Aligned orthotropic materials

Symmetry and invariance arguments were invoked in Section 3.5 to conclude that 

The shear stress component 76 also vanishes in this case due to the assumed S5rmmetry of the configuration and the absence of a mismatch in shear strain. 

The total strain field in the system is not compatible, in general, due to the mismatch. However, the incompatibility is represented explicitly by the mismatch strain, which is the elastic strain which must be imposed on the film material to fully counteract the incompatibility. The total strain minus the mismatch strain is compatible, from which it follows that the 

The response of the material is linearly elastic with orthotropic symmetry, so that 

The stress in (3.77) is next expressed in terms of and by means of (3.76). A consequence of imposing the equilibrium conditions (3.75) on the resultant is a set of linear algebraic equations for the curvature and the strain of the substrate midplane, 

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In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

Because of the analytical complications involving the stiffnesses Ai6, A26, D g, and D26, a laminate is sometimes desired that does not have these stiffnesses. Laminates can be made with orthotropic layers that have principal material directions aligned with the laminate axes. If the thicknesses, locations, and material properties of the laminae are symmetric about the middle surface of the laminate, there is no coupling between bending and extension. A general example is shown in Table 4-2. Note that the material property symmetry requires equal [Q j], of the two layers that are placed at the same distance above and below the middle surface. Thus, both the orthotropic material properties, [Qjjlk. of the layers and the angle of the principal material directions to the laminate axes (i.e., the orientation of each layer) must be identical. 

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. 

The orthorhombic crystal lattice itself is not too important technically because there are only a small number of materials crystallising in this structure. Composites (chapter 9), however, frequently have the same symmetry because they may contain aligned fibres. Materials with the same symmetry as an orthorhombic crystal are called orthotropic. 

Hill [15] has developed a generalisation of the von Mises criterion for anisotropic materials. Anisotropy is defined with respect to specific axes fixed within the material which, in the case of orthotropic materials, are mutually perpendicular. Then, a 1-2-3 axes set can be chosen to align with the directions of orthotropy and the yield criterion defined with respect to the stresses in this axis set. This precludes the use of principal stresses as the principal directions do not in general coincide with the directions of orthotropy. Therefore, Hill s criterion is a generalisation of Equation (12.9)... 

A so-called specially orthotropic lamina is an orthotropic lamina whose principal material axes are aligned with the natural body axes ... 

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

Each layer is orthotropic (but the principal material directions of each layer need not be aligned with the plate axes), linear elastic, and of constant thickness (so the entire plate is of constant thickness). 

Pagano studied cylindrical bending of symmetric cross-ply laminated composite plates [6-21]. Each layer is orthotropic and has principal material directions aligned with the plate axes. The plate is infinitely long in the y-direction (see Figure 6-16). When subjected to a transverse load, p(x), that is, p is independent of y, the plate deforms into a cylinder ... 

The basic approaches as summarized by Ashton and Whitney [6-31] will now be discussed. First, a symmetric laminate with orthotropic laminae having principal material directions aligned with the plate axes will be treated. The transverse normal strain can be found from the orthotropic stress-strain relations, Equation (2.15), as... 

For all of the cases of substrate curvature induced by film stress that were considered in Section 2.1, it was assumed that both the film and substrate materials were isotropic. This provided a basis for a relatively transparent discussion of curvature phenomena and it led to results which have proven to be broadly useful. However, there are situations for which some understanding of the influence of material anisotropy of the film material or the substrate material, or perhaps of both materials, is important to know. Therefore, in this section, representative results on the influence of material anisotropy on substrate curvature are included. Results are established for two particular curvature formulas. In the first case, the film is presumed from the outset to be very thin compared to the substrate. Furthermore, the substrate is assumed to be isotropic and the film is considered to be generally anisotropic. In the second case discussed, no restriction is placed on the thickness of the film relative to the substrate, but both materials are assumed to be anisotropic. However, to obtain expressions for curvature which are not too complex to be interpretable, attention is limited to cases for which both the film and substrate materials are orthotropic, that their axes of orthotropy aligned with each other and that one axis of or-thotropy of each material is normal to the film-substrate interface. There is no connection between the values of orthotropic elastic constants of the two materials. Consideration of these two cases illustrates the most useful approaches for anisotropic materials generalizations for cases of greater complexity are evident. 

Remark 5.7. For composites in accordance with Remark 5.2 consisting of aligned transversely isotropic fiber and matrix materials, the overall material behavior is expected to be at most orthotropic and, in the case of identical directional fiber fractions i/i and 1/2, to be transversely isotropic. 

The magnitude of the elastic moduli obtained for an anisotropic material will depend on the orientation of the coordinates used to describe the material elastic response. However, if the material elastic moduli are known for coordinates aligned with the principal material directions, then the elastic moduli for any other orientation can be determined through appropriate transformation equations. Thus, only four elastic constants are needed in order to fully characterize the in-plane maaoscopic elastic response of an orthotropic lamina. The reference coordinates in the plane of the lamina are aligned with longitudinal axis (L) parallel to the fibers, and the transverse axis (7) perpendicnlar to the fibers. The engineering orthotropic elastic moduli of the lamina defined earher are... 

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