Optimal structures
made of two elastic materials and void
Grzegorz Dzierżanowski
in collaboration with
Nathan Briggs and Andrej Cherkaev
Introduction
In this talk, some novel, recently obtained, exact results in optimal design of
three-phase elastic structures are discussed. The problem is formulated as follows: Two isotropic materials, the “strong” and the “weak” one, are laid out
with void in a given two-dimensional domain so that the compliance plus weight
of a structure is minimized. As in the classical two-phase problem, the optimal
layout of three phases is also determined on two levels: macro- and microscopic.
On the macrolevel, the design domain is divided into several subdomains. Some
are filled with pure phases, and others with their mixtures (composites). The
main aim of the talk is to discuss the non-uniqueness of the optimal macroscopic multiphase distribution. This phenomenon does not occur in the twophase problem, and in the three-phase design it arises only when the moduli of
material isotropy of “strong” and “weak” phases are in certain relation.
Statement of the problem
Let K1 , K2 , K1 < K2 denote the inverses of the bulk moduli of respective elastic
materials and assume that K3 = +∞ in void. Similarly, write L1 , L2 , L1 < L2
and L3 = +∞ for the inverses of the shear moduli. Equilibrium conditions of
linearized elasticity are
∇ · τ = 0 in Ω, τ n = f on ∂Ωf , τ = τ T ,
where τ stands for the symmetric 2nd order tensor field (elastic stress field), f
denotes the vector field on ∂Ωf (loading of the structure) and n is a vector field
normal to ∂Ω. Stress energy accumulated in e-th phase (e = 1, 2) is given by
8We (τ ) = Ke (τ I + τ II )2 + Le (τ I − τ II )2 ,
where τ I , τ II stand for the eigenvalues of τ . In void it is assumed that
(
0
if τ = 0,
W3 (τ ) =
+∞ otherwise.
1
Compliance + weight of a structure (for fixed division of Ω) is calculated according to
Z J(χ1 , χ2 ) = min
Φ1 (τ )χ1 + Φ2 (τ )χ2 dx τ ∈ Σ ,
Ω
with Φ1 (τ ) = 2W1 (τ ) + γ1 , Φ2 (τ ) = 2W2 (τ ) + γ2 , where γ1 , γ2 , γ2 < γ1 , are
the coefficients associated to the areas occupied by phases 1 and 2.
The problem of optimal phase distribution in Ω
R
1
(P1 ) : J0 = inf J(χ1 , χ2 ) χi : |Ω|
χ
dx
≤
V
,
V
+
V
≤
1
i
i
1
2
Ω
is equivalent to a non-convex variational problem
(P2 ) : J0 = inf F (τ ) τ ∈ Σ , F (τ ) = min Φ1 (τ ), Φ2 (τ ), 0 .
and γ1 , γ2 in (P2 ) denote the Lagrange multipliers for restrictions in (P1 ).
Results
Introduce
α=
γ2
,
γ1
α ∈ (0, 1).
Results discussed in the talk shows that for certain values of α the macroscopic
distribution of phases cannot be uniquely defined. It turns out that optimal
volume fractions of phases are not unique in two following cases:
Case A:
For
K1 + L1
α = αA , αA =
K2 + L1
and
det τ > 0.
Case B:
For
α = αB ,
αB =
K1 + L1
K1 + L2
and
det τ < 0.
2
1/17
Optimal structures
made of two materials and void
Grzegorz DZIERŻANOWSKI
Faculty of Civil Engineering, Warsaw University of Technology (Poland)
Nathan BRIGGS, Andrej CHERKAEV
Department of Mathematics, University of Utah
Continuum Models Discrete Systems – 13
Salt Lake City, UT, USA, 21-25 July, 2014
Summary
2/17
◮
Equations of 2D three-phase elasticity
◮
Variational problem of optimal phase distribution
◮
Discussion on the uniqueness of solution
◮
Example of optimal design
Equations of 2D three-phase elasticity
3/17
◮ Design domain
Ω = Ω1 ∪ Ω2 ∪ Ω3 – 2D domain,
∂Ω – boundary of Ω.
◮ Statically admissible elastic stresses (tensor fields)
Σ = τ : div τ = 0 in Ω, τ n = f on ∂Ωf ⊂ ∂Ω ,
τ – symmetric 2nd order tensor field,
f – vector field on ∂Ωf (loading of the structure),
n – vector field normal to ∂Ω.
◮ Kinematically admissible displacements (vector fields)
V = u : u = 0 on ∂Ωu ⊂ ∂Ω
◮ Small strains in the elastic body (tensor fields)
ε=
1
∇u + (∇u)T
2
Equations of 2D three-phase elasticity
4/17
◮ Constitutive equation
ε = E : τ,
E – 4th order tensor field: Eijkl = Eklij = Ejikl , i, j, k, l = 1, 2,


