62 )σ2σ6 . Note that the effect of the plane stress assumption is to eliminate the terms F3 , F4 , F5 , F13 , F14 , K F55 , thus reducing the expression to the form shown. Now assume, without loss of generality, that the strength tensor is a symmetric tensor. Thus, Fij = Fji . Therefore, the failure criterion can be expressed as f (σ ) = F1σ1 + F2 σ 2 + F6 σ6 + F11σ1 + F22 σ 2 + F66 σ 6 + 2 F12 σ1σ 2 + 2 F16σ1σ6 + 2 F26σ 2σ 6 2 2 2 . Consider two stress states (σ1 , σ2 , σ6 ) , and (σ1 , σ2 ,− σ6 ) that are both found to cause failure. Thus, T T 1 = F1σ1 + F2 σ2 + F6 σ6 + F11σ1 + F22 σ2 + F66 σ6 + 2 F12 σ1σ2 + 2 F16 σ1σ6 + 2 F26 σ2 σ6 2 2 2 and 1 = F1σ1 + F2 σ2 − F6 σ6 + F11σ1 + F22 σ2 + F66 σ6 + 2 F12 σ1σ2 − 2 F16 σ1σ6 − 2 F26 σ2 σ6 2 2 2 . Taking the difference between these equations one finds that 0 = −2 F6 σ6 − 4 F16 σ1σ6 − 4 F26 σ2 σ6 , or 0 = −2(F6 − 2 F16 σ1 − 2 F26 σ2 )σ6 . In these load states σ6 ≠ 0 by definition, thus, the term in the brackets, F6 − 2 F16 σ1 − 2 F26 σ2 , must be zero. However, the applied stresses, σ1 , σ2 , are independent variables, that is this expression must hold true for all stress states. Therefore, this expression can be zero if and only if, F6 = F16 = F26 = 0 . Consequently, the quadratic form of the tensor polynomial failure criterion simplifies to: f (σ ) = F1σ1 + F2 σ 2 + F11σ1 + F22 σ 2 + F66 σ6 + 2 F12 σ1σ 2 . 2 Revised: 10 February, 2000 2 2 M. V. Bower COMPOSITE MATERIALS STRENGTH OF LAMINA 68 The elements of the strength tensors can be related to the strengths measured in standard tensile tests. These relations yield: F1 = 1 1 1 1 1 1 1 − − , F11 = , F2 = , F22 = , and F66 = 2 , XT XC YT YC XT XC YT YC S where X T and YT are the tensile strengths in the first and second principal material directions, respectively, X C and YC are the compressive strengths in the first and second principal material directions, respectively, and S is the principal material direction shear strength. Unfortunately, the interaction term, F12 , cannot be determined from a simple uniaxial tensile test. Various interaction terms are presented in the literature. In the Tsai-Hill failure criterion, the tensile and compressive strengths are assumed to be equal and the interaction term is given as − 1 2 . Hahn presented the interaction term for unequal tensile and compressive 2X strengths as − 1 . Fundamental constraints on the form of the quadratic 2 X T X C YT YC tensor polynomial require that F11F22 ≥ F12 . Hahn’s interaction term satisfies this requirement. 2 The plot shown in Figure 3 is a plot of a quadratic tensor polynomial failure criterion. Note the characteristically ellipsoidal shape of the failure surface, this is directly related to the quadratic form of the failure function. Further note, that the failure surface is concave, this also is directly related to the form of the failure function. In three-dimensional stress space the failure surface resembles a cigar or thin rugby ball (a football is too pointed at the ends). The function value shown in the figure represents the criterion value at the “current stress” point. QUADRATIC TENSOR POLYNOMIAL TRANSFORMATION TO OFF-AXIS FORM Noting that a stress state can be transformed from one coordinate system to another through the orthogonal transformation 2 n2 2mn  σ x   σ1   m      2 2 m − 2mn  σ y  , σ 2  =  n σ   − mn mn m 2 − n 2  σ   6   s  where m = cosθ and n = sin θ , the result of the tensor polynomial failure criterion must be invariant with respect to coordinate transformation. Another way to state this principle is: any stress state that causes failure in one coordinate system must also cause failure in any other stress state obtained through simple coordinate system rotation. Substituting structural direction stresses for the principal material direction stresses in the quadratic tensor polynomial yields: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS STRENGTH OF LAMINA 69 f (σ ) = Fx σ x + Fy σ y + Fs σ s + Fxx σ x + Fyy σ y + Fss σ s + 2 Fxy σ x σ y + 2 Fxs σ x σ s + 2 Fys σ y σ s 2 2 2 , where 2 n2   Fx   m  F     2 m2  1  Fy  =  n  F  2mn − 2mn  F2   s   and  Fxx   m 4 F   4  yy   n  Fxy   m 2 n 2  = 2 2  Fss   4m n  Fxs   2m 3n     Fys   2mn 3 n4 2m 2 n 2 m4 2m 2 n 2 m2n2 m4 + n4 − 8m 2 n 2 4m 2 n 2 − 2mn3 − 2mn m 2 − n 2 − 2m 3n 2mn m 2 − n 2 ( ( ) ) m 2n2 m 2n2 − m 2n2 2 m2 − n 2 − mn m 2 − n 2 mn m 2 − n 2 ( ( ( ) )     F11   F22    ,   F12   F66    ) again, m and n are defined as before. As with the stiffness properties the strength tensors can be expressed in terms of invariant properties. For the strength tensors the invariants are 1 [F1 + F2 ] 2 1 u2f = [F1 − F2 ] 2 u1f = 1 [3F11 + 3F22 + 2 F12 + F66 ] 8 1 = [F11 − F22 ] 2 1 = [F11 + F22 − 2 F12 − F66 ] . 8 1 = [F11 + F22 + 6 F12 − F66 ] 8 1 = [F11 + F22 − 2 F12 + F66 ] 2 U1f = U 2f U 3f U 4f U 5f The strength tensors in terms of the strength invariants are: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS STRENGTH OF LAMINA 70  Fx  1 cos 2θ  f     u1  Fy  = 1 − cos 2θ  f   F  0 2 sin 2θ  u 2   s    Fxx  U1f F   f  yy  U1  Fxy  U 4f  = f  Fss  U 5  Fxs   0     Fys   0 cos2θ cos4θ   - cos2θ cos4θ   1  0 - cos4θ   f   U 2  . 0 - 4cos4θ  f  U 3  sin 2θ 2sin 4θ   sin 2θ - 2sin 4θ  It should be noted here that the stresses in the principal material directions are not invariant with respect to rotation. Therefore, the failure criterion formulated in terms of the principal material direction stresses is not necessarily invariant with respect to coordinate rotation, a primary requirement for a failure criterion. However, Bower and Koedam developed a proof to establish that the failure criterion formulated in terms of the principal stresses is invariant. Therefore, the analyst is free to use whichever formulation is most convenient, provided of course the appropriate application of the required transformations. STRAIN FORMULATION OF THE TENSOR POLYNOMIAL FAILURE CRITERION Given the Tsai-Wu second order form of the tensor polynomial failure criterion in terms of stress and the plane stress constitutive relation σi = Qij ε j , where the Qij are the so-called reduced stiffnesses, with i, j = 1, 2,&6 , the criterion, g (ε ) , can be expressed in terms of strain such that 1 ≥ g (ε ) = Fi Qij ε j + Fij Qik Q jl ε k ε l or 1 ≥ g (ε ) = Gi εi + Gij εi ε j , where G j = Fi Qij and Gkl = Fij Qik Q jl . Given a state of plane stress, the failure criterion is rewritten as: 1 ≥ g (ε ) = G1ε1 + G2 ε 2 + G11ε1 + G22 ε 2 + G66 ε 6 + 2G12 ε1ε 2 . 2 Revised: 10 February, 2000 2 2 M. V. Bower COMPOSITE MATERIALS STRENGTH OF LAMINA 71 The same arguments are used in this development regarding symmetry with respect to shear as in the preceding development. In this expression the coefficients are: G11 = F11Q11 + 2 F12 Q11Q12 + F22 Q12 2 2 G22 = F11Q12 + 2 F12 Q12 Q22 + F22 Q22 2 ( 2 ) G12 = F11Q11Q12 + F12 Q11Q22 + Q12 + F22 Q12 Q22 G66 = F66 Q66 2 . 2 G1 = F1Q11 + F2 Q12 G2 = F1Q12 + F2Q22 OFF-AXIS FORM OF THE STRAIN FORMULATION OF THE TENSOR POLYNOMIAL FAILURE CRITERION As with the previous development, note that a strain state can be transformed from one coordinate system to another through the orthogonal transformation: 2 n2 mn  ε x   ε1   m      2 2 − mn  ε y  , m ε 2  =  n ε  − 2mn 2mn m 2 − n 2  ε   6   s  where m and n are as defined previously. Substituting structural direction strains for the principal material direction strains in the strain form of the quadratic tensor polynomial yields: g (ε ) = Gx ε x + G y ε y + Gs ε s + Gxx ε x + G yy ε y + Gss ε s + 2Gxy ε x ε y + 2Gxs ε x ε s + 2G ys ε y ε s 2 2 2 , where 2 n2  Gx   m  G     2 m2   1  G y  =  n G   mn − mn  G2   s   and Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS STRENGTH OF LAMINA Gxx   m 4 G   4  yy   n Gxy  m 2 n 2  = 2 2 Gss  m n Gxs   m 3n    3 G ys   mn 72 n4 2m 2 n 2 m4 2m 2 n 2 m 2n2 m4 + n4 − 2m 2 n 2 m 2n2 − mn3 − mn m 2 − n 2 − m 3n mn m 2 − n 2 ( ( ) ) 4m 2 n 2 4m 2 n 2 − 4m 2 n 2 2 m2 − n 2 − 2mn m 2 − n 2 2mn m 2 − n 2 ( ( ( ) )    G11   G22    ,  G12   G66    ) again, m and n are defined as before. As with the stiffness properties and strength tensors can be expressed in terms of invariant properties. For the strength tensors the invariants are 1 [G1 + G2 ] 2 1 u2g = [G1 − G2 ] 2 u1g = U1g = U 2g = U 3g = U 4g = U 5g = 1 [3G11 + 3G22 + 2G12 + 4G66 ] 8 1 [G11 − G22 ] 2 1 [G11 + G22 − 2G12 − 4G66 ] . 8 1 [G11 + G22 + 6G12 − 4G66 ] 8 1 [G11 + G22 − 2G12 + 4G66 ] 2 The strain form of the strength tensors in terms of the invariants are: Gx  1 cos 2θ  g     u1  G y  = 1 − cos 2θ  g  G  0 sin 2θ  u2    s  Gxx  U1g G   g  yy  U1 g Gxy  U 4 =    g Gss  U 5 Gxs   0    G ys   0 cos2θ cos4θ   - cos2θ cos4θ  1  0 - cos4θ  g   U 2  . 