A computationally efficient hybrid formulation for viscoelastic–viscoplastic polymer solids and structures under large numbers of loading cycles

The numerical simulation of the high cycle response of solids and structures made of thermoplastic polymers is challenging because those materials exhibit a complex viscoelastic–viscoplastic (VEVP) behavior and even under large numbers of loading cycles, they continue to dissipate energy and feature a frequency dependent response. On the one hand classical simplified methods based on linear elasticity are not applicable, and on the other hand direct structural analyses with VEVP material models are so computationally prohibitive that they are not possible in practice. In this article, a computationally efficient hybrid formulation is proposed. The structure is first computed as being purely VE, using a recently proposed formulation based on Laplace-Carson transform (LCT) and its numerical inversion, and enabling to compute accurate strain and stress fields at a very reduced cost, which is also independent of the number of cycles. Next, the VEVP solution at any points of interest is computed with a time homogenization formulation which uses fast and slow time scales and asymptotic time expansions to compute complete solutions at extremely limited cost. An experimentally identified TPU material and a 3D lattice are used for the numerical simulations. Predictions of the hybrid formulation are compared against reference VEVP solutions and their accuracy verified. Numerical simulations for one million cycles are presented and the low computational cost of the hybrid formulation illustrated. The underlying assumptions of the hybrid formulation linking the VE results with the VEVP calculations are discussed. The proposal lays the foundation for the time and space multiscale modeling and simulation of the high cycle fatigue of thermoplastic solids and structures.