Anisotropic material stress-strain relations

The macromechanical behavior of a lamina was quantitatively described in Chapter 2. The basic three-dimensional stress-strain relations for elastic anisotropic and orthotropic materials were examined. Subsequently, those relations were specialized for the plane-stress state normally found in a lamina. The plane-stress relations were then transformed in the plane of the lamina to enable treatment of composite laminates with different laminae at various angles. The various fundamental strengths of a lamina were identified, discussed, and subsequently used in biaxial strength criteria to predict the off-axis strength of a lamina. 

The complete stress-strain relation requires the six as to be written in terms of the six y components. The result is a 6 x 6 matrix with 36 coefficients in place of the single constant, Twenty-one of these coefficients (the diagonal elements and half of the cross elements) are needed to express the deformation of a completely anisotropic material. Only three are necessary for a cubic crystal, and two for an amorphous isotropic body. Similar considerations prevail for viscous flow, in which the kinematic variable is y. 

When a composite is treated as an anisotropic elastic material, the stress-strain relations are based on a general approach, which is valid in a region of small deformations. These theoretical relations were developed for application... 

Equations (2.9) and (2.10) are representative of all isotropic, homogeneous solids, regardless of the stress-strain relations of a solid. What is strongly materials specific and uncertain is the appropriate value for shear stress, particularly if materials are in an inelastic condition or anisotropic, inhomogeneous properties are involved. The limiting shear stress controlled by strength is termed r. ... 

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. 

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... 

Theory and Physics of Piezoelectricity. The discussion that follows constitutes a very brief introduction to the theoretical formulation of the physical properties of crystals. If a solid is piezoelectric (and therefore also anisotropic), acoustic displacement and strain will result in electrical polarization of the solid material along certain of its dimensions. The nature and extent of the changes are related to the relationships between the electric field (E) and electric polarization (P). which are treated as vectors, and such elastic factors as stress Tand strain (S), which are treated as tensors. In piezoelectric crystals an applied stress produces an electric polarization. Assuming Ihe dependence is linear, the direct piezoelectric effect can be described by the equation ... 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

Material Functions—Linear Viscoelasticity. One of the most important aspects of both the phenomenological and the molecular theories of viscoelasticity is the ability to characterize the material functions. The material functions are the properties that allow one to relate the stress response to a strain (deformation) history and vice versa through a constitutive equation. In linear viscoelasticity theory, generally isotropic descriptions are dealt with that is, the properties are the same in all directions. However, a material may be anisotropic and still have properties that vary with the direction of the test (7). Here only the isotropic case is considered and it is recognized that straight forward extensions can be made to the anisotropic case. In addition, only homogeneous materials, for which the properties are the same at all points within the material, only are discussed. 

Finally, in the macromechanics of a laminate, a representative volume made up of several layers of composite material, is replaced by an equivalent homogeneous anisotropic elanent as shown in Figure 8.4. Constitutive laws are then derived which relate the overall (macroscopic) deformation of the laminate to the applied stress resultants, or thennomechanical loads. These relationships provide a means of determining the strain distribution in a laminated composite structure when subjected to known loads and displacements. The strain distribution can then be used to calculate stresses in individnal plies by nsing the stress-strain laws obtained from the macro-mechanics of a single layer. 

The transformed reduced compliance and stiffness matrices [5] and [G1 relate off-axis stress and strain in an orthotropic lamina. Since these matrices are fully populated, the material responds to off-axis stresses as though it was fully anisotropic. Some consequences of the anisotropic nature of a unidirectional lamina are discussed in the following. 

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