应变是固体力学中最重要的基本概念之一,直接影响几何方程和物理方程的建立,本文从基于变形梯度分解的基本运动学关系和变形张量入手,阐述讨论了Seth-Hill广义应变度量函数的本质和含义,推导了相同构型不同应变度量之间以及不同构型应变度量之间的一般变换关系,为应变度量的变换和应用提供了算法依据。 Strain, one of the most important fundamental concepts in mechanics, directly affects the estab-lishment of geometric and constitutive equations. This paper reviews the essential kinematic rela-tions and the different stretch tensors within continuum based on the decomposition of the de-formation gradient. According to the definition of strain, the essence of Seth-Hill general strain measures is discussed. Furthermore, the general transformation relations between different strain measures in the same configuration are derived as well as those in different configurations to provide an algorithm basis for the transformation and application of strain measures.
应变是固体力学中最重要的基本概念之一,直接影响几何方程和物理方程的建立,本文从基于变形梯度分解的基本运动学关系和变形张量入手,阐述讨论了Seth-Hill广义应变度量函数的本质和含义,推导了相同构型不同应变度量之间以及不同构型应变度量之间的一般变换关系,为应变度量的变换和应用提供了算法依据。
应变度量,物质应变,空间应变,Seth-Hill应变
Fengwu Lyu, Jinming Ma
College of Civil Engineering, Tongji University, Shanghai
Received: Nov. 24th, 2020; accepted: Dec. 22nd, 2020; published: Dec. 29th, 2020
Strain, one of the most important fundamental concepts in mechanics, directly affects the establishment of geometric and constitutive equations. This paper reviews the essential kinematic relations and the different stretch tensors within continuum based on the decomposition of the deformation gradient. According to the definition of strain, the essence of Seth-Hill general strain measures is discussed. Furthermore, the general transformation relations between different strain measures in the same configuration are derived as well as those in different configurations to provide an algorithm basis for the transformation and application of strain measures.
Keywords:Strain Measure, Material Strain, Spatial Strain, Seth-Hill Strain
Copyright © 2020 by author(s) and Hans Publishers Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
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应变是固体力学中最重要的基本概念之一,它直接影响几何方程和物理方程的建立。基于变形梯度可在初始构型和现时构型上定义不同的应变度量 [
目前,在非线性固体力学中,基于不同变形和应变度量的计算是一个传统但仍然非常活跃的研究领域。Xiao [
物体 B 在 t 0 时刻填满区域 B 0 ⊂ ℝ 3 ,运动后t时刻填满区域 B t ⊂ ℝ 3 ,其运动可用如下映射表示:
