考虑初始缺陷的网格结构的非线性稳定性理论及其应用
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摘要
在材料的加工过程中及外载荷和环境的作用下，材料内部存在大量的微裂纹和微孔洞等微观缺陷，这些微观缺陷不断地发展、扩大、汇合，使材料的力学性质逐渐劣化，直至破坏，这些微观缺陷引起的材料性质劣化可以用初始损伤来模拟。对于工程中常用的薄壁结构，如冷弯薄壁结构和网格结构等，从其施工到服役期间都不可避免地会产生一定的静变形。结构的初始缺陷（结构静变形）和材料的物理缺陷是网格结构的两大初始缺陷，而网格结构对初始缺陷较为敏感。本文以初始缺陷为切入点，把轴对称和非轴对称网壳结构（扁球面网壳、扁锥面网壳和扁柱面网壳）作为主要研究对象，系统地研究了网壳这类薄壁结构的非线性动力学行为；并利用突变理论研究了具有材料初始缺陷（初始损伤）网格结构整体稳定性问题。为推动薄壁结构的安全性和寿命评估奠定了必要的理论基础。
本文首先分析了网格结构近些年来发生的工程事故，回顾了网格结构稳定性的国内外研究现状和研究方法，阐明了本研究的理论意义和应用前景。其次，建立了考虑结构初始缺陷的网格结构的动力学模型，并进行了动力学分析。然后，构建了建筑用钢的损伤本构关系，研究了具有材料初始缺陷的网格结构的非线性弯曲问题，建立了考虑材料初始缺陷的动力学模型，利用分叉理论研究了系统在不动点附近的局部稳定性质，利用混沌理论解释系统混沌运动特性和对混沌运动的抑制，较为系统地研究了考虑材料初始缺陷网壳的非线性动力学稳定性。此外，研究了结构的初始缺陷和材料的初始缺陷耦合情况下的网格结构的非线性动力学特性。最后，利用突变理论建立了考虑材料初始缺陷的网格结构的尖点突变模型。本文主要研究了系统的局部稳定性和整体稳定性，揭示了系统的混沌运动性态。本文的主要工作和研究成果包括：
1）本文将网格结构的静变形作为结构的初始缺陷，建立了结构的非线性动力学模型，利用Floquet指数法研究了系统分叉问题，并给出了平衡点的局部稳定状态，利用计算机模拟绘出了系统在不同初始条件和不同参数下的分叉图。最后利用Melnikov函数给出了系统产生混沌运动的临界载荷，并在给定的混沌域内进行了计算机仿真模拟，证明了系统产生了混沌运动，由此得到了结构初始缺陷对结构局部稳定性和混沌运动的影响。
2）从不可逆热力学理论出发建立了建筑用钢损伤本构关系。该本构关系较为真实地反应了结构的实际工作状态，物理意义明确、形式简单，能较方便地用于钢结构设计。文中基于该损伤本构关系推导出具有材料初始缺陷的一类网格结构的大挠度方程，利用修正迭代法求解了具有损伤的网格结构在均布载荷作用下的非线性弯曲问题。
3）利用Galerkin方法推导出一个含二次项和三次项的非线性微分振动方程，给出了具有材料初始缺陷的网格结构的非线性自由振动方程的精确解。用Floquet指数法研究了系统分叉问题，给出了不动点附近的局部稳定状态，并且利用计算机模拟仿真绘出了不同初始条件和不同参数下的分叉图。
4）在假设薄膜张力有两项组成后，将几何协调方程化为两个独立方程，选取中心最大振幅为摄动参数，采用摄动变分法将变分方程和微分方程线性化，对夹紧固定边界条件下具有材料初始缺陷的扁球面网壳非线性振动方程进行求解，得出了以静载荷为参数的最低固有频率与中心最大振幅之间的特征关系。然后利用Galerkin方法得到具有初始损伤网格结构（扁球面网壳、扁锥面网壳和扁柱面网壳）的一个含二次和三次的非线性振动微分方程，求解了具有初始损伤网格结构的非线性自由振动方程，并且研究了该类网格结构的非线性动力稳定问题。
5）由于网格结构对初始缺陷的敏感性，在研究其稳定性时必须考虑结构的静变形和材料的物理缺陷。尤其是在狂风、暴雨、暴雪和地震等强激励作用下，网格结构更容易失稳而坍塌，实际工程中也发生过很多这样的事故。因此，本文考虑了结构的初始和材料的物理缺陷耦合作用，建立了网格结构在强激励作用下的非线性动力学模型，揭示了其丰富的动力学特性。
6）利用突变理论研究了具有材料初始缺陷的单层扁球面网壳的非线性动力稳定性问题，建立了系统静态和动态的尖点突变模型，研究了该系统的整体稳定性问题。
Damage, in its mechanical sense in solid matericals with initial micro-imperfectionsuch as microvids or microcracks subjected to unfavorable mechanical andenvironmental conditions, is the creation and growth of microvids or microcracks underexternal load and makes deterioration of the material. On the other hand, for thin-walledstructure as cold-formed thin-walled structure and lattice structure and so on, it occurredstatic deformation (i.e. intinal imperfection) from construction to occupation. Initialimperfection included intial physical imperfection and intial structure imperfection. Thelattice structure was sensitive structure to imperfection. So this dissertationsystematically studied non-linear dynamic stability of axial symmetrical and non-axialsymmetrical lattices taking the initial imperfection as research breakthrough point.Finally, using catastrophe theory, this dissertation studie d the overall stability of thelattices with the initial damage.Thus a series of useful throretical results were obtained,which was essential in the safety and durability of lattice structure.
