弹性圆柱壳动力和热屈曲中的辛方法
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摘要
结构的稳定性问题一直以来都是近代固体力学研究的热点,也是工程问题中关注的焦点问题之一。作为固体力学中最基本的力学模型,圆柱壳的失稳问题一直备受关注。已经有许多学者针对圆柱壳在不同环境下的稳定性问题进行了研究,并解决了诸多实际的问题。实验中的一些局部屈曲现象需要由基本理论加以研究和解释,因此提出一种新的方法是必要的。
冲击荷载和热荷载是两种比较常见的荷载形式,由于这两种荷载的特性不同,圆柱壳在这两种荷载分别作用下屈曲时具有一定的差异性。在冲击荷载作用下,由于圆柱壳的惯性和荷载作用的时效性,冲击荷载会在壳体内以应力波形式传播和反射等。由于应力波是局部的,圆柱壳在冲击载荷和热耦合作用下的屈曲问题呈现出整体屈曲中伴随着局部的屈曲。在拉格朗日体系下一类变量求解圆柱壳的前屈曲问题时,面临高阶的偏微方程,传统的分离变量法等在解决这类高阶偏微方程时失去了效用。因此在新的体系研究该类问题是必要的。
本文针对弹性圆柱壳在热荷载、冲击荷载以及两种荷载耦合作用下的屈曲问题,通过引入原变量与对偶变量组成的全状态变量,建立了系统的哈密顿体系。从而将问题从欧几里德空间过渡到辛空间。在辛空间中,圆柱壳的临界屈曲荷载和屈曲模态归结为辛本征值和本征解问题。在辛体系下,零本征值本征解和非零本征值本征解分别对应圆柱壳屈曲时的轴对称屈曲模态和非轴对称屈曲模态。对于壳体的前屈曲问题可采用小变形理论,而后屈曲问题则需采用几何大变形理论。借助于辛本征解的完备性,后屈曲问题屈曲模态通过辛本征解的展开逼近,并以前屈曲模态作为初始模态进行研究和讨论。这样就将前屈曲和后屈曲问题有机统一起来。从而揭示了结构从前屈曲到后屈曲整个屈曲发展过程。同时形成一种求解非线性问题方法。
在讨论了应力波在圆柱壳内部传播、反射和透射等之后,研究了应力波对圆柱壳屈曲的影响。给出了在轴向冲击载荷、热荷载以及它们的耦合作用下和各种端部支承条件下圆柱壳屈曲的临界荷载和屈曲模态。结果表明在轴向冲击载荷下的屈曲必然伴随应力波效应,因而会产生局部的屈曲;而圆柱壳发生热屈曲表现为整体的屈曲形式。在圆柱壳受到冲击载荷作用时,应力波传播的初期,壳体容易发生的局部屈曲表现为冲击端形如“喇叭口”的张开型;由于应力波在另一端发生反射后,则反射端的张开型屈曲容易出现;在脉冲载荷下圆柱壳会发生中部“竹节”形状的屈曲。脉冲载荷往往是由冲击弹性质量块(子弹)中应力波的传播、反射和透射形成的。文中对这种现象进行了理论上的分析。事实上,这些局部屈曲现象是能量集中的表现。研究结果得出结论,应力波在圆柱壳内部传播、反射和透射是产生该壳局部屈曲的主要因素。圆柱壳局部屈曲形式还与圆柱壳的物理参数和几何参数有关。脉冲的长短和圆柱壳的波速与波阻抗对局部屈曲现象起着相当重要的作用。
为了探讨圆柱壳后屈曲的发展对初始模态的敏感程度,采用各种前屈曲模态作为圆柱壳后屈曲问题中的初始模态(包括初始模态的幅值、环向各阶模态和母线方向的各分支模态等)。数值结果表明,在圆柱壳热后屈曲问题中,对初始模态是不敏感的。此外,对于比较薄的圆柱壳,其后屈曲过程发展中的模态在环向倾向于高阶;而对于比较长的圆柱壳的后屈曲,会在母线方向上产生比较多的屈曲波纹。这些结果和结论为工程结构稳定性设计提供了重要数据和依据。
The stability of structure is an important research subject in modern solid mechanics and much attention has been focused on the problem in industry equipments.Since cylindrical shell is a basic structure,its buckling problems play an important role in the theories of structural stability.Many researches have been carried out and great progress has been made in solving many practical problems.Some phenomena which deal with local buckling of a cylindrical shell in experiments are reported.To explain these,it is necessary to consider basic theories and a new method is needed.
