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# Invariant Properties for Finding Distance in Space of Elasticity Tensors

Authors:

## Abstract

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor, hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogonal group. These expressions are stated in terms of the eigenvalues of both the given tensor and the projected one. The process of minimization is facilitated by the fact that, as we prove, the traces of the corresponding Voigt and dilatation tensors are invariant under these orthogonal projections. For isotropy, cubic symmetry and transverse isotropy, we formulate algorithms to find both the orientation and the eigenvalues of the elasticity tensor that is endowed with a particular symmetry and is closest to the given elasticity tensor.
arXiv:0712.1082v2 [cond-mat.mtrl-sci] 11 Sep 2008
c
sym
c
Sym
c
d (c, c
sym
)
c d (c, Sym)
c Sym
c
d (c, c
sym
)
R
3
O (3)
R
3
A O(3)
c
sym
(A) c Sym
A
d (c, c
sym
(A))
c c
sym
(A)
d (c, Sym) d (c, c
sym
(A)) d (c, Sym)
d (c, c
sym
(A)) A O(3)
d (c, c
sym
(A)) f (λ
sym
α
(A)) λ
sym
α
(A)
c
sym
(A)
f (λ
sym
α
(A))
O(3) A
e
c 7→ c
iso
f
λ
iso
1
(A), λ
iso
2
(A)
d
c, c
iso
= d (c, Iso)
A c c
cube
(A)
λ
cube
1
(A) λ
cube
2
(A) λ
cube
3
(A) 1
2 3
c 7→ c
cube
d
c, c
cube
(A)
f
λ
cube
2
(A)
d (c, Cube) .
A
c c
T I
(A)
λ
T I
1
(A)
λ
T I
2
(A)
λ
T I
3
(A)
λ
T I
4
(A)
1 1 2 2
c 7→ c
T I
d
c, c
T I
(A)
f
λ
T I
1
(A), λ
T I
2
(A), λ
T I
3
(A)
d (c, T I) .
R
3
σ
ij
= c
ijkl
ε
kl
.
i, j, . . .
α, β, . . .
R
3
R
3
L
2,s
R
3
ε
1
· ε
2
:= Tr
ε
1
ε
t
2
,
Tr t
c : L
2,s
R
3
L
2,s
R
3
1
· ε
2
= ε
1
·
2
, ε
1
, ε
2
L
2,s
R
3
,
· ε > 0, ε L
2,s
R
3
, ε 6= 0.
c R
3
{e
1
, e
2
, e
3
}
c
ijkl
= c (e
i
e
j
) · (e
k
e
l
) ,
c
ijkl
= c
jikl
= c
klij
.
R
3
V
ij
= c
ikjk
and D
ij
= c
ijkk
,
O (3)
R
3
B = { e
1
, e
2
, e
3
}
R
3
B
= {f
1
, f
2
, f
3
}
A O(3) Ae
i
= f
i
i {1, 2, 3}
A
A O (3)
ε L
2,s
R
3
(A ε) (u, v) := ε (Au, Av) , u, v R
3
.
A O (3)
(A c) (ε) = A (c (A ε)) , ε L
2,s
R
3
.
B = {e
1
, e
2
, e
3
} A c
(A c)
ijkl
= c (Ae
i
Ae
j
) · (Ae
k
Ae
l
) .
c
A
{e
1
, e
2
, e
3
} R
3
L
2,s
R
3
ε
α(i,j)
= 2
1
2δ
ij
(e
i
e
j
+ e
j
e
i
) ,
α : {(i, j) , 1 i < j 3} {1, 2, . . . , 6},
α (i, j) =
ij
+ (1 δ
ij
) (9 i j) δ
ij
ε
ij
= ε (e
i
, e
j
)
{e
1
, e
2
, e
3
} R
3
ε
11
, ε
22
, ε
33
,
2ε
23
,
2ε
13
,
2ε
12
,
L
2,s
R
3
C
αβ
= Cε
α
· ε
β
.
C
αβ
6 × 6
c
1111
c
1122
c
1133
2c
1123
2c
1113
2c
1112
c
1122
c
2222
c
2233
2c
2223
2c
2213
2c
2212
c
1133
c
2233
c
3333
2c
3323
2c
3313
2c
3312
2c
1123
2c
2223
2c
3323
2c
2323
2c
2313
2c
2312
2c
1113
2c
2213
2c
3313
2c
2313
2c
1313
2c
1312
2c
1112
2c
2212
2c
3312
2c
2312
2c
1312
2c
1212
.
c · c
:= c
ijkl
c
ijkl
= C
αβ
C
αβ
,
kck
2
=
X
ijkl
c
2
ijkl
=
X
αβ
C
2
αβ
.
λ
1
, λ
2
, . . . , λ
r
m
1
, m
2
, . . . , m
r
m
1
+ m
2
+ ··· + m
r
= 6
kck
2
= m
1
λ
2
1
+ m
2
λ
2
2
+ ··· + m
r
λ
2
r
.
O (3)
{e
1
, e
2
, e
3
} c
ijkl
λ =
1
15
[c
1111
+ c
2222
+ c
3333
+ 4 (c
1122
+ c
1133
+ c
2233
)
2 (c
1212
+ c
1313
+ c
2323
)] ,
µ =
1
15
[c
1111
+ c
2222
+ c
3333
(c
1122
+ c
1133
+ c
2233
)
+3 (c
1212
+ c
1313
+ c
2323
)] .