0 - cos4θ  g  U 3  1 sin 4θ  2 sin 2θ  1 - sin 4θ  2 sin 2θ Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS STRENGTH OF LAMINA 73 R-FACTOR ANALYSIS In the analysis of traditional engineering materials it is common practice to determine a factor of safety. For the traditional engineering materials one method of determining the factor of safety is to take the ratio of the Von Mises-Henke stress for the load to the tensile yield stress for the material. In the analysis of composite materials an analogous process is used, the R-factor analysis. An R-factor is not a safety factor. An R-factor is the factor that scales the initial stress state to failure. Strictly speaking a factor of safety would describe how far away from failure the system is at the prescribed load. For a composite material this would require finding the shortest distance to the failure surface from the prescribed stress state in stress space. An R-factor scales the stress state along a radial in stress space, which in most cases is not the shortest distance to the failure surface. Based on the previous discussion define a new stress state, {σ′} , that is a multiple, R, of the initial stress state: σ x  σ′x      σ′y  = R σ y  . σ  σ′   s  s This stress state is defined to be a stress state that causes failure. Therefore: f (σ′) = 1 = Fx σ′x + Fy σ′y + Fs σ′s + , Fxx σ′x + Fyy σ′y + Fss σ′s + 2 Fxy σ′x σ′y + 2 Fxs σ′x σ′s + 2 Fys σ′y σ′s 2 2 2 note the above is the off-axis form of the failure function, by appropriate exchange of indices it can also be used to show the on-axis form. In this case the initial stress state is known. Therefore, the unknown scale factor, R, can be found through the solution of the quadratic equation: 0 = −1 + R(Fx σ x + Fy σ y + Fs σ s ) + ( R 2 Fxx σ x + Fyy σ y + Fss σ s + 2 Fxy σ x σ y + 2 Fxs σ x σ s + 2 Fys σ y σ s 2 2 2 ). Fortuitously, the form of this equation is such that we are guaranteed to get a real solution. Thus, the R-factor is: R1, 2 = − b ± b 2 + 4a , 2a where Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS STRENGTH OF LAMINA ( 74 a = Fxx σ x + Fyy σ y + Fss σ s + 2 Fxy σ x σ y + 2 Fxs σ x σ s + 2 Fys σ y σ s 2 2 2 b = (Fx σ x + Fy σ y + Fs σ s ) ). Note this produces two real values. One should be positive and the other negative. The factor of interest is the positive value. The negative value corresponds to the factor needed to intersect the failure surface along the opposite radial in stress space. An analogous form exists for the strain form of the failure function. It is: R1, 2 = − b ± b 2 + 4a , 2a where ( a = Gxx ε x + G yy ε y + Gss ε s + 2Gxy ε x ε y + 2Gxs ε x ε s + 2G ys ε y ε s 2 2 b = (Gx ε x + G y ε y + Gs ε s ) Revised: 10 February, 2000 2 ). M. V. Bower CLASSICAL LAMINATION THEORY HISTORY The history of plate analysis began with Leonard Euler (1707-1783). In his approach he presented two methods: a direct method, which is an application of the equilibrium principle; and the method of final causes, which is a variational principle. In this work, the plate was treated as a membrane made up of strings. Jacque Bernoulli (1759-1789) was the next to contribute to the development of the plate theory. In his attempt, he assumed that a plate is made up of beams. This caused him to miss the twist term and have an incorrect stiffness constant due to missing the Poisson stiffening. Next, Ernst Florens Fredric Chadni (1756-1827) approached the development of a plate theory by using acoustical vibration with sand. In 1809, at the direction of Napoleon, the French Academy of Science announced a competition to develop a more accurate plate theory. October 1811 was established as the closing date for submission of the theories. The contest was won by Mlle. Sophie Germain (1776-1831). In this theory, she proposed that the energy in the plate was 2 1 1  A∫  +  dA  ρ1 ρ 2  . This theory was the best to date and only missed by failure to include the twist energy. LaGrange then proposed a plate theory in which ∂ 4w ∂4w ∂4w  ∂2w k 4 + 2 2 2 + 4  + 2 = 0 . ∂x ∂y ∂y  ∂t  ∂x The French Academy of Science recognized the available plate theory was inadequate and announced a second closure date of October 1813. No better theory was put forward and so the Academy announced a third closure date, October 1816. Revised: 10 February, 2000 Page 75 COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY was 76 S. D. Poisson (1781-1840) put forth a theory in which the energy in the plate  1 1  2  1 1   A∫  +  + m  2 + 2  dA .  ρ1 ρ 2   ρ1 ρ 2   The form of the energy is correct. However, Poisson had the wrong values for A and m. On August 14, 1820, Navier (1785-1836) recognized that plane sections remain plane in a plate in bending. This fundamental observation was published in 1823. Navier was responsible for the first successful solutions for plate bending. PRELIMINARIES A laminate is an engineering structure formed by bonding two or more laminae together. The laminae are assumed to be perfectly bonded at their interfaces. Thus, the displacements, and hence the strains, are continuous throughout the laminate. The laminae of the structure may have isotropic, transversely isotropic, orthotropic, anisotropic properties, differing thicknesses, or differing principal material direction orientations. Mid-plane h/2 h/2 y x z Figure 4. Schematic drawing of a laminated plate element showing coordinate system and plate thickness. Figure 4 shows a schematic drawing of a typical laminated plate element. The thickness of the plate shown is h and the origin of the coordinate system is on Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 77 the mid-plane of the plate. The coordinate system used is a typical aerospace coordinate system with x- and y-directions oriented in the plane of the plate and the z-direction perpendicular to the plane in a downward direction. FORCE -- MOMENT RESULTANTS The internal force and moment resultants are derived from the stresses in the plate element. These resultants are expressed in units of force per unit length and force times length per unit length, respectively. The normal force resultant in the x-direction on an internal face with normal in the x-direction is N x ( x, y ) and is an integral through the thickness of the plate of the x-direction normal stress, σ x ( x, y, z ) , on that face. Therefore, N x = N x ( x, y ) = h/ 2 ∫ σ (x, y, z )dz . x 29 - h/2 The normal force resultant in the y-direction is similarly defined as an integral of the y-direction normal stress, σ y ( x, y, z ) . It is N y = N y ( x, y ) = h/ 2 ∫ σ (x, y, z )dz . y 30 - h/2 There is no normal force resultant in the z-direction due to the consideration of an element that includes the total plate thickness. The shear force resultant in the y-direction on an internal face with normal in the x-direction is N xy ( x, y ) . This force resultant is defined as an integral through the thickness of the x-y shear stress, σ s (x, y , z ) , on that face. Therefore, N xy = N xy ( x, y ) = h/2 ∫ σ (x, y, z )dz . s 31 -h/2 The shear force resultant in the x-direction on an internal face with normal in the y-direction is N yx ( x, y ) . This force resultant is defined as an integral through the thickness of the y-x shear stress, σ yx ( x, y , z ) , on that face. Noting that the x-y and y-x shear stresses are equal, i.e., σ s ( x, y , z ) = σ xy ( x, y , z ) = σ yx ( x, y , z ) , then the x-y and y-x shear resultant forces must also be equal, i.e., N xy ( x, y ) = N yx ( x, y ) . The shear force resultants in the z-direction on the internal faces with normals in the x- and y-directions are N xz ( x, y ) and N yz ( x, y ) , respectively. They are defined as Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY N xz = N xz ( x, y ) = 78 h/2 ∫ σ (x, y, z )dz 32 xz -h/2 and N yz = N yz ( x, y ) = h/2 ∫ σ (x, y, z )dz . 33 yz -h/2 The moment resultant in the x-direction on an internal face with normal in the x-direction is defined to be M xx ( x, y ) . This resultant is found by integrating the differential moment in the x-direction through the thickness of the plate. To develop the differential moment in the x-direction consider the drawing shown in Figure 5. A differential element located at (y,z) is indicated by the shaded area in the figure. The stresses on this element are also shown in the figure. The normal stress in the x-direction does not contribute to the moment about the x-axis while the shear stresses σ xy ( x, y, z ) and σ xz ( x, y , z ) have moment arms z and y, respectively. Therefore, the moment resultant M xx ( x, y ) is M xx = M xx ( x, y ) = − h/2 ∫ {zσ (x, y, z ) − yσ (x, y, z )}dz . s 34 xz -h/2 y z x y σxy σxz z σx Figure 5. Schematic showing the stresses on a face with normal in the x-direction. Similarly, the moment resultant y-direction on an internal face with a normal in the x-direction is M xy = M xy ( x, y ) = h/2 ∫ σ (x, y, z )zdz x 35 - h/2 Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 79 and the moment resultant z-direction on an internal face with a normal in the x-direction is M xz = M xz ( x, y ) = h/ 2 ∫ σ (x, y, z )ydz . 