Modeling Ligaments as Composite Materials
Keywords:
Composites, rule of mixture, Halpin Tsai equations, Eshelby method, ACL ligament, elastic modulusAbstract
The anterior cruciate ligament (ACL) is one of the four major ligaments of the human knee. It was studied to compute its mechanical properties in 2D and 3D as a composite. ACL ligament structure was studied based on equal stress in the fiber and matrix (Rule of Mixture) and Halpin Tsai model treatment and its effects when modeled as a laminate. The Eshelby method was employed for the calculation of expressions for stresses and strains in composites, and hence their elastic constants. The study looked at how differently it behaved when loading at an angle (Φ) to the fiber axis to understand the way composite behaves when loaded uniaxially and when loaded in other directions. Assuming that there is a perfect bonding between the fiber and matrix at the interface, the ACL has similar behavior to a transversely isotropic composite. Rule of Mixture and Halpin Tsai model was carried out. A value of unity was selected for the adjustable parameter (ξ=1), [1]. The Eshelby method was utilized to compute the Esehlby tensor and the stiffness matrix. The study procedure starts with knowing the Young’s modulus and the Poisson ratio of the ground matrix and the fiber [2,3], then the behavior of the Young’s modulus of the composite in axial and transverse directions were computed. Besides elastic modulus, Poisson ratio and the shear modulus for a varying fiber volume fraction were also computed. For all additional calculations, a fiber volume fraction of 50% was assumed and computation was performed. The stiffness tensor Cpq and the compliance matrix Spq were appropriately computed for 2D and 3D cases for both Rule of Mixture and Halpin-Tsai models. From the components of the Eshelby tensor matrix, calculations were done to obtain the Eshelby tensor for the composite. It was observed that Rule of Mixture and Halpin-Tsai models followed the same trend with a slight offset in the data. The Eshelby matrices and the stiffness matrix based on Eshelby calculation are presented. This study gives us the opportunity to know how the ACL ligaments elasticity behaves when modeled as a composite comprised of the ground matrix and unidirectional continuous collagen fibers. Results and plots obtained for various parameters against the loading angle can give an estimate of the trend consistent with the change in the loading angle. Variations in the three methods for five transversely isotropic elastic parameters are presented in this study.
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