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 for 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 and 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.

Development of an Efficient Variable Step-Size Gradient Method Utilizing Variable Fractional Derivatives

“…. Obviously, the target point here is at point (1,5). In this simulation, the iterati point of all algorithms is set to the same point (20,20), the iteration step size 𝜌 is set to parameters can be selected to be approximately the same as the previous Simula such as 𝑣 = 1 and 𝑢 = −1.5.…”

“…. Obviously, the target point here is at point (1,5). In this simulation, the iterati point of all algorithms is set to the same point (20,20), the iteration step size 𝜌 is set to Experiment 3.…”

“…For a long time, due to the lack of corresponding physical background, fractional calculus was regarded as a purely theoretical study of mathematics. With the development of science and technology, researchers found that fractional calculus can provide a good description of modeling some abnormal behaviors such as diffusion process [1], cybernetics [2,3], viscoelasticity [4,5], biology [6], etc. The long-memory properties of fractional calculus not only reflect local characteristics at the time of evaluation but also reflect the entire history of the function.…”

“…Here, we consider a two-dimensional objective function 𝑓(𝑥, 𝑦) = 2(𝑥 − (𝑦 − 5). Obviously, the target point here is at point(1,5). In this simulation, the iterati…”

Polyurea–Graphene Nanocomposites—The Influence of Hard-Segment Content and Nanoparticle Loading on Mechanical Properties

“…The storage and loss modulus master curves showed broad transition regions, indicating a wide distribution of relaxation times. Tzelepis et al found that such a distribution was well-described by the so-called fractional Maxwell model (FMM) [23][24][25][26][27][28][29]-or, to be more precise, a sum of two fractional Maxwell gels (FMG), with one FMG element describing the continuous soft phase (with dispersed hard domains and dissolved hard segments) and the second FMG element representing the percolated hard phase. The plateau modulus of the first element was found to be nearly independent of the HSWF, while the plateau modulus of the second element was a strong function of HSWF, consistent with earlier experiments and theories [10].…”