Lamina analysis

Shear-stress-shear-strain curves typical of fiber-reinforced epoxy resins are quite nonlinear, but all other stress-strain curves are essentially linear. Hahn and Tsai [6-48] analyzed lamina behavior with this nonlinear deformation behavior. Hahn [6-49] extended the analysis to laminate behavior. Inelastic effects in micromechanics analyses were examined by Adams [6-50]. Jones and Morgan [6-51] developed an approach to treat nonlinearities in all stress-strain curves for a lamina of a metal-matrix or carbon-carbon composite material. Morgan and Jones extended the lamina analysis to laminate deformation analysis [6-52] and then to buckling of laminated plates [6-53]. 

Naik NK, Shembekar PS. Elastic behaviour of woven fabric composites I - lamina analysis. J Compos Mater 1992 26 2196-225. 

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. 

The previous section illustrated how to obtain the elastic properties of a unidirectional lamina. In practice considerably more information may be required about the behavioural characteristics of a single lamina. To obtain details of the stresses and strains at various orientations in a single ply the following type of analysis is required. 

The previous analysis has shown that the properties of unidirectional fibre composites are highly anisotropic. To alleviate this problem, it is common to build up laminates consisting of stacks of unidirectional lamina arranged at different orientations. Clearly many permutations are possible in terms of the numbers of layers (or plies) and the relative orientation of the fibres in each... 

The preceding stress-strain and strain-stress relations are the basis for stiffness and stress analysis of an individual lamina subjected to forces in its own plane. Thus, the relations are indispensable in laminate analysis. 

The fundamental analysis of a laminate can be explained, in principle, by use of a simple two-layered cross-ply laminate (a layer with fibers at 0° to the x-direction on top of an equal-thickness layer with fibers at 90° to the x-direction). We will analyze this laminate approximately by considering what conditions the two unbonded layers in Figure 4-3 must satisfy in order for the two layers to be bonded to form a laminate. Imagine that the layers are separate but are subjected to a load in the x-direction. The force is divided between the two layers such that the x-direction deformation of each layer is identical. That is, the laminae in a laminate must deform alike along the interface between the layers or else fracture must existl Accordingly, deformation compatibility of layers is a requirement for a laminate. Because of the equal x-direction deformation of each layer, the top (0°) layer has the most x-direction ress because it is stiffer than the bottom (90°) layer in the x-direction./ Trie x-direction stresses in the top and bottom layers can be shown to have the relation... 

The laminate stress-analysis elements are affected by the state of the material and, in turn, determine the state of stress. For example, the laminate stiffnesses are usually a function of temperature and can be a function of moisture, too. The laminae hygrothermomechanical properties, thicknesses, and orientations are important in determining the directional characteristics of laminate strength. The stacking sequence... 

The analysis of stresses in the laminae of a laminate is a straight-fonvard, but sometimes tedious, task. The reader is presumed to be familiar with the basic lamination principles that were discussed earlier in this chapter. There, the stresses were seen to be a linear function of the applied loads if the laminae exhibit linear elastic behavior. Thus, a single stress analysis suffices to determine the stress field that causes failure of an individual lamina. That is, if all laminae stresses are known, then the stresses in each lamina can be compared with the lamina failure criterion and uniformly scaled upward to determine the load at which failure occurs. 

The overall procedure of laminate-strength analysis, which simultaneously results in the laminate load-deformation behavior, is shown schematically in Figure 4-36. There, load is taken to mean both forces and moments similarly, deformations are meant to include both strains and curvatures. The analysis is composed of two different approaches that depend on whether any laminae have failed. 

If no laminae have failed, the load must be determined at which the first lamina fails (so-called first-ply failure), that is, violates the lamina failure criterion. In the process of this determination, the laminae stresses must be found as a function of the unknown magnitude of loads first in the laminate coordinates and then in the principal material directions. The proportions of load (i.e., the ratios of to Ny, to My,/ etc.) are, of course, specified at the beginning of the analysik The loaa parameter is increased until some individual lamina fails. The properties, of the failed lamina are then degraded in one of two ways (1) totally to zero if the fibers in the lamina fail or (2) to fiber-direction properties if the failure is by cracking parallel to the fibers (matrix failure). Actually, because of the matrix manipulations involved in the analysis, the failed lamina properties must not be zero, but rather effectively zero values in order to avoid a singular matrix that could not be inverted in the structural analysis problem. The laminate strains are calculated from the known load and the stiffnesses prior to failure of a lamina. The laminate deformations just after failure of a lamina are discussed later. 