E1111 = E2222 = K+L


4 ,


K−L
E
1122 = E2211 =
4 ,
isotropy
L
E1212 = ,


2


E
=
E
1211
1222 = 0.
◮ Mechanical properties of phases
K(x) =



K1
K
2


+∞
if x ∈ Ω1 ,
if x ∈ Ω2 ,
if x ∈ Ω3 ,



 L1
if x ∈ Ω1 ,
L(x) = L2
if x ∈ Ω2 ,


+∞ if x ∈ Ω ,
3
such that K1 < K2 and L1 < L2 .
Equations of 2D three-phase elasticity
5/17
◮ Stress energy We accumulated in e-th phase (e = 1, 2)
8We (τ ) = Ke (τ I + τ II )2 + Le (τ I − τ II )2 ,
τ I , τ II – eigenvalues of τ .
◮ In void it is assumed that
W3 (τ ) =
(
0
if τ = 0,
+∞ otherwise.
◮ Compliance + weight of a structure (for fixed division of Ω)
J(χ1 , χ2 ) = min
Z Ω
Φ1 (τ )χ1 + Φ2 (τ )χ2
dx τ ∈ Σ ,
Φ1 (τ ) = 2W1 (τ ) + γ1 , Φ2 (τ ) = 2W2 (τ ) + γ2 ,
γ1 , γ2 – arbitrary coefficients associated to areas
occupied by phases 1 and 2.
Variational problem of optimal phase distribution
6/17
◮ The problem of optimal phase distribution in Ω
(P1 ) :
J0 = inf
J(χ1 , χ2 ) χi :
1 R
|Ω| Ω
χi dx ¬ Vi , V1 + V2 ¬ 1
is equivalent to a non-convex variational problem
(P2 ) :
J0 = inf
F (τ ) τ ∈ Σ ,
F (τ ) = min Φ1 (τ ), Φ2 (τ ), 0 .
◮ γ1 , γ2 in (P2 ) – Lagrange multipliers for restrictions in (P1 ).
Variational problem of optimal phase distribution
7/17
◮ Solution exists if the problem is “relaxed” by allowing
mixtures of pure phases (limits of classical designs).
◮ “Relaxation” means quasiconvexification of F (τ ) in (P2 )
(P3 ) :
QJ0 = min QF (τ ) τ ∈ Σ ,
QF (τ ) = min 2 W ∗ (τ, m1 , m2 , m3 ) + γ1 m1 + γ2 m2
0 ¬ me ¬ 1, e = 1, 2, 3, m1 + m2 + m3 = 1 .
τ – average stress, m1 , m2 , m3 – volume fractions,
W ∗ – minimal stress energy, i.e. energy stored in a mixture
composed of optimally stressed phases.
◮ Sufficient optimality conditions of microstresses are
expressed in terms of τ I and τ II .1
1
A. Cherkaev, GD (2013), Int. J. Solids Struct., 50, 4145-4160.
Variational problem of optimal phase distribution
8/17
◮ Composite region
R(α) = τ : QF (τ ) < F (τ )
for given
α=
◮ QF (τ ) is supported by Φ1 , Φ2 and W3 (0) = 0.
◮ QF (τ ) ¬ F (τ ),
QJ0 = J0 .
γ2
.
γ1
Discussion on the uniqueness of solution
9/17
◮ In a locally optimal microstructure, volume fractions
of phases: m1 , m2 , m3 , m1 + m2 + m3 = 1, depend
on the quotient α and average stress τ ,
◮ Optimal stress fields in phases: τ (1) , τ (2) , τ (3) = 0,
have to be statically admissible. Thus it is necessary
that they are in rank-1 connection on phase interfaces:
det(τ (i) − τ (j) ) = 0,
i, j = 1, 2, 3.
◮ Optimal stress fields in phases and the average stress
are univalent :
det τ ­ 0 ⇒ det τ (i) ­ 0 (works also for “¬”).
◮ QUESTION:
Are the volume fractions and microstresses uniquely
determined for all possible choices of α and τ ?
Discussion on the uniqueness of solution. Large values of α
10/17
◮ Assume that α is greater than a certain threshold, α > αA ,
which results in m2 = 0.
◮ Quasiconvex envelope is supported by Φ1 and W3 (0) = 0.
Thus QF = QF13 , such that
For τ ∈ R(α):
QF13 (τ ) =
p
(K1 + L1 )γ1 |τ I | + |τ II | −
For τ ∈
/ R(α):

