连续介质力学基本理论物理量的不变性表示
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Institute of Engineering Mechanics
Derivatives of invariants
The derivatives of the principal invariants
I k 1 I Ik I AT k A A
I k k 1 (1) i I k i 1 A i A i 0
wk.baidu.com
连续介质力学 是研究连续介质的宏观力学行为。连续介质力学用统一的观点 来研究固体和流体的力学问题，因此也有人把连续介质力学狭义地理解为理 性力学。 研究内容﹕ 1变形几何学：研究连续介质变形的几何性质, 这里包括诸如运动、构形﹑变 形梯度﹑应变张量﹑变形的基本定理﹑极分解定理等重要概念。 2运动学：主要研究连续介质力学中各种量的时间率 , 这里包括诸如速度梯度、 变形速率和旋转速率, Rivlin-Ericksen张量等重要概念。 3基本方程: 根据适用于所有物质的守恒定律建立的方程﹐例如：连续性方程、 运动方程、能量方程、熵不等式等。 4本构关系。 5特殊理论：例如弹性理论、粘性流体理论、塑性理论、粘弹性理论、热弹性 固体理论、热粘性流体理论等。 6问题的求解。 主要数学工具： 张量
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
非线性连续介质力学中通常其物理量可表示为两种不同形 式, 一种是Hill在应变主轴标架思想基础上系统发展起来的主轴 表示。但主轴法是在主标架给出的，不仅需要求解特征值而且 需要计算相应的特征向量，主轴法将耗费很大的计算量。为了 避免计算特征值及其特征向量, 物理量还可以表示为与坐标无关 的不变性表示形式, 不变性表示理论及其在连续介质力学中的应 用吸引了众多学者的研究兴趣。 例如
•4. Stress
Conjugate stress, Stress rate
•5 Applications in computational mechanics
Eigenvalues and eigenvectors, Tangent stiffness, Anisotropic materials
Another important set of invariants is the moments I k tr A k Its derivative two trace identities
I k 1 (k 1)( A k ) T A I k 1 I k 1 tr (k 1) I k tr (n k ) I k A A
T
Truesdell and Noll (1965) gave an elegant derivation for the gradients of the principal invariants in the case when A is an invertible tensor. By a new approach, Carlson and Hoger (1986) proved that the formulae are valid for arbitrary tensor A. Some different proofs have been reported in Guo (1989) and Jaric (1996). Truesdell and Noll’s method was modified for arbitrary tensor A by Dui, et. al (2004). (Dui, Jin, Huang, J. Elast. 75: 2004.)
where the scalars I 1 , I 2 , , and I n are the principal invariants of A. Then, we have (Gantmakher, 1959 )
(1) k 1 Ik tr( A k I1 A k 1 I 2 A n 2 (1) k 1 I k 1 A) k
E ( 0 ) ln U lni Ni Ni
i
工欲善其事，必先利其器
兑关锁 Dui Guansuo
Contents
•1. Tensor analysis
Beijing Jiaotong University
Institute of Engineering Mechanics
Tensor identities, Representation of tensor functions, Tensor equations, Derivative of tensor function
陈至达
Beijing Jiaotong University
Institute of Engineering Mechanics
高玉臣院士
兑关锁 Dui Guansuo
0引
言
Beijing Jiaotong University
Institute of Engineering Mechanics
连续介质力学中物理量的不变性表示
兑关锁
北京交通大学工程力学研究所
兑关锁
Dui Guansuo
发展历史
Beijing Jiaotong University
Institute of Engineering Mechanics
①奠基时期。牛顿的《自然哲学的数学原理》是理性力学的第一部著 作。J.le R.达朗贝尔1743年提出理性力学的框架。D.Hilbert 1900年提出的 23个问题中的第6个问题就是公理化问题。 ②停滞时期。约从20世纪初到1945年。这段时期形成了以从事线性力 学及其相关数学的研究为主的局面。非线性理论的研究没有多大进展， 理性力学也因此处于停滞时期。 ③复兴时期。巨大的变化发端于1945年M.Reiner和1948年R.S.Rivlin的 工作。C.Truesdell 1953年提出低弹性体的概念。同年，J.L.Ericksen发表 了各向同性不可压缩弹性物质中波的传播理论。 ④发展时期。1966年以后，理性力学进入发展时期。 Rivlin, Pipkin, Smith, Carroll, Ting, Gurtin, Fosdick, Carlson Ericksen, Beatty, Batra, Y-C Chen
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
Rivlin’s identities
For any positive integer m, it is well known that The derivative of Cayley-Hamilton theorem yields
A 2 CA ACA 2 I1 ( A) ACA I 3 ( A)C tr( AC) A 2 [I 1 ( A) tr( AC) tr( A 2 C)]A I 3 ( A) tr( C)I A 2 CA 2 I 2 ( A) ACA I 3 ( A)( AC CA )
Beijing Jiaotong University
Institute of Engineering Mechanics
工程力学研究所 (IEM)
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
Trusdell, Noll, Coleman, CC Wang, C-s Man
Hill, Ogden 国内教材
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
郭仲衡
兑关锁 Dui Guansuo
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
Cayley–Hamilton theorem
Let A be a second order tensors in n dimensions.
