Comparative analysis of creep curves properties generated by linear and nonlinear heredity theories under multi-step loadings

The general equation and basic properties of theoretic creep curves generated by the linear integral constitutive relation of viscoelasticity or by the Rabotnov nonlinear (quasi-linear) constitutive relation under arbitrary multi-step uni-axial loadings have been studied analytically with the implication that material functions are arbitrary.
The Rabotnov constitutive relation generalizes the Boltzmann–Volterra linear relation in a uni-axial case by introducing the second material function (the non-linearity function) beside the creep compliance. The Rabotnov equation is aimed at adequate modeling of the rheological phenomena set which is typical for isotropic rheonomic materials exhibiting non-linear hereditary properties, strong strain rate sensitivity and tension-compression asymmetry. The model is applicable for simulation of mechanical behaviour of various polymers, isotropic composites, metals and alloys, ceramics at high temperature, biological tissues and so on.
The qualitative features of the theoretic creep curves produced by the relations mentioned above are examined and compared to each other and to basic properties of typical test creep curves of viscoelastoplastic materials under multi-step uni-axial loadings in order to find out inherited (from linear viscoelasticity) properties and additional capabilities of the non-linear relation, to elucidate and compare the applicability (or nonapplicability) scopes of the linear and quasi-linear relations, to reveal their abilities to provide an adequate description of basic rheological phenomena related to creep, recovery and cyclic creep, to find the zones of material functions influence and necessary phenomenological restrictions on material functions and to develop techniques for their identification and tuning. Assuming the material functions are arbitrary, we study analytically the theoretic creep curves properties dependence on creep compliance function, the nonlinearity function and parameters of loading programs. We analyze monotonicity and convexity intervals of creep curves, conditions for existence of extrema or flexure points, asymptotic behavior at infinity and deviation from the associated creep curve at constant stress, conditions for memory fading, the formula for plastic strain after complete unloading (after recovery), influence of a stress steps permutation and the asymptotic commutativity phenomenon, ratcheting rate and a criterion for non-accumulation of plastic strain under cyclic loadings, the relations for strain and strain rate jumps produced by given stress jumps and the phenomenon of elastic strain drift due to creep, etc. A number of effects are pointed out that the nonlinear model (and the linear theory as well) can’t simulate whatever material functions are taken.