The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity
The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity
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DOI：
10.1016/0022-0396(69)90118-1
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发表时间：
1969-07
影响因子：
2.4
通讯作者：
C. Dafermos
中科院分区：
文献类型：
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作者：
C. Dafermos
Lax [1]，MacCamy-Mizel[2]证明了一维非线性弹性力学方程在一般情况下不存在光滑解。然而，如果应力以适当的方式依赖于运动的历史，则存在光滑解。具有历史依赖性的固体的最简单模型由一维粘弹性提供，其中应力a是变形梯度u及其时间导数zi的函数。Greenberg，MacCamy和Mizel [3]考虑了半线性情形，cr（uZ，zi.）=~ J（zL~）+ ti，其中θ 3是严格增函数。他们证明了唯一的解决方案，这是渐近稳定的存在。本文考虑一般情况下的牵引边值问题，其中O（U，，ti，）在u，zi，. CT（U~，2%）对ti的依赖性的形式受到粘度远离零的要求的限制。相反，对U的依赖性，除了某些有界性要求之外，基本上是不受限制的。
It has been proved (Lax [I], MacCamy-Mizel[2]) that the equations of one-dimensional nonlinear elasticity do not admit, in general, smooth solutions in the large. It is expected though, that if the stress depends on the history of motion in an appropriate fashion, then smooth solutions exist.The simplest model of a solid with history dependence is provided by one-dimensional viscoelasticity, where the stress a is a function of the deformation gradient u, and its time derivative zi,. Greenberg, MacCamy and Mizel [3] have considered the semilinear case, cr (uZ, zi.)=~ J (zL~)+ ti, where 93 is a strictly increasing function. They prove the existence of a unique solution which is asymptotically stable. In this paper we consider the traction boundary value problem in the general case where O (U,, ti,) may be nonlinear in both u,, zi,. The form of the dependence of CT (U~, 2%) on ti, is restricted by the requirement that the viscosity be bounded away from zero. On the contrary, the dependence on U, is essentially unrestricted apart from certain requirements of boundedness.