By Apel N.
The current paintings offers with basically macroscopic descriptions of anisotropic fabric behaviour. Key features are new advancements within the thought and numerics of anisotropicplasticity. After a quick dialogue of the category of solids through symmetry ameliorations a survey approximately illustration concept of isotropic tensor services and tensor polynomials is given. subsequent substitute macroscopic ways to finite plasticity are mentioned. while contemplating a multiplicative decomposition of the deformation gradient into an elastic half and a plastic half, a 9 dimensional °ow rule is got that enables the modeling of plastic rotation. an alternate process bases at the advent of a metric-like inner variable, the so-called plastic metric, that money owed for the plastic deformation of the fabric. during this context, a brand new classification of constitutive versions is got for the alternative of logarithmic traces and an additive decomposition of the complete 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 thought. at the numerical facet, implicit and particular integration algorithms and pressure replace algorithms for anisotropic plasticity are built. Their numerical e+ciency crucially bases on their cautious building. distinct concentration is wear algorithms which are compatible for variational formulations. because of their (incremental) capability estate, the corresponding algorithms may be formulated by way of symmetric amounts. a discounted garage eRort and no more required solver ability are key merits in comparison to their usual opposite numbers.
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Additional info for Approaches to the Description of Anisotropic Material Behaviour at Finite Elastic and Plastic Deformations. Theory and Numerics
Schoenflies Symbols. Another notation for the crystals and quasi-crystal classes goes back to Schoenflies. This notion is commonly used in chemistry. A summary is given in table 6. 5. Icosahedral, Cylindrical and Spherical Symmetry Classes If we consider quasi crystals or engineering materials like composites or biological materials like soft tissues, further symmetry operations than those so far considered will have to be taken into account. Here discrete rotations around an axis A(n) with α = 2π/n, n ∈ N are possible as well as their corresponding rotoinversions.
53) Balance of kinetic energy is equivalent to the equilibrium condition. 45)2 with the velocity x˙ and integrating over the volume, one obtains K˙ = P − S . 55) Bp Sp with respect to unit-volume of the actual and reference configuration, respectively. 2. Balance of Internal Energy (First Law of Thermodynamics). The total energy can be additively decomposed into the kinetic part and a remaining part U := E − K, denoted as internal energy. The latter is related to the specific internal energy density per unit mass u according to ρ0 u dV .
Aes (x), W e1 (x), . . , W et (x)} . e. Ξ(Q x) = Q (Ξ(x)) ∀Q ∈ G . 11) The key observation is as follows. Any G-invariant tensor function of an argument x can be represented in terms of an isotropic tensor function with an extended set of arguments. 6) poses the following restriction on the function f (x, Ξ(x)) = f (Q x, Q (Ξ(x)) ∀Q ∈ O(3) . e. 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.
Approaches to the Description of Anisotropic Material Behaviour at Finite Elastic and Plastic Deformations. Theory and Numerics by Apel N.