Material compliance tensor

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

Since there are only 6 independent components of stress and strain there are 36 components to S, the compliance tensor and C, the stiffness tensor. These 36 components may be further reduced using thermodynamic arguments so that there are 21 independent constants for triclinic symmetry, 13 for monodinic, 7 for tetragonal, 5 for hexagonal, 3 for cubic and 2 for isotropic materials. It is consequently more convenient to use die simplified notation of Voigt where ... 

The compliance tensor for background rock matrix is a general expression however, in the current work, it is defined by elastic constants. For an assumed transversely anisotropic material, the tensor is defined by five elastic constants (Ej, E2, Vi, V2, and Gt -Young s modulus in the horizontal plane. Young s modulus in the vertical plane, Poisson s ratio in the horizontal plane, Poisson s ratio in the vertical plane, and shear modulus in the vertical plane of the background rock mass, respectively). The compliance tensor for fractures is defined by ... 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

The rigidity tensor C or the compliance tensor S of the homogenized material are simply defined ly the relation ... 

From the homogeneous stress hypothesis, the elastic compliance tensor of the aggregate can be expressed by equation (84). Similarly, from the hypothesis of homogeneous strain, the elastic stiffness tensor of the aggregate can be expressed by equation (85), where tjp, for example, is the direction cosine between the X axis of the aggregate and the Up axis of the crystal unit cell and are the elastic compliances and stiffnesses of the unit respectively. These expressions for averaging are, of course, two extreme approximations, which neglect the interaction between the units and do not correspond to the real situation of polycrystalline materials, as has been pointed out by several authors. ... 

The susceptibility tensors measure the macroscopic compliance of the electrons. Since the second order polarization is a second rank tensor, SHG is zero in a centrosymmetric or randomly oriented system. To make the material capable of SHG, the NLO dopants must be oriented noncentrosymmetricaly in the polymer matrix (2-3). When modeling the poled, doped films using a free gas approximation, the poled second order susceptibilities are given by (2.19)... 

Chapter 4 outlines operations of symmetry on ideal solids that show how the number of independent components of the modulus tensor diminishes as the number of symmetry elements in the solid increases. This analysis leads to the formulation of the generalized Hooke s law utilizing both elastic modulus and elastic compliances for amorphous solid materials. These relationships, conveniently modified, are further used in viscoelasticity. In this chapter the generalized law of Newton for ideal liquids is also stated. 

Here S (g) and C (g) are fourth-rank tensors describing the elastic behavior of the crystallites in the polycrystalline material [not necessarily the single-crystal compliance and stilfness tensors as in Equation (73)]. If Equations (84) are true then the average strain measured by diffraction has the following expression ... 

Tables 4.4-3-4.4-21 are arranged according to piezoelectric classes in order of decreasing symmetry (see Table 4.4-2), and alphabetically within each class. They contain a number of columns placed on two pages, even and odd. The following properties are presented for each dielectric material density q, Mohs hardness, thermal conductivity k, static dielectric constant Sij, dissipation factor tanS at various temperatures and frequencies, elastic stiffness Cmn, elastic compliance s n (for isotropic and cubic materials only), piezoelectric strain tensor di , elastooptic tensor electrooptic coefficients r k (the lat-...

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... 

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