� ( X , t ) : B 0 × ℝ → B t or ( X , t ) ↦ x (1)
位移场的Lagrange描述和Euler描述分别为
u ( X , t ) = � t ( X ) − X (2.1)
u ( x , t ) = x − � t − 1 ( x ) (2.2)
相应的变形梯度为初始构型切线空间 T B 0 到现时构型切线空间 T B t 的切线映射:
F = ∂ � ∂ X : T B 0 → T B t with det ( F ) > 0 (3)
引入 T B 0 和 T B t 的对偶空间 T * B 0 和 T * B t 后,存在以下协变和逆变度规张量:
G : T B 0 → T * B 0 , G − 1 : T * B 0 → T B 0 (4.1)
g : T B t → T * B t , g − 1 : T * B t → T B t (4.2)
考虑完备性,变形梯度和度规张量满足以下逆协变二阶恒等式:
I = F − 1 ⋅ F = G − 1 ⋅ G : T B 0 → T B 0 (5.1)
i = F ⋅ F − 1 = g ⋅ g − 1 : T B t → T B t (5.2)
根据极分解定理,有
F = R ⋅ U = v ⋅ R (6)
其中,旋转张量 R 正交, det ( R ) = 1 ;右拉伸张量 U 和左拉伸张量 v 均对称正定; F 、 U 、 v 具有相同的特征值 λ i ( i = 1 , 2 , 3 ) 。
协变和逆变度规张量存在如下关系:
R T ⋅ g ⋅ R = G (7)
应用谱分解定理,式(6)中的各个张量可用特征值 λ i 和特征向量 N i 、 N i 、 n i 、 n i 表示为
F = ∑ i = 1 3 λ i n i ⊗ N i , R = ∑ i = 1 3 n i ⊗ N i , U = ∑ i = 1 3 λ i N i ⊗ N i , v = ∑ i = 1 3 λ i n i ⊗ n i (8)
由式(4)的映射可知,特征向量存在关系:
N i = G − 1 ⋅ N i , N i = G ⋅ N i (9.1)
n i = g − 1 ⋅ n i , n i = g ⋅ n i (9.2)
而且满足:
N i ⋅ G − 1 ⋅ N i = N i ⋅ G ⋅ N i = 1 (10.1)
n i ⋅ g − 1 ⋅ n i = n i ⋅ g ⋅ n i = 1 (10.2)
根据极分解定理,得
n i = R ⋅ N i , N i = R T ⋅ n i (11.1)
n i = R ⋅ N i , N i = R T ⋅ n i (11.2)
在上述关系的基础上,基于线元平方长度的变化,可以导出以下不同的变形张量:
C = F T ⋅ g ⋅ F = U T ⋅ G ⋅ U = ∑ i = 1 3 λ i 2 N i ⊗ N i (12.1)
B = F − 1 ⋅ g − 1 ⋅ F − T = U − 1 ⋅ G − 1 ⋅ U − T = ∑ i = 1 3 λ i − 2 N i ⊗ N i (12.2)
c = F − T ⋅ G ⋅ F − 1 = v − T ⋅ g ⋅ v − 1 = ∑ i = 1 3 λ i − 2 n i ⊗ n i (12.3)
b = F ⋅ G − 1 ⋅ F T = v ⋅ g − 1 ⋅ v T = ∑ i = 1 3 λ i 2 n i ⊗ n i (12.4)
式(12)定义的变形张量均对称正定,采用主轴表述法,用特征值表征主方向上的主伸长量,而特征向量是与构型对应的变形主轴方向,这种表示法为变形给出了一个很形象的几何解释。显然,
B ⋅ C = I , b ⋅ c = i (13)
这些变形张量之间的关系如表1。
变形张量 | C | B | c | b |
---|---|---|---|---|
C | / | INV | EV−1 | EV |
B | INV | / | EV | EV−1 |
c | EV−1 | EV | / | INV |
b | EV | EV−1 | INV | / |
表1. 变形张量之间的关系
说明:表中INV表示互逆,EV表示特征值相同,EV−1表示特征值互为倒数。
与变形度量相对应,应变度量同样也可以表达在不同构型上,相应应变度量的几何解释当然也就不同,由此产生了不同的应变张量。
根据Hill-Seth通类应变度量函数的定义原则,物质应变定义在初始构型 B 0 上,采用Lagrange描述,与右伸长张量 U 共轴。物质应变既可以采用变形张量 C 定义协变的应变张量 E ( m ) ( U ; C ; G ) ,也可以采用变形张量 B 定义逆变的应变张量 K ( m ) ( U ; B ; G − 1 ) ,应变度量函数如下:
E ( m ) ( U ; C ; G ) = { ∑ i = 1 3 1 m ( λ i m − 1 ) N i ⊗ N i = 1 m [ C m 2 − G ] = 1 m ( U m − I ) ⋅ G with m > 0 ∑ i = 1 3 ln λ i N i ⊗ N i = 1 2 ln C = ln U ⋅ G if m = 0 (14)
