Andrej Cherkaev, cherk@math.utah.edu
Optimal multimaterial composites: Bounds and structures
The paper suggests a method for funding exact bounds for the effective conductivity moduli of multimaterial
composites. These bounds expand and refine Hashin–Shtrikman and Nesi bounds. We prove that the fields in
the materials within optimal structures vary in restricted domains and take this into account, obtaining more
restricted bounds. The new bounds are solutions of a formulated relaxed finite-dimensional constrained optimization problem. For two-dimensional conducting three-material composites, bounds for effective conductivity are explicitly computed. These bounds are exact: Three-material isotropic microstructures of extremal
conductivity are found that realize the bounds for all values of parameters. The optimal structures are laminates
of a finite rank, their parameters vary with the volume fractions and they experience two topological transitions:
For large values of material of minimal conductivity, its subdomain percolates (is connected), for intermediate
values of that fraction, no material forms a connected domain, and for small values of that fraction, the domain
of intermediate material percolates. Another type of isotropic optimal three–material structures is the “wheel
assemblages” that replaces the Hashin–Shtrikman coated circles.
Optimal multimaterial composites: Bounds and
structures
Andrej Cherkaev
Department of Mathematics, University of Utah
The talk is based on a joint project with Grzegorz
Dzierżanowski (Warsaw Polytechnic University) and Nathan
Briggs (University of Utah).
An optimal design problem
• Given three materials
– an expensive and strong one,
– a cheap and weak one,
– void,
find their layout in a fixed loaded from the boundary domain, that minimizes
the stress energy, i.e. maximizes the stiffness with a fixed total cost (weight).
If the layout is periodic, the problem is to find the best (stiffest) composite.
The two-material problem has been solved decades ago
(Hashin & Shtrikman 1963, Lurie & Ch 1982,
Gibiansky & Ch 1986 (elasticity)
by variational “translation” method”
(Kohn and Strang, 1983, Lurie & Ch, 1986, Milton 1982, 2001) of
bounds.
Multimaterial design
The standard variational method, however, does not give the correct result
for multimaterial problem.
Example (Hashin-Shtrikman bound, 2d conductivity)
n
1
mi

,
kL  k1 i1 ki  k1

lim kL  f (k1)
m1  0
Optimal bounds for effective properties and structures of multimaterial
composites were studied in the last 30 years, and some partial results
were obtained in Milton 1983, Lurie & Ch., 1986, Milton & Kohn 1988,
Nesi 1995, Albin, Ch, Nesi 2004, Liping Lui 2009. et al.
This talk is based on the papers Ch, and Dzergzanovsky, 2013, Briggs, Ch, and
Dzergzanovsky, 2013.
Earlier development of the approach: Ch, 09, Ch. & Zhang, 2011, Ch. 2012
Design of minimal compliance
• Given two materials – the expensive strong and the cheap weak
ones -- and the void, find their layout in a fixed domain loaded from
the boundary that minimizes the stress energy, i.e. maximizes the
stiffness.
• The problem is reduced to a variational with a nonconvex piecewise quadratic Lagrangian:
inf

 F(,g)dx 


f (uT  n) ,  :    T ,   0,
F  min(W i ( )  gi ), W i  kiTr( 2 ) /2
k   S u,   RT  R,  W 22 (O).
k1  k 2  k3  , g1  1, g3  0, g2   ,
Quasiconvex envelope –
local problem of the best structure.
1
QF( ) 
inf
| | 
 F(   )dx,   dx,

   T ,    0.

 is   periodic.
Due to non-convexity of F, the solution - optimal stress in
a structure – oscillates in an infinite fine scale,
alternating the values that are called supporting
points or supports of the Young’s measure.
1

inf
| | 
 (   )dx,
    supporting points.

Compatibility: The supporting points of the envelope must be
compatible: Any two neighboring stresses satisfy
det(1  2 )  0.
Fields in optimal structures and bounded and
ordered
Bounds problem  multiwell variational problem
inf
u U
 F(u, )dx,
F  minW i (u,ki   i 

i
Materials (wells) is an optimal structure are identified by the range fields (gradients).
Fields in materials are ordered and bounded

