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In this work, we deal with a one-dimensional stress-rate type model for the response of viscoelastic materials, in relation to the strain-limiting theory. The model is based on a constitutive relation of stress-rate type. Unlike classical models in elasticity, the unknown of the model under consideration is uniquely the stress, avoiding the use of the deformation. Here, we treat the case of periodic boundary conditions for a linearized model. We determine an optimal function space that ensures the local existence of solutions to the linearized model around certain steady states. This optimal space is known as the Gevrey-class 3/2 , which characterizes the regularity properties of the solutions. The exponent 3/2 in the Gevrey-class reflects the specific dispersion properties of the equation itself. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'. [ABSTRACT FROM AUTHOR]