Adaptive Composite Materials: Bionics Principles, Abnormal Elasticity, Moving Interfaces

Abstract

The systematic analysis of the problem of developing adaptive composites has enabled us to trace evolution of structural organization of artificial materials, to clarify the mechanisms of adaptation to the external action, and to disclose, to a certain degree, the effect of structure on formation of the optimum back reaction.
In above considered examples of composites, description of adaptive structures is formulated as a problem on localizing moving interfaces. The study of synergetic phenomena in the nonliving nature and analogous processes in biological objects will, in our opinion, provide a possibility to find structural-and-functional prototypes of adaptive composites. The proposed analytical and numerical models predict self-reinforcing in composites and joints made of auxetics under uniaxial and contact compression. The role of friction, previous stress history, multimodule solid phase and adhesion forces acting between the walls of concave and rectangular cells were shown in the formation of auxetic properties of the porous materials. The limiting values of compression deformation on the stability criterion of cellular structures under compression and the possibility of energetically preferable self-assembly of auxetic porous nano-sized materials have been predicted. It’s seemed that realization of self-healing in composites made of auxetic and multimodule materials is a perspective goal of further studies.

Full text

21
Adaptive Composite
Materials: Bionics Principles,
Abnormal Elasticity, Moving Interfaces
Shilko Serge
V.A. Belyi Metal-Polymer Research Institute of NASB,
Belarus
1. Introduction
Requirements imposed on artificial materials are constantly rising with time. Along with
lately requisite properties, including stability of physical and mechanical characteristics,
linearity of the equation of state and unambiguity of response to disturbance, there arose a
problem of a complex active response to varying outer conditions. In other words, a
tendency is observed of increasing number of material functions acquiring the features of
intellectual systems.
So, obvious prototypes of these materials turn to be biosystems, from the one hand, and
computer monitored technical systems able to reproduce intellectual behavior using sensor,
processor and executive functions (including effector function and response action), from
the other hand, plus feedforward and feedback. Although means of these properties
realization can’t be similar in artificial materials and above mentioned natural prototypes,
generalizations obtained at the junction of the materials science, bionics and cybernetics
allow to formulate the conceptual principles and to consider probable ways of the named
interdisciplinary problem solution.
Recent reviews and terminological discussions in the field have confirmed actuality of the
structural and functional analyses of smart composites, including functional nanomaterials
[Bergman & Inan, 2004]. However papers, devoted to such materials (e.g., self-controlled
membranes on hydrogel base [Galaev, 1995]) are commonly reduced to creation of sensors
and actuators. Less attention have received principles and models of adaptive reactions in
composites. The adaptive mode of reinforcing and self-assembling in smart materials
[Schwartz, 2007] has been studied below in the form of phenomena caused unusual elastic
properties of auxetic and multimodule materials. The development of adaptive composites
allows us to hamper the failure process and promotes reliability and service life of products
for different technical applications.
2. Adaptive composites in classification of materials
2.1 Classification of materials
The first stage of the present study is classification of materials with account of interrelations
found between structure and functions as well as analysis and modeling of a subclass of

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intellectual systems, namely adaptive composite materials (ACM). Some of the assumptions
put forward by the authors are based on the theory of functional systems and synergism
[Prigogine & Stengers, 1984]. Three generations of materials which can be discriminated in
the proposed classification, are given in Table 1.
Generation
of materials
Structural-and-
functional
characteristics
Means of property
regulation
Factor
determining
optimum result
Traditional
material
Monofunctional
single-component
material
Properties determined a
priori by the origin of
component material
Initial property of
monocomponent
Composite
material
Monofunctional
polycomponental
material with fixed
boundaries between
components
Properties are
efficiently regulated
technologically based
on principles of
additivity and
synergism
Initial property of
components and
intermediate
layers
Smart (adaptive)
composites
Polyfunctional
polycomponental
material with
movable boundaries
between components
Self-regulation of
structure based on
sensor, processor and
effector functions and
feedforward and
feedback channels
Efficiency of
sensing extreme
effects and
elimination of
refusals
Table 1. Evolution of structure and properties of materials
The first generation is traditional materials including monofunctional medium whose
properties are determined by the nature and initial quality of a single component. The next
are traditional composites with a prominent structural hierarchy, being also monofunctional.
They are characterized by stability of inner and external boundaries, i.e. fixed structure of
components, intermediate layers and the composite as a whole.
Adaptive materials with coordinated functions and active behavior belong to the third
advanced generation of materials. These systems perceive outer effects at unchanged
function owing to, presumably, structural self-organization. In this connection, the mobility
of the component boundaries should be remembered as an indispensable property of smart
materials, which is not present in traditional composites.
The qualitative transition of materials from the passive to active functioning is shown in
Table 2. Naturally, prerequisites of such a transition are formed at the levels of two preceding
generations. Thus, transformation of one physical field into another (e.g., piezo- or photo
effects) is probable at the stage of monofunctional material. The creation of qualitatively
new (emerged), including forecast properties, is a logical continuation of the additive and
synergetic principles of composite production. This precedes the development of adaptive
composites, being a subclass of smart systems with the dominating adaptive strategy.

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The suggested classification makes it possible to forecast other unknown materials of the
intellectual type, for example, capable of self-destruction “kamikaze”, those ensuring partial
or full restoration “regenerators” and materials offering programmed control of the
environment (“cyber”) and implicit (“incognito”) ones. These subclasses constitute a new
type of “ecophilous” materials which behavior supports homeostasis of the environment.
Functional
evolution
Degree of
activity
Degree of
intellect
Functioning
quality
Mode of
behavior
mono-
functional passive “trivial” material “predictable”
active
“wit”
(functional)
poly-
functional active smart
(adaptive) material = part “indefinite”
“egoist”
material =
system “time-server”
“wise”
(ecophilous)
material =
medium “kamikaze”
“regenerate”
“cyber”
“incognito”
Table 2. Systematization of materials by general criteria
2.2 Adaptive composites
Relative simplicity of ACM is due to their orientation aimed to fulfill only the adaptive
function of the part or a system in contrast to a higher status of the material-medium
subclass (Table 2). However the adaptive composite is formed rather in time than by a
mechanical mixing of structural components, and evolutionizes as a specific unit by
coordinating interrelated physical processes based on an imparted optimum criterion. In
this case, the emergence of macrostructure is specified by origination of collective modes
under the action of fluctuations, there competing and, finally, by selection of the most
accommodated mode or their combination [Prigogine & Stengers, 1984]. The structures
themselves could be described in physical terms as types of adaptation to outer
conditions.
2.2.1 Self-organization of material structure
Reaction of a material due to mutual coordination of structural and functional parameters of
microsystems characterizes it as an open self-regulating system. Selection of the mode of

