几类轴对称超弹性橡胶结构的有限变形分析
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摘要
橡胶材料组成的轴对称结构(如橡胶圈、橡胶管、橡胶垫等)在社会生产和生活中的应用非常广泛,由于上述结构都是在一定环境和载荷下使用的,因此都会遇到变形、失稳、破坏以及使用寿命有限等问题,相关材料和结构有限变形的稳定性问题一直是国内外专家和学者关注的焦点。本文利用非线性弹性理论以及分岔理论等理论的基本观点和结论,研究了基于超弹性本构关系的橡胶材料组成的几类轴对称结构在轴向或径向加载下的静、动力学变形问题,得到了一些新的结论。具体内容如下：
1.研究了几类不可压缩超弹性材料组成的矩形橡胶圈在端部轴向压缩载荷作用下的有限变形问题。对于各向同性neo-Hookean材料、关于径向横观各向同性的neo-Hookean材料和各向同性Mooney-Rivlin材料,利用材料的不可压缩约束及边界条件分别求得了问题的隐式解析解。结合数值算例讨论了轴向压缩载荷、橡胶圈的径向厚度以及轴向高度对橡胶圈有限变形的影响。从理论上验证了：轴向压缩载荷越大、橡胶圈径向相对越薄、轴向相对越高,橡胶圈外表面沿径向的膨胀越大,其两端沿轴向的收缩也越大。对于关于径向横观各向同性的neo-Hookean材料,材料的各向异性参数越大,橡胶圈的变形相对越大；对于各向同性Mooney-Rivlin材料,材料的剪切模量之比越小,结构的变形相对越大。三种材料模型下,轴向压缩率均在中截面处最小而在橡胶圈两端处最大,轴向压缩率和轴向位移同样受轴向载荷和结构参数的影响。
2.分别研究了不可压缩超弹性材料组成的层合橡胶圆管的轴向压缩问题以及径向膨胀问题。
对于由两类不可压缩neo-Hookean材料组成的有限长的层合橡胶圆管,当其两端受轴向载荷压缩作用时,利用材料的不可压缩约束、应力和应变的连续性条件、边界条件求得了问题的隐式解析解。结合数值算例验证了：随着轴向载荷的增大或外层材料与内层材料的剪切模量之比的减小,圆管的外表面和材料的层合面沿径向的膨胀均越来越大,沿轴向的收缩也越来越大；橡胶圆管外表面和层合面在圆管中间大部分区域的变形都趋于一致,而在结构的两端附近,二者的变形较为明显。橡胶圆管中间部分的轴向压缩率基本保持不变而在两端附近变化较快,两端的轴向压缩率最大；轴向位移的绝对值从橡胶圆管的中截面开始逐渐增大,在两端处轴向位移的绝对值达到最大。
对于由两类关于径向横观各向同性的不可压缩幂率型材料组成的无限长的层合橡胶圆管,当其内表面受径向均布压力作用时,通过对描述结构径向膨胀的方程的定性分析分别得到了圆管静态膨胀和动态膨胀的机理。证明了：若圆管内、外层材料的应变能函数中幂率参数均未超过1,则存在一个临界压力,当径向压力小于临界压力时,圆管的径向膨胀随时间的演化做非线性周期振动；当径向压力大于临界压力时,圆管将无限膨胀。若两层材料的应变能函数中幂率参数至少有一个超过1,则对任意给定的径向压力,橡胶圆管的径向膨胀随时间演化都是非线性周期振动；对于某些特殊的材料参数,层合橡胶圆管振动的振幅会出现跳跃性的增长。
3.分别研究了不可压缩超弹性材料组成的两类轴对称结构的空穴问题。将两类问题的数学模型分别简化为关于描述结构径向运动的二阶非线性常微分方程,通过对两组方程的定性分析证明了：
对于由各向同性不可压缩Ogden材料组成的中心含有微孔的无限长圆柱体,当其外表面受突加径向拉伸载荷作用时,随着径向拉伸载荷的增加,微孔半径的增长开始比较缓慢,存在临界载荷,当拉伸载荷达到临界载荷时,微孔半径会突然快速增长。当材料参数满足一定条件时,对于任意给定的拉伸载荷,微孔的运动始终是非线性周期振动；存在特殊的材料参数,使得运动方程的相图中存在同宿轨道,结构振动的振幅同样会出现跳跃性的增长,此类情形对于各向同性不可压缩超弹性材料并不多见。
对于由一类各向同性不可压缩超弹性材料组成的在径向预拉伸状态下的圆板,当其外表面受突加径向拉伸载荷作用时,只有材料的幂率参数满足一定条件时,圆板轴向中心处才会出现空穴；径向预拉伸的值越大,结构中的空穴出现的越早,材料的幂率参数越大,空穴出现越晚。当圆板中心处出现空穴以后,空穴将随时间的演化做经典的非线性周期振动,且振动的振幅随着外表面拉伸载荷或径向预拉伸的增加而增大。
Axisymmetric structures composed of rubber materials (such as rubber rings, rubber tubes, rubber blankets. etc.) are widely used in social production and life under certain circumstance and load, and so they often encounter the problems of deformation, instability, destroy and limited service life, and so on. The stability problems of finite deformation of relative materials and structures are always the focuses of experts and scholars at home and abroad. In this dissertation. the static and dynamical problems are examined for several axisymmetric structures composed of rubber materials based on the hyperelastic constitutive relations via using the basic viewpoints and conclusions of the nonlinear elasticity and the bifurcation theory, and so forth. Some new conclusions are obtained, as follows,
