Gen Nakamura | Relaxation tensor for extended Burgers model

Days on Diffraction 2024. Mini-symposium “Inverse Problems”. Tuesday, 11 June, 2024
Gen Nakamura (Dept. Math. and Research Institute of Electronic Science)
Relaxation tensor for extended Burgers model

We provide a new method for deriving the anisotropic relaxation tensor and its exponentially decaying property for the extended Burgers model (abbreviated by EBM), which is an important model in rheology, and Earth and planetary sciences. Upon having this tensor, the EBM can be converted to a Boltzmann-type viscoelastic system of equations (abbreviated by BVS). Historically, the relaxation tensor for the EBM is derived by solving the constitutive equation using the Laplace transform. We refer to this by L-method. Since inverting the inverse Laplace transform needs the partial fraction expansion, the L-method needs to assume the commuta tivity for the elastic tensors of the EBM. This is a very strong assumption. The new method not only avoids using this assumption but also can derive several important properties of the relaxation tensor, which are the positivity, C1-smoothness with respect to the time variable, the exponentially decaying property together with its derivative, and the causality. Further, we will show that for the initial boundary value problem for the BVS with homogeneous boundary data and source term, any solution to this problem decays exponentially with respect to time as it goes to infinity.