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- W2055086354 abstract "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. It turns out that the viscoelastic part dominates the elastic part and secures the existence of a unique solution in the large. This solution is smooth enough so that all derivatives entering the equation of motion are Holder continuous. The tools of the proof are certain “energy” estimates combined with known a priori bounds from the theory of parabolic equations, and the Leray-Schauder fixed point theorem. In the final part of the paper, we investigate the asymptotic stability of the solution. The mechanism which provokes the decay of the solution is induced by the viscosity. On the other hand, the number and the nature of all possible static configurations depend entirely on the elastic part of" @default.
- W2055086354 created "2016-06-24" @default.
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- W2055086354 date "1969-07-01" @default.
- W2055086354 modified "2023-09-29" @default.
- W2055086354 title "The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity" @default.
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- W2055086354 doi "https://doi.org/10.1016/0022-0396(69)90118-1" @default.
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