Stiffness tensors

The stiffness tensor Cy is a tensor of fourth rank, with 81 elements. To enable it to be written as a matrix, a reduced notation for the independent elements of stress and strain is used,... 

For isotropic materials there are only two independent constants, which may be taken as Cn and c44 (the relationship between the various isotropic elastic constants is given in Table 6.1 at the end of this section). The isotropic stiffness tensor may be obtained by substituting c12 = 0n — 2c44 in the cubic stiffness matrix. 

The summation sign has been included for emphasis. In the first line of (6.30) there do not appear to be any repeated subscripts, but that is because they have been subsumed in the abbreviated notation. If the stiffness tensor elements are written out in full, as in (6.24), this is immediately apparent. The selection rules in (6.30) correspond directly to the components of stress and strain that are related by each of the three stiffness constants. 

In many cases considerable simplification is possible, because of the constraints imposed on the number of independent elastic constants. For cubic symmetry, for which the elastic stiffness tensor has only three independent constants as given in (6.29), the elements of T, are given in Table 11.1(b), and for hexagonal symmetry the elements are given in Table 11.1(c). If c12 = Cn - 2c44 were to be substituted in Table 11.1(b) the isotropic elements would... 

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 ... 

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

Estimates for the macroscopic drained stiffness tensor Chom(f) as a function of the morphological parameter can be derived from various micromechanical techniques. The micromechanical approach classically refers to the concept of strain concentration tensor, denoted here by A. By definition, in an evolution... 

The macroscopic drained stiffness tensor Chom(f) and Biot s coefficient B( ) can then be expressed as a function of the average of A over the fluid saturated pore space (I=fourth identity tensor) ... 

In noncubic materials, A must be replaced by the 3x3 exchange-stiffness tensor A v, and the energy is X J A v dM/dxy dM/dxv dV. Here the indices ju and v denote the spatial coordinates x, y, and z of the bonds. The energy is anisotropic with respect to the nabla operator = d/d (bond anisotropy) but isotropic with respect to the magnetization M. By contrast, the relativistic anisotropic exchange Xa i Ha(i VMa VMP dV is isotropic with respect to V but anisotropic with respect to M. 

These simplifications reduce the size of the elasticity tensors from [9 x 9] to [6 x 6], with 36 elastic coefficients. The shorthand notation normally used for the elasticity tensors are now introduced, namely, that the subscripts become 1 11 2 22 3 33 4 23, 32 5 31, 13 and 6 12, 21. With this change, the elastic stiffness tensor may be written in matrix form as ... 

Draw, in matrix notation, the elastic-stiffness tensor for a tetragonal monocrystal in the 422 class. 

The four indices of the elastic stiffness constants, c,y /, result in the possibility of as many as 3 = 81 elements in the stiffness tensor. Because the stress and strain... 

Taking into account a progressive plasticization of the liner involves to arbitrarily considering ui layer through the metal thickness. Consequently, the structure is assumed to be made of (ni + Us) layers. Moreover, to be rigorous in the present analysis the same relations should have been expressed under an increment form. The symmetric stiffness tensor of the k layer takes the following expression in the cylindrical coordinate system ... 

To find the stress tensor in the same system we place Equation (87) into Equation (73a) and one obtains an expression similar to Equation (84b). To obtain the macroscopic stress we must integrate this expression over the Euler space. The integral acts only on the single-crystal stiffness tensor elements Equation (74a) and can be calculated analytically. The macroscopic stress is ... 

In Equation (91), Sy is the compliance tensor obtained by inverting the stiffness tensor Ch ... 

Checking the effect of fictitious Joints on the equivalent stiffness tensor... 

We have evaluated the effect of considering the fictitious joints as real joints on the equivalent stiffness tensor T,ju. We can notice that the terms of the equivalent stiffness tensor terms are smaller if we consider 3DEC fictitious joints are real. For the WP3 case (where high joint stiffness values are considered) the differences remain small. It increased up to 30 % for Kn = 4.34 lO Pa/m. 

Table 3 Equivalent stiffness tensor for formation 1 at 2 m scale...

The equivalent stiffness tensor Tijn has been computed up to 5 m scale. No clear conclusion can be given for the mechanical REV. We have notice... 

The effective elastic stiffness tensor of damaged material C(d) is given by ... 

Here we consider only high-symmetry orientations, such as [001] and [111]. In this case, the forbidden orientation implies that the surface-stiffness tensor [24, 8],... 

The quantities are the components of a fourth-rank tensor called the stiffness tensor and the quantities Syi i are the components of a fourth-rank tensor called the compliance tensor (note the confusing relationship between the names and the symbols). Specification of all values of either c,y / or Sy i allows all values of the other set to be calculated. 

Using properties of the reinforcement, assign stiffness tensors to the inclusions (microhomogenisation may he performed on this step)... 

Table 2.3 Form of the compliance and stiffness tensors for the crystal classed... 

---