Abstract
We study analytically the Boltzmann - Volterra linear constitutive equation for isotropic non-aging viscoelastic media in order to elucidate its capabilities to provide a qualitative simulation of rheological phenomena related to different types of evolution of triaxial strain state and of the lateral contraction ratio (the Poisson ratio) observed in uni-axial tests of viscoelastic materials under tension or compression at constant stress rate. In particular, we consider such effects as increasing, decreasing or non-monotone dependences of lateral strain and Poisson's ratio on time, sign changes and negativity of Poisson's ratio (auxeticity effect) and its stabilization at large times. The viscoelasticity equation implies that the hydrostatic and deviatoric parts of stress and strain tensors don't depend on each other. It is governed by two material functions of a positive real argument (that is shear and bulk creep compliances). Assuming both creep compliances are arbitrary positive, differentiable, increasing and convex up functions on time semi-axis, we analyze general expressions for the Poisson ratio and strain triaxiality ratio (which is equal to volumetric strain divided by deviatoric strain) generated by the viscoelasticity relation under uni-axial tension or compression. We investigate qualitative properties and peculiarities of their evolution in time and their dependences on material functions characteristics. We obtain the universal accurate two-sided bound for the Poisson ratio range and criteria for the Poisson ratio increase or decrease and for extrema existence. We derive necessary and sufficient restrictions on shear and bulk creep compliances providing sign changes of the Poisson ratio and negative values of Poisson's ratio on some interval of time. The properties of the Poisson ratio under tension at constant stress rates found in the study we compare to properties the Poisson ratio evolution under constant stress (in virtual creep tests) and illustrate them using popular classical and fractal models with shear and bulk creep functions each one controlled by three parameters. The analysis carried out let us to conclude that the linear viscoelasticity theory (supplied with common creep functions which are non-exotic from any point of view) is able to simulate qualitatively the main effects associated with different types of the Poisson ratio evolution under tension or compression at constant stress rate except for dependence of Poisson's ratio on stress rate. It is proved that the linear theory can reproduce increasing, decreasing or non-monotone and convex up or down dependences of lateral strain and Poisson's ratio on time and it can provide existence of minimum, maximum or inflection points and sign changes from minus to plus and vice versa and asymptotic stabilization at large times.
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