1. The problems of finite deformation of several rectangular rubber rings composed of incompressible hyperelastic materials are studied, where both ends of the rings are subjected to axial compressive loads. Systems of implicit analytical solutions are derived respectively by using the incompressible constraint and the boundary conditions for the isotropic neo-Hookean material, the transversely isotropic neo-Hookean material and the isotropic Mooney-Rivlin material. Combining with numerical examples,the influences of axial compressive loads, radial thickness and axial height of the rubber rings on finite deformation of the rings are discussed in detail. It is proved theoretically that with the increasing axial compressive loads, the decreasing radial thickness and the increasing axial height, the lateral surfaces of these rings along the radial direction inflate and both ends along the axial direction shrink more and more. For the transversely isotropic neo-Hookean material, the larger the value of the anisotropy parameter is, the bigger the deformation of the ring is. For the isotropic Mooney-Rivlin material, the smaller the ratio of the shear moduli is, the bigger the deformation of the ring is. For the three kinds of material models, the ratios of axial compression are all the smallest at the central cross-sections of these rings and are the biggest at the ends. The ratios of axial compression and the axial displacements are also influenced by axial loads and structure parameters.
2. The problems of axial compression and radial inflation are examined respectively for the composite rubber tubes composed of incompressible hyperelastic materials.
For a finitely long composite rubber tube subjected to static axial compressive loads at its both ends, where the tube is composed of two classes of incompressible neo-Hookean materials, a system of implicit analytical solutions is derived by using the incompressible constraint, the continuous conditions of stress and strain, the boundary conditions. It is proved with numerical examples that the lateral surface and the interface of the tube along the radial direction inflate and both ends along the axial direction shrink more and more as the axial loads increase or as the ratio of the shear moduli of the outer and the inner materials decreases. Moreover, the deformation models of both the lateral surface and the interface of the composite rubber tube are nearly uniform at most of the middle portion of the tube, whereas, are obvious near the two ends. In the middle part, the ratio of axial compression maintains almost the same, while it changes very fast near the two ends and achieves the maximum at the ends. The absolute value of axial displacement increases gradually from the central cross-section and reaches the maximum at the two ends of the tube.