λ = k 2µ/3
λ
µ
λ =
1
15
(2TrD TrV ) and µ =
1
30
(3TrV TrD) ,
O(3)
c
iso
=
λ + 2µ λ λ 0 0 0
λ λ + 2µ λ 0 0 0
λ λ λ + 2µ 0 0 0
0 0 0 2µ 0 0
0 0 0 0 2µ 0
0 0 0 0 0 2µ
λ µ
c
iso
λ
iso
1
= 3λ + 2µ =
1
3
TrD
λ
iso
2
= 2µ =
1
15
(3TrV TrD) ,
m
1
= 1 m
2
= 5
c c
iso
· c
iso
= 0
c c
iso
c
iso
c
iso
:= c c
iso
,
kck
2
= kc
iso
k
2
+ kc
iso
k
2
.
d
2
c, c
iso
= kc
iso
k
2
= kck
2
kc
iso
k
2
.
c 7→ c
iso
O(3)
d
2
(c, Iso) = d
2
c, c
iso
= kc
iso
k
2
= kck
2
kc
iso
k
2
.
k
µ
d
2
(c, Iso) = kck
2
3
h
(λ + 2µ)
2
+ 2
λ
2
+ 2µ
2
i
.
d
2
(c, Iso) = kck
2
1
15
2Tr
2
D + 3Tr
2
V 2TrV TrD
.
d
2
(c, Iso) =
6
X
α=1
λ
2
α
λ
iso
1
2
+ 5
λ
iso
2
2
=
λ
2
1
λ
iso
1
2
+
6
X
α=2
λ
2
α
λ
iso
2
2
.
λ
iso
1
λ
iso
2
B = {e
1
, e
2
, e
3
}
R
3
A O(3) AB :=
{Ae
1
, Ae
2
, Ae
3
}
A
c
ijkl
A
c
cube
(A) =
c
cube
1111
c
cube
1122
c
cube
1122
0 0 0
c
cube
1122
c
cube
1111
c
cube
1122
0 0 0
c
cube
1122
c
cube
1122
c
cube
1111
0 0 0
0 0 0 2c
cube
1212
0 0
0 0 0 0 2c
cube
1212
0
0 0 0 0 0 2c
cube
1212
,
c
cube
1111
=
1
3
(c
1111
+ c
2222
+ c
3333
) ,
c
cube
1122
=
1
3
(c
1122
+ c
1133
+ c
2233
)
c
cube
1212
=
1
3
(c
1212
+ c
1313
+ c
2323
) .
c
cube
(A)
λ
cube
1
(A) = c
cube
1111
+ 2c
cube
1122
,
λ
cube
2
(A) = c
cube
1111
c
cube
1122
λ
cube
3
(A) = 2c
cube
1212
,
m
1
= 1 m
2
= 2 m
3
= 3
c
A
a c b
c c
cube
(A)
·c
cube
(A) = 0
c
cube
(A) := c c
cube
(A)
kck
2
= kc
cube
(A)k
2
+ kc
cube
(A)k
2
.
c
c
cube
(A)
d
2
c, c
cube
(A)
= kc
cube
(A)k
2
= kck
2
kc
cube
(A)k
2
.
a b c
d
2
c, c
cube
(A)
d
2
c, c
cube
(A)
=
6
X
α=1
λ
2
α
λ
cube
1
(A)
2
+ 2
λ
cube
2
(A)
2
+ 3
λ
cube
3
(A)
2
=
λ
2
1
λ
cube
1
(A)
2
+
3
X
α=2
λ
2
α
λ
cube
2
(A)
2
+
6
X
α=4
λ
2
α
λ
cube
3
(A)
2
,
d (c, Cube)
A O(3)
f
λ
cube
1
(A), λ
cube
1
(A), λ
cube
1
(A)
=
λ
cube
1
(A)
2
+ 2
λ
cube
2
(A)
2
+ 3
λ
cube
3
(A)
2
.
λ
cube
1
(A)
λ
cube
2
(A)
λ
cube
3
(A)
TrV TrD
c 7→ c
cube
c c
cube
(A) V V
cube
(A) D D
cube
(A)
Tr V = Tr V
cube
(A) and Tr D = Tr D
cube
(A), A O(3).
Tr D
c 7→ c
cube
c 7→ c
iso
λ
cube
1
(A) = λ
iso
1
=
1
3
Tr D,
λ
cube
1
λ
cube
2
(A)
λ
cube
3
(A)
2λ
cube
2
(A) + 3λ
cube
3
(A) = 5λ
iso
2
= Tr V
1
3
Tr D.
λ
iso
2
m
iso
2
= 5
λ
cube
2
λ
cube
3
m
cube
2
= 2
m
cube
3
= 3
Tr V
c 7→ c
cube
c 7→ c
iso
Tr V
cube
(A) = λ
cube
1
(A) + 2λ
cube
2
(A) + 3λ
cube
3
(A)
= Tr V
iso
= λ
iso
1
+ 5λ
iso
2
= Tr V, A O(3).
O(3)
A
e
A
e
O(3)
d (c, Cube) = d
c, c
cube
(A
e
)
.
c
cube
(A
e
) c
λ
cube
1
(A
e
) λ
cube
2
(A
e
) λ
cube
3
(A
e
)
A
e
c
R
3
λ
cube
1
d
2
c, c
cube
(A)
= d
2
(c, Iso)
10
3
λ
iso
2
λ
cube
2
(A)
2
= d
2
(c, Iso)
15
2
λ
iso
2
λ
cube
3
(A)
2
.
