Exponential Constitutive Material Law
JaeHoon
Chung
Auckland Bioengineering Institute, The University of Auckland
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
$\mathrm{Tdev11}=2.0\mathrm{c1}\mathrm{c2}e^{2.0\mathrm{c2}(\mathrm{E11}+\mathrm{E22}+\mathrm{E33})}$
$\mathrm{Tdev22}=2.0\mathrm{c1}\mathrm{c2}e^{2.0\mathrm{c2}(\mathrm{E11}+\mathrm{E22}+\mathrm{E33})}$
$\mathrm{Tdev33}=2.0\mathrm{c1}\mathrm{c2}e^{2.0\mathrm{c2}(\mathrm{E11}+\mathrm{E22}+\mathrm{E33})}$
$\mathrm{Tdev12}=0.0\mathrm{E12}$
$\mathrm{Tdev13}=0.0\mathrm{E13}$
$\mathrm{Tdev23}=0.0\mathrm{E23}$
7278185
In this simple model we only have one component, which holds the
six equations.
The University of Auckland
Auckland Bioengineering Institute
1981
H
Demiray
The exponential constitutive material law
keyword
biological tissues
deformation
mechanical constitutive laws
keyword
Journal of Biomechanical Engineering
Large deformation analysis of some soft biological tissues
103
73
78
We'll use this component as the "interface" to the model, all
other components are hidden via encapsulation in this component.
2004-11-10
This is a CellML version of the Exponential constitutive material law,
defining the relation between the eight independent strain components
and the stress components. It is assumed that the strain components
will be controlled externally by the application using this CellML
model.
JaeHoon
Chung
jh.chung@auckland.ac.nz