It is well known that an object can be impacted or heated.The buckling phenomena of a cylindrical shell subjected to an impact load or a thermal load are not the same because of their different characters.When a shell is impacted,the internal forces caused by the impact will be propagated and reflected in a stress wave form.Becuase stress waves are affected locally,the shell will be overall bucked and accompanied with local wrinkles,if both loads are applied. High-order partial differential equations often occur when one solves problems which deal with pre-buckling of a cylindrical shell.Sometimes traditional separation of variables is not work anymore if the problem is presented in a Lagrangian system.When this happens,it is necessary to introduce a new system.
This paper deals with pre-buckling of a shell subjected to thermal load,impact load or coupling load.By introducing a Hamiltonian function,the Hamiltonian system for the problem is established.This means that the transformation from a Lagrangian system to a Hamiltonian system is finished.In a symplectic space,eigenvalues and eigenfunctions take the place of the critical buckling loads and buckling modes of the problem.The zero-eigenvalues describe axisymmetric buckling and non-zero eigenvalues mean non-axisymmetric bucking.In pre-buckling problems,small deformation theories are used,while lager deformation theories are applied in solving post-buckling problems.Based on the complete solutions obtained in the pre-buckling problems,a symplectic eigenfunction expansion method is developed and the whole progress from prebuckling to postbuckling are described.A new method for solving such nonlinear problems is presented.
Stress waves play an important role in the local buckling of a shell when the waves propogate,reflect and transmit in the shell.After studying these effects,critical loads and buckling modes of a shell subjected to impact loads,thermal loads and coupling loads with different boundary condtions are presented.Numerical results show that when a shell is impacted,local buckling is unavoidable due to the local effects of the stress waves.There will be overall buckling if a shell is heated.When a shell is impacted,a big open is tend to occur at the impact end in the beginning and when the stress waves are reflected,the big open may appear at the reflect end.If a pulse is imposed,a bamboo-node type will come up in the middle of the shell.The pulse usually is pruduced by the propagation,reflection and transmission of the stress waves in the bullet.In fact,local buckling of a shell is a type of energy concentration in the shell.It can be concluded that local buckling in a shell are mainly caused by the propagation, reflection and transmission of stress waves.Local buckling have something to do with not only the phisical properties of the shell but also its geometric feature parameters.The length of a pulse and the wave impedence are of major signifance on a bamboo node-type.
In post-buckling problems,an intial mode which is in the form of pre-buckling solutions is adopted,and the effects of the initial mode to the post-buckling are discussed in three ways,that is,its amplitudes,circular orders and axial branches.It is found that the shell is not sensitive to the intial mode.In post-buckling process,a thinner shell is inclined to a higher circular order and a longer shell tends to a higher axial order.These provide important rules for structure stability design in engineering.
冲击荷载和热荷载是两种比较常见的荷载形式,由于这两种荷载的特性不同,圆柱壳在这两种荷载分别作用下屈曲时具有一定的差异性。在冲击荷载作用下,由于圆柱壳的惯性和荷载作用的时效性,冲击荷载会在壳体内以应力波形式传播和反射等。由于应力波是局部的,圆柱壳在冲击载荷和热耦合作用下的屈曲问题呈现出整体屈曲中伴随着局部的屈曲。在拉格朗日体系下一类变量求解圆柱壳的前屈曲问题时,面临高阶的偏微方程,传统的分离变量法等在解决这类高阶偏微方程时失去了效用。因此在新的体系研究该类问题是必要的。
本文针对弹性圆柱壳在热荷载、冲击荷载以及两种荷载耦合作用下的屈曲问题,通过引入原变量与对偶变量组成的全状态变量,建立了系统的哈密顿体系。从而将问题从欧几里德空间过渡到辛空间。在辛空间中,圆柱壳的临界屈曲荷载和屈曲模态归结为辛本征值和本征解问题。在辛体系下,零本征值本征解和非零本征值本征解分别对应圆柱壳屈曲时的轴对称屈曲模态和非轴对称屈曲模态。对于壳体的前屈曲问题可采用小变形理论,而后屈曲问题则需采用几何大变形理论。借助于辛本征解的完备性,后屈曲问题屈曲模态通过辛本征解的展开逼近,并以前屈曲模态作为初始模态进行研究和讨论。这样就将前屈曲和后屈曲问题有机统一起来。从而揭示了结构从前屈曲到后屈曲整个屈曲发展过程。同时形成一种求解非线性问题方法。
在讨论了应力波在圆柱壳内部传播、反射和透射等之后,研究了应力波对圆柱壳屈曲的影响。给出了在轴向冲击载荷、热荷载以及它们的耦合作用下和各种端部支承条件下圆柱壳屈曲的临界荷载和屈曲模态。结果表明在轴向冲击载荷下的屈曲必然伴随应力波效应,因而会产生局部的屈曲;而圆柱壳发生热屈曲表现为整体的屈曲形式。在圆柱壳受到冲击载荷作用时,应力波传播的初期,壳体容易发生的局部屈曲表现为冲击端形如“喇叭口”的张开型;由于应力波在另一端发生反射后,则反射端的张开型屈曲容易出现;在脉冲载荷下圆柱壳会发生中部“竹节”形状的屈曲。脉冲载荷往往是由冲击弹性质量块(子弹)中应力波的传播、反射和透射形成的。文中对这种现象进行了理论上的分析。事实上,这些局部屈曲现象是能量集中的表现。研究结果得出结论,应力波在圆柱壳内部传播、反射和透射是产生该壳局部屈曲的主要因素。圆柱壳局部屈曲形式还与圆柱壳的物理参数和几何参数有关。脉冲的长短和圆柱壳的波速与波阻抗对局部屈曲现象起着相当重要的作用。