36 x - h/2 The moment resultants on an internal face with a normal in the y-direction are developed in the same manner as above. The moment resultants on a face with normal in the y-direction are h/ 2 M yx = M yx ( x, y ) = − ∫ σ y ( x, y, z )zdz , 37 - h/2 M yy = M yy ( x, y ) = ∫ {zσ (x, y, z ) − xσ (x, y, z )}dz , h/2 s yz 38 -h/2 and M yz = M yz ( x, y ) = h/2 ∫ σ (x, y, z )xdz . y 39 - h/2 EQUILIBRIUM OF A PLATE ELEMENT Consider an infinitesimal plate element as shown in Figures 6 and 7. Figure 6 shows the internal force resultants acting on the element along with the externally applied force, p∆x∆y , due to a distributed load over the element. Note that the infinitesimal element is shown in the undeformed configuration. Thus, the various internal force resultants lie in their respective coordinate directions. Figure 7 shows the internal moment resultants acting on the element. The moments are indicated with two-headed arrows following the right-hand rule. Again, the element is shown in the undeformed configuration. The forces and moments shown in the figures are as defined previously. Now assume that the in-plane accelerations of the plate are small when compared to the out of plane accelerations. Applying the equilibrium conditions to the element, summation of the forces in the x-direction yields: − N x ∆y + ( N x + ∆N x )∆y − N yx ∆x + (N yx + ∆N yx )∆x = 0 , or ∆N x ∆y + ∆N yx ∆x = 0 . Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY Then noting, ∆N x = 80 ∂N x ∂N x ∆y , the sum of the forces in the ∆x and ∆N xy = ∂x ∂y x-direction reduces to ∂N yx ∂N x ∆x∆y + ∆x∆y = 0 , ∂x ∂y which further reduces to ∂N x ∂N yx + = 0. ∂x ∂y 40 N yx ∆x N xy ∆y p∆x∆y N xz ∆y N yz ∆x N x ∆y N y ∆x ∆x y (N (N y + ∆N y )∆x yz + ∆N yz )∆x ∆y (N x + ∆N x )∆y (N xy + ∆N xy )∆y (N z yx + ∆N yx )∆x x (N xz + ∆N xz )∆y Figure 6. Schematic drawing of an infinitesimal element showing the internal force resultants acting on the element. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 81 The summation of forces in the y-direction reduces to ∂N xy ∂x + ∂N y ∂y =0. 41 As drawn in Figure 6 the summation of the forces in the z-direction is − N xz ∆y + ( N xz + ∆N xz )∆y − N yz ∆x + (N yz + ∆N yz )∆x + p ( x, y , t )∆x∆y = ρ( x, y )h∆x∆y ∂ 2w . ∂t 2 42 However, this equation does not allow for the inclusion of buckling effects. To develop the buckling terms one must apply equilibrium in the deformed configuration. When the effect of deformation is considered, the normal force in the x-direction will have a component in the z-direction equal to ∂2w ∂N ∂w N x 2 ∆x∆y + x ∆x∆y , ∂x ∂x ∂x 43 where N x is the applied buckling force in the x-direction on the x-face. Similarly, the normal force in the y-direction will have a component in the z-direction equal to Ny ∂N y ∂w ∂ 2w ∆x∆y + ∆x∆y , 2 ∂y ∂y ∂y 44 where N y is the applied buckling force in the y-direction on the y-face, and the in-plane shear forces have a component in the z-direction equal to ∂N yx ∂w ∂N ∂w ∂2w 2 N xy ∆x∆y + ∆x∆y + xy ∆x∆y , ∂x∂y ∂y ∂y ∂x ∂x 45 where N xy is the applied buckling force in the y-direction on the x-face, and N yx is the applied buckling force in the x-direction on the y-face. Then, combining the z-direction terms in equations 43, 44, and 45, including the results from equations 40 and 41 yields an additional z-direction term, which is  ∂2w ∂ 2w ∂2w  + N y 2 ∆x∆y .  N x 2 + 2 N xy ∂x∂y ∂y   ∂x 46 Thus, the summation of the forces in the z-direction, including effects of buckling is obtained by combining 42 and 46. The result on simplification is ∂N yz ∂N xz ∂ 2w ∂2w ∂ 2w ∂ 2w + + N x 2 + 2 N xy + N y 2 + p ( x, y, t ) = ρ( x, y )h 2 . ∂y ∂x ∂x ∂x∂y ∂y ∂t Revised: 10 February, 2000 47 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 82 M yx ∆x M xz ∆y M xy ∆y M yz ∆x ∆ M xx ∆y ∆y ∆x (M yy + ∆M yy )∆x (M xx + ∆M xx )∆y y (M M yy ∆x x yz + ∆M yz )∆x (M xy + ∆M xy )∆y (M z yx + ∆M yx )∆x (M xz + ∆M xz )∆y Figure 7. Schematic drawing of infinitesimal element showing internal moment resultants acting on the element. Equilibrium of a body requires that the summation of the moments about the center of mass be equal to the rate of change of the moment of momentum. Assuming that the rotary inertia of the plate element is small in all directions, the sum of the moments about the center of mass must be zero. The summation of the moments in the x-direction about the mid-point of the element yields N yz + ∂M xx ∂M yx =0. + ∂x ∂y 48 Summation of the moments in the y-direction about the mid-point of the element yields N xz + ∂M yy ∂y + ∂M xy ∂x =0. Revised: 10 February, 2000 49 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 83 Finally, summation of the moments in the z-direction about the mid-point of the element yields ∂M xz ∂M yz + + N xy − N yx = 0 . ∂x ∂y 50 Substituting for the shear force resultants, N xz and N yz , from equations 48 and 49 into equation 47 yields ∂ 2 M xy ∂ 2 M yy ∂ 2 M yx ∂2w ∂2w ∂ 2w ∂2w ( ) ( ) 2 2 , , , p x y t x y h N N N + = ρ + + + − + , x xy y ∂t 2 ∂y 2 ∂x∂y ∂x 2 ∂x 2 ∂x∂y ∂y 2 51 which is analogous to the more familiar moment -- distributed load equation for beams. This equation is complete for classical plate theory. It contains four sub-cases: • Complete -- Buckling with dynamic effects, • Dynamic -- Neglect buckling effects, • Buckling -- Neglect dynamic effects, • Static -- Neglect buckling and dynamic effects. DISPLACEMENT FIELD MODEL The next step in development of a complete plate theory for laminated plates is the assumption of a displacement field. The Kirchoff-Love model for the displacement of a plate element is based on observation of simple bending of a plate. In this model, the first assumption is the normal strain through the plate thickness is much much less than the normal strains in the plane of the plate, i.e., ε z >> ε x , ε y . The second assumption in this model is based on Navier's observation. This assumption is sections plane before loading remain plane after loading. The third assumption is plane sections initially perpendicular to the mid-plane of the plate before loading remain perpendicular to the mid-plane after loading. The impact of these assumptions on the deformation is shown in Figure 8. The consequences of the second and third assumptions on the through thickness shear strains are Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 84 x-z Plane Before Deformation y-z Plane After Deformation x After Deformation y x ⇒ z Before Deformation y ⇒ and z z z Figure 8. Sections perpendicular before and after deformation. ε xz = 0 , 52 and ε yz = 0 . 53 Figure 9 contains a schematic of the displacement in the x and z plane of an element of the plate in the undeformed and deformed configurations. From this figure we determine that the displacement in the x-direction of point Q is: u (Q ) = u (R ) − z sin θ . R θ u(R) z x w(R) R' w(Q) Q z θ zcosθ Q' z u(Q) zsinθ Figure 9. Movement of a plate element from the undeformed configuration to the deformed position. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY If we assume that the angles are small, then sin θ ≈ expression becomes: u (Q ) = u0 ( x, y ) − z 85 ∂w0 ( x, y ) and the above ∂x ∂w0 ( x, y ) . ∂x 54 The tensile strain in the x-direction is: ∂ 2 w0 ( x, y ) ∂u (Q ) ∂u0 ( x, y ) −z = ε x (Q ) = , ∂x 2 ∂x ∂x or ε x ( x, y, z ) = ε 0x ( x, y ) + z κ 0x ( x, y ) , 55 where κ 0x ( x, y ) is the mid-plane curvature. Similarly, for the y and z plane the displacement in the y-direction of point Q is: v(Q ) = v(R ) − z sin θˆ . Again, applying the assumption that the angles are small, then sin θˆ ≈ ∂w0 ( x, y ) and ∂y the above expression becomes: v(Q ) = v0 ( x, y ) − z ∂w0 ( x, y ) . ∂y 56 The tensile strain in the y-direction is: ε y (Q ) = ∂v (Q ) ∂v0 ( x, y ) ∂ 2 w0 (x, y ) = −z , ∂y ∂y ∂y 2 or ε y ( x, y, z ) = ε 0y ( x, y ) + z κ 0y ( x, y ) , 57 where κ 0y ( x, y ) is the mid-plane curvature. Now recall that the x-y shear strain, ε s , is defined as: εs = ∂u ∂v + , ∂y ∂x Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY ε s ( x, y , z ) = ∂w ( x , y )  ∂w ( x, y )  ∂  ∂  ,  u0 ( x, y ) − z 0  +  v0 ( x, y ) − z 0 ∂y  ∂x  ∂x  ∂y  ε s ( x, y , z ) = ∂ 2 w0 ( x, y ) ∂u0 ( x, y ) ∂v0 ( x, y ) − 2z + , ∂x∂y ∂x ∂y 86 ε s ( x, y , z ) = ε 0s (x, y ) − z κ 0xy ( x, y ) , 58 where κ 0xy ( x, y ) is the mid-plane cross curvature. Equations 55, 57, and 58, can be combined into vector form as: 0 0 ε x  ε x   κ x     0  0  ε y  = ε y  − z  κ y  . ε  ε0  κ 0   s   s   xy  59 Applying the three dimensional orthotropic constitutive relation to equations 52 and 53 yields: σ xz = 0 , 60 and σ yz = 0 , 61 which fits with previous results. Now if one assumes that the plate is thin, then σz = 0 , 62 which effectively assumes a plane stress state. ORTHOTROPIC CONSTITUTIVE RELATION As discussed in the development of the mechanics of an orthotropic lamina, the off-axis stress-strain relationship for an orthotropic lamina is: σ x  Qxx    σ y  = Qxy σ  Q  s   xs Qxy Qyy Qys Qxs  ε x    Qys  ε y  , Qss   ε s  63 where Qij , i, j = x, y , & s are the off-axis properties of the lamina. For a laminated structure the mechanical properties are assumed to be constant throughout each individual lamina. Therefore, the mechanical properties of the structure vary Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 87 through the thickness of the plate, but are constant through the thickness of the individual lamina. Note that use of the off-axis properties does not preclude a lamina from being oriented with principal material directions aligned with the structural directions. The off-axis constitutive relation is used as it is more general than the on-axis relationship. Figure 10 shows a schematic drawing of a laminated plate with N plies, the ply numbering system, ply coordinates, and ply thicknesses. The ply numbers range from 1 to N, with ply 1 located on the top surface of the laminate (minimum z-coordinate) and ply N located on the bottom surface of the laminate (maximum z-coordinate). The ith ply has thickness ti, with top coordinate zi-1 and bottom coordinate zi. The total laminate thickness is h. Consequently, z0 = − h and 2 zN = h . 2 1 2 z0 z1 z 2 t2 -h/2 zi ti i zi+1 x zN-1 zN tN-1 z h/2 N-1 N Figure 10. Schematic of a Laminate with N ply showing the ply numbers, corresponding coordinates and thicknesses. For the purposes of this development we write equation 63 with a second subscript, k, that indicates the lamina number. This expression is:    σ x  Qkxx  k  σ y  = Qxy  k  k σ s  Qxs  k  k Qxy k Q yy k Q ys k   Qxs  ε x  k  k  Q ys  ε y  , k  k  Qss  ε s  k  k  Revised: 10 February, 2000 64 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 88 where Qij , i, j = x, y, & s , k = 1,2,K N are the off-axis properties of the kth lamina and k σij and ε ij are the associated stresses and strains. k Now, considering that the k stresses in each lamina are dependent on the properties in that layer and the properties from one layer to the next are not necessarily the same, equation 29 becomes: z1 h/2 Nx = z2 zN ∫ σ (x, y, z )dz = ∫ σ (x, y, z )dz + ∫ σ (x, y, z )dz + L + ∫ σ (x, y, z )dz x -h/2 N Nx = ∑ x 1 z0 z1 x 2 z N −1 x N zk ∫ σ (x, y, z )dz . 65 x k k =1 zk −1 Now, substituting equations 59 and 64 into 65 yields: N Nx = ∑ zk  ∫ Q xx k k =1 zk −1 N Nx = ∑ zk  ∫ Q k =1 zk −1   N x = ∑  Qxx k =1   k   xx k   ε x   k  Qxs ε y dz , k   k ε s   k Qxy k  ε0x   κ 0x         Qxs  ε 0y  + z  κ 0y  dz , k   ε 0s  κ 0xy        Qxy k N Qxy k  ε 0x  zk   N   0  Qxs ε y  ∫ dz  + ∑  Qxx k  ε 0  zk −1  k =1   k  s   Qxy k   κ 0x  zk   0  Qxs  κ y  ∫ zdz  . k  κ 0 zk −1   xy   66 Now define the in-plane laminate stiffnesses, Aij , i, j = x, y, & s as zk   N  Axx = ∑ Qxx ∫ dz  = ∑ Qxx ( z k − z k −1 )   k =1 k k =1  k zk −1  N zk N   N Axy = ∑  Qxy ∫ dz  = ∑ Qxy ( z k − z k −1 )   k =1 k k =1  k zk −1  Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 89 zk   N  Axs = ∑ Qxs ∫ dz  = ∑ Qxs ( z k − z k −1 )   k =1 k k =1  k zk −1  N or in general N Aij = ∑ Qij (z k − z k −1 ) k =1 67 k and the coupling stiffnesses, Bij , i, j = x, y, & s as zk   1 N 2 2  Bxx = ∑ Qxx ∫ zdz  = ∑ Qxx z k − z k −1   2 k =1 k k =1  k zk −1  ) zk N   1 N 2 2 Bxy = ∑  Qxy ∫ zdz  = ∑ Qxy z k − z k −1   k =1  k zk −1  2 k =1 k ) ( N ( zk   1 N 2 2  Bxs = ∑ Qxs ∫ zdz  = ∑ Qxs z k − z k −1   2 k =1 k k =1  k zk −1  ( N ) or in general Bij = ( ) 1 N 2 2 Qij z k − z k −1 . ∑ 2 k =1 k 68 Then Nx becomes: N x = {Axx Axy ε 0x    Axs }ε 0y  + {Bxx ε 0   s Bxy  κ 0x    Bxs } κ 0y  . κ0   xy  69 By a similar process the relationship between the other in-plane forces and the strains and curvatures are derived to be:  N x   Axx     N y  =  Axy N   A  xy   xs Axy Ayy Ays Axs  ε 0x   Bxx    Ays  ε0y  +  Bxy Ass  ε 0s   Bxs Bxy Byy Bys Bxs   κ 0x    Bys   κ 0y  , Bss  κ 0xy  70 where Aij and Bij are as defined previously. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 90 The relationship between the moments and the strains and curvatures is derived by applying the same considerations of the laminate behavior used to derive the force-strain curvature relationship to equation 35. This gives: z1 h/ 2 M xy = z2 zN ∫ σ (x, y, z )zdz = ∫ σ (x, y, z )zdz + ∫ σ (x, y, z )zdz + L + ∫ x -h/2 N M xy = ∑ z0 x 1 z1 x 2 z N −1 σ x ( x, y , z )zdz N zk ∫ σ (x, y, z )zdz . 71 x k k =1 zk −1 Now, substituting equations 59 and 64 into 71 yields: N M xy =∑ zk  ∫ Q xx k k =1 zk −1 N M xy = ∑ zk  ∫ Q k =1 zk −1   = ∑  Qxx k =1   k   xx k Qxy k  ε 0x   κ 0x         Qxs  z ε 0y  + z 2  κ 0y  dz , k  κ 0    ε 0s   xy      Qxy k N M xy   ε x   k  Qxs ε y  zdz , k   k ε s   k Qxy k   ε 0x  zk  N   0  Qxs ε y  ∫ zdz  + ∑  Qxx k  ε 0  zk −1  k =1   k  s   Qxy k   κ 0x  zk  0  2  Qxs  κ y  ∫ z dz  . k   κ 0 zk −1  xy   72 Note in the above expression the presence of the coupling stiffnesses, Bij , i, j = x, y, & s . Now define the out-of-plane laminate stiffnesses, Dij , i, j = x, y, & s as zk   1 N 3 3  Dxx = ∑ Qxx ∫ z 2dz  = ∑ Qxx zk − z k −1   3 k =1  k zk −1 k =1 k  ) zk   1 N 3 3  Dxy = ∑ Qxy ∫ z 2 dz  = ∑ Qxy z k − z k −1   k =1  k zk −1  3 k =1 k ) zk   1 N 3 3  Dxs = ∑ Qxs ∫ z 2dz  = ∑ Qxs zk − z k −1   3 k =1  k zk −1 k =1 k  ) N N N Revised: 10 February, 2000 ( ( ( M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 91 or in general Dij = ( ) 1 N 3 3 ∑ Qij zk − zk −1 . 3 k =1 k 73 Then Mxy becomes: M xy = {Bxx Bxy ε 0x    Bxs }ε 0y  + {Dxx ε 0   s Dxy  κ 0x    Dxs } κ 0y  . κ0   xy  74 By a similar process the relationship between the moments and the strains and curvatures are derived to be:  M xy   Bxx    − M yx  =  Bxy  M  B  yy   xs Bxy Byy Bys Bxs  ε 0x   Dxx    Bys  ε 0y  +  Dxy Bss  ε 0s   Dxs Dxy Dyy Dys Dxs   κ 0x    Dys   κ 0y  , Dss  κ 0xy  75 where Bij and Dij are as defined previously. A constitutive relation between the forces and moments and the strains and curvatures is written by combining equations 70 and 75. The constitutive relationship for a laminate is:  N x   Axx  N  A  y   xy  N xy   Axs  M  = B  xy   xx − M yx   Bxy     M yy   Bxs Axy Ayy Ays Bxy Byy Bys Axs Ays Ass Bxs Bys Bss Bxx Bxy Bxs Dxx Dxy Dxs Bxy Byy Bys Dxy Dyy D ys Bxs   ε 0x    Bys   ε 0y  Bss   ε 0s  , Dxs   κ 0x    Dys   κ 0y    Dss  κ 0xy  76 or  N  [A]  = M  [B ] [B ] ε 0  . [D]κ0  77 Note that the A, B, D, matrix, the laminate stiffness, is a partitioned matrix that is also symmetric. THE LAMINATED PLATE EQUATIONS The three force balance equations, equations 40, 41, and 51, are the foundation for the governing system of equations for the displacement of the plate. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 92 THE IN-PLANE EQUATIONS The governing in-plane force balance equations for the laminated plate in terms of the deflections are derived by substituting the constitutive equation into the in-plane force balance equations; equation 76 and equations 40 and 41. Using ∂N x , and assuming that the laminate is homogeneous in equation 76 or 69 to write ∂x the x- and y-directions gives: ∂N x = {Axx ∂x Axy  ∂ε 0x   ∂x   0  ∂ε  Axs } y  + {Bxx  ∂x0   ∂ε s   ∂x    Bxy  ∂κ 0x   ∂x   0  ∂κ  Bxs } y  .  ∂x0   ∂κ xy   ∂x    Now substituting the definitions of the strains and curvatures of the mid-plane into the equation above yields: ∂N x = {Axx ∂x Axy   ∂ 2u 0   2 ∂x   2 ∂ v0   Axs }  + {Bxx  2 ∂x∂y 2   ∂ u0 ∂ v0   ∂x∂y + ∂x 2    Bxy  ∂ 3w0   − 3  ∂x    ∂ 3w0  . Bxs } − 2   ∂x∂3 y  ∂ w0   − 2 ∂x 2∂y    78 ∂N xy , and again assuming that the laminate ∂x is homogeneous in the x- and y-directions gives: Similarly, using equation 76 to write ∂N xy ∂y = {Axs Ays  ∂ε 0x   ∂y   0  ∂ε  Ass } y  + {Bxs  ∂y0   ∂ε s   ∂y    Bys  ∂κ 0x   ∂y   0  ∂κ  Bss } y  .  ∂y0   ∂κ xy   ∂y    Then substituting the definitions of the strains and curvatures of the mid-plane into the equation above yields: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY ∂N xy ∂y = {Axs Ays   ∂ 2u 0    ∂x2∂y  ∂ v0   Ass }  + {Bxs 2  2 ∂y 2   ∂ u0 ∂ v0   ∂y 2 + ∂x∂y    93 Bys  ∂ 3 w0  − 2   ∂x3 ∂y   ∂w  Bss } − 30  . ∂y   ∂ 3 w0   − 2  ∂x∂y 2    79 Substituting equations 78 and 79 into equation 40 gives: ∂ 2v0 ∂ 2v0 ∂ 2v0 ∂ 2u 0 ∂ 2 u0 ∂ 2u 0 ( ) 2 + + + + + + A A A A A A xs ss xs xy ss ys ∂y 2 ∂x∂y ∂x 2 ∂y 2 ∂x∂y ∂x 2 Axx ∂ 3 w0 ∂ 3 w0 ∂ 3w0 ∂ 3w0 ( ) 3 2 =0 − − + − − Bxx B B B B xs xy ss ys ∂y 3 ∂x∂y 2 ∂x 2∂y ∂x 3 Similarly for the y-direction sum of forces ∂N y ∂y = {Axy Ayy   ∂ 2 u0   ∂x∂y   ∂ 2 v0   Ays }  + {Bxy 2  2 ∂y 2   ∂ u0 ∂ v0   ∂y 2 + ∂x∂y    Ays   ∂ 2u 0   2 ∂x   ∂ 2 v0   Ass }  + {Bxs x y ∂ ∂  2   ∂ u0 ∂ 2 v0   ∂x∂y + ∂x 2    . 80 ∂N y ∂N xy and are: ∂y ∂x Byy  ∂ 3 w0   − ∂ 2∂  x y    ∂ 3 w0  Bys } − 3  . ∂y   ∂ 3 w0   − 2 ∂x∂y 2    81 Bys  ∂ 3 w0   − 3  ∂x    ∂ 3 w0  Bss } − . 