A n I1 A n 1 I 2 A n 2 (1) n I n I 0
•2. Deformation gradient
Polar decomposition, S-R decomposition, Relations between two decompositions.
3. Kinematics
Time rate of the invariants, Time rate of the rotation and stretch tensor, Spin tensor, Time rate of logarithmic strain tensor, Time rate of logarithmic strain tensor
兑关锁 Dui Guansuo
1. Tensors analysis
1.1 Tensors identities
Beijing Jiaotong University
Institute of Engineering Mechanics
As a useful tool, tensor identities involving multi-variables are successfully used to solve many problems in continuum mechanics, such as, the solutions of tensor equations (Scheidler 1994, Hoger and Carlson 1984) and representation theorems for isotropic tensor functions (Rivlin and Ericksen 1955). Several identities have been derived by Rivlin (1955) through matrix calculations. In the present work, several tensor identities involving second order tensors C, A and its transpose are presented in 3-dimensions. Then they are extended to n-dimensions. By using the properties of the derivatives of the principal invariants, some tensor identities, such as Rivlin’s identities, are derived directly by a new method.
A2C CA2 ACA I1 (A)(AC CA) I2 (A)C
2
m 1 A m [C] A i CA m 1i A i 0
(Dui and Chen, J. Elast. 76:
2004)
tr(C)A [I1 ( A) tr(C) tr( AC)]A [I1 ( A) tr( AC) I2 ( A) tr(C) tr( A 2C)]I
In particularly, for n=3
A 3 I1A 2 I 2 A I 3I 0
The number of useful results can be extracted from the Cayley– Hamilton theorem.
兑关锁 Dui Guansuo
Beijing Jiaotong University
Derivatives of invariants
The derivatives of the principal invariants
I k 1 I Ik I AT k A A
I k k 1 (1) i I k i 1 A i A i 0
wk.baidu.com
连续介质力学 是研究连续介质的宏观力学行为。连续介质力学用统一的观点 来研究固体和流体的力学问题，因此也有人把连续介质力学狭义地理解为理 性力学。 研究内容﹕ 1变形几何学：研究连续介质变形的几何性质, 这里包括诸如运动、构形﹑变 形梯度﹑应变张量﹑变形的基本定理﹑极分解定理等重要概念。 2运动学：主要研究连续介质力学中各种量的时间率 , 这里包括诸如速度梯度、 变形速率和旋转速率, Rivlin-Ericksen张量等重要概念。 3基本方程: 根据适用于所有物质的守恒定律建立的方程﹐例如：连续性方程、 运动方程、能量方程、熵不等式等。 4本构关系。 5特殊理论：例如弹性理论、粘性流体理论、塑性理论、粘弹性理论、热弹性 固体理论、热粘性流体理论等。 6问题的求解。 主要数学工具： 张量
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
非线性连续介质力学中通常其物理量可表示为两种不同形 式, 一种是Hill在应变主轴标架思想基础上系统发展起来的主轴 表示。但主轴法是在主标架给出的，不仅需要求解特征值而且 需要计算相应的特征向量，主轴法将耗费很大的计算量。为了 避免计算特征值及其特征向量, 物理量还可以表示为与坐标无关 的不变性表示形式, 不变性表示理论及其在连续介质力学中的应 用吸引了众多学者的研究兴趣。 例如
•4. Stress
Conjugate stress, Stress rate
•5 Applications in computational mechanics
Eigenvalues and eigenvectors, Tangent stiffness, Anisotropic materials
Another important set of invariants is the moments I k tr A k Its derivative two trace identities
I k 1 (k 1)( A k ) T A I k 1 I k 1 tr (k 1) I k tr (n k ) I k A A
T
Truesdell and Noll (1965) gave an elegant derivation for the gradients of the principal invariants in the case when A is an invertible tensor. By a new approach, Carlson and Hoger (1986) proved that the formulae are valid for arbitrary tensor A. Some different proofs have been reported in Guo (1989) and Jaric (1996). Truesdell and Noll’s method was modified for arbitrary tensor A by Dui, et. al (2004). (Dui, Jin, Huang, J. Elast. 75: 2004.)
