热超弹性材料中的空穴生成问题

研究热超弹性材料中的空穴生成问题,讨论了温度对空穴生成的影响．球体的材料为考虑温度影响的不可压Gent-Thomas材料,或者说是一种与不可压Gent—Thomas材料对应的热超弹性材料,得到了在表面死载荷作用下球体中空穴生成时的分叉曲线及临界载荷,给出了球体中的应力分布,讨论了温度对临界载荷、分叉曲线和应力分布的影响。     

热超弹性材料中空穴增长的分岔问题

研究了受外加均布拉伸死载荷作用的不可压热超弹性材料中空穴突然快速增长的分岔问题.不同于外载荷较小的情况,不可压热超弹性材料球体中的预存微空在外载荷足够大时可以发生突然的快速增长,可视为一类分岔问题.给出了不同温度场下,不同初始半径的微空的增长曲线.预存微空的增长曲线相应于初始半径的极限是实心球体中空穴突然生成的分岔曲线.讨论了均匀和非均匀,升高或降低的温度场对空穴增长问题的影响;给出了预存微空能够发生突然的快速增长的临界载荷,得到了临界载荷与实心球体中空穴突然生成时的临界载荷之间的关系及临界载荷与温度变化之间的关系.     

可压超弹性-塑性材料中的空穴生成

本文研究了材料的弹塑性性质对球体中空穴生成问题的影响，材料的弹性用一种可压超弹性材料的本构关系来描述，材料的塑性用满足材料的不可压条件和Tresca屈服条件的理想塑性材料的本构关系来描述。这类超弹性．塑性材料中可以发生空穴的生成现象，得到了在表面拉伸作用下球体中空穴生成时空穴半径与临界拉伸之间的关系式和临界拉伸。球体的变形可分为弹-塑性变形阶段和完全塑性变形阶段，球体中心首先形成塑性变形区域，并有空穴的突然生成；塑性变形区域能够快速增长，并且使球体很快进入完全塑性变形阶段；空穴在弹-塑性变形阶段迅速增长，但进入完全塑性变形阶段后增长较慢。同时给出了不同变形阶段球体中的应力分布。数值结果表明材料的塑性性质对材料中的空穴生成有明显的影响。     

超弹性材料中空穴的动态生成

本文在有限变形动力学的框架下研究了一种不可压超弹性材料圆柱体在表面突加均布拉伸载荷作用下空穴的动态生成问题,除一个相应于均匀变形状态的平凡解外,当外加载荷超过其临界值时,柱体内部还有空穴的突然生成,得到了空穴半径和表面载荷之间的一个精确的微分关系,证明了空穴随时间的演化是非线性的周期性振动,给出了空穴振动的相图、最大振幅、临界载荷及近似的周期。     

组合超弹性材料中的空穴生成与增长

本文研究了一种组合不可压超弹性材料圆柱体中空穴的生成与增长问题，得到了这种材料受表面均布拉伸死荷载和轴向拉压共同作用下空穴生成问题的解析解，得到了不同组合情况下圆柱体中空穴生成时的临界载荷及分叉曲线，发现组合材料可以发生右分叉，也可以发生左分叉；给出了空穴生成后的应力分布，并讨论了所存在的应力间断和应力集中问题；通过能量比较分析了解的稳定性，讨论了发生右分叉或左分叉的条件，并分析了材料中预存微孔的增长情况。     

湿热作用下热超弹性材料在电子封装中的分层失效问题

研究了具有Gent-Thomas特征的热超弹性材料构成的高聚物电子封装件在回流焊过程中由于吸湿所引发的蒸汽压力以及由于材料的热失配引发的热应力共同作用下而导致的“爆米花”式的分层失效问题.利用超弹性材料空穴生成和增长的理论给出了此类封装材料在回流焊过程中孔穴的生成及增长与蒸汽压力和热应力之间的解析关系.分析结果表明,当...     

