Yury Grabovsky, yury@temple.edu
A complete list of exact relations for effective elastic tensors of fiber-reinforced composites
Microstructure-independent relations for effective tensors of composites have been attracting the attention of
materials scientists throughout the history of the subject. In the context of fiber-reinforced elastic composites,
the confluence of fully 3-dimensional elastic tensors with an inherently 2D microstructure yields an enormous
number of exact relations ranging from physically obvious to bewildering. Invariably, these relations can be
represented by unexpectedly elegant formulas. I will describe a long and exciting journey starting from the
general theory, through the efforts of Ph.D. and undergraduate students, to the complete list of exact relations.
The general theory identifies exact relations that are closed under lamination with special algebras, whose
multiplication (called Jordan product) is commutative, but not assiciative. Moreover, these algebras are closed
with respect to several such multiplications, and hence, are called Jordan multialgebras. Some of these objects
can also be constructed in a natural way from the associative multialgebras, in which case we can prove that
they produce exact relations closed under homogenization.
In order to identify all polycrystalline exact relations one needs to begin by identifying all rotationally invariant Jordan multialgebras. This was done by Meredith Hegg in her Ph.D. dissertation. The resulting list of
203 rotationally invariant Jordan multialgebras was highly redundant, since it contained all intersections of all
the multialgebras in the list. Tatyana Nuzhnaya, a 2013 Summer Research Assistant, developed and implemented a Matlab algorithm to eliminate all intersections from the list. During 2013-2014 academic year two
undergraduate researchers Mark Mikida and Andrew Schneider took the reduced list of 23 multialgebras and
converted them into the language of elastic tensors. Finally, Adam Jacobi, a 2014 Summer Research Assistant,
verified that all of these Jordan multialgebras can indeed be constructed from associative ones, proving that all
23 lamination exact relations for fiber reinforced elstic composites are valid for all microstructures.
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
A complete list of exact relations for effective
elastic tensors of fiber-reinforced composites
Yury Grabovsky
CMDS-13
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Exact relations for 2D and 3D elastic polycrystals
ε—strain, σ—stress, σ = Cε—Hooke’s law, n = 2 or 3
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Exact relations for 2D and 3D elastic polycrystals
ε—strain, σ—stress, σ = Cε—Hooke’s law, n = 2 or 3
Lurie, Cherkaev and Fedorov ’84
C(x)In = nκ0 In
⇒
C∗ In = nκ0 In
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Exact relations for 2D and 3D elastic polycrystals
ε—strain, σ—stress, σ = Cε—Hooke’s law, n = 2 or 3
Lurie, Cherkaev and Fedorov ’84
C(x)In = nκ0 In
⇒
C∗ In = nκ0 In
1
Hill ’63: C(x)ε = κ(x)(Tr ε)In + 2µ0 ε − (Tr ε)In
n
1
∗
∗
C ε = κ (Tr ε)In + 2µ0 ε − (Tr ε)In ,
n
nκ∗
1
1
