Multiscale Modeling of Ligaments and Tendons - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 116

defines the Cauchy stress, r, to be the inner product of the engineering strain, e,
and the fourth order elasticity tensor, : e. Due to the inherent

nonlinearity of ligament and tendon tissue, strain energy approaches (referred to as
hyperelasticity) based on the invariants of the deformation tensor (I 1 , I 2 , I 3 , I 4 , I 5 ),
are commonly utilized. Such an approach is particularly attractive because it
automatically satisfies a number of constraints, such that the formulation will be
objective (i.e., invariant to rigid body rotation and displacement) and the tangent
elasticity tensor (i.e., the linearization) will be positive definite for a polyconvex
strain energy function [ 108 ]. In this approach a scalar strain energy function (W)is
defined which is typically (but not necessarily) a function of the strain invariants.
The Cauchy stress tensor is computed by taking the derivative of the strain energy
function with respect to the right Cauchy deformation tensor C:

C , such that: r ¼ C

F T :

r ¼ 2

J F oW

ð 6 Þ

oC

The fourth order elasticity tensor (necessary for the linearization and sub-
sequent nonlinear analysis in numerical methods) is found by taking the second
derivative:

¼ 4 o 2 W

C ð 7 Þ

oCoC ;

where is the elasticity tensor in the material frame, which is pushed forward to

the spatial frame in most practical implementations.
Hyperelastic, invariant-based, anisotropic continuum models have proved
successful in modeling the macroscale behavior of ligament and tendon [ 233 , 235 ].
One such formulation has been used to model the macroscopic stress-strain
behavior of ligament [ 65 , 75 , 76 , 232 , 233 , 235 , 237 ]:

C

W ¼ W m ð I 1 ; I 2 Þþ W f ðÞþ U ðÞ : ð 8 Þ

Here, W is the total strain energy, W m is the strain energy for the inter-fiber
matrix, W f is the strain energy for the collagen fibers, and U represents a volu-
metric strain energy. I 1 and I 2 are the invariants of the right Cauchy deformation
tensor C, k is the stretch along the fiber direction, and J ¼ det ðÞ is the volume
ratio. The fiber strain energy term (W f ) is defined to capture the toe region and
linear region of the stress-strain curve for ligaments, and to represent the relatively
small compressive stiffness:

< =

0 k 1

k o W f c 2 e c 3 ð k 1 Þ 1 1\k\k

ok ¼

ð 9 Þ

:

: ;

k [ k c 4 k þ c 5

This formulation represents a structurally motivated constitutive model, as it
specifies strain energy terms for the collagen fiber family and the inter-fiber
matrix. Numerical implementation of hyperelastic constitutive models in finite