Anisotropic residual stresses in arteries

Abstract

The paper provides a deepened insight into the role of anisotropy in the analysis of residual stresses in arteries. Residual deformations are modelled following Holzapfel and Ogden (Holzapfel and Ogden 2010, J. R. Soc. Interface 7 , 787–799. ( doi:10.1098/rsif.2009.0357 )), which is based on extensive experimental data on human abdominal aortas (Holzapfel et al. 2007, Ann. Biomed. Eng. 35 , 530–545. ( doi:10.1007/s10439-006-9252-z )) and accounts for both circumferential and axial residual deformations of the individual layers of arteries—intima, media and adventitia. Each layer exhibits distinctive nonlinear and anisotropic mechanical behaviour originating from its unique microstructure; therefore, we use the most general form of strain-energy function (Holzapfel et al. 2015, J. R. Soc. Interface 12 , 20150188. ( doi:10.1098/rsif.2015.0188 )) to derive residual stresses for each layer individually. Finally, the systematic experimental data (Niestrawska et al. 2016, J. R. Soc. Interface 13 , 20160620. ( doi:10.1098/rsif.2016.0620 )) on both mechanical and structural properties of the different layers of the human abdominal aorta facilitate our discussion on (i) the importance of anisotropy in modelling residual stresses; (ii) the variability of residual stresses within the same class of tissue, the abdominal aorta; (iii) the limitations of conventional opening angle method to account for complex residual deformations; and (iv) the effect of residual stresses on the loaded configuration of the aorta mimicking in vivo conditions.

Multi-sector approximation method for arteries: the residual stresses of circumferential rings with non-trivial openings

“…Here θ (i) is the conveniently chosen subsidiary circumferential coordinate used to locate the position of the deformed sectors i = 1 … N. When i = 1, the sector is located at u [ [0, 2a (1) d ]; when i = 2, it is located at u [ [2a (1) d ; 2a (1) d þ 2a (2) d ], etc. Clearly, the following condition must hold:…”

“…Owing to the absence of mechanical data for the individual sectors, we have to assume that the material along the circumference of the aorta is homogeneous, i.e. μ (1) = μ (2) = μ (3) = μ. Therefore, our choice of strain energy density functions (3.1) makes our solution independent of the shear modulus μ, which may thus be removed from the analysis by way of non-dimensionalization, i.e.…”

“…Hence, in our numerical simulations we have to assume that the undeformed sectors opened asymmetrically either because of variations in wall thickness H (i) or because of differences in residual axial deformations l (i) z (i = 1, 2, 3). In the first scenario, we would have to assume that the l (i) z are the same for all sectors (l (1) z ¼ l (2) z ¼ l (3) z ¼ l z , say); in the second scenario, that the H (i) are the same for all three sectors (H (1) = H (2) = H (3) = H, say). One may argue that, in general, the l (i) z are harder to measure than the H (i) .…”

“…In particular in arteries, residual stresses are known to optimize the distribution of transmural stresses due to internal pressure in order to achieve better functionality [1,2]. The conventional experimental approach for detecting and evaluating residual stress in arteries is the opening angle method, where thin circumferential rings from an artery are cut axially/radially and open into single sectors, revealing that the rings were under stress; see [3,4] and figure 1a.…”

“…An estimation of the residual stresses based on the experimental observations of residual deformations can be done numerically with finite-element simulations and analytically using the methodology of large deformations universal to all nonlinear incompressible isotropic materials [11]. The models used to estimate the residual stress vary from the simplest homogeneous, isotropic ones to more advanced inhomogeneous, anisotropic ones that account for each layer's mechanics, microstructure and both circumferential and axial residual deformations [2]. Our focus in this work is the analytical approach.…”