=h
i
+ 2(n − 1)µ0
nκ(x) + 2(n − 1)µ0
⇒
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Exact relations for 2D and 3D elastic polycrystals
ε—strain, σ—stress, σ = Cε—Hooke’s law, n = 2 or 3
Lurie, Cherkaev and Fedorov ’84
C(x)In = nκ0 In
C∗ In = nκ0 In
⇒
1
Hill ’63: C(x)ε = κ(x)(Tr ε)In + 2µ0 ε − (Tr ε)In
n
1
∗
∗
C ε = κ (Tr ε)In + 2µ0 ε − (Tr ε)In ,
n
nκ∗
⇒
1
1
=h
i
+ 2(n − 1)µ0
nκ(x) + 2(n − 1)µ0
Rank-1 plus null-Lagrangian (Grabovsky and Milton’ 98)
C(x) = 2µ0 T + C(x) ⊗ C(x)
⇒
C∗ = 2µ0 T + C ∗ ⊗ C ∗
Tr ((Tε)ε) = Tr (ε2 ) − (Tr ε)2
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Special 2D results
Lurie-Cherkaev exact relation (1984)
C(x)I2 = 2κ0 I2
If 2µ1 (x) and 2µ2 (x) are two other eigenvalues of C(x) and
1
1
1
1
+
+
= ρ0
κ0 µ1 (x)
κ0 µ2 (x)
C∗ I2 = 2κ0 I2
1
1
1
1
+
+
= ρ0
κ0 µ∗1
κ0 µ∗2
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Special 2D results
Lurie-Cherkaev exact relation (1984)
C(x)I2 = 2κ0 I2
If 2µ1 (x) and 2µ2 (x) are two other eigenvalues of C(x) and
1
1
1
1
+
+
= ρ0
κ0 µ1 (x)
κ0 µ2 (x)
C∗ I2 = 2κ0 I2
1
1
1
1
+
+
= ρ0
κ0 µ∗1
κ0 µ∗2
Global Link (Cherkaev, Lurie and Milton ’92)
−1
If C′ (x) = α(C(x)−1 + βT)−1 then C′∗ = α(C−1
∗ + βT)
Tr ((Tε)ε) = Tr (ε2 ) − (Tr ε)2
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Exact relations count
3D Elastic polycrystals: 3
Theory of exact relations
Computing exact relations
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Exact relations count
3D Elastic polycrystals: 3
2D Elastic polycrystals: 4
Theory of exact relations
Computing exact relations
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Exact relations count
3D Elastic polycrystals: 3
2D Elastic polycrystals: 4
Fiber-reinforced elastic polycrystals: 203 (modulo global link)
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Exact relations count
3D Elastic polycrystals: 3
2D Elastic polycrystals: 4
Fiber-reinforced elastic polycrystals: 203 (modulo global link)
Minimal non-redundant exact relations: 23
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Stress and strain in fiber-reinforced elastic materials
e3 —fiber direction
F = {αe3 : α ∈ R}—line parallel to the fibers
T = {αe1 + βe2 : {α, β} ⊂ R}—transversal plane
R3 = T ⊕ F.
Strain and stress
ε ∈ Sym(R3 ),
σ ∈ Sym(R3 ).
Sym(R3 ) = Sym(T ⊕ F) = Sym(T) ⊕ Sym(F) ⊕ Sym(T, F)
Sym(T, F) = {ε ∈ Sym(R3 ) : ε : T → F,
F → T}
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Geometric meaning of strain tensor block-components
Sym(R3 ) = Sym(T ⊕ F) = Sym(T) ⊕ Sym(F) ⊕ Sym(T, F)
ε=
"
ε⊤ ε⋌
ε⋌
εk
#
= T
F
T F
ε⊤ ε⋌
ε ⋌ εk
ε⊤ ∈ Sym(R2 )—strain in the transversal plane
εk ∈ R—strain in the fibers
(1)
(2)
ε⋌ = (ε⋌ , ε⋌ ) ∈ R2 —tilt of the transversal plane:
(1)
(2)
(ε , ε , 1)
e = p⋌ ⋌
n = (0, 0, 1) 7→ n
1 + |ε⋌ |2
|ε⋌ | = α—tilt angle
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Elastic tensors of fiber-reinforced elastic materials
Sym(R3 ) = Sym(T ⊕ F) = Sym(T) ⊕ Sym(T, F) ⊕ Sym(F)
C ∈ Sym(Sym(R3 ))
C=
Sym(T)
Sym(T, F)
Sym(F)
Sym(T) Sym(T, F) Sym(F)
C⊤
C⊤⋌
C⊤k
⋆
C⊤⋌
C⋌
C⋌k
⋆
⋆
C⊤k
C⋌k
Ck
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
An example from the Lurie, Cherkaev, Fedorov family