Firstly, the development in nonlinear theorier of stability and accident analysis forthe lattice structures was reviewed, which indicated theoretical significance andapplication prospect of this research. Secondly, taking account of the effects of theinitial imperfection, dynamic model of the lattices with initial imperfection wasestablished to analyse the characteristic nonlinear dynamic. And, considering the effectof the initial damaged, damage model of structure steel was presented. Then nonlinearbending of the lattices with initial damage was solved. Thirdly, nonlinear dynamic model of the lattices with initial damage was obtained to systematically study thenonlinear dynamic stability of the lattice with initial damage. This dissertation studiedlocal stability of the lattice with initial damage at the fixed point using bifurcationtheory and explained the chaotic characteristic and control chaotic motion by chaotictheory. Meanwhile, using the catastrophe theory, this dissertation studied the overallstability of the system with initial damage. Then, in order to take account of couplingfactor of the initial structure imperfection and the initial materials damage, nonlineardynamic model of the lattices with coupling the intial structure imperfection and intialdamage was estabilished. Finally, a cusp catastrophic mode of the lattices with theinitial damage was established using catastrophe theory. Summing up the above, thisdissertation studied systematically local stability and overall stability. The main worksand achievements of this dissertation were as follows:
(1) This dissertation took deformation as initial imperfection and nonlineardynamic model of the lattices with initial imperfection was given. The problem of localstability at the equilibrium of the system was discussed by exponent Floquet underdifferent intial conditions and parameter. So theoretically critical condition of chaoticmotion was given using the Melnikov function method. The chaos motion of theshallow reticulated spherical shell with initial imperfection was simulated by computernumerical emulation under nonlinear forced vibration. And it was founded that theinitial imperfection made the chaos motion of the system easily occur.
(2) The damage model of the structure steel was presented by theory of irreversiblethermodynamics. This model had clear physical meaning, simple form, providing anexpedient usage in steel structure design. Then the nonlinear large deflection equation ofthe initial damage was given by the model and the problem of nonlinear bending wassolved by the method of modified iteration.
(3) The nonlinear dynamical equations of the shallow reticulated spherical shellwith initial damage were given by Galerkin method. An accurate solution of the shallowreticulated spherical shell with initial damage was solved. The problem of local stabilityat the equilibrium of the system was discussed by exponent Floquet under differentintial conditions and parameter. It was found that the initial damage had effect on localstability of the lattices.
(4) Assuming the thin film tension consists of two items, the compatible equationswere simplified to two independent equations. Selecting the maximal amplitude in thecenter of the shallow reticulated spherical shell with damage as the perturbationparameter, the nonlinear vibration equation of the system with damage under theboundary conditions of fixed and clamped was solved by perturbation variation. So therelation of the lowest natural frequency and the maximal amplitude was obtained.Then,the nonlinear dynamical equations of the shallow reticulated spherical shell with initialdamage were given by Galerkin method. An accurate solution of the shallow reticulatedspherical shell with initial damage was obtained. Finally, theoretical critical condition ofchaotic motion was given by the Melnikov function method. The chaos motion of theshallow reticulated spherical shell with initial damage was simulated by computernumerical emulation under nonlinear forced vibration. And it was founded that theinitial damage makes the chaos motion of the system easily occur.
