Model reduction by mean-field homogenization in viscoelastic composites. I. Primal theory

Area of Science:

• Continuum Mechanics
• Materials Science
• Computational Mechanics

Background:

• The Lahellec & Suquet (2007) homogenization scheme for viscoelastic composites uses an incremental variational formulation and Legendre transforms.
• This scheme approximates the composite's microscopic state using internal variables based on strain moments.

Purpose of the Study:

• To revisit and provide an alternative derivation of the Lahellec & Suquet homogenization scheme.
• To expose the mathematical structure of the approximation.
• To generalize the scheme to fully anisotropic material systems.

Main Methods:

• Revisiting the incremental variational formulation of the homogenization scheme.
• Employing the Cauchy-Schwarz inequality as an alternative to Legendre transforms.
• Analyzing the mathematical structure of the resulting approximation.

Main Results:

• An alternative derivation of the homogenization scheme is presented.
• The mathematical underpinnings of the approximation are clarified.
• The scheme is generalized to accommodate fully anisotropic viscoelastic composites.

Conclusions:

• The Cauchy-Schwarz inequality provides a robust alternative for deriving the homogenization scheme.
• The revised scheme offers a more general framework for analyzing viscoelastic composites.
• This work enhances the understanding and applicability of mean-field homogenization techniques for complex materials.