By Apel N.

The current paintings bargains with in basic terms macroscopic descriptions of anisotropic fabric behaviour. Key elements are new advancements within the conception and numerics of anisotropicplasticity. After a quick dialogue of the category of solids via symmetry changes a survey approximately illustration idea of isotropic tensor services and tensor polynomials is given. subsequent replacement macroscopic methods to finite plasticity are mentioned. whilst contemplating a multiplicative decomposition of the deformation gradient into an elastic half and a plastic half, a 9 dimensional °ow rule is got that permits the modeling of plastic rotation. an alternate strategy bases at the creation of a metric-like inner variable, the so-called plastic metric, that debts for the plastic deformation of the fabric. during this context, a brand new type of constitutive types is bought for the alternative of logarithmic traces and an additive decomposition of the whole pressure degree into elastic and plastic components. The popularity of this type of types is because of their modular constitution in addition to the a+nity of the constitutive version and the algorithms contained in the logarithmic pressure area to versions from geometric linear conception. at the numerical facet, implicit and specific integration algorithms and tension replace algorithms for anisotropic plasticity are constructed. Their numerical e+ciency crucially bases on their cautious development. distinctive concentration is wear algorithms which are compatible for variational formulations. as a result of their (incremental) power estate, the corresponding algorithms may be formulated by way of symmetric amounts. a discounted garage eRort and not more required solver skill are key merits in comparison to their normal opposite numbers.

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**Example text**

In the second case, the relative orientation of v and A is unspecified. 25) can be dropped and the coset is completely described by the fundamental invariants only. 2. Smith’s Approach A systematic treatment of deriving representations for isotropic scalar-, first-order and second-order tensor-valued functions of first-order and symmetric and skew-symmetric second-order tensors is presented in Smith [130]. The basic idea goes back to the work of Rivlin & Ericksen [108] who derived a representation for two symmetric second-order tensors.

Here we consider elastic material behaviour and a functional dependence on the deformation gradient F . The principle of material objectivity demands that the free energy function ψ = ρ0 Ψ is independent of the choice of reference frame. For two arbitrary Cartesian frames linked with orthogonal rotation tensor Q, it reads ψ(F ) = ψ(QF ) . 64) Thus the free energy function has to be invariant with respect to superimposed rigid body motions. A common way to satisfy this restriction in a Lagrangian setting is to assume a functional dependence on the right Cauchy-Green tensor C := F T gF .

3). Consequently 5-fold axes are not possible for a space lattice. PSfrag replacements A(1) A(2) A(3) A(4) A(6) Figure 12: Stereographic projections of points on a sphere visualizing 1-, 2-, 3-, 4- and 6-fold symmetry. The outlined symbols at the centers denote the different rotation axis. 2. Rotation-Inversions Let M ∈ A(n) denote a point which is left unchanged when applying a symmetry operation. A rotation-inversion or rotoinversion is obtained by the composition of a rotation and a central inversion on M .