For Linear Elastic or Nonlinear Elastic materials, each Fiber node adds a contribution to the stress tensor of the type
Here, Efib − E is the difference between the stiffness of the isotropic matrix
E and the fibers’ stiffness
Efib,
vfib is the fiber volume fraction,
a is the direction vector for the fiber orientation in the undeformed geometry, and
is the elastic strain in the fiber direction
a.
For Nonlinear Elastic Materials, the same contribution to the stress tensor is added, although the stress-strain relation in the isotropic matrix could be nonlinear.
For Hyperelastic Materials, the fibers are represented by additional contributions to the strain energy density (
Ref. 28 and
Ref. 29).
Here, Wiso is the strain energy density of the isotropic hyperelastic material. In
Ref. 28 Wiso is represented by a nearly incompressible
Neo-Hookean material; in COMSOL Multiphysics it is possible to select any of the built-in hyperelastic material models.
The second and third terms, Wfib,1 and
Wfib,2 are anisotropic additions that describe the mechanical contribution of collagen fiber networks. These terms contribute with a strain energy density of the type
where the parameter k1 represents the fiber stiffness (SI unit: Pa),
k2 is a dimensionless tuning parameter, and
k3 is the fiber dispersion (dimensionless). A value of
k3 = 0 recovers the formulation in
Ref. 28, where all the fibers are perfectly oriented in the
a direction. Setting
k3 = 1/3 means that the fibers are completely dispersed, thus recovering a similar formulation as given in the
Delfino material model.
The invariant represents the squared value of the isochoric elastic stretch in the fiber direction a. A common modeling assumption for biological tissues is that fibers cannot sustain compression, so the fiber stiffness is added only for tensile stretches such as
.
The anisotropic strain energy density for the fiber family Wfib can also be specified as a user-defined expression of the strain invariants
,
,
,
,
, and
, where
and b and
c are orthogonal directions to the main fiber direction
a.
where Efib − E is the difference between the stiffness of the isotropic matrix
E and the fibers’ stiffness
Efib,
vfib is the fiber volume fraction,
a is the direction vector for the fiber orientation in the undeformed geometry, and
is the elastic strain in the fiber direction
a. When considering the thermal expansion, the stress reads
where εth,fib is the thermal strain in the fibers.
For Nonlinear Elastic Materials, the same contribution to the stress tensor is added, although the stress-strain relation in the isotropic matrix could be nonlinear.
The thermal strain in the fibers, εth,fib, can be specified by different means. When the secant coefficient of thermal expansion is used, it reads
Here, the secant coefficient of thermal expansion αs can be temperature-dependent. The reference temperature
Tref is the temperature at which there are no thermal strains in the fibers, and
Tfib is the current fiber temperature.