Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials
A theoretical attempt is proposed to explain of auxeticity of elastic materials by use of nonlinear models of elastic deformations for wide range of strain values up to moderate level. The analytical expressions are obtained that corresponds to three kinds of universal deformations (simple shear, un...
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Rushchitsky, J.J. 2023-02-15T16:32:33Z 2023-02-15T16:32:33Z 2019 Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials / J.J. Rushchitsky // Прикладная механика. — 2019. — Т.55, № 6. — С. 115-134. — Бібліогр.: 46 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/188198 A theoretical attempt is proposed to explain of auxeticity of elastic materials by use of nonlinear models of elastic deformations for wide range of strain values up to moderate level. The analytical expressions are obtained that corresponds to three kinds of universal deformations (simple shear, uniaxial tension, omniaxial tension) within the framework of three well-known in the nonlinear theory of elasticity models – two-constant Neo-Hookean model, three-constant Mooney-Rivlin model, five-constant Murnaghan model. A most interesting novelty consists in that the sample from elastic material is deformed as the conventional material for small values of strains whereas as the auxetic with increasing to moderate values of strains. Запропоновано теоретичну спробу пояснення ауксетичності матеріалів за допомогою нелінійних моделей пружного деформування у широкому діапазоні значень деформацій від малого до помірного рівня. Одержано аналітичні вирази, які відповідають трьом видам універсальної деформації (простий зсув, односторонній розтяг, всесторонній розтяг) і рамках трьох відомих в класичній теорії нелінійної пружності моделей - двоконстантної неогукової моделі, триконстантної моделі Муні-Рівліна, п'ятиконстантної моделі Мурнагана. Найбільш цікавий новий результат полягає у тому, що як в показаних експериментах, так і в запропонованих теоретичних розрахунках зразок з пружного матеріалу деформується як традиційний матеріал для малих значень деформацій, тоді як при збільшенні деформацій до помірних він деформується як нетрадиційний (ауксетичний) матеріал.Запропоновано теоретичну спробу пояснення ауксетичності матеріалів за допомогою нелінійних моделей пружного деформування у широкому діапазоні значень деформацій від малого до помірного рівня. Одержано аналітичні вирази, які відповідають трьом видам універсальної деформації (простий зсув, односторонній розтяг, всесторонній розтяг) і рамках трьох відомих в класичній теорії нелінійної пружності моделей - двоконстантної неогукової моделі, триконстантної моделі Муні-Рівліна, п'ятиконстантної моделі Мурнагана. Найбільш цікавий новий результат полягає у тому, що як в показаних експериментах, так і в запропонованих теоретичних розрахунках зразок з пружного матеріалу деформується як традиційний матеріал для малих значень деформацій, тоді як при збільшенні деформацій до помірних він деформується як нетрадиційний (ауксетичний) матеріал. This article corresponds to results reported partially at three international scientific conferences (Poland, 2016; Ukraine, 2017; China, 2018) and the short communication in the scientific journal of the NAS of Ukraine “Dopovidi NAN Ukrainy” (2018, N7). en Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials Нелинейность и умеренность упругих деформаций как фактор, объясняющий ауксетичность материалов Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials |
| spellingShingle |
Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials Rushchitsky, J.J. |
| title_short |
Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials |
| title_full |
Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials |
| title_fullStr |
Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials |
| title_full_unstemmed |
Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials |
| title_sort |
nonlinearity of elastic deformations and moderateness of strains as a factor explaining the auxeticity of materials |
| author |
Rushchitsky, J.J. |
| author_facet |
Rushchitsky, J.J. |
| publishDate |
2019 |
| language |
English |
| container_title |
Прикладная механика |
| publisher |
Інститут механіки ім. С.П. Тимошенка НАН України |
| format |
Article |
| title_alt |
Нелинейность и умеренность упругих деформаций как фактор, объясняющий ауксетичность материалов |
| description |
A theoretical attempt is proposed to explain of auxeticity of elastic materials by use of nonlinear models of elastic deformations for wide range of strain values up to moderate level. The analytical expressions are obtained that corresponds to three kinds of universal deformations (simple shear, uniaxial tension, omniaxial tension) within the framework of three well-known in the nonlinear theory of elasticity models – two-constant Neo-Hookean model, three-constant Mooney-Rivlin model, five-constant Murnaghan model. A most interesting novelty consists in that the sample from elastic material is deformed as the conventional material for small values of strains whereas as the auxetic with increasing to moderate values of strains.
