向量式有限元薄壳单元的理论与应用
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摘要
薄壳结构在实际工程中的应用十分广泛,在外荷载作用下,薄壳结构往往会出现大变形大变位、弹塑性材料非线性以及碰撞接触、破裂破碎和穿透等严重不连续行为。传统有限元在处理这类复杂结构行为分析时需事先区分行为类型并采用专用的公式和程序来处理,对于结构行为预测和破坏全过程模拟往往难以得到满意的结果。
本文基于向量式有限元这一新型分析方法,以向量力学为理论基础,通过理论分析、程序开发和数值模拟等手段实现金属薄壳结构的复杂力学行为分析。研究工作沿两条主线：一是单元发展,包括薄膜单元、薄板单元和薄壳单元；二是复杂结构行为实现,包括大变形大转动、屈曲、碰撞、断裂和穿透。
第二章介绍了向量式有限元法的基本概念、假设、原理和推导思路,并对中央差分公式、共转坐标和逆向运动、材料(非)线性、静力和动力求解、等效质量和惯量矩阵以及误差分析和内力自平衡机制等问题进行了探讨；针对大变形大转动运动以及屈曲、碰撞、断裂和穿透等复杂结构行为问题及其求解思路进行了简单讨论；总结了向量式有限元法的分析步骤与流程。
第三章建立了三角形CST常应变薄膜单元和四节点等参薄膜单元的向量式有限元基本公式,描述运动解析的原理及变形坐标系下单元节点内力的求解方法,同时对四节点膜单元的位置模式和内力计算的数值积分等问题提出处理方法；编制了相应的分析程序并通过算例进行验证。在复杂结构行为分析方面,针对膜材大变形大转动问题,采用本文方法跟踪获得其运动变形全过程；针对膜材的碰撞接触行为,采用“点-三角形”检测和基于中央差分式的罚接触力方法处理膜材的碰撞检测和碰撞响应问题,跟踪获得其碰撞接触全过程；针对膜材的破裂破碎行为,采用失效应力判断准则和质点分裂的向量式有限元处理,跟踪获得其破裂和破碎的全过程。
第四章建立了三角形DKT薄板单元的向量式有限元基本理论,同时对质点的质量矩阵与惯量矩阵、应力计算的数值积分及插值方法、时间步长及阻尼参数的取值等问题提出处理方式；并通过平板结构的静、动力算例进行分析验证。
第五章基于三角形CST薄膜单元和三角形DKT薄板单元的叠加组合,建立了三角形薄壳单元的向量式有限元基本理论,同时对质点位移和内力的合并和分离、单元应变和应力的合并和分离以及单元节点内力积分方案的处理等关键问题提出处理方法；基于C-S粘塑性材料本构模型和动态Mises屈服准则,推导获得C-S材料本构模型的弹塑性增量分析步骤,并将其引入向量式有限元薄壳单元的基本理论推导,实现金属薄壳结构同时考虑塑性硬化和应变率硬化效应的非线性分析。
第六章研究了向量式有限元在薄壳结构复杂行为分析中的应用。针对薄壳结构的屈曲和后屈曲行为,采用位移和力控制处理方法来跟踪获得其变形全过程,并分析了两种控制方法的特点和应用范围；针对薄壳结构的碰撞接触行为,仍采用“点-三角形”检测和罚接触力法处理碰撞检测和碰撞响应问题,跟踪获得其碰撞接触的全过程；针对薄壳结构的破裂破碎行为,采用失效应变断裂判断准则和质点分裂的向量式有限元处理,跟踪获得其破裂和破碎的全过程；针对薄壳结构的穿透行为,结合碰撞检测、碰撞响应处理机制以及基于失效应变的断裂准则,实现刚体对薄壳结构的穿透过程。
第七章综合本文基于向量式有限元的薄壳结构非线性分析基本理论,包括薄壳单元、材料本构、破裂破碎的理论公式与有效算法,实现了钢储罐结构在爆炸冲击荷载作用下的动力响应分析和破坏全过程分析,分析了内部三角形简化形式爆炸荷载作用下平顶罐、锥顶罐和拱顶罐的应力变形发展以及破坏模式。
本文成果体现了向量式有限元分析方法在处理结构大变形、大变位、碰撞、断裂和穿透等强非线性和不连续力学行为中的独特优势,推动了向量式有限元的理论发展与工程应用,为薄膜结构、金属薄壳结构的复杂行为分析提供了一种新的有效方法。
Thin shell structures are widely applied in practical engineering. Under the externnal loading, the behaviors of the thin shell structures usually contain large deformation and large deflection, elastoplastic nonlinear material and complex discontinuous behaviors such as collision-contact, crack-fracture and penetration. For these complex structural behaviors, the traditional finite element analysis need to distinguish the type of the behavior first, then adopt the corresponding special formulas ande process. Thus, for the problems of the structural behavior prediction and the whole process simulation of structural failure, it is difficult to obtain satisfactory results by using the traditional finite element method.