(5) As the lattices had sensitive to the initial imperfection, this dissertation tookaccount of structure imperfections and phycisal imperfection. The lattices with twoimperfections were frail to collapse under rainstorm, blizzard and earthquake. So anonlinear dynamic model of the lattices with coupling the intial imperfection and initialdamage was obtained to reveal the plenty dynamics characteristic.
(6) The nonlinear dynamical stability of the lattices with initial damage wasanalyzed by the method of catastrophe. Firstly, potential functionΠof the globecharacter was obtained. Then the static and dynamic cusp catastrophic model of thesystem was given separately. The equilibrium state of the system was dissussed.
本文首先分析了网格结构近些年来发生的工程事故，回顾了网格结构稳定性的国内外研究现状和研究方法，阐明了本研究的理论意义和应用前景。其次，建立了考虑结构初始缺陷的网格结构的动力学模型，并进行了动力学分析。然后，构建了建筑用钢的损伤本构关系，研究了具有材料初始缺陷的网格结构的非线性弯曲问题，建立了考虑材料初始缺陷的动力学模型，利用分叉理论研究了系统在不动点附近的局部稳定性质，利用混沌理论解释系统混沌运动特性和对混沌运动的抑制，较为系统地研究了考虑材料初始缺陷网壳的非线性动力学稳定性。此外，研究了结构的初始缺陷和材料的初始缺陷耦合情况下的网格结构的非线性动力学特性。最后，利用突变理论建立了考虑材料初始缺陷的网格结构的尖点突变模型。本文主要研究了系统的局部稳定性和整体稳定性，揭示了系统的混沌运动性态。本文的主要工作和研究成果包括：
1）本文将网格结构的静变形作为结构的初始缺陷，建立了结构的非线性动力学模型，利用Floquet指数法研究了系统分叉问题，并给出了平衡点的局部稳定状态，利用计算机模拟绘出了系统在不同初始条件和不同参数下的分叉图。最后利用Melnikov函数给出了系统产生混沌运动的临界载荷，并在给定的混沌域内进行了计算机仿真模拟，证明了系统产生了混沌运动，由此得到了结构初始缺陷对结构局部稳定性和混沌运动的影响。
2）从不可逆热力学理论出发建立了建筑用钢损伤本构关系。该本构关系较为真实地反应了结构的实际工作状态，物理意义明确、形式简单，能较方便地用于钢结构设计。文中基于该损伤本构关系推导出具有材料初始缺陷的一类网格结构的大挠度方程，利用修正迭代法求解了具有损伤的网格结构在均布载荷作用下的非线性弯曲问题。
3）利用Galerkin方法推导出一个含二次项和三次项的非线性微分振动方程，给出了具有材料初始缺陷的网格结构的非线性自由振动方程的精确解。用Floquet指数法研究了系统分叉问题，给出了不动点附近的局部稳定状态，并且利用计算机模拟仿真绘出了不同初始条件和不同参数下的分叉图。
4）在假设薄膜张力有两项组成后，将几何协调方程化为两个独立方程，选取中心最大振幅为摄动参数，采用摄动变分法将变分方程和微分方程线性化，对夹紧固定边界条件下具有材料初始缺陷的扁球面网壳非线性振动方程进行求解，得出了以静载荷为参数的最低固有频率与中心最大振幅之间的特征关系。然后利用Galerkin方法得到具有初始损伤网格结构（扁球面网壳、扁锥面网壳和扁柱面网壳）的一个含二次和三次的非线性振动微分方程，求解了具有初始损伤网格结构的非线性自由振动方程，并且研究了该类网格结构的非线性动力稳定问题。
5）由于网格结构对初始缺陷的敏感性，在研究其稳定性时必须考虑结构的静变形和材料的物理缺陷。尤其是在狂风、暴雨、暴雪和地震等强激励作用下，网格结构更容易失稳而坍塌，实际工程中也发生过很多这样的事故。因此，本文考虑了结构的初始和材料的物理缺陷耦合作用，建立了网格结构在强激励作用下的非线性动力学模型，揭示了其丰富的动力学特性。
6）利用突变理论研究了具有材料初始缺陷的单层扁球面网壳的非线性动力稳定性问题，建立了系统静态和动态的尖点突变模型，研究了该系统的整体稳定性问题。
Damage, in its mechanical sense in solid matericals with initial micro-imperfectionsuch as microvids or microcracks subjected to unfavorable mechanical andenvironmental conditions, is the creation and growth of microvids or microcracks underexternal load and makes deterioration of the material. On the other hand, for thin-walledstructure as cold-formed thin-walled structure and lattice structure and so on, it occurredstatic deformation (i.e. intinal imperfection) from construction to occupation. Initialimperfection included intial physical imperfection and intial structure imperfection. Thelattice structure was sensitive structure to imperfection. So this dissertationsystematically studied non-linear dynamic stability of axial symmetrical and non-axialsymmetrical lattices taking the initial imperfection as research breakthrough point.Finally, using catastrophe theory, this dissertation studie d the overall stability of thelattices with the initial damage.Thus a series of useful throretical results were obtained,which was essential in the safety and durability of lattice structure.