Запропоновано теоретичну спробу пояснення ауксетичності матеріалів за допомогою нелінійних моделей пружного деформування у широкому діапазоні значень деформацій від малого до помірного рівня. Одержано аналітичні вирази, які відповідають трьом видам універсальної деформації (простий зсув, односторонній розтяг, всесторонній розтяг) і рамках трьох відомих в класичній теорії нелінійної пружності моделей - двоконстантної неогукової моделі, триконстантної моделі Муні-Рівліна, п'ятиконстантної моделі Мурнагана. Найбільш цікавий новий результат полягає у тому, що як в показаних експериментах, так і в запропонованих теоретичних розрахунках зразок з пружного матеріалу деформується як традиційний матеріал для малих значень деформацій, тоді як при збільшенні деформацій до помірних він деформується як нетрадиційний (ауксетичний) матеріал.Запропоновано теоретичну спробу пояснення ауксетичності матеріалів за допомогою нелінійних моделей пружного деформування у широкому діапазоні значень деформацій від малого до помірного рівня. Одержано аналітичні вирази, які відповідають трьом видам універсальної деформації (простий зсув, односторонній розтяг, всесторонній розтяг) і рамках трьох відомих в класичній теорії нелінійної пружності моделей - двоконстантної неогукової моделі, триконстантної моделі Муні-Рівліна, п'ятиконстантної моделі Мурнагана. Найбільш цікавий новий результат полягає у тому, що як в показаних експериментах, так і в запропонованих теоретичних розрахунках зразок з пружного матеріалу деформується як традиційний матеріал для малих значень деформацій, тоді як при збільшенні деформацій до помірних він деформується як нетрадиційний (ауксетичний) матеріал.
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| issn |
0032-8243 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/188198 |
| citation_txt |
Nonlinearity of Elastic Deformations and Moderateness of Strains as a Factor Explaining the Auxeticity of Materials / J.J. Rushchitsky // Прикладная механика. — 2019. — Т.55, № 6. — С. 115-134. — Бібліогр.: 46 назв. — англ. |
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2019 П Р И К Л А Д Н А Я М Е Х А Н И К А Том 55, № 6
ISSN0032–8243. Прикл. механика, 2019, 55, № 6 115
J . J . R u s h c h i t s k y
NONLINEARITY OF ELASTIC DEFORMATIONS AND MODERATENESS OF
STRAINS AS A FACTOR EXPLAINING THE AUXETICITY OF MATERIALS
S.P. Timoshenko Institute of Mechanics of the NAS,
Nesterov str. 3, 03057, Kiev, Ukraine; e-mail: rushch@inmech.kiev.ua
Abstract. A theoretical attempt is proposed to explain of auxeticity of elastic materials
by use of nonlinear models of elastic deformations for wide range of strain values up to
moderate level. The analytical expressions are obtained that corresponds to three kinds of
universal deformations (simple shear, uniaxial tension, omniaxial tension) within the frame-
work of three well-known in the nonlinear theory of elasticity models – two-constant Neo-
Hookean model, three-constant Mooney-Rivlin model, five-constant Murnaghan model. A
most interesting novelty consists in that the sample from elastic material is deformed as the
conventional material for small values of strains whereas as the auxetic with increasing to
moderate values of strains.
Key words: аuxeticity, universal deformation, moderate strain, nonlinear hyperelastic
model.
Motivation and essential.
