
S. Leelavanichkul, A. Cherkaev
An impact protective structure with bistable links. International Journal of Engineering Science, 2010 (submitted)

S. Leelavanichkul, A. Cherkaev
Elastic cylinder with helicoidal orthotropy: Theory and applications. International Journal of Engineering Science, 2009

A. Cherkaev, Y. Zhang
Optimal structures of multiphase elastic composites 8th World Congress on Structural and Multidisciplinary Optimization
June 1  5, 2009, Lisbon, Portugal

A. Cherkaev, E. Cherkaev, S. Leelavanichkul
Principles of optimization of structures against an impact, 8th World Congress on Structural and Multidisciplinary Optimization
June 1  5, 2009, Lisbon, Portugal

Andrej Cherkaev
Bounds for effective properties of multimaterial twodimensional conducting composites
and fields in optimal composites  pdf Mechanics of Materials 41 (2009) 411433.
 A.Cherkaev, A, Kouznetzov, A.Panchenko
STILL STATES OF BISTABLE LATTICES, COMPATIBILITY, AND
PHASE TRANSITION 2008
 E.Cherkaev, A.Cherkaev,
Minimax optimization problem of structural design
Computers and Structures, 2007

A. Cherkaev *, V. Vinogradov, S. Leelavanichkul
The waves of damage in elasticāplastic lattices with waiting
links: Design and simulation. Mechanics of Materials, 2006

Andrej Cherkaev and Seubpong Leelavanichkul
Structural optimization and spiralling of the grain in a pine
trunk (2006)  pdf Submitted to "Advanced Composites"
From the introduction:
We observe that some trees have grains spiralling around their
trunk. An example is Ponderosa pine that grows in rocky windy
terrains in South West of the United State (Utah, Anizona,
Nevada). One wonders what an evolution significance of such design
is. The paper concerns with morphology of tree's trunk from
structural optimization viewpoint. Specifically, we investigate
the reasons behind spiral grows of the Ponderosa pine trunk in
southern Utah. The question is why they twist. The
considered problem is an example of the inverse optimization
problem that arrives in evolution biology.
Studying morphology like bones or trees'trunks that are critical
for the survival of the species, we may postulate that they are
optimally adapted to the environment. Trees' trunks should stay
unbroken and be able to sustain extreme wind loads applied from
all directions. If a natural design becomes more complex, there
must be a good reason for this. We treat the evolutionary
development of the species as the minimizing sequence of an
optimization problem with unknown objective.

N. Albin, A. Cherkaev and V. Nesi
A class of optimal twodimensional multimaterial conducting laminates (2005) pdf
In: IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials
Status and Perspectives,
SERIES = {Solid Mechanics and Its Applications}, AMS 2006.
VOLUME = {137},

N. Albin, A. Cherkaev
Optimality conditions on fields in microstructures and controllable differential schemes (2006) pdf
Postscript version
In: Inverse Problems, MultiScale Analysis, and Effective Medium Theory.
Habib Ammari and Hyeonbae Kang  editors. Series: Contemporary Mathematics number 408, pp 137150.

Cherkaev, A., Vinogradov, V., Leelavanichkul, S. 2006. The waves of damage in elasticplastic lattices with waiting links: design and simulation. Mechanics of Materials 38, 748756
Abstract
We consider protective structures with elasticplastic links of a special
morphology. The links are bistable: they are designed to arrest the
development of localized damage (a neck) in an element, initiating
the damage instead in another sequential element. A wave of
``partial damage" propagates through the chain, as all the links
develop necks but do not fail. When subjected to an impact, such a
structure absorbs several times more energy than a conventional
structure, which makes it more protective against collision.
A threedimensional finite element analysis is used to compute the
forceelongation relation for a bistable element. Another numerical
procedure analyzes the dynamics of a chain or a lattice undergoing
a collision with a heavy projectile. The procedure describes the
nonlinear waves, determines the places of failure and the critical
loading conditions. A bistable chain is compared with a chain of
conventional links, of the same length, mass, and material. The
bistable chain is capable to withstand a collision with a mass of
kinetic energy, several times greater. Even when the bistable chain
breaks it is able to more effectively reduce the speed of the
projectile.

