Study on cavitated bifurcation problems for spheres composed of hyper-elastic materials

Abstract

In this paper, spherical cavitated bifurcation problems are examined for incompressible hyper-elastic materials and compressible hyper-elastic materials, respectively. For incompressible hyper-elastic materials, a cavitated bifurcation equation that describes cavity formation and growth for a solid sphere, composed of a class of transversely isotropic incompressible hyper-elastic materials, is obtained. Some qualitative properties of the solutions of the cavitated bifurcation equation are discussed in the different regions of the plane partitioned by material parameters indicating the degree of radial anisotropy in detail. It is shown that the cavitated bifurcation equation is equivalent, by use of singularity theory, to a class of normal forms with single-sided constraint conditions at the critical point. Stability and catastrophe of the solutions of the cavitated bifurcation equation are discussed by using the minimal potential-energy principle. For compressible hyper-elastic materials, a group of parameter-type solutions for the cavitated deformation for a solid sphere, composed of a class of isotropic compressible hyper-elastic materials, is obtained. Stability of the solutions is also discussed.

On the near-critical behavior of cavitation in elastic plane membranes

“…When the material is compressible, the two-point boundary value problem can be solved by a shooting procedure in the most general case (see, e.g., [8]), but many studies have focused on finding closed-form solutions for specific material models (see, e.g., [9][10][11][12]). There also exists a large body of literature concerned with the effects of anisotropy, material inhomogeneity, surface tension, and plastic behavior; see, e.g., [13][14][15][16][17][18][19]. We refer to [20] and [21] for a comprehensive review of the literature.…”

Positive solutions of periodic boundary value problems for second-order differential equations with the nonlinearity dependent on the derivative

“…For compressible hyper clastic materials, the existence of solution to the corresponding problems is strictly dependent on the form of strain energy function. For details, the readers are referred to [11,14] and the references therein. (1.1) describes a class of growth problem for preexisting microvoid in column composed of compressible hyper elastic materials under given radial extension.…”

Dynamic cavitations in incompressible hyperelastic pre-strained circular sheets

“…For cavitation in nonlinearly elastic spheres, Polignone and Horgan [2] described cavity formation and growth in a solid sphere as a bifurcation problem. Recently, Ren and Cheng [3], Yuan et al [4,5] respectively studied the static and dynamic cavitated bifurcations in transversely isotropic incompressible hyperelastic spheres, and obtained some interesting results, such as secondary turning bifurcation, and singular nonlinearly periodic oscillations. For hyperelastic sheets, Haddow et al [6] studied the problem of wave propagation resulting from a suddenly punched circular hole in a thin sheet, which is initially subjected to a finite equibiaxial stretch; David and Humphrey [7] investigated stress and strain concentration at holes embedded in anisotropic soft bio-tissues, which is modeled as nonlinear elastic Fung materials in a biomechanics context; Wessler and Mazza [8] investigated the model of a pre-strained circular actuator made of dielectric elastomers.…”