Linear elasticity tensor

Cartesian components of the 2nd Piola-Kirchhoff stress tensor Cartesian components of the Green-Lagrange strain tensor Components of the linear elasticity tensor Increment in the /th displacement component... 

Let C i be a bounded domain with a smooth boundary L, and n = (ni,n2,n3) be a unit outward normal vector to L. Introduce the stress and strain tensors of linear elasticity (see Section 1.1.1),... 

Using relations (2.5) and (2.6) we can determine the elasticity tensor which describes the linear relation between components of the stress and strain tensors. 2 slr.ss = CEstta n is therefore an expression of Hooke s law for anisotropic crystals... 

If the solid is linear elastic (stiffness tensor C = S,v ), the potential energy Fj 1 takes the form (Deude et al., 2002) ... 

It is important to use the exact strain tensor definition, Eq. (6), to achieve rotational invariance with respect to lattice rotation the conventional linear strain tensor only provides differential rotational invariance of u in Eq. (7).hierarchy of approximations may be used for the elastic tensor 7. The most rigorous approach is to transform the bulk elastic tensor c according to... 

What this equation tells us is that a particular state of stress is nothing more than a linear combination (albeit perhaps a tedious one) of the entirety of components of the strain tensor. The tensor Cijn is known as the elastic modulus tensor or stiffness and for a linear elastic material provides nearly a complete description of the material properties related to deformation under mechanical loads. Eqn (2.52) is our first example of a constitutive equation and, as claimed earlier, provides an explicit statement of material response that allows for the emergence of material specificity in the equations of continuum dynamics as embodied in eqn (2.32). In particular, if we substitute the constitutive statement of eqn (2.52) into eqn (2.32) for the equilibrium case in which there are no accelerations, the resulting equilibrium equations for a linear elastic medium are given by... 

Compatibility conditions address the fact that the various components of the strain tensor may not be stated independently since they are all related through the fact that they are derived as gradients of the displacement fields. Using the statement of compatibility given above, the constitutive equations for an isotropic linear elastic solid and the definition of the Airy stress function show that the compatibility condition given above, when written in terms of the Airy stress function, results in the biharmonic equation = 0. 

In the present setting p is the mass density while the subscript i identifies a particular Cartesian component of the displacement field. In this equation recall that Cijki is the elastic modulus tensor which in the case of an isotropic linear elastic solid is given by Ciju = SijSki + ii(5ikSji + SuSjk). Following our earlier footsteps from chap. 2 this leads in turn to the Navier equations (see eqn (2.55))... 

Material Parameters. The key means whereby material specificity enters continuum theories is via phenomenological material parameters. For example, in describing the elastic properties of solids, linear elastic models of material response posit a linear relation between stress and strain. The coefficient of proportionality is the elastic modulus tensor. Similarly, in the context of dissipative processes such as mass and thermal transport, there are coefficients that relate fluxes to their associated driving forces. From the standpoint of the sets of units to be used to describe the various material parameters that characterize solids, our aim is to make use of one of two sets of units, either the traditional MKS units or those in which the e V is the unit of energy and the angstrom is the imit of length. 

While we do not want to give a sophisticated model including all the effects found in the mechanical behavior of polymers, we restrict ourselves to the simplest case, namely to an elastic small-strain model at constant temperature. Therefore, the governing variables are the linear strain tensor [Eq. (13)] derived from the spatial gradient of the displacement field u, and the microstructural parameter k and its gradient. The free energy density is assumed to be a function of the form of Eq. (14). 

The linear elasticity of an anistropic solid is a tensor property of rank 4. It... 

Although bone is a viscoelastic material, at the quasi-static strain rates in mechanical testing and even at the ultrasonic frequencies used experimentally, it is a reasonable first approximation to model cortical bone as an anisotropic, linear elastic solid with Hooke s law as the appropriate constitutive equation. Tensor notation for the equation is written as ... 

Figure 2.28 shows the orientation conventions for single crystals relative to orthogonal stress and strain axes for description of the elastic constants. Table 2.3 shows the form of the elastic constants for these crystal types. Comparing this table with Eq. (2.55), one can determine which elastic constants are equivalent and which are zero. For example, in the triclinic system, one can see that C2i=Ci2, etc., indicating the tensor is symmetric. Thus 21 elastic constants are needed to describe the linear elastic behavior of a triclinic crystal. In the hexagonal system one finds, in addition to symmetry of the tensor, that C, =C22, 13 23 44 55 I many other elastic constants are zero. 

For both magnesite and calcite, the elastic bulk modulus Bq was computed straightforwardly by the Murnaghan interpolation formula, while of the elasticity tensor only the C33 component and the C + C 2 linear combination could be calculated in a simple way. The relations used are C = (l/Vo)c (d L /crystal structure. To derive other elastic constants, the symmetry must be lowered with a consequent need of complex calculations for structural relaxation. A detailed account of how to compute the Ml tensor of crystal elasticity by use of simple lattice strains and structure relaxation was given previously[10, 11]. For the present deformations only the c-o ( ) relaxation need be considered. The results are reported in Table 6, together with the corresponding values extrapolated to 0 K from experimental data (Table 2). For calcite, the mea-... 

The first concept is the linear elasticity, that is, the linear relationship between the total stress and infinitesimal strain tensors for the filler and matrix as expressed by the following constitutive equations ... 

According to the statistical mechanics of rock-mass (Wu, 1993), linear elastic stress-strain relationship of fractured rock-mass is stated as follows in tensor form... 

Book content is otganized in seven chapters and one Appendix. Chapter 1 is devoted to the fnndamental principles of piezoelectricity and its application including related histoiy of phenomenon discoveiy. A brief description of crystallography and tensor analysis needed for the piezoelectricity forms the content of Chap. 2. Covariant and contravariant formulation of tensor analysis is omitted in the new edition with respect to the old one. Chapter 3 is focused on the definition and basic properties of linear elastic properties of solids. Necessary thermodynamic description of piezoelectricity, definition of coupled field material coefficients and linear constitutive equations are discussed in Chap. 4. Piezoelectricity and its properties, tensor coefficients and their difierent possibilities, ferroelectricity, ferroics and physical models of it are given in Chap. 5. Chapter 6. is substantially enlarged in this new edition and it is focused especially on non-linear phenomena in electroelasticity. Chapter 7. has been also enlarged due to mary new materials and their properties which appeared since the last book edition in 1980. This chapter includes lot of helpful tables with the material data for the most today s applied materials. Finally, Appendix contains material tensor tables for the electromechanical coefficients listed in matrix form for reader s easy use and convenience. 

The elasticity tensor (7 is a tensor of fourth order. It can be considered as a four-dimensional matrix with three components in each of its 4 directions. Its 3 = 81 components Cjjfc are the material parameters that completely describe the (linear) elastic behaviour. 

The full tensor form of the piezoelectric constitutive equations can be written by adding the linear elastic (Hook s law) and dielectric responses to Equation (16.1) [1] ... 

The states of stress and strain in a deformed crystal being idealized as a continuum are characterized by symmetric second-rank tensors and Cjj, respectively, each comprising six independent components. Hooke s law of linear elasticity for the most general anisotropic solid expresses each component of the stress tensor linearly in terms of all components of the strain tensor in the form... 

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