A.Cherkaev and L.Zhornitskaya.
Protective Structures with waiting links and
their damage elovulion. Multibody System Dynamics Journal. Volume 13, Number 4 May, 2005

Leonid Slepyan, Andrej Cherkaev, Elena Cherkaev
Transition waves in bistable structures. I. Delocalization of damage
J. Mech. Phys. Solids, Vol 53/2 pp 383405, 2005.
We consider chains of dimensionless masses connected by breakable bistable
links.
A nonmonotonic piecewise linear constitutive relation for each link
consists of two stable branches separated by a
gap of zero resistance. Mechanically, this model can be envisioned as a
"twinelement"
structure which consists of two links (rods or strands) of different lengths
joined by
the ends. The longer link does not resist to the loading until the shorter
link
breaks. We call this construction the {\em waiting link structure}. We show
that the
chain of such strongly nonlinear elements has an increased inthelarge
stability
under extension in comparison with a conventional chain, and can absorb a large
amount
of energy. This is achieved by two reasons. One is an increase of dissipation
in the form of highfrequency waves transferring the mechanical energy to heat; this
is
manifestation of the inner instabilities of the bonds. The other is
delocalization of
the damage of the chain. The increased stability is a consequence of the
distribution
of a partial damage over a large volume of the body instead of its localization, as
in the case of a single neck formation in a conventional chain. We optimize
parameters
of the structure in order to improve its resistance to a slow loading and
show that
it can be increased significantly by delocalizing a damage process. In
particular, we
show that the dissipation is a function of the gap between the stable branches
and find
an optimal gap corresponding to maximum energy consumption under quasistatic
extension. The results of numerical simulations of the dynamic behavior of
bistable
chains show that these chains can withstand
without breaking the force which is several times larger than the force
sustained by a conventional chain.
The formulation and results are also related to the modelling
of compressive destruction of a porous material or a frame construction which
can be described by a twobranched diagram with a
large gap between the branches.
We also consider an extension of the model to multilink chain that could
imitate
plastic behavior of material.

Leonid Slepyan, Andrej Cherkaev, Elena Cherkaev.
Transition waves in bistable structures.
II. Analytical solution: Wave speed and energy dissipation.
J. Mech. Phys. Solids, Vol 53/2 pp 407436, 2005.
We consider dynamics of chains of rigid masses connected by links described
by
irreversible, piecewise linear constitutive relation: the forceelongation
diagram
consists of two stable branches with a jump discontinuity at the transition
point. The
transition from one stable state to the other propagates along the chain
and excites a
complex system of waves. In the first part of the paper
Transition waves in bistable structures. I. Delocalization of damage, the branches
could be separated by a gap where the tensile force is zero,
the transition wave was treated as a wave of partial damage.
Here we assume that there is no zeroforce
gap between the branches. This allows us to obtain steadystate
analytical solutions for a general piecewise linear trimeric diagram
with parallel and nonparallel branches
and an arbitrary jump at the transition. We derive necessary
conditions for the existence of the transition waves and compute
the speed of the wave. We also determine the energy of dissipation
which can be significantly increased in a structure characterized
by a nonlinear discontinuous
constitutive relation. The considered chain model reveals
some phenomena typical for waves of failure or crushing in
constructions and materials under collision, waves in a
structure specially designed as a dynamic energy absorber
and waves of phase transitions in artificial and natural
passive and active systems.