For an infinitely long composite rubber tube subjected to a suddenly applied radial pressure at its inner surface, where it is composed of two classes of transversely isotropic incompressible power-law materials, the mechanisms of static inflation and dynamic inflation are given by qualitatively analyzing the equation describing the radial inflation of the tube. It is proved that if the power-law parameters of the strain energy functions associated with the two materials do not exceed1, then there exists a critical pressure such that the radial inflation mode of the tube with time is a nonlinearly periodic oscillation as the radial pressure does not exceed the critical pressure, otherwise, the tube will inflate infinitely. If at least one of the power-law parameters of the strain energy functions is larger than1, then for any given pressure, the radial inflation mode of the tube with time is always a nonlinearly periodic oscillation. Moreover, for some special material parameters, the oscillation amplitude will increase discontinuously.
3. The bifurcation problems are studied for two classes of axisymmetric structures composed of isotropic incompressible hyperelastic materials. The mathematical models are reduced to two classes of second order nonlinear ordinary differential equations which are used to describe the radial motions of the structures with time, respectively. Through qualitatively analyzing the two equations, it yields the following conclusions:
For an infinitely long cylinder composed of the isotropic incompressible Ogden material with a microvoid at its center, when it is subjected to a suddenly applied tensile load at the outer surface, the radius of the microvoid grows very slowly at the beginning with the increasing tensile load. However, there exists a critical load, when the tensile load exceeds the critical load the radius of the microvoid grows rapidly. Moreover, for the material parameters satisfying certain conditions, the motion of the microvoid is always a nonlinear periodic oscillation for any given load. As the material parameters take certain special values, the phase diagrams of the motion equation have homoclinic orbits, which means that the oscillation amplitude increases discontinuously with the increasing tensile load, such case is very rare for isotropic incompressible hyperelastic materials.
For a pre-strained circular sheet composed of a class of isotropic incompressible hyperelastic materials, when it is subjected to a suddenly applied tensile load at the radial surface, it is shown that cavitation depends exactly on the power-law parameter. The larger the pre-strained value is prescribed, the earlier cavitation occurs; the larger the power-law parameter is, the later cavitation occurs. Dynamically, once a cavity forms at the axial line of the circular sheet, the motion of the formed cavity with time is a classical nonlinear periodic oscillation. Moreover, the oscillation amplitude increases with the increasing tensile load or with the increasing pre-strained value.
1.研究了几类不可压缩超弹性材料组成的矩形橡胶圈在端部轴向压缩载荷作用下的有限变形问题。对于各向同性neo-Hookean材料、关于径向横观各向同性的neo-Hookean材料和各向同性Mooney-Rivlin材料,利用材料的不可压缩约束及边界条件分别求得了问题的隐式解析解。结合数值算例讨论了轴向压缩载荷、橡胶圈的径向厚度以及轴向高度对橡胶圈有限变形的影响。从理论上验证了：轴向压缩载荷越大、橡胶圈径向相对越薄、轴向相对越高,橡胶圈外表面沿径向的膨胀越大,其两端沿轴向的收缩也越大。对于关于径向横观各向同性的neo-Hookean材料,材料的各向异性参数越大,橡胶圈的变形相对越大；对于各向同性Mooney-Rivlin材料,材料的剪切模量之比越小,结构的变形相对越大。三种材料模型下,轴向压缩率均在中截面处最小而在橡胶圈两端处最大,轴向压缩率和轴向位移同样受轴向载荷和结构参数的影响。
