Ciprian D. Coman

Bending instabilities of soft biological tissues

Rubber components and soft biological tissues are often subjected to large bending deformations while ‘‘in service”. The circumferential line elements on the inner face of a bent block can contract up to a certain critical stretch ratio kcr (say) before bifurcation occurs and axial creases appear. For several models used to describe rubber, it is found that kcr ¼ 0:56, allowing for a 44% contraction. For models used to describe arteries it is found, somewhat surprisingly, that the strain-stiffening effect promotes instability. For example, the models used for the artery of a seventy-year old human predict that kcr ¼ 0:73, allowing only for a 27% contraction. Tensile experiments conducted on pig skin indicate that bending instabilities should occur even earlier there.

Solitary wave interaction phenomena in a strut buckling model incorporating restabilisation

The archetypal model of the buckling of a compressed long elastic strut resting on a nonlinear elastic foundation is studied. Localised buckling is investigated when the foundation has both quadratic and cubic terms which initially destabilise but subsequently restabilise the structure. The primary solution can be detected by a double-scale perturbation procedure and is reminiscent of a solitary wave: essentially, it consists of a fast periodic oscillation which is slowly modulated and decays exponentially in both directions. Particular interest is paid to the process of adapting the procedure to account for the post-buckling behaviour of two-packet or double-humped solitary waves in this model. We employ the methods of beyond-all-orders asymptotics to reveal terms formally exponentially small in the perturbation parameter which have macroscopic effects on the post-buckling behaviour of the system including the interaction phenomenon of interest. The analysis is reinforced by direct numerical computations which reveal the so-called snaking behaviour in the subsidiary homoclinic orbits as is observed in the case of primary solutions. However, additional phenomena arise for these subsidiary forms including the formation of bridges linking solution paths and the appearance of a multitude of closed isolated loops disconnected from other features of the bifurcation diagram.

Asymptotic results for bifurcations in pure bending of rubber blocks

The bifurcation of an incompressible neo-Hookean thick block with a ratio of thickness to length η, subject to pure bending, is considered. The two incremental equilibrium equations corresponding to a nonlinear pre-buckling state of strain are reduced to a fourth-order linear eigenproblem that displays a multiple turning point. It is found that for 0 < η < ∞, the block experiences an Euler-type buckling instability which in the limit η → ∞ degenerates into a surface instability. Singular perturbation methods enable us to capture this transition, while direct numerical simulations corroborate the analytical results.

Wrinkling of Pre-stressed Annular Thin Films under Azimuthal Shearing

The wrinkling instabilities of a pre-tensioned annular thin film undergoing azimuthal shearing are investigated within the framework of the linearized Donnell—von Kármán bifurcation equation for thin plates. The main objective here is to provide a rational understanding of the role played by the presence of finite bending stiffness and to explain the nature of the localized deformation patterns observed in experiments. In order to achieve this, the eigenvalue problem is formulated as a differential equation with variable coefficients depending on a large parameter. The singular perturbation nature of this equation arises from a combination involving both the pre-stress and the geometrical features of the annular domain. The localization mechanism of the corresponding eigenmodes is then unravelled with the help of a WKB analysis motivated by the qualitative behavior of the neutral stability curves. We show that the asymptotic findings are in very good agreement with the results of direct numerical simulations of the original bifurcation equation.

Classifying the role of trade-offs in the evolutionary diversity of pathogens

In this paper we use a system of non-local reaction–diffusion equations to study the effect of host heterogeneity on the phenotypic evolution of a pathogen population. The evolving phenotype is taken to be the transmission rate of the pathogen on the different hosts, and in our system there are two host populations present. The central feature of our model is a trade-off relationship between the transmission rates on these hosts, which means that an increase in the pathogen transmission on one host will lead to a decrease in the pathogen transmission on the other. The purpose of the paper is to develop a classification of phenotypic diversity as a function of the shape of the trade-off relationship and this is achieved by determining the maximum number of phenotypes a pathogen population can support in the long term, for a given form of the trade-off. Our findings are then compared with results obtained by applying classical theory from evolutionary ecology and the more recent adaptive dynamics method to the same host–pathogen system. We find our work to be in good agreement with these two approaches.

Boundary layers and stress concentration in the circular shearing of annular thin films

This work addresses a generalization of Deans classical problem, which sought to explain how an annular thin elastic plate buckles under uniform shearing forces applied around its edges. We adapt the original setting by assuming that the outer edge is radially stretched while the inner rim undergoes in-plane rotation through some small angle. Boundary-layer methods are used to investigate analytically the deformation pattern which is set up and localized around the inner hole when this angle reaches a well-defined critical wrinkling value. Linear stability theory enables us to identify both the critical load and the preferred number of wrinkles appearing in the deformed configuration. Our asymptotic results are compared with a number of direct numerical simulations.

Asymptotic phenomena in pressurized thin films

An asymptotic study of the wrinkling of a pressurized circular thin film is performed. The corresponding boundary-value problem is described by two non-dimensional parameters; a background tension μ and the applied loading P. Previous numerical studies of the same configuration have shown that P tends to be large, and this fact is exploited here in the derivation of asymptotic descriptions of the elastic bifurcation phenomena. Two limiting cases are considered; in the first, the background tension is modest, while the second deals with the situation when it is large. In both instances, it is shown how the wrinkling is confined to a relatively narrow zone near the rim of the thin film, but the mechanisms driving the bifurcation are different. In the first scenario, the wrinkles are confined to a region which, though close to the rim, is asymptotically separate from it. By contrast, when μ is larger, the wrinkling is within a zone that is attached to the rim. Predictions are made for the value of the applied loading P necessary to generate wrinkling, as well as details of the corresponding wrinkling pattern, and these asymptotic results are compared to some direct numerical simulations of the original boundary-value problem.

Buckling-resistant thin annular plates in tension

Motivated by the localized nature of elastic instabilities in radially stretched thin annular plates, we investigate the resistance to buckling of such configurations in the case when their mechanical properties are piecewise constant. By considering a plate consisting of two sub-annular regions perfectly bonded together and with different linear elastic properties, the neutral stability envelope corresponding to the case when radial constant displacement fields are applied on the inner and outer edges of the plate is investigated numerically in considerable detail. These results are complemented by an asymptotic reduction strategy that provides a greatly simplified eigenproblem capable of describing the original buckling problem in the limit of very thin plates.

Instabilities of Highly Anisotropic Spinning Disks

This work investigates the asymptotic structure of a boundary-value problem proposed recently in connection with in-plane instabilities of spinning disks. Assuming an orthotropic elastic material with cylindrical symmetry we consider a perturbation with respect to the constitutive behavior. The material is assumed to be very stiff in the azimuthal direction, a situation which is commonly encountered in the case of composite flywheels based on hoop-wound carbon fibers in a flexible polyurethane resin. The accuracy of the asymptotic strategy is confirmed by a number of direct computer simulations of the original problem.

Elasto-plastic localised responses in one-dimensional structural models

This work complements recent developments concerning the buckling of beams lying on a nonlinear (non-convex) elastic foundation, and also reports on some investigations on the role of material nonlinearity. Two structural models are studied using a simple elasto-plastic constitutive relationship, and buckling problems are formulated as reversible fourth-order differential equations. It is demonstrated that modulated responses are possible under certain circumstances. Some numerical simulations are presented supporting the analytical findings.