基于基面力的弹性大变形拟变分原理
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摘要
基于基面力的非保守系统弹性大变形问题是一个新兴的课题。基面力作为一种描述应力状态的新方法,较传统的应力张量表示方法简单。将基面力应用于大变形问题,这种特点就体现得十分明显。对于大变形问题,体系又常常是非保守的。本文研究的非保守系统是指“伴生力”系统,即作用于系统的非保守力随物体的变形而变化。
本文从基于基面力的弹性大变形的基本方程出发,采用变积方法和Lagrange乘子法,研究了非保守系统弹性大变形的拟变分原理。
首先,研究了基于基面力的非保守系统弹性大变形的拟变分原理。推导了基于基面力的非保守系统弹性大变形的虚功原理、拟势能原理,余虚功原理、拟余能原理。论述了虚功原理和余虚功原理的宽广适用性,并给出拟势能原理和拟余能原理的其它表示形式。建立了第一类两类变量、第二类两类变量的广义拟势能原理和广义拟余能原理；一对有先决条件的两类变量的广义拟变分原理。建立了三类变量的广义拟势能原理和广义拟余能原理。提出了基于基面力的非保守系统弹性大变形问题的拟驻值条件的概念,通过推导拟变分原理的拟驻值条件对拟变分原理进行了检验。建立了承受伴生力作用的薄壁曲梁大变形的拟势能原理,通过推导其拟驻值条件,得到了非保守大变形薄壁圆弧曲梁的平衡方程；建立了承受伴生力作用的大挠度矩形薄板的第一类两类变量的广义拟势能原理,通过推导其拟驻值条件,得到了非保守大挠度矩形薄板的平衡条件和连续条件。
其二,研究了基于基面力的非保守系统弹性大变形时域边值问题的Hamilton型拟变分原理。建立了基于基面力的非保守系统弹性大变形的拟Hamilton原理、拟余Hamilton原理,以及它们的其它表示形式。建立了第一类两类变量、第二类两类变量的广义拟Hamilton原理和广义拟余Hamilton原理,以及有先决条件的两类变量的广义拟Hamilton型变分原理。建立了三类变量的广义拟Hamilton原理和广义拟余Hamilton原理。建立了承受伴生力作用的大挠度悬臂梁的拟Hamilton原理,通过推导其拟驻值条件,得到了非保守大挠度悬臂梁的动态平衡条件；建立了承受伴生力作用的大挠度非保守矩形扁壳的三类变量的广义拟Hamilton原理,并推导出其拟驻值条件,它的拟驻值条件正是该问题的全部基本方程。
其三,研究了基于基面力的非保守系统弹性大变形初值问题的卷积型拟变分原理。建立了基于基面力的非保守系统弹性大变形卷积型拟势能原理和拟余能原理。建立了第一类两类变量、第二类两类变量的卷积型广义拟势能原理和广义拟余能原理,以及有先决条件的两类变量的卷积型广义拟变分原理。建立了卷积型三类变量的广义拟势能原理和广义拟余能原理。
其四,根据基面力Ti与第一类Piola-Kirchhoff应力张量τ和第二类Piola-Kirchhoff应力张量Σ的对应关系,根据位移梯度gi与ui、Green有限应变张量εG的对应关系,给出以第一类Piola-Kirchhoff应力张量τ和位移梯度ui为变量的非保守系统弹性大变形的各级拟变分原理；给出以第二类Piola-Kirchhoff应力张量Σ和Green有限应变张量εG为变量的非保守系统弹性大变形的各级拟变分原理。
其五,以静力学为例,研究了建立适用于有限元计算的基于基面力的非保守系统弹性大变形的拟变分原理、广义拟变分原理以及修正的拟变分原理、广义拟变分原理的方法。
此外,说明了本文所建立的基于基面力的非保守系统弹性大变形各级拟变分原理的含义非常广泛,可以将其退化到基于基面力的保守系统弹性大变形的各级变分原理。
The problem of large elastic deformation in non-conservative systems on the base forces is a jumped-up topic. As a new description of the stress state at a point, the base forces is more simple than traditional stress tensors. Applying the base forces to the problems of large deformation, its virtue is obvious. The systems are usually non-conservative for large deformation problems. In present paper, non-conservative systems are specified as " follower force " system, in which the force changes with the deformation of the object.
Started from the basic equations of large elastic deformation on the base forces, the quasi-variational principles of large elastic deformation in non-conservative systems were studied by the variational integral method and Lagrange Multiplier Method.
First of all, the quasi-variational principles of large elastic deformation in non-conservative systems on the base forces were studied. The virtual work principle, quasi-potential energy principle, complementary virtual work principle and quasi-complementary energy principle were deduced. The broad applicability of virtual work and complementary virtual work principle was discussed, and the other forms of quasi-potential energy principle and quasi-complementary energy principle were introduced. The first and second types generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds of variables were established, and so were a couple of quasi-variational principles with two kinds of variables which had precedent conditions. The generalized quasi-potential energy principle and quasi-complementary energy principle with three kinds of variables were established. The concept of quasi-stationary value condition of large elastic deformation in non-conservative systems on the base forces was proposed, and the quasi-variational principles were examined by deriving their quasi-stationary value conditions. The quasi-potential energy principle for thin-walled circular curved beams which bore follower force was established. Then equilibrium equations were gained by deducing its quasi-stationary value conditions. The first type generalized quasi-potential energy principle with two kinds of variables for large deflection rectangular sheet which bore follower force was established. Then equilibrium conditions and continuity conditions were gained by deducing its quasi-stationary value conditions.
Secondly, the quasi-Hamilton variational principles of large elastic deformation time boundary value problem in non-conservative systems on the base forces were studied. The quasi-Hamilton principle, quasi-complementary Hamilton principle and their other forms were established. The first and second types generalized quasi-Hamilton principles and quasi-complementary Hamilton principles with two kinds of variables were established, and so were the quasi-Hamilton variational principles with two kinds of variables which had precedent conditions. The generalized quasi-Hamilton principle and quasi-complementary Hamilton principle with three kinds of variables were established. The quasi-Hamilton principle for large deformation overhanging beam which bore follower force was established. Then dynamical equilibrium conditions were gained by deducing its quasi-stationary value conditions. The generalized quasi-Hamilton principle with three kinds of variables for large deflection rectangular flat shell which bore follower force was established. Then all of its basic equations were gained by deducing its quasi-stationary value conditions.
Thirdly, the convolutional quasi-variational principles of large elastic deformation time initial value problem in non-conservative systems on the base forces were studied. The convolutional quasi-potential energy principle and convolutional quasi-complementary energy principle of large elastic deformation in non-conservative systems on the base forces were established. The first and second types convolutional generalized quasi-potential energy principle and convolutional generalized quasi-complementary energy principle with two kinds of variables were established, and so were the convolutional quasi-variational principles with two kinds of variables which had precedent conditions. The convolutional generalized quasi-potential energy principle and quasi-complementary energy principle with three kinds of variables were established.
Fourthly, according to corresponding relations between base forces Ti and the first Piola-Kirchhoff stress tensorτ、the second Piola-Kirchhoff stress tensorΣ, and according to corresponding relations between displacement gradient gi and ui、Green finite strain tensorεG, the corresponding quasi-variational principles of large elastic deformation in non-conservative systems which only consist of the first Piola-Kirchhoff stress tensorτand displacement gradient ui, were obtained. The corresponding quasi-variational principles of large elastic deformation in non-conservative systems which only consist of the second Piola-Kirchhoff stress tensorΣand Green finite strain tensorεG were obtained.
Fifthly, taking elastostatics for example, it was worked out the quasi-variational principles and generalized quasi-variational principles of large elastic deformation in non-conservative systems on the base forces which were applicable for the calculation of finite element method.
Furthermore, it was concluded that the meaning of the quasi-variational principles of large elastic deformation in non-conservative systems on the base forces in the paper was very abundant. They could be degenerated to the corresponding quasi-variational principles of large elastic deformation in conservative systems on the base forces.
本文从基于基面力的弹性大变形的基本方程出发,采用变积方法和Lagrange乘子法,研究了非保守系统弹性大变形的拟变分原理。
首先,研究了基于基面力的非保守系统弹性大变形的拟变分原理。推导了基于基面力的非保守系统弹性大变形的虚功原理、拟势能原理,余虚功原理、拟余能原理。论述了虚功原理和余虚功原理的宽广适用性,并给出拟势能原理和拟余能原理的其它表示形式。建立了第一类两类变量、第二类两类变量的广义拟势能原理和广义拟余能原理；一对有先决条件的两类变量的广义拟变分原理。建立了三类变量的广义拟势能原理和广义拟余能原理。提出了基于基面力的非保守系统弹性大变形问题的拟驻值条件的概念,通过推导拟变分原理的拟驻值条件对拟变分原理进行了检验。建立了承受伴生力作用的薄壁曲梁大变形的拟势能原理,通过推导其拟驻值条件,得到了非保守大变形薄壁圆弧曲梁的平衡方程；建立了承受伴生力作用的大挠度矩形薄板的第一类两类变量的广义拟势能原理,通过推导其拟驻值条件,得到了非保守大挠度矩形薄板的平衡条件和连续条件。
其二,研究了基于基面力的非保守系统弹性大变形时域边值问题的Hamilton型拟变分原理。建立了基于基面力的非保守系统弹性大变形的拟Hamilton原理、拟余Hamilton原理,以及它们的其它表示形式。建立了第一类两类变量、第二类两类变量的广义拟Hamilton原理和广义拟余Hamilton原理,以及有先决条件的两类变量的广义拟Hamilton型变分原理。建立了三类变量的广义拟Hamilton原理和广义拟余Hamilton原理。建立了承受伴生力作用的大挠度悬臂梁的拟Hamilton原理,通过推导其拟驻值条件,得到了非保守大挠度悬臂梁的动态平衡条件；建立了承受伴生力作用的大挠度非保守矩形扁壳的三类变量的广义拟Hamilton原理,并推导出其拟驻值条件,它的拟驻值条件正是该问题的全部基本方程。
其三,研究了基于基面力的非保守系统弹性大变形初值问题的卷积型拟变分原理。建立了基于基面力的非保守系统弹性大变形卷积型拟势能原理和拟余能原理。建立了第一类两类变量、第二类两类变量的卷积型广义拟势能原理和广义拟余能原理,以及有先决条件的两类变量的卷积型广义拟变分原理。建立了卷积型三类变量的广义拟势能原理和广义拟余能原理。
其四,根据基面力Ti与第一类Piola-Kirchhoff应力张量τ和第二类Piola-Kirchhoff应力张量Σ的对应关系,根据位移梯度gi与ui、Green有限应变张量εG的对应关系,给出以第一类Piola-Kirchhoff应力张量τ和位移梯度ui为变量的非保守系统弹性大变形的各级拟变分原理；给出以第二类Piola-Kirchhoff应力张量Σ和Green有限应变张量εG为变量的非保守系统弹性大变形的各级拟变分原理。
其五,以静力学为例,研究了建立适用于有限元计算的基于基面力的非保守系统弹性大变形的拟变分原理、广义拟变分原理以及修正的拟变分原理、广义拟变分原理的方法。
此外,说明了本文所建立的基于基面力的非保守系统弹性大变形各级拟变分原理的含义非常广泛,可以将其退化到基于基面力的保守系统弹性大变形的各级变分原理。
The problem of large elastic deformation in non-conservative systems on the base forces is a jumped-up topic. As a new description of the stress state at a point, the base forces is more simple than traditional stress tensors. Applying the base forces to the problems of large deformation, its virtue is obvious. The systems are usually non-conservative for large deformation problems. In present paper, non-conservative systems are specified as " follower force " system, in which the force changes with the deformation of the object.
Started from the basic equations of large elastic deformation on the base forces, the quasi-variational principles of large elastic deformation in non-conservative systems were studied by the variational integral method and Lagrange Multiplier Method.
First of all, the quasi-variational principles of large elastic deformation in non-conservative systems on the base forces were studied. The virtual work principle, quasi-potential energy principle, complementary virtual work principle and quasi-complementary energy principle were deduced. The broad applicability of virtual work and complementary virtual work principle was discussed, and the other forms of quasi-potential energy principle and quasi-complementary energy principle were introduced. The first and second types generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds of variables were established, and so were a couple of quasi-variational principles with two kinds of variables which had precedent conditions. The generalized quasi-potential energy principle and quasi-complementary energy principle with three kinds of variables were established. The concept of quasi-stationary value condition of large elastic deformation in non-conservative systems on the base forces was proposed, and the quasi-variational principles were examined by deriving their quasi-stationary value conditions. The quasi-potential energy principle for thin-walled circular curved beams which bore follower force was established. Then equilibrium equations were gained by deducing its quasi-stationary value conditions. The first type generalized quasi-potential energy principle with two kinds of variables for large deflection rectangular sheet which bore follower force was established. Then equilibrium conditions and continuity conditions were gained by deducing its quasi-stationary value conditions.
Secondly, the quasi-Hamilton variational principles of large elastic deformation time boundary value problem in non-conservative systems on the base forces were studied. The quasi-Hamilton principle, quasi-complementary Hamilton principle and their other forms were established. The first and second types generalized quasi-Hamilton principles and quasi-complementary Hamilton principles with two kinds of variables were established, and so were the quasi-Hamilton variational principles with two kinds of variables which had precedent conditions. The generalized quasi-Hamilton principle and quasi-complementary Hamilton principle with three kinds of variables were established. The quasi-Hamilton principle for large deformation overhanging beam which bore follower force was established. Then dynamical equilibrium conditions were gained by deducing its quasi-stationary value conditions. The generalized quasi-Hamilton principle with three kinds of variables for large deflection rectangular flat shell which bore follower force was established. Then all of its basic equations were gained by deducing its quasi-stationary value conditions.
Thirdly, the convolutional quasi-variational principles of large elastic deformation time initial value problem in non-conservative systems on the base forces were studied. The convolutional quasi-potential energy principle and convolutional quasi-complementary energy principle of large elastic deformation in non-conservative systems on the base forces were established. The first and second types convolutional generalized quasi-potential energy principle and convolutional generalized quasi-complementary energy principle with two kinds of variables were established, and so were the convolutional quasi-variational principles with two kinds of variables which had precedent conditions. The convolutional generalized quasi-potential energy principle and quasi-complementary energy principle with three kinds of variables were established.
Fourthly, according to corresponding relations between base forces Ti and the first Piola-Kirchhoff stress tensorτ、the second Piola-Kirchhoff stress tensorΣ, and according to corresponding relations between displacement gradient gi and ui、Green finite strain tensorεG, the corresponding quasi-variational principles of large elastic deformation in non-conservative systems which only consist of the first Piola-Kirchhoff stress tensorτand displacement gradient ui, were obtained. The corresponding quasi-variational principles of large elastic deformation in non-conservative systems which only consist of the second Piola-Kirchhoff stress tensorΣand Green finite strain tensorεG were obtained.
Fifthly, taking elastostatics for example, it was worked out the quasi-variational principles and generalized quasi-variational principles of large elastic deformation in non-conservative systems on the base forces which were applicable for the calculation of finite element method.
Furthermore, it was concluded that the meaning of the quasi-variational principles of large elastic deformation in non-conservative systems on the base forces in the paper was very abundant. They could be degenerated to the corresponding quasi-variational principles of large elastic deformation in conservative systems on the base forces.
引文
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