Determination of the strain energy

function using stress-strain response of

a single fascicle for the modeling of ligaments and tendons

Md Asif Arefeen

ABSTRACT

A

review and analysis of the strain energy function by using the distribution of crimp angles of the fibrils to

determine the stress-strain response of single fascicle. (Kastelic, Palley et al. 1980) gave a

non-linear stress-strain relationship based on the radial variation of the

fibril crimp. By correcting this relationship Tom

Shearer derived a new strain energy function and compared it with the commonly

used model HGO. The relative and absolute errors related to the new model are

less than 9% and 40% than of that HGO model. Undoubtedly

new model gives a better performance than the HGO model. But

it

is mandatory to measure the

and

o separately for

the ligament or tendon in order to authenticate this model.

1. Introduction

A fascicle

is the main subunit of the ligaments and tendons which are the soft collagenous

tissue. These tissues are the fundamental

structures of in the musculoskeletal systems and play a significant role in biomechanics.

Ligaments provide stability and also make the joints work perfectly by

connecting bone to bone, on the other hand, tendons transfer force to a skeleton which is generated by muscle by

connecting bone to muscle. The collagenous fibers

like fascicle consist of crimped pattern

fibrils and this crimp are called the waviness

of the fibrils(see fig.1) which provides the non-linear stress-strain response

for ligaments and tendons.As an anisotropic tissue, the characteristic of

stress-strain of ligaments and tendons within a non-linear elastic framework

occur in the toe region where mechanically loading up to 2% strain(see fig.2).

Fig 1. Tendon hierarchy

Fig 2. Model within a non-linear framework

(Fung 1967) gave an

exponential stress-strain relationship based on rabbit

mesentery which was only in a phenomenological

sense but there was no microstructural basis for the choice of the

exponential function. Based on his work (Gou 1970) proposed a strain energy function for isotropic tissues

that also gave an exponential stress-strain relationship but was not valid for

tissue like tendons and ligaments. (Kastelic, Palley et al. 1980) gave a

non-linear stress-strain relationship based on the radial variation of the fibril crimp. But there was an error in the

Hook’s law which leads his relationship incorrect.

The strain energy function which has used

for modeling biological tissue for a long time is Holzapfel-grasser-Ogden (HGO)

model, given by

W

=

(I1-3) +

(

-1), where, I1= trC, I4=

M. (CM), C=

I1

and I4 are the strain invariants where I4 has a

direct interpretation as the square of the stretch in the direction of the fiber.More explanations about invariants can be found

in the (Holzapfel et al. 2010).”C is the right Cauchy-Green tensor, F is the

deformation gradient tensor and M is a unit vector pointing in the direction of

the tissue’s fibers before any deformation has taken place, c, k1and

k1 are material parameters and the above expression is only valid

when I4?1(when I4>1,

W =

(I1-3)). As a

phenomenological model, the parameters are not directly linked to measurable

quantities”.So this model has some limitations.

A large number SEF model has been proposed so far by

different researchers like( Humphrey and Lin 1987),(

Humphrey et al.1990), (Fung et al. 1993),( Taber 2004), (Murphy

2013) but none of them were valid for ligaments and tendons.In 2014 Tom Shearer

proposed a model by correcting the work done by Kastelic based on the fibril

crimp angle.This new model is more efficient than the HGO model.

2.Development of new

stress-strain relationship

A new stress-strain response has given by the Tom Shearer

based on the radial variation in the crimp angle of a fascicle’s fibrils by

correcting the Hook’s law in that paper.The Hook’s law stated by Kastelic et

al.(1980) is given by

?p(?)=E*. ??p (?), where ??p (?)= ? – ?p (?)

Here ??p (?)(elastic-deformation)

is not the fibril strain and differs from the fibril strain by a quantity that is

dependent on ?.All fibrils should have same Young’s modulus.So E* is not valid

for all ?.New Hook’s law was given by Tom Shearer in his paper which

can be derived from the figure-3 below.

?p(?)=E.

(?)