Firstly, the development in nonlinear theorier of stability and accident analysis forthe lattice structures was reviewed, which indicated theoretical significance andapplication prospect of this research. Secondly, taking account of the effects of theinitial imperfection, dynamic model of the lattices with initial imperfection wasestablished to analyse the characteristic nonlinear dynamic. And, considering the effectof the initial damaged, damage model of structure steel was presented. Then nonlinearbending of the lattices with initial damage was solved. Thirdly, nonlinear dynamic model of the lattices with initial damage was obtained to systematically study thenonlinear dynamic stability of the lattice with initial damage. This dissertation studiedlocal stability of the lattice with initial damage at the fixed point using bifurcationtheory and explained the chaotic characteristic and control chaotic motion by chaotictheory. Meanwhile, using the catastrophe theory, this dissertation studied the overallstability of the system with initial damage. Then, in order to take account of couplingfactor of the initial structure imperfection and the initial materials damage, nonlineardynamic model of the lattices with coupling the intial structure imperfection and intialdamage was estabilished. Finally, a cusp catastrophic mode of the lattices with theinitial damage was established using catastrophe theory. Summing up the above, thisdissertation studied systematically local stability and overall stability. The main worksand achievements of this dissertation were as follows:
(1) This dissertation took deformation as initial imperfection and nonlineardynamic model of the lattices with initial imperfection was given. The problem of localstability at the equilibrium of the system was discussed by exponent Floquet underdifferent intial conditions and parameter. So theoretically critical condition of chaoticmotion was given using the Melnikov function method. The chaos motion of theshallow reticulated spherical shell with initial imperfection was simulated by computernumerical emulation under nonlinear forced vibration. And it was founded that theinitial imperfection made the chaos motion of the system easily occur.
(2) The damage model of the structure steel was presented by theory of irreversiblethermodynamics. This model had clear physical meaning, simple form, providing anexpedient usage in steel structure design. Then the nonlinear large deflection equation ofthe initial damage was given by the model and the problem of nonlinear bending wassolved by the method of modified iteration.
(3) The nonlinear dynamical equations of the shallow reticulated spherical shellwith initial damage were given by Galerkin method. An accurate solution of the shallowreticulated spherical shell with initial damage was solved. The problem of local stabilityat the equilibrium of the system was discussed by exponent Floquet under differentintial conditions and parameter. It was found that the initial damage had effect on localstability of the lattices.
(4) Assuming the thin film tension consists of two items, the compatible equationswere simplified to two independent equations. Selecting the maximal amplitude in thecenter of the shallow reticulated spherical shell with damage as the perturbationparameter, the nonlinear vibration equation of the system with damage under theboundary conditions of fixed and clamped was solved by perturbation variation. So therelation of the lowest natural frequency and the maximal amplitude was obtained.Then,the nonlinear dynamical equations of the shallow reticulated spherical shell with initialdamage were given by Galerkin method. An accurate solution of the shallow reticulatedspherical shell with initial damage was obtained. Finally, theoretical critical condition ofchaotic motion was given by the Melnikov function method. The chaos motion of theshallow reticulated spherical shell with initial damage was simulated by computernumerical emulation under nonlinear forced vibration. And it was founded that theinitial damage makes the chaos motion of the system easily occur.
(5) As the lattices had sensitive to the initial imperfection, this dissertation tookaccount of structure imperfections and phycisal imperfection. The lattices with twoimperfections were frail to collapse under rainstorm, blizzard and earthquake. So anonlinear dynamic model of the lattices with coupling the intial imperfection and initialdamage was obtained to reveal the plenty dynamics characteristic.
(6) The nonlinear dynamical stability of the lattices with initial damage wasanalyzed by the method of catastrophe. Firstly, potential functionΠof the globecharacter was obtained. Then the static and dynamic cusp catastrophic model of thesystem was given separately. The equilibrium state of the system was dissussed.
引文
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