Andrej Cherkaev
Approaches to nonconvex
variational problems of mechanics In: Nonlinear homogenization and its
applications to composites, polycrystals and smart materials.
P.Ponte Castaneda, J.J.Telega and B. Gamblin eds., Kluwer 2004. pp. 65106.
(NATO Science Series, Mathematics, Physics and Chemistry v. 170)
Abstract:
The paper reviews recent development of mathematical methods for
nonconvex variational problems of mechanics, particularly,
problems of optimal layouts of material in a heterogeneous medium.
We discuss variational formulations of these problems, properties of
their solutions and several approaches to address them:
minimizing sequences and technique of laminates, laminate closure,
and differential scheme; necessary conditions by structural
variations and minimal extension technique; the lower bounds and
bounds for the variety of effective tensors of properties.
Particularly, the bound for the tensor of
thermal expansion coefficients is found. Special attention is paid
to the use of duality for reformulation of minimax problems as
minimal ones.
 Seubpong Leelavanishkul and Andrej Cherkaev
Why grains in the tree's trunk spiral: mechanical perspective
Structural and Multidisciplinary
Optimization, 28, 23, 2004, pp. 127  135.
Abstract:
We discuss whether the spiraling of the grain has a biological significance.
We perform the analysis of the stresses in the trunk and argue why the trees are "designed" the way they are.
 Elena Cherkaev Andrej Cherkaev
Principal compliance and robust optimal design
Journal of Elasticity, 2003, 72, 7198. (also in The Rational Spirit in Modern Continuum Mechanics, Essays and
Papers Dedicated to the Memory of Clifford Ambrose Truesdell
III, 2003)
Abstract:
The paper addresses a problem of robust optimal
design of elastic structures
when the loading is unknown, and only an integral constraint for
the loading is given. We suggest to minimize the principal
compliance of the domain equal to the maximum of the stored
energy over all admissible loadings.
The principal compliance is the maximal compliance under
the extreme, worst possible loading. The robust optimal
design is formulated as a minmax problem for the energy stored
in the structure. The maximum of the energy is chosen over
the constrained class of loadings, while the
minimum is taken over the design parameters.
It is shown that the problem for
the extreme loading can be reduced to an elasticity problem
with mixed nonlinear boundary condition; the last problem
may have multiple solutions. The optimization with respect to
the designed structure
takes into account the possible multiplicity of extreme
loadings and divides resources (reinforced material) to
equally resist all of them. Continuous change of the loading
constraint causes bifurcation of the solution of the optimization
problem. It is shown that an invariance of the constraints under
a symmetry transformation leads to a symmetry of the optimal
design. Examples of optimal design are investigated; symmetries
and bifurcations of the solutions are revealed.

Leonid
Slepyan, Andrej Cherkaev,
Elena Cherkaev,
Vladimir Vinogradov
Transition waves in controllable cellular structures with high structural
resistance (submitted)
Abstract:
The paper suggests an approach for optimization of morphology of
mechanical structures subjected to an
impact.
We consider chains or lattices with breakable bistable
links. A nonmonotonic constitutive relation for
each link consists of two stable branches separated by an unstable branch. M
echanically, this model can be envisioned as a
"twinelement"
structure which consists of two elasticbrittle or elasticplastic
links (rods or strands) of different lengths
joined by the ends. The longer link
does not resist to the loading until the shorter rod breaks or
develops a neck. When a chain or lattice of these elements is elongated
they excite waves of damage that carry the energy away.
We analytically describe and simulate transition waves
of damage in bistable
structures. We show that strength against an impact
of the chain or lattice with nonmonotonic links increases several
times; such structure can absorb much more energy before breakage
than a conventional structure.

Andrej Cherkaev,
Vladimir Vinogradov
Bounds for expansion coefficients of composites (submitted)
Abstract:
A method is suggested to bound the anisotropic
effective stiffness and extension tensors
of a multiphase composite made of expandable materials. The bounds are
valid for composites of any microstructure. It is shown that the expansion
coefficients vary an ellipsoid which parameters depend on properties of
the phases, their fractions, and the effective stiffness of a
composite. The obtained tensorial inequalities generalize bounds by
Schapery, Rozen and Hashin, and Gibiansky and
Torquato. Particularly, the bounds for the mixtures with voids are obtained.

Andrej Cherkaev, Liya Zhornitskaya
Dynamics of damage in twodimensional structures with waiting links
In: Asymptotics, Singularities and Homogenisation in Problems of Mechanics, A.B.Movchan editor, Kluwer 2004. pp 273  284,
Abstract:
The paper deals with simulation of damage spread in special structures
with ``waiting links." These structures are stable against dynamic impacts
due to their morphology. They are able to transform ``partial damage"
through a large region thereby dissipating the energy of the impact.
We simulate various structures with waiting links and compare their
characteristics with conventional designs.

Andrej Cherkaev, Ismail Kucuk
Detecting stress fields in an Optimal Structure I:
Twodimensional Case and Analyzer Int.J Struct. Opt. January 2004 pp 115
Abstract:
In this paper, we investigate the stress field in optimal elastic structures
by the necessary conditions of optimality, studying twophase elastic composites
in two dimensions. We also suggest a visualization tool that shows how
close is a design to an optimal one.

Andrej Cherkaev,
Ismail Kucuk
Detecting stress fields in an Optimal Structure II:
Threedimensional Case, Int. J Struct. Opt. January 2004 pp 1524.
Abstract:
The paper extends the results of the previous one to the threedimensional
case.

Alexander Balk, Andrej Cherkaev, and Leonid
Slepyan
Dynamics of chains with nonmonotone stressstrain relations.
J. of Phys. Mech. Solids, 49 (2001) pp. 131148
Abstract:
We discuss the dynamical processes in the materials with nonmonotonic
constitutive relation. We introduce a model of a chain of masses joined
by springs with a nonmonotone strainstress relation. Numerical experiments
are conducted to find dynamics of that chain under slow external excitation.
We find that the dynamics leads either to a vibrating steady state ({\it
twinkling phase}) with radiation of the energy, or (if a dissipation is
introduced) to a hysteresis, rather than to an unique stressstrain dependence
that would correspond to the energy minimization.

Alexander Balk, Andrej Cherkaev, and Leonid
Slepyan
Nonlinear Waves and Waves of Phase Transition,
J. of Phys. Mech. Solids, J. Mech. Phys. Solids. 49 (2001) pp. 149172
Abstract:
We investigate the dynamics of one dimensional massspring chain with
nonmonotone dependence of the spring force vs.\ spring elongation. For
this strongly nonlinear system we find a family of exact solutions that
represent the nonlinear waves. We have found numerically that this system
displays a dynamical phase transition from the stationary phase (when all
masses are at rest) to the twinkling phase (when the masses oscillate
in a wave motion). This transition has two fronts, which propagate with
different speeds. We study this phase transition analytically and derive
the relations between its quantitative characteristics.
& Recent Papers

Equally stressed structures: Twodimensional perspective (with
Ismail
Kucuk)
in: Topology Optimization of Structures and Composite Contunia, G.I.N.Rozvany
and N.Olhoff eds.
Kluwer 2000, 251266.
Abstract:
The widely used engineering principle of optimality requires that a
material in a design should be not understressed. Here we extend this principle
to twodimensional designs made of two materials. We show that a norm of
the stress twnsor in a strong material is bounded from below, and a norm
in a weak but cheap material is bounded from above. We alse find an analog
of this principle for badly ordered materials, and we apply it to suboptimal
designs.
Dynamics of chains with nonmonotone stressstrain relations.
I. Model and Numerical experiments. (with Alexander Balk and
Leonid
Slepyan) J. of Phys. Mech. Solids 49 (2001) 131148
Abstract: We discuss the dynamical processes
in the materials with nonmonotonic constitutive relation. We introduce
a model of a chain of masses joined by springs with a nonmonotone strainstress
relation. Numerical experiments are conducted to find dynamics of that
chain under slow external excitation. We find that the dynamics leads either
to a vibrating steady state ({\it twinkling phase}) with radiation of the
energy, or (if a dissipation is introduced) to a hysteresis, rather than
to an unique stressstrain dependence that would correspond to the energy
minimization.
Dynamics of chains with nonmonotone stressstrain relations. II. Nonlinear
waves and waves of phase transition. (with Alexander Balk and Leonid
Slepyan) . J. of Phys. Mech. Solids 49 (2001) 149171

Abstract: We investigate the dynamics of one dimensional massspring
chain with nonmonotone dependence of the spring force vs.\ spring elongation.
For this strongly nonlinear system we find a family of exact solutions
that represent the nonlinear waves. We have found numerically that this
system displays a dynamical phase transition from the stationary phase
(when all masses are at rest) to the twinkling phase (when the masses
oscillate in a wave motion). This transition has two fronts, which propagate
with different speeds. We study this phase transition analytically and
derive the relations between its quantitative characteristics.

