Fractional-order uniaxial visco-elasto-plastic models for structural analysis

We propose two fractional-order models for uniaxial large strains and visco-elasto-plastic behavior of materials in structural analysis. Fractional modeling seamlessly interpolates between the standard elasto-plastic and visco-elasto-plastic models, taking into account the history (memory) effects of the accumulated plastic strain to specify the state of stress. To this end, we develop two models, namely M1 and M2, corresponding to visco-elasto-plasticity considering a rate-dependent yield function and visco-plastic regularization, respectively. Specifically, we employ a fractional-order constitutive law that relates the Kirchhoff stress to the Caputo time-fractional derivative of the strain with order β∈(0,1). When β→0 the standard rate-independent elasto-plastic model with linear isotropic hardening is recovered by the models for general loading, and when β→1, the corresponding classical visco-plastic model of Duvaut-Lions (Perzyna) type is recovered by the model M2 for monotonic loading. Since the material behavior is path-dependent, the evolution of the plastic strain is achieved by fractional-order time integration of the plastic strain rate with respect to time. The plastic strain rate is then obtained by means of the corresponding plastic slip and proper consistency conditions. Finally, we develop the so called fractional return-mapping algorithm for solving the nonlinear system of the equilibrium equations developed for each model. This algorithm seamlessly generalizes the standard return-mapping algorithm to its fractional counterpart. We test both models for convergence subject to prescribed strain rates, and subsequently we implement the models in a finite element truss code and solve for a two-dimensional snap-through instability problem. The simulation results demonstrate the flexibility of fractional-order modeling using the Caputo derivative to account for rate-dependent hardening and viscous dissipation, and its potential to effectively describe complex constitutive laws of engineering materials and especially biological tissues.