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behavior in response to outer effect does not arise from the principle of the least action,
neither from the principle of compulsion (Gauss principle) nor from that of the utmost
probability. Active response systems eliminate (or subordinate) contingency. This makes
grounds to speak about a programmed behavior of the system, i.e. the decision is made
according to the inner criteria determined by the structure itself and system parameters,
which substantiates the necessity of direct and reverse connection channels.
It follows from the above said that to form a more complex processor function of ACM it is
possible to use the universe phenomenon of self-organization, which is not limited to only
systems of higher organization and functional complexity and isn’t a monopoly of bio- or
social systems. A self-organizing system is understood as a system capable of stabilizing
parameters under varying outer conditions through directed ordering of its structural and
functional relations aimed at withstanding entropic factors of the environment, which helps
to preserve its characteristics as an integral formation [Prigogine & Stengers, 1984].
The material formed by combining its components acquires the characteristics of a
composite structure, which is a notion nonequivalent to the structure of its constituents. This
fact raises composite materials to a higher structural level and admits the probability of per
layer differentiation of the functions in order to reach the integral control system. In our
view, to realize adaptation mechanism to outer conditions in composite materials, it’s
worthwhile considering the combination of different scale physical processes, where we
single out at least 4 structural levels: molecular, mesoscopic, macroscopic and
polycomponental (Figure 1).
The molecular level is the basic one at programming material behavior. This is because its scale
in polymer composites corresponds to cooperative effects of segmental mobility and
conformal rebuilding that provide conditions for self-organization in high-molecular bodies.
Just here the processor function is realized as a capacity for estimating variations due to outer
effects and as a tool formulating the character and force of response based on stationary
characteristics of the microsystem. Also, the effector function is fulfilled here for exciting
reverse reactions by varying characteristics of the microsystem on a self-organization base.
The mesoscopic level performs the sensor function as an ability to perceive outer effects.
Non-equilibrium processes are initiated at this level changing molecular structure and
supporting the interaction of direct and reverse channels between the levels.
The macroscopic level makes provision for the mobile function as a reorganization of the
initial subsystems (components) aimed at preserving the behavior model.
The mobile function is also realized at the polycomponental level, though intention in this
case to provide the system (material = article) functioning as a whole.
To organize control, the processes relating to the mention levels should be coordinated
using functional links between them.
It is to be remembered that polymer composites are potential carriers of intellectual
properties. Namely, they are sensitive to physical fields, i.e. show a sensor function; make it
possible to carry out the actuator function (shape memory of thermosetting resins, etc) and,
finally, among all other artificial material media they most closely approach the living
nature (biotissues are usually built of high-molecular compounds).
The study of synergetic phenomena in nonliving nature as a linking element between
analogous processes in original objects will, in our opinion, provide a possibility to find
structural-and-functional bioprototypes of adaptive composites.

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Fig. 1. Differentiation of structural levels at ACM development

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3. Self-reinforcing in auxetic composites
The effort towards improving the performance of novel devices based upon realisation of
non-linear and non-trivial (anomalous) deformation properties of materials is the aim of
many current investigations. First, we shall consider the materials with a negative Poisson’s
ratio, ν, termed ‘auxetics’. Data structuring, examination of the mechanisms of generating
the negative Poisson’s ratio and analysis of likely applications for auxetics have been
discussed recently in [Wojciechowski et al, 2007] and reviewed particularly in [Koniok et al,
2004]. Poisson’s ratio affects a very important mechanical property, i.e. compressibility of a
material. Under a uniaxial stress, auxetics expand/contract at the direction perpendicular to
the tension/compression direction, respectively as shown in Figure 2.
(a) (b)
Fig. 2. The deformation mode of an auxetic material under uniaxial stress: (a) compression,
(b) tension. Initial configuration before loading has been shown by dashed lines
This property should influence stiffness and slip under contact loading, and in this way
allow control over deformability and friction characteristics of composites and joints based
on auxetics. As will be shown, the contact characteristics vary dramatically with variation of
the sign of Poisson’s ratio. In the classical elasticity theory for isotropic bodies [Landau &
Lifshitz, 1986] Poisson’s ratio
ν
μμ(3 2 )/(6 2 )KK
=
−+, where μ, К are the shear and volume
moduli respectively, the Poisson’s ratio of isotropic bodies can vary in the limits
10.5
ν
−≤ ≤ . The upper limit corresponds to incompressible materials, e.g. rubber, whose
volume remains constant at significant shape variations, the lower one belongs to the
materials preserving their geometrical form with changing volume.
Several natural and artificial auxetic materials have been described to date, but experimental
and theoretical studies of the adaptive frictional and mechanical properties of these
materials are not still well developed [Baughman & Galvao, 1993]. For example, there exists
the possibility for realisation of self-reinforcing or self-locking effect in contact joints
containing auxetic components. As a result, this effect would bring about a significant
increase in the bearing capacity of frictional joints or shear strength of the fibre – matrix
interface under mechanical or thermo-mechanical load.
Of specific interest here is the study of the self-locking effect under contact deformation of
anisotropic auxetics based on directionally reinforced composites. This is because such
materials may posses Poisson’s ratios of much less than -1 (ν < -50) and considerable
strength due to their directional reinforcement.
The approaches available for creation of composites with ν < 0 assume either the use of
individual auxetic components or formation of an auxetic composite – a combination of

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structural units of mesoscopic level (pores, granules, permolecular formations of polymers,
etc). To study friction effects under contact loading the existing estimates of elastic
properties of quasi-isotropic and anisotropic composites should be taken into account.
3.1 Auxetic inclusions (quasi-isotropic auxetics)
In [Wei & Edvards, 1999] the mechanical characteristics of a composite with ellipsoidal and
spherical particles were calculated for the case of randomly distributed filler particles.
Simulation results under different ratios of filler stiffness to matrix stiffness, for 45% volume
fraction, are presented in Table 3.
Inclusion geometry 0.1 1.0 10
Disc (2D) -0.3020 -0.2856 0.1216
Disc (3D) -0.0385 -0.3575 -0.7387
Sphere -0.0624 -0.2081 0.0650
Wedge (2D) -0.2679 -0.2266 -0.0508
Needle (3D) -0.0555 -0.1714 -0.0562
Table 3. Effective Poisson’s ratio νс of the composite at ν = 0
The possibility of obtaining auxetic composites using filled polymers has been considered in
[Kolupaev et al, 1996]. The authors have obtained such composites using thermoplastic
polyurethane with ultra-dispersed (0.3-1 μm) particles of tungsten, iron and molybdenum
having ν ≈ -0.2-0.4. The composite possessed auxetic properties due to internal stresses σin
produced by the inclusions in the matrix in the range 0.97 MPa < σin < 7.11 MPa.
3.2 Non-auxetic inclusions (anisotropic auxetics)
Let us consider a composite formed by the oblique packing of fibres in an elastic
incompressible elastomeric matrix (Figure 3a).
(a) (b)
Fig. 3. Structure (a) and the mesofragment (b) of obliquely reinforced auxetic composite
OX
OY
u

