Abstract
The present paper proposes a new Strain Energy Function (SEF) for modeling incompressible orthotropic hyperelastic materials with a specific application to the mechanical response of passive ventricular myocardium. In order to build our SEF, we have followed a classical strategy based on exponential functions, but we have chosen to work with polyconvex invariants instead of the standard ones. Actually, in the context of hyperelastic problems, the polyconvexity of the strain energy density is considered as a prerequisite for ensuring the existence of solutions. By selecting a set of polyconvex invariants, we demonstrate that our model can predict the experimental data with 6 different shear modes applied to passive ventricular myocardium.
Highlights
Understanding the behavior of anisotropic hyperelastic materials is of major importance for scientists because their modeling has a wide range of applications in engineering biosciences such as in health therapeutic, medical prosthesis, ergonomics or virtual surgery
This replacement provides consistent numerical results with [2] and our model can perfectly match the experimental data obtained by Dokos et al [1] with 6 different shear modes applied to passive ventricular myocardium
Beyond working with polyconvex invariants, the interest of Eq (30) is to account for the invariant I4n = ⟨Cn0, n0⟩ and to distinguish explicitly the effect of the stretch in the myocyte axis direction f0, in the direction s0 lying within the muscle layer and transverse to f0, and in the direction n0 normal to the muscle layer
Summary
Understanding the behavior of anisotropic hyperelastic materials is of major importance for scientists because their modeling has a wide range of applications in engineering biosciences such as in health therapeutic, medical prosthesis, ergonomics or virtual surgery. The mechanical study of the shear deformation of myocardial layers is for example useful because these deformations are considered to play an important role in the mechanical behavior of the heart [1] These past ten years, many works have been performed to investigate the structurally based model originally proposed by Holzapfel and Ogden in [2] in relation to the tests carried out in [1] where the orthotropic nature of the ventricular myocardium has been proven. Working with the set of polyconvex invariants exhibited in [10] allows to replace the classical mixed invariant I8, which is non polyconvex (the proof is in Section 4), by the polyconvex invariant L4 defined by Eq (10) This replacement provides consistent numerical results with [2] and our model can perfectly match the experimental data obtained by Dokos et al [1] with 6 different shear modes applied to passive ventricular myocardium. The standard Euclidean inner product ⟨., .⟩ in a n vector space dimension, its related norm ‖.‖, and the product .⊗. between two vectors a and b, are respectively defined by:
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