10
λ
iso
2
λ
cube
2
(A)
2
/3
15
λ
iso
2
λ
cube
3
(A)
2
/2
c
cube
(A)
d
2
c, c
cube
(A)
= d
2
(c, Iso) d
2
c
cube
(A), Iso
.
A A
e
d
2
(c, Cube) = d
2
(c, Iso) d
2
c
cube
(A
e
), Iso
,
A A
e
d
2
(c, Cube) =
6
X
α=1
λ
2
α
λ
cube
1
(A
e
)
2
+ 2
λ
cube
2
(A
e
)
2
+ 3
λ
cube
3
(A
e
)
2
=
λ
2
1
λ
cube
1
(A
e
)
2
+
3
X
α=2
λ
2
α
λ
cube
2
(A
e
)
2
+
6
X
α=4
λ
2
α
λ
cube
3
(A
e
)
2
,
λ
cube
1
λ
cube
2
(A
e
)
λ
cube
3
(A
e
)
c
ijkl
A
c
T I
(A) =
c
T I
1111
c
T I
1122
c
T I
1133
0 0 0
c
T I
1122
c
T I
1111
c
T I
1133
0 0 0
c
T I
1133
c
T I
1133
c
T I
3333
0 0 0
0 0 0 2c
T I
2323
0 0
0 0 0 0 2c
T I
2323
0
0 0 0 0 0 c
T I
1111
c
T I
1122
,
c
T I
1111
=
1
8
(3c
1111
+ 3c
2222
+ 2c
1122
+ 4c
1212
) ,
c
T I
1122
=
1
8
(c
1111
+ c
2222
+ 6c
1122
4c
1212
) ,
c
T I
1133
=
1
2
(c
1133
+ c
2233
) ,
c
T I
3333
= c
3333
c
T I
2323
=
1
2
(c
2323
+ c
1313
) .
c
T I
(A)
λ
T I
1
(A) =
c
T I
1111
+ c
T I
1122
+ c
T I
3333
+
q
c
T I
1111
+ c
T I
1122
c
T I
3333
2
+ 8
c
T I
1133
2
2
,
λ
T I
2
(A) =
c
T I
1111
+ c
T I
1122
+ c
T I
3333
q
c
T I
1111
+ c
T I
1122
c
T I
3333
2
+ 8
c
T I
1133
2
2
,
λ
T I
3
(A) = c
T I
1111
c
T I
1122
λ
T I
4
(A) = 2c
T I
2323
,
m
1
= 1 m
2
= 1 m
3
= 2 m
4
= 2
c
A
λ
T I
1,2
(A) =
a + b ±
q
(a b)
2
+ c
2
2
, λ
T I
3
(A) = f, λ
T I
4
(A) = g.
c c
T I
(A)
· c
T I
(A) = 0,
c
T I
(A) := c c
T I
(A)
kck
2
= kc
T I
(A)k
2
+ kc
T I
(A)k
2
.
c
c
T I
(A)
d
2
c, c
T I
(A)
= kc
T I
(A)k
2
= kck
2
kc
T I
(A)k
2
.
a b c f g d
2
c, c
T I
(A)
d
2
c, c
T I
(A)
=
6
X
α=1
λ
2
α
λ
T I
1
(A)
2
+
λ
T I
2
(A)
2
+ 2
λ
T I
3
(A)
2
+ 2
λ
T I
4
(A)
2
=
λ
2
1
λ
T I
1
(A)
2
+
λ
2
2
λ
T I
2
(A)
2
+
4
X
α=3
λ
2
α
λ
T I
3
(A)
2
+
6
X
α=5
λ
2
α
λ
T I
4
(A)
2
,
d
2
c, c
T I
(A)
= d
2
(c, Iso) +
λ
iso
1
2
+ 5
λ
iso
2
2
λ
T I
1
(A)
2
+
λ
T I
2
(A)
2
+ 2
λ
T I
3
(A)
2
+ 2
λ
T I
4
(A)
2
.
d (c, T I)
A O(3)
f
λ
T I
1
(A), λ
T I
2
(A), λ
T I
3
(A), λ
T I
4
(A)
=
λ
T I
1
(A)
2
+
λ
T I
2
(A)
2
+ 2
λ
T I
3
(A)
2
+ 2
λ
T I
4
(A)
2
.
Tr V
Tr D
c c
T I
(A) V V
T I
(A) D D
T I
(A)
Tr V = Tr V
T I
(A) and Tr D = Tr D
T I
(A), A O(3),
Tr V
c 7→ c
T I
c 7→ c
iso
Tr V
T I
(A) = λ
T I
1
(A) + λ
T I
2
(A) + 2λ
T I
3
(A) + 2λ
T I
4
(A)
= Tr V
iso
= λ
iso
1
+ 5λ
iso
2
= Tr V, A O(3).
Tr D
T I
λ
T I
1
, λ
T I
2
γ
Tr D
T I
d
2
c, c
T I
(A)
= d
2
(c, Iso) +
λ
iso
1
2
+ 5
λ
iso
2
2
h
λ
T I
1
(A)
2
+
λ
T I
2
(A)
2
+ 2
λ
T I
3
(A)
2
+
1
2
λ
iso
1
+ 5λ
iso
2
λ
T I
1
(A) + λ
T I
2
(A) + 2λ
T I
3
(A)

2
,
c
O(3)
A
e
A
e
O(3)
d (c, T I) = d
c, c
T I
(A
e
)
.
c
T I
(A
e
)
c λ
T I
1
(A
e
) λ
T I
2
(A
e
) λ
T I
3
(A
e
) λ
T I
4
(A
e
)
A
e
c
A
A
e
d
2
(c, T I) = d
2
(c, Iso) d
2
c
T I
(A
e
), Iso
.
