This lecture surveys recent advances and open problems in the analysis of nonlinear evolution equations of solid mechanics, especially those describing the dynamics of viscoelastic and viscoplastic materials. The governing equations studied here form systems of quasilinear partial differential equations of "parabolic-hyperbolic" type involving singular differential operators of monotone type. Since the unknown in these equations represents a deformation of a body in Euclidean space, it should be locally one-to-one. This requirement is the source of severe analytical difficulties, the treatment of which surprisingly depends on the nature of the dissipation. The role of dissipation is likewise central in dealing with nonlinear effects due to rotation and with the behavior of the spectrum for Hopf bifurcation problems. The general theory is illustrated with a variety of concrete problems.

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