Functional and Numerical Methods in Viscoplasticity

This book presents some mathematical methods used in the study of initial and boundary problems for Bingham fluids and rate-depedent viscoplasticity. Existence and uniqueness results are presented and the behaviour of the solution with respect to the data is also studied. For some problems, the numerical approximation of the solution is also given and numerical results are presented. Elliptic variational inequalities, convex functions, monotone operators, semigroups of operators, and nonlinear evolution equations are used as mathematical instruments. The book presents a sudy of some problems of viscoplasticity using various and different mathematical methods, the mathematical results are interpreted from a mathematical point of view, and the theory is developed to deal with numerical results from possible applications in industry and technology.

1 - Kinematics of continuous media
1.1 - Material and spatial description
1.2 - Deformation and strain tensors
1.3 - The rate of deformation tensor
2 - Balance laws and stress tensors
2.1 - The balance law of mass
2.2 - The balance law of momentum
2.3 - The Cauchy stress tensor
2.4 - The Piola-Kirchhoff stress tensors and the linearized theory
3 - Some experiments and models for solids
3.1 - Standard tests and elastic laws
3.2 - Loading and unloading tests. Plastic laws
3.3 - Long-range tests and viscoplastic laws
1 - Functional spaces of scalar-valued functions
1.1 - Test functions, distributions, and L* spaces
1.2 - Sobolev spaces of integer order
2 - Functional spaces attached to some linear differential operators of first order
2.1 - Linear differential operators of first order
2.2 - Functional spaces associated with the deformation operator
2.3 - A Hilbert space associated with the divergence operator
3 - Functional spaces of vector-valued functions defined on real intervals
3.1 - Weak and strong measurability and L* spaces
3.2 - Absolutely continuous vectorial functions and A** spaces
3.3 - Vectorial distributions and W** spaces
1 - Discussion of a quasistatic elastic-viscoplastic problem
1.1 - Rate-type constitutive equations
1.2 - Statement of the problems
1.3 - An existence and uniqueness result
1.4 - The dependence of the solution upon the input data
2 - Behaviour of the solution in the viscoelastic case
2.1 - Asymptotic stability
2.2 - Periodic solutions
2.3 - An approach to elasticity
2.4 - Long-term behaviour of the solution
3 - An approach to perfect plasticity
3.1 - A convergence result
3.2 - Quasistatic processes in perfect plasticity
3.3 - Some 'pathological' examples
4.1 - Error estimates over a finite time interval
4.2 - Error estimation over an infinite time interval in the viscoelastic case
4.3 - Numerical examples
5 - Quasistatic processes for rate-type viscoplastic materials with internal state variables
5.1 - Rate-type constituve equations with internal state variables
5.2 - Problem statement
5.3 - Existence, uniqueness, and continuous dependence of the solutions
5.4 - A numerical approach
6 - An application to a mining engineering problem
6.1 - Constitutive assumptions and material constants
6.2 - Boundary conditiions and initial data
6.3 - Numerical results
6.4 - Failure
1 - Discussion of a dynamic elastic-viscoplastic problem
1.1 - Problem statement
1.2 - An existence and uniqueness result
1.3 - The dependence of the solution upon the input data
1.4 - Weak solutions
2 - The behaviour of the solution in the viscoelastic case
2.1 - The energy function
2.2 - An energy bound for isolated bodies
2.3 - An approach to linear elasticity
3 - An approach to perfect plasticity
3.1 - A convergence result
3.2 - Dynamic processes in perfect plasticity
4 - Dynamic processes for rate-type elastic-viscoplastic materials with internal state variables
4.1 - Problem statement and constitutive assumptions
4.2 - Existence, uniqueness and continuous dependence of the solution
4.3 - A local existence result
5 - Other functional methods in the study of dynamic problems
5.1 - Monotony methods
5.2 - A fixed point method
6 - Perturbations of homogeneous simple shear and strain localization
6.1 - Problem statement
6.2 - Existence and uniqueness of smooth solutions
6.3 - Perturbations of the homogeneous solutions
6.4 - Numerical results
1 - Boundary value problems for the Bingham fluid with friction
1.1 - The constitutive equations of the Bingham fluid
1.2 - Statement of the problems and friction laws
1.3 - An existence and uniqueness result in the local friction law case
1.4 - An existence result in the non-local friction law case
2 - The blocking property of the solution
2.1 - Problem statements and blocking property
2.2 - The blocking property for abstract variational inequalities
2.3 - The blocking property in the case without friction
2.4 - The blocking property in the case with friction
3 - A numerical approach
3.1 - The penalized problem
3.2 - The discrete and regularized problem
3.3 - A Newton iterative method
3.4 - An application to the wire drawing problem
1 - Elements of linear analysis
1.1 - Normed linear spaces and linear operators
1.2 - Duality and weak topologies
1.3 - Hilbert spaces
2 - Elements of non-linear analysis
2.1 - Convex functions
2.2 - Elliptic variational inequalities
2.3 - Maximal monotone operators in Hilbert spaces
3 - Evolution equations in Banach spaces
3.1 - Ordinary differential equations in Banach spaces
3.2 - Linear evolution equations
3.3 - Lipschitz perturbation of linear evolution equations
3.4 - Non-linear evolution equations in Hilbert spaces
4 - Some numerical methods and complements
4.1 - Numerical methods for elliptic problems
4.2 - Euler's methods for ordinary differential equations in Hilbert spaces
4.3 - A numerical method for non-linear evolution equation
4.4 - Some technical results