Laminated Composites Theory

Classical laminated plate tlieoiy is used to determine the stiffness of laminated composites. Details of the Kitchoff-Love hypothesis on which the theory is based can be found in standard texts (1,7,51). Essentially, the strains in each ply of the laminate ate represented as middle surface strains plus... 

The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

Actually, because of the stress and deformation hypotheses that are an inseparable part of classical lamination theory, a more correct name would be classical thin lamination theory, or even classical laminated plate theory. We wiiruS ffi bmmon term classical lamination theory, but recognize that it is a convenient oversimplification of the rigorous nomenclature. In the composite materials literature, classical laminationtheoryls en abbreviated as CLT. 

J. N. Reddy, Mechanics of Laminated Composite Piates Theory and Analysis, CRC Press, Boca Raton, Florida, 1997. 

During the processing of composite materials in a hot press or an autoclave, the laminate is usually kept flat until cure is complete. If the platen surfaces are assumed frictionless, the effect of the constraints is to require that the curvatures K] and k2 be zero throughout cure. To develop the elastic solution under these constrained conditions, the laminated plate theory may be used with conditions of N = 0 and jc, = 0. The resulting midplane strains are given by... 

Use of Products), Vol. 15, American Society for Testing and Materials, Philadelphia, PA. (1998). ABAQUS/Standard User s Manual. Hibbitt, Karlsson Sorensen, Pawtucket. RI. Reddy, H. N. (1997). Mechanics of Laminated Composite Plates Theory and Analysis, CRC Press, Inc., Boca Raton, Florida. 

Fiber-reinforced composites are also highly anisotropic when fibers are aligned. In addition, they are nonhomogeneous and usually laminated with individual plies oriented in several different directions. Because of these difficulties, once again the isotropic failure criteria are of little use. However, reasonably successful failure criteria for individual plies have been developed (9-11) and applied to the failure of laminates by laminated plate theory ( , 12). 

Alternatively, tests can be used to obtain the basic stiffness properties of the material form and their corresponding range measured by some statistical property such as the standard deviation. In two-dimensional cases where there are no significant loads in the out-of-plane direction, the basic orthotropic stiffness properties in Eqn (6.1) can be measured experimentally. Then, the classical laminated plate theory described in previous sections for determination of stiffness can be used effectively to model these sttuctures. Alternatively, the four basic stiffiiesses for 3-D woven composites can be... 

A multilayer laminate is composed of several laminas (or layers, or plies). Classical laminate theory (CLT) describes the linear elastic response of a thin laminated composite subjected to in-plane loads and bending moments see, e.g., Eckold (1994) and Herakovich (1998). Individual layers are assumed to be homogeneous, orthotropic, or transversely isotropic and in a state of plane stress. The constitutive relation for a thin multilayer laminated composite is given as ... 

This chapter is devoted to the analysis of the elastic properties and their characterization for laminated advanced composites. It starts with a general overview of composite stiffness and then moves to lamina analysis focused on unidirectional reinforced composites. The analysis of laminated composites is addressed through the classical lamination theory (CLT). The last section describes full-field techniques coupled with inverse identification methods that can be employed to measure the elastic constants. 

In (Pao et al, 1980), the stretching tests of the strips circumferentially cut and longitudinally cut from the left ventricular wall were reported. The calculated instantaneous stress and strain data revealed that significant differences in the wall muscle properties existed between these two groups of strips because of the different directions along which they were cut. It points out the fact that the myocardial fibrous characteristics should be incorporated into cardiodynamics studies. When the stress and strain data were treated in (Pao et al, 1980) by application of the theory for laminated composites, the following stess-dependent stiffness equations for the circumferentially cut strips were obtained ... 

Pao YC, Nagendra GK, Padiyar R, Ritman EL (1980) Derivation of myocardial stiffness equation based on theory of laminated composites. J Biomech Eng 102 252-257 Peskin CS (1975) Mathematical aspects of heart physiology. Courant Institute of Mathematical Sciences, New York... 

