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

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.

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:

A sample call of the model would look as follows:

mate,1solid

finite

mixed

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.

The deformed shear block should look like: