Analysis of the bulk creep influence on stress-strain curves under tensile loadings at constant rates and on Poisson's ratio evolution based on the linear viscoelasticity theory

Abstract

The Boltzmann-Volterra linear constitutive equation for isotropic non-aging viscoelastic materials is studied analytically in order to elucidate its abilities to provide a qualitative simulation of rheological effects related to different behavior types of lateral strain and the Poisson's ratio (i.e. lateral contraction ratio) observed in uni-axial tests under tension or compression at constant stress rate. The viscoelasticity equation is controlled by two material functions of a positive real argument (that is shear and bulk creep compliances); they are implied to be positive, differentiable, increasing and convex up functions. General properties of the volumetric, longitudinal and lateral strain-time curves, stress-strain curves and the Poisson's ratio evolution in time generated by the viscoelasticity relation (with an arbitrary shear and bulk creep functions) are examined, their dependence on stress rate and on qualitative characteristics of two creep functions are analyzed, conditions for their monotonicity and convexity or for existence of extrema, inflection points and sign changes are studied. Taking into account compressibility and volumetric creep (governed by a time-dependent bulk creep function) is proved to affect strongly the qualitative behavior of lateral strain and the Poisson's ratio. In particular, it is proved that the linear theory can reproduce increasing, decreasing or non-monotone and convex up or down dependencies of lateral strain and Poisson's ratio on time under tension or compression at constant stress rate, it can provide existence of minimum, maximum or inflection points and sign changes from minus to plus and vice versa. It is shown, that the Poisson's ratio at any moment of time is confined in the interval from $-1$ to 0.5 and the restriction on creep compliancies providing negative values of the Poisson's ratio is derived. Criteria for the Poisson's ratio increase or decrease and for extrema existence are obtained. The analysis revealed the set of characteristic features of the theoretic volumetric, axial and lateral strain-time curves, stress-strain curves families and the Poisson's ratio dependence on time which are convenient to check in tensile tests at constant stress rates and should be employed as indicators of the linear viscoelasticity theory applicability (or non-applicability) for simulation of a material behavior before identification. The specific properties of the two models are considered based on the assumption that the Poisson's ratio is time-independent or the assumption that bulk creep function is constant which neglects bulk creep and simulates purely elastic volumetric strain dependence on a mean stress. This assumptions reduce the number of material function to the single one and one scalar parameter and are commonly (and very often) used for simplification of viscoelasticity problems solutions. A number of restrictions and additional applicability indicators are found for this models. In particular, it is proved that elastic volumetric deformation assumption does not cut the overall range of the Poisson's ratio values and does not demolish the Boltzmann-Volterra relation ability to describe non-monotonicity and sign changes of lateral strain and to produce negative values of the Poisson's ratio, but neglecting bulk creep restricts this ability significantly and reduces drastically the variety of possible behavior modes of lateral strain-time curves and the Poisson's ratio evolution and so contracts applicability field of the model. The model with constant bulk compliance generates only convex-up lateral strain-time curves which can not have minima or inflection points and can change sign from minus to plus only and the Poisson's ratio is increasing convex-up function of time (without any extrema or inflection points which are possible in general case) and can not change sign from positive to negative.