Functional and Numerical Methods in Viscoplasticity

Hardcover | April 30, 1999

byIoau R. Ionescu, Mircea Sofonea

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This book studies problems of viscoplasticity using various and different mathematical methods. The mathematical results obtained are carefully interpreted from a mechanical point of view. The theory is developed to deal with numerical results from practical problems in industry andtechnology.

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This book studies problems of viscoplasticity using various and different mathematical methods. The mathematical results obtained are carefully interpreted from a mechanical point of view. The theory is developed to deal with numerical results from practical problems in industry andtechnology.

Ioau R. Ionescu, Senior Research Fellow, Laboratoire de Mecaniques et Technologie, ENS DE Cachan/CNRS/Universite de Paris. Mircea Sofonea, Associated Professor, Blaise Pascal University, Clermont Ferrand, Aubiere.
Format:HardcoverDimensions:282 pages, 9.21 × 6.14 × 0.01 inPublished:April 30, 1999Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0198535902

ISBN - 13:9780198535904

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Table of Contents

Chapter 1. Preliminaries on Mechanics of Continuous Media1. Kinematics of continuous media1.1. Material and spatial description1.2. Deformation and strain tensors1.3. The rate of deformation tensor2. Balance laws and stress tensors2.1. The balance law of mass2.2. The balance law of momentum2.3. The Cauchy stress tensor2.4. The Piola-Kirchhoff stress tensors and the linearized theory3. Some experiments and models for solids3.1. Standard tests and elastic laws3.2. Loading and unloading tests. Plastic laws3.3. Long-range tests and viscoplastic lawsBibliographical notesChapter 2. Functional Spaces in Viscoplasticity1. Functional spaces of scalar-valued functions1.1. Test functions, distributions, and L* spaces1.2. Sobolev spaces of integer order2. Functional spaces attached to some linear differential operators of first order2.1. Linear differential operators of first order2.2. Functional spaces associated with the deformation operator2.3. A Hilbert space associated with the divergence operator3. Functional spaces of vector-valued functions defined on real intervals3.1. Weak and strong measurability and L* spaces3.2. Absolutely continuous vectorial functions and A** spaces3.3. Vectorial distributions and W** spacesBibliographical notesChapter 3. Quasistatic Processes for Rate-Type Viscoplastic Materials1. Discussion of a quasistatic elastic-viscoplastic problem1.1. Rate-type constitutive equations1.2. Statement of the problems1.3. An existence and uniqueness result1.4. The dependence of the solution upon the input data2. Behaviour of the solution in the viscoelastic case2.1. Asymptotic stability2.2. Periodic solutions2.3. An approach to elasticity2.4. Long-term behaviour of the solution3. An approach to perfect plasticity3.1. A convergence result3.2. Quasistatic processes in perfect plasticity3.3. Some 'pathological' examples44.1. Error estimates over a finite time interval4.2. Error estimation over an infinite time interval in the viscoelastic case4.3. Numerical examples5. Quasistatic processes for rate-type viscoplastic materials with internal state variables5.1. Rate-type constituve equations with internal state variables5.2. Problem statement5.3. Existence, uniqueness, and continuous dependence of the solutions5.4. A numerical approach6. An application to a mining engineering problem6.1. Constitutive assumptions and material constants6.2. Boundary conditiions and initial data6.3. Numerical results6.4. FailureBibliographical notesChapter 4. Dynamic Processes for Rate-Type Elastic-Viscoplastic Materials1. Discussion of a dynamic elastic-viscoplastic problem1.1. Problem statement1.2. An existence and uniqueness result1.3. The dependence of the solution upon the input data1.4. Weak solutions2. The behaviour of the solution in the viscoelastic case2.1. The energy function2.2. An energy bound for isolated bodies2.3. An approach to linear elasticity3. An approach to perfect plasticity3.1. A convergence result3.2. Dynamic processes in perfect plasticity4. Dynamic processes for rate-type elastic-viscoplastic materials with internal state variables4.1. Problem statement and constitutive assumptions4.2. Existence, uniqueness and continuous dependence of the solution4.3. A local existence result5. Other functional methods in the study of dynamic problems5.1. Monotony methods5.2. A fixed point method6. Perturbations of homogeneous simple shear and strain localization6.1. Problem statement6.2. Existence and uniqueness of smooth solutions6.3. Perturbations of the homogeneous solutions6.4. Numerical resultsBibliographical notesChapter 5. The Flow of the Bingham Fluid with Friction1. Boundary value problems for the Bingham fluid with friction1.1. The constitutive equations of the Bingham fluid1.2. Statement of the problems and friction laws1.3. An existence and uniqueness result in the local friction law case1.4. An existence result in the non-local friction law case2. The blocking property of the solution2.1. Problem statements and blocking property2.2. The blocking property for abstract variational inequalities2.3. The blocking property in the case without friction2.4. The blocking property in the case with friction3. A numerical approach3.1. The penalized problem3.2. The discrete and regularized problem3.3. A Newton iterative method3.4. An application to the wire drawing problemBibliographical notesAppendix1. Elements of linear analysis1.1. Normed linear spaces and linear operators1.2. Duality and weak topologies1.3. Hilbert spaces2. Elements of non-linear analysis2.1. Convex functions2.2. Elliptic variational inequalities2.3. Maximal monotone operators in Hilbert spaces3. Evolution equations in Banach spaces3.1. Ordinary differential equations in Banach spaces3.2. Linear evolution equations3.3. Lipschitz perturbation of linear evolution equations3.4. Non-linear evolution equations in Hilbert spaces4. Some numerical methods and complements4.1. Numerical methods for elliptic problems4.2. Euler's methods for ordinary differential equations in Hilbert spaces4.3. A numerical method for non-linear evolution equation4.4. Some technical resultsBibliographical notesReferences

Editorial Reviews

`a very successful piece of work ... It may be of interest to a wide spectrum of readers, starting from students interested in applied mathematics and engineering, and researchers in various applied mathematics fields. It is recommended to all libraries of universities, possessing anengineering and/or mathematical department.'N. Cristescu, Zbl. Math. 787