软物质力学

In the considered case of Cartesian coordinates, the tensor can be interpreted as a matrix of its components. In the case of curvilinear coordinates, the situation is subtler and
free. Computing (1.10) for varying i we get c1 a2b3 a3b2 , c2 a3b1 a1b3 , c3 a1b2 a2b1 .
(1.11)
1.2 Second-order tensors
To define a second-order tensor we introduce dyadic or tensor product, , of base vectors
(1.1)
Within this coordinate system we define arbitrary vector a as follows
a a1e1 a2e 2 a3e 3 ai e i ,
i 1
3
(1.2)
where ai are the components of the vector. Notation in (1.2) is excessive and it is worth simplifying it by using the Einstein rule
ci e i c e i (a b) e i {(a j e j ) (bk e k )} {e i (e j e k )}a j bk ijk a j bk .
c
(1.10)
ijk
It is important that there is no summation over index i in (1.10). Such index is called
Amj Dmj
.
(1.17)
By using the double dot product we can calculate components of a second-order tensor as follows e i e j : A e i e j : ( Amne m e n ) Amn (e i e m )(e j e n ) Aij .
1, ijk 123; 231; 312 e i (e j e k ) 1, ijk 321; 213; 132 ijk . 0, ijk ...
(1.9)
The permutation symbol allows us to write the components of the vector product in a short way
1 0 0 1 e1 e1 e e 0 1 0 0 0 0 0 , 0 0 0 0
T 1 1
Mechanics of Soft Materials
3
Volokh 2010
0 1 0 1 e1 e 2 e e 0 0 1 0 0 0 0 , 0 0 0 0
(1.7)
(1.8)
The Kronecker delta was introduced through the scalar products of the Cartesian base vectors. It is also very convenient to introduce the permutation (Tulio Levi-Civita) symbol by using triple product of base vectors
jm
bj ci
(1.15)
or
c1 A11 ci Aij b j , c2 A21 c A 3 31 A12 A22 A32 A13 b1 A23 b2 . A33 b3
Product of two second-order tensors is defined as follows
Mechanics of Soft Materials
Volokh 2010
1 Tensors
1.1 Vectors
Vectors are tensors of the first order/rank, by definition, while scalars are zero-order tensors.
Mechanics of Soft Materials
4
Volokh 2010
F AD ( Aij e i e j )( Dmn e m e n ) Aij Dmn (e j e m )e i e n
jm
Aij Dmn jm e i e n Aij D jn e i e n
MECHANICS OF SOFT MATERIALS
K. Y. Volokh Faculty of Civil and Environmental Engineering Technion – Israel Institute of Technology
Contents 1. Tensors 2. Kinematics 3. Balance laws 4. Isotropic elasticity 5. Anisotropic elasticity 6. Viscoelasticity 7. Chemo-mechanical coupling 8. Electro-mechanical coupling 2 16 28 37 49 56 64 71
(1.5)
where we introduced the (Leopold) Kronecker delta for short notation. Substituting (1.5) in (1.4) we have
a b ai b j e i e j ai b j ij ai bi a j b j a1b1 a2b2 a3b3 ,
Fin
,
(1.16)
or
F11 Fin Aij D jn , F21 F 31
F12 F22 F32 F13 A11 F23 A21 F33 A31 A12 A22 A32 A13 D11 A23 D21 A33 D31 D12 D22 D32 D13 D23 . D33
(1.4)
Mechanics of Soft Materials
2
Volokh 2010
The scalar product of base vectors is zero for different base vectors and one for the same vector
1, i j ei e j ij , 0, i j
A Aij e i e j Aij e i e j .
j 1 i 1 3 3
(1.13)
(1.14)
The components of the second-order tensor can be written in the matrix form
A11 A21 A 31 A12 A22 A32 A13 A23 . A33
ij
(1.6)
where b j ij b1 i1 b2 i 2 b3 i 3 bi . By using the dot product of base vector ei with vector a we find ai
e i a e i (a j e j ) a j e i e j a j ij ai .
a e
i 1
3
i i
ai e i ,
Байду номын сангаас
(1.3)
which means that the symbol of the sum can be dropped when the summation is performed over two repeated indices. Such indices are called dummy because they can be designated by any character
Double dot product of two tensors is a scalar
A : D ( Aij e i e j ) : ( Dmn e m e n ) Aij Dmn (e i e m )(e j e n )
im jn
Aij im Dmn jn Amj Dmj A11 D11 A12 D12 .... A33 D33
T 1 2
…
0 0 0 0 e 3 e 3 e e 0 0 0 1 0 0 0 , 0 0 1 1
T 3 3
(1.12)
ei e j e j ei . By analogy with vectors, we define second-order tensors as a linear combination of base dyads A A11e1 e1 A12e1 e 2 A13e1 e3 A21e 2 e1 A22e 2 e 2 A23e 2 e3 . A31e 3 e1 A32e3 e 2 A33e 3 e 3 By using short notation we can rewrite (1.13) as follows
ai e i a j e j am e m .
Using Einstein’s rule we can write down the scalar or dot product of two vectors a and b as follows
a b (ai e i ) (b j e j ) ai b j e i e j .
various matrices of components can represent the same tensor. The latter will be discussed
below. A second-order tensor (or matrix) maps one vector into another as follows c Ab ( Aij e i e j )(bm e m ) Aij bm e i (e j e m ) Aij bm jm ei Aij b j ei ,
x3
a
e3
e1
b
e2 x2
x1
We consider Cartesian coordinate system with mutually orthogonal axes, xi , and base vectors
0 0 1 e1 0 , e 2 1 , e 3 0 . 1 0 0