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At E2 << E1 we get 2
1
xz ct
g
ν
θ
≈− , where E1, E2 are Young’s moduli of the fibres and matrix
respectively. For small fibre packing angles, θ, Poisson’s ratio νxz has negative values. The
deformation results in a pantographic change in orientation of the fibres, which elongate
insignificantly compared to the low-modulus matrix, thus promoting its contraction normal
to the reinforcement direction. To this class of auxetics belong the laminates produced by
the oblique superposing of the layers (Figure 3b). Investigations into laminates made of
prepregs with carbon fibres and epoxy matrix have shown that νxz of the composite
obtained at small packing angles of the layers (10º-40º) is negative.
3.3 Analysis of contact deformation of auxetic composites
The stress state parameters were determined for the double-lap type joint in conditions of
initial compression δy and compression with shear δx (Figure 4a). The analysis of the auxetic
element 1 interaction with two conjugated and located symmetrically non-deformable
bodies 2 and 3 (Figure 4b) has been carried out using the finite element solution of contact
problem with friction.
δ
S
a
S
s
2
3
1
δ
y
u
x
Fig. 4. General view (a) and calculation scheme (b) of a frictional joint with auxetic element
A peculiarity of this problem is a considerable nonlinear deformation brought about in
conditions of unlimited shear by formation of the zones of adhesion Sa and slippage Ss with
nonzero tangential contact displacements u (Figure 4b). Under compression of the joint the
slippage zones are located symmetrically to the central zone of adhesion. Shear application
leads to violations of this symmetry. The limiting load capacity of the joint is dependent on
slippage onset over the whole contact area which, in its turn, is dependent upon the material
compressibility.
For the case of a quasi-isotropic material Poisson’s ratio was varied within theoretically
acceptable values of the isotropic elastic medium, i.e. 10,5
ν
−
≤≤ . The extreme values of the
contact stresses tend to localise near to the right edge of the junction. The contact parameters
vary insignificantly for the positive Poisson’s ratios typical of isotropic materials, except for
the limiting values characteristic for practically incompressible elastomers. Incompressibility
(ν = 0.48-0.5) results in the elastic compression of the material and contact slippage. The
stress strain state parameters including the maximal equivalent stress σeqv, contact pressure
p, tangential stress τ and slippage u, have been studied as a function of Poisson’s ratio for
the quasi-isotropic materials and the reinforcement angle for the anisotropic ones (Figure 5).
An abrupt leap in the maximal contact parameters is observed when ν < 0, not seen with the
positive Poisson’s ratios. This increase is most marked when ν < -0.9. The adaptive mode of
friction has been studied in the form of a self-locking effect under contact loading in

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isotropic and anisotropic auxetic cases [Shilko et al., 2008a]. This effect suggests that the
strength of such a joint rises with increasing shear load.
Similar calculations have been made for a joint with a deformable element of the anisotropic
auxetic composite on the low-modular matrix base (see section 3.2, Fig. 3) under varying
reinforcement angles that determine the elastic moduli Ex, Ey, Ez, νxy, νyz, νxz, Gxy, Gyz, Gxz.
These elastic constants (Table 4) were calculated based on the volume fraction of the fibrous
filler with μ = 0.1, elasticity modulus and Poisson’s ratio of the matrix and filler, respectively
Em = 4 MPa, νm =0.5, Ef = 1.5 GPa, νf = 0.4 using the formulas
11 22 33 66 55 44
12 23 13
111111
; ; ; ; ; ;
; ; .
xyzxy yz xz
xy x yz y xz x
EEEG G G
aaa a a a
Ea Ea Ea
ννν
=== = = =
=− =− =−
(1)
where i
j
a are compliance coefficients of a unidirectional composite:
()
()
()()
()
()
()
()
()
()
()
()
()
()
()
()
()
()
()
()
()()
()
()
()
()
()
()
()
()
()
()
()
2
11 22 33
2
12 13 23
66 55 44
11 1 1
1, ,
11 11
111 1
1, ,
11 11
11111 111
2, 2
11 11
mf
mf
fm mf
mf
mf
mm f fm
mfm
nnn
aaa
nE nE
nnn
aa a
nE nE
nn
aa a
nE
μμ μννμμ
μμ
ν
μν μ μ ν ν μ μ
νμνμ
μμ
ννμνμ νμ ν
νμνμ
⎛⎞
+− +− − − −
⎜⎟
⎝⎠
===
+− +−
+−+−+− −
−+
==− =−
+− +−
++−+++ +++ −
== =
++++−
()
f
E
μ
(2)
where /
f
m
nE E=.
The minimal Poisson’s ratio νxz = -2.142 is attained when the angle between the
reinforcement direction and OY axis is 70° and deformation u is directed along OX axis
(Figure 3).
0
5
10
15
20
25
-1 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4
σ
eqv
, p, τ, MPa
σ
eqv
p
τ
ν
Fig. 5. Maximal values of equivalent stresses, contact pressures and tangential stresses as
dependent on Poisson’s ratio
The maximal contact parameters were determined for different surface geometries of the
conjugated bodies, namely plane, cylindrical (curvature radius r = 100 mm) and wedge-like

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(wedge aperture angle α = 174°). It is seen in Figs. 6-10 that the extreme dependence of contact
stresses is characteristic for all geometries with a minimum at reinforcement angle 45°.
ϕ, degree 0 15 30 45 60 70 80 90
Ex, MPa 6.627 6.286 5.619 7.788 38.47 133.4 266.8 303.2
Ey, MPa 303.2 207.5 38.47 7.788 5.619 6.051 6.471 6.627
Ez, MPa 6.627 7.259 11.96 81.50 11.96 7.95 6.88 6.627
vxy 0.010 0.082 0.336 0.945 2.242 2.793 0.912 -0.48
vyz 0.480 -1.727 -1.316 0.048 0.662 0.858 0.958 0.989
vxz 0.989 0.918 0.662 0.048 -1.316 -2.142 -0.743 0.48
Gxy, MPa 1.996 21.1 59.58 78.17 63.31 42.49 16.98 1.996
Gyz, MPa 1.996 2.285 3.181 3.754 2.753 2.521 1.771 1.666
Gxz, MPa 1.666 1.910 2.753 3.754 3.181 2.117 2.122 1.996
Table 4. Elastic constants for obliquely reinforced composite as a function of reinforcement
angle
ϕ
0
10
20
30
40
50
60
0 102030405060708090100
σ
eqv
, p, τ, MPa
σ
eqv
p
τ
ϕ,
degree
Fig. 6. Dependence of maximal values of equivalent stress, contact pressure and tangential
stress on reinforcement angle at compression: δy = -1 mm (plane surface)
With a plane surface of the conjugated bodies (Figures 6, 7) the dependencies of stresses
σeqv(ϕ), p(ϕ),τ(ϕ) have local maxima. Their location varies with increasing shear due to
slipping.
For cylindrical conjugated bodies, the local minima are absent under pure compression of
the auxetic section, although shear promotes their appearance in the region of large
reinforcement angles (Figures 8, 9).
It is peculiar that the auxetic body in a junction with a wedge surface shows a rather weak
shear effect upon the contact stress state. Similarly to the case of planar surfaces, the local