热载荷作用下梯度多孔材料梁弯曲及过屈曲力学行为的研究

基于Bernoulli-Euler梁理论,引入物理中面解耦了复合材料结构的面内变形与横向弯曲特性,研究了梯度多孔材料矩形截面梁在热载荷作用下的弯曲及过屈曲力学行为．假设沿梁厚度方向材料的性质是连续变化的,利用能量法推导了矩形截面梁的控制微分方程和边界条件,并用打靶法对无量纲化的控制方程进行数值求解．利用计算得到的结果分析了材料的性质、热载荷、边界条件对矩形截面梁非线性力学行为的影响．结果表明,对称材料模型下,固支梁与简支梁均显示出了典型的分支屈曲行为特征,而其临界屈曲热载荷值均会随着孔隙率系数的增加而单调增加．非对称材料模型下,固支梁仍显示出分支屈曲行为特征,但其临界屈曲热载荷不再随着孔隙率系数的变化而单调变化;而对于两端简支梁,发生了弯曲变形,弯曲挠度随载荷的增大而增大．     

可压缩条件下超弹性电子封装材料中孔穴的增长问题

考虑了温度改变对高聚物材料体积变化的影响,将材料的不可压缩假定修正为可压缩假定。对具有neo-Hookea特征的高聚物电子封装材料在回流焊过程中由于湿热所引发的“爆米花”式的孔穴破裂现象进行了理论研究。利用有限变形的理论给出此类材料在计及体积改变效应下的孔穴增长和吸湿产生的蒸气压力与热应力之间的广义解析关系。该广义解析关系包含了不可压缩条件下的解析关系。分析结果表明：当温度改变引起的可压缩效应较大时,利用可压缩假定分析得到的极限载荷值与利用不可压缩假定分析得到的极限载荷值相比有所提高。但当温度改变引起的可压缩效应较小时,利用两种假定分析得到的极限载荷值相差不大。在温度变化范围不大的情况下,采用不可压缩的假定是合理的。     

蜂窝材料的非线性剪切行为

将梁的弹性大挠度弯曲理论应用于蜂窝壁板,研究了大变形条件下蜂窝材料的非线性剪切变形行为．研究中将椭圆积分形式的解答应用于蜂窝胞元的壁板,并利用平衡和变形谐调条件建立了相应的非线性代数方程组,然后利用牛顿-拉夫森迭代法求解．在上述数值解法的基础上,确定了等效剪应力和剪应变间的非线性曲线,并给出了剪切模量的非线性修正因子,该因子只与蜂窝形状和变形情况有关,而与细长比无关,因而能描述一类蜂窝材料的剪切行为．与有限元数值结果的比较表明,此方法具有较好的精度．     

超弹性材料的不稳定性问题

超弹性材料是一类性能独特、不可替代且有广泛工程应用的高分子材料,对其独特的材料不稳定性问题的研究极大地推动了连续介质力学有限变形理论和超弹性理论的发展.综述了超弹性材料中的材料不稳定性问题的研究成果和最新进展,包括Rivlin立方块问题、薄壁球壳和薄壁圆筒的内压膨胀问题、圆柱的扭转问题、块体的表面不稳定性问题、空穴的生成、增长和闭合问题等.阐述了这类材料中各类非线性不稳定性问题的特点、问题的求解、主要结果及今后进一步的研究方向等.      

DYNAMICAL FORMATION OF CAVITY FOR COMPOSED THERMAL HYPERELASTIC SPHERES IN NONUNIFORM TEMPERATURE FIELDS

Dynamical formation and growth of cavity in a sphere composed of two incompressible thermal-hyperelastic Gent-Thomas materials were discussed under the case of a non-uniform temperature field and the surface dead loading. The mathematical model was first presented based on the dynamical theory of finite deformations. An exact differential relation between the void radius and surface load was obtained by using the variable transformation method. By numerical computation, critical loads and cavitation growth curves were obtained for different temperatures. The influence of the temperature and material parameters of the composed sphere on the void formation and growth was considered and compared with those for static analysis. The results show that the cavity occurs suddenly with a finite radius and its evolvement with time displays a non-linear periodic vibration and that the critical load decreases with the increase of temperature and also the dynamical critical load is lower than the static critical load under the same conditions.     

