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Development of singularities in the motion of materials with fading memory. (English) Zbl 0595.73026
As the author says, the aim of the paper is ”to compare the behavior of elastic materials with the behavior of materials with fading memory in which the destabilizing influence of nonlinear elastic response coexists and competes with the damping action of memory response”.
A one-dimensional simple material with homogeneous reference configuration, the axis \((-\infty,\infty)\) and reference density \(\rho_ 0=1\) is considered. The Piola-Kirchhoff stress \(\sigma\) is given by \[ \sigma (x,t)={\mathcal S}\{u(x,t),u^{(t)}(x,\cdot)\}, \] where \(x\in {\mathbb{R}}\) is the typical material particle, \(u(x,t)>0\) is the present value of the deformation gradient at x, \(u^{(t)}(x,\tau)=u(x,t- \tau),\quad \tau (-\infty,0)\) is the past history of u up to time t, and \({\mathcal S}\) is a continuously Fréchet differentiable functional on Banach space \({\mathbb{R}}\times L^ 1_ h(0,\infty).\)
Here \(L^ 1_ h\) denotes the \(L^ 1\) space weighted by the influence function h: [0,\(\infty)\to {\mathbb{R}}^+.\)
If \(D{\mathcal S}\{u,w(\cdot);\) \(f,g,(\cdot)\}= {\mathcal S}_ u\{u,w(\cdot)\}f+ \int^{\infty}_{0} {\mathcal K}\{u,w(\cdot),\tau\} g(\tau)d\tau\) is the Riesz representation of the Fréchet derivative of \({\mathcal S}\) then equations of motion, in the absence of body force, can be written in the form \[ \begin{cases}\partial_ tu-\partial_ xv=0 \\ \partial_ tv-{\mathcal S}_ u\{u,u^{(t)}\} \partial_ xu= \int^{t}_{-\infty} {\mathcal K}\{u,u^{(t)},t-\tau\} \partial_ xud\tau, \end{cases} \tag{1} \] where v is the velocity. It is assumed that the body has been in a state of homogeneous strain (2) \(u(x,t)=\bar u=\) const, \((x,t)\in {\mathbb{R}}\times (- \infty,0]\), and the motion is generated by an impulse at time \(t=0\) that induces the initial conditions (3) \(u(x,0)=\bar u,\) \(v(x,0)=v_ 0(x),\) \(x\in {\mathbb{R}}.\)
Assuming that \({\mathcal S}\) satisfies some smoothness properties, too complicated to be stated here, the author is proving the following result regarding the development of singularities in the motion of materials with fading memory: ”Given any positive numbers N and T, there is a positive number \(\epsilon\), depending only of \(\bar u,\) and a positive number M, depending on \(\bar u,\) \(\epsilon\), N, T, such that when \(v_ 0\) is a \(C^ 2\) function with compact support in \({\mathbb{R}}\) which satisfies \(| v_ 0(x)| <\epsilon,-\partial_ xv_ 0(x)<N,\) \(x\in {\mathbb{R}}\), and \(\max_{{\mathbb{R}}}\partial_ xv_ 0(\cdot)>M\) then the length of maximal time interval of existence of any classical solution of (1), (2), and (3) cannot exceed T”.
Reviewer: Gh.Gr.Ciobanu

MSC:
74J99 Waves in solid mechanics
74B20 Nonlinear elasticity
35L67 Shocks and singularities for hyperbolic equations
35L05 Wave equation
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