2  x y ∂ ∂   ∂ 3 w0   − 2 ∂x 2 ∂y    82 and ∂N xy ∂x = {Axs Substituting equations 81 and 82 into equation 41 gives: Axs ∂ 2 u0 ∂ 2u 0 ∂ 2 u0 ∂ 2v0 ∂ 2v0 ∂ 2v0 ( ) + A + A + A + A + 2 A + A xy ss ys ss ys yy ∂x 2 ∂x∂y ∂y 2 ∂x 2 ∂x∂y ∂y 2 ∂ 3 w0 ∂ 3 w0 ∂ 3 w0 ∂ 3 w0 ( ) − Bxs − B + 2 B − 3 B − B =0 xy ss ys yy ∂x 3 ∂x 2∂y ∂x∂y 2 ∂y 3 Revised: 10 February, 2000 . 83 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 94 Equations 80 and 83 are the in-plane force balance equations expressed in terms of the displacements. They represent two simultaneous coupled partial differential equations in u0 , v0 , and w0 . Note that if the coupling stiffnesses, Bij , are zero, then the out-of-plane displacement, w0 , is decoupled from the in-plane force balance equations. Note that the buckling loads, N x , N y , and N xy , are constant throughout the plate. Consequently, they do not appear in the in-plane force balance equations. THE OUT-OF-PLANE EQUATION The out-of-plane force balance equation for the laminated plate in terms of the deflections is derived by substituting the constitutive equation into the out-ofplane force balance equation; equation 76 and equation 51. Using equation 76 to ∂ 2 M xy ∂ 2 M yx ∂ 2 M yy , and write , , as was done previously for the forces, leads to: ∂y 2 ∂x∂y ∂x 2 ∂ 2 M xy ∂x 2 ∂ 2 M yx ∂y 2 = {Bxx = {Bxy Bxy   ∂ 3 u0   3 ∂x   3 ∂ v0   Bxs }  + {Dxx 2  3 ∂x ∂y 3   ∂ u0 ∂ v0   ∂x 2∂y + ∂x 3    Byy   ∂ 3u0  ∂x∂y 2    ∂ 3v0   Bys }  + {Dxy 3  3 ∂y 3   ∂ u0 ∂ v0   ∂y 3 + ∂x∂y 2    Bys   ∂ 3 u0   2 ∂x ∂y   ∂ 3v0   Bss }  + {Dxs 2  3 ∂x∂y 3  ∂ v0   ∂ u0  ∂x∂y 2 + ∂x 2∂y    Dxy  ∂ 4 w0   − 4  ∂x    ∂ 4 w0  Dxs } − 2 2  ,  ∂x 4∂y  ∂ w0   − 2 ∂x 3∂y    84 Dyy  ∂ 4 w0   − ∂x 2∂y 2     ∂ 4 w0  Dys } − 4  , ∂y   ∂ 4 w0   2 −  ∂x∂y 3    85 and ∂ 2 M yy ∂x∂y = {Bxs D ys  ∂ 4 w0   − ∂x 3∂y     ∂ 4 w0  . Dss } − 3   ∂x∂4 y  ∂ w0   − 2 ∂x 2∂y 2    86 Substitution of the results in equations 84, 85, and 86 into equation 51 yields the full out-of-plane plate displacement equation with dynamic and buckling effects: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY Bxx 95 ∂ 3u0 ∂ 3u0 ∂ 3 u0 ∂ 3u0 ( ) B B B B 3 2 + + + + xs ss xy ys ∂y 3 ∂x∂y 2 ∂x 2 ∂y ∂x 3 + Bxs ∂ 3v0 ∂ 3v0 ∂ 3v0 ∂ 3v0 ( ) B B B B 2 3 + + + + ss xy ys yy ∂y 3 ∂x∂y 2 ∂x 2∂y ∂x 3 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ( ) D D D D D 4 2 4 4 − − − + − − Dxx xs xy ss ys yy ∂y 4 ∂x∂y 3 ∂x 2 ∂y 2 ∂x 3∂y ∂x 4 +Nx . 87 ∂ 2 w0 ∂ 2 w0 ∂ 2 w0 ∂ 2 w0 h N N 2 − p (x , y , t ) = ρ + + xy y ∂t 2 ∂y 2 ∂x∂y ∂x 2 Equations 80, 83, and 87 form the complete system of coupled simultaneous partial differential equations that describe the displacement of the mid-plane of the laminate with dynamic and buckling loads included. SIMPLIFICATIONS OF THE PLATE EQUATION As noted earlier, there are four loading cases that can be analyzed with equations 80, 83, and 87. They are: • Complete -- Buckling with dynamic effects, • Dynamic -- Neglect buckling effects, • Buckling -- Neglect dynamic effects, • Static -- Neglect buckling and dynamic effects. These assumptions do not directly impact equations 80 and 83. system of equations is equations 80, 83, and 87. The complete Dynamic Plate Equation Applying the assumption of no buckling loads to equation 87 yields: Bxx ∂ 3u0 ∂ 3 u0 ∂ 3u0 ∂ 3u0 ( ) + 3 B + 2 B + B + B xs ss xy ys ∂x 3 ∂x 2 ∂y ∂x∂y 2 ∂y 3 + Bxs ∂ 3v0 ∂ 3v0 ∂ 3v0 ∂ 3v0 ( ) + 2 B + B + 3 B + B ss xy ys yy ∂x 3 ∂x 2 ∂y ∂x∂y 2 ∂y 3 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ( ) − Dxx − 4 D − 2 D + 4 D − 4 D − D xs xy ss ys yy ∂x 4 ∂x 3∂y ∂x 2 ∂y 2 ∂x∂y 3 ∂y 4 = ρh . 88 ∂ 2 w0 − p ( x, y, t ) ∂t 2 Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS CLASSICAL LAMINATION THEORY 96 Buckling Plate Equation Applying the assumption of no dynamic effects to equation 87 yields: Bxx ∂ 3u0 ∂ 3u0 ∂ 3 u0 ∂ 3u0 ( ) B B B B 3 2 + + + + xs ss xy ys ∂y 3 ∂x∂y 2 ∂x 2 ∂y ∂x 3 + Bxs ∂ 3v0 ∂ 3v0 ∂ 3v0 ∂ 3v0 ( ) B B B B 2 3 + + + + ss xy ys yy ∂y 3 ∂x∂y 2 ∂x 2∂y ∂x 3 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ( ) D D D D D 4 2 4 4 − − − + − − Dxx xs xy ss ys yy ∂y 4 ∂x∂y 3 ∂x 2 ∂y 2 ∂x 3∂y ∂x 4 +Nx . 89 ∂ 2 w0 ∂ 2 w0 ∂ 2 w0 N N 2 = − p( x, y ) + + xy y ∂y 2 ∂x∂y ∂x 2 Static Plate Equation Applying the assumption of no buckling and no dynamic effects to equation 87 yields: Bxx ∂ 3u0 ∂ 3 u0 ∂ 3u0 ∂ 3 u0 3 B 2 B B B ( ) + + + + xs ss xy ys ∂x 3 ∂x 2 ∂y ∂x∂y 2 ∂y 3 + Bxs ∂ 3v0 ∂ 3v0 ∂ 3v0 ∂ 3v0 ( ) 2 B B 3 B B + + + + ss xy ys yy ∂x 3 ∂x 2 ∂y ∂x∂y 2 ∂y 3 − Dxx ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ∂ 4 w0 ( ) 4 D 2 D 4 D 4 D D − − + − − = − p( x, y ) xs xy ss ys yy ∂x 4 ∂x 3∂y ∂x 2 ∂y 2 ∂x∂y 3 ∂y 4 Revised: 10 February, 2000 . 90 M. V. Bower LAMINATES INTRODUCTION In the previous section the laminate stiffnesses Aij , Bij , and Dij and the governing system of equations for a laminated plate were derived. In this section, alternate expressions for the laminate stiffnesses are presented, the impact of various simplifying assumptions, the distribution of stress through a laminate, and laminate failure theories are discussed. ALTERNATE EXPRESSIONS FOR LAMINATE STIFFNESSES Figure 10 shows a schematic drawing of a laminate cross-section. In this figure, the laminate is shown as constructed of N laminae of thicknesses tk . The laminae thicknesses are directly related to the laminae coordinates. This relationship is: tk = z k − z k −1 . 91 Recognizing this relationship, the in-plane extensional stiffnesses, Aij , can be written in terms of the individual laminae thicknesses: N Aij = ∑ Qij t k . k =1 92 k This expression is an alternative to that presented previously and is, at times, more convenient to use. The lamina centroid, z k , is defined as: zk = zk + z k −1 . 2 93 Now using the laminae centroids and thicknesses the coupling stiffnesses, Bij , can be written as: Bij = N 1 N 1 N (z + zk −1 ) , 2 2 − = ( − )( + ) = Q z z Q z z z z Qij ( z k − z k −1 ) k ∑ ∑ ∑ ij k k −1 ij k k −1 k k −1 2 k =1 k 2 k =1 k 2 k =1 k ( ) or Revised: 10 February, 2000 Page 97 COMPOSITE MATERIALS LAMINATES 98 N Bij = ∑ Qij t k z k . k =1 94 k It is arguable whether the alternate expression is simpler than the original form of the coupling stiffnesses. The alternate form does demonstrate the relationship between the coupling stiffnesses and the individual laminae thicknesses and their relative location within the laminate. The bending stiffnesses, Dij , can also be expressed in terms of the laminae centroids and thicknesses. This alternate expression is: N 1 3  2 Dij = ∑ Qij  tk z k + tk  . 12  k =1 k  95 Again, the relative simplicity of the alternate expression to that of the original form of the bending stiffnesses is arguable. Nevertheless, this expression is important because of its instructive nature. Recall from classical isotropic plate theory that Et 3 the plate stiffness, D, is , where E is the modulus of elasticity, ν is the 12 1 − ν 2 Poisson’s ratio, and t is the plate thickness. Further recall that the plate stiffness is analogous to the bending stiffness of a beam, EI yy , where I yy is the second area ( ) moment of inertia about the y-axis. Next, note that the second area moment of bt 3 . inertia of a rectangular beam, with height t and width b, about the centroid is 12 Ebt 3 Therefore, the beam bending stiffness is . Comparing the beam bending 12 stiffness and the isotropic plate bending stiffness, the plate bending stiffness appears to be a beam bending stiffness per unit width and including Poisson’s effects. Now remember the Parallel Axis Theorem for the second area moment of inertia is I yyO = I yyC + d z2 A , where point C is the centroid of the cross-section, point O OC is the point where the moment is being calculated, d z2 is the distance along the OC z-axis between point O and point C, and A is the area of the cross-section. Now 3 Qij tk examine equation 95, we observe the bending stiffness of each lamina, k , and 12 2 the shifting factor of each lamina, Qij t k zk . Therefore, we conclude that the laminate k stiffness is the summation of the bending stiffnesses of the individual lamina about the mid-plane. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 99 SIMPLIFYING ASSUMPTIONS ON LAMINATE STRUCTURE There are several simplifying assumptions that can be made that have a significant impact on the force-moment strain-curvature relationship and governing system of equations. These assumptions deal directly with the structure of the laminate, in particular the lay-up or stacking sequence. In this discussion we will consider only laminated structures, i. e., N > 1, of orthotropic materials. SYMMETRIC LAMINATES Recall that a symmetric laminate is defined as a laminate that is symmetric in both geometry and material properties about the mid-plane. The stiffness equations, equations 92, 94, and 95, are expressed in terms of the number of laminae in the laminate and the laminae reduced stiffnesses, thicknesses and centroids. Symmetry of geometry means: tk = t N −k ∀ ∈1 ≤ k ≤ k N 2 96 and − z k = z N −k ∀ ∈1 ≤ k ≤ k N . 