Dynamical formation of cavity in a composed hyper-elastic sphere

The dynamical formation of cavity in a hyper-elastic sphere composed of two materials with the incompressible strain energy function, subjected to a suddenly applied uniform radial tensile boundary dead-load, was studied following the theory of finite deformation dynamics. Besides a trivial solution corresponding to the homogeneous static state, a cavity forms at the center of the sphere when the tensile load is larger than its critical value. An exact differential relation between the cavity radius and the tensile land was obtained. It is proved that the evolution of cavity radius with time displays nonlinear periodic oscillations. The phase diagram for oscillation, the maximum amplitude, the approximate period and the critical load were all discussed.     

Dynamical formation of cavity in transversely isotropic hyper-elastic spheres

The cavity formation in a radial transversely isotropic hyper-elastic sphere of an incompressible Ogden material, subjected to a suddenly applied uniform radial tensile boundary dead-load, is studied fllowing the theory of finite deformation dynamics. A cavity forms at the center of the sphere when the tensile load is greater than its critical value. It is proved that the evolution of the cavity radius with time follows that of nonlinear periodic oscillations. The project supported by the National Natural Science Foundation of China (10272069) and Shanghai Key Subject Program     

Cavity formation and singular periodic oscillations in isotropic incompressible hyperelastic materials

In this paper, a dynamical problem is considered for an incompressible hyperelastic solid sphere composed of the classical isotropic neo-Hookean material, where the sphere is subjected to a class of periodic step radial tensile loads on its surface. A second-order non-linear ordinary differential equation that describes cavity formation and motion is proposed. The qualitative properties of the solutions of the equation are examined. Correspondingly, under a prescribed constant dead-load, it is proved that a cavity forms in the sphere as the dead-load exceeds a certain critical value and the motion of the formed cavity presents a class of singular periodic oscillations. Under periodic step loads, the existence conditions for periodic oscillation of the formed cavity are determined by using the phase diagrams of the motion equation of cavity. In each section, numerical examples are also carried out.     

Cavitation in finite elasticity with surface energy effects

Cavity formation in incompressible as well as compressible isotropic hyperelastic materials under spherically symmetric loading is examined by accounting for the effect of surface energy. Equilibrium solutions describing cavity formation in an initially intact sphere are obtained explicitly for incompressible as well as slightly compressible neo-Hookean solids. The cavitating response is shown to depend on the asymptotic value of surface energy at unbounded cavity surface stretch. The energetically favorable equilibrium is identified for an incompressible neo-Hookean sphere in the case of prescribed dead-load traction, and for a slightly compressible neo-Hookean sphere in the case of prescribed surface displacement as well as prescribed dead-load traction. In the presence of surface energy effects, it becomes possible that the energetically favorable equilibrium jumps from an intact state to a cavitated state with a finite cavity radius, as the prescribed loading parameter passes a critical level. Such discontinuous cavitation characteristics are found to be highly dependent on the relative magnitude of the surface energy to the bulk strain energy.     

Dynamical analysis of cavitation for a transversely isotropic incompressible hyper-elastic medium: Periodic motion of a pre-existing micro-void

In this paper, the dynamical cavitation behavior is analyzed for a sphere composed of a class of transversely isotropic incompressible hyper-elastic materials, where there is a pre-existing micro-void in the interior of the sphere. A second-order non-linear ordinary differential equation that governs the motion of the initial micro-void is obtained by using the boundary conditions. On analyzing the qualitative properties of the solutions of the differential equation, some interesting conclusions are proposed. It is proved that the number of equilibrium points of the differential equation depends on the values of the material parameters, and that the phase diagrams of the equation are closed, smooth and convex trajectories. For any prescribed surface tensile dead-loads, the motion of the initial micro-void undergoes a non-linear periodic oscillation. The dependence of the periodic motion of the initial micro-void on material parameters and the radius of the initial micro-void is examined, and numerical results are also provided. It is worth pointing out that the conclusions in this paper can be used to describe approximately the physical implications of the dynamical formation of a cavity in the sphere.