C⊤
 ⋆
C=
 C⊤⋌
⋆
C⊤k
Moreover,
C⊤⋌ C⊤k
C⋌
⋆
C⋌k


C⋌k 
,
Ck
C⊤ I2 = 2κ0 I2 ,
∗
Tr C⊤k
= Tr hC⊤k i.
⋆
C⊤⋌
I2 = 0,
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
An example from the Hill family



C=

Moreover,
Ck∗ −
2µ0 I + 2λI2 ⊗ I2
t ⊗ I2
νI2
I2 ⊗ t
σ 0 I2 +
t⊗t
2(λ+2µ0 )
p0 t⊥ +(ν+ν0 )t
2(λ+2µ0 )
νI2
p0 t⊥ +(ν+ν0 )t
2(λ+2µ0 )
Ck



,

t
t∗
=h
i
∗
λ + 2µ0
λ + 2µ0
p02 + (ν ∗ + ν0 )2
4µ20 |t∗ |2
−
=
2(λ∗ + 2µ0 )
σ0 (λ∗ + 2µ0 )2
4µ20 |t|2
p 2 + (ν + ν0 )2
−
i
hCk − 0
2(λ + 2µ0 )
σ0 (λ + 2µ0 )2
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
An example from the RPN family

2µ0 T + C ⊗ C
C ⊗t
νC + ν0 I2
t⊗C
σ 0 I2 + t ⊗ t
C⊤k  ,
C =
νC + ν0 I2
C⊤k
Ck

Moreover,
∗
C⊤k
− ν ∗ t∗ = hC⊤k i − hνti
Tr ((Tε)ε) = Tr (ε2 ) − (Tr ε)2
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
An example from the Lurie-Cherkaev family
1
2
3
4
5
6
⋆ I =0
C⊤ I2 = 2κ0 I2 , C⊤⋌
2
For constants σ0 > 0 and 0 < τ0 < κ0 , define Shur
complements S⊤ , S⋌ and tensor Λ by the formula
#−1 "
"
#
C⊤ + τ0 I
C⊤⋌
(S⊤ + τ0 I)−1
Λ
=
⋆
C⊤⋌
C⋌ + σ0 I2
Λ⋆
(S⋌ + σ0 I2 )−1
Λ must belongs to one of the two connected components of
the solution set of dev(Λ⋆ Λ) = 0.
det(S⋌ ) = σ02 .
Let 2µ1 and 2µ2 be the eigenvalues (shear moduli) of S⊤ (its
third eigenvalue is 2κ0 ). We require that
!
−1 −1
κ−1
1
1
1
1
1
0 + ρ0
τ0 =
+
+
= 2,
κ0 µ1
κ0 µ2
2
ρ0
∗ = Tr hC i.
Moreover, Tr C⊤k
⊤k
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Global link (Hegg 2013)
Affine part

q1 I
Q0 =  0
q4 I2
C′ = Q0 CQT
0 + F0 ,



0
0
0
0 f1 I 2
q2 I2 0  ,
0
0 
F0 =  0
0
q3
f1 I 2 0 f2
Fully non-linear part
C′ = C − CΛa0 (C)C,