2.分别研究了不可压缩超弹性材料组成的层合橡胶圆管的轴向压缩问题以及径向膨胀问题。
对于由两类不可压缩neo-Hookean材料组成的有限长的层合橡胶圆管,当其两端受轴向载荷压缩作用时,利用材料的不可压缩约束、应力和应变的连续性条件、边界条件求得了问题的隐式解析解。结合数值算例验证了：随着轴向载荷的增大或外层材料与内层材料的剪切模量之比的减小,圆管的外表面和材料的层合面沿径向的膨胀均越来越大,沿轴向的收缩也越来越大；橡胶圆管外表面和层合面在圆管中间大部分区域的变形都趋于一致,而在结构的两端附近,二者的变形较为明显。橡胶圆管中间部分的轴向压缩率基本保持不变而在两端附近变化较快,两端的轴向压缩率最大；轴向位移的绝对值从橡胶圆管的中截面开始逐渐增大,在两端处轴向位移的绝对值达到最大。
对于由两类关于径向横观各向同性的不可压缩幂率型材料组成的无限长的层合橡胶圆管,当其内表面受径向均布压力作用时,通过对描述结构径向膨胀的方程的定性分析分别得到了圆管静态膨胀和动态膨胀的机理。证明了：若圆管内、外层材料的应变能函数中幂率参数均未超过1,则存在一个临界压力,当径向压力小于临界压力时,圆管的径向膨胀随时间的演化做非线性周期振动；当径向压力大于临界压力时,圆管将无限膨胀。若两层材料的应变能函数中幂率参数至少有一个超过1,则对任意给定的径向压力,橡胶圆管的径向膨胀随时间演化都是非线性周期振动；对于某些特殊的材料参数,层合橡胶圆管振动的振幅会出现跳跃性的增长。
3.分别研究了不可压缩超弹性材料组成的两类轴对称结构的空穴问题。将两类问题的数学模型分别简化为关于描述结构径向运动的二阶非线性常微分方程,通过对两组方程的定性分析证明了：
对于由各向同性不可压缩Ogden材料组成的中心含有微孔的无限长圆柱体,当其外表面受突加径向拉伸载荷作用时,随着径向拉伸载荷的增加,微孔半径的增长开始比较缓慢,存在临界载荷,当拉伸载荷达到临界载荷时,微孔半径会突然快速增长。当材料参数满足一定条件时,对于任意给定的拉伸载荷,微孔的运动始终是非线性周期振动；存在特殊的材料参数,使得运动方程的相图中存在同宿轨道,结构振动的振幅同样会出现跳跃性的增长,此类情形对于各向同性不可压缩超弹性材料并不多见。
对于由一类各向同性不可压缩超弹性材料组成的在径向预拉伸状态下的圆板,当其外表面受突加径向拉伸载荷作用时,只有材料的幂率参数满足一定条件时,圆板轴向中心处才会出现空穴；径向预拉伸的值越大,结构中的空穴出现的越早,材料的幂率参数越大,空穴出现越晚。当圆板中心处出现空穴以后,空穴将随时间的演化做经典的非线性周期振动,且振动的振幅随着外表面拉伸载荷或径向预拉伸的增加而增大。
Axisymmetric structures composed of rubber materials (such as rubber rings, rubber tubes, rubber blankets. etc.) are widely used in social production and life under certain circumstance and load, and so they often encounter the problems of deformation, instability, destroy and limited service life, and so on. The stability problems of finite deformation of relative materials and structures are always the focuses of experts and scholars at home and abroad. In this dissertation. the static and dynamical problems are examined for several axisymmetric structures composed of rubber materials based on the hyperelastic constitutive relations via using the basic viewpoints and conclusions of the nonlinear elasticity and the bifurcation theory, and so forth. Some new conclusions are obtained, as follows,
1. The problems of finite deformation of several rectangular rubber rings composed of incompressible hyperelastic materials are studied, where both ends of the rings are subjected to axial compressive loads. Systems of implicit analytical solutions are derived respectively by using the incompressible constraint and the boundary conditions for the isotropic neo-Hookean material, the transversely isotropic neo-Hookean material and the isotropic Mooney-Rivlin material. Combining with numerical examples,the influences of axial compressive loads, radial thickness and axial height of the rubber rings on finite deformation of the rings are discussed in detail. It is proved theoretically that with the increasing axial compressive loads, the decreasing radial thickness and the increasing axial height, the lateral surfaces of these rings along the radial direction inflate and both ends along the axial direction shrink more and more. For the transversely isotropic neo-Hookean material, the larger the value of the anisotropy parameter is, the bigger the deformation of the ring is. For the isotropic Mooney-Rivlin material, the smaller the ratio of the shear moduli is, the bigger the deformation of the ring is. For the three kinds of material models, the ratios of axial compression are all the smallest at the central cross-sections of these rings and are the biggest at the ends. The ratios of axial compression and the axial displacements are also influenced by axial loads and structure parameters.
