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We prove the existence of a unique large-data global-in-time weak solution toa class of models of the form $\mathbf{u}_{tt} = \mathrm{div}(\mathbb{T}) +\mathbf{f}$ for viscoelastic bodies exhibiting strain-limiting behaviour, wherethe constitutive equation, relating the linearised strain tensor$\boldsymbol{\epsilon}(\mathbf{u})$ to the Cauchy stress tensor $\mathbb{T}$,is assumed to be of the form $\boldsymbol{\epsilon}(\mathbf{u}_t) +\alpha\boldsymbol{\epsilon}(\mathbf{u})= F(\mathbb{T})$, where we define$F(\mathbb{T}) = (1 + |\mathbb{T}|^a)^{-\frac{1}{a}}\mathbb{T}$, for constantparameters $\alpha \in (0, \infty)$ and $a\in (0, \infty)$, in any number $d$of space dimensions, with periodic boundary conditions. The Cauchy stress$\mathbb{T}$ is show to belong to $L^1(Q)^{d\times d}$ over the space-timedomain $Q$. In particular, in three space dimensions, if $a\in (0,\frac{2}{7})$, then in fact $\mathbb{T}\in L^{1+\delta}(Q)^{d\times d}$ for a$\delta>0$, the value of which depends only on $a$.

Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body