The object of study can be defined as the auxetic materials as some subclass of nontra-
ditional materials. This class includes the metamaterials, which include the mechanical met-
amaterials, which in turn include the auxetic materials. It should be noted that metamaterials
(the materials engineered to have a property that is not found in naturally occurring materi-
als; they derive their properties not from the properties of the base materials, but from their
newly designed structures) [30] and smart materials (the designed materials that have one or
more properties that can be significantly changed in a controlled fashion by external stimuli;
smart materials are the basis of many applications, including sensors and actuators) [11] are
the modern kinds of materials. To the
point, the journal “Prikladnaya me-
khanika – International Applied Mechan-
ics” is publishing regularly the articles
on this topic [23, 46]. Auxetic materials
are deformed elastically exhibiting the
unconventional property of increasing
the cross-section (growing swollen) of
cylindrical or prismatic sample under
uniaxial tension, whereas in the con-
ventional materials this cross-section
decreases (grows thin). This is shown in
Figure 1 [3]. The point is that the prop-
erty of the decrease is described in the
linear theory of elasticity by use of the Poisson’s ratio. A change of the decrease of cross-
section on the increase of one means a change of positive values of Poisson’s ratio on the
negative ones.
Fig. 1
Note. This article corresponds to results reported partially at three international scientific conferences (Poland,
2016; Ukraine, 2017; China, 2018) and the short communication in the scientific journal of the NAS of Ukraine
“Dopovidi NAN Ukrainy” (2018, N7).
116
But the auxetic materials can not be associated only with description in the framework
of linear theory of elasticity, where the Poisson’s ratio is the elastic constant. Recently, the
term “negative ratio of the trans-verse strain to the longi tudinal one” is often used instead the
term “negative Poisson’s ratio”.
The main partial component of author’s motivation in this study is the often used (de-
scribed verbally or by the picture) demonstration of auxeticity of a foam as increasing the
volume of sample from a foam under tension. It is shown in Fig. 2 [10 (left), 20 (right)].
Fig. 2
These pictures are really very demonstrative, because they show two basic features. The
feature 1 consists in that the sample length is possibly not sufficient to create the classical
conditions of the test on the universal deformation of uniaxial tension-compression. The fea-
ture 2 can be meant as follows: the longitudinal and transverse strains are seemingly not
sufficiently small.
An essential part of studies of auxetics consists in finding of diverse variants of internal
structure that is further studied by methods of molecular physics and computational simula-
tions. The most popular is so called hexagonal system (It is shown in Figure 2; left – before
stretching, right – after stretching). Just this structure is given by different authors to illus-
trate the auxeticity.
Fig. 3
The linear theory admits the swelling of sample only in the case of negative values of
Poisson’s ratio. But materials with such characteristics nobody observed during centuries
and nobody recalled in textbooks on elasticity a possibility of their existence. The exception
can be seen in only two classical books – Love’s [27] and Lurie’s [28].
Let us cite first [27, page 244]: “If were 1 , would be negative, and the function
W would not be a positive quadratic function. We may show that this would also be the case
if k were negative. Negative values for are not exclude by the condition of stability, but
such values have not been found for any isotropic material.”
And now a few sentences from the Lurie’s book [28, page 117]: “A tension of the rod
with negative (but the more than 1 ) would be accompanied by increasing of transverse
sizes. Energetically, an existence of such elastic materials is not excluded.” “In hypothetic
material with 1 , the hydrostatic compression of the cube would accompanied by in-
creasing its volume.”
Note that the Poisson/s ratio is denoted in the theory of elasticity by “sigma” and
“nu”. Love uses , whereas Lurie uses . In this paper, the Love’s denotation is used.
The first publications on auxetics linked the auxeticity with negativity of Poisson’s ra-
tio. A prevalent majority of scientists identify up to this time auxeticity with negativity of
Poisson/s ratio. A few publications only exist, where the nonlinear models are used (for ex-
ample, [39] and experiments with large strains in auxetics (for example, [4, 9, 43, 45]).
