Equivalent histories, minimal state and initial value problem in viscoelasticity. Math. Methods Appl. Sci. 28 (2005), no. 2, 233--251.

References: 10 [-] Reference Citations: 0 Review Citations: 0 In the standard theory of viscoelasticity the constitutive equation for the stress is $$ T(x,t)=G_0(x)E(x,t)+int_0^infty G'( x,xi)E^t(x,xi),dxi tag1 $$ where $E$ is the infinitesimal strain tensor, $ G(x,·)$ is the relaxation function, defined by $$ G(x,s)coloneq G_0(x)-int_0^s G'(x,xi),dxi, $$ and assumed to be such that $G_infty(x)=lim_{stoinfty} G(x,s)$ is positive definite, and $E^t(x,·)$ is the history. The main idea of this paper is to replace (1) by $$ T(x,t)=int_{t_0}^t G(x,t-xi)widehat E( x,xi),dxi+I^{t_0}(x,t-t_0) tag2 $$ where $I^{t_0}(x,t-t_0)$ denotes the contribution to $T( x,t)$ by the stress at a previous time $t_0$. It follows that $$lim_{ttoinfty}I^{t_0}(x,t-t_0)=G_infty(x) E(x,t_0)$$ and moreover $I^t(x,t-t_0)$ has the operative meaning of the stress tensor at time $t$ once the solid is held at a constant strain since $t_0$, i.e., $E(·,xi)=E(·,t_0)$, $xiin[t_0,t)$. Representation (2) involves the relation function $G(x,·)$, which is required only to have a limit value $G_infty$. Then, the authors consider the following initial-boundary value problem: $$ Lu=0 text{ in } Omegatimes{Bbb R}^{++}, tag3 $$ $$ u=widehat u text{ on }partialOmega_utimes{Bbb R}^+,,, u(0)=u_0, dot u(0)=dot u_0 text{ in }overlineOmega, tag4 $$ $$ Tn=widehat t-tilde I^0n text{ on }partialOmega_Ttimes{Bbb R}^+. $$ Here $partialOmega_ucappartialOmega_T=varnothing$, $ partialOmega_ucuppartialOmega_T=partialOmega$, $widehat u$ and $widehat t$ are known functions on the pertinent domain, and $$ Lucoloneq rhoddot u-nabla·(G'_0nabla u+ Gtimesnabla u)-rho f=0 text{ in }Omegatimes{Bbb R}^{++},tag5$$ $$rho f(tau)=rho b(tau)+nablawidehat I^0(tau), $$ the body force $b$ and $tilde I^0$ are known on $overlineOmegatimes{Bbb R}^{++}$, and $$ tilde I^t(tau)=int_0^infty G(tau+s)E^t(s),ds,tag6$$ $$(G'timesnabla u)(tau)=int_0^tau G'(t-s)nabla u(s),ds. $$ Using the variational approach they prove that the functional associated with (3)--(6) has a stationary point $uin C^{2,2}(Omegatimes{Bbb R}^+,V)$ if and only if $u$ is a solution to the mixed problem (3)--(4). Next, the authors consider a virtual-work type formulation which provides a weak form of the initial-value problem for the system of the dynamics of viscoelastic solids. Applying the Fourier transform to the initial-boundary value problem which represents the weak formulation of the problem $$ rho dot v(t)=nabla·int_0^t G(t-tau)nabla v(tau),dt+b(t) text{ in }{Bbb R}times{Bbb R}^+, tag7 $$ $$ v(x,0)=0, v(x,t)|_{partialOmega}=0 tag8 $$ where $v(x,t)=dot u(x,t)$, under some assumption about $b(t)$ they prove that there exists a unique virtual-work solution of problem (7)--(8) in a certain class of functions.