By IBM Redbooks, Saida Davies

This e-book describes the theoretical foundations of inelasticity, its numerical formula and implementation. The subject material defined herein constitutes a consultant pattern of state-of-the- paintings method at present utilized in inelastic calculations. one of the a number of themes lined are small deformation plasticity and viscoplasticity, convex optimization concept, integration algorithms for the constitutive equation of plasticity and viscoplasticity, the variational environment of boundary worth difficulties and discretization by way of finite aspect tools. additionally addressed are the generalization of the idea to non-smooth yield floor, mathematical numerical research problems with normal go back mapping algorithms, the generalization to finite-strain inelasticity thought, goal integration algorithms for fee constitutive equations, the idea of hyperelastic-based plasticity types and small and massive deformation viscoelasticity. Computational Inelasticity can be of significant curiosity to researchers and graduate scholars in a variety of branches of engineering, specially civil, aeronautical and mechanical, and utilized arithmetic.

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4) ⇒ midpoint rule ϑ ⎪ 2 ⎪ ⎪ ⎪ ϑ 1 ⇒ backward (implicit) Euler. , Gear [1971] or Hairer, Norsett and Wanner [1987], for a discussion of this class of algorithms and, in particular, for relevant notions of consistency, stability, and accuracy. A complete numerical analysis of the class of methods discussed below for the general three-dimensional problem 34 1. One-Dimensional Plasticity and Viscoplasticity is deferred to Chapter 6. Here, we merely recall that second-order accuracy is 1 whereas unconditional (linearized) stability requires obtained only for ϑ 2 1 ϑ ≥ 2.

3. One-Dimensional, Rate-Independent Plasticity. Combined Kinematic and Isotropic Hardening. i. Elastic stress-strain relationship E ε − εp . σ ii. Flow rule ε˙ p iii. iv. γ sign(σ − q). Isotropic and kinematic hardening laws q˙ γ H sign(σ − q), α˙ γ. Yield condition and closure of the elastic range f σ, q, α : Eσ v. { σ, q, α ∈ R × R+ × R | f σ, q, α ≤ 0}. Kuhn–Tucker complementarity conditions γ ≥ 0, vi. σ − q − [σY + Kα] ≤ 0, f σ, q, α ≤ 0, Consistency condition γ f˙ σ, q, α 0 (if γf σ, q, α f (σ, q, α) 0.

40) γ yields the following result: trial fn+1 > 0. 34). 5. 5. Return-Mapping Algorithm for One-Dimensional, Rate-Independent Plasticity. Combined Isotropic/Kinematic Hardening. p 1. Database at x ∈ B : εn , αn , qn . εn + 2. Given strain ﬁeld at x ∈ B : εn+1 εn . 3. Compute elastic trial stress and test for plastic loading trial : σn+1 E εn+1 − εnp trial : ξn+1 trial σn+1 − qn trial fn+1 : trial ξn+1 − σY + Kαn trial ≤ 0 THEN IF fn+1 Elastic step: set (•)n+1 (•)trial n+1 & EXIT ELSE Plastic step: Proceed to step 4.