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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >A doubly nonlinear problem associated with a mathematical model for piezoelectric material behavior
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A doubly nonlinear problem associated with a mathematical model for piezoelectric material behavior

机译：与压电材料行为数学模型相关的双非线性问题

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We consider a mathematical model, which describes piezoelectric material behavior. This model is similar to models of plasticity theory. However, piezoelectric models describe coupled mechanical and electrical material behavior. Therefore they contain additional nonlinearities in the piezoelectric tensor and in the enthalpy function, which is non quadratic. These nonlinearities cause difficulties in the proof of existence theorems. Under the assumption that the piezoelectric tensor is constant (i.e. independent of P), we show how the system of model equations can be reduced to a doubly nonlinear evolution equation of the form z _t ∈ G(-Mz-Φ (z) + f), which contains a composition of two monotone operators. The monotone mapping G is a subdifferential of the indicator function of some convex set while the second monotone mapping Φ is the Nemyckii operator of a monotone function. We prove existence of strong solutions, if Φ is replaced by a regularization. If in addition Φ is Lipschitz continuous we can show that the solution is unique.
机译：我们考虑一个数学模型，该模型描述了压电材料的行为。该模型类似于可塑性理论的模型。但是，压电模型描述了机械和电气材料的耦合行为。因此，它们在压电张量和焓函数中包含其他非线性，这是非二次的。这些非线性在存在性定理的证明中造成困难。在压电张量恒定的前提下（即与P无关），我们展示了如何将模型方程组简化为z _t∈G（-Mz-Φ（z）+ f ），其中包含两个单调运算符。单调映射G是某个凸集的指示符函数的次微分，而第二个单调映射Φ是单调函数的Nemyckii算符。如果用正则化替换Φ，我们证明存在强解。如果Φ是Lipschitz连续的，我们可以证明该解是唯一的。

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