Identifying relaxation kernels of linearly viscoelastic bodies (Q2785322)

Let us consider a sample of compressible material which occupies a bounded domain \(\Omega\subset\mathbb{R}^3\) for any \(t\in[0,T]\), \(T>0\) being given. We suppose further that the material is homogeneous. We indicate by \({\mathbf u}(x,t):= (u^1(x,t),u^2(x,t),u^3(x,t))\) its displacement vector with respect to the unstressed state at point \(x\in\Omega\), at time \(t\). Also, we denote by \(\sigma=[\sigma_{ij}]\) the stress tensor and by \(\varepsilon({\mathbf u})= D^s{\mathbf u}\) the linearized strain tensor, where \(D^s{\mathbf u}\) stands for the symmetric part of the Jacobian matrix \(D{\mathbf u}\). Assume that the mechanical behavior of the body is described by the linear stress-strain law NEWLINE\[NEWLINE\sigma(t)= A\otimes\varepsilon({\mathbf u}_t)+ B\otimes\varepsilon({\mathbf u})+ \int^t_0 G(s)\otimes\varepsilon({\mathbf u}_t(t- s))ds,\tag{1}NEWLINE\]NEWLINE where \(A=[A_{ijkh}]\), \(B=[B_{ijkh}]\) are fourth-order tensors with constant components and \(G=[G_{ijkh}]\) is a time dependent relaxation tensor termed relaxation kernel. Here \(\otimes\) stands for the usual tensor product and the standard symmetry properties \(A_{ijkh}= A_{khij}= A_{jikh}= A_{ijhk}\), \(B_{ijkh}= B_{jikh}= B_{khij}= B_{ijhk}\), and \(G_{ijkh}= G_{khij}= G_{jikh}= G_{ijhk}\) hold. Suppose now that \(A\) and \(B\) vanish, i.e., NEWLINE\[NEWLINEA_{ijkh}= B_{ijkh}=0\quad \forall i,j,k,h\in \{1,2,3\}\tag{2}NEWLINE\]NEWLINE and, besides, let \(G\) be differentiable and NEWLINE\[NEWLINEG_{ijkh}(0)\xi_{ij} \xi_{kh}\geq \alpha\xi_{ij} \xi_{ij}\tag{3}NEWLINE\]NEWLINE for some positive constant \(\alpha\) and any second-order symmetric tensor \(\xi=[\xi_{ij}]\). In this case, the relation (1) characterizes the so-called linear viscoelasticity of Boltzmann type. Stress-strain laws like (1) also come out from homogenization of composite materials with periodic structure at a microscopic level. To be more precise, if the microscopic mechanical behavior is described by a viscoelastic stress-strain relationship of rate type, then the homogenized macroscopic stress-strain law may contain a damping term with memory. For instance, if the stress-strain relation at a microscopic level is of Kelvin-Voight type, then the homogenization process leads to (1) where both \(A\) and \(B\) are coercive, that is NEWLINE\[NEWLINEA_{ijkh} \xi_{ij} \xi_{kh}\geq \beta\xi_{ij} \xi_{ij}, \quad B_{ijkh}\xi_{ij}\xi_{kh}\geq \beta\xi_{ij} \xi_{ij}\tag{4}NEWLINE\]NEWLINE for some positive constant \(\beta\) and any second-order symmetric tensor \(\xi=[\xi_{ij}]\). Here \(G\) needs not to be differentiable.NEWLINENEWLINENEWLINEThe author studies an inverse problem for the model (1), (4). A material with heterogeneous periodic structure at a microscopic level is considered. The microscopic mechanical behavior is described by a stress-strain law of Kelvin-Voight type. It is shown that a homogeneous process leads to a macroscopic stress-strain relation containing a time convolution term. This term is characterized by a time dependent relaxation kernel which accounts for memory effects. The relaxation kernel is identified by boundary tractions, provided that the displacement vector field solves a suitable Cauchy- Dirichlet problem for the motion equation.