The Vector Form Intrinsic Finite Element (VFIFE) is a new analysis method on the basis of vector mechanics theory. This thesis achieves the complex mechanical behaviors for thin metal shell structures by using theoretical analysis, program development and numerical simulation. The study of this thesis along the following two main lines:one is the element development including thin membrane element, thin plate element and thin shell element, and the other is the realization of the complex behaviors including large deformation and large rotation, buckling and post buckling, collision, fracture and penetration.
Chapter2introduces the basic concepts, assumptions, principles and deduction ideas of VFIFE. And the related problems such as center differential formula, co-rotational coordinate and reverse movement, nonlinear material, static and dynamic calculation, equivalent mass matrix and inertia matrix, error analysis and internal force balance mechanism are studied. Then the analysis and solving ideas for complex behavior problems such as large deformation and large rotation, buckling and post buckling, collision-contact, crack-fracture and penetration are simply discussed. Finally, the analysis process of VFIFE method is summarized.
Chapter3establishes the basic formulas of VFIFE for the triangle CST constant strain membrane element and4-node quadrilateral isoparametric membrane element, describes the principles of motion analysis and the solving method of element node internal forces in deformation coordinate system. Then, some special problems such as the position modes and the numerical integration of the internal force calculation for the4-node quadrilateral isoparametric membrane element are presented, and the corresponding treatment methods are proposed. Computer analysis programs are then developed, and the derived theory and the developed programs are verified through numerical examples finially. In the analysis of complex structural behaviors, for the large deformation and large rotation problems, VFIFE method is used to track the whole process of structural motion and deformation. For the collision-contact problems, the "point-triangle" detection and the penalty contact force method based on the central difference method are adopted separately for the problems of collision detection and collision response, and the whole process of structural collision-contact is tracked. For the crack-fracture problems, the judgment criterion based on the failure stress and the particle splitting method based on VFIFE are adopted to track the whole process of structural crack-fracture.
Chapter4establishes the basic theory of VFIFE for the triangle DKT plate element. Then, some special problems such as the mass matrix and inertia matrix of particle, the numerical integration of the stress calculation and interpolation method, and the parameters of time step and damp for the triangle DKT plate element are presented, and the corresponding treatment methods are proposed. All of above are verified by the static and dynamic analysis of plate structural examples finally.
Chapter5establishes the basic theory of VFIFE for the triangle shell element based on the combination of the triangle CST constant strain membrane element and the triangle DKT plate element. Then, the processsing methods for the problems are proposed, including the combination and the division of the particle displacements and the internal forces, the combination and the division of the element stresses and strains, and the element node integration scheme. Based on the C-S viscoplastic constitutive model and the dynamic Mises yield criterion, the elastoplastic incremental analysis steps of the C-S constitutive model are deducted, and introduced into the theoretical derivation of the thin shell element of VFIFE. Thus, the plastic hardening effect and the strain rate hardening effect for the nonlinear analysis of the thin metal shell structures are both considered in this material model.
Chapter6carries out the research of complex structural behaviors for thin shell structures. For the buckling and post buckling problems, the displacement and the force control methods are used to track the whole process of structural deformation, and the characteristics and applications of the two control methods are then analysed. For the collision-contact problems, the "point-triangle" detection and the penalty contact force method based on the central difference method are still adopted separately for the problems of collision detection and collision response, and the whole process of structural collision-contact is tracked. For the crack-fracture problems, the judgment criterion based on the failure strain and the particle splitting method based on VFIFE are adopted to track the whole process of structural crack-fracture. For the penetration problems, the combination of the collision detection mechanisms, the collision response mechanisms and the fracture criterion based on the failure strain are adopted, and the process of penetration for rigid body-thin shell structure is achieved.
Chapter7synthesizes the nonlinear analysical basic theorys for the thin shell structures based on VFIFE, including the theory formulas and the effective algorithms of thin shell element, material constitutive, crack-fracture. And the dynamic response analysis and the whole failure process for the steel tank structures under the explosive loading are achieved. Then, the development of the stresses and the deformationes and the failure modes for flat-roof tank, cone-roof tank and dome-roof tank under the explosive loadings with simplified triangle forms are analysed.
