  We consider a system of evolutionary equations that is capable of describing
certain viscoelastic effects in linearized yet nonlinear models of solid
mechanics. The essence of the paper is that the constitutive relation,
involving the Cauchy stress, the small strain tensor and the symmetric velocity
gradient, is given in an implicit form. For a large class of implicit
constitutive relations we establish the existence and uniqueness of a
global-in-time large-data weak solution. We then focus on the class of
so-called limiting strain models, i.e., models for which the magnitude of the
strain tensor is known to remain small a priori, regardless of the magnitude of
the Cauchy stress tensor. For this class of models, a new technical difficulty
arises, which is that the Cauchy stress is only an integrable function over its
domain of definition, resulting in the underlying function spaces being
nonreflexive and thus the weak compactness of bounded sequences of elements of
these spaces is lost. Nevertheless, even for problems of this type we are able
to provide a satisfactory existence theory, as long as the initial data have
finite elastic energy and the boundary data fulfill natural compatibility
conditions.