A
A
e
d
2
(c, T I) =
6
X
α=1
λ
2
α
λ
T I
1
(A
e
)
2
+
λ
T I
2
(A
e
)
2
+ 2
λ
T I
3
(A
e
)
2
+ 2
λ
T I
4
(A
e
)
2
=
λ
2
1
λ
T I
1
(A
e
)
2
+
λ
2
2
λ
T I
2
(A
e
)
2
+
4
X
α=3
λ
2
α
λ
T I
3
(A
e
)
2
+
6
X
α=5
λ
2
α
λ
T I
4
(A
e
)
2
,
λ
T I
1
(A
e
) λ
T I
2
(A
e
)
λ
T I
3
(A
e
)
λ
T I
4
(A
e
)
56.60 8.98 3.45 0 0 0
8.98 56.60 3.45 0 0 0
3.45 3.45 16.43 0 0 0
0 0 0 3.60 0 0
0 0 0 0 3.60 0
0 0 0 0 0 47.62
km
2
s
2
,
γ =
6.12
δ = 0.39 ε = 1.22
66.07 15.95 47.62
3.60
λ
iso
1
= 53.80 λ
iso
2
= . . . =
λ
iso
6
= 26.13
λ = 9.22
µ = 13.06
d (c, Iso) = 53.59 km
2
/s
2
c λ
cube
1
= 53.80
λ
cube
2
= 37.92
λ
cube
3
= 18.27
A c
cube
(A)
d (c, Iso) λ
iso
2
d
2
c, c
cube
(A)
= d
2
(c, Iso)
10
3
λ
iso
2
λ
cube
2
(A)
2
= 53.59
2
10
3
26.13 λ
cube
2
(A)
2
.
λ
cube
2
(A)
d (c, Cube) = 49.07 km
2
/s
2
c
A
e
=
0 0 1
0.82 0.58 0
0.58 0.82 0
.
c
cube
(A
e
)
c
λ
cube
1
(A
e
) = 53.80
λ
cube
2
(A
e
) = 12.64
λ
cube
3
(A
e
) = 18.27
λ
T I
1
λ
T I
2
λ
T I
3
λ
T I
4
d (c, T I) = 0
4.00 2.06 2.10 0.07 0.01 0.03
2.06 3.83 1.96 0.17 0.07 0.18
2.10 1.96 3.96 0.16 0.04 0.13
0.07 0.17 0.16 2.00 0.22 0.14
0.01 0.07 0.04 0.22 1.76 0.02
0.03 0.18 0.13 0.14 0.02 2.22
km
2
s
2
,
8.02
2.39 2.16 1.86 1.82 1.52
λ
iso
1
= 8.01
λ
iso
2
= . . . = λ
iso
6
= 1.95
λ = 2.02 µ = 0.98
0.724 km
2
/s
2
c c
iso
d (c, Cube) = 0.64 km
2
/s
2
A
e
=
0.77 0.36 0.53
0.60 0.13 0.79
0.21 0.93 0.31
.
λ
cube
1
(A
e
) =
8.01
λ
cube
2
(A
e
) = 0.59 λ
cube
3
(A
e
) = 2.08
d (c, T I) = 0.57 km
2
/s
2
A
e
=
0.76 0.53 0.38
0.58 0.29 0.76
0.29 0.80 0.53
.
λ
T I
1
(A
e
) = 8.01
λ
T I
2
(A
e
) = 1.62
λ
T I
3
(A
e
) = 2.05
λ
T I
4
(A
e
) = 2.02
... La variabilité du tenseur homogénéisé a également fait l'objet d'une étude attentive dans ce cas de l'homogénéisation en élasticité et nous sommes parvenus à la conclusion qu'un échantillon de n = 10 VER est suffisant. [18,55,13] : ...
... Le tenseur d'Eshelby pour l'inclusion I noté S I Esh est un tenseur d'ordre 2 dont les composantes écrites dans la base (e i ⊗ e j ) sont : S I Esh = S ij e i ⊗ e j . On retrouve des expressions analogues aux précédentes : [18,55,13]. Dans les articles cités, les auteurs définissent en particulier la distance à l'isotropie dans le cas de l'élasticité linéaire. ...
