Modeling and simulation of viscoelastic solids under large numbers of loading cycles

Abstract

An efficient modeling procedure is proposed for viscoelastic (VE) solids subjected to large numbers of loading cycles. While the Laplace–Carson transform (LCT) is often used to solve VE creep or relaxation problems, the originality here is an efficient extension of the approach to a plethora of cycles, based on some key ingredients. The time history of the cyclic loading is decomposed into transient and periodic signals, leading to two subproblems. Each one is transformed into a finite number of linear elastic analyses in the L–C domain. A method to choose the number and positioning of the L–C domain sampling points for each one of the two subproblems is detailed. Specific LCT inversion methods are applied to each subproblem in order to reconstruct the displacement, strain, and stress fields in the time domain. For the transient subproblem, Schapery’s collocation method based on exponential basis functions is used, while a new LCT inversion method is proposed for the periodic subproblem based on sinusoidal basis functions and a Newton–Gauss algorithm. After a verification on well-known 1D functions, the accuracy of the proposed method is assessed on two structural problems with large numbers of cycles. Comparison with reference finite element analyses conducted directly in the time domain shows that the proposed methodology provides excellent predictions, both at local scale (displacement, strain, and stress components at various points) and macroscale (global energy indicator). The important speedup factor (e.g., 32 for 10 k cycles) will increase significantly with the number of cycles, enabling the proposed method to be extended to high cycle fatigue of thermoplastic polymer structures in future work.

Keywords:

• Laplace–Carson numerical inversion
• high cycle simulation
• periodic basis extension
• viscoelastic structure

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.

Figure A1. Dependence of the two reconstruction methods on the number of sampling points for the sinusoidal function of Eq. (16)(16) $$\tilde{f}(t)=a\hspace{0.17em}\text{sin}\hspace{0.17em}(\omega t+\varphi )$$(16) .

Funding

This work was conducted and funded within the framework of the European Union’s Horizon 2020 research and innovation programme for the project “Multi-scale Optimisation for Additive Manufacturing of fatigue resistant shock-absorbing MetaMaterials (MOAMMM)”, grant agreement No. 862015, of the H2020 European Research Council (H2020-EU.1.2.1.) – FET OpenProgramme.