The results of this thesis reflect the unique advantages of VFIFE method in dealing with the strong nonlinear and discontinuous mechanics behaviors such as large deformation, large deflection, collision, fracture and penetration. The theory development and the engineering application of VFIFE are pushed, and a new effective method for complex behaviors of thin membrane structures and thin metal shell structures.
本文基于向量式有限元这一新型分析方法,以向量力学为理论基础,通过理论分析、程序开发和数值模拟等手段实现金属薄壳结构的复杂力学行为分析。研究工作沿两条主线：一是单元发展,包括薄膜单元、薄板单元和薄壳单元；二是复杂结构行为实现,包括大变形大转动、屈曲、碰撞、断裂和穿透。
第二章介绍了向量式有限元法的基本概念、假设、原理和推导思路,并对中央差分公式、共转坐标和逆向运动、材料(非)线性、静力和动力求解、等效质量和惯量矩阵以及误差分析和内力自平衡机制等问题进行了探讨；针对大变形大转动运动以及屈曲、碰撞、断裂和穿透等复杂结构行为问题及其求解思路进行了简单讨论；总结了向量式有限元法的分析步骤与流程。
第三章建立了三角形CST常应变薄膜单元和四节点等参薄膜单元的向量式有限元基本公式,描述运动解析的原理及变形坐标系下单元节点内力的求解方法,同时对四节点膜单元的位置模式和内力计算的数值积分等问题提出处理方法；编制了相应的分析程序并通过算例进行验证。在复杂结构行为分析方面,针对膜材大变形大转动问题,采用本文方法跟踪获得其运动变形全过程；针对膜材的碰撞接触行为,采用“点-三角形”检测和基于中央差分式的罚接触力方法处理膜材的碰撞检测和碰撞响应问题,跟踪获得其碰撞接触全过程；针对膜材的破裂破碎行为,采用失效应力判断准则和质点分裂的向量式有限元处理,跟踪获得其破裂和破碎的全过程。
第四章建立了三角形DKT薄板单元的向量式有限元基本理论,同时对质点的质量矩阵与惯量矩阵、应力计算的数值积分及插值方法、时间步长及阻尼参数的取值等问题提出处理方式；并通过平板结构的静、动力算例进行分析验证。
第五章基于三角形CST薄膜单元和三角形DKT薄板单元的叠加组合,建立了三角形薄壳单元的向量式有限元基本理论,同时对质点位移和内力的合并和分离、单元应变和应力的合并和分离以及单元节点内力积分方案的处理等关键问题提出处理方法；基于C-S粘塑性材料本构模型和动态Mises屈服准则,推导获得C-S材料本构模型的弹塑性增量分析步骤,并将其引入向量式有限元薄壳单元的基本理论推导,实现金属薄壳结构同时考虑塑性硬化和应变率硬化效应的非线性分析。
第六章研究了向量式有限元在薄壳结构复杂行为分析中的应用。针对薄壳结构的屈曲和后屈曲行为,采用位移和力控制处理方法来跟踪获得其变形全过程,并分析了两种控制方法的特点和应用范围；针对薄壳结构的碰撞接触行为,仍采用“点-三角形”检测和罚接触力法处理碰撞检测和碰撞响应问题,跟踪获得其碰撞接触的全过程；针对薄壳结构的破裂破碎行为,采用失效应变断裂判断准则和质点分裂的向量式有限元处理,跟踪获得其破裂和破碎的全过程；针对薄壳结构的穿透行为,结合碰撞检测、碰撞响应处理机制以及基于失效应变的断裂准则,实现刚体对薄壳结构的穿透过程。
第七章综合本文基于向量式有限元的薄壳结构非线性分析基本理论,包括薄壳单元、材料本构、破裂破碎的理论公式与有效算法,实现了钢储罐结构在爆炸冲击荷载作用下的动力响应分析和破坏全过程分析,分析了内部三角形简化形式爆炸荷载作用下平顶罐、锥顶罐和拱顶罐的应力变形发展以及破坏模式。
本文成果体现了向量式有限元分析方法在处理结构大变形、大变位、碰撞、断裂和穿透等强非线性和不连续力学行为中的独特优势,推动了向量式有限元的理论发展与工程应用,为薄膜结构、金属薄壳结构的复杂行为分析提供了一种新的有效方法。
Thin shell structures are widely applied in practical engineering. Under the externnal loading, the behaviors of the thin shell structures usually contain large deformation and large deflection, elastoplastic nonlinear material and complex discontinuous behaviors such as collision-contact, crack-fracture and penetration. For these complex structural behaviors, the traditional finite element analysis need to distinguish the type of the behavior first, then adopt the corresponding special formulas ande process. Thus, for the problems of the structural behavior prediction and the whole process simulation of structural failure, it is difficult to obtain satisfactory results by using the traditional finite element method.
