Approaches to the Description of Anisotropic Material by Apel N.

By Apel N.

The current paintings bargains with merely macroscopic descriptions of anisotropic fabric behaviour. Key features are new advancements within the idea and numerics of anisotropicplasticity. After a quick dialogue of the class of solids via symmetry differences a survey approximately illustration conception of isotropic tensor services and tensor polynomials is given. subsequent replacement macroscopic ways 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 acquired that permits the modeling of plastic rotation. another 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 classification of constitutive types is acquired for the alternative of logarithmic traces and an additive decomposition of the full pressure degree into elastic and plastic elements. The reputation of this category of versions is because of their modular constitution in addition to the a+nity of the constitutive version and the algorithms contained in the logarithmic pressure house to versions from geometric linear conception. at the numerical aspect, implicit and specific integration algorithms and pressure replace algorithms for anisotropic plasticity are constructed. Their numerical e+ciency crucially bases on their cautious development. particular concentration is wear algorithms which are appropriate for variational formulations. as a result of their (incremental) capability estate, the corresponding algorithms might be formulated by way of symmetric amounts. a discounted garage eRort and no more required solver skill are key benefits in comparison to their usual opposite numbers.

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Extra info for Approaches to the Description of Anisotropic Material Behaviour at Finite Elastic and Plastic Deformations. Theory and Numerics

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5) The number of elements in G is termed order of the group. Thereupon one speaks of a finite group, if its order is finite, otherwise of an infinite group. Both types play an important role in continuum mechanics. 3. The 14 Bravais Lattices and 7 Crystal Systems Different lattices can be distinguished due to their inherent symmetry. We will see that there exist seven different crystal systems, each having its own symmetry group. Within these systems, several grid metrics are possible. We differentiate 14 Bravais lattices belonging to the seven crystal systems according to the shape of their unit-cells.

F (x, Ξ(x)) = f (Q x, Ξ(Q x)) ∀Q ∈ G . 13) Representations of Anisotropic Tensor Functions 39 Using this isotropic extension method, the problem of finding representations for anisotropic functions is shifted to the problem of finding representations for isotropic functions. 3. 10). 1: In literature this extension method with constant structural tensors is referred to as the isotropicization theorem. It can easily be extended to higher-order structural tensors. Zheng [162] specifies a single structural tensor characterizing the symmetry group G for each crystal class.

50)   ρ0 r dV  −Q · N dA + ρr dv = −q · n da + Q := ∂Sp Sp ∂Bp Bp 18 Fundamentals of Continuum Mechanics respectively. r is the heat supply per unit mass and unit time. With these definitions, the balance of total energy reads E˙ = P + Q . 51) The corresponding local equations related to the unit-volume of the actual and reference configuration run as follows ¯ + ρr ρe˙ = div[x˙ · σ − q] + ρx˙ · γ ∀x∈S ¯ + ρ0 r ∀ X ∈ B . 52) The total energy can be additively split up into the kinetic energy and an internal energy, which is discussed in the next two subsections.

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