In this paper, the strain energy function proposed by Shang and Cheng was generalized by introducing a nonlinear term. Void formation and growth in the interior of a sphere composed of compressible hyper-elastic mater...In this paper, the strain energy function proposed by Shang and Cheng was generalized by introducing a nonlinear term. Void formation and growth in the interior of a sphere composed of compressible hyper-elastic material, subjected to a prescribed uniform displacement, was examined. A parametric cavitated bifurcation solution for the radial deformed function was obtained. Stability of the solution of the cavitated bifurcation equation was discussed. With the appearance of a cavity, an interesting feature of the radial deformation near the deformed cavity wall is the transition from extension to compression.展开更多
The problem of spherical cavitated bifurcation was examined for a class of incompressible generalized neo-Hookean materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean...The problem of spherical cavitated bifurcation was examined for a class of incompressible generalized neo-Hookean materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. The condition of void nucleation for this problem was obtained. In contrast to the situation for a homogeneous isotropic neo-Hookean sphere, it is shown that not only there exists a secondary turning bifurcation point on the cavitated bifurcation solution which bifurcates locally to the left from trivial solution, and also the critical load is smaller than that for the material with no perturbations, as the parameters belong to some regions. It is proved that the cavitated bifurcation equation is equivalent to a class of normal forms with single-sided constraints near the critical point by using singularity theory. The stability of solutions and the actual stable equilibrium state were discussed respectively by using the minimal potential energy principle.展开更多
This paper is concerned with strictly k-convex large solutions to Hessian equations Sk(D2u(x))=b(x)f(u(x)),x∈Ω,whereΩis a strictly(k-1)-convex and bounded smooth domain in Rn,b∈C∞(Ω)is positive inΩ,but may be v...This paper is concerned with strictly k-convex large solutions to Hessian equations Sk(D2u(x))=b(x)f(u(x)),x∈Ω,whereΩis a strictly(k-1)-convex and bounded smooth domain in Rn,b∈C∞(Ω)is positive inΩ,but may be vanishing on the boundary.Under a new structure condition on f at infinity,the author studies the refined boundary behavior of such solutions.The results are obtained in a more general setting than those in[Huang,Y.,Boundary asymptotical behavior of large solutions to Hessian equations,Pacific J.Math.,244,2010,85–98],where f is regularly varying at infinity with index p>k.展开更多
文摘In this paper, the strain energy function proposed by Shang and Cheng was generalized by introducing a nonlinear term. Void formation and growth in the interior of a sphere composed of compressible hyper-elastic material, subjected to a prescribed uniform displacement, was examined. A parametric cavitated bifurcation solution for the radial deformed function was obtained. Stability of the solution of the cavitated bifurcation equation was discussed. With the appearance of a cavity, an interesting feature of the radial deformation near the deformed cavity wall is the transition from extension to compression.
文摘The problem of spherical cavitated bifurcation was examined for a class of incompressible generalized neo-Hookean materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. The condition of void nucleation for this problem was obtained. In contrast to the situation for a homogeneous isotropic neo-Hookean sphere, it is shown that not only there exists a secondary turning bifurcation point on the cavitated bifurcation solution which bifurcates locally to the left from trivial solution, and also the critical load is smaller than that for the material with no perturbations, as the parameters belong to some regions. It is proved that the cavitated bifurcation equation is equivalent to a class of normal forms with single-sided constraints near the critical point by using singularity theory. The stability of solutions and the actual stable equilibrium state were discussed respectively by using the minimal potential energy principle.
基金the National Natural Science Foundation of China(No.11571295)RP of Shandong Higher Education Institutions(No.J17KA173)。
文摘This paper is concerned with strictly k-convex large solutions to Hessian equations Sk(D2u(x))=b(x)f(u(x)),x∈Ω,whereΩis a strictly(k-1)-convex and bounded smooth domain in Rn,b∈C∞(Ω)is positive inΩ,but may be vanishing on the boundary.Under a new structure condition on f at infinity,the author studies the refined boundary behavior of such solutions.The results are obtained in a more general setting than those in[Huang,Y.,Boundary asymptotical behavior of large solutions to Hessian equations,Pacific J.Math.,244,2010,85–98],where f is regularly varying at infinity with index p>k.