Note that the lamina failure criterion was not mentioned explicitly in the discussion of Figure 4-36. The entire procedure for strength analysis is independent of the actual lamina failure criterion, but the results of the procedure, the maximum loads and deformations, do depend on the specific lamina failure criterion. Also, the load-deformation behavior is piecewise linear because of the restriction to linear elastic behavior of each lamina. The laminate behavior would be piecewise nonlinear if the laminae behaved in a nonlinear elastic manner. At any rate, the overall behavior of the laminate is nonlinear if one or more laminae fail prior to gross failure of the laminate. In Section 2.9, the Tsai-Hill lamina failure criterion was determined to be the best practical representation of failure... 

The procedure of laminate strength analysis outlined in Section 4.5.2, with the Tsai-Hill lamina failure criterion will be illustrated for cross-ply laminates that have been cured at a temperature above their service or operating temperature in the manner of Tsai [4-10]. Thus, the thermal effects discussed in Section 4.5.3 must be considered as well. For cross-ply laminates, the transformations of lamina properties are trivial, so the laminate strength-analysis procedure is readily interpreted. 

The example considered to illustrate the strength-analysis procedure is a three-layered laminate with a [4-15°/-15°/+15°] stacking sequence [4-10]. The laminae are the same E-glass-epoxy as in the cross-ply laminate example with thickness. 005 in (.1270 mm), so that the total laminate thickness is. 015 in (.381 mm). In laminate coordinates, the transformed reduced stiffnesses are... 

The Tsai-Hill criterion governs failure of a lamina (the strength-analysis procedure could, of course, involve another criterion). 

The analysis of such a laminate by use of classical lamination theory revolves about the stress-strain relations of an individual orthotropic lamina under a state of plane stress in principal material directions... 

Obviously, the foregoing description of problems in the mechanics of composite materials is incomplete. Some topics do not fit well within the logical framework just described. Other topics are too advanced for an introductory book, even at the graduate level. Thus, the rest of this chapter is devoted to a brief discussion of some basic lamina and laminate analysis and behavior characteristics that are not included in preceding chapters. 

The second special case is an orthotropic lamina loaded at angle a to the fiber direction. Such a situation is effectively an anisotropic lamina under load. Stress concentration factors for boron-epoxy were obtained by Greszczuk [6-11] in Figure 6-7. There, the circumferential stress around the edge of the circular hole is plotted versus angular position around the hole. The circumferential stress is normalized by a , the applied stress. The results for a = 0° are, of course, identical to those in Figure 6-6. As a approaches 90°, the peak stress concentration factor decreases and shifts location around the hole. However, as shown, the combined stress state at failure, upon application of a failure criterion, always occurs near 0 = 90°. Thus, the analysis of failure due to stress concentrations around holes in a lamina is quite involved. 

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

Fibers are often regarded as the dominant constituents in a fiber-reinforced composite material. However, simple micromechanics analysis described in Section 7.3.5, Importance of Constituents, leads to the conclusion that fibers dominate only the fiber-direction modulus of a unidirectionally reinforced lamina. Of course, lamina properties in that direction have the potential to contribute the most to the strength and stiffness of a laminate. Thus, the fibers do play the dominant role in a properly designed laminate. Such a laminate must have fibers oriented in the various directions necessary to resist all possible loads. 

Goel VK et al (1995) Interlaminar shear stresses and laminae separation in a disc finite element analysis of the L3-L4 motion segment subjected to axial compressive loads. Spine (Phila Pa 1976) 20(6) 689-698... 

The next level of complexity in the treatment is to orient the apphed stress at an angle, 6, to the lamina fiber axis, as illustrated in Figure 5.119. A transformation matrix, [T], must be introduced to relate the principal stresses, o, 02, and tu, to the stresses in the new x-y coordinate system, a, ay, and t j, and the inverse transformation matrix, [T]- is used to convert the corresponding strains. The entire development will not be presented here. The results of this analysis are that the tensile moduli of the composite along the x and y axes, E, and Ey, which are parallel and transverse to the applied load, respectively, as well as the shear modulus, Gxy, can be related to the lamina tensile modulus along the fiber axis, 1, the transverse tensile modulus, E2, the lamina shear modulus, Gu, Poisson s ratio, vn, and the angle of lamina orientation relative to the applied load, 6, as follows ... 

The final level of complexity is to arrange layers of unidirectional laminae to form the laminate composite, as shown in Figure 5.121. The lower surface of the /th lamina is assigned the coordinates hf relative to the midplane, so the thickness of the /th lamina is /i/ — /i/ 1. The mathematics of stress analysis are an extension of those used in the foregoing description, and they include force and moment analysis on the bending of the laminate. The complete development will not be presented here, but only outlined. The interested reader is referred to two excellent descriptions of the results [9,10]. 

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