K ( m ) ( U ; B ; G − 1 ) = { ∑ i = 1 3 1 m ( 1 − λ i − m ) N i ⊗ N i = 1 m [ G − 1 − B m 2 ] = 1 m ( I − 1 − U − m ) ⋅ G − 1 with m > 0 ∑ i = 1 3 ln λ i N i ⊗ N i = − 1 2 ln B = − ln ( U − 1 ) ⋅ G − 1 if m = 0 (15)
空间应变定义在现时构型 B t 上,采用Euler描述,与左伸长张量 v 共轴。空间应变可以分别采用变形张量 c 或 b 定义协变的应变张量 e ( m ) ( v ; c ; g ) 和逆变的应变张量 k ( m ) ( v ; b ; g − 1 ) ,应变度量函数如下:
e ( m ) ( v ; c ; g ) = { ∑ i = 1 3 1 m ( 1 − λ i − m ) n i ⊗ n i = 1 m [ g − c m 2 ] = 1 m ( i − v − m ) ⋅ g with m > 0 ∑ i = 1 3 ln λ i n i ⊗ n i = − 1 2 ln c = − ln ( v − 1 ) ⋅ g if m = 0 (16)
k ( m ) ( v ; b ; g − 1 ) = { ∑ i = 1 3 1 m ( λ i m − 1 ) n i ⊗ n i = 1 m [ b m 2 − g − 1 ] = 1 m ( v m − i − 1 ) ⋅ g − 1 with m > 0 ∑ i = 1 3 ln λ i n i ⊗ n i = 1 2 ln b = ln v ⋅ g − 1 if m = 0 (17)
式(14)~(17)中的m是非负整数。
Hill-Seth应变度量函数本质上表示了线元的幂次(广义几何长度)改变相对于不同参考构架的变化率。具体来讲, E ( m ) ( U ; C ; G ) 表示线元的幂次改变相对于初始构型的协变度规 G 的变化率; e ( m ) ( v ; c ; g ) 表示线元的幂次改变相对于现时构型的协变度规 g 的变化率;而 K ( m ) ( U ; B ; G − 1 ) 则表示线元幂次变化相对于初始构型的逆变度规 G − 1 的变化率;同理, k ( m ) ( v ; b ; g − 1 ) 表示线元幂次变化相对于现时构型的逆变度规 g − 1 的变化率。Hill-Seth通类应变度量函数是单调函数,显然, λ i = 1 时对应主轴方向的主应变为零。
前述Hill-Seth应变度量函数族中,定义在同一构型上相同m值的应变族存在协变张量与逆变张量之间的转换关系。
因为 B 与 C 互逆,可得
C m 2 ⋅ K ( m ) ⋅ G = 1 m C m 2 ⋅ [ G − 1 − B m 2 ] ⋅ G = 1 m [ C m 2 ⋅ G − 1 ⋅ G − C m 2 ⋅ B m 2 ⋅ G ] = 1 m [ C m 2 ⋅ I − I ⋅ G ] = 1 m [ C m 2 − G ]
即,
E ( m ) = C m 2 ⋅ K ( m ) ⋅ G (18.1)
类似地,因为 b 与 c 互逆,可得
c m 2 ⋅ k ( m ) ⋅ g = 1 m c m 2 ⋅ [ b m 2 − g − 1 ] ⋅ g = 1 m [ c m 2 ⋅ b m 2 ⋅ g − c m 2 ⋅ g − 1 ⋅ g ] = 1 m [ i ⋅ g − c m 2 ⋅ i ] = 1 m [ g − c m 2 ]
即,
e ( m ) = c m 2 ⋅ k ( m ) ⋅ g (18.2)
关系式(18.1)和(18.2)也可以写成以下等价形式:
K ( m ) = B m 2 ⋅ E ( m ) ⋅ G − 1 (19.1)
k ( m ) = b m 2 ⋅ e ( m ) ⋅ g − 1 (19.2)
将式(12.1)、(12.3)分别代入式(18.1)、(18.2), E ( m ) 与 K ( m ) 、 e ( m ) 与 k ( m ) 的变换关系可以分别用右拉伸张量 U 和左拉伸张量 v 表示为
E ( m ) = ( U m 2 ) T ⋅ G ⋅ U m 2 ⋅ K ( m ) ⋅ G = ( U m 2 ) T ⋅ G ⋅ K ( m ) ⋅ ( U m 2 ) T ⋅ G = ( U m 2 ) T ⋅ G ⋅ K ( m ) ⋅ G ⋅ U m 2 (20.1)
e ( m ) = ( v − m 2 ) T ⋅ g ⋅ v − m 2 ⋅ k ( m ) ⋅ g = ( v − m 2 ) T ⋅ g ⋅ k ( m ) ⋅ ( v − m 2 ) T ⋅ g = ( v − m 2 ) T ⋅ g ⋅ k ( m ) ⋅ g ⋅ v − m 2 (20.2)
同理,式(19.1)和(19.2)也可以用拉伸张量表示为
K ( m ) = U − m 2 ⋅ G − 1 ⋅ E ( m ) ⋅ G − 1 ⋅ ( U − m 2 ) T (21.1)
