Elastic constants fourth-order

Ctjki is a fourth order tensor that linearly relates a and e. It is sometimes called the elastic rigidity tensor and contains 81 elements that completely describe the elastic characteristics of the medium. Because of the symmetry of a and e, only 36 elements of Cyu are independent in general cases. Moreover only 2 independent rigidity constants are present in Cyti for linear homogeneous isotropic purely elastic medium Lame coefficient A and /r have a stress dimension, A is related to longitudinal strain and n to shear strain. For the purpose of clarity, a condensed notation is often used... 

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) ... 

Equations (A.7) also show that, in general, the prediction of a property that depends on a tensor of rank I will require knowledge of orientation averages of order /. The elastic constants of a material are fourth-rank tensor properties thus the prediction of their values for a drawn polymer involves the use of both second- and fourth-order averages, in the simplest case P ico O)) and P (x>s6)), and thus provides a more severe test of the models for the development of orientation. The elastic constants are considered in section 11.4. 

In a perfectly harmonic crystal the elastic constants would be strictly independent of temperature. However, due to the existence of third- and fourth-order anharmonic terms in the crystal potential there is a coupling between the homogeneous strains and the phonon coordinates. This will lead to a background temperature dependence of the elastic constants. It can be described within a quasiharmonic approximation (Ludwig 1967), in which the anharmonic contributions to the crystal potential are implicitly included by assuming a strain dependence of the phonon frequencies which can be characterized by the... 

In this system of equations the piezoelectric charge constant d indicates the intensity of the piezo effect is the dielectric constant for constant T and is the elastic compliance for constant E eft is the transpose of matrix d. The mentioned parameters are tensors of the first to fourth order. A simplification is possible by using the symmetry properties of tensors. Usually, the Cartesian coordinate system in Fig. 6.12a is used, with axis 3 pointing in the direction of polarization of the piezo substance (see below) [5,6]. 

The s are then the generalized elastic constants. The third and fourth order s are directly proportional to the matrix elements for the three- and four-phonon processes respectively. 

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... 

They calculated the coefficients of an expansion of the Kij(S, T) up to fourth order in the order parameter S and the degree of biaxiality T. In case of weak biaxiality Telastic moduli (/= /, m, n)) are predominant and the deformation state may be described satisfactorily with three bulk and one surface elastic constant, as in the uniaxial case. Recently, these three quasi-uni-axial bulk elastic constants of slightly biaxial nematic copolyesteramide have been determined by De Neve et al. [313] from an optical observation of the Freedericksz transition in different geometries. 

When it is assumed, as is usually done, that the stiffiiess and compliance tensors are additionally symmetric with respect to their diagonals, the total number of independent components is reduced from 36 to 21 (so-called Green elasticity or hyperelasticity, in contrast to the so-called Cauchy elasticity, where this is not the case). Thus in the most general case of well-defined anisotropy (triclinic monociystals) the (6 x 6) stiffness or compliance matrices or, alternatively, the fourth-order stiffness or compliance tensors, have 36 elastic constants or coefficients, respectively, 21 of which can be assumed to be independent, cf. [Pabst Gregorova 2003a]. For reasons of convenience we confine ourselves to the stiffiiess matrices in the sequel. It is understood, however, that completely analogous relations and symmetry considerations are valid in the case of the compliance matrices. 

---