Plasticity: Mathematical Theory and Numerical Analysis: v. 9 by Weimin Han;B. Daya Reddy

By Weimin Han;B. Daya Reddy

The speculation of elastoplastic media is now a mature department of sturdy and structural mechanics, having skilled major improvement throughout the latter 1/2 this century. This monograph specializes in theoretical points of the small-strain thought of hardening elastoplasticity. it really is meant to supply a fairly accomplished and unified therapy of the mathematical conception and numerical research, exploiting specifically the nice merits to be received through putting the idea in a convex analytic context. The ebook is split into 3 components. the 1st half offers a close creation to plasticity, within which the mechanics of elastoplastic habit is emphasised. the second one half is taken up with mathematical research of the elastoplasticity challenge. The 3rd half is dedicated to blunders research of assorted semi-discrete and completely discrete approximations for variational formulations of the elastoplasticity. The paintings is meant for a large viewers: this is able to comprise experts in plasticity who desire to comprehend extra concerning the mathematical concept, in addition to people with a history within the mathematical sciences who search a self-contained account of the mechanics and arithmetic of plasticity thought.

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Continuum Mechanics and Linear Elasticity ¨ corresponding to a displaceThe velocity field u˙ and acceleration field u ment field u(x, t) are defined by ∂u(x, t) , ∂t 2 ∂ u(x, t) ¨ (x, t) = . u ∂t2 ˙ u(x, t) = Thus, the linear momentum of the subset Ωt of Ωt at time t is defined by ρu˙ dx, Ω and its angular momentum by x ∧ ρu˙ dx, Ω in which ρ denotes the mass density of the body, that is, the mass per unit reference volume of the body. The body is subjected to a system of forces, which are of two kinds.

There is a wide range of materials, however, that respond in an essentially rateindependent fashion for slow processes, and there is likewise a wide range of practical situations in which such slowly varying processes occur. 16) to be neglected. We will develop a theory of plasticity for quasistatic situations in which the material is assumed to be rate-independent. The appropriate extension to rate-dependent behavior is the theory of viscoplasticity (see, for example, [75, 80, 114] for accounts of viscoplasticity).

This phenomenon is known as elastic unloading. Elastic behavior continues until the new yield stress −σ1 is reached, after which the curve CD would be followed if the stress were to be decreased further. We thus have an initial elastic range, that is, σ ∈ (−σ0 , σ0 ), which includes the unstressed, undeformed state (the origin). We also have subsequent elastic ranges, such as the interval (−σ1 , σ1 ), that are reached only as a result of plastic deformation having taken place. It is the feature of irreversibility that sets an elastoplastic material apart from an elastic one; the nonlinear behavior described before is not a feature peculiar to plastic materials, since nonlinearly elastic behavior is possible, and indeed common.

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