/*
 * Exponential Constitutive Material Law
 * 
 * Model Status
 * 
 * This model contains no ODEs so cannot be solved in available
 * software (OpenCell or COR). However the model is still valid
 * CellML with full unit consistency. The model is intended to
 * interface with a CMISS model at a larger spatial scale.
 * 
 * Model Structure
 * 
 * This exponential material law for isotropic soft biological
 * material was first proposed by H. Demiray in 1981 (see full
 * publication details below). From the stochastic models for the
 * elastic behaviour of biological tissues proposed by Soong and
 * Huang (A Stochastic Model for Biological Tissue Elasticity in
 * Simple Elongation, Journal of Biomechanics, 6, p451-458, 1973)
 * and Lanir (A Structural Theory for the Homogenous Biaxial Stress-Strain
 * Relationships in Flat Collagenous Tissues, Journal of Biomechanics,
 * 12, p423-436, 1979), Demiray observed the form of the strain
 * energy density function to be an exponential function of the
 * first invariant (trace) of the right Cauchy-Green deformation
 * tensor, I1. This argument is further supported by Fung's statement
 * that the strain energy function of soft biological tissues in
 * a one-dimensional case is an exponential function of the stretch
 * ratio (Elasticity of Soft Tissues in Simple Elongation, American
 * Journal of Physiology, 213, p1532-1544, 1967).
 * 
 * Demiray assumed that biological materials were elastic, homogenous,
 * isotropic and incompressible. Like the Neo-Hookean material
 * law (a special case of the The Mooney-Rivlin Constitutive Material
 * Law, 1951, this exponential constitutive law is a function of
 * only the axial Green-Lagrange strain components, and hence it
 * does not penalise shear deformations.
 * 
 * The complete original paper reference is cited below:
 * 
 * Large deformation analysis of some soft biological tissues,
 * H. Demiray, 1981, Journal of Biomechanical Engineering ,103,
 * 73-78. (Unfortunately this article doesn't appear to be available
 * as an online version). PubMed ID: 7278185
 */

import nsrunit;
unit conversion on;
unit strain=1  dimensionless;
unit stress=1  dimensionless;
unit pole=1  dimensionless;
unit curvature=1  dimensionless;
unit scale=1  dimensionless;

math main {
	//Warning:  the following variables were set 'extern' or given
	//  an initial value of '0' because the model would otherwise be
	//  underdetermined:  E11, E22, E33, E12, E13, E23
	extern real E11 strain;
	extern real E22 strain;
	extern real E33 strain;
	extern real E12 strain;
	extern real E13 strain;
	extern real E23 strain;
	real c1 strain;
	c1=0;
	real c2 strain;
	c2=0;
	real Tdev11 stress;
	real Tdev22 stress;
	real Tdev33 stress;
	real Tdev12 stress;
	real Tdev13 stress;
	real Tdev23 stress;

	// <component name="interface">

	// <component name="equations">
	Tdev11=(2*c1*c2*exp(2*c2*(E11+E22+E33)));
	Tdev22=(2*c1*c2*exp(2*c2*(E11+E22+E33)));
	Tdev33=(2*c1*c2*exp(2*c2*(E11+E22+E33)));
	Tdev12=(0*E12);
	Tdev13=(0*E13);
	Tdev23=(0*E23);
}