Morphology of residually stressed tubular tissues: Beyond the elastic multiplicative decomposition

Abstract

Many interesting shapes appearing in the biological world are formed by the onset of mechanical instability. In this work we consider how the build-up of residual stress can cause a solid to buckle. In all past studies a fictitious (virtual) stress-free state was required to calculate the residual stress. In contrast, we use a model which is simple and allows the prescription of any residual stress field. We specialize the analysis to an elastic tube subject to a two-dimensional residual stress, and find that incremental wrinkles can appear on its inner or its outer face, depending on the location of the highest value of the residual hoop stress. We further validate the predictions of the incremental theory with finite element simulations, which allow us to go beyond this threshold and predict the shape, number and amplitude of the resulting creases.

On the existence of elastic minimizers for initially stressed materials

“…In the next sections, we prove the local existence of a relaxed state around each material point and a theorem on the existence of elastic minimizers for a strain energy of the form given by (13).…”

“…Under the incompressibility constraint, it has been shown that only eight invariants are independent [59]. This method has been widely used to model initially stressed materials; applications of this theory include wave propagation in soft media [58,41], the modeling of residual stress in living tissues [66] and the stability of residually stressed materials [13,56,46]. The main advantage of this approach is that the initial stress tensor Σ belongs to the group of the divergence-free symmetric tensors satisfying the boundary condition in the given reference configuration, whilst it is still unclear which physical restrictions must be imposed for the well-posedness of the elastic problem.…”

Morpho-elastic model of the tortuous tumour vessels

“…The incremental form of the balance of the linear momentum and of the incompressibility constraint are given by (16) div δP 0 = 0, in Ω,…”

“…We denote with u, v and w the components of δu in cylindrical coordinates.To reduce the system of partial differential equations (16) to a system of ordinary differential equations, we assume the following ansatz [28]:…”

“…Subsequently O'Keeffe et al [33] addressed the problem of the stability of a solid cylinder growing along the axial direction, embedded into an elastic inert matrix and confined between two parallel, rigid planes. The stability of residually stressed cylindrical structures has been further studied by exploiting an alternative approach, prescribing the residual stress field instead of the growth tensor [29,16].…”