A GENERAL RETURN-MAPPING FRAMEWORK FOR FRACTIONAL VISCO-ELASTO-PLASTICITY

Abstract

We develop a fractional return-mapping framework for power-law visco-elasto-plasticity. In our approach, the fractional viscoelasticity is accounted through canonical combinations of Scott-Blair elements to construct a series of well-known fractional linear viscoelastic models, such as Kelvin-Voigt, Maxwell, Kelvin-Zener and Poynting-Thomson. We also consider a fractional quasi-linear version of Fung's model to account for stress/strain nonlinearity. The fractional viscoelastic models are combined with a fractional visco-plastic device, coupled with fractional viscoelastic models involving serial combinations of Scott-Blair elements. We then develop a general return-mapping procedure, which is fully implicit for linear viscoelastic models, and semi-implicit for the quasi-linear case. We find that, in the correction phase, the discrete stress projection and plastic slip have the same form for all the considered models, although with different property and time-step dependent projection terms. A series of numerical experiments is carried out with analytical and reference solutions to demonstrate the convergence and computational cost of the proposed framework, which is shown to be at least first-order accurate for general loading conditions. Our numerical results demonstrate that the developed framework is more flexible, preserves the numerical accuracy of existing approaches while being more computationally tractable in the visco-plastic range due to a reduction of 50% in CPU time. Our formulation is especially suited for emerging applications of fractional calculus in bio-tissues that present the hallmark of multiple viscoelastic power-laws coupled with visco-plasticity.

Keywords: fast convolution; fractional quasi-linear viscoelasticity; power-law visco-elasto-plasticity.

Fig. 1:

Fractional linear viscoelastic models employed in this work, constructed from serial/parallel combinations of SB elements. The SB building blocks naturally account for an infinite fractal arrangement of Hookean/Newtonian elements. The employed fractional quasi-linear model is not represented by a mechanical analogue although the time-dependent component of the relaxation function has a SB-like representation.

Fig. 2:

Fractional visco-elasto-plastic model diagram. Here, any of the linear and quasi-linear fractional viscoelastic models can be separately coupled with a fractional visco-plastic rheological device.

Fig. 3:

Convergence analysis for the fractional viscoelastic models with known analytical solutions. (a) A stress relaxation test with non-smooth step-strains, and material parameters $\left({\mathbb{E}}_1,{\mathbb{E}}_2,{\mathbb{E}}_3\right)=(1,1,1)$ and $\left({\beta }_1,{\beta }_2,{\beta }_3\right)=(0.3,0.7,0.1)$, yielding first-order convergence. (b) Convergence for the FQLV model with a fabricated solution of linearly increasing strains and material properties $(E,\beta ,A,B)=(1,0.3,1,1)$. The slopes of the error curves are q ≈ 2 − β.

Fig. 4:

Convergence analysis for all fractional viscoelastic models with $\left({\mathbb{E}}_1,{\mathbb{E}}_2,{\mathbb{E}}_3\right)=(1,1,1)$ and $\left({\beta }_1,{\beta }_2,{\beta }_3\right)=(0.3,0.7,0.1)$. A cubic strain function was employed with a reference solution using the time step size $\text{Δ}t=2^{-17}$. Monotone loading test with the convergence rate of q ≈ 1.3 was applied for all models.

Fig. 5:

CPU times for the developed fractional return-mapping algorithm and the original one [31] for an SB viscoelastic part. The black line has slope q = 2.

Fig. 6:

Comparison between the presented return-mapping algorithm and the reference approach from [31] under low and high frequency loading.

Fig. 7:

Convergence analysis for the fractional visco-elasto-plastic models under cyclic loads. Due to the particular choice of fractional orders (with β₂ = β_K = 0.7 being dominant), we observed the convergence rate off q ≈ 1.3 for all models except for the FKV. In the latter case, we observe a linear convergence to the reference solution.

Fig. 8:

Visco-elasto-plastic reference solutions for the employed models for the first 30 loading cycles. We notice a similar behavior for most models under the choice of material parameters except for the FPT and FKV models. The FKV particularly yielded a very stiff response due to the combination of high fractional order values and high strain rates.