The radial symmetric motion problem was examined for a spherical shell composed of a class of imperfect incompressible hyper-elastic materials, in which the materials may be viewed as the homogeneous incompressible is...The radial symmetric motion problem was examined for a spherical shell composed of a class of imperfect incompressible hyper-elastic materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. A second-order nonlinear ordinary differential equation that describes the radial motion of the inner surface of the shell was obtained. And the first integral of the equation was then carded out. Via analyzing the dynamical properties of the solution of the differential equation, the effects of the prescribed imperfection parameter of the material and the ratio of the inner and the outer radii of the underformed shell on the motion of the inner surface of the shell were discussed, and the corresponding numerical examples were carded out simultaneously. In particular, for some given parameters, it was proved that, there exists a positive critical value, and the motion of the inner surface with respect to time will present a nonlinear periodic oscillation as the difference between the inner and the outer presses does not exceed the critical value. However, as the difference exceeds the critical value, the motion of the inner surface with respect to time will increase infinitely. That is to say, the shell will be destroyed ultimately.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
An isolated cylindrical void is located inside an incompressible nonlinearelasticmediumwhose constitutive behaviour is governed by a neo-Hookean strain energy function.In-plane hydrostatic pressure is applied in the f...An isolated cylindrical void is located inside an incompressible nonlinearelasticmediumwhose constitutive behaviour is governed by a neo-Hookean strain energy function.In-plane hydrostatic pressure is applied in the far-field so that the void changes its radius and an inhomogeneous region of deformation arises in the vicinity of the void.We consider scattering from the void in the deformed configuration due to an incident field(of small amplitude)generated by a horizontally polarized shear(SH)line source,a distance r0(R0)away from the centre of the void in the deformed(undeformed)configuration.We show that the scattering coefficients of this scattered field are unaffected by the pre-stress(initial deformation).In particular,they depend not on the deformed void radius a or distance r0,but instead on the original void size A and original distance R0.展开更多
The problem of radial symmetric motion for a solid sphere composed of a class of generalized incompressible neo-Hookean materials, subjected to a suddenly applied surface tensile dead load, is examined.The ana...The problem of radial symmetric motion for a solid sphere composed of a class of generalized incompressible neo-Hookean materials, subjected to a suddenly applied surface tensile dead load, is examined.The analytic solutions for this problem and the motion equation of cavity that describes cavity formation and growth with time are obtained. The e?ect of radial perturbation of the materials on cavity formation and its motion is discussed. The plane of the perturbation parameters of the materials is divided into four regions. The existential conditions and qualitative properties of solutions of the motion equation of the cavity are studied in di?erent parameters’ regions in detail. It is proved that the cavity motion with time is a nonlinear periodic vibration. The vibration center is then determined.展开更多
The problem of finite deformation of an incompressible rectangular rubber ring with an internal rigid body, where the ring is subjected to equal axial loads at its two ends, is examined. A reasonable mathematical mode...The problem of finite deformation of an incompressible rectangular rubber ring with an internal rigid body, where the ring is subjected to equal axial loads at its two ends, is examined. A reasonable mathematical model is formulated by using the nonlinear field theory and the implicit analytical solutions are derived. Then numerical simulations are implemented to further illustrate the results and obtain some meaningful conclusions. The deformation of the lateral surface of the ring becomes larger with the increasing axial loads, the decreasing ratio of the inner and outer radii and the increasing height of the ring.展开更多
In this paper,the problem of axially symmetric deformation is examined for a composite cylindrical tube under equal axial loads acting on its two ends,where the tube is composed of two different incompressible neo-Hoo...In this paper,the problem of axially symmetric deformation is examined for a composite cylindrical tube under equal axial loads acting on its two ends,where the tube is composed of two different incompressible neo-Hookean materials.Significantly,the implicit analytical solutions describing the deformation of the tube are proposed.Numerical simulations are given to further illustrate the qualitative properties of the solutions and some meaningful conclusions are obtained.In the tension case,with the increasing axial loads or with the decreasing ratio of shear moduli of the outer and the inner materials,it is proved that the tube will shrink more along the radial direction and will extend more along the axial direction.Under either tension or compression,the deformation along the axial direction is obvious near the two ends of the tube,while in the rest,the change is relatively small.Similarly,for a large domain of the middle part,the axial elongation is almost constant;however,the variation is very fast near the two ends.In addition,the absolute value of the axial displacement increases gradually from the central cross-section of the tube and achieves the maximum at the two endpoints.展开更多
Nonlinear vibration with axisymmetric 3:1 internal resonance is investigated for an incompressible neo-Hookean hyperelastic cylindrical shell under both axial and radial harmonic excitations.A full nonlinear strain-di...Nonlinear vibration with axisymmetric 3:1 internal resonance is investigated for an incompressible neo-Hookean hyperelastic cylindrical shell under both axial and radial harmonic excitations.A full nonlinear strain-displacement relation is derived from the large deflection theory of thin-walled shells.A set of nonlinear differential equations describing the large deflection vibration are formulated by the Lagrange equation and the assumption of small strains.Steady-state responses of the system are predicted via the harmonic balance method with the arc length continuation,and their stabilities are determined via the modified sorting method.The effects of excitations on the steady-state responses are analyzed.The results reveal a crucial role played by the phase difference in the structural response,and the phase difference can effectively control the amplitude of vibration.展开更多
基金国家自然科学基金，Municipal Key Subject Program of Shanghai
文摘The radial symmetric motion problem was examined for a spherical shell composed of a class of imperfect incompressible hyper-elastic materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. A second-order nonlinear ordinary differential equation that describes the radial motion of the inner surface of the shell was obtained. And the first integral of the equation was then carded out. Via analyzing the dynamical properties of the solution of the differential equation, the effects of the prescribed imperfection parameter of the material and the ratio of the inner and the outer radii of the underformed shell on the motion of the inner surface of the shell were discussed, and the corresponding numerical examples were carded out simultaneously. In particular, for some given parameters, it was proved that, there exists a positive critical value, and the motion of the inner surface with respect to time will present a nonlinear periodic oscillation as the difference between the inner and the outer presses does not exceed the critical value. However, as the difference exceeds the critical value, the motion of the inner surface with respect to time will increase infinitely. That is to say, the shell will be destroyed ultimately.