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minima of maximal stress σeqv and contact pressure p correspond to the reinforcement angle
of ϕ = 15°.
0
10
20
30
40
50
0 102030405060708090
σ
eqv
p
τ
ϕ,
degree
σeqv, p, τ, MPa
Fig. 7. Dependence of maximal values of equivalent stress, contact pressure and tangential
stress on reinforcement angle at compression with shear: δy = -1 mm, δx = 5 mm (plane
surface)
σ
eqv
p
τ
ϕ,
degree
σ
eqv
, p, τ, MPa
0
5
10
15
20
25
30
0 102030405060708090100
Fig. 8. Dependence of maximal values of equivalent stress, contact pressure and tangential
stress on reinforcement angle at compression: δy = -1 mm (cylindrical surface)
So, promising functional materials with negative Poisson’s ratios (auxetics) have been
considered. The results reported here help to quantitatively evaluate the influence of
Poisson’s ratio (in the isotropic materials) and reinforcing angle (in the anisotropic
composites) for compression and compression with shear contact interactions. The
adaptive mode of friction has been studied in the form of self-reinforcing under contact
loading in isotropic and anisotropic auxetics. This effect suggests that the bearing capacity
of such a frictional joint rises with increasing shear [Shilko & Stolyarov, 1996]. It is shown
that the use of auxetic materials is an efficient means of improving the mechanical and
frictional characteristics of composites.

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σ
eqv
p
τ
ϕ,
degree
σ
eqv
, p, τ, MPa
0
5
10
15
20
25
020406080100
Fig. 9. Dependence of maximal values of equivalent stress, contact pressure and tangential
stress on reinforcement angle at compression with shear: δy = -1 mm, δx = 0.1 mm
(cylindrical surface)
4. Self-structuring of multimodule materials
As it was mentioned above, in contrast to traditional composites, which display stable
interfaces between components determining invariability of technologically specified complex
of properties (Table 1), the adaptive material admits mobility of interfaces.
As a consequence, description of able to regulate its structure ACM proceeding from the
given optimal criterion presumes statement of the moving boundaries problem and use of
methods for its solution. In this connection, let’s turn to investigation results of elastic
(reversible) remodelling of a physically nonlinear multimodular metals such as Fe, Al, Cu,
Mg, etc [Bell, 1968] (particularly, in Figure 10 these data are given for Fe). Softening of bulk
modulus near the volume phase transition has been observed in polymer gels too [Hirotsu,
1991]. In the paper [Baughman & Galvao, 1993] it has been shown that crystalline networks
demonstrates unusual mechanical and thermal properties.
The hypothesis is to be introduced into the course of the model development according to
which the adaptive reaction reduces is a specific transfer process to an optimum control
over intrinsic to ACM moving boundaries.
Let a composite at the polycomponental level be formed by bonded elastic particles
(Fig. 11a). Being initially homogeneous and isotropic, each particle is characterized under
the force action by stress concentration in the contact zone with the neighboring particle
(Fig. 11b). The role of the sensor function is presumed to be played by characteristic for a
number of substances multimodule ability [Bell, 1968], that is, the presence of a set of n
discrete values of Young’s modulus E depending on stress
σ
, being in this case the control
parameter. For numeric modeling of adaptation to extreme external loads, which are by far
higher the acceptable one for the initial material, discretization of each particle by finite
elements and block relaxation algorithm, are used. A system of inequalities is taken as a
processor function of the multimodule composite, where the lower and upper estimates of
stresses are corrected at model “exposure” proceeding from the optimum criterion, namely,
the condition of the minimal equivalent stress
σ
eq

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0
5
10
15
20
250
0 0,4 0,8
E = 200,9 GPа
E = 113,7 GPа
e
*
10
-3
σ
,
MPа
Fig. 10. Stress–strain curve of multimodule material (Fe) having two values of elastic
modulus E
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
≤≤
−−−−−−−−−−−−−
≤≤
−−−−−−−−−−−−−
≤≤
=
−
−
, ,
; ,
; ,
1
1
10
1
nn
n
ll
l
ifE
ifE
ifE
E
σσσ
σσσ
σσσ
(3)
Simulation results prove that the adaptive reaction consists in transformation of the initial
homogeneous structure into a reversible inhomogeneous one (Fig. 11c,d) ensuring
perception of extreme loads through effective reduction of stress concentration due to
dynamically optimal distribution of elastic modulus E.
So, the statement and systematic analysis of the problem of developing adaptive composites
has enabled to trace evolution of structural organization of artificial materials, to clarify
mechanisms of adaptation to external media and to disclose, to a certain degree, the effect of
structure on formation of the optimum back reaction. In above considered example
simulation of composite materials adaptivity is formulated as a problem on localizing
moving internal boundaries, while differentiation of material functions is related to the
changing scale level of the structure.
5. Self-assembling of auxetic porous composites with multimodular solid
phase
Porous or cellular materials like «solid – gas» inhomogeneous systems are efficient
composite structures in respect to optimizing strength and stiffness for a given weight.

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These materials are useful for cushioning, insulating, damping, absorbing the kinetic energy
from impact, packing, etc. Stiff and strong ones are preferable in load-bearing structures
such as a lightweight core in sandwich panels. The term cellular is appropriate when the
material contains polyhedral closed cells, as if it had resulted from solidification of a liquid
foam.
(a) (b)
34
68
102
136
171
205
239
278
307
10
20
30
40
50
60
70
80
90
σeqv MPa
,
(c) (d)
Fig. 11. ACM response after loading: typical composite structure (a); mesomechanical model
(b); initial and final (after remodeling) distribution of equivalent stresses σeq. (c);
dynamically optimal distribution of elastic modulus E (d)
A new means of improving the mechanical characteristics may be realized using abnormal
deformation properties of auxetic porous materials having a negative Poisson’s ratio ν
[Lakes, 1987]. Auxetic porous materials, including auxetic porous nanomaterials (APN),
having very high mechanical properties, are suitable for creating adaptive contact joints and
for replacing natural materials such as damaged bone and tooth biotissues.
Examination of the mechanisms of generating a negative Poisson’s ratio has been discussed
and published in the last years, including special issues of physica status solidi (b) journal
[Wojciechowski et al, 2007]. It is known that the inverted or re-entrant cell structure of