Thesis
Ce mémoire aborde les questions relatives à la mise en place de procédures de conception rapide, fiable et automatisée des volumes élémentaires représentatifs (VER) d’un matériau composite à microstructure complexe (matrice/inclusions), et de la détermination de leurs propriétés homogénéisées ou effectives. Nous avons conçu et développé des algorithmes conduisant à des outils efficaces permettant la génération aléatoire de tels matériaux à inclusions sphériques, cylindriques, elliptiques ou toute combinaison de celles-ci. Ces outils sont également capables d’altérer les inclusions : inflation, déflation, arrachements aléatoires, ondulation et de les pelliculer permettant ainsi de générer des VER s’approchant des matériaux composites fabriqués. Un soin particulier a été porté sur la génération de VER périodiques. Les caractéristiques homogénéisées ou propriétés effectives de matériaux constitués de tels VER périodiques peuvent alors être déterminées selon le principe d’homogénéisation périodique, soit par une méthode basée sur un schéma itératif utilisant la FFT (Transformation de Fourier Rapide) via l’équation de Lippmann-Schwinger, soit par une méthode d’éléments finis. Le caractère aléatoire de la génération nous amène à réaliser des études en moyenne à partir d’un ensemble de paramètres morphologiques déterminé : nombre d’inclusions, type et forme, fraction volumique, orientation des inclusions, prise en compte d’une éventuelle altération. Deux études particulières sur la conductivité thermique apparente ont été menées, la première sur les composites à inclusions sphériques pelliculées de façon à déterminer l’influence de l’épaisseur de la pellicule et la seconde sur les composites de type stratifié en polymère et fibre de carbone, cousu par un fil de cuivre pour évaluer l'apport de la couture en cuivre selon la fibre de carbone utilisée.
... Some algebraic relations, called syzygies, between the invariants are already apparent ( see [14][15][16][17][18] for other results). ...
Article
Full-text available
In this paper, we constructed relationships with the differents 2D elasticity tensor invariants. Indeed, let ${\bf A}$ be a 2D elasticity tensor. Rotation group action leads to a pair of Lax in linear elasticity. This pair of Lax leads to five independent invariants chosen among six. The definite positive criteria are established with the determined invariants. We believe that this approach finds interesting applications, as in the one of elastic material classification or approaches in orbit space description.
... In this study, we use the notion of mechanical isotropy since the goal is the evaluation of the effective elastic properties, and a deviation in geometric anisotropy (which is inherently present in our unit-cells) cannot be translated directly to deviation in mechanical isotropy. Several authors (see for instance Zener and Siegel (1949), Spoor et al. (1995), Bucataru and Slawinski (2008), Moussaddy et al. (2013), Ghossein and Lévesque (2014)) have proposed methods which can be used to estimate the deviation from mechanical isotropy. They differ from each other by the measure of the amplitude of the stiffness tensor (represented in matrix form adopting Voigt notation), as well as the number of the coefficients used in that measure. ...
Thesis
This thesis deals with the 3D-printing, numerical simulation and experimental testing of porous materials with random isotropic microstructures. In particular, we attempt to assess by means of well-chosen examples the effect of partial statistical descriptors (i.e., porous volume fraction or porosity, two-point correlation functions and chord-length distribution) upon the linear effective elastic response of random porous materials and propose (nearly) optimal microstructures by direct comparison with available theoretical mathematical bounds. To achieve this, in the first part of this work, we design ab initio porous materials comprising single-size (i.e. monodisperse) and multiple-size (polydisperse) spherical and ellipsoidal non-overlapping voids. The microstructures are generated using a random sequential adsorption (RSA) algorithm that allows to reach very high porosities (e.g. greater than 80%). The created microstructures are then numerically simulated using finite element (FE) and Fast Fourier Tranform (FFT) methods to obtain representative isotropic volume elements in terms of both periodic and kinematic boundary conditions. This then allows for the 3D-printing of the porous microstructures in appropriately designed dog-bone specimens. An experimental setup for uniaxial tension loading conditions is then developed and the 3D-printed porous specimens are tested to retrieve their purely linear elastic properties. This process allows, for the first time experimentally, to show that such polydisperse (multiscale) microstructures can lead to nearly optimal effective elastic properties when compared with the theoretical Hashin-Shtrikman upper bounds for a very large range of porosities spanning values between 0-82%. To understand further the underlying mechanisms that lead to such a nearly optimal response, we assess the influence of several statistical descriptors (such as the one- and two-point correlation functions, the chord-length distribution function) of the microstructure upon the effective elastic properties of the porous material. We first investigate the ability of the two-point correlation function to predict accurately the effective response of random porous materials by choosing two different types of microstructures, which have exactly the same first (i.e., porosity) and second-order statistics. The first type consists of non-overlapping spherical and ellipsoidal pores generated by the RSA process. The second type, which uses the thresholded Gaussian Random Field (GRF) method, is directly reconstructed by matching the one- and two-point correlation functions from the corresponding RSA microstructure. The FFT-simulated effective elastic properties of these two microstructures reveal very significant differences that are in the order of 100% in the computed bulk and shear moduli. This analysis by example directly implies that the two-point statistics can be highly insufficient to predict the effective elastic properties of random porous materials. We seek to rationalize further this observation by introducing controlled connectivity in the original non-overlapping RSA microstructures. The computed effective elastic properties of these microstructures show that the pore connectivity does not change neither the two-point correlation functions nor the chord-length distribution but leads to a significant decrease in the effective elastic properties. In order to quantify better the differences between those three microstructures, we analyze the link between the local geometry of the porous phase and the corresponding computed elastic fields by computing the first (average) and second moments of the elastic strain fluctuations. This last analysis suggests that partial statistical information of the microstructure (without any input from the corresponding elasticity problem) might be highly insufficient even for the qualitative analysis of a porous material and by extension of any random composite material.
... (1. 14) In the following, we use A to describe the volume average of A(x) over volume V , A = A(x) V . Proposed by Hill [46,47], for two phases composite, the homogenized stiffness tensor is given by: ...
