Transcript Document
REPRESENTATIONS OF FOURTH-ORDER CARTESIAN
TENSORS OF STRUCTURAL MECHANICS
András Lengyel ― Tibor Tarnai
Budapest University of Technology and Economics
Department of Structural Mechanics
Budapest, Hungary
Tensor: system-independent mapping
(Cartesian tensors in the 3-dimensional Euclidean space)
Tensor A of order n is an array of entries denoted by Aijk…m with n indices i, j, k, …, m.
Hooke’s law: stresses ― strains
4-index notation (tensor notation)
Voigt’s notation (matrix notation)
ij Cijkl kl
ij Sijkl kl
σ Cε
σ ij Cijkl : ε kl
ε ij S ijkl : σ kl
3
3
11 c11
c
22 21
33 c31
23 c41
13 c51
12 c61
3
ij Cijkl kl
k 1 l 1
3
ij Sijkl kl
C: stiffness tensor
S: compliance tensor
σ: stress tensor
ε: strain tensor
k 1 l 1
ε S σ
c12
c22
c13
c23
c14
c24
c15
c25
c32
c33
c34
c35
c42
c52
c43
c53
c44
c54
c45
c55
c62
c63
c64
c65
c16 11
c26 22
c36 33
c46 2 23
c56 213
c66 212
C: stiffness matrix
S: compliance matrix
σ: stress vector
ε: strain vector
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Objectives:
• to create representations of fourth-order tensors (e.g. tensors of Hooke’s law)
in a way to show their structure, properties, operations, etc.
• to clarify technical terms involved in representations (e.g. identity, inverse,
definiteness, etc.)
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Representation:
entries of a tensor in an n-dimensional grid
vector of lower-order tensors
111
i
112
j
11
12
132
122
13
211
221
113
123
133
212
21
22
222
232
23
311
213
31
32
321
223
312
3
231
i
i
2
131
j
k
1
121
331
233
322
332
33
313
323
333
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Representation:
entries of a tensor in an
n-dimensional grid
1111
k
1211
i
1121
l
1221
vector of lower-order tensors
1112
1231
1122
2121
1331
2221
2112
1232
1123
3111
1132
2131
2231
2331
3221
3131
3122
3331
1322
2321
2212
1332
1223
3123
1323
2213
3212
2313
3312
1333
2223
3113
2323
3322
3213
3313
2333
3332
3233
1313
2312
2322
2113
2332
3222
1213
3311
2233
3232
3133
1113
3321
2123
2133
3132
2222
1312
2311
1233
2232
3231
1222
3112
1133
2132
1212
3211
2122
3121
1321
2311
2211
1131
1311
j
3223
3323
3333
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Representation:
entries of a tensor in an n-dimensional grid
2nd order tensor of 2nd order tensors: matrix of matrices
Aijkl (Aij )kl
Aijkl
1111
2111
3111
1121
2121
3121
1131
2131
3131
1211 1311 1112
2211 2311 2112
3211 3311 3112
1221 1321 1122
2221 2321 2122
3221 3321 3122
11
21
31
1213 1313
2213 2313
3213 3313
1223 1323
2223 2323
3223 3323
1133 1233 1333
2133 2233 2333
3133 3233 3333
1212 1312 1113
2212 2312 2113
3212 3312 3113
1222 1322 1123
2222 2322 2123
3222 3322 3123
12
22
32
1231 1331 1132 1232 1332
2231 2331 2132 2232 2332
3231 3331 3132 3232 3332
13
23
33
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Representation: matrix of matrices
2-dot product (double contraction)
3
Cijmn Aijkl : Bklmn
3
3
cijmn aijkl bklmn
k 1 l 1
3
c2331 a23kl bkl 31
k 1 l 1
:
=
7
Representation: mapping to a 9-by-9 matrix
(rearranging 81 entries of the tensor)
Aijkl A(ij ),( kl )
Aijkl
1111
1211
1311
2111
2211
2311
3111
3211
3311
ij, kl
11
12
13
21
22
23
31
32
33
F
1
2
3
4
5
6
7
8
9
1112 1113 1121 1122 1123 1131 1132 1133
1212 1213 1221 1222 1223 1231 1232 1233
1312 1313 1321 1322 1323 1331 1332 1333
2112 2113 2121 2122 2123 2131 2132 2133
2212 2213 2221 2222 2223 2231 2232 2233
2312 2313 2321 2322 2323 2331 2332 2333
3112 3113 3121 3122 3123 3131 3132 3133
3212 3213 3221 3222 3223 3231 3232 3233
3312 3313 3321 3322 3323 3331 3332 3333
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Representation: mapping to a 9-by-9 matrix
2-dot product (double contraction)
3
Cijmn Aijkl : Bklmn
3
3
c2331 a23kl bkl 31
k 1 l 1
3
cijmn aijkl bklmn
k 1 l 1
9
c67 a6 mbm7
m 1
:
=
9
Representation: quadratic form
x x1 ,x2 ,x3
A Aij
T
3
Q xT A x 1
3
Q x A x xi Aij x j x1 x1 A11 x1 x2 A12 x1 x3 A13 x2 x1 A21 x2 x2 A22
T
i 1 j 1
x2 x3 A23 x3 x1 A31 x3 x2 A32 x3 x3 A33
2nd-order stress tensor: Lamé’s ellipsoid
11 12 13
σ 21 22 23
31 32 33
3
3
Q x σ x xi ij x j x1 x111 2 x1 x212 2 x1 x313 x2 x2 22 2 x2 x3 23 x3 x3 33 1
T
i 1 j 1
10
Representation: quartic form
4th-order stiffness/compliance tensor: quartic surface
xi x1 ,x2 ,x3
X ij xi x j
3
3
3
3
Q Xij : Aijkl : Xkl xi x j Aijkl xk xl
i 1 j 1 k 1 l 1
x1 x1 x1 x1 A1111 x1 x1 x1 x2 A1112 x1 x1 x1 x3 A1113 x1x1x2 x1 A1121
x3 x3 x3 x3 A3333
Q Xij : Aijkl : Xkl 1
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Representation: quartic form
corundum (Al2O3)
c11 497 ,5 GPa
c12 162 , 7 GPa
c13 115,5 GPa
c14 22 ,5 GPa
c33 503,3GPa
c44 147 , 4 GPa
GLADDEN J.R., MAYNARD J.D., SO H., SAXE P., LE PAGE Y.,
Reconciliation of ab initio theory and experimental elastic properties
of Al2O3. Appl. Phys. Lett. Vol. 85, 2004, pp. 392-394.
