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AG 6: Partial Differential Equations and Applications

Mathematical Investigations of Time Dependent Equations in the Theory of Inelastic Material Behavior

For the description of history dependent material behavior of metals in addition to the partial differential equations derived from the balance laws of moment and mass constitutive equations expressing the dependence of the stress from the deformation history are used. In the case of metals, the formulation of these constitutive equations is based on the use of internal variables, and the constitutive equations consist of a system of nonlinear ordinary differential equations, called evolution equations, for these internal variables. In the SFB-project A1 these systems of partial and ordinary differential equations are investigated. In particular, existence, uniqueness, and properties of solutions are studied.

Sketch of the results obtained in this project and of the present state of the work: It was shown in the project A1 that for the initial-boundary value problems to a special class of constitutive equations existence and uniqueness of solutions can be proved by the theory of evolution equations to monotone operators. Constitutive equations from this class shall be called of `monotone type'. It was also shown that for many constitutive equations, which are not of monotone type, the interior variables can be transformed so that the transformed variables satisfy equations of monotone type, which means that the boundary value problem for the transformed constitutive equations can be solved. By the inverse transformation we then get solutions for the original equations.

By this transformation we can also classify constitutive equations. One of the new classes introduced is the class of constitutive equations transformable to monotone type, for which existence and uniqueness of solutions can be proved. We hope that these researches lead to a general theory for boundary value problems for inelastic behavior of metals.

But there are many open questions concerning this transformation theory. In particular, the theory is not sufficiently general. On one side a number of constitutive equations with practical meaning can be transformed to monotone type, but on the other side most constitutive models from engineering science do not belong to this class. In the planned investigations we are going to study a general approach which should allow most constitutive equations to be transformed into a form, for which existence and uniqueness of solutions can be proved.

Besides this development of a general theory, also special constitutive equations are investigated.