Thesis
To circumvent the meshing difficulty of the existing numerical methods for composites homogenization, an original finite element method,named Phantom domain Finite Element Method (PFEM), is proposed in this thesis. The PFEM relies on computations of integrals with independent meshes based on a fictitious domain principle. In other words, one structured mesh is used for the entire domain, and independent meshes are used for the inclusions. The inclusion meshes will be related to the structured mesh through a substitution matrix. The PFEM is not only capable of calculating effective properties in homogenization technique with KUBC, SUBC and periodic condition, but also can be used in all the problems which can be solved by the FEM, such as the Dirichlet or Neumann boundary value problems. Numerical experiments in two or three dimensional cases, with inclusions of elementary geometry such as disk, square, sphere,cube and ellipsoid, have been performed to validate the PFEM method. Linear convergences of relative errors with respect to reference solutions such as the Mori-Tanaka model and the Fast Fourier Transform method are shown for thermal and elastic effective properties. We have illustrated some interesting features of the PFEM, such as the total flexibility concerning the inclusions meshes, by showing an example with a very thin pellicle sphere.
... Several authors (see for instance Zener and Siegel (1949), Spoor et al. (1995), Bucataru and Slawinski (2008), Moussaddy et al. (2013), Ghossein and Lévesque (2014)) have proposed methods which can be used to estimate the deviation from mechanical isotropy. They differ from each other by the measure of the amplitude of the stiffness tensor (represented in matrix form adopting Voigt notation), as well as the number of the coefficients used in that measure. ...
Article
Full-text available
The present study introduces a methodology that allows to combine 3D printing, experimental testing, numerical and analytical modeling to create random closed-cell porous materials with statistically controlled and isotropic overall elastic properties that are extremely close to the relevant Hashin-Shtrikman bounds. In this first study, we focus our experimental and 3D printing efforts to isotropic random microstructures consisting of single-sized (i.e. monodisperse) spherical voids embedded in a homogeneous solid matrix. The 3D printed specimens are realized by use of the random sequential adsorption method. A detailed FE numerical study allows to define a cubic representative volume element (RVE) by combined periodic and kinematically uniform (i.e. average strain or affine) boundary conditions. The resulting cubic RVE is subsequently assembled to form a standard dog-bone uniaxial tension specimen, which is 3D printed by use of a photopolymeric resin material. The specimens are then tested at relatively small strains by a proper multi-step relaxation procedure to obtain the effective elastic properties of the porous specimens.
... As shown by Gazis et al. [7], projection (7) ensures that a positive-definite tensor is projected to another positive-definite tensor, as required by Hookean solids. As shown by Moakher and Norris [9] and Bucataru and Slawinski [3]—in a fixed orientation of the rotation-symmetry axis that coincides with the x 3 -axis of the coordinate system—the Frobenius-norm effective transversely isotropic tensor derived from expression (7) has the form of tensor (3) with components given byˆc ...
Article
Full-text available
A generally anisotropic elasticity tensor, which might be obtained from physical measurements, can be approximated by a tensor belonging to a particular material-symmetry class; we refer to such a tensor as the effective tensor. The effective tensor is the closest to the generally anisotropic tensor among the tensors of that symmetry class. The concept of closeness is formalized in the notion of norm. Herein, we compare the effective tensors belonging to the transversely isotropic class and obtained using two different norms: the Frobenius norm and the L2 operator norm. We compare distributions of the effective elasticity parameters and symmetry-axis orientations for both the error-free case and the case of the generally anisotropic tensor subject to errors. © 2009 - 2014. Universita degli Studi di Padova - Padova University Press - All Rights Reserved.
Article
The problem of determining the transversely isotropic tensor closest in Euclidean norm to a given anisotropic elastic modulus tensor is considered. An orthonormal basis in the space of transversely isotropic tensors for any given axis of symmetry was obtained by decomposition of a transversely isotropic tensor in the general coordinate system into one isotropic part, two deviator parts, and one nonor part. The closest transversely isotropic tensor was obtained by projecting the general anisotropy tensor onto this basis. Equations for five coefficients of the transversely isotropic tensor were derived and solved. Three equations describing stationary conditions were obtained for the direction cosines of the axis of rotation (symmetry). Solving these equations yields the absolute minimum distance from the transversely isotropic tensor to a given anisotropic elastic modulus tensor. The transversely isotropic elastic modulus tensor closest to the cubic symmetry tensor was found.
Article
The aim of this short paper is to provide, for elasticity tensors, generalized Euclidean distances that preserve the property of invariance by inversion. First, the elasticity law is expressed under a non-dimensional form by means of a gauge, which leads to an expression of elasticity (stiffness or compliance) tensors without units. Based on the difference between functions of the dimensionless tensors, generalized Euclidean distances are then introduced. A subclass of functions is proposed, which permits the retrieval of the classical log-Euclidean distance and the derivation of new distances, namely the arctan-Euclidean and power-Euclidean distances. Finally, these distances are applied to the determination of the closest isotropic tensor to a given elasticity tensor.
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This work proposes an extension of the well-known random sequential adsorption (RSA) method in the context of non-overlapping random mono- and polydisperse ellipsoidal inclusions. The algorithm is general and can deal with inclusions of different size, shape and orientation with or without periodic geometrical constraints. Specifically, polydisperse inclusions, which can be in terms of different size, shape, orientation or even material properties, allow for larger volume fractions without the need of additional changes in the main algorithm. Unit-cell computations are performed by using either the fast Fourier transformed-based numerical scheme (FFT) or the finite element method (FEM) to estimate the effective elastic properties of voided particulate microstructures. We observe that an isotropic overall response is very difficult to obtain for random distributions of spheroidal inclusions with high aspect ratio. In particular, a substantial increase (or decrease) of the aspect ratio of the voids leads to a markedly anisotropic response of the porous material, which is intrinsic of the RSA construction. The numerical estimates are probed by analytical Hashin-Shtrikman-Willis (HSW) estimates and bounds.
