The maximal advance path constraint for the homogenization of materials with random network microstructure

Abstract

This paper presents a non-affine homogenization scheme for materials with a random network microstructure. It is based on a newly developed kinematic constraint that links the microscopic deformation of the network to the macroscopic strain of the material. This relation accounts for the network functionality and is established by means of maximal advance paths that are long enough to reach the macroscopic scales of the continuous body and deform accordingly but are also composed of the microscopic fibres that follow the network deformation. The exact distribution of the variable fibre stretch is determined by the principle of minimum averaged free energy, which ultimately allows one to derive the homogenized elastic response of the network at equilibrium. Besides the general formulation, the model is presented in detail for the case of tetrafunctional networks, for which the micro–macro relation and the expression for the homogenized elastic stress are derived in a compact and interpretable tensorial form. The performance of the model as well as the convexity and stability of the obtained homogenized response of the material is examined for networks composed of two different types of fibres, namely flexible chains and stiff filaments. The qualitative behaviour of the networks predicted for the two considered cases agrees with experimentally observed phenomena for soft materials. This includes a consistent explanation for the difference in the stiffness of elastomers at uniaxial and equibiaxial extension as well as a validation of recent experimental investigations of atypical normal stress amplitudes in biopolymer gels under shear loading.

Keywords:

• random networks
• micromechanics
• homogenization
• soft matter
• elastomers
• biopolymers

Acknowledgements

Financial support for this research was provided by the ‘Juniorprofessorprogramm des Landes Baden-Württemberg’ and the German Research Foundation (DFG) within the Cluster of Excellence in Simulation Technology (ExC 310/1) at the University of Stuttgart. This support is gratefully acknowledged.