Riemann-Cartan Geometry of nonlinear dislocation mechanics

We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold – where the body is stress free – is a Weitzenbock manifold, i.e. a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan’s moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance

Dynamics of mechanically induced fiber reorientation in\ud the material reinforced by two families of fibers

The specific fiber alignment and its content in biological tissues are created and maintained by the cells, which respond to mechanical stimuli arising from properties of the surrounding material. This coupling between mechanical anisotropy and tissue remodeling can be modeled in the theory of nonlinear elasticity as a fiber-reinforced hyperelastic material where the remodeling is represented as the change in the fiber orientation and/or amount. Here, we study analytically a simple model of fiber reorientation in a rectangular elastic tissue reinforced by two symmetrically arranged families of fibers subject to constant external loads. In this model, the fiber direction tends to align with the maximum principal stretch. We characterize the global behaviour of the system for all material parameters and applied loads, and show that provided the fibers are tensile initially, the system converges to a stable equilibrium, which corresponds to either complete or intermediate fiber alignment

On the mechanical stability of growing arteries

Arteries are modeled, within the framework of nonlinear elasticity, as incompressible two-layer cylindrical structures that are residually stressed through differential growth. These structures are loaded by an axial force, internal pressure and have nonlinear, anisotropic, hyperelastic response to stresses. Parameters for this model are directly related to experimental observations. The possible role of axial residual stress in regulating stress in arteries and preventing buckling instabilities is investigated. It is shown that axial residual stress lowers the critical internal pressure leading to buckling and that a reduction of axial loading may lead to a buckling instability which may eventually lead to arterial tortuosity

Spontaneous rotational inversion in Phycomyces

The filamentary fungus Phycomyces blakesleeanus undergoes a series of remarkable transitions during aerial growth. During what is known as the Stage IV growth phase, the fungus extends while rotating in a counterclockwise manner when viewed from above (Stage IVa) and then, while continuing to grow, spontaneously reverses to a clockwise rotation (Stage IVb). This phase lasts for 24 - 48 hours and is sometimes followed by yet another reversal (Stage IVc) before the overall growth ends. Here, we propose a continuum mechanical model of this entire process using nonlinear, anisotropic, elasticity and show how helical anisotropy associated with the cell wall structure can induce spontaneous rotation and, under appropriate circumstances, the observed reversal of rotational handedness

Self-diffusion in remodelling and growth

Self-diffusion, or the flux of mass of a single species within itself, is viewed as an independent phenomenon amenable to treatment by the introduction of an auxiliary field of diffusion velocities. The theory is shown to be heuristically derivable as a limiting case of that of an ordinary binary mixture

Rotation, inversion, and perversion in anisotropic elastic cylindrical tubes and membranes

Cylindrical tubes and membranes are universal structural elements found in biology and engineering over a wide range of scales. Working in the framework of nonlinear elasticity we consider the possible deformations of elastic cylindrical shells reinforced by one or two families of anisotropic fibers. We consider both small and large deformations and the reduction from thick cylindrical shells (tubes) to thin shells (cylindrical membranes). In particular, a number of universal regimes can be identified including the possibility of inversion and perversion of rotation

Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials

Motivated by recent experiments on biopolymer gels whereby the reverse of the usual (positive) Poynting effect was observed, we investigate the effect of the so-called “adscititious inequalities” on the behaviour of hyperelastic materials subject to shear. We first demonstrate that for homogeneous isotropic materials subject to pure shear, the resulting deformation consists of a triaxial stretch combined with a simple shear in the direction of the shear force if and only if the Baker-Ericksen inequalities hold. Then for a cube deformed under pure shear, the positive Poynting effect occurs if the “sheared faces spread apart”, whereas the negative Poynting effect is obtained if the “sheared faces draw together”. Similarly, under simple shear deformation, the positive Poynting effect is obtained if the “sheared faces tend to spread apart”, whereas the negative Poynting effect occurs if the “sheared faces tend to draw together”. When the Poynting effect occurs under simple shear, it is reasonable to assume that the same sign Poynting effect is obtained also under pure shear. Since the observation of the negative Poynting effect in semiflexible biopolymers implies that the (stronger) empirical inequalities may not hold, we conclude that these inequalities must not be imposed when such materials are described

Stability estimates for a twisted rod under terminal loads: a three-dimensional study

The stability of an inextensible unshearable elastic rod with quadratic strain energy density subject to end loads is considered. A self-contained proof in terms of local energy minimizers is presented and optimal bounds are obtained for the problem

Anticavitation and differential growth in elastic shells

Elastic anticavitation is the phenomenon of a void in an elastic solid collapsing on itself. Under the action of mechanical loading alone, very few materials admit anticavitation. We study the possibility of anticavitation as a consequence of an imposed differential growth.Working in the geometry of a spherical shell, we seek radial growth functions which cause the shell to deform to a solid sphere. It is shown, surprisingly, that most materials do not admit full anticavitation, even when infinite growth or resorption is imposed at the inner surface of the shell. However, void collapse can occur in a limiting sense when radial and circumferential growth are properly balanced. Growth functions which diverge or vanish at a point arise naturally in a cumulative growth process