Determination of the new strain energy function using stress-strain response of a single fascicle for the hyperelastic modeling of ligaments and tendons
Md Asif Arefeen
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 10% and 41% 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 validate this model.
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 contributes significantly to 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 of the tendon 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 suitable 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 implementation of 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
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.
In this section, a comparison of the stress-strain relationship among new 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 among the model 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 0.053 (new model)