Adaptive Composite Materials: Bionics Principles, Abnormal Elasticity, Moving Interfaces
511
porous auxetics may be obtained by isotropic permanent volumetric compression of the
conventional foam, resulting in non-reversible micro-buckling of the cell walls.
There is interest in compression-driven self-assembly as a means to create auxetic porous
structures at the nanoscale. Below we predict the deformation behaviour of porous materials
under uniaxial tension and compression (by an analytical method) and contact compression
(by the finite element method).
5.1 Analytical determination of porous material elastic modulus
For open-cell flexible cellular materials, Poisson’s ratio can be determined by a rod type
structural unit with chaotically oriented cubic cells, as presented in Figure 12. It is worth
mentioning that such a kind of unit cell model has been simulated in reference [Gibson &
Ashby, 1982]. However, a cubic, not a spherical, structural unit had been used. Also, shear
deformation of the rods had not been taken into account.
z
0
y
0
x
0
1
2
3
4
5
6
(a)
σ
σ
F
F
∫
∫
∫
∫
∫
∫
u
u
ε
ε
∫
∫
∫
∫
∫
∫
(b)
Fig. 12. Structural unit (a) and simulation procedure (b) of flexible cellular plastics: σ, ε are
stress and strain tensor components; F is force acting on the rod end; u is displacement of
force application point relatively to the rods; ∫∫∫ is averaged over direction
The rods of this structural unit are directed normally to the cubic planes. Symmetry of the
element allows one to represent the displacement of the force application points (ends of
rods) relatively to the rod joints through the deformation tensor components.

Advances in Composite Materials - Analysis of Natural and Man-Made Materials
512
22
00 00
00
22
00 00 00
22
00 00
00
,,
11 1
2
,,
33 3
2
,,
55 5
2
L
xxLL y xz yz
LL L
zz
L
xxLL y
LL L
yy xy yz
L
xxLL y x
y
xz
LL L
xx
εγγ
εγγ
εγγ
Δ= −= = +
Δ=−= = +
Δ= −= = +
(4)
where L is the structural unit rod length; xLi, yLi are the coordinates for the end of the i-th rod
(i = 1..6) in the xy coordinate system; the x axis is directed longitudinally to the i-th rod in
the non-deformed state (Figure 13).
x
y
l
θ
+
ω
Fx
Fy
Fig. 13. Scheme of cantilever beam under large bending
Eq. (4) refers to deformations for which the Cauchy relations are satisfied. Here, the
parameter L can be related to the solid state volumetric fraction by the following equation
2
9
2
Vm
VfV
q
π
== , (5)
where Vm is the volume of rods in the structural unit; V is the structural unit total volume
before deformation; q is the rod length L to its cross sectional side length r ratio. For
simplification we neglected the volume of the nodes (rod joints) and assumed that the rods
have a square cross-section. During further calculations we have estimated that the
simulation results do not depend on the r value. So we may assume that r = 1, L = q.
Let us assume that in the coordinate system XYZ uniaxial strain is defined as ()
nm
f
t
ε
=
(other components of strain are equal to zero). The system XYZ position relative to the
system x0y0z0 is defined by Euler’s angles
β
1,
β
2,
β
3. Once the function f(t) and Euler’s angles
are known, these define the deformation components in the x0y0z0 system (Fig. 12). Then, the
displacements (4) can be written as follows:

Adaptive Composite Materials: Bionics Principles, Abnormal Elasticity, Moving Interfaces
513
123 123
() ( , ), () ( , ).
Li nm i Li nm i
yt xt
ε
ηβ ββ ε ξβ ββ
=
Δ= (6)
Here
η
i(
β
1,
β
2,
β
3),
ξ
i(
β
1,
β
2,
β
3) are the Euler’s angle functions which are related by
recalculating the tensor components under coordinate axis rotation. For the determination of
forces i
F
G
acting at the ends of the rods by the set deflections, it is necessary to solve a large
flexure problem of a cantilever beam taking into account material viscosity. At the same
time, to describe deformation of the low-density porous materials (Vf < 0.1) it can be
assumed that the rod is deformed equally over all length L. The viscoelastic behaviour of the
rod material is described by Rzhanitsyn’s relaxation function
1
() t
Rt Ae t
βα
−−
=, (7)
where t is time, s; and A,
α
,
β
are the kernel parameters.
The stress/strain relations are determined by the following equation
0
2()(),3
t
ff
sG Rt d K
ρχ ρχ ρχ
υ
τυ τ τ σ ε
⎛⎞
=−− =
⎜⎟
⎜⎟
⎝⎠
∫. (8)
where sρχ,
υ
ρχ,
σ
,
ε
are the deviatoric and spherical parts of the stress and strain tensors; Gf,
Kf are the shear and bulk moduli of the material.
For the beam deformations, let us assume
0
1
() (), ()
2
ll l
ll l
λ
ε
ελθεω
′
=+ = . (9)
where l is the coordinate referred along the rod median in the deformed state;
λ
is the
coordinate referred perpendicularly to l;
θ
is the rotation of the rod cross-section connected
with flexural strain;
θ
′–
θ
is the derivative of the l coordinate;
ω
is the rod cross-section
turning angle as a function of shear strain;
ε
0 is the deformation of the centre line passing
through the centre of gravity under tension or compression.
The allowance for flexural, shear and tensile-compression strains helps to describe
deformation of “short” rods when their length is commensurable with the cross-sectional
side length. For an arbitrary cross-sectional shape, the following expressions are valid
,,
ll ll l
SSS
M
dS P dS Q dS
λ
σλ σ σ
===
∫∫ ∫∫ ∫∫ . (10)
where M is the bending moment; Q, P are the transverse and longitudinal forces. Therefore,
the equilibrium Eqs. for the cantilever rod for the large deflection case will take the form
cos( ) sin( ),
cos( ) sin( ),
()().
yx
xy
yL xL
QF F
PF F
MFx x Fy y
θ
ωθω
θω θω
=
+− +
=+++
=−−−
(11)
Substituting Eqs. (8) and (9) into (10) gives

