In non-linear continuum mechanics, in contrast to linear continuum mechanics, a reference configuration and a deformed configuration of the considered continuum are distinguished. The reference configuration is denoted by K0 and the actual deformed configuration is denoted by K. The actual configuration and the reference configuration are related by a displacement vector u(X), where X is the position vector in the undeformed or reference configuration.
We define the deformation gradient F, the Green-Lagrange deformation tensor E and the right Cauchy-Green deformation tensor C by
The deformation gradient F is related to the displacement vector u(X) by
The infinitesimal volume element of dv in the deformed configuration and the corresponding volume element dV in the reference configuration of a material undergoing a deformation are related by
For the soft tissue models we use the concept of hyperelasticity, i.e. we define a strain energy function W depending on the three invariants I1, I2, I3 of the right Cauchy-Green deformation tensor C
The second Piola-Kirchhoff stress tensor S is obtained from the strain energy function by
We use the following strain energy functions in this study
Since the water content of soft biological tissues is very high tissues can be modelled as incompressible materials. From a numerical point of view it is better not to require
(which states that the material is totally incompressible) but to require
To model the viscoelastic material properties of soft biological tissues a quasi-linear viscoelastic model is used, i.e. the stresses in the material are superimposed linearly as regarding their time history.
where Nd+1 is the number of reduced exponential relaxation functions
used to approximate the viscoelastic material properties. ci and ti are constants used to adjust the spectrum approximation given by the above formulation.
The material parameters mi, a, g, and the weighting factors ci are summarized in the target parameter vector p for the inverse finite element paramter determination.
Fung Y.C., 1993, Biomechanics: Mechanical Properties of Living Tissues, Springer-Verlag, New York, Sec. Ed.
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