But the linear theory (model) has an important restriction on the value of rod deforma-
tion: it must be small (for traditional materials, it is restricted to 3% from the initial length of
rod). The experiment shown in Fig. 2 demonstrates that deformations of rod are not small
and elongation reach tens percents of the initial length of rod. Such elongations can be cor-
rectly described only in the nonlinear theory of elasticity.
117
Let us turn on once again the fact that the first observed auxetic materials were the
foams which are characterized by the small value of density and the porous internal struc-
ture. In the next studies, the new auxetics were revealed, the density of which was also
small. But it was shown later that small density is not the defining property of auxetics, be-
cause the significant part of foams has not the property of auxeticity. The defining character-
istics of auxetics is the special internal structure of this material.
Note that in mechanics the internal structure of materials can be appeared on two differ-
ent stages of modeling the materials. First, on the stage of changing the discrete structure of
material by the continuous one (that is, when the notion of continuum is introduced accord-
ing to the principle of continualization). Second, on the stage of modeling the piece-wise
inhomogeneous continuum by the homogeneous continuum (that is, when the principle of
homogenization is applied).
The first stage is usually associated
with methods of molecular physics,
whereas the second stage is standard one
in mechanics od composite materials.
This is peculiar to all the materials that
are studied in mechanics. For the pres-
ence in material a property of auxeticity,
its internal structure have to change un-
der deformation by the special way ex-
hibiting the un-usual (nontraditional)
mechanical effects. Note here that me-
chanics of materials studied traditionally
first the elastic deformation and this con-
cern both traditional (non-auxetic), and
nontraditional (auxetic) materials. As far
as the number of known nonauxetic ma-
terials exceeds the number of auxetic
ones on many orders, then the term “unu-
sual effect” is looking appropriate. In
contrast to the traditional effects that
count tens, the effects of auxeticity are
observed as now in the identical me chanical problems in three types of such problems that
are realized experimentally and described theoretically. An identity consists in that the sam-
ples from material must be compared, when the internal structure of material in cases
“auxetic-nonauxetic” is differing by only geometrical shape of pores. This case is shown on
Figure 4 for the sample from the polyuretane foam (left – traditional structure, right – auxe-
tic structure) [3].
Thus, the auxeticity is generated by the special kind of internal structure of material and
appears in three basic mechanical tests on deformation of material – swelling under tension,
hardening under indentation, synclastic and anticlastic deformation of thin flexible plate).
The test 1 is described above and shown in Figure 1.
The test 2 on indentation (statical Hertz’s problem, problem on hardness by Rockwell -
Brinell-Wikkers) and impact (dynamical Hertz’s problem) shows the effect of hard-ness of
auxetics in the contact zone. Within the framework of the theory of elasticity, this problem is
solving nume rically with the given exactness. A scheme of test that exhibits the essential
difference in the degree of indentation of the spherical indentor into the traditional (left) and
auxetic (right)materials is shown in Figure 5 [3].The test 3 on synclastic and anticlastic de-
formation of deformation of flexible elastic plate is stated within the assumption that the
plate is quadratic in plan and is loaded by the balanced system of three forces – one force is
applied at the center of plate and directed upward, whereas two other identical forces are
applied at the centers of two opposite ends of plate and directed downward. Within the
framework of theory of flexible plates, this problem is solving numerically with the given
exactness. The simple experiment that exhibits the essential difference in deformation of
plate from the traditional and auxetic materials is shown in Figure 6 [2] (left- traditional
material, right – auxetic material).
Fig. 4
Fig. 5
118
Fig. 6
.The shown above information on auxetics permits to state that their definition is based
on the secondary fact – negativity of Poisson’s ratio that in addition is the term from only
the linear theory of elasticity. The primary fact consists in existence of the special internal
structure and shown above basic mechanical effects.