2. The problems of axial compression and radial inflation are examined respectively for the composite rubber tubes composed of incompressible hyperelastic materials.
For a finitely long composite rubber tube subjected to static axial compressive loads at its both ends, where the tube is composed of two classes of incompressible neo-Hookean materials, a system of implicit analytical solutions is derived by using the incompressible constraint, the continuous conditions of stress and strain, the boundary conditions. It is proved with numerical examples that the lateral surface and the interface of the tube along the radial direction inflate and both ends along the axial direction shrink more and more as the axial loads increase or as the ratio of the shear moduli of the outer and the inner materials decreases. Moreover, the deformation models of both the lateral surface and the interface of the composite rubber tube are nearly uniform at most of the middle portion of the tube, whereas, are obvious near the two ends. In the middle part, the ratio of axial compression maintains almost the same, while it changes very fast near the two ends and achieves the maximum at the ends. The absolute value of axial displacement increases gradually from the central cross-section and reaches the maximum at the two ends of the tube.
For an infinitely long composite rubber tube subjected to a suddenly applied radial pressure at its inner surface, where it is composed of two classes of transversely isotropic incompressible power-law materials, the mechanisms of static inflation and dynamic inflation are given by qualitatively analyzing the equation describing the radial inflation of the tube. It is proved that if the power-law parameters of the strain energy functions associated with the two materials do not exceed1, then there exists a critical pressure such that the radial inflation mode of the tube with time is a nonlinearly periodic oscillation as the radial pressure does not exceed the critical pressure, otherwise, the tube will inflate infinitely. If at least one of the power-law parameters of the strain energy functions is larger than1, then for any given pressure, the radial inflation mode of the tube with time is always a nonlinearly periodic oscillation. Moreover, for some special material parameters, the oscillation amplitude will increase discontinuously.
3. The bifurcation problems are studied for two classes of axisymmetric structures composed of isotropic incompressible hyperelastic materials. The mathematical models are reduced to two classes of second order nonlinear ordinary differential equations which are used to describe the radial motions of the structures with time, respectively. Through qualitatively analyzing the two equations, it yields the following conclusions:
For an infinitely long cylinder composed of the isotropic incompressible Ogden material with a microvoid at its center, when it is subjected to a suddenly applied tensile load at the outer surface, the radius of the microvoid grows very slowly at the beginning with the increasing tensile load. However, there exists a critical load, when the tensile load exceeds the critical load the radius of the microvoid grows rapidly. Moreover, for the material parameters satisfying certain conditions, the motion of the microvoid is always a nonlinear periodic oscillation for any given load. As the material parameters take certain special values, the phase diagrams of the motion equation have homoclinic orbits, which means that the oscillation amplitude increases discontinuously with the increasing tensile load, such case is very rare for isotropic incompressible hyperelastic materials.
For a pre-strained circular sheet composed of a class of isotropic incompressible hyperelastic materials, when it is subjected to a suddenly applied tensile load at the radial surface, it is shown that cavitation depends exactly on the power-law parameter. The larger the pre-strained value is prescribed, the earlier cavitation occurs; the larger the power-law parameter is, the later cavitation occurs. Dynamically, once a cavity forms at the axial line of the circular sheet, the motion of the formed cavity with time is a classical nonlinear periodic oscillation. Moreover, the oscillation amplitude increases with the increasing tensile load or with the increasing pre-strained value.
引文
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