Structural optimization
and rationality of biological "designs". ( with Elena
Cherkaev)
Abstract
The amazing rationality of biological "constructions" excites the interest
to modelling them by using the mathematical tools developed in the theory
of structural optimization. The structural optimization solves a geometrical
problem of the ``best" displacements of different materials in a given
domain, under certain loadings. Of course, this approach simplifies the
real biological problem, because the questions of the mechanism of the
building and maintaining of structures are not addressed. The main problem
is to guess a functional for the optimization of a living organism. The
optimal designs are highly inhomogeneous; their microstructures may be
geometrically different, but possess the same effective properties. Therefore
the comparing of the various optimal geometries is not trivial. We show,
that the variety of optimal geometries shares the same characteristics
of the stress tensor in any optimal structure. Namely, special norm of
this tensor stay constant within each phase of the optimal mixture.
The paper also adresses the uncertainty of the ultimate load in biological
``structures". We discuss the corresponding minmax formulation of the
optimal design problem. The design problem is formulated as minimization
of the stored energy of the project under the most unfavorable loading.
The problem is reduced to minimization of Steklov eigenvalues.

Stable optimal
design of twodimensional elastic structures
(with Lars A. Krog and
Ismail
Kucuk)
Abstract
We study optimal layout of piecewise periodic structures of linearly elastic
materials. The effective tensors of these structures are constant within
prespecified regions, the optimality is understood as the minimum of complementary
energy. The suggested formulation leads to a construction that is stable
under variation of the loading and which does not degenerates into checkerboard
type structures. We derive necessary conditions of optimality of such layouts
and analyze them. Numerically, we find optimal structures for a number
of examples, which are analyzed.
Keywords
structural optimization, composites, optimality conditions,
optimal design.

Optimal design
for uncertain loading conditions (with
Elena
Cherkaev)
Abstract
Optimal design problems are usually formulated as problems of minimization
of the energy, stored in the design under a prescribed loading. Solutions
of these problems are unstable to perturbations of the loading. We suggest
a new minmax formulation of the optimal design problem which has a stable
solution. The loading is not prescribed, but only a set of admissible loadings
is given. The stable optimal design problem is formulated as minimization
of the stored energy of the project under the most unfavorable loading.
This most dangerous loading is one that maximizes the stored energy over
the class of admissible functions. The problem is reduced to minimization
of Steklov eigenvalues. Several stable solutions of various optimal design
problems are demonstrated; among them are the optimal structure of a material
stable to variations in uniaxial loading, the optimal specific stiffness
of an uncertainly loaded beam, and the stable design of an optimal wheel.

Variational approach
to the structural optimization
Abstract
The variational approach to structural optimization is considered. The
problem asks for the optimal layout of several conducting and elastic materials
throughout an inhomogeneous body. The simplest problem of the minimization
of the total energy is reviewed. A problem of minimization of a more general
functional is considered and a method is suggested to reduce it to the
simplest variational problem of the minimization of the sum of the energy
and the complimentary energy corresponding to two orthogonal external fields.
The characteristic properties of optimally designed media are discussed.
Particularly, it is shown that optimal microstructures possess the maximal
anisotropy. Examples of optimal structures and of optimal designs are discussed.

The cavity of
the optimal shape under the shear stress
(with Yury Grabovsky, Alexander
Movchan, Sergey Serkov)
Abstract
The problem of optimal shape of a single cavity in an infinite 2D elastic
domain is analyzed. An elastic plane is subjected to a uniform load at
infinity. The cavity of fixed area is said to be optimal if it provides
the minimal energy change between the homogeneous plane and the plane with
the cavity. We show that for the case of shear loading the contour of the
optimal cavity is not smooth but is shaped as a curved quadrilateral. The
shape is specified in terms of conformal mapping coefficients, and explicit
analytical representations for components of the dipole tensor associated
with the cavity are employed. We also find the exact values of angles at
the corners of the optimal contour. The applications include the problems
of optimal design for dilute composites.

Which elasticity
tensors are realizable? (with
Graeme
Milton)
Abstract
It is shown that any given positive definite fourth order tensor satisfying
the usual symmetries of elasticity tensors can be realized as the effective
elasticity tensor of a twophase composite comprised of a sufficiently
compliant isotropic phase and a sufficiently rigid isotropic phase configured
in an suitable microstructure. The building blocks for constructing this
composite are what we call extremal materials. These are composites of
the two phases which are extremely stiff to a set of arbitrary given stresses
and, at the same time, are extremely compliant to any orthogonal stress.
An appropriately chosen subset of the extremal materials are layered together
to form the composite with elasticity tensor matching the given tensor.