The term “auxetic material” was proposed by Evans in 1991 [12] for materials with
negative Poisson’s ratio. Nowadays, the short term “auxetics” is used. Both names come
from the Greek word (that which tends to increase). One of the first mentions
(1982) of materials with negative Poisson’s ratio can be found in [17, 18]. The works of La-
kes [24] and Wojciechowski [44] of 1987 can be considered as the fundamental ones in the
area of the auxetic materials. In the following years, the scientific journals regularly reported
on new mate rials with Poisson’s ratio with value less than -1. The state-of-the-art in the sci-
ence on auxetic materials is shown in the review articles [3, 5, 6, 19, 23, 30, 31, 39 – 41] and
the monograph [26].
It is worth to note that the auxetic materials are studies mainly by methods of molecular
physics, computer simulations and material science. The contribution of specialists from the
mechanics of materials looks very small. Further more, only the methods of linear models of
mechanics of materials are used. This situation allows to predict that using the notions and
methods of nonlinear mechanics of materials can be promising what constitute the main ge-
neral component of the author’s motivation.
The facts stated above permit to formulate the following goal: let us try to use the non-
linear models of deformation of elastic materials that will allow to take into account the non-
small strains within the framework of classical experiments on three universal deformations:
simple shear, uniaxial tension, multiaxial tension.
Therefore the short information on the universal deformation and the basic notions of
non linear theory of elasticity seems to be expedient.
1. Universal deformations.
Universal deformations (uniform deformations, universal states) occupy the special
place in the theory of elasticity just owing to their universality. It consists in that the theoreti-
cally and experimentally determined elastic constants of material in samples, in which the
universal deformation are created purposely, are valid also for all other deformed states both
samples and any different production made of this material. It is considered therefore that the
particular importance of universal deformation (their fundamentality) consists in a possibil-
ity to use them in determination of properties of materials from tests [9, 13, 15, 21, 26, 28].
To realize the universal deformation, two conditions have to be fulfilled: 1. Uniformity of
deformation must not depend on the choice of material. 2. Deformation of material has to
occur by using only the surface loads.
In the theory of infinitesimal deformations, the next kinds of universal deformations are
studied in detail: simple shear, simple (uniaxial) tension-compression, uniform volume (om-
niaxial) tension-compression. In the linear theory of elasticity, the experiment with a sample,
in which the simple shear is realized, allows to determine the elastic shear modulus . The
experiment with a sample, in which the uniaxial tension is realized, allows to determine the
Young’s elastic modulus E and Poisson’s ratio . The experiment with a sample, in which
the uniform compression is realized, allows determining the elastic bulk modulus k .
While being passed from the linear model of very small deformations to the models of
non-small (moderate or large) ones, that is, from the linear mechanics of materials to
119
nonlinear mechanics of materials, the universal states permits to describe theoretically and
experimentally many nonlinear phenomena. The history of mechanics testifies the experi-
mental observation in XIX century of the nonlinear effects that arose under the simple shear
and were named later by names of Poynting and Kelvin [12, 37]. After about hundred years
in XX century, these effects were described theoretically within the framework of the non-
linear Mooney-Rivlin model [16, 31, 33, 35].
The mechanics of composite materials is one more area of application of universal de-
formations. The simplest and most used model in this case is the model of averaged (effec-
tive, reduced) moduli. In the theory of effective moduli, the composite materials of the com-
plex internal structure with internal links are treated usually as the homogeneous elastic me-
dia. A possibility to create in such media the states with universal deformations was used in
evaluation of effective moduli by different authors and different methods. It was found that
it is sufficient for isotropic composites to study the energy stored in the elementary volumes
of composites under only two kinds of universal deformations: simple shear and omniaxial
compression [36 – 38, 42].
2. Theoretical description of experiments where the universal deformations of sim-
ple shear, uniaxial tension, and omniaxial compression are realized.
2.1 Essentials of nonlinear mechanics of materials [16, 21, 22, 28, 37]. A body is termed
some area V of 3D space
