Determination of the strain energyfunction using stress-strain response ofa single fascicle for the modeling of ligaments and tendonsMd Asif Arefeen ABSTRACTAreview and analysis of the strain energy function by using the distribution of crimp angles of the fibrils todetermine the stress-strain response of single fascicle. (Kastelic, Palley et al.
1980) gave anon-linear stress-strain relationship based on the radial variation of thefibril crimp. By correcting this relationship TomShearer derived a new strain energy function and compared it with the commonlyused model HGO. The relative and absolute errors related to the new model areless than 9% and 40% than of that HGO model. Undoubtedlynew model gives a better performance than the HGO model. Butitis mandatory to measure the and o separately forthe ligament or tendon in order to validate this model.
1. IntroductionA fascicleis the main subunit of the ligaments and tendons which are the soft collagenoustissue. These tissues are the fundamentalstructures of in the musculoskeletal systems and play an essential role in biomechanics.Ligaments provide stability and also make the joints work perfectly byconnecting bone to bone, on the other hand, tendons transfer force to a skeleton which is generated by muscle byconnecting bone to muscle. The collagenous fiberslike fascicle consist of crimped patternfibrils and this crimp are called the wavinessof the fibrils(see fig.
1) which provides the non-linear stress-strain responsefor ligaments and tendons.As an anisotropic tissue, the characteristic ofstress-strain of ligaments and tendons within a non-linear elastic frameworkoccur 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 anexponential stress-strain relationship based on rabbitmesentery which was only in a phenomenologicalsense but there was no microstructural basis for the choice of theexponential function.
Based on his work (Gou 1970) proposed a strain energy function for isotropic tissuesthat also gave an exponential stress-strain relationship but was not suitablefor tissue like tendons and ligaments. (Kastelic, Palley et al. 1980) gave anon-linear stress-strain relationship based on the radial variation of the fibril crimp. But there was an error in theimplementation of the Hook’s law which leadshis relationship incorrect. The strain energy function which has used for modeling biological tissue for along time is Holzapfel-grasser-Ogden (HGO) model,given byW= (I1-3) + ( -1), where, I1= trC, I4=M.
(CM), C= I1and I4 are the strain invariants where I4 has adirect interpretation as the square of the stretch in the direction of the fiber.More explanations about invariants can be foundin the (Holzapfel et al. 2010).”C is the right Cauchy-Green tensor, F is thedeformation gradient tensor and M is a unit vector pointing in the direction ofthe tissue’s fibers before any deformation has taken place, c, k1andk1 are material parameters and the above expression is only validwhen I4?1(when I4>1,W = (I1-3)). As aphenomenological model, the parameters are not directly linked to measurablequantities”.
So this model has some limitations.A large number SEF model has been proposed so far bydifferent researchers like( Humphrey and Lin 1987),(Humphrey et al.1990), (Fung et al. 1993),( Taber 2004), (Murphy2013) but none of them were valid for ligaments and tendons.In 2014 Tom Shearerproposed a model by correcting the work done by Kastelic based on the fibrilcrimp angle.This new model is more efficient than the HGO model.2.Development of newstress-strain relationshipA new stress-strain response has given by the Tom Shearerbased on the radial variation in the crimp angle of a fascicle’s fibrils bycorrecting the Hook’s law in that paper.
The Hook’s law stated by Kastelic etal.(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 isdependent on ?.
All fibrils should have same Young’s modulus.So E* is not validfor all ?.New Hook’s law was given by Tom Shearer in his paper whichcan 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 theaverage traction in the direction of the fascicle = 2 WherePp is the tensile load faced by the fascicle.
Taking p=1,2 andsimplifying few things Tom Shearer derived a new stress-strain relationship which is given by = (2?-1+ ) = ( ? +1)-1, ?= = E(??-1), ?> TomShearer used this form to derive the new strain energy function.3. Strain Energy FunctionInthis section, a derived strain energy function will be shown for the ligamentsand tendons. For the details, the readeris referred to Tom Shearer (2014).
His strain energy function is valid for both ofthe isotropic and anisotropic tissue.Foranisotropic tissue SEFW= (4 I4 -3log (I4)- -3)”Theneo-Hookean model is still reasonable for isotropictissue”. Based on this an isotropic SEFcan be derivedW=(1-?) (I2-3) Nowfull form of strain energy function can be given asW=(1-?) (I2-3) + (4 I4 -3log(I4)- -3), I4 W=(1-?) (I2-3) + (? I4 – log(I4)+?), I4 Where isthe collagen volume fraction, E is thefibril 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 seemsquite unusual for isotropic SEF but it happens due to the inability of the linear term in their stress-strain relationshipfor small strains of fascicles.4.
ResultInthis section, a comparison of the stress-strain relationship among Tom Shearer’smodel, HGO model, an experimental model will be shown. The existing data were takenfrom 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 ligamentand tendon matrix is negligible 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 apredicted value. Based on this Tom Shearer measured the stress-strain responsewhich is given below Fig4: Comparison stress-strain curves ofthe new model and HGO model withexperimental data.
Black: new model, Blue: HGO model, Red: experimental data. From the above graph,an average relative error and absoluteerror among the model can be calculated.Calculation of the Tom Shearer suggested that average relative error andabsolute error of new model is less than the HGO model respectively 0.052 (TomShearer’s model) <0.59(HGO) and 0.13MPa (Tom Shearer) < 0.33 MPa (HGO). 5.
ConclusionUndoubtedly Tom Shearer’s model gives a betterperformance than the HGO model. But after reviewing and analyzing different kinds of literature it is mandatory to measurethe and o separately for the ligamentor tendon in order to authenticate this model. 6.
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