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It is common to assume that a Hookean solid is isotropic. For a generally anisotropic elasticity tensor, it is possible to obtain its isotropic counterparts. Such a counterpart is obtained in accordance with a given norm. Herein, we examine the effect of three norms: the Frobenius 36-component norm, the Frobenius 21-component norm and the operator norm on a general Hookean solid. We find that both Frobenius norms result in similar isotropic counterparts, and the operator norm results in a counterpart with a slightly larger discrepancy. The reason for this discrepancy is rooted in the very definition of that norm, which is distinct from the Frobenius norms and which consists of the largest eigenvalue of the elasticity tensor. If we constrain the elasticity tensor to values expected for modelling physical materials, the three norms result in similar isotropic counterparts of a generally anisotropic tensor. To examine this important case and without loss of generality, we illustrate the isotropic counterparts by commencing from a transversely isotropic tensor obtained from a generally anisotropic one. Also, together with the three norms, we consider the L 2 slowness-curve fit. Upon this study, we infer that-for modelling physical materials-the isotropic counterparts are quite similar to each other, at least, sufficiently so that-for values obtained from empirical studies, such as seismic measurements-the differences among norms are within the range of expected measurement errors.
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equations Following Fermat's principle, the time during which the light covers its trajectory from a given point A to another given point B satisfies the condi-tion 0 = = ∫ ∫ B A B A Nds dt δ δ (I) where N denotes the index of refraction. By trans-formation, which is unnecessary to restate here 3 , one writes the above equation in this form: B A B A x ds dx N d ds x x N 0 ... ... δ δ , from which results the well-known differential equations 4 , 0 = ∂ ∂ −       x N ds dx N ds d 1 Original French title: Essai d'application du principe de Fermat aux milieux anisotropes; original Polish title: Próba zastosowania zasady Fermata do osrodków nieizotropowych. 2 Memoir presented on May 5, 1913. 3 The derivation follows the standard approach of Calculus of Varia-tions, which is described in numerous textbooks of mathematical physics (translator's comment). 4 Euler-Lagrange equations (translator's comment). , 0 = ∂ ∂ −       y N ds dy N ds d . 0 = ∂ ∂ −       z N ds dz N ds d All of the above relate to isotropic media. In aniso-tropic media, N no longer denotes the index of re-fraction. In such media, one must distinguish be-tween the speed of the propagation of light in the direction of the ray s and the speed in the direction normal to the wave surface q 5 . The index of refrac-tion is inversely proportional to q, whereas N is inversely proportional to s. There is more; N de-pends also on direction, and it is a function of not only x, y, z but also of direction cosines ds
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Vertical seismic profiling (VSP), an established technique, can be used for estimating in‐situ anisotropy that might provide valuable information for characterization of reservoir lithology, fractures, and fluids. The P‐wave slowness components, conventionally measured in multiazimuth, walkaway VSP surveys, allow one to reconstruct some portion of the corresponding slowness surface. A major limitation of this technique is that the P‐wave slowness surface alone does not constrain a number of stiffness coefficients that may be crucial for inferring certain rock properties. Those stiffnesses can be obtained only by combining the measurements of P‐waves with those of S (or PS) modes. Here, we extend the idea of Horne and Leaney, who proved the feasibility of joint inversion of the slowness and polarization vectors of P‐ and SV‐waves for parameters of transversely isotropic media with a vertical symmetry axis (VTI symmetry). We show that there is no need to assume a priori VTI symmetry or any other specific type of anisotropy. Given a sufficient polar and azimuthal coverage of the data, the polarizations and slownesses of P and two split shear (S1 and S2) waves are sufficient for estimating all 21 elastic stiffness coefficients c ij that characterize the most general triclinic anisotropy. The inverted stiffnesses themselves indicate whether or not the data can be described by a higher‐symmetry model. We discuss three different scenarios of inverting noise‐contaminated data. First, we assume that the layers are horizontal and laterally homogeneous so that the horizontal slownesses measured at the surface are preserved at the receiver locations. This leads to a linear inversion scheme for the elastic stiffness tensor c. Second, if the S‐wave horizontal slowness at the receiver location is unknown, the elastic tensor c can be estimated in a nonlinear fashion simultaneously with obtaining the horizontal slowness components of S‐waves. The third scenario includes the nonlinear inversion for c using only the vertical slowness components and the polarization vectors of P‐ and S‐waves. We find the inversion to be stable and robust for the first and second scenarios. In contrast, errors in the estimated stiffnesses increase substantially when the horizontal slowness components of both P‐ and S‐waves are unknown. We apply our methodology to a multiazimuth, multicomponent VSP data set acquired in Vacuum field, New Mexico, and show that the medium at the receiver level can be approximated by an azimuthally rotated orthorhombic model.
Article
A concise overview of a series of papers following [the author, J. Appl. Math. Mech. 48, 303-314 (1935; Zbl 0581.73015); translation from Prikl. Mat. Mekh. 48, 420-435 (1984)] has been presented; their principal feature consists in spectral decomposition of the stiffness and compliance tensors: the stiffness tensor is determined by six stiffness moduli and six mutually orthogonal stress-strain states, called proper elastic states. Numerous consequences of such a description have been analyzed.
