By Weimin Han
This ebook makes a speciality of the theoretical elements of small pressure concept of elastoplasticity with hardening assumptions. It presents a finished and unified therapy of the mathematical conception and numerical research. it truly is divided into 3 elements, with the 1st half supplying a close creation to plasticity, the second one half overlaying the mathematical research of the pliability challenge, and the 3rd half dedicated to blunders research of assorted semi-discrete and entirely discrete approximations for variational formulations of the elastoplasticity. This revised and accelerated version comprises fabric on single-crystal and strain-gradient plasticity. moreover, the complete publication has been revised to make it extra obtainable to readers who're actively all for computations yet much less so in numerical research. reports of previous version: “The authors have written a good publication that are steered for experts in plasticity who desire to be aware of extra in regards to the mathematical idea, in addition to people with a heritage within the mathematical sciences who search a self-contained account of the mechanics and arithmetic of plasticity theory.” (ZAMM, 2002) “In precis, the booklet represents a powerful accomplished review of the mathematical method of the speculation and numerics of plasticity. Scientists in addition to teachers and graduate scholars will locate the publication very beneficial as a reference for examine or for getting ready classes during this field.” (Technische Mechanik) "The booklet is professionally written and may be an invaluable connection with researchers and scholars drawn to mathematical and numerical difficulties of plasticity. It represents an incredible contribution within the zone of continuum mechanics and numerical analysis." (Math Reviews)
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Extra resources for Plasticity: Mathematical Theory and Numerical Analysis
We deﬁne an elastic material to be one for which the constitutive equations take the form ψ = ψ( ), σ = σ( ). 48) That is, the free energy and stress depend only on the current strain; there is no dependence on the history of behavior, for example. It should be remarked that the more general point of departure is to take the free energy and stress to be functions of the displacement gradient ∇u rather than the strain. That these variables in fact depend on ∇u through its symmetric part, the strain , is a consequence of the principle of material frame indiﬀerence (see ).
This slope will continue to decrease, and eventually a variety of phenomena may take place. For example, the material may rupture, at which point the experiment will necessarily be regarded as concluded. 1(c)). 1(d)). All of these features, and others yet, are important, the importance of any particular feature depending on the application in question and on the range of stress that is expected to be experienced. 1(b), in which the curve continues to rise, albeit at a slope less than that when σ < σ0 , is known as hardening behavior.
We are using here the result from calculus that dy = Jdx, where dx and dy denote the volume elements in Ω and Ωt . A suﬃcient condition for the motion y to be invertible is that there exist a constant c(Ω) > 0, depending only on Ω, such that sup |∇y − I| < c(Ω). Ω This result, as well as others on the invertibility of the motion, may be found in . Instead of adopting the function y as the primary unknown variable, it is more convenient to introduce the displacement vector u by u(x, t) = y(x, t) − x and to replace the motion by the displacement as the primary unknown.