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  Cε
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   213 
 

c66   212 
C: stiffness matrix
S: compliance matrix
σ: stress vector
ε: strain vector
2
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.)
3
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
4
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
5
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
6
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
8
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 x111  2 x1 x212  2 x1 x313  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
11
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
12
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
+
+
+
∞
∞
∞
13
Tensorial notation
3 3 3 3
Aijkl

?
mapping to 9×9
definiteness
Matrix representation
Aij

99
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
15
Tensorial notation
positive definite
3 3 3 3
Aijkl

All crystal groups
mapping to 9×9
All crystal groups
Matrix representation
Aij

99
definiteness
positive semidefinite
(3 zero eigenvalues)
eliminating
redundancies
Voigt’s notation
Aij
6 6
All crystal groups
positive definite
16
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
17
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








18
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
×
×





×
19
Tensorial notation
Hooke tensors: I, IT, IS
3 3 3 3
Aijkl

general tensor: I
mapping to 9×9
Matrix representation
Aij

99
identity
identity matrix 9×9
eliminating
redundancies
Voigt’s notation
Aij
identity matrix 6×6
6 6
20
4th-order tensor – inverse tensor
operation: 2-dot product (double contraction)
B  A1
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!
21
Tensorial notation
Hooke tensors: IS (multiple)
3 3 3 3
Aijkl

general tensor: I (unique)
mapping to 9×9
Matrix representation
Aij

99
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!
23
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)
24
Thank you for your attention!
Our research was supported by OTKA grant K81146
25