Matrix stiffness tensor

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. 

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. 

This stiffness tensor has monoclinic symmetry rather than pseudo-hexagonal, since the underlying pseudo-hexagonal crystal is tilted away from the c-axis. The uncertainty in each of the tensile stiffnesses is 0.03 GPa the uncertainty in each of the shear stiffnesses is 0.06 GPa, with the exception of C44, where the uncertainty is closer to 0.1 GPa. Within the accuracy of our sampling, C44 is zero within errors. As a consequence the system is at best only marginally stable, since the determinant of the stiffness matrix in (14.13) is close to zero. 

Since Ci and 8 are symmetric tensors, each of them has 6 independent components, the fourth order stiffness tensor Cyia contains at most 36 independent constants, such that it can be displayed as a 6 x 6 matrix of components using contracted notation, Cto), where m,n — 1,2,3,4,5,6. There is a unique correspondence between of and Cijki- The index m is related to ij, and n is related to Id, as shown in Table B.l. For instance, Cu22 = C12, C1323 = C54- Since C = C the number of independent constants is generally 21. For orthotropic materials, the the number of independent constants further reduces to 9. When the fourth order tensor is transformed to the principal axes, all Qj — 0, except for Cn, C22, C33, C12. 13. 23. 44, 55, and... 

The distinct feature of elastic-plastic finite element computations is the presence of two iteration levels. In a standard displacement based finite element implementation, constitutive driver at each integration (Gauss) point iterates in stress and internal variable space, computes the updated stress state, constitutive stiffness tensor and delivers them to the finite element functions. Finite element functions then use the updated stresses and stiffness tensors to integrate new (internal) nodal forces and element stiffness matrix. Then, on global level, nonlinear equations are iterated on until equilibrium between internal and external forces is satisfied within some tolerance. 

Curly brackets represent an average over all possible orientations of term ( ). In Eq. (1), the terms Cj and C2 are the elastic stiffness tensors of the matrix material and the particle, respectively, is the particle volume fraction and A2 is the strain concentration tensor defined as ... 

Because the particles are assumed randomly oriented in the isotropic matrix material, the effective elastic stiffness tensor is a fourth-rank isotropic tensor, which may be expressed in terms of the effective bulk and shear moduli, k and n, as follows ... 

Where C represents the stiffness tensor, superscript m and f indicate a quantity associated with matrix and fibers respectively, Vf is Poisson ratio. 

Equation (5.7) related the stress tensor to the strains throngh the modulus matrix, [Q, also known as the reduced stiffness matrix. Oftentimes, it is more convenient to relate the strains to the stresses through the inverse of the rednced stiffness matrix, [2] called the compliance matrix, [5]. Recall from Section 5.3.1.3 that we introduced a quantity called the compliance, which was proportional to the inverse of the modulus. The matrix is the more generalized version of that compliance. The following relationship then holds between the strain and stress tensors ... 

Once the inclusion assembly has been constmcted, the homogenised stiffness matrix of the composite is calculated as follows. First, calculate Eshelby tensors S,- for the inclusions [97,98] in local coordinates CS,. Transform the result in the global coordinate system GCS. Then calculate the strain concentration tensors for all the inclusions ... 

The matrix a , defined in Appendix B, is a contravariant metric tensor in the infinitely stiff coordinates. Each det (a ) term can be evaluated directly using a recursion relation [102,123] or via the Fixman relation, Eq. (82). In a quantum mechanics-based statistical mechanics model, the effect of infinite stiffness on each partition function is not ... 

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... 

Using the above concepts and equations, the average composite stiffness can be obtained from the strain concentration tensor A and the filler and matrix properties ... 

Due to the symmetry of T and E, the number of components of the stiffness and compliance tensors is reduced from a total of 81 to 36 independent ones. Thus, it is possible to represent these fourth-order tensors alternatively in the form of (6 x 6) matrices (which, of course, do not have the transformation properties of a tensor), and to express Hooke s law and inverse Hooke s law in direct matrix notation (engineering notation) as... 

Equation (9) ean express the average composite stiffness in terms of the strain-coneentration tensor A and the fiber (filler) and matrix properties (Tucker and Liang, 1999) ... 

In order to prediet the stiffness, the strain concentration tensor A is needed. Mori and Tanaka (1973) eonsidered a composite model where the heterogeneities are diluted in the matrix. This model takes into account an interaction between the inclusion and the surroimdings (inelusions and polymer matrix) in their original model, the inclusions were considered to have the same shape and orientation. Benveniste (1987) made a reconsideration and reformulation of the Mori-Tanaka s theory in its application to the computation of the effective properties of composite. In this model the inclusions can be considered either aligned or randomly oriented. This formulation is more suitable for the morphology of clays dispersed in a polymer. The expression of the strain concentration tensor A is written as follows ... 

---