k ( m ) = v m 2 ⋅ g − 1 ⋅ e ( m ) ⋅ g − 1 ⋅ ( v m 2 ) T (21.2)
初始构型和现时构型上应变张量之间的变换称为“拉回”和“推前”操作。协变张量的“拉回”和“推前”分别记为 ϕ ← ( ⋅ ) 和 ϕ → ( ⋅ ) ,逆变张量的“拉回”和“推前”分别记为 χ ← ( ⋅ ) 和 χ → ( ⋅ ) ,具体操作运算如下:
ϕ ( F ) ← ( ⋅ ) = F T ⋅ ( ⋅ ) ⋅ F , ϕ ( F ) → ( ⋅ ) = F − T ⋅ ( ⋅ ) ⋅ F − 1 (22)
χ ( F ) ← ( ⋅ ) = F − 1 ⋅ ( ⋅ ) ⋅ F − T , χ ( F ) → ( ⋅ ) = F ⋅ ( ⋅ ) ⋅ F T (23)
由此可得,同为协变张量的 E ( m ) 与 e ( m ) “拉回”和“推前”关系为
E ( m ) = ϕ ( F ) ← ( e ( m ) ) = F T ⋅ e ( m ) ⋅ F (24.1)
e ( m ) = ϕ ( F ) → ( E ( m ) ) = F − T ⋅ E ( m ) ⋅ F − 1 (24.2)
同为逆变张量的 K ( m ) 与 k ( m ) “拉回”和“推前”关系为
K ( m ) = χ ( F ) ← ( k ( m ) ) = F − 1 ⋅ k ( m ) ⋅ F − T (25.1)
k ( m ) = χ ( F ) → ( K ( m ) ) = F ⋅ K ( m ) ⋅ F T (25.2)
比较式(14)~(17)中用主轴法表示的谱分解应变度量函数, E ( m ) 与 k ( m ) 、 K ( m ) 与 e ( m ) 具有完全相同的特征值,只是主轴方向不同,前后二者之间的“拉回”和“推前”变换只涉及旋转张量 R 。
将(7)和(9.1)代入(14)用主轴表示的应变度量函数中,有
E ( m ) = ∑ i = 1 3 1 m ( λ i m − 1 ) G ⋅ N i ⊗ N i ⋅ G = ∑ i = 1 3 1 m ( λ i m − 1 ) R T ⋅ g ⋅ R ⋅ N i ⊗ N i ⋅ R T ⋅ g ⋅ R
E ( m ) 做 R 旋转推前操作,
R − T ⋅ E ( m ) ⋅ R − 1 = ∑ i = 1 3 1 m ( λ i m − 1 ) R − T ⋅ R T ⋅ g ⋅ R ⋅ N i ⊗ N i ⋅ R T ⋅ g ⋅ R ⋅ R − 1 = ∑ i = 1 3 1 m ( λ i m − 1 ) g ⋅ R ⋅ N i ⊗ N i ⋅ R T ⋅ g = ∑ i = 1 3 1 m ( λ i m − 1 ) g ⋅ n i ⊗ n i ⋅ g = g ⋅ k ( m ) ⋅ g
从而得到 E ( m ) 与 k ( m ) 的变换关系为
类似地,对
对
式(26.1)~(26.4)所表示的不同构型协变与逆变应变张量的转换关系也可以通过式(22)和(23)所表示的完整“拉回”和“推前”操作,用左右伸长张量写成如下形式:
应变度量通常引入参考架构用应变张量的分量组合表示,应变张量与参考架构无关,但应变张量分量与参考架构有关。引入度规张量以后,式(14)~(17)的应变度量对初始构型和现时构型采用何种参考架构没有约束和限制,采用不同的坐标系,实际上就是选择了不同的基矢和度规,由此推导的上述变换关系不受参考构架的影响,既可以采用直线坐标系,也可以选择曲线坐标系,坐标轴可以正交,也可以斜交。
根据度规张量的几何意义,其分量等于两个对应坐标轴的切向基矢的点积,它反映了非正交曲线坐标系的坐标轴夹角和弯曲程度。正交坐标系度规张量的副元分量都为零,斜角坐标系则不全为零;工程实用中常用的参考架构主要是正交坐标系,比如直角坐标系、球坐标系、柱坐标系,此时,度规张量除了主元分量非零以外,其他分量都为零。球坐标系和柱坐标系是曲线坐标系,度规张量的主元分量不全为1,大小与坐标轴的曲率有关,协变度规张量与逆变度规张量对应的主元分量呈倒数关系。
对于笛卡尔直角坐标系,每个坐标轴上的量度一致,对应的构型空间为欧氏空间,均匀且各向同性,协变基矢和逆变基矢完全一致,都是正交单位矢量,度规张量为单位张量,即
此时,考虑到变形张量
所以对于笛卡尔应变张量,协变张量
此时无需再把应变张量分成协变和逆变,通常仅考虑基于初始构型的应变度量
本文基于Seth-Hill广义应变度量函数,推导了由此衍生的同一构型上的应变度量,即,
非线性连续介质力学的本构理论中,应变度量和应力度量存在能量共轭关系,二者成对用于建立能量泛函。本文仅从理论上探索了不同应变度量之间的变换,各类应变度量共轭应力与之配套的相应变换有待于进一步研究。
国家自然科学基金项目(51578405)资助。
吕凤悟,马锦明. 应变度量及其一般变换关系Strain Measures and Their General Transformation Relations[J]. 力学研究, 2020, 09(04): 150-158. https://doi.org/10.12677/IJM.2020.94018