Material 1
Material 2
Material 3
Alessandrini-Nesi inequality
det e  0 s12  s22  d12  d22
(S 2  D2 ) x, if det E 0  0
An open question:
The range of fields that
support the quasiconvex
envelope
Ch, 2009: (for optimal isotropic
composites)
det(e(x)  e(x')  0
if x  1, x' 3,det E 0  0
Comment on types of optimal partitions
Fields in optimal structures are bounded.
Therefore three or more materials cannot meet
in isolated points
Optimal structures either do not have meeting points (coated spheres)
Or the meeting points are dense (laminates)
Expected structure of optimal fields
• 1. Piece-wise constancy of the fields -- all structures are compared
and “the best” is chosen. The stress depends on only several
parameters and does not depend on a structural point. It is either
piece-wise constant, or it varies within a manifold so that this
does not change the effective stiffness.
• 2. This constancy forces a structure to become fractals with
infinitely thin ligaments, etc. We a priori expect that the optimal
solution does not exist but the minimizing sequences tend to a
distribution and the fields become differentially constrained
Young’s measures.
• The boundary conditions are satisfied:
det  A   B   0, det SA A  SB B   0.
Technique for calculating/estimating the
quasiconvex envelope
Upper bound:
Necessary conditions on fields: Structural (Weierstrass
type) variations – K.Lurie
Direct methods: Building minimizing sequences.
Lower bound: Sufficient conditions:
Translation method + inequalities
To find the lower bound for the quasiconvex envelope, we
-- replace differential constrains on the minimizers with several weaker
algebraic conditions and
-- constract the convex envelope, or solve
a finite-dimensional constrained optimization problem
Sufficient conditions:
Translation method + Point-wise constraints
• Constraints
(A)
det   det   :    T ,   0, is 1- periodic
W i ( )  t det  =  T (ki I  tT)
(B)   ,    C() : det( - ) = 0 - compatibility
(C) If k1  k2 ..., then det( -  0 )  0    1 - optimality cond.
The algebraic expression for the lower bound depends on
-- which constraints (B) and (C) are active
-- singularity of the matrix
(ki I  tT)
for some i
To constraint (B)
Necessary condition for for all supporting points
(B)  ,   C(/ ) : det( - ) = 0
det        0 on the lines of discontinuity
at a multiscale boundary,  k  c i i , c i  1, c i  0,  i i
i:ik
i:ik
To constraint (c)
Conditions on supports in relation to the average field
(C) If k1  k2 ..., then det( -  0 )  0   1 - optimality cond.
• Example: if a strong material envelopes the structure of forms an
“exterior layer”, the inequality is obvious.
Admissible
region of stress
in mat. 1
Optimal structures
Knowing the positions of optimal fields (supports of
Young measure), we build laminate structure that
possess these fields in the layers.
In the construction, we satisfy the boundary conditions
between the layers.

Example. Two materials
1
Zones optimality of the pure materials and
zone of non-quasiconvexity
Observe, that both inequalities (B) and (C)
are “naturally” satisfied in optimal structures
 2
a)-c): the trace of stress tensor in P1
is constant everywhere
Tr  (x)  21 x 1
c) The stress in P2 is isotropic
d) One component of stress in P1 and P2

is equal to the external stress.

(x)  2I x 2
det( (x)   0 )  0
x  .
Optimal geometries
are not unique
Optimal three-material structures
• Below, we present the types of optimal structures, in dependence of their
volume fraction and anisotropy of the external field.
• Each composites minimizes the stored stress energy in a given average stress
field.
• The optimality of the structures is proven by sufficient conditions (bounds).
• The results for general elasticity case (two constants) were obtained in 2013.
• Optimal three-material structures depend to the anisotropy of the external
field and to the volume fractions.
• The structure is independent of the magnitude of the external field.
• In each point of an optimal structure, the stress satisfies sufficient
conditions of optimality.
Color code:
orange – strong material
blue -- weak material,
white -- void.
Moderate m1, isotropic
Equivalent optimal structures
Periodic assemblage
Stress in P1 is always in rank-one connection with the stress in P3
Stress in P2 is isotropic
det( (x))  0 x 1, (x)  I x 2
Moderate m1, anisotropic stress
det( (x))  0 x 1,
det( (x)   0 )  0 x 2



