## Soft Tissue Models

### Measures of Deformation
and Strain

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 `K`^{0}
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

### Hyperelastic Material Models

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 `I`_{1}, `I`_{2}, `I`_{3}
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

instead.

### Modelling Viscoelasticity

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 `N`_{d}`+1`
is the number of reduced exponential relaxation functions

used to approximate the viscoelastic material
properties. `c`_{i}
and t_{i} are
constants used to adjust the spectrum approximation given by the
above formulation.

The material parameters m_{i}, a, g, and the weighting factors `c`_{i} are summarized in the target parameter vector `p` for the inverse finite element paramter
determination.

### References

**Fung Y.C.**, 1993, *Biomechanics:
Mechanical Properties of Living Tissues*, Springer-Verlag,
New York, Sec. Ed.

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