Proc Symp on Appl Comput Methods, pp 477-486 Pao YC, Mayendra KK, Padiyar RR, Ritman EL (1980) Derivation of myocardial fiber stiffness equation based on theory of laminated composite. Trans ASME 103 202-259 Parmley WW, Tyberg JV (1976) Determination of myocardial oxygen demand. Prog Cardiol 5 19-36 Pierce WH (1981) Body forces and pressures in elastic models of the myocardium. Biophys J 34 35-39 Pollack OH, Krueger JW (1978) Myocardial sarcomere mechanics some parallels with skeletal muscle. In Baan Y, Noordegraaf A, Raines J (eds) Cardiovascular System Dynamics, Cambridge, pp 3-10... 

The sequence of theories developed throughout the foregoing chapters, leading the way from piezoelectric composites via adaptive laminated composite shells to adaptive thin-walled beams, allows for examinations of almost arbitrary complexity. The attempt to provide examples for the full spectrum of capabilities will be dropped in favor of an application-oriented approach. Hence, the subsequent investigations will be carried out in view of the integral actuation of structures with certain similarity to helicopter rotor blades. 

A two-component composite consists of a matrix and a filler, which may be in the form of long or short fibers, large or small particles, or laminates. The theory of composite materials is well documented. For this discussion it is sufficient to simply describe composites by the so-called rule of mixtures, stated as... 

Tang S., A boundary layer theory - Part I Laminated composites in plane stress, J. Compos. Mater., 1975, 9, pp. 33 - 41. 

The failure of laminate composites is a topic that has been researched for over 50 years. Currently, there are many competing theories on laminate failnre. Typically, the failure theories are used to estimate the nominal layup of a laminate followed by experimental verification. The most noteworthy attempt to reconcile the various composite failure theories was The World-Wide Failure Exercise. This exercise had its genesis in a meeting held in 1991. After the meeting the authors conceived of the selection of test samples and biaxial loading cases for a pressurized tubular test specimen (see Table 9.1). 

It was assmned that the complex 0 wound layers could be treated as two separate layers. It was also assmned that the matrix infiltrated the fibre layers completely and could be used as the matrix material for both the 0 wound layers. The composite laminated plate theory was used... 

From a fundamental imderstanding of the mechanics of winding fibers onto a cylinder, it is possible to predict, nsing micromechanics and the classical composite laminate plate theory, the effective elastic properties of the finished filament wormd catheter/tubing. 

Thermal residual stresses on each ply, due to thermal expansion mismatch between adjacent plies from high cure temperatures, are derived from the lamina constitutive equation associated with extension, bending, and thermal strains based on the laminated composite plate theory as shown in Equation (2) [13]. 

The characteristic features of a cord—mbber composite have produced the netting theory (67—70), the cord—iaextensible theory (71—80), the classical lamination theory, and the three-dimensional theory (67,81—83). From stmctural considerations, the fundamental element of cord—mbber composite is unidirectionaHy reinforced cord—mbber lamina as shown in Figure 5. From the principles of micromechanics and orthotropic elasticity laws, engineering constants of tire T cord composites in terms of constitutive material properties have been expressed (72—79,84). The most commonly used Halpin-Tsai equations (75,76) for cord—mbber single-ply lamina L, are expressed in equation 5 ... 

Note that no assumptions involve fiber-reinforced composite materials explicitly. Instead, only the restriction to orthotropic materials at various orientations is significant because we treat the macroscopic behavior of an individual orthotropic (easily extended to anisotropic) lamina. Therefore, what follows is essentially a classical plate theory for laminated materials. Actually, interlaminar stresses cannot be entirely disregarded in laminated plates, but this refinement will not be treated in this book other than what was studied in Section 4.6. Transverse shear effects away from the edges will be addressed briefly in Section 6.6. 

Study of transverse shearing stress effects is divided in two parts. First, some exact elasticity solutions for composite laminates in cylindrical bending are examined. These solutions are limited in their applicability to practical problems but are extremely useful as checl oints for more broadly applicable approximate theories. Second, various approximations for treatment of transverse shearing stresses in plate theory are discussed. 

Bowles, D.E. and Griffin, O.H. (1991a). Micromechanics analysis of space simulated thermal stresses in composites, part I Theory and unidirectional laminates. J. Reinforced Plast. Composites 10, 504-521. 

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