*International Journal of Solids and Structures, 41 (2004), 511-557.*
## A FRAMEWORK FOR MICROPLANE MODELS AT LARGE STRAIN, WITH APPLICATION TO
HYPERELASTICITY

Ignacio Carol, Milan Jirásek
and Zdeněk P. Bažant

### Abstract

A general framework is proposed for the formulation of microplane models
at large strain. It is based on the thermodynamic approach to microplane formulation
recently presented by the authors, which defines the macroscopic free energy
of the material as an integral of a microplane free-energy potential over
all possible orientations. By simple differentiation with respect to strain,
it is possible to obtain the consistent definition of microplane stresses
and integral expressions for evaluation of the macroscopic stress tensor.
To apply this approach to large strains, new microplane strain measures need
to be defined, including volume change, stretch of fibers, ``thickening''
of planes, deviatoric parts of the stretch and thickening, and distortion
(shear) angles. Based on these, various microplane formulations are developed.
Each formulation starts with the definition of microplane stresses and the
derivation of the integral expressions which are valid for the general case
of dissipative materials. Then, these expressions are particularized to specific
forms of hyperelastic potentials leading to various hyperelastic models. The
simplest model, with a quadratic microplane potential in terms of the fiber
stretch, corresponds to the classical Gaussian statistical theory of long-chain
molecules and leads to the neo-Hookean type of macroscopic free-energy potential.
Many other, more complex forms of the microplane potential are investigated
and their relation to existing models for rubber elasticity is analyzed.
It is shown that, in the small-strain limit, they collapse into well-known
small-strain microplane formulations, either with restricted or with unrestricted
values of Poisson's ratio.

EPFL / 1 March 2004 / milan.jirasek@epfl.ch