where the scalars I 1 , I 2 , , and I n are the principal invariants of A. Then, we have (Gantmakher, 1959 )
(1) k 1 Ik tr( A k I1 A k 1 I 2 A n 2 (1) k 1 I k 1 A) k
E ( 0 ) ln U lni Ni Ni
i
工欲善其事，必先利其器
兑关锁 Dui Guansuo
Contents
•1. Tensor analysis
Beijing Jiaotong University
Institute of Engineering Mechanics
Tensor identities, Representation of tensor functions, Tensor equations, Derivative of tensor function
陈至达
Beijing Jiaotong University
Institute of Engineering Mechanics
高玉臣院士
兑关锁 Dui Guansuo
0引
言
Beijing Jiaotong University
Institute of Engineering Mechanics
连续介质力学中物理量的不变性表示
兑关锁
北京交通大学工程力学研究所
兑关锁
Dui Guansuo
发展历史
Beijing Jiaotong University
Institute of Engineering Mechanics
①奠基时期。牛顿的《自然哲学的数学原理》是理性力学的第一部著 作。J.le R.达朗贝尔1743年提出理性力学的框架。D.Hilbert 1900年提出的 23个问题中的第6个问题就是公理化问题。 ②停滞时期。约从20世纪初到1945年。这段时期形成了以从事线性力 学及其相关数学的研究为主的局面。非线性理论的研究没有多大进展， 理性力学也因此处于停滞时期。 ③复兴时期。巨大的变化发端于1945年M.Reiner和1948年R.S.Rivlin的 工作。C.Truesdell 1953年提出低弹性体的概念。同年，J.L.Ericksen发表 了各向同性不可压缩弹性物质中波的传播理论。 ④发展时期。1966年以后，理性力学进入发展时期。 Rivlin, Pipkin, Smith, Carroll, Ting, Gurtin, Fosdick, Carlson Ericksen, Beatty, Batra, Y-C Chen
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
Rivlin’s identities
For any positive integer m, it is well known that The derivative of Cayley-Hamilton theorem yields
A 2 CA ACA 2 I1 ( A) ACA I 3 ( A)C tr( AC) A 2 [I 1 ( A) tr( AC) tr( A 2 C)]A I 3 ( A) tr( C)I A 2 CA 2 I 2 ( A) ACA I 3 ( A)( AC CA )
Beijing Jiaotong University
Institute of Engineering Mechanics
工程力学研究所 (IEM)
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
Trusdell, Noll, Coleman, CC Wang, C-s Man
Hill, Ogden 国内教材
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
郭仲衡
兑关锁 Dui Guansuo
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
Cayley–Hamilton theorem
Let A be a second order tensors in n dimensions.
A n I1 A n 1 I 2 A n 2 (1) n I n I 0
•2. Deformation gradient
Polar decomposition, S-R decomposition, Relations between two decompositions.
3. Kinematics
Time rate of the invariants, Time rate of the rotation and stretch tensor, Spin tensor, Time rate of logarithmic strain tensor, Time rate of logarithmic strain tensor
兑关锁 Dui Guansuo
1. Tensors analysis
1.1 Tensors identities
Beijing Jiaotong University
Institute of Engineering Mechanics
As a useful tool, tensor identities involving multi-variables are successfully used to solve many problems in continuum mechanics, such as, the solutions of tensor equations (Scheidler 1994, Hoger and Carlson 1984) and representation theorems for isotropic tensor functions (Rivlin and Ericksen 1955). Several identities have been derived by Rivlin (1955) through matrix calculations. In the present work, several tensor identities involving second order tensors C, A and its transpose are presented in 3-dimensions. Then they are extended to n-dimensions. By using the properties of the derivatives of the principal invariants, some tensor identities, such as Rivlin’s identities, are derived directly by a new method.
A2C CA2 ACA I1 (A)(AC CA) I2 (A)C
2
m 1 A m [C] A i CA m 1i A i 0
(Dui and Chen, J. Elast. 76:
2004)
tr(C)A [I1 ( A) tr(C) tr( AC)]A [I1 ( A) tr( AC) I2 ( A) tr(C) tr( A 2C)]I
In particularly, for n=3
A 3 I1A 2 I 2 A I 3I 0
The number of useful results can be extracted from the Cayley– Hamilton theorem.
兑关锁 Dui Guansuo
Beijing Jiaotong University