Modelling of residually stressed materials with application to AAA

Abstract

Residual stresses are generated in living tissues by processes of growth and adaptation and they significantly influence the mechanical behaviour of the tissues. Thus, to effectively model the elastic response of the tissues relative to a residually stressed configuration the residual stresses need to be incorporated into the constitutive equations. The purposes of this paper are (a) to summarise a general elastic constitutive formulation that includes residual stress, (b) to specify the tensors needed for the three-dimensional implementation of the theory in a nonlinear finite element code, and (c) to use the theory and its implementation to evaluate the wall stress distribution in an abdominal aortic aneurysm (AAA) using patient specific geometry and material model parameters. The considered material is anisotropic with two preferred directions indicating the orientation of the collagen fibres in the aortic tissue. The method described in this paper is general and can be used, by specifying appropriate energy functions, to investigate other residually stressed biological systems.

Pulsatile flow measurements and wall stress distribution in a patient specific abdominal aortic aneurysm phantom

“…Following established practice, see for example, [], the von Mises stress has been used to characterize the stress distributions in the walls of AAAs, and here we also adopt this as our stress measure, which we denote by σ vm . The wall stress distribution on the inner and outer surfaces of the AAA wall is then visualized on the geometry of the reference configuration.…”

“…The three‐dimensional incremental formulation appropriate for the implementation of the constitutive equations in a nonlinear finite element code, including the elasticity tensor, was developed in []. Here we summarize the most important characteristics.…”

“…The finite element layout was the same as in [] and used 63,654 hexahedral elements connected at 127,926 nodes, with four elements through the wall thickness. The traction boundary conditions at the inner surface were given by the pressure measurements on the geometrically equivalent phantom, as described in Section 3.…”

“…To define the ‘corresponding’ orientations in the AAA model, we followed the method developed in []. Briefly, the tensor generates images of and projected onto the AAA wall surface defined by the local normal unit vector .…”

“…Location of the piecewise linear center line, which is used to define a local cylindrical coordinate system with unit basis vectors []…”