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\begin{document}



\title{EFFECTIVE RELATIONS FOR NONLINEAR DYNAMICS OF
CRACKED SOLIDS}
\author{V.P.Smyshlyaev}{School of Mathematical Sciences, University of
Bath,\\ Claverton Down, Bath BA2 7AY, U.K.} 
\coauthor{J.R.Willis}{Department
of Applied Mathematics and Theoretical Physics,\\ Cambridge University,\\ 
Cambridge CB3 9EW, U.K.}


The ``effective" nonlinear dynamic response is studied for a solid containing
a random distribution of cracks with nonlinear properties. The nonlinearity
arises due to unilateral constraints, which correspond to closure of each
individual crack during compression and opening during tension. Via
statistical averaging, we derive and study a system of effective nonlinear
relations  for dilutely cracked solids.  The system
consists of the averaged equation of motion coupled to {\it two} constitutive
relations. One is the
effective stress-strain relation which involves an internal variable
which reflects the extent to which the cracks are open. The other relation
governs the evolution of this internal variable, which is found by
considering the scattering of the ambient field by a single crack situated
in it. The resulting system is both \underline{nonlinear} and
 \underline{nonlocal} system of partial differential equations and
integral inequalities involving the Green's operator.
We also develop and discuss an equivalent formulation of the full model
in the form of a variational inequality with a hereditary term. Example
problems are solved numerically, both for a more simple ``bilinear"
(local) model and for the new nonlocal model. The nonlocal term introduces
some new features,
in particular damping of shock singularities that  bilinear model would
predict, transforming them instead into localized bands. Question of
well- posedness of the nonlinear nonlocal problem and
future prospects shall also be discussed.

\vspace*{-0.5cm}
\begin{thebibliography}{99}
\vspace*{-0.2cm}
\bibitem{sm1}  V.P. Smyshlyaev and J.R. Willis, Effective relations for
nonlinear dynamics of cracked solids,
{\it Journal of Mechanics and Physics of Solids}, {\bf 44}(1), 49--75 (1996).
\end{thebibliography}

\end{document}