Advances in Composite Materials - Analysis of Natural and Man-Made Materials
514
()
0
00
0
0
(sin( ) cos( )) ( )(),
1(sin( ) cos( )) ( )(),
1()()()(),
cos( ),
sin( ).
t
xy
f
t
yx
f
t
ynm xnm
f
kFF Rtd
GS
FF Rtd
ES
FL x F
y
Rt d
EJ
x
y
ωθωθωτωττ
ε
θω θω τεττ
θ
εξ εη τθττ
θω
θω
−
=+−++−
=++++−
′′
=+−−−+−
′=+
′=+
∫
∫
∫ (12)
where J, S are the second moments of the area and the cross-sectional area of the rod,
correspondingly; Ef is Young’s modulus of rod material; k is the coefficient complying with
non-uniformity of tangential stress distribution over the cross-sectional area. In our
calculations we assumed k = 1.
Therefore, a system of Eqs. was obtained for the five unknown coordinates l and time
functions. Let us apply the following boundary conditions: (0, ) (0, ) (0, ) 0txtyt
θ
=
==
. In (12)
η
,
ξ
are constants. Solution of these combined Eqs. using the finite difference method allows
us to obtain the coordinates of the free end of rod as a function of five variables, viz:
(,) ( , ,,,),
(,) ( , ,,,).
Lxxy
Lyxy
xxLt
f
FF t
yy
Lt
f
FF t
ηξ
ηξ
==
== (13)
During computation of (12) it was taken into account that the l coordinate differentiation is
made in the deformed state. Therefore, the increment of the l parameter was assumed equal
to 0
0
(1 ) L
dl n
ε
=+ . Here, n0 is a discretization number. The solution of (12) was carried out for
a specified t. It should be mentioned that the structure of Rzhanitsyn’s relaxation function
(7) causes the integral terms in (12) to contain
θ
, γ and
ε
0 functions which were defined
during the previous steps.
The conditions for calculation of the required forces are of the type
(,,,,) (),
( , ,,,) ().
xxy nm
yxy nm
fFF t L t
fFF t t
η
ξεξ
η
ξεη
=+
⎧
⎪
⎨=
⎪
⎩ (14)
The solution of Eqs. (12) and (14) was obtained numerically with the help of MathCad® 7.0
software. The system of nonlinear Eqs. was solved using Newton’s method. As the initial
approximation we took the solution of the previous step. Therefore, we obtain the functions
(,,), (,,)
xy
FtFt
η
ξηξ
which can be presented as follows
22
123 4 5
223322
678910
.
xx x x x x
xxxxx
FC C C C C
CCCCC
ξηξηξ η
ξ
ηξη ξ η ξη
=++ + + +
++ +++ (15)
At a given t, the coefficients C
xj, Cyj (j = 1..10) can be defined by standard regression
procedures. The stress tensor components are related through the forces (15) as follows:

Adaptive Composite Materials: Bionics Principles, Abnormal Elasticity, Moving Interfaces
515
00
00
1
23
2
13
1,
()()
1,
()()
x
xx nm nm
x
yy nm nm
FLL
FLL
σπεξ εξ
σπεξ εξ
=++
=++
(16)
()
()
()
00
00
00
00 00
00
00
00 00
00
00
00 00
3
12
11/2
22
23
11/2
22
23
21/2
22
13
1,
()()
1,
()()
1,
()()
1.
()()
x
zz nm nm
xy
y
xy nm nm
xy xz
xz
y
xz nm nm
xy xz
yz
y
yz nm nm xy yz
FLL
FLL
FLL
FLL
σπεξ εξ
ε
σπεξ εξ
εε
ε
σπεξ εξ
εε
ε
σπεξ εξ
εε
=++
=++ +
=++ +
=++ +
(17)
The stresses for the XYZ system were then redefined. Because of the chaotic orientation of
the unit cells, the stress tensor components should be averaged over direction (Euler’s
angles)
22
3
123 1 2 3
2
00 0
sin
(,,) .
8
nm nm ddd
πππ β
σ
σβββ βββ
π
=∫∫∫ (18)
Therefore, for the known stress to time dependence, we defined time dependencies of the
stresses in a representative volume of the material.
5.2 Calculation example: Uniaxial stress
As an example of using above technique, let us examine the stress-strain state of an elastic
porous material based on high density polyethylene (HDPE). Experimental data for HDPE
were obtained from [Goldman, 1979]: Gf = 237 МPа; Kf = 1402 МPа; A = 0.022 s-β;
β
= 2.995 ⋅
10-5 s-1;
α
= 0.175.
Averaging in all possible loading directions (14) makes the simulated material isotropic at
the macroscopic level. The
τ
(
γ
) function therefore characterizes the dependence of stress on
strain deviator components () (2 )
nm nm
s
τ
γυ
=
. Thus, if functions
τ
(
γ
) and p(Θ) are known, it is
possible to simulate isotropic material behaviour at an arbitrary homogeneous stress-strain
state. Hence, for a uniaxial stress ( 0
ZZ
σ
≠
) the following relations are true
2
4
3
2
3
1
3
2(1);
(1 )(1 ) 1;
;
.
ZZ ZZ
ZZ ZZ
ZZ ZZ
ZZ
s
υεμ
εεμ
σ
σσ
=+
Θ
=+ − −
=
=
(19)

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516
We introduce the transverse strain factor xx
zz
ε
μ
ε
=− , which is analogous to Poisson’s ratio in
the linear elasticity region. Making allowance for a large bending flexure of the ribs where
μ
depends on strain
ε
ZZ, this dependence is determined by the following Eq.
()
2
4(1 ) 2 (1 )(1 ) 1 .
3ZZ ZZ ZZ
p
τε μ ε εμ
⎛⎞
+= + − −
⎜⎟
⎝⎠ (20)
Under stretching,
μ
also decreases rapidly when the strain reaches
ε
cr. In addition, the
μ
(
ε
ZZ)
dependence rapidly passes on the horizontal plateau 0
()
ZZ const
μ
εν
=
=, where Poisson’s
ratio
ν
0 is defined as
0
0
0
32
62
,
KG
KG
ν
−
=+ (21)
here K is the foam bulk modulus defined by the initial part of the p(Θ) curve; G0 is the shear
modulus defined by the
τ
(
γ
) curve.
It was found that the
μ
(
ε
ZZ) function does not depend on the strain rate. In Figure 14 the
dependence of the factor
μ
on the longitudinal strain
ε
ZZ at stretching (а) and compression
(b) of an elastic cellular plastic based on HDPE (Vf = 0.01) is presented. Under compression
the strain reaches some critical value
ε
cr and
μ
rapidly decreases and becomes negative at
ε
ZZ
> 0.9%. Such an anomaly of the elastic behaviour was experimentally observed in polymer
foams [Lakes, 1987]. Our investigations showed that this effect may occur in cellular
materials with a tetrahedronal cell form when the cell ribs buckle inward or in a honeycomb
microstructure.
0,12
0,14
0,16
0,18
0,2
0,22
0,24
0,26
0246810
-0,46
-0,37
-0,28
-0,19
-0,1
-0,01
0,08
0,17
0,26
0 0,4 0,8 1,2 1,6 2
µ
µ
εzz
, %
εzz
, %
(a) (b)
Fig. 14. Dependence of transverse strain factor μ on longitudinal strain εZZ under stretching
(а) and compression (b) of flexible cellular plastics

