On the identifiability of the stored energy function of hyperelastic materials from sensor data at the boundary
On the identifiability of the stored energy function of hyperelastic materials from sensor data at the boundary
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DOI：
10.1088/0266-5611/30/10/105002
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发表时间：
2014-09
期刊：
影响因子：
2.1
通讯作者：
T. Schuster;A. Wöstehoff
中科院分区：
文献类型：
--
作者：
T. Schuster;A. Wöstehoff
本文讨论了由部分柯西数据重建一类超弹性材料的储能函数的反问题。它是由所谓的结构健康监测系统推动的，其想法是从施加在物体边界的传感器上记录的数据中揭示弹性各向异性结构中的缺陷。损伤会影响储能函数的空间变化，因此它的重建揭示了结构的缺陷。超弹性材料的动力学行为用柯西ʼS运动方程来描述，这是一个二阶非线性偏微分方程组，从而得到了这个偏微分方程组的辨识问题。我们证明了该辨识问题唯一可解的条件，并且解连续地依赖于测量数据和结构的初始条件。一个重要的假设是储能函数是有限多个给定函数的二次组合。此外，我们证明了在线性情况下，如果弹性张量的项在连续函数空间的有限维子空间中，并且满足某些谱条件，则存在这样的弹性张量的圆锥分解。在均匀各向同性介质的情况下，这种分解是明确已知的。因此，我们得到的结果是，两个传感器足以识别这种介质的两个独立的材料参数--LAMé系数。
This article addresses the inverse problem of reconstructing the stored energy function of a certain class of hyperelastic materials from partial Cauchy data. It is motivated by so-called structural health monitoring systems, whose idea is to disclose defects in elastic, anisotropic structures from data which are recorded at sensors that are applied at the boundary of the object. Damage affects the spatially varying stored energy function and thus its reconstruction reveals defects of the structure. The dynamic behavior of hyperelastic materials is described by Cauchyʼs equation of motion, a second order, non-linear system of partial differential equations, and thus we get an identification problem for this PDE system. We prove conditions under which this identification problem is uniquely solvable and that the solution continuously depends on the measured data as well as on the initial conditions of the structure. An important assumption is that the stored energy function is a conic combination of finitely many given functions. Moreover, we show that in the linear case such a conic decomposition of the elasticity tensor exists if its entries are in a finite dimensional subspace of the space of continuous functions and some spectral conditions are satisfied. In the case of a homogeneous, isotropic medium this decomposition is explicitly known. As a consequence, we get the result that two sensors are sufficient to identify the two independent material parameters, the Lamé coefficients, of such a medium.