A model of a hyperelastic body via hyperbolic conservation laws and lower semicontinuity of a matrix kinetic energy functional

• Ein Modell eines hyperelastischen Körpers mittels hyperbolischer Erhaltungsgleichungen und Unterhalbstetigkeit eines matriziellen Kinetische-Energie-Funktionals

Lotterstedt, Stefan; Westdickenberg, Michael (Thesis advisor); Noelle, Sebastian (Thesis advisor)

Aachen : RWTH Aachen University (2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

The compressible isentropic Euler equations are a system of hyperbolic partial differential equations depending on time and space that model the dynamics of a given compressible fluid such as an (ideal) gas, where, for simplicity, the entropy is assumed to be constant in time and space. In "A variational time discretization for compressible Euler equations" (Cavalletti, Sedjro, Westdickenberg; 2019) ([CSW19]), the authors have shown that measure-valued solutions of this system exist in the distributional sense for arbitrary spatial dimension and for any given initial data (see Theorem 1.8 there). In the first part of this thesis, we study a modified version of this system which is again a set of hyperbolic differential equations. We do not model a fluid any more, but moving objects: Consider for instance an object in the universe passing a black hole and thus experiencing stress and strain (this effect is called spaghettification since the part of the object which is closer to the black hole receives stronger gravitational forces than the side which is more remote) and a change in velocity, hence a change in kinetic energy. We consider hyperelastic materials, meaning that the object returns to its original state in absence of applied forces. We retract to this case since the evolution of such objects has been under firm investigation in elasticity theory. Mathematically, this change of setting necessarily leads to the (reasonable) restriction to spatial dimension d < 4 and to a new internal energy of the hyperelastic material memorizing the stress and strain that it experiences in terms of its so-called deformation gradient. A good choice for the new internal energy functional is motivated by elasticity theory and a special case of the so-called strain energy density functions investigated in "On the convexity of Nonlinear Elastic Energies in the Right Cauchy-Green Tensor" (Yang, Neff, Roventa, Thiel; 2017). Like in [CSW19], we construct solutions of the modified system by discretizing the time variable and by solving, in every time step, an optimal-transport problem by minimizing the sum of the so-called minimal work functional plus the internal energy of the material at the end of the given time step in order to gain an upper estimate of the total energy of the hyperelastic material. After thereby achieving discrete approximate solutions of the system, convex interpolation of the discrete approximate solutions as presented in [CSW19] and in "Minimal acceleration for the multi-dimensional isentropic euler equations" (Westdickenberg; 2020) makes it possible to establish the existence of exact measure-valued solutions for any given initial data. In [CSW19] the authors construct a sequence of approximate solutions (\rho_\tau,u_\tau)_\tau, \tau > 0, of the Euler equations, where every \rho_\tau is a finite, nonnegative Borel measure modeling the mass distribution and every u_\tau is an \R^d-valued vector field square-integrable with respect to \rho_\tau modeling the velocity distribution of the fluid, respectively. Consider the approximate sequence of momenta \left(m_\tau:=\rho_\tau u_\tau\right)_\tau which is a sequence of \R^d-valued Borel measures absolutely continuous with respect to \rho_\tau. Since the Euler equations particularly model the evolution of the momentum of the fluid, it is of importance to conserve the structure of the sequence in the limit in the following sense: Assume that (m_\tau)_\tau narrowly converges to a limit measure m for \tau -> 0. If we prescribe a uniform bound on the kinetic energy of the sequence of approximate solutions (which, mathematically speaking, is a uniform bound on the L^2(\rho_\tau) norm of the vector fields u_\tau), then there holds m=:\rho u for a suitable Borel measure \rho and a velocity field u, where (\rho,u) is the measure-valued solution of the Euler equations. An essential ingredient of the proof is lower semicontinuity of the functional (\rho,m=\rho u) |-> \|u\|_{L^2(\rho)} with respect to narrow convergence. In the second part of the thesis, we transfer this result to a new setting: if one transforms the minimization problem (minimal work functional plus internal energy of the fluid) mentioned earlier into a saddle point problem, there naturally arise \R^{d x d}-valued Borel measures, i. e. \R^{d x d}-valued mappings such that every component is a signed, finite Borel measure. We replace the real-valued Borel measures \rho_\tau by \R^{d x d}-valued Borel measures M_\tau, the \R^d-valued velocity fields u_\tau by \R^{d x d}-valued fields F_\tau and the L^2(\rho_\tau) norm by the yet to be defined L^2(M_\tau) norm. Picking up the concept of a matrix version of the Radon-Nikodým theorem in "The square-integrability of matrix-valued functions with respect to a nonnegative Hermitian measure" (Rosenberg; 1964) by defining P_\tau:=F_\tau M_\tau in a suitable way, P_\tau actually becomes a sequence of \R^{d x d}-valued Borel measures. Similarly to the primary case, we obtain lower semicontinuity with respect to narrow convergence of the functional (F,P:=FM) |-> \|F\|_{L^2(M)} and, provided the L^2(M_\tau) norm of the sequence \left(F_\tau\right)_\tau is uniformly bounded, the limit measure P of a narrowly converging sequence (P_\tau:=F_\tau M_\tau)_\tau is absolutely continuous in the matrix sense, i. e. there holds P=FM for a suitable \R^{d x d}-valued function F and an \R^{d x d}-valued Borel measure M. In the second part of the chapter, we give an equivalent notion of narrowly converging sequences (M_\tau,F_\tau M_\tau)_\tau |-> (M,FM) for \tau -> 0.