a (T + aC⊤ )−1 0 0
Λa (C) = 
Tr ((Tε)ε) = Tr (ε2 )−(Tr ε)2
0
0 0 ,
0
0 0
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Exact relations: formal definitions
Definition. A submanifold M of Sym+ (Sym(R3 )) is called an
exact relation if C∗ ∈ M for any composite with C(x) ∈ M for all
x. We will also say that M is stable under homogenization.
Definition. A submanifold M of Sym+ (Sym(R3 )) is stable under
lamination if C∗ ∈ M for any laminate of {C1 , C2 } ⊂ M.
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Lamination formula (with vertical lamina)
En = {a ⊗ n + n ⊗ a : a ∈ R3 },
n = (n1 , n2 , 0).
Jn = {ε ∈ Sym(R3 ) : εn = 0}.
C0 —arbitrary transversely isotropic elastic tensor
Γ′ (n)—non-orthogonal projection operator onto C0 En along Jn .
′
3
Γ0 (n) = C−1
0 Γ (n) ∈ Sym(Sym(R )).
−1
Wn (C) = (C0 − C)−1 − Γ0 (n)
Theorem (Milton 1990): Let C(x) be the laminate with normal n.
Then
Wn (C∗ ) = hWn (C)i
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Exact relations theorem (necessary conditions)
Define: A = Span{Γ0 (n) − Γ0 (n0 ) : n = (n1 , n2 , 0)}.
Theorem (YG, Milton, Sage 2000): If M is stable under
lamination then for any C0 ∈ M there exists a subspace
Π ⊂ Sym(Sym(R3 )) with dim Π = dim M and such that
Wn (C) ∈ Π for any n = (n1 , n2 , 0). Moreover, for every
{K1 , K2 } ⊂ Π and any A ∈ A
1
K1 ∗A K2 = (K1 AK2 + K2 AK1 ) ∈ Π
2
Π is a special Jordan A-multialgebra
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Exact relations theorem (sufficient conditions)
Theorem. Suppose the subspace Π ⊂ Sym(Sym(R3 )) is such that
K1 A1 K2 + K2 A1 K1 ∈ Π
K1 A1 K2 A2 K3 + K3 A2 K2 A1 K1 ∈ Π
K1 A1 K2 A2 K3 A3 K4 + K4 A3 K3 A2 K2 A1 K1 ∈ Π
∀{K1 , K2 , K3 , K4 } ⊂ Π, ∀{A1 , A2 , A3 } ⊂ A. Then
M = {C ∈ Sym+ (Sym(R3 )) : Wn0 (C) ∈ Π},
is independent of the choice of n0 and is stable under
homogenization.
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
How can one compute all exact relations?
Polycrystalline exact relations: C ∈ M ⇒ R · C ∈ M for all
rotations R around the fiber. This implies that R · K ∈ Π for
all rotations R ∈ SO(2) and all K ∈ Π, provided C0 is
transversely isotropic
Use “inversion key” M: Instead of Wn (C) use
−1
WM (C) = (C0 − C)−1 − M
where M is such that for all K ∈ Π
K(Γ − M)K ∈ Π,
1
Γ=
4π
Z
S2
Γ0 (n)dS(n)
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Meredith Hegg: Ph.D. dissertation 2012
Identified all SO(2)-invariant subspaces of Sym(Sym(R3 ))
Computed Γ0 (n), Γ and A
Computed the “multiplication table” of SO(2)-invariant
subspaces of Sym(Sym(R3 ))
Developed a procedure for finding all SO(2)-invariant Jordan
A-multialgebras
Computed Global Link (Comptes Rendus Mécanique 2013)
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Tatyana Nuzhnaya (Summer 2013 RA)
Computed all SO(2)-invariant Jordan A-multialgebras and
volume fraction relations in algebraic form
Eliminated all intersections from the list of all SO(2)-invariant
Jordan A-multialgebras
Computed all inversion keys
Identified all links in algebraic form (468)
All calculations were completely automated in Matlab
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Andrew Schneider and Mark Mikida: URP 2013-2014
Converted the reduced list of 23 SO(2)-invariant Jordan
A-multialgebras into exact relations theorems by inverting
K = WM (C)
Computed all additional volume fraction relations
All calculations had to be done by hand to present every
exact relation in a mathematically beautiful form
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Adam Michael Jacoby (Summer 2014 RA)
Verified sufficient conditions for all exact relations and links
Eliminated all intersections from the list of links
Computed inversion keys for essential links (25)
All calculations were completely automated in Maple
2D and 3D polycrystals
Fiber-reinforced materials
Exact relations
Theory of exact relations
Computing exact relations
Remaining work (URP)
Computing beatiful frormulations of 25 essential links