Large m1, close to isotropy
(Hashin-Shtrikman or Translation bounds)
Isotropic stress or pure shear
Vertical stress is bigger
The structures are not unique.
However, in all optimal structures
-- the trace of the stress is constant in P1
-- the stress is isotropic in P2.
Tr (x)  21 x 1,  (x)  2I x 2
W1()  t1 det  = T (k1I  t1T)
Large m1, anisotropic
The stress in P1 either is in rank-one connection
with P3, or with rank-one connection with external field
det( (x)) det((x)   0 )  0 x 1
A periodic assemblage
A different type of structure
and a different expression
for the optimal energy
Small m1, isotropic stress
det( (x))  0 x 1, Tr((x))  22 x 1
Small m1, anisotropy varies
Both types of structure correspond to the same bound
det( (x))  0 x  1,
Tr( (x))  22 x  1
W2 ()  t2 det  = T (k2I  t2T)
Structures at a glance
Close to isotropy
large m1
moderate m1
small m1
Anisotropic
Optimal structures are not unique:
Assemblage elements, “wheels”, vary depending on
the fractions of materials. Ch. 2011
Assemblage elements: the topology depends on m1
m1>m11
Hashin Shtrikman
bound. k1 (white)
is connected
m12<m1<m11
No connected phases
m1<m12
k2 (gray)
is connected
Distribution of the structures
• An optimal distribution of the materials in an optimal structure depends
on the costs of the materials. We assume that cost g1 of P1 is equal one,
• g1=1,
• cost of P3 (void) is equal zero, g3=0,
• and the cost g2 of P2 varies.
• The problem is to minimize the stored energy plus the total cost by
optimization of the structures’ volume fractions and anisotropy level.
min W
str
(,m1,m2 )  m1  gm2 
m1,m2
• It the stress is very high, only stiff material P1 is optimal (m1=1).
Multimaterial happen when the stress density is low enough and it is
beneficially
to mix the materials.

Distribution of optimal structures (general case)
Notice that the optimal structures
for more intensive fields correspond
to P1-P2 composites.
After the first transition, P1 always enters the 1-2-3 structures (gray areas)
via 1-3 laminate.
Taxi cab geometry of the quasiconvex envelope
These rules can be heuristically
generalized to more materials,
to 3d problem, to polycrystalls,
etc.
We would obtain the upper bound
(perhaps exact) of the quasiconvex
envelope
Supporting sets: boundaries of domains of
optimality of the pure materials
2
.
1
1
2
3
Obtained by structural variations
and verified by sufficient conditions
1
2
Region of optimality of the second well
Dependence on the cost g.
The region is bounded by
two symmetric ellipses and two
dependent on g symmetric
hyperbolas with the independent of g
asymptotes.
Asymptotic: “blue” region of optimality of pure second
material shrinks to a point:
Hashin-Shtrikman case.
The graph shows optimal structures in
dependence of eigenvalues of the
external stress field.
Large stresses correspond to the
optimality of pure stiff material. Gray
fields show three–material structures.
First transition
• The effective stiffness of an optimal 1-3 structure in an isotropic field is
equal to the stiffness of P2, the costs are equal too.
W 2 (I)  g2  QW13 (I)  m1, g2 
2k1
k1  k2
=
=
=
Optimal structures at this point
are not unique.
Well 2 touches the
1-3-quasiconvex envelope
Volume fraction of m2
immediately after the transition
The amount of P2 depends on the relative area of the 13 optimal structure that can be occupied with the
isotropic field of a given density. This area can be filled
by P2 without changing the total stiffness and cost.
m2 decreases when the external field is too large, too small, or too anisotropic. It is
equal one in the point where the second well touches the quasiconvex envelope
The second asymptotic: fields and geometries
These structures correspond to a range of
total volume fractions.
Second transition
• After the first transition, P1 always enters the1-2-3 structure through a 13 laminate.
• Second transition occurs when the effective stiffness and cost of a 1-3laminate structure in a unidirectional stress becomes equal to the stiffness
and cost of 1-3-laminate.
=
Optimal structures are not unique.
In 1-2-3 laminate, one can replace any portion of P2 with an equivalent 1-3
laminate
Low cost of P2: fields and geometries
After the second transition,
well 2 separates two
components of 1-3quasiconvex envelope.
The optimality domain of the
pure P2 cuts the domain of the
optimal composites into two.
After the transition, the three materials do not meet in an optimal structure.
For smaller fields, P2 is an enveloping material (P3 – inclusions) ,
for larger fields P2 forms inclusions, P1 –an envelope.

Quasiconvex envelope. First transition.
Quasiconvex envelope,
first quadrant
Nonlinear part of the envelope
1 4k1 | 1 2 | 2 2k1 (| 1 |  |  2 |), 2k1(| 1 |  |  2 |) 2 1,
QW  
2 
1 2k1 (12   2 2 )
otherwise
First quadrant
T
2k1 (1   2 )  1, i  0
1 4k1R R  2 2k1 I,


2 
2k1
2k1 (1   2 )  1, i  0
Tr   4k1Tr   4 2k1
Euler - Lagrange equation is identically zero.
Cantilever, loaded at the middle (computations
by Grzegorz Dzierżanowski and Nathan Briggs)
We expect novel unexpected suboptimal designs when penalties for mixing is introduced.
Conclusion
Quotient of design problems:
Multimaterial design
-----------------------------Two-material design
Color TV
= ------------------------Black-and-white TV
Thank you!