K1 |τ I ||τ II |
L |τ ||τ |
1 I
II
if τ I τ II < 0,
if τ I τ II > 0.
QF13 (τ ) = Φ1 (τ ).
n
o
R(α) = τ : |τ I | + |τ II | ¬ ξ0 ,
ξ0 = 2
r
γ1
.
K1 + L1
Discussion on the uniqueness of solution: Large values of α
11/17
◮ Optimal volume fractions of phases are uniquely defined
m1 =
|τ I | + |τ II |
,
ξ0
m2 = 0,
m3 = 1 − m1 .
Discussion on the uniqueness of solution: Special values of α
12/17
◮ Let α = αA
αA =
K1 + L1
.
K2 + L1
Then γ2 = αA γ1 and energy well Φ2 touches QF13 , i.e.
Φ2 (τ ) = QF13 (τ ),
at two points
τ1 = (ξ, ξ),
τ2 = (−ξ, −ξ),
1
ξ = αA ξ0 .
2
◮ Both tensors above correspond to pure spherical stress.
Discussion on the uniqueness of solution: Special values of α
13/17
Discussion on the uniqueness of solution: Special values of α
14/17
◮ Optimal volume fractions of phases are not uniquely defined
for α = αA
mmax
2
=

ξ0 −|τ I |−|τ II |


 ξ0 −2 ξ





ξ −|τ I |−|τ II |

 0ξ0 −ξ−|τ
II |

ξ0 −|τ I |−|τ II |



ξ0 −|τ I |−ξ





 |τ I | |τ II |
if τ ∈ R1 ,
|τ II |
ξ
if τ ∈ R2.1 ,
|τ I |
ξ
if τ ∈ R2.2 ,
ξ2
if τ ∈ R3 ,
,
0 ¬ m2 ¬ mmax
2
m1 + m3 = 1 − m2 .
Discussion on the uniqueness of solution: Special values of α
15/17
◮ Let α = αB
αB =
K1 + L1
.
K1 + L2
Then γ2 = αB γ1 and energy well Φ2 touches QF13 , i.e.
Φ2 (τ ) = QF13 (τ ),
at two points
τ3 = (−η, η),
τ4 = (η, −η),
1
η = αB ξ0 .
2
◮ Both tensors above correspond to pure deviators.
◮ Optimal volume fractions of phases are not uniquely defined
for α = αB .
Discussion on the uniqueness of solution: Special values of α
16/17
17/17
Example of optimal design
2
Thank you !
2
N. Briggs, A. Cherkaev, GD (2014), accepted for publication in Struct.
Multidiscip. Optimiz., see also http://arxiv.org/abs/1401.7652v2