2 97 Note that equation 91 does not imply that the laminae thicknesses are constant. Symmetry of material properties means: Qij = Qij k N −k i , j = x, y, & s , ∀ ∈1 ≤ k ≤ k N . 2 98 Applying the symmetry specialization to the in-plane stiffnesses, Aij , from equation 92 leads to:  N2 2 Q t N ∈ Even  ∑ ij k Aij =  kN=1+1 k .  2  2 ∑ Qij tk N ∈ Odd  k =1 k 99 Applying the symmetry specialization to the coupling stiffnesses, Bij , from equation 94 leads to: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 100  N2 N  Q t z + Q t z N ∈ Even ∑N ijk k k ij k k ∑ k k=  k =1 2 Bij =  N +1 N  2  ∑ Qij t k zk + ∑ Qij t k zk N ∈ Odd N +1 k  k =1 k k= 2  N  N2 2  Qt z + Q t z N ∈ Even ∑ ∑ ij k k ij N −k N −k  k =1 k N −k k =1 Bij =  N +1 N +1 2  2  ∑ Qij t k z k + ∑ Qij t N −k z N −k N ∈ Odd N −k k =1  k =1 k  N2  Q t ( z − z ) N ∈ Even k ∑ ij k k Bij =  kN=1+1 k =0  2  ∑ Qij tk ( z k − zk ) N ∈ Odd  k =1 k ∀ N . 100 The impact of the results shown in equation 100 is significant. When the structure is a symmetric laminated plate, the in-plane problem is decoupled from the out-ofplane problem. Therefore, equations 87, 88, 89, and 90 are independent of the inplane displacement and the system of equations reduces to a single partial differential equation for the out-of-plane displacement. This simplifies the analysis significantly. Applying the symmetry specialization to the bending stiffnesses, Dij , from equation 95 leads to:  N2 2 Q  t z 2 + 1 t 3  N ∈ Even  ∑ ij k k 12 k  . Dij =  kN=1+1 k   2 1 3  2  2 ∑ Qij  tk z k + tk  N ∈ Odd 12   k =1 k  101 Therefore, the laminate stiffness matrix for a symmetric laminate is: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES  N x   Axx  N  A  y   xy  N xy   Axs  M = 0  xy   − M yx   0     M yy   0 Axy Axs Ayy Ays Ays Ass 0 0 0 0 0 0 101 0 0 0   ε 0x    0 0 0   ε0y  0 0 0   ε 0s  . Dxx Dxy Dxs   κ 0x     Dxy D yy D ys   κ 0y    Dxs D ys Dss  κ 0xy  102 This makes the previous statement regarding the decoupling of the in-plane problem and the out-of-plane even more clear. Recognize that plates of isotropic materials, laminated or homogeneous, are necessarily symmetric. Therefore, for all plates of isotropic materials (the traditional engineering materials of the last century) the in-plane problem is decoupled from the out-of-plane problem. This is one of the reasons that the majority of laminated composite structures are symmetric laminates. Note that this does not necessarily take full advantage of the structural capabilities of laminated composite materials. CONSIDERATION OF THE MATERIAL PROPERTIES Note that in the application of the symmetry specialization no further specialization regarding the material properties have been made. Therefore, equations 99, 100, and 101 are general for all materials. Further specializations for the laminate stiffnesses can be made by applying restrictions to the material properties. These restrictions or specializations are to angle ply and cross ply laminates. From review of lamina mechanics, the off-axis stiffness properties were expressed in terms of the invariants of the lamina properties. Now applying this to the laminate properties, the lamina stiffness invariants, U lQ , are: k  1 U1Q = 3Q11 + 3Q22 + 2Q12 + 4Q66  8 k k k k k   1 U 2Q = Q11 − Q22  2 k k k   1 U 3Q = Q11 + Q22 − 2Q12 − 4Q66  8 k k k k k   1 U 4Q = Q11 + Q22 + 6Q12 − 4Q66  8 k k k k k  U 5Q = k 103  1 Q11 + Q22 − 2Q12 + 4Q66   2 k k k k  and the off-axis stiffness properties are related to the invariant properties by: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES Qxx  U Q  k   1k Qyy  U Q  k   1k Qxy   Q  k  U4 Q  =  k  ssk  U Q Q   5k  kxs   0 Q    kys   0 102 cos4θk   cos4θk     1  - cos4θk   Q   U 2  . k - cos4θk   Q   U 3  sin 4θk   k   - sin 4θ k  cos2θk - cos2θ k 0 0 1 2 1 2 sin 2θk sin 2θk 104 Using equation 104 the laminate stiffnesses can be written as:   U1Q  Axx     k  A    U1Q  yy     k  Axy   N  U Q 4    = ∑  t k  k A ss    k =1  U Q  Axs     5k      0  Ays       0 cos2θ k - cos2θk 0 0 sin 2θk 1 2 sin 2θ k 1 2   U1Q    k  Bxx  U1Q B     yy   k    U Q  Bxy   N  4 = t z   ∑  k k  k B U Q  ss   k =1   5k  Bxs     0      Bys       0 cos2θ k - cos2θk 0 0 sin 2θk 1 2 sin 2θ k 1 2  cos4θk     cos4θk      1    - cos4θk   Q    U 2   ,  k - cos4θk   Q    U 3    θ sin 4 k   k     - sin 4θ k    cos4θ k     cos4θ k      1    - cos4θk   Q    U 2   ,  k - cos4θk   Q    U 3    sin 4θk   k     - sin 4θ k   105 106 and   U1Q    k  Dxx  U1Q D     yy   k     Q 1  Dxy   N   2 3  U4 = t z + t   ∑   k k k  k D 12   U Q ss k = 1      5k  Dxs     0       D ys       0 Revised: 10 February, 2000 cos2θ k - cos2θ k 0 0 sin 2θk 1 2 sin 2θ k 1 2  cos4θk     cos4θk       1    - cos4θk   Q     U 2   .  k - cos4θk   Q     U 3    sin 4θ k   k       - sin 4θ k   107 M. V. Bower COMPOSITE MATERIALS LAMINATES 103 Note that symmetry of material properties means: θk = θ N −k ∀ ∈1 ≤ k ≤ k N . 2 108 Symmetric Cross-ply Laminates Note in equation 104, that if the lamina angle, θk , is 0° or an integer multiple of 90° that coupling stiffnesses between extension and shear, Qxs and Qys , are zero. Recall k k that a cross-ply laminate is defined as one in which all lamina are oriented such that their orientation angles are either 0° or 90°. Therefore, all coupling stiffnesses between extension and shear, Qxs and Qys , are zero. Further, cos 4θk = 1 , ∀k and k k cos 2θk = ±1 . Now define an alternating function, δ, such that: 1 δ(θ ) =  − 1 θ = 0° . θ = 90° 109 Thus, for a symmetric cross-ply laminate the stiffnesses are:  N Mid  Q  Q Q  ∑ t k U1 + U 3 + U 2 δ(θ k ) k k   k =1  k  Axx   N Mid    A   ∑ tk U1Q + U 3Q − U 2Q δ(θ k ) yy k k    k =1  k  N Mid  Axy      t k U 4Q − U 3Q  ,  = ∑ k  k =1  k   Ass   N Mid   Axs    Q  t k U 5 − U 3Q      ∑ k  k =1  k  Ays    0     0   110  Bxx  B   yy   Bxy    = {0},  Bss   Bxs     Bys  111 as before, and Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 104  N Mid   1 3  Q 2 Q Q  ∑  tk z k + tk U1 + U 3 + U 2 δ(θk )  12  k k k   k =1   Dxx   N Mid    1  D   ∑  t k z k 2 + tk 3 U1Q + U 3Q − U 2Q δ(θ k )  yy 12  k k k    k =1   N Mid  Dxy    1 3  Q  2 Q  tk z k + tk U 4 − U 3  ,  = ∑ D 12 k k   ss k =1       N Mid  Dxs     1 3  Q  2 Q  tk z k + tk U 5 − U 3      ∑ 12  k k  k =1   Dys    0     0   112 where N Mid N N ∈ Even  =2 N +1  N ∈ Odd  2 Therefore, the force-moment strain-curvature relationship for a symmetric cross-ply laminate is:  N x   Axx  N    y   Axy  N xy   0  M = 0  xy   − M yx   0     M yy   0 Axy Ayy 0 0 0 0 0 0 0 0 0 Ass 0 0 0   ε 0x    0 0 0   ε 0y  0 0 0   ε 0s  . Dxx Dxy 0   κ 0x    Dxy Dyy 0   κ 0y    0 0 Dss  κ 0xy  113 Note that the response is further decoupled, with the normal and shear responses mutually decoupled. Balanced Regular Symmetric Angle-ply Laminates For angle-ply laminates θk , is restricted to ± α alternating through the laminate. Letting α be either a positive or negative angle, then θ1 = α , θ2 = −α , θ3 = α , θ4 = −α , et cetera, or in general θk = (− 1) α . Because the presence of multiple angle trigonometric functions in the invariant material property expression it is necessary to consider the off-axis material properties as expressed in equation 21, which is now written for an individual lamina of a laminate: k +1 Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES Qxx   k  4 Qyy   mk  k   nk4 Qxy   2 2  k   mk nk  = 2 2 Qssk   mk nk    m3n Qkxs   k k3    mk nk Q ys   k  nk4 mk4 mk2nk2 mk2nk2 − mk nk3 − mk3nk 105 2mk2 nk2 2mk2 nk2 mk4 + nk4 − 2mk2 nk2 − mk nk mk2 − nk2 mk nk mk2 − nk2 ( ( ) ) 4mk2 nk2 4mk2 nk2 − 4mk2nk2 2 mk2 − nk2 − 2mk nk mk2 − nk2 2mk nk mk2 − nk2 ( ( ( ) )    Q11 k      Q22 k    ,  Q12 k   Q    66 k    ) Since θk = (− 1) α , then k +1 note in this expression that mk = cos θ k and nk = sin θk . ( ) mk = cos (− 1) α = cos α = m ( k +1 ) and ( ) nk = sin (− 1) α = (− 1) sin α = (− 1) n . ( k +1 ) k +1 nk2 = (− 1) n = n 2 , and similarly nk4 = (− 1) n = n 4 . becomes k +1 2 Qxx   k  m4 Qyy    k   n4  Qxy  2 2  k   mn  = 2 2 Qkss   m n   (− 1)k +1 m3n Qkxs   k +1 3   (− 1) mn Qys   k  n4 m4 m2 n 2 m2 n 2 114 k +1 2m 2 n 2 2m 2 n 2 m4 + n4 − 2m 2 n 2 (− 1)k mn3 (− 1)k mn m2 − n 2 (− 1)k m3n (− 1)k +1 mn m2 − n 2 ( ( ) ) 4 k +1 Therefore, equation 114    Q11 k     Q22 k    . 2 m2 − n 2  Q12k  k 2(− 1) mn m 2 − n 2  Q66   k  k +1 2(− 1) mn m 2 − n 2    ( Then 4m 2 n 2 4m 2 n 2 − 4m 2 n 2 ( ( ) 115 ) ) Note in the above expression that the transformation coefficients of the on-axis lamina stiffnesses for Qxx , Qyy , Qxy , and Qss are independent of the lamina number, k k k k k. Therefore, in the calculation of the laminate stiffnesses these terms will be distributed over the summations. Recall that the in-plane stiffnesses for the laminate are: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 106   Qxx      k    Axx    Q    A     kyy    yy        Axy   N  Qkxy     = ∑  t k  Q    .  