Article
Shortly after his appointment to the first geophysical professorship (1895 at the Jagiellonian University of Cracow), Rudzki had published two papers in which he made a strong case for anisotropy of crustal rocks [Beitr. Geophys. 2 (1898); Bull. Acad. Sci. Crac. (1899)]. He had solved the Christoffel equation for transversely isotropic media in terms of what we today would call the ‘‘slowness surface’’. Rudzki regarded this as the representation of the wave surface in line coordinates. The conversion to point coordinates lead to an equation of 12th degree. Rudzki had determined a few points, but this was not sufficient to obtain an impression of the wavefront. Costanzi [Boll. Soc. Sismol. Ital. 7 (1901)] had suggested to simplify the coordinate conversion by expressing the solution of the Christoffel equation in a parameter form. The first part of the current paper describes the implementation of this idea. For the first time, the cusps in the wave surface became visible. The results of this first part have been discussed and expanded by Helbig [Beitr. Geophys. 67 (1958) 177; Bull. Seismol. Soc. Am. 56 (1966) 527; Helbig, K., 1994. Foundations of Anisotropy for Exploration Geophysics. Pergamon] and Khatkevich [Isv. Akad. Nauk. SSSR, Ser. Geofiz. 9 (1964) 788]. In a second part, Rudzki applied the ideas to orthorhombic media. The process is straightforward: the elements of the characteristic determinant are of order 2 in the three line coordinates (the three slowness components), with squares of coordinates in the diagonal elements and products of two coordinates in the off-diagonal elements. The elements are easily manipulated so that they are expressed in terms of squares only. Next, the determinant is expanded in terms of rows. This leads to three (equivalent) expressions. The vanishing of any of the three expressions means that the characteristic determinant vanishes, i.e., it corresponds to a solution of the Christoffel equation. Each of the equations can be used to determine one of the sheets of the line coordinates of the wave surface (point coordinates of the slowness surface). To this end, it is expressed in terms of two parameters, which have been chosen strictly for mathematical convenience. After conversion of the line coordinates to point coordinates (formation of the envelopes), one obtains a parameter expression for the wave surface. Until today, the second part of the Rudzki’s paper has not been closely studied. However, a blind test of the equations showed that they indeed describe the wave surface of orthorhombic media. The final sections discuss a few interesting aspects, among them the stability conditions for orthorhombic media and the condition under which a transversely isotropic medium transmits pure P- and S-waves.
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An insightful, structurally appealing and potentially utilitarian formulation of the anisotropic form of the linear Hooke's law due to Lord Kelvin was independently rediscovered by Rychlewski (1984, Prikl. Mat. Mekh.48, 303) and Mehrabadi and Cowin (1990, Q. J. Mech. appl. Math.43, 14). The eigenvectors of the three-dimensional fourth-rank anisotropic elasticity tensor, considered as a second-rank tensor in six-dimensional space, are called eigentensors when projected back into three-dimensional space. The maximum number of eigentensors for any elastic symmetry is therefore six. The concept of an eigentensor was introduced by Kelvin (1856, Phil. Trans. R. Soc.166, 481) who called eigentensors “the principal types of stress or of strain”. Kelvin determined the eigentensors for many elastic symmetries and gave a concise summary of his results in the 9th edition of the Encyclopaedia Britannica (1878). The eigentensors for a linear isotropic elastic material are familiar. They are the deviatoric second-rank tensor and a tensor proportional to the unit tensor, the spherical, hydrostatic or dilatational part of the tensor. Mehrabadi and Cowin (1990, Q. J. Mech. appl. Math.43, 14) give explicit forms of the eigentensors for all of the linear elastic symmetries except monoclinic and triclinic symmetry. We discuss two approaches for the determination of eigentensors and illustrate these approaches by partially determining the eigentensors for monoclinic symmetry. With the nature of the eigentensors for monoclinic symmetry known, a rather complete table of the structural properties of all linear elastic symmetries can be constructed. The purpose of this communication is to give the most specifically detailed presentation of the eigenvalues and eigentensors of the Kelvin formulation to date.
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Presenting a comprehensive introduction to the propagation of high-frequency body-waves in elastodynamics, this volume develops the theory of seismic wave propagation in acoustic, elastic and anisotropic media to allow seismic waves to be modelled in complex, realistic three-dimensional Earth models. The book is a text for graduate courses in theoretical seismology, and a reference for all academic and industrial seismologists using numerical modelling methods. Exercises and suggestions for further reading are included in each chapter.
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The fourth-rank tensors that embody the elastic or other properties of crystalline anisotropic substances can be partitioned into a number of sets in order that each shall acknowledge the symmetry of one or more of the crystal classes and moreover shall make up a closed linear associative algebra of hypercomplex numbers for the purpose of calculating the sums, products and inverses of its constituent tensors, to which end coordinate-invariant expressions of the tensors are adopted. The calculations are simplified immensely, and ensuing physical analyses are well prepared for, once the structure of every algebra is unravelled completely in terms of a number of separate subalgebras isomorphic to familiar algebras such as the binary one of the complex numbers, the quaternary one of the 2 x 2 matrices and the octonary one of the complex quaternions. The fourth-rank tensors do not seem to have been submitted previously to the present algebraic point of view, and nor do those of any other rank: a parallel, but less intricate, development can be provided for the second-rank ones.