Critical stretch for formation of a cylindrical void in a compressible hyperelastic material

Formation of a cylindrical void in an infinitely long compressible hyperelastic cylinder under axial and radial stretches is examined. The cavitation phenomenon is viewed here as a bifurcation of a solution with a cavity from the homogeneously deformed configuration, taking place when the applied radial stretch reaches a certain critical value. This amounts to modeling the underlying phenomenon as a kind of elastic instability. A formulation of shooting-method type is presented to derive an equation which gives the critical radial stretch for a prescribed axial stretch. For a special class of hyperelastic solids, called modified Blatz-Ko material, the obtained equation leads to an explicit expression for the critical stretches and stresses. Some analytical as well as numerical calculations are carried out to explicitly obtain the critical values for cavitation. The results are summarized in the form of cavitation curves in the two-dimensional space of axial and radial stretches or stresses. Influence of the axial stretch on the critical radial stretch is discussed. Throughout the paper, the corresponding results for a spherically symmetric void formation are referred to when appropriate and compared with the cylindrical case of the present interest. It is then indicated that in the state of equitriaxial stretch, cavitation into a cylindrical shape is likely to occur at lower stretch than into a spherical one.