Abstract
A new anisotropic viscoelastic model is developed for application to the aortic valve (AV). The directional dependency in the mechanical properties of the valve, arising from the predominantly circumferential alignment of collagen fibres, is accounted for in the form of transverse isotropy. The rate dependency of the valve's mechanical behaviour is considered to stem from the viscous (η) dissipative effects of the AV matrix, and is incorporated as an explicit function of the deformation rate (). Model (material) parameters were determined from uniaxial tensile deformation tests of porcine AV specimens at various deformation rates, by fitting the model to each experimental dataset. It is shown that the model provides an excellent fit to the experimental data across all different rates and satisfies the condition of strict local convexity. Based on the fitting results, a nonlinear relationship between η and is established, highlighting a ‘shear-thinning’ behaviour for the AV with increase in the deformation rate. Using the model and these outcomes, the stress–deformation curves of the AV tissue under physiological deformation rates in both the circumferential and radial directions are predicted and presented. To verify the predictive capabilities of the model, the stress–deformation curves of AV specimens at an intermediate deformation rate were estimated and validated against the experimental data at that rate, showing an excellent agreement. While the model is primarily developed for application to the AV, it may be applied without the loss of generality to other collagenous soft tissues possessing a similar structure, with a single preferred direction of embedded collagen fibres.
Highlights
Based on the abovementioned attributes, the mechanical behaviour of aortic valve (AV) tissue may be broadly classified as ‘anisotropic viscoelastic’, to reflect both directional and rate dependency of the mechanical properties of the tissue
Most mathematical continuum-based AV models developed to date have been derived under the assumption of hyperelasticity [7,12,13,14,15], and in spite of providing a good fit to experimental stress–strain data obtained at any specific deformation rate, such models cannot, by definition, account for rate effects, nor model AV mechanics over a range of rate-dependent loading conditions
Parameters α, β, k1 and k2 are the ‘elastic’ parameters and by definition are independent of the deformation rate; η1 and η2 are the parameters related to the viscous behaviour of the continuum, and are rate-dependent
Summary
Following Pioletti et al [22], the second Piola–Kirchhoff stress tensor S for a viscoelastic material undergoing large deformations, with strain rate as an explicit variable, may be expressed as. Where C is the right Cauchy–Green tensor, which is related to the deformation gradient tensor F by. We note that Cis the time derivative of C. In the presence of viscous effects, the stress tensor S in equation (2.1) may be derived [22] as. Where We and Wv are referred to as the elastic strain and the viscous dissipation energy functions, respectively. Where p is the arbitrary Lagrange multiplier, enforcing the constraint of incompressibility
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