The Vector Form Intrinsic Finite Element (VFIFE) is a new analysis method on the basis of vector mechanics theory. This thesis achieves the complex mechanical behaviors for thin metal shell structures by using theoretical analysis, program development and numerical simulation. The study of this thesis along the following two main lines:one is the element development including thin membrane element, thin plate element and thin shell element, and the other is the realization of the complex behaviors including large deformation and large rotation, buckling and post buckling, collision, fracture and penetration.
Chapter2introduces the basic concepts, assumptions, principles and deduction ideas of VFIFE. And the related problems such as center differential formula, co-rotational coordinate and reverse movement, nonlinear material, static and dynamic calculation, equivalent mass matrix and inertia matrix, error analysis and internal force balance mechanism are studied. Then the analysis and solving ideas for complex behavior problems such as large deformation and large rotation, buckling and post buckling, collision-contact, crack-fracture and penetration are simply discussed. Finally, the analysis process of VFIFE method is summarized.
Chapter3establishes the basic formulas of VFIFE for the triangle CST constant strain membrane element and4-node quadrilateral isoparametric membrane element, describes the principles of motion analysis and the solving method of element node internal forces in deformation coordinate system. Then, some special problems such as the position modes and the numerical integration of the internal force calculation for the4-node quadrilateral isoparametric membrane element are presented, and the corresponding treatment methods are proposed. Computer analysis programs are then developed, and the derived theory and the developed programs are verified through numerical examples finially. In the analysis of complex structural behaviors, for the large deformation and large rotation problems, VFIFE method is used to track the whole process of structural motion and deformation. For the collision-contact problems, the "point-triangle" detection and the penalty contact force method based on the central difference method are adopted separately for the problems of collision detection and collision response, and the whole process of structural collision-contact is tracked. For the crack-fracture problems, the judgment criterion based on the failure stress and the particle splitting method based on VFIFE are adopted to track the whole process of structural crack-fracture.
Chapter4establishes the basic theory of VFIFE for the triangle DKT plate element. Then, some special problems such as the mass matrix and inertia matrix of particle, the numerical integration of the stress calculation and interpolation method, and the parameters of time step and damp for the triangle DKT plate element are presented, and the corresponding treatment methods are proposed. All of above are verified by the static and dynamic analysis of plate structural examples finally.
Chapter5establishes the basic theory of VFIFE for the triangle shell element based on the combination of the triangle CST constant strain membrane element and the triangle DKT plate element. Then, the processsing methods for the problems are proposed, including the combination and the division of the particle displacements and the internal forces, the combination and the division of the element stresses and strains, and the element node integration scheme. Based on the C-S viscoplastic constitutive model and the dynamic Mises yield criterion, the elastoplastic incremental analysis steps of the C-S constitutive model are deducted, and introduced into the theoretical derivation of the thin shell element of VFIFE. Thus, the plastic hardening effect and the strain rate hardening effect for the nonlinear analysis of the thin metal shell structures are both considered in this material model.
Chapter6carries out the research of complex structural behaviors for thin shell structures. For the buckling and post buckling problems, the displacement and the force control methods are used to track the whole process of structural deformation, and the characteristics and applications of the two control methods are then analysed. For the collision-contact problems, the "point-triangle" detection and the penalty contact force method based on the central difference method are still adopted separately for the problems of collision detection and collision response, and the whole process of structural collision-contact is tracked. For the crack-fracture problems, the judgment criterion based on the failure strain and the particle splitting method based on VFIFE are adopted to track the whole process of structural crack-fracture. For the penetration problems, the combination of the collision detection mechanisms, the collision response mechanisms and the fracture criterion based on the failure strain are adopted, and the process of penetration for rigid body-thin shell structure is achieved.
Chapter7synthesizes the nonlinear analysical basic theorys for the thin shell structures based on VFIFE, including the theory formulas and the effective algorithms of thin shell element, material constitutive, crack-fracture. And the dynamic response analysis and the whole failure process for the steel tank structures under the explosive loading are achieved. Then, the development of the stresses and the deformationes and the failure modes for flat-roof tank, cone-roof tank and dome-roof tank under the explosive loadings with simplified triangle forms are analysed.
The results of this thesis reflect the unique advantages of VFIFE method in dealing with the strong nonlinear and discontinuous mechanics behaviors such as large deformation, large deflection, collision, fracture and penetration. The theory development and the engineering application of VFIFE are pushed, and a new effective method for complex behaviors of thin membrane structures and thin metal shell structures.
引文
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