Time-domain asymptotic homogenization for linear-viscoelastic composites: mathematical formulae and finite element implementation

Abstract

Asymptotic homogenization (AH) is a rigorous method used for modeling multiscale mechanical behavior in periodic composites. Unlike conventional AH methods, which predict the effective viscoelasticity using the elastic–viscoelastic correspondence principle, an AH method was proposed to directly calculate the effective material properties in the time domain, thereby enabling the complex transform to be omitted between the time and Laplace domains. This time-domain AH method is reformulated based on the integral form rather than the differential form of the Kelvin–Voigt viscoelastic model. Thus, multiple time-domain-based characteristic elasticity and viscosity displacement tensors are replaced by one integral form characteristic displacement tensor in the polymer matrix composite. Using mathematical formulae, a numerical implementation method was then developed by establishing location- and time-related stress loads. An in-house Python subroutine was also programmed to compute the effective viscoelastic properties from the finite element implementation results. Unidirectional and woven fabric composite homogenization results were comparable to those obtained using the conventional representative volume element method, indicating high effectiveness and accuracy. The results suggest that the proposed time-domain AH method is a powerful tool for modeling the two-scale mechanical behavior of viscoelastic composites and is compatible with the existing commercial software.

Keywords

• Asymptotic homogenization
• Periodic composite
• Linear viscoelasticity
• Thermal expansion strain
• Time domain
• Finite element