Adaptive Composite Materials: Bionics Principles, Abnormal Elasticity, Moving Interfaces
517
At small strains,
μ
remains constant and coincides with Poisson’s ratio. Under compression
the strain reaches some critical value
ε
cr and
μ
rapidly decreases and becomes negative at
ε
ZZ
> 0.9%. Under stretching,
μ
decreases rapidly when the strain reaches
ε
cr and
μ
(
ε
ZZ)
dependence passes on the horizontal plateau.
By defining the
μ
(
ε
ZZ) function, the dependence of stress
σ
ZZ on strain
ε
ZZ can be obtained as
[]
34
() 1().
23
ZZ ZZ ZZ ZZ
σε τε με
⎛⎞
=+
⎜⎟
⎝⎠
(22)
At small strains, stability of
μ
allows us to determine the correlation between
ε
cr, Θcr and
γ
cr
cr cr cr cr
14(1)
,.
12 3(12)
ν
εγ
νν
+
=
Θ= Θ
−−
(23)
5.3 Comparison with experimental data
To examine the applicability of the theoretical model for foam deformation properties, we
compared the calculated and experimental values of the relative Young’s modulus E/Ef and
critical strains
ε
cr proceeding from the following considerations: the majority of
experimental data on deformation of elastic foams are based on their uniaxial compression
behaviour; the calculated stress/strain dependence and the experimental behaviour are
almost linear at
ε
<
ε
cr. As it was shown in [Hilyard & Cunningham, 1987],
σ
ZZ(
ε
ZZ)
dependence at
ε
ZZ >
ε
cr to a certain degree is conditioned by inhomogeneity of the inner
structure of the material.
During definition of Young’s modulus E of an elastic cellular plastic, we considered that the
rod cross-section turning angles are small ( cos( ) 1, sin( ) 0
θω θω
+
≈+≈
) and we do not
consider rod viscosity. In this case, the solutions can be obtained in the analytical form. For
the relative Young’s modulus, we have
36 (7 4 )
216 3 (9 8 )
ff
f
fff
V
EV
EV
πν
π
ν
++
=++
, (24)
where
ν
f is the solid phase Poisson’s ratio. In particular, for the elastic polymer material we
assume that
ν
f = 0.49. Equations (10) and (15) yield an approximate expression for the critical
strain
ε
cr
(
)
()
372 9 8
72 36 7 4
fff
cr
ff
VVv
Vv
ππ
επ
⎡
⎤
++
⎣
⎦
=
⎡
⎤
++
⎣
⎦
. (25)
The dependence of the relative Young’s modulus E/Ef on the relative solid volume fraction
Vf for the elastic foam is presented in Figure 15.
In Figure 15, curve 1 corresponds to Eqs. (23). Curve 2 agrees with the results obtained in
[Warren & Kraynik, 1987]. Curve 3 meets the results obtained in [Beverte & Kregers, 1987]
using the semi-axes hypothesis. Curve 4 corresponds to the analytical expression

Advances in Composite Materials - Analysis of Natural and Man-Made Materials
518
()
/
12 0,16
3
f
f
f
V
EV
E
ν
=−= , (26)
obtained in [Gibson & Ashby, 1982]. Here
ν
/ is the Poisson’s ratio of the material dependent
on the number of rods in structural unit N. For simulation of mechanical behaviour of the
rubber foam [Gibson & Ashby, 1982], we used a structural element with 4 < N < 8, when
ν
/
= 0.26. Curve 5 in Figure 15 corresponds to the empirical relation for the relative Young’s
modulus of foam rubbers [Hilyard & Cunningham, 1987]
()
2
27 3
12
f
ff
f
V
EVV
E=++ . (27)
The circles in Figure 15 reflect experimental data for the foam rubber [Lederman, 1971]. This
comparison proves that the proposed technique makes it possible to predict quite accurately
elastic properties of the material at Vf < 0,15.
0
0,01
0,02
0,03
0,04
0,05
0,06
0 4 8 12 16 20
1
2
3
4
5
Vf, %
E/E
f
Fig. 15. The dependence of relative Young modulus E/Ef on the solid phase volumetric
fraction Vf for the flexible foam: curve 1 corresponds to Eqs. (23); curve 2 corresponds to
results obtained in [Warren & Kraynik, 1987]; curve 3 corresponds to results obtained in
[Beverte & Kregers, 1987]; curve 4 corresponds to results obtained in [Gibson & Ashby,
1982]; curve 5 corresponds to results obtained in [Hilyard & Cunningham, 1987]; circles
correspond to experimental data [Lederman, 1971]

Adaptive Composite Materials: Bionics Principles, Abnormal Elasticity, Moving Interfaces
519
5.4 Construction of the mesomechanical model
Mesomechanical (in the scale of the separate cells) description of cellular structure is time-
consuming but a very informative method. A possibility for the determination of Poisson’s
ratio during special thermomechanical treatment of basic porous material when convex
structural cells transform into concave ones as shown in Figure 16, is an advantage of the
mesoscopic model.
Cooling
Three-axial
compression
Basic
material
Heating
Auxetic
Unloading
0_________100
μ
m
0_________100 μm
Fig. 16. A scheme of obtaining the auxetic material using transformation of ther basic structure
into the inverted one with concave cells. Electron microscopy of a porous polyurethane
fragment with magnification 50*
Previously, the determination of v as a function of the transverse and longitudinal strain
was achieved for the case of compression of the sample made of a one-phase material with
known values of Poisson’s ratio (Figure 1a). The geometrical sizes of the rectangular sample
are Lx = 50 μm, Ly = 250 μm; the compressive strain is
ε
y = 0.5%. The calculated results are
shown in Table 5. It should be noted that the technique has an acceptable accuracy which

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520
increases as the friction between the sample and the plates decreases. This fact is explained
by a free slip of the contact surfaces.
For calculation of the effective elastic characteristics of the porous material mesofragments
we replace the real structure by a system of cells of regular polyhedrons. The transformation
of the porous material into the auxetic one appears to be possible at bulk compression
Vin/Vtr equal to 1.4÷4.8 where Vin, Vtr are the volume of the initial and transformed structure
respectively. The best results are achieved at Vin/Vtr = 3.3÷3.7. This agrees with the data
derived for foamed polyurethane and copper sponge.
The simulation allows us to describe cell transformation at the expense of free volume due
to connection of structural units providing the required deformation mode.
ux, μm
f = 0.1 f = 0.5
The number of
node
Left side Right side Left side Right side
1 -0.0507 0.0498 -0.0510 0.0493
2 -0.0507 0.0498 -0.0510 0.0493
3 -0.0507 0.0498 -0.0510 0.0494
4 -0.0507 0.0498 -0.0509 0.0494
5 -0.0506 0.0499 -0.0508 0.0495
6 -0.0506 0.0499 -0.0508 0.0496
7 -0.0506 0.0500 -0.0507 0.0498
8 -0.0506 0.0503 -0.0508 0.0503
9 -0.0504 0.0483 -0.0504 0.0499
10 -0.0373 0.0370 -0.0471 0.0405
ux, average -0.04929 0.04846 -0.05045 0.0487
ux, total average 0.048875 0.049575
Table 5. The calculation results of the transverse displacements
5.5 Mesomechanical analysis
According to the mesomechanical approach, some systems of regular polyhedrons,
presented in Figure 17, were constructed for calculation of Poisson’s ratio v during
structural transformation under compression (Figure 16). In the numerical example, we give
the following initial data for the solid phase of the porous material E = 1 GPa, v = 0.1; the
sizes of the fragment 240*280 μm and the periodic cell 34*34 μm, the friction coefficient on
the contacting surface with the rigid plates is f = 0.5.
Besides the linear elastic solid phase, we have assumed a physically nonlinear multimodular
solid phase. In the last case, the stepwise dependence of Young’s modulus on the stress
component has been used (1).

