Elliptic-Spline Solutions for Large Localizations in a Circular Blatz-Ko Cylinder Due to Geometric Softening

It has been known that geometric softening can induce strain localizations in solids. However, it is very difficult to analytically capture the localized deformation states within a threedimensional framework, especially when the deformation is large. In this paper, we introduce a novel approach, which resembles the use of a spline to approximate a curve, to construct analytical (asymptotic) solutions for large localizations in a circular cylinder composed of a Blatz-Ko material due to geometric softening. The asymptotic normal form equation (in the form of an ODE) valid for the axial stretch in a small neighborhood is first derived and then a set of these equations, each valid in a small neighborhood, can be obtained. The union of these small neighborhoods can cover a large range of the axial stretch and as a result this set of equations governs the deformation states for the axial stretch in a large interval. Through a phase-plane analysis on this set of ODEs we manage to obtain the analytical solutions (in the form of a spline of elliptic integrals) for the large strain localizations. Both a force-controlled problem and a displacement-controlled problem are solved and the analytical results capture well the non-uniqueness of the stress-displacement relation and the snap-through phenomenon, which are often observed in experiments when strain localizations happen. And, some insightful information on the bifurcation points, is obtained. The important geometric size effect is also discussed through the analytical solutions.