Ass   k =1   ssk    Axs           Qkxs    Ays         Q ys      k   116 Then substituting equation 115 gives:  Axx     m4    A  n4  yy      Axy   N   m 2 n 2   = ∑  tk  2 2  Ass   k =1   m n  Axs    (− 1)k +1 m3 n      k +1 3  Ays    (− 1) mn   2m 2 n 2 4m 2 n 2 Q   11  k  2m 2 n 2 4m 2 n 2       4 4 2 2  Q22 − 4m n m +n k   . (m2 − n 2 )2 Q12k   − 2m 2 n 2 (− 1)k mn(m 2 − n 2 ) 2(− 1)k mn(m2 − n 2 )  Q66      (− 1)k +1 mn(m 2 − n 2 ) 2(− 1)k +1 mn(m2 − n2 )  k   n4 m4 m 2n2 m 2n2 (− 1)k mn3 (− 1)k m3n 117 Now recognizing that m and n are independent of lamina number, then the in-plane laminate stiffnesses Axx , Ayy , Axy , and Ass can be written:  Axx   m A   4  yy   n  = 2 2  Axy  m n  Ass  m 2 n 2 4 4 n m4 m2 n 2 m2 n 2 2 2 2m n 2m 2 n 2 m4 + n4 − 2m 2 n 2    Q11   4 m n     k      Q   4m 2 n 2   N   k22   ∑  tk    . − 4m 2 n 2   k =1  Q12   k (m 2 − n 2 )2    Q66      k   2 2 118 Now make two further specializations. First, that the laminate is regular, this requires that all lamina have the same thickness, t. Second, that the laminate is homogeneous, this requires that the same material is used throughout the laminate. The consequence of the second specialization is that the on-axis material properties are independent of lamina number. Applying these specializations to equation 118 gives:  m4  Axx   4 A  n  yy    = Nt  2 2 m n  Axy   2 2  Ass   m n n4 2m 2 n 2 m4 2m 2 n 2 m 2n 2 m4 + n4 m 2n 2 − 2m 2 n 2 Revised: 10 February, 2000 4m 2 n 2  Q11    4m 2 n 2  Q22   . − 4m 2 n 2  Q12  2 m 2 − n 2  Q66  ( 119 ) M. V. Bower COMPOSITE MATERIALS LAMINATES 107 Now consider specifically the Axs term: N    k +1 k k k Axs = ∑  t k  (− 1) m3 nQ11 + (− 1) mn3Q22 + (− 1) mn m 2 − n 2 Q12 + 2(− 1) mn m 2 − n 2 Q66   k k k k  k =1   ( ) ( ) Again recognizing that m and n are independent of lamina number, then Axs becomes: N N     k +1 k Axs = m 3n ∑  t k (− 1) Q11  + mn3 ∑  t k (− 1) Q22  k  k  k =1  k =1    k + mn m − n ∑  t k (− 1) Q12  + 2mn m 2 − n 2 k  k =1  ( 2 2 ) ( N   k ∑  t k (− 1) Q66  k  k =1  ) . N Applying the specializations of a regular balanced lay-up to Axs gives: N N Axs = m 3n t Q11 ∑ (− 1) + mn3 t Q22 ∑ (− 1) k +1 k =1 k k =1 + mn(m − n )t Q12 ∑ (− 1) + 2mn(m − n )t Q66 ∑ (− 1) N 2 2 k =1 k N 2 2 . k k =1 Note that the summations in the above are summations of integers (1) with alternating sign. Making a third specialization, that the laminate is balanced, thus the number of laminae in the laminate is even, N ∈ Even . Thus, N N k =1 k =1 ∑ (− 1)k +1 = ∑ (− 1)k = 0 and therefore Axs = 0 . 120 By a similar process, for a homogeneous regular balanced symmetric angle-ply laminate the in-plane laminate stiffness Ays can be shown to be: Axs = 0 . 121 Summarizing, the laminate in-plane stiffnesses for a homogeneous regular balance symmetric angle-ply laminate are: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES  Axx   m4  4 A   yy   n m 2 n 2  Axy    = Nt  2 2 m n  Ass   0  Axs      0  Ays  n4 m4 m 2n2 m 2n2 0 0 108 2m 2 n 2 2m 2 n 2 m4 + n 4 − 2m 2 n 2 0 0 4m 2 n 2   4m 2 n 2  Q11    − 4m 2 n 2  Q22  . (m 2 − n 2 )2  Q12   Q66  0  0  122 At this point, the use of material property invariants may be applied to equation 122, which gives: U 1Q  Axx   Q A  U 1  yy  U 4Q  Axy    = Nt  Q U 5  Ass   0  Axs      Ays   0 cos4α   cos4α   1  - cos4α  Q   U 2  . - cos4α  Q  U 3  0   0  cos2α - cos2α 0 0 0 0 123 The laminate coupling stiffnesses are also zero because the laminate in consideration is symmetric. Therefore,  Bxx  0 B     yy  0  Bxy  0   =  .  Bss  0  Bxs  0      Bys  0 124 The development of the laminate bending stiffnesses for a homogeneous regular balance symmetric angle-ply laminate parallels the development of the inplane stiffnesses. For the laminate bending stiffnesses Dxx , Dyy , Dxy , and Dss , the lamina stiffnesses and m and n are independent of lamina number, then can be written as:  Dxx   D   N 2  yy   N 3 2  t t zk 2 = +    ∑ k =1  Dxy   12  Dss      m4    n 4  2 2  m n  m 2 n 2  ( ) n4 m4 m 2n2 m 2n2 2m 2 n 2 2m 2 n 2 m4 + n 4 − 2m 2 n 2 4m 2 n 2  Q11      4m 2 n 2  Q22      . − 4m 2 n 2  Q12   (m2 − n2 )2 Q66   125 Alternately, these stiffnesses can be written in terms of the invariant properties of the material: Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 109  Dxx   D   N 2  yy   N 3 2   =  t + 2t ∑ z k k =1  Dxy   12  Dss     cos2α cos4α    1  - cos2α cos4α   Q  U 2  . 0 - cos4α  Q   U 3   0 - cos4α  U 1Q   U 1Q  Q  U 4  Q U 5 ( ) 126 Now, as with the in-plane stiffness development, consider specifically the Dxs term. However, in this case it is easier to consider the initial form of the bending stiffness, which is: Dxs = ( 1 N  3  zk − z k −13 ∑ 3 k =1  ) (− 1) k +1   k k k m 3 nQ11 + (− 1) mn 3Q22 + (− 1) mn m 2 − n 2 Q12 + 2(− 1) mn m 2 − n 2 Q66   k k k k  ( ) ( ) Again recognizing that m and n are independent of lamina number, then Dxs becomes: Dxs = + ( )  mn3 m3n N  3 3 k +1  z k − zk −1 (− 1) Q11  + ∑ 3 k =1  3 k  ( mn m 2 − n 2 3 ) ∑  (z N  k =1 3 k − z k −1 3 ∑  (z  )  3 k − z k −1 (− 1) Q22  k  k =1  .  2 mn m 2 − n 2 N  3  k k 3 (− 1) Q12  +  zk − z k −1 (− 1) Q66  ∑ 3 k  k  k =1  N 3 k ( ) ) ( ) Applying the specializations of a regular balanced lay-up to Dxs gives: Dxs = (( ) ) m3n N mn3 3 3 k +1 1 − − + ( ) z z Q ∑ k k −1 11 3 k =1 3 ∑ ((z N k =1 3 k ) − zk −1 (− 1) Q22 3 k ) 2mn(m 2 − n 2 ) N mn(m 2 − n 2 ) N 3 3 3 3 k k + z k − zk −1 (− 1) Q12 + zk − z k −1 (− 1) Q66 ∑ ∑ 3 3 k =1 k =1 (( ) ) (( ) ) , or ( ) ( ) ( ) ( ) N N m 3 nQ11  N mn3Q22  N 3 k +1 3 k +1  3 k 3 k   ∑ zk (− 1) − ∑ zk −1 (− 1)  +  ∑ zk (− 1) − ∑ z k −1 (− 1)  3  k =1 3  k =1 k =1 k =1   2 2 N N mn(m − n )Q12  3 k 3 k  . +  ∑ z k (− 1) − ∑ zk −1 (− 1)  3 k = 1 k = 1   Dxs = ( ) + ( ) N 2mn(m 2 − n 2 )Q66  N 3 k 3 k   ∑ zk (− 1) − ∑ z k −1 (− 1)  3 k =1  k =1  ( ) ( ) Now examine the summations in the first bracket above, Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES ∑ (z )(− 1) N 3 k +1 k k =1 N 110 ( ) − ∑ zk −1 (− 1) k =1 k +1 3 = z1 − z 2 + z3 − z 4 + z5 − z6 + L + z N −3 − z N −2 + z N −1 − z N 3 3 3 3 3 3 3 3 3 3 . − z0 + z1 − z2 + z3 − z4 + z5 − L − z N − 4 + z N −3 − z N − 2 + z N −1 3 3 3 3 3 3 3 3 3 3 Note for a regular balanced symmetric laminate that: z0 = − z N , z2 = − z N −2 , z3 = − z N −3 , K z k = − z N − k , z1 = − z N −1 , . Therefore, the summations become: ∑ (z )(− 1) N k =1 3 k k +1 N ( ) − ∑ z k −1 (− 1) k =1 k +1 3 = z1 − z2 + z3 − z4 + z5 − z6 + L − z3 + z2 − z1 + z0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 − z0 + z1 − z 2 + z3 − z 4 + z5 − L + z 4 − z3 + z2 − z1 . = z0 − z0 3 3 =0 Through a similar process, the three remaining brackets in the expression for Dxs are also shown to be zero. Therefore, the coupling bending stiffness between normal and shear, Dxs , is zero. The derivation for the other coupling bending stiffness between normal and shear, D ys , is the similar, with identical results. Consequently, the bending stiffness for a regular balanced symmetric angle-ply laminate in terms of the lamina stiffnesses is:  Dxx   D    yy   N 2  Dxy   N 3 2   =  t + 2t ∑ zk k =1  Dss   12  Dxs       Dys     m4    n4   m 2 n 2   2 2  m n   0  0  ( ) n4 m4 m 2n2 m 2n2 0 0 2m 2 n 2 2m 2 n 2 m4 + n 4 − 2m 2 n 2 0 0  4m 2 n 2   2 2  4m n  Q11     − 4m 2 n 2  Q22   , (m2 − n2 )2 Q12    Q66   0    0   127 or, in terms of the invariant properties,  Dxx   D    yy   N 2  Dxy   N 3 2  t 2 t zk = +    ∑ k =1  Dss   12  Dxs       Dys   U1Q  Q U1  U 4Q   Q  U 5  0   0 ( ) Revised: 10 February, 2000  cos4α    - cos2α cos4α   1  0 - cos4α   Q   U 2  . 0 - cos4α   Q  U 3  0 0    0 0   cos2α 128 M. V. Bower COMPOSITE MATERIALS LAMINATES 111 When the results for a regular balanced symmetric angle-ply laminate are collected, the force-moment strain-curvature relationship is:  N x   Axx  N    y   Axy  N xy   0  M = 0  xy   − M yx   0     M yy   0 Axy 0 Ayy 0 0 Ass 0 0 0 0 0 0 0 0 0   ε 0x    0 0 0   ε 0y  0 0 0   ε 0s  . Dxx Dxy 0   κ 0x    Dxy Dyy 0   κ 0y    0 0 Dss  κ 0xy  129 By way of commentary on the specializations presented above, they provide engineering insight into the behavior of the various components of the stiffness terms. As such, they are important engineering developments. However, in practical application, direct calculation of the stiffnesses for the laminate in question is less restrictive and no more complicated. The preprocessing components of various finite element programs have the capability to calculate the laminate stiffnesses based on the laminate structure. Alternately, spreadsheets can be used to perform the stiffness calculations. Stress Distributions σ (ksi) sx sy -0.0500 ss -0.0400 -0.0300 -0.0200 z (in) -0.0100 -20 -15 -10 -5 0 5 10 15 0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 Figure 11. A Plot of the Stress Distribution Through a 4[± 45]S T300/N5208 Laminate Under a Mixed Force-Moment Load. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 112 STRESS DISTRIBUTION IN A LAMINATE The Kirchoff-Love model for the displacement of a plate leads to a linear distribution of the strain through the thickness of the plate. From a practical viewpoint, the assumption that the laminae are perfectly bonded together at their interfaces requires that the displacements of adjacent laminae are equal at the interface. Consequently, the strain distribution is continuous through the laminate. However, continuity of strain in a laminate does not imply continuity of stress. The laminae stresses are calculated from the laminae strains and the laminae properties. The discontinuity of the laminae properties produces the discontinuity of the stress through a laminate. Figure 11 shows a plot of the normal and shear stresses in a regular balanced symmetric angle-ply laminate due to a mixed force-moment loading. Note in this figure the significant discontinuities in all of the stresses. Between the first and second ply, the x-direction normal stress has a 17.8 kpsi jump. Figure 12 shows a plot of the normal and shear stresses in another angle-ply laminate due to a mixed force-moment loading. Note in this figure the jump in the x-direction normal stress between the first and second ply is approximately 22 kpsi. When a laminate experiences such a large discontinuity in any of the stresses the interfacial bond is highly stressed and may fail producing a delamination. Delamination of a laminate Stress Distributions σ (ksi) sx sy -0.0500 ss -0.0400 -0.0300 -0.0200 z (in) -0.0100 -20 -15 -10 -5 0 5 10 15 0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 Figure 12. A Plot of the Stress Distribution Through a 4[± 10]S T300/N5208 Laminate Under a Mixed Force-Moment Load. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 113 is a potentially hazardous situation that is not, in general, addressed by laminate failure theories. LAMINATE FAILURE THEORIES Laminate failure analysis is based primarily on lamina failure analysis. Laminates by their definition are inhomogeneous. Consequently, laminate failure theories are largely empirical. Alternately, lamina failure theories, though also empirical in origin, have a phenomenological connection. Knowing that stress distribution in a laminate it is possible to apply lamina failure theories to the individual lamina. This leads to two laminate failure “theories”, First-Ply-Failure (FPF) and Last-Ply-Failure (LPF). Drawing an analogy to metallic material behavior, ductile materials display two critical points in their stress-strain plot the yield point and the ultimate load. The first-ply-failure load is analogous to the yield point of a metal. When one ply fails, the load carrying capability of the laminate is degraded; however, it does not generally fail catastrophically at that load. Many laminates can continue to carry increasing loads safely after first-plyfailure, though at reduced stiffness. The last-ply-failure load is analogous to the ultimate load for a metal. At the last-ply-failure load, the structure fails catastrophically. FIRST-PLY-FAILURE To determine the first-ply-failure load the analyst must know the laminate stacking sequence, laminae properties, and the laminate loading. The laminate loading may be either applied forces and moments or the resulting mid-plane strains and curvatures. The latter is easier than the former since laminae strains can be calculated directly from the mid-plane strains and curvatures. For the former case, in which the forces and moments are known, the laminate stiffnesses and the inverse (compliance) must be calculated and the mid-plane strains and curvatures calculated in turn. Once the mid-plane strains and curvatures are known, they are used to calculate the strains in the individual laminae. At this point, there is some flexibility in the process. If the off-axis strain formulation of the quadratic failure criterion is used, the first-ply-failure analysis goes directly into the calculation of the R-factors. If the on-axis strain formulation of the quadratic failure criterion is used, the lamina strains must be transformed to the laminae principal material directions and the failure criterion applied. If either stress form of the quadratic failure criterion is used, the lamina strains are used to calculate the lamina stresses. Then, if the off-axis form of the quadratic failure criterion is used, the analysis goes to calculation of the R-factors. The final option, using the on-axis form of the quadratic failure criterion, requires transforming the lamina stresses to the lamina principal material directions and then applying the failure criterion. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 114 Laminate Stacking Sequence Laminae Material Properties Laminate Loading Yes Mid-plane Strains and Curvatures Known? No Calculate Laminate Stiffnesses Calculate Laminate Compliance Calculate Mid-plane Strains and Curvatures Calculate Laminae Strains Calculate Laminae Stresses Calculate Laminae R-Factors Select Minimum R-Factor Scale Laminate Load by Minimum R-Factor Figure 13. Flowchart for First-Ply-Failure Calculation. The first-ply-failure load is determined by scaling the original applied load by the minimum of the R-factors calculated for all of the laminae. This process is Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 115 shown in the flowchart presented in Figure 13. Note in this flowchart, as described in the paragraph above, the additional steps that must be performed in the event the mid-plane strains and curvatures are not known. LAST-PLY-FAILURE Last-ply-failure analysis begins with a first-ply-failure analysis. After the first-ply-failure load is determined, the assumption is made that the failed ply (or plies) makes no contribution to the stiffness of the laminate. Therefore, the laminate stiffness must be recalculated. The flowchart shown in Figure 14 describes a last-ply-failure analysis. In this process the forces and moments are used to determine the failure load. The analysis loops through the same steps until all lamina have been determined to have failed. Key to the process is the setting to zero of the failed lamina stiffnesses. Note that the failed laminae are not removed from the laminate, only the stiffnesses are set to zero. This is significant to the laminate bending stiffness. Recall that the bending stiffnesses are dependent on the centroidal coordinate of the laminae. If the failed laminae were totally removed from the laminate, the laminate stiffness calculation would not accurately reflect the vertical spacing of the remaining laminae. Therefore, only the stiffnesses of the failed laminae are set to zero. As an analysis proceeds through the LPF calculations, the intermediate values of the failure loads do not necessarily increase or decrease. Figure 15 shows a plot of critical bending moment values versus mid-plane curvature from a lastply-failure analysis. Note that the curve is not monotonic. There is a generally decreasing trend from the maximum load, which for this laminate was the initial failure load. However, there is a region, after the initial failure, in which the laminate sustains increasing loads. It is important to note that a last-ply-failure analysis presupposes that the laminate will not fail catastrophically after the first-ply-failure load. It is not at all impossible in a dynamic situation for the failure to cascade through the laminate although a last-ply-failure analysis might indicate that the laminate should not fail. In experimental investigation of the fracture of materials investigators characterize the text machines used as hard and soft. This characterization has to do with the ability of the machine to continue to apply loads to a specimen after a maximum load is applied without catastrophic failure. In this application, if the structure supporting the laminate maintains a constant load (force-moment) the laminate is more likely to fail catastrophically. Conversely, if the structure supporting the laminate maintains a constant displacement (strain-curvature), the laminate is less likely to fail catastrophically. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 116 First-Ply-Failure Analysis Mid-plane Strains and Curvatures Yes No Calculate Laminate Calculate FPF Forces and Moments Set Failed Laminae Stiffnesses to Zero Re-calculate Laminate Stresses Calculate Mid-plane Strains and Curvatures Calculate Laminae Strains Calculate Laminae R-Factors Select Minimum R-Factor Scale Laminate Load by Minimum R-Factor End Yes All Plies Failed? No Figure 14. Flowchart for Last-Ply-Failure Calculation. Revised: 10 February, 2000 M. V. Bower COMPOSITE MATERIALS LAMINATES 117 6 5 Mx (lb. in./in.) 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 kx 1/in. Figure 15. A Plot of Applied Bending Moment Versus Mid-Plane Curvature for an Angle-Ply Laminate From a Last-Ply-Failure Analysis. Revised: 10 February, 2000 M. V. Bower