Search

This work provides a mathematical understanding of the elastic fingering provoked by a large axial extension of a soft solid cylinder bonded between rigid plates. In this prototypical system model, a topological transition from a ground axis-symmetric meniscus is quasi-statically controlled by the applied displacement, which acts as the order parameter of a pitchfork bifurcation. Since the isotropic elastic energy becomes nonconvex under finite strains, geometric nonlinearity is of paramount importance for the loss of uniqueness of the solution of the boundary value problem. Nonetheless, physical nonlinearity in the elastic energy is found to exert an opposite stabilizing effect. It indeed penalizes the local stretching at the free boundary that would arise as a consequence of any change of its Gaussian curvature. The theoretical and numerical results are in agreement with recent experimental observations, showing that elastic fingering is strongly affected by the aspect ratio of the disc and can be even suppressed in soft materials with physical nonlinearity. [ABSTRACT FROM AUTHOR]

Shock waves in a structure can result in the detachment of an interface and induce microcracks. In a recent study [Huang et al., R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 468 (2012), pp. 3882-3901], it was shown that for certain nonlinearly elastic materials it is possible to generate a phenomenon in which a tensile wave can catch the first transmitted compressive wave (so the former can be undermined) in an initially stress-free two-material bar. In this study, we consider the wave catching-up phenomenon in a nonlinearly elastic prestressed two-material bar. We use the same method as that used by Huang et al. in the previously mentioned paper to construct solutions. Our main focus is on proving the global structure stability of the solutions in a prestressed (or initially stress-free) two-material bar. We first reduce the corresponding initial boundary value problem into several typical free boundary problems based on the formulation of Riemann invariants. Then, using a constructive method and carefully treating the complexity arising from multiple reflections of waves at the interface in the two-material bar, we successfully prove the global structure stability of the wave catching-up phenomenon. [ABSTRACT FROM AUTHOR]

In the two experiments by Beatty and his coauthors on the compressions of elastic rectangles/bars, it was found that there is a transition region of the aspect ratio which separates buckling from barrelling. Friction, which prevents the lateral movement of the end cross section, might be the cause. Here, we study the compressions with clamped end conditions. One of the purposes is to show, with this setting in which the lateral movement of the end cross section is limited, that there is indeed such a transition region. By using combined series-asymptotic expansions, we derive two decoupled nonlinear ordinary differential equations (ODEs). By phase plane analysis, the leading-order axial strain can be obtained from one of the ODEs. Then an eigenvalue problem can be formulated from another ODE, which is solved by the WKB (Wentzel–Kramers–Brillouin) method. It is found that when the aspect ratio is relatively large there is only a bifurcation to the barrelling which leads to a corner-like profile on the lateral boundaries of the rectangle. When the aspect ratio is relatively small there are only bifurcation points which lead to the buckled profiles. A lower bound of the aspect ratio for barrelling and a different upper bound for buckling are found, which implies the existence of the above-mentioned transition region. Another finding is that, after the barrelling, no further bifurcation to buckling can occur. The critical buckling loads obtained from our asymptotic solutions are also compared with those obtained from the Euler buckling formula. [ABSTRACT FROM AUTHOR]