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Nonuniqueness for a hyperbolic system: cavitation in nonlinear elastodynamics. (English) Zbl 0651.73005
The authors consider the initial boundary value problem of nonlinear elastodynamics posed in a ball $$\Omega$$ of $${\mathbb{R}}^ n$$, $$n\geq 3:$$ Find $${\mathfrak u}:\Omega \times [0,T]\to {\mathbb{R}}^ n$$ such that $(1)\quad div {\mathfrak S}(\nabla {\mathfrak u})={\mathfrak u}_{tt}\quad in\quad \Omega \times]0,T[,\quad {\mathfrak u}({\mathfrak x},t)=\lambda {\mathfrak x}\quad on\quad \partial \Omega \times [0,T],$ ${\mathfrak u}({\mathfrak x},0)=\lambda {\mathfrak x}\quad in\quad \Omega,\quad {\mathfrak u}({\mathfrak x},0)_ t=0\quad in\quad \Omega,$ for the special constitutive law (in terms of the first Piola- Kirchhoff stress tensor) $${\mathfrak S}({\mathfrak F})={\mathfrak F}+h'(\det {\mathfrak F})adj F$$, where h(v) is a convex function satisfying additional conditions. They are interested in radial cavitation similarity solutions of the form (2) $${\mathfrak u}({\mathfrak x},t)=(\phi (s)/s){\mathfrak x}$$, $$s=| {\mathfrak x}| /t$$, where $$\phi$$ is continuous, piecewise $$C^ 2$$ and such that $$\phi (0)>0$$, $$\phi (s)=\lambda s$$ for $$s\geq \sigma >0$$. Such a solution opens a spherical hole of radius $$t\phi$$ (0) in the material for $$t>0$$. The authors show that there exist $$\lambda_ i\to +\infty$$ such that problem (1) has a nontrivial (weak) solution of the form (2) for $$\lambda =\lambda_ i$$ (apart from the trivial solution $$\bar{\mathfrak u}({\mathfrak x},t)=\lambda_ ix)$$. This is not in contradiction with known uniqueness theorems [C. M. Dafermos, ibid. 70, 167-179 (1979; Zbl 0448.73004)] and extends to elastodynamics the fundamental results of J. M. Ball [Phil. Trans. R. Soc. London A 306, 557-611 (1982; Zbl 0513.73020)] on cavitation in elastostatics.
The proof goes as follows. Insertion of (2) into equation (1) yields an ODE for $$\phi$$ which is solved for s in a maximal interval $$[0,s_ M]$$ with $$s_ M<+\infty$$. Then two cases arise: Either $$\lambda_ M:=\phi (s_ M)/s_ M={\dot \phi}(s_ M)$$ in which case $$\phi$$ is extended to $$[s_ M,+\infty [$$ in a $$C^ 1$$ fashion by $$\lambda_ Ms$$, or $$\lambda_ M>{\dot \phi}(s_ M)$$ in which case there exists a $$s_ J<s_ M$$ such that $$\phi$$ extended to $$[s_ J,+\infty [$$ by $$\lambda_ Js_ J$$ $$(\lambda_ J=\phi (s_ J)/s_ J)$$ satisfies the Rankine- Hugoniot condition for a shock obtained from equation (1). The proof is completed by showing that the set of such $$\lambda_ M,\lambda_ J$$ is not bounded from above.
Admissibility of the solutions thus found is then discussed. It is shown that the total energy of the cavitation solutions does not exceed that of the trivial solutions. In fact, if $$\phi$$ is $$C^ 1$$ these energies are equal, while the energy of a cavitation solution is strictly less than that of the corresponding trivial solution if $$\phi$$ is not $$C^ 1$$, i.e., if $$u$$ has a shock. Thus shocks dissipate energy and, were they do exist, then it could be possible to ever decrease the energy by superposing many cavitation solutions, that is opening many holes in the material. In this sense, this work can be relevant to the study of the onset of fracture (i.e., by coalescence of microvoids). The solutions are also admissible according to the Lax criterion.
Reviewer: H.Le Dret

##### MSC:
 74B20 Nonlinear elasticity 35L15 Initial value problems for second-order hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations 74R99 Fracture and damage 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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