准静态弹性损伤演化过程的识别（Ⅰ）--分析
Identifying Processes Governing Damage Evolution in Quasi-Static Elasticity Part I -- Analysis
摘要：
基于Kachanov（1958）和Rabotnov（1968）的思想，提出了一个考虑损伤演化的准静态弹性模型。我们分析了由此产生的强非线性系统的微分方程的适定性。其特点是位移通过Nemytskii算子或叠加算子与损伤发展方程相联系。在这项工作中的新颖之处在于，我们提出了一个参数识别设置中的逆问题，在该逆问题中，我们能够识别该Nemytskii算子的形状。从材料建模的角度来看，材料损伤和位移的关系是由该算子的形状建模。我们建立的Fréchet导数的前向运营商以及伴随的衍生物和通过系统的线性微分方程的特点。证明了反问题的不适定性，并给出了经典非线性Landweber方法收敛的充分条件。
英文摘要：
We present a quasi-static elasticity model that accounts for damage evolution based on the ideas of Kachanov 1958 and and Rabotnov 1968. We analyze the resulting strongly nonlinear system of differential equations in view of well-posedness. The specific feature is that the displacements are connected to the damage evolution equation via a Nemytskii- or superposition- operator. The novelty in this work is that we present an inverse problem in a parameter identification setting, in which we are able to identify the shape of this Nemytskii-operator. From the material modelling point of view, the relation of material damage and displacements is modelled by the shape of this operator. We establish the Fréchet-derivative of the forward operator as well as the adjoint of the derivative and characterize both via systems of linear differential equations. We prove ill-posedness of the inverse problem and provide a sufficient condition for the classical nonlinear Landweber method to converge.
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