Q 497,5 x14 497,5 x24 501 x34 90 x23 x3 995 x12 x22 820,6 x12 x32 820,6 x22 x32 270 x12 x2 x3 1
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Representation: quartic form
Plane of symmetry
Crystal
n-fold rotational symmetry around axis
system
x
y
z
x
y
z
triclinic
–
–
–
–
–
–
monoclinic
–
+
–
–
2
–
orthorombic
+
+
+
2
2
2
hexagonal
+
+
+
2
2
∞
trigonal
+
–
–
2
–
3
tetragonal
+
+
+
2
2
4
cubic
+
+
+
4
4
4
isotropic
+
+
+
∞
∞
∞
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Tensorial notation
3 3 3 3
Aijkl
?
mapping to 9×9
definiteness
Matrix representation
Aij
99
identity
?
inverse
eliminating
redundancies
Voigt’s notation
Aij
6 6
14
4th-order tensor – definiteness
operation: 2-dot product (double contraction)
general tensor (81 elements):
A is positive definite if the quadratic form is positive for all nonzero X:
Q Xij : Aijkl : Xkl 0
Hooke tensors (21 elements):
Stiffness tensor C is positive definite if the quadratic form is positive for all nonzero X,
where X is symmetric (or not?):
Q Xij : Aijkl : Xkl 0
xi x1 ,x2 ,x3 , Xij xi x j
Q ij : Cijkl : kl 0
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Tensorial notation
positive definite
3 3 3 3
Aijkl
All crystal groups
mapping to 9×9
All crystal groups
Matrix representation
Aij
99
definiteness
positive semidefinite
(3 zero eigenvalues)
eliminating
redundancies
Voigt’s notation
Aij
6 6
All crystal groups
positive definite
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4th-order tensor – identity tensor
operation: 2-dot product (double contraction)
I:A A:I A
I eijkl ik jl ei e j ek el
low symmetry
I T eijkl il jk ei e j ek el
low symmetry
ijkl
ijkl
IS
1
I IT
2
high symmetry
general tensor (81 elements)
I, IT, IS ?
Hooke tensors (21 elements)
I, IT, IS ?
aijkl aklij a jikl aijlk
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Symmetry
all permutation of ijkl
aijkl aklij a jikl aijlk
aijkl a jikl aijlk a jilk
aijkl a jilk
aijkl aklij
aijkl a jikl
aijkl aijlk
none
Different types of symmetry
Correspondence to
symmetry types
Type
15
21
36
45a
45b
54a
54b
81
A15
A 21
N max Type
15
15
21
21
36
36
45
45a
45
45b
54
54a
54
54b
81
81
A36 A 45 a A 45b A54 a A54b I I T
IS
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4th-order tensor – identity tensor
operation: 2-dot product (double contraction)
I:A A:I A
I eijkl ik jl ei e j ek el
ijkl
I T eijkl il jk ei e j ek el
ijkl
IS
1
I IT
2
Hooke tensors (21 elements)
general tensor (81 elements)
15 21 36 45a 45b 54a 54b 81
I
×
×
×
×
×
×
×
×
IT ×
×
×
×
×
IS
×
×
×
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Tensorial notation
Hooke tensors: I, IT, IS
3 3 3 3
Aijkl
general tensor: I
mapping to 9×9
Matrix representation
Aij
99
identity
identity matrix 9×9
eliminating
redundancies
Voigt’s notation
Aij
identity matrix 6×6
6 6
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4th-order tensor – inverse tensor
operation: 2-dot product (double contraction)
B A1
B:A A:B I
inhomogeneous linear equation system
162 equations (81+81, left+right inverse)
81 unknowns
(81 elements of the inverse tensor)
15 21 36 45a 45b 54a 54b 81
I
×
×
×
IT
IS
!
!
!
general tensor (81 elements)
I
unique solution!
Hooke tensors (21 elements)
I
no solution!
aijkl aklij a jikl aijlk
IS
rank deficiency 9
multiparameter solution
no inverse!
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Tensorial notation
Hooke tensors: IS (multiple)
3 3 3 3
Aijkl
general tensor: I (unique)
mapping to 9×9
Matrix representation
Aij
99
inverse
rank 6
no inverse matrix 9×9
eliminating
redundancies
Voigt’s notation
Aij
inverse matrix 6×6
6 6
22
Stiffness and compliance tensors of Hooke’s law – inverse tensor
Mathematical problem statement on
the inverse tensor
Physical problem statement on the
inverse tensor
inhomogeneous equation system
in 81 variables
solution sought in 81D space
multiple solution!
no inverse!
mathematical problem statement
+
conditions applied by physical properties
solution sought in 21D space
unique solution!
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Conclusions:
• there are a few ways to represent fourth-order tensors (advantages and
disadvantages)
• concepts of inverse, definiteness, etc. should be handled carefully
(mathematical vs. physical, 4-index vs. Voigt’s notation)
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Thank you for your attention!
Our research was supported by OTKA grant K81146
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