Elastic anisotropic bodies

In this section, we will briefly address the homogeneous, bulk (i.e., 3D), primarily crystalline materials in uniform stressed states within the limits of reversible small deformations. 

Physical-Chemical Mechanics of Disperse Systems and Materials 

In the most general case, aU nine components of the strain tensor are linearly related to all nine components of the stress tensor. This would yield Hooke s law with 81 parameters Ey = Sgitiaw = Sgucrid with the summation taking place over the repeating indices and with the inverse transform matrix of Ojj = CijyEy. Obviously, there is no inverse proportionality between individual s and c. The 81 s and c parameters transform linearly upon turning the coordinate system, that is. 

summation takes place over the four repeating indices. The coefficients a constitute the direction cosines to the fourth power the parameters (or represent a fourth-order tensor, referred 

Thermodynamics yields a particular symmetry of pairwise components Cijki (or iijki), so there are only 36 independent variables. These variables can be written down in the form of a square 6x6 matrix. Further simplification is achieved by a pairwise combination of indices according to the following rule  

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The fundamental quantities in elasticity are second-order tensors, or dyadicx the deformation is represented by the strain thudte. and the internal forces are represented by Ihe stress dyadic. The physical constitution of the defurmuble body determines ihe relation between the strain dyadic and the stress dyadic, which relation is. in the infinitesimal theory, assumed lo be linear and homogeneous. While for anisotropic bodies this relation may involve as much as 21 independent constants, in the euse of isotropic bodies, the number of elastic constants is reduced lo two. 

Lekhnitskii, S. G. Theory of elasticity of an anisotropic body. Holden-Day, San Francisco, 1963... 

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

The first Lame constant (A) has no physical interpretation. However, both Lame constants are related to other elastic moduli. To see this, recall that the Young s modulus, E, is defined as the ratio of normal stress to normal strain. Hence, for an elastically isotropic body, E is given by (cn-Ci2)(cn-b 2ci2)/(cn-I-C12), or /r(3A-b 2/r)/(A-b/r). It should be emphasized that the Young s modulus is anisotropic for all crystal classes, including the cubic class, so this relation would never apply to any monocrystal. 

Leknitskii, S.G. (1981). Theory of Elasticity of an Anisotropic Body. Mir Publ. 

S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Government Publishing House for Technical-Theoretical Works, Moscow and Leningrad, 1950. Also P. Fern (Translator), Holden-Day, San Francisco, 1963. 

It can be shown that for the cross-terms 221 = 2i2, 2si = 2b. and so on, so that of the initial 36 values, there are only 21 independent elastic constants necessary to completely define an anisotropic volume without any geometrical symmetry (not to be confused with matrix symmetry). The number of independent elastic constants decreases with increasing geometrical symmetry. For example, orthorhombic symmetry has 9 elastic constants, tetragonal 6, hexagonal 5, and cubic only 3. If the body is isotropic, the number of independent moduli can decrease even fmther, to a limiting... 

Eqn (2.92) is the culmination of our efforts to compute the displacements due to an arbitrary distribution of body forces. Although this result will be of paramount importance in coming chapters, it is also important to acknowledge its limitations. First, we have assumed that the medium of interest is isotropic. Further refinements are necessary to recast this result in a form that is appropriate for anisotropic elastic solids. A detailed accounting of the anisotropic results is spelled out in Bacon et al. (1979). The second key limitation of our result is the fact that it was founded upon the assumption that the body of interest is infinite in extent. On the other hand, there are a variety of problems in which we will be interested in the presence of defects near surfaces and for which the half-space Green function will be needed. Yet another problem with our analysis is the assumption that the elastic constants... 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require cjxy=(jyx, stress components in the symmetric stress matrix are only six. Similarly there are only six independent strain components. Therefore there can only be six stress and six strain components for an elastic body which has unequal elastic responses in x, y and z directions as in a completely anisotropic solid. The representation of elastic properties become simple by following the well known Einstein convention. The subscript xx, yy, zz, yz, zx and xy are respectively represented by 1, 2, 3, 4, 5 and 6. Therefore Hooke s law may now be written in a generalized form as. 

Lekhniitski, 1963. Elasticity of an anisotropic elastic body, Holden Day. 

Lekhnitskii SG. Theory of elasticity of an anisotropic elastic body [Fem P, Trans.]. San Francisco Holden Day Inc. 1963. 

Elastic properties, chareicteristic for an isotropic solid polymer body, are a specific feature of gels. There exist and find practical use anisotropic gels resembling the structure of liquid crystals, but this subsection will disetiss isotropic gels only. 

Bone is mineralized tissue that constitutes part of the vertebral skeleton. Its function is to transmit and bear the loads to which the body is constantly subjected, protect the inner organs, and produce blood cells. From a mechanic point of view, the osseous tissue is an anisotropic viscoelastic material with properties that depend on direction and velocity of the applied load, as well as on the mineral content. Indeed, although the minerals confer rigidity and hardness to this tissue, the collagen imparts some elasticity, which ultimately results in its limited tensile strength and resilience. As a consequence of its composition, bone is an essentially brittle material (Fig. 17.1). Several pathologies can affect bone, including fractures, arthritis, infections, osteoporosis, and tumors, and may require adjuvant biomaterial devices. 

The solution of the 3-D elastic problem of a nonlinear nonhomogeneous anisotropic material presents a major engineering challenge. The interaction between the geometry of the moving dynamic muscle and muscle mechanics can be evaluated by different approaches. The obvious difficulty lies in the uncertainty of the spatial values of the material properties in a nonhomogeneous body. 

Lekhnitskii, S.G., Theory of an Anisotropic Elastic Body, Holden Day, SF,... 

According to the theory for infinitesimal deformation of an anisotropic elastic body, the mechanical properties can be represented by 36 independent elastic compliances Sij or by the same number of independent elastic stiffnesses Cij. The number of constants can be reduced to nine on the basis of the orthogonal anisotropy of the body with respect to Cartesian coordinates 0-XiX2X3Jn the bulk specimen, and can be further reduced to five on the basis of the transverse isotropy about the X3 axis. 

Figure 17 Comparison of mechanical anisotropy of the bulk specimen observed with that calculated from the theory of infinitesimal deformation of an orthogonally anisotropic elastic body...

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