Computational Inelasticity by IBM Redbooks, Saida Davies

By IBM Redbooks, Saida Davies

This ebook 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 technique at present utilized in inelastic calculations. one of the a number of subject matters lined are small deformation plasticity and viscoplasticity, convex optimization idea, integration algorithms for the constitutive equation of plasticity and viscoplasticity, the variational surroundings of boundary price difficulties and discretization by way of finite aspect equipment. additionally addressed are the generalization of the idea to non-smooth yield floor, mathematical numerical research problems with common go back mapping algorithms, the generalization to finite-strain inelasticity concept, aim integration algorithms for expense constitutive equations, the idea of hyperelastic-based plasticity types and small and massive deformation viscoelasticity. Computational Inelasticity may be of significant curiosity to researchers and graduate scholars in numerous branches of engineering, specially civil, aeronautical and mechanical, and utilized arithmetic.

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3 by applying an implicit backward Euler difference scheme. 34) αn + γ , αn+1 ⎪ ⎪ ⎪ ⎪ qn + γ H sign(ξn+1 ), qn+1 ⎪ ⎪ ⎪ ξ − (σ + Kα ) 0, ⎭ : f n+1 n+1 Y n+1 where σn+1 − qn+1 . 35) relies on exploiting an expression for ξn+1 obtained as follows. 35), ξn+1 trial (σn+1 − qn ) − trial : Now we use the fact that ξn+1 obtain ξn+1 + γ (E + H )sign(ξn+1 ). 36) to γ (E + H ) sign(ξn+1 ) trial trial sign(ξn+1 ξn+1 ). 37) must be positive. 38) along with the condition ξn+1 + γ [E + H ] trial ξn+1 . 4. 34)3 : fn+1 trial ξn+1 − (E + H ) γ − [σY + Kαn+1 ] trial ξn+1 − (E + H ) γ − σY + Kαn − K αn+1 − αn trial − fn+1 γ [E + (K + H )] Solving this algebraic equation for 0.

9) γ ≥ 0, ⎪ ⎪ ⎭ 0. 8b) is regarded merely as the definition for εn+1 . Further, we note that, by applying the implicit backward Euler algorithm, we have transformed the initial constrained problem of evolution into p a discrete constrained algebraic problem for the variable {εn+1 , αn+1 }. 9) the optimality conditions of a discrete constrained optimization problem. 9). 2 Return-Mapping Algorithms. 9) is unique. 3. 9) is the introduction of the following auxiliary problem. 1 The trial elastic state.

3 Discrete Variational Formulation. 4 possesses a more fundamental interpretation which is the manifestation of a basic variational structure underlying classical rate-independent plasticity. We show below that this algorithm is interpreted as the Kuhn–Tucker optimality conditions of a convex-optimization problem which is, in fact, the discrete counterpart of a classical postulate known as the principle of maximum plastic dissipation (or entropy production). A discussion of the role played by this principle is given in Chapter 2.

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