Adaptive Composite Materials: Bionics Principles, Abnormal Elasticity, Moving Interfaces
521
We then simulated deformation of the initial structure with rectangular cells to analyse the
formation of auxetic properties under compression of traditional porous material. To
increase the accuracy, Poisson’s ratio was determined by averaging the displacements for
the left and right sides of the model structure fragment. According to Table 6, the results in
the case of a multimodule solid phase seems to be more stable than for E = const and at less
expressed auxetic properties (stability loss of the porous fragment made of multimodular
material is absent at compression displacement uy = 14.0 μm).
Fig. 17. Distribution of contact pressure under deformation of the porous structure (the
vertical compressive displacement uy = 0.1 μm)
uy, μm 1.4 2.8 7.0 14.0* 21.0 28.0 35.0 42.0
E = const -0.040 -0.054 -0.085 -0.49 -0.180 -0.222 -0.130 -0.291
E = E(σ) -0.0146 -0.0195 -0.0340 -0.076 -0.080 -0.100 -0.118 -
*stability loss of porous material fragment.
Table 6. The calculation results of Poisson’s ratio v
The dependences in Figure 18 were shown in a dimensionless form (compression level was
taken as a ratio of normal displacements to the height of the porous material fragment uy/b)
for comparison under different conditions of loading. It can be seen from Figure 18 that
instability of solution is observed at a step-by-step loading of the porous material with
hexagonal cells at a deformation level 5%. The porous material with hexagonal cells and
multimodule solid phase coincide closely. The solution is not converged for the porous
material with square cells at deformation level more than 15% with local unstability in the
range 2.5-7.5% and for multimodule solid phase at deformation level more than 10%. The
solution is not converged at the deformation level more than 5% for the concave cells with

Advances in Composite Materials - Analysis of Natural and Man-Made Materials
522
linear elastic and multimodular solid phase. According to the previous stress history the
solution is not converged at the deformation levels greater than 7.5% and 3% for the square
and concave cells respectively.
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
0,3
0,4
0 5 10 15 20
-0,5
-0,4
-0,3
-0,2
-0,1
0
02468
u
y
/b, %
νu
y
/b, %
(a) (b)
Fig. 18. Dependence of Poisson’s ratio on compression level: (a) step-by-step loading;
(b) accounting the previous stress history for the porous material with square cells (squares),
hexagonal cells (generated angle α = 600) (triangles), circular cells (stars) and with
multimodule solid phase (hatches)
The analysis of the stress-strain state of these cellular structures for various deformation
levels shows that Poisson’s ratio is near to zero at the initial stress state but decreases
significantly under compression of the material, which its solid phase has a constant
elasticity modulus. The predicted auxetic behaviour is due to generation of the concave cells
at the determinative compression level. Poisson’s ratio decreases for the structure with the
given concave cells transferring into a plateau. At significant deformation, the solution is not
converged due to closing of the cell edges.
At the macroscale the model of the cell structure is unstable. This may result in the a
displacement of the fragment (in the given example this takes place at compression level
uy = 14 μm). For obtaining a stable solution, it is necessary to take into consideration the
previous stress history of the contact friction process (Fig. 18b). The account of the previous
stress history is also important for calculating the auxetic self-lock mode at the conditions of
contact compression and shear [Shilko et al, 2008a].
5.6 Prediction of auxetic effects in porous materials with nano-sized cells
Self-assembling high-strength and rigid materials of small density are of great interest like
Langmuir films. This may be reached by the auxetic porous material “construction” on the
micro- and nano-size level. It is important that the value of adhesion forces F increases
essentially at decreasing of the gap H between solid surfaces. The values of the adhesion
force are shown in Table 3 for two pairs of polymers and three values of the gap H
according to

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523
12
3,
6
A
FH
π
= (28)
where 12
A is the Hamaher constant and H is the distance between surfaces.
It is seen that a sharp increase of the adhesion force takes place in nano-sized cells of the
porous material.
F, MPa
Material
H = 10A H = 5A H = 4A
A12, Erg
Polytetrafluorethylene - Polyimide 7.38 51.1 115.3 1.39*10-12
Polycaproamide - Polycaproamide 7.30 58.3 113.9 1.37*10-12
Table 3. Estimation of adhesion forces in nano-sized cells on the basis of polymers
The calculations of the deformed state of the porous material subject to adhesion forces and
multimodule effect simultaneously, show a possibility of self-assembling of a spontaneous,
energetically preferable auxetic nano-sized structure as shown in Figure 19.
So, the auxetic porous materials with micro- and nano-sized cells, having good combination
of density, deformational and strength properties, seem quite preferable for many technical
and biomedical applications. Analytical and numerical modelling describes the cellular solid
transformation resulting in microbuckling of the cell walls under certain loading conditions
and providing the auxetic deformation mode.
Geometrically simple mesomechanical models of the porous material based on cubic,
rectangular and concave structural units have been investigated in the present paper taking
into account such important factors as large strains, history of loading, physical
nonlinearities of solid phase, adhesive interaction and so on.
The limitations of the effective finite element simulation are caused by stability loss of the
representative fragment of structure. The possibility of compression-driven self-assembly of
nano-sized auxetics due to the increasing adhesion force between the cell walls has been
predicted.
6. Conclusion
• The systematic analysis of the problem of developing adaptive composites has enabled
us to trace evolution of structural organization of artificial materials, to clarify the
mechanisms of adaptation to the external action, and to disclose, to a certain degree, the
effect of structure on formation of the optimum back reaction.
• In above considered examples of composites, description of adaptive structures is
formulated as a problem on localizing moving interfaces. The study of synergetic
phenomena in the nonliving nature and analogous processes in biological objects will,
in our opinion, provide a possibility to find structural-and-functional prototypes of
adaptive composites.
• The proposed analytical and numerical models predict self-reinforcing in composites
and joints made of auxetics under loading. The role of friction, previous stress history,

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524
multimodule solid phase and adhesion forces acting between the walls of cells were
shown in the formation of auxetic properties of the porous materials as the composites
with the gas phase.
• The limiting values of compression deformation on the stability criterion of cellular
structures under compression and the possibility of energetically preferable self-
assembly of auxetic porous nano-sized materials have been predicted.
• It’s seemed that realization of self-healing in composites made of auxetic and
multimodule materials is a perspective goal of further studies.
(a)
(b)
Fig. 19. Deformation modes of the porous material with initially rectangular (a) and concave
(b) shape of the cells under the action of adhesion forces

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525
7. Acknowledgement
The author is grateful for assistance to Prof. Yu. Pleskachevsky, Prof. R.D. Adams, Dr. D.
Chernous and K. Petrokovets.
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