An implementation of the non-affine micro-sphere model of rubber elasticity in FEAP

The micro-sphere model

The non-affine micro-sphere model of rubber elasticity (C. Miehe, S. Goektepe and F. Lulei, A micro-macro approach to rubber-like materials - Part I: The non-affine micro-sphere model of rubber elasticity, Journal of the Mechanics and Physics of Solids 52 (2004), pp. 2617-2660.) is a microscopically motivated finite deformation model for rubberlike materials. The model combines three special features: Firstly, there is a non-affine stretch part, where micro and macro stretches are linked through a fluctuation field on a micro-sphere. The fluctuation field is determined by a minimization of the macroscopic free energy. Secondly, polymer cross-links and entanglements are also considered by using the so called tube model of rubber elasticity, where the movement of a single chain is restricted by a tube-like constraint. Thirdly, since closed-form solutions to the averaging integrals over a sphere are not available, a 21-point integration scheme (Z.P. Bazant and B.H. Oh, Efficient numerical integration on the surface of a sphere, Z.A.M.M. 66 (1986), pp. 37-49.) is used.

FEAP implementation

The model is implemented in FEAP (Version 8.1) using the user defined material interface. To use the model the following three source files have to be copied into the main folder of FEAP:

  • umati1.f
  • umatl1.f
  • microsphere.f
  • A sample call of the model would look as follows:

      ucon micro 9.31 3.273 0.1475 0.567 9.94 100.0

    where the parameters are defined as mentioned in the microsphere.f source file.

    Non-homogeneous shear example

    The non-homogeneous shear test as mentioned in Miehe et al. (2004) is calculated using the following FEAP input file:

  • Itest3d0
  • The deformed shear block should look like: