若干弹性力学问题解的唯一性定理
Some uniqueness theorems of solutions for the problems of elasticity
查看参考文献161篇
|
文摘
|
弹性力学问题解的适定性,包括存在性、唯一性以及稳定性(对边界条件的连续依赖性).其中解的唯一性定理是求解定解问题的一个有力工具,为各种求解方法提供理论依据,解是否满足唯一性是弹性理论的一个基本问题.然而,在物理上三维弹性问题存在非唯一解的例子广泛存在,其相应的数学模型并不能要求解的唯一性定理无条件成立.因此解的唯一性需在一定的条件下成立,如弹性张量、应变能函数和变形范围的限制条件.本文回顾了解的唯一性定理在弹性理论中的背景和发展历史,重点介绍了线性弹性理论、有限变形非线性弹性理论和具有初始应力场的弹性理论中边值问题的解的唯一性定理,给出了其中重要定理的证明方法.在此基础上,结合研究进展提出了弹性力学问题解的唯一性定理的待解决问题. |
|
其他语种文摘
|
In order to construct the well-posed mathematical models of the elastic problems, it is necessary to study the mathematical properties of solutions, including existence, uniqueness, and stability (the continuous dependence on boundary conditions). The uniqueness theorem of solutions provides the methods for solving the problem. The uniqueness of the solutions is one of the most basic and important issues of elasticity theory. In the three-dimensional elasticity theory, unconditional uniqueness is not expected. Therefore, the uniqueness theorem of solutions is established under certain conditions, such as the restrictions on the elasticity tensor, the strain energy function and elastic deformation range. This study reviews the background and history of the uniqueness theorem in elasticity. The focuses are on the uniqueness theorems of solutions in the boundary-value problems of the linear elasticity theory, nonlinear elasticity theory of finite deformation, and elasticity theory with initial stress field. The classical proofs of uniqueness theorems are also given. We also present some unsolved problems on the solution uniqueness in elasticity. |
|
来源
|
中国科学. 物理学
, 力学, 天文学,2020,50(8):084601 【核心库】
|
|
DOI
|
10.1360/SSPMA-2020-0042
|
|
关键词
|
弹性理论
;
边值问题
;
唯一性定理
;
本构关系
;
应变能函数
;
凸性
|
|
地址
|
1.
中国科学院力学研究所, 非线性力学国家重点实验室, 北京, 100190
2.
中国科学院大学工程科学学院, 北京, 100049
|
|
语种
|
中文 |
|
文献类型
|
研究性论文 |
|
ISSN
|
1674-7275 |
|
学科
|
力学 |
|
基金
|
国家自然科学基金
;
中国科学院前沿科学重点研究计划
;
中国科学院战略性先导科技专项
|
|
文献收藏号
|
CSCD:6807936
|
参考文献 共
161
共9页
|
|
1.
Hadamard J. Sur les problemes aux derivees partielles et leur signification physique.
Princeton Univ Bull,1902,13:49-52
|
CSCD被引
1
次
|
|
|
|
|
2.
Saint-Venant B D. Memoire sur la torsion des prismes.
Mem Savants Etrangers,1855,14:233-560
|
CSCD被引
1
次
|
|
|
|
|
3.
Kirchhoff G. Uber das gleichgewicht und die bewegung eines unendlich dunnen elastischen stabes.
J Reine Angew Math,1859,56:285-313
|
CSCD被引
11
次
|
|
|
|
|
4.
Neumann F.
Vorlesung uber die Theorie des Elasticitat der festen Korper und des Lichtaters,1885
|
CSCD被引
1
次
|
|
|
|
|
5.
Knops R J.
Uniqueness Theorems in Linear Elasticity,2012
|
CSCD被引
1
次
|
|
|
|
|
6.
Gurtin M E. The Linear Theory of Elasticity.
Linear Theories of Elasticity and Thermoelasticity,1973:1-295
|
CSCD被引
3
次
|
|
|
|
|
7.
Truesdell C. The non-linear field theories of mechanics.
The Non-Linear Field Theories of Mechanics,2004:1-579
|
CSCD被引
2
次
|
|
|
|
|
8.
Valent T.
Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data,1988
|
CSCD被引
1
次
|
|
|
|
|
9.
Ogden R W.
Non-Linear Elastic Deformations,1997
|
CSCD被引
31
次
|
|
|
|
|
10.
Ciarlet P G.
Mathematical Elasticity,1988
|
CSCD被引
1
次
|
|
|
|
|
11.
Ciarlet P G. Injectivity and self-contact in nonlinear elasticity.
Arch Rational Mech Anal,1987,97:171-188
|
CSCD被引
2
次
|
|
|
|
|
12.
Ciarlet P G. Unilateral problems in nonlinear, three-dimensional elasticity.
Arch Rational Mech Anal,1985,87:319-338
|
CSCD被引
2
次
|
|
|
|
|
13.
Ball J M. Does rank-one convexity imply quasiconvexity?.
Metastability and Incompletely Posed Problems,1987:17-32
|
CSCD被引
1
次
|
|
|
|
|
14.
Silhavy M. Energy minimization for isotropic nonlinear elastic bodies.
Multiscale Modeling in Continuum Mechanics and Structured Deformations,2004:1-51
|
CSCD被引
1
次
|
|
|
|
|
15.
Hill R. On uniqueness and stability in the theory of finite elastic strain.
J Mech Phys Solids,1957,5:229-241
|
CSCD被引
4
次
|
|
|
|
|
16.
Ball J M. Convexity conditions and existence theorems in nonlinear elasticity.
Arch Rational Mech Anal,1976,63:337-403
|
CSCD被引
21
次
|
|
|
|
|
17.
Morrey C B. Quasi-convexity and the lower semicontinuity of multiple integrals.
Pac J Math,1952,2:25-53
|
CSCD被引
3
次
|
|
|
|
|
18.
Coral M. On the necessary conditions for the minimum of a double integral.
Duke Math J,1937,3:585-592
|
CSCD被引
1
次
|
|
|
|
|
19.
Graves L M. Problems.
Duke Math J,1939,5:656-660
|
CSCD被引
1
次
|
|
|
|
|
20.
Qi L. Conditions for strong ellipticity and m-eigenvalues.
Front Math China,2009,4:349-364
|
CSCD被引
22
次
|
|
|
|
|
了解气象卫星更多信息，请访问