(1)

where

(?)=

cos(

( ? – ?p (?))= ( ? +1) cos(

-1= ( ? +1) cos(

-1

Fig 3: Stretching of fibril of initial length lp(?)

within a fascicle of initial length L

Using the equation (1) he derived an expression for the

average traction in the direction of the fascicle

= 2

Where

Pp is the tensile load faced by the fascicle. Taking p=1, 2 and

simplifying few things Tom Shearer derived a new stress-strain relationship which is given by

=

(2?-

1+

)

=

( ? +1)-1, ?=

= E(??-1), ?>

Tom

Shearer used this form to derive the new strain energy function.

3. Strain Energy Function

In

this section, a derived strain energy function will be shown for the ligaments

and tendons. For the details, the reader

is referred to Tom Shearer (2014).His strain energy function is valid for both of

the isotropic and anisotropic tissue.

For

anisotropic tissue SEF

W=

(4

I4 -3log (I4)-

-3)

“The

neo-Hookean model is still reasonable for isotropic

tissue”. Based on this an isotropic SEF

can be derived

W=

(1-?)

(I2-3)

Now

full form of strain energy function can be given as

W=

(1-?)

(I2-3) +

(4

I4 -3log(I4)-

-3),

I4

W=

(1-?)

(I2-3) +

(?

I4 –

log(I4)+?), I4

Where

is

the collagen volume fraction, E is the

fibril stiffness and

is the average out fibril crimp angle. Here

cannot be measured directly. As a result, it was taken based on assumptions.

Finally, the above SEF gives stress-strain response for both isotropic and anisotropic tissues. It seems

quite unusual for isotropic SEF but it happens due to the inability of the linear term in their stress-strain relationship

for small strains of fascicles.

4. Result

In

this section, a comparison of the stress-strain relationship among Tom Shearer’s model, HGO model, an

experimental model will be shown. The existing data were taken from the (Johnson, Tramaglini et al. 1994), Parameter values: c=(1-?)

=0.01MPa, k1=25MPa, k2=183MPa,

=552 MPa,

=0.19 rad=10.7?.As stiffness of ligament

and tendon matrix is insignificant compared with that of its fascicles, (1-?)

were chosen to be small,

cannot be measured directly , it was taken based on assumptions like 0.11

1. Also

was not available so it was taken as a

predicted value. Based on this Tom Shearer measured the stress-strain response

which is given below

Fig

4: Comparison stress-strain curves of

the new model and HGO model with

experimental data. Black: new model, Blue: HGO model, Red: experimental data.

From the above graph,

an average relative error and absolute

error can be calculated. Calculation of

the Tom Shearer suggested that average relative error and absolute error of new

model is less than the HGO model respectively 0.051 (Tom Shearer model) <0.59(HGO)
and 0.13MPa (Tom Shearer) < 0.35 MPa (HGO).
5. Conclusion
Undoubtedly Tom Shearer's model gives a better
performance than the HGO model. But after reviewing and analyzing different kinds of literature it is mandatory to measure
the
and
o separately for the
ligament or tendon in order to validate this model.
6. Reference
Fung, Y. C. (1967). "Elasticity of
soft tissues in simple elongation." Am J Physiol 213(6): 1532-1544.
Gou, P. F. (1970). "Strain energy function
for biological tissues." J Biomech 3(6): 547-550.
Johnson, G. A., et al. (1994). "Tensile and
viscoelastic properties of human patellar tendon." J Orthop Res 12(6): 796-803.
Kastelic, J., et al. (1980). "A structural
mechanical model for tendon crimping." J Biomech 13(10): 887-893.
Johnson, G.A., Rajagopal, K.R., Woo, S.L-Y.,1992."A
single integral finite strain(SIFS) model of ligaments and tendons".Adv.Bioeng.22,245–248.
Holzapfel, G.A., Gasser, T.C., Ogden, R.W.,2000."A
new constitutive framework for arterial wall mechanics and a comparative study
of material models". J.Elast.61,1–48
Shearer, T., Rawson, S., Castro, S.J., Ballint,
R., Bradley, R.S., Lowe, T., Vila-ComamalaJ., Lee, P.D., Cartmell, S.H., 2014."
X-ray computed tomography of the anterior cruciate ligament and patellar
tendon. Muscles Ligaments Tendons". J.4, 238–244.