A complete characterization
of the possible bulk and shear moduli of planar polycrystal (with Marco
Avellaneda,
Leonid
Gibiansky,
Graeme
Milton, and Mark Rudelson.
Abstract
This paper investigates the range of possible elastic moduli of twodimensional
isotropic polycrystals. The polycrystals are comprised of grains obtained
from a single orthotropic material. The overall elastic properties are
described by an effective bulk modulus $K_0$ and an effective shear modulus
$\Gm_0$. The pair $(K_0,\Gm_0)$ is shown to be confined to a rectangle
in the $(K,\Gm)$ plane. Microstructures are identified which correspond
to every point within the rectangle, and in particular to the corner points.
Optimal bounds on the effective Poisson's ratio and Young's modulus follow
immediately. Under a certain constraint on the crystal moduli, the rectangle
degenerates to a line segment: the effective shear modulus of such a polycrystal
is microstructure independent. This extends earlier work of Hill and Lurie
and Cherkaev who established microstructure independence for polycrystals
constructed from square symmetric crystals.

Extremal structures
of multiphase heat conducting composites (with Leonid
Gibiansky)
Abstract
In this paper we construct microstructures of multiphase composites with
unusual properties: their heat conductivity in one direction is equal to
the harmonic or arithmetic mean of the phases' heat conductivities and
the conductivity in an orthogonal direction does not equal either arithmetic
or harmonic mean. Two dimensional threephase structures are studied, but
the results can be easily generalized for the threedimensional composites
with arbitrary number of phases.

Reducing of optimal
design problems to minimal variational problems.
Abstract
The problems of optimal design of conducting and elastic inhomogeneous
bodies is considered; these problems are described as variational problem
with differential restrictions which are the equations of the state of
the body. A method is suggested for reducing the problem of restricted
minimum to the simplest variational problem of the minimization of the
sum of the energy and the complimentary energy correspondent to two orthogonal
external fields; it is based on the Legandre transformation of slightly
reformulated initial problem. The characteristic properties of optimally
designed bodies are discussed. It is shown, in particular, that the optimal
microstructures possess the maximal anisotropy in the sense that they minimize
the difference between the energy densities caused by two orthogonal loadings.

Variational
principles for complex conductivity... (with Leonid
Gibiansky).
Abstract
We consider linear processes in media with dissipation arising in conductivity,
optics, viscoelasticity, etc. Timeperiodic fields in such media are described
by linear differential equations for complexvalued potentials. The properties
of the media are characterized by complex valued tensors, for example,
by complex conductivity or complex elasticity tensors. We suggest variational
formulations for such problems: we construct the functionals whose Euler
equations coincide with the original ones. We get four equivalent variational
principles: two minimax and two minimal ones. The functionals of the obtained
minimal variational principles are proportional to the energy dissipation
averaged over the period of oscillation. The last principles can be used
in the homogenization theory to obtain the bounds on the effective properties
of composite materials with complex valued properties tensors.

Waiting element
structures and stability under extention
(with
Leonid
Slepyan).
Abstract
In this paper the following questions are considered: What are the phenomena
which limit the total fracture energy of a structure under the extension
before is breaks? When is the limiting energy level more important than
the stress limit? What are possible ways to increase the required fracture
energy of a sample before it breaks? It is shown that the required features
of a material or of a construction can be achieved by using special structures
of ordinary elements. The possibilities are discussed for increasing the
fracture energy density in a sample, and increasing the total fracture
energy in a construction. The dynamic process of damaging discussed as
well.

Stability of optimal
structures of elasic composites.

Invariant
properties of the stress in plane elasticity and equivalence classes of
composites (with Konstantin
Lurie and Graeme
Milton).

Coupled bounds
for elastic moduli of a composite (with Leonid
Gibiansky).

The exact
coupled bounds for electrical and magnetic properties of a composite
(with Leonid Gibiansky).