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.
文摘An isolated cylindrical void is located inside an incompressible nonlinearelasticmediumwhose constitutive behaviour is governed by a neo-Hookean strain energy function.In-plane hydrostatic pressure is applied in the far-field so that the void changes its radius and an inhomogeneous region of deformation arises in the vicinity of the void.We consider scattering from the void in the deformed configuration due to an incident field(of small amplitude)generated by a horizontally polarized shear(SH)line source,a distance r0(R0)away from the centre of the void in the deformed(undeformed)configuration.We show that the scattering coefficients of this scattered field are unaffected by the pre-stress(initial deformation).In particular,they depend not on the deformed void radius a or distance r0,but instead on the original void size A and original distance R0.
基金Project supported by the National Natural Science Foundation of China (No. 10272069) and Shanghai Key Project Program.
文摘The problem of radial symmetric motion for a solid sphere composed of a class of generalized incompressible neo-Hookean materials, subjected to a suddenly applied surface tensile dead load, is examined.The analytic solutions for this problem and the motion equation of cavity that describes cavity formation and growth with time are obtained. The e?ect of radial perturbation of the materials on cavity formation and its motion is discussed. The plane of the perturbation parameters of the materials is divided into four regions. The existential conditions and qualitative properties of solutions of the motion equation of the cavity are studied in di?erent parameters’ regions in detail. It is proved that the cavity motion with time is a nonlinear periodic vibration. The vibration center is then determined.
基金supported by the National Natural Science Foundation of China (Nos. 10872045, 10721062 and 10772104)the Program for New Century Excellent Talents in University (No. NCET-09-0096)the Fundamental Research Funds for the Central Universities
文摘The problem of finite deformation of an incompressible rectangular rubber ring with an internal rigid body, where the ring is subjected to equal axial loads at its two ends, is examined. A reasonable mathematical model is formulated by using the nonlinear field theory and the implicit analytical solutions are derived. Then numerical simulations are implemented to further illustrate the results and obtain some meaningful conclusions. The deformation of the lateral surface of the ring becomes larger with the increasing axial loads, the decreasing ratio of the inner and outer radii and the increasing height of the ring.
基金supported by the National Natural Science Foundation of China(Grant Nos.10872045 and 11232003)the Program for New Century Excellent Talents in University(Grant No.NCET-09-0096)+1 种基金the Fundamental Research Funds for the Central Universities(Grant No.DC120101121)the Program for Liaoning Excellent Talents in University(Grant No.LR2012044)
文摘In this paper,the problem of axially symmetric deformation is examined for a composite cylindrical tube under equal axial loads acting on its two ends,where the tube is composed of two different incompressible neo-Hookean materials.Significantly,the implicit analytical solutions describing the deformation of the tube are proposed.Numerical simulations are given to further illustrate the qualitative properties of the solutions and some meaningful conclusions are obtained.In the tension case,with the increasing axial loads or with the decreasing ratio of shear moduli of the outer and the inner materials,it is proved that the tube will shrink more along the radial direction and will extend more along the axial direction.Under either tension or compression,the deformation along the axial direction is obvious near the two ends of the tube,while in the rest,the change is relatively small.Similarly,for a large domain of the middle part,the axial elongation is almost constant;however,the variation is very fast near the two ends.In addition,the absolute value of the axial displacement increases gradually from the central cross-section of the tube and achieves the maximum at the two endpoints.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11672069,11872145,11872159,12172086,and 12101106).
文摘Nonlinear vibration with axisymmetric 3:1 internal resonance is investigated for an incompressible neo-Hookean hyperelastic cylindrical shell under both axial and radial harmonic excitations.A full nonlinear strain-displacement relation is derived from the large deflection theory of thin-walled shells.A set of nonlinear differential equations describing the large deflection vibration are formulated by the Lagrange equation and the assumption of small strains.Steady-state responses of the system are predicted via the harmonic balance method with the arc length continuation,and their stabilities are determined via the modified sorting method.The effects of excitations on the steady-state responses are analyzed.The results reveal a crucial role played by the phase difference in the structural response,and the phase difference can effectively control the amplitude of vibration.