为了探讨圆柱壳后屈曲的发展对初始模态的敏感程度,采用各种前屈曲模态作为圆柱壳后屈曲问题中的初始模态(包括初始模态的幅值、环向各阶模态和母线方向的各分支模态等)。数值结果表明,在圆柱壳热后屈曲问题中,对初始模态是不敏感的。此外,对于比较薄的圆柱壳,其后屈曲过程发展中的模态在环向倾向于高阶;而对于比较长的圆柱壳的后屈曲,会在母线方向上产生比较多的屈曲波纹。这些结果和结论为工程结构稳定性设计提供了重要数据和依据。
The stability of structure is an important research subject in modern solid mechanics and much attention has been focused on the problem in industry equipments.Since cylindrical shell is a basic structure,its buckling problems play an important role in the theories of structural stability.Many researches have been carried out and great progress has been made in solving many practical problems.Some phenomena which deal with local buckling of a cylindrical shell in experiments are reported.To explain these,it is necessary to consider basic theories and a new method is needed.
It is well known that an object can be impacted or heated.The buckling phenomena of a cylindrical shell subjected to an impact load or a thermal load are not the same because of their different characters.When a shell is impacted,the internal forces caused by the impact will be propagated and reflected in a stress wave form.Becuase stress waves are affected locally,the shell will be overall bucked and accompanied with local wrinkles,if both loads are applied. High-order partial differential equations often occur when one solves problems which deal with pre-buckling of a cylindrical shell.Sometimes traditional separation of variables is not work anymore if the problem is presented in a Lagrangian system.When this happens,it is necessary to introduce a new system.
This paper deals with pre-buckling of a shell subjected to thermal load,impact load or coupling load.By introducing a Hamiltonian function,the Hamiltonian system for the problem is established.This means that the transformation from a Lagrangian system to a Hamiltonian system is finished.In a symplectic space,eigenvalues and eigenfunctions take the place of the critical buckling loads and buckling modes of the problem.The zero-eigenvalues describe axisymmetric buckling and non-zero eigenvalues mean non-axisymmetric bucking.In pre-buckling problems,small deformation theories are used,while lager deformation theories are applied in solving post-buckling problems.Based on the complete solutions obtained in the pre-buckling problems,a symplectic eigenfunction expansion method is developed and the whole progress from prebuckling to postbuckling are described.A new method for solving such nonlinear problems is presented.
Stress waves play an important role in the local buckling of a shell when the waves propogate,reflect and transmit in the shell.After studying these effects,critical loads and buckling modes of a shell subjected to impact loads,thermal loads and coupling loads with different boundary condtions are presented.Numerical results show that when a shell is impacted,local buckling is unavoidable due to the local effects of the stress waves.There will be overall buckling if a shell is heated.When a shell is impacted,a big open is tend to occur at the impact end in the beginning and when the stress waves are reflected,the big open may appear at the reflect end.If a pulse is imposed,a bamboo-node type will come up in the middle of the shell.The pulse usually is pruduced by the propagation,reflection and transmission of the stress waves in the bullet.In fact,local buckling of a shell is a type of energy concentration in the shell.It can be concluded that local buckling in a shell are mainly caused by the propagation, reflection and transmission of stress waves.Local buckling have something to do with not only the phisical properties of the shell but also its geometric feature parameters.The length of a pulse and the wave impedence are of major signifance on a bamboo node-type.
In post-buckling problems,an intial mode which is in the form of pre-buckling solutions is adopted,and the effects of the initial mode to the post-buckling are discussed in three ways,that is,its amplitudes,circular orders and axial branches.It is found that the shell is not sensitive to the intial mode.In post-buckling process,a thinner shell is inclined to a higher circular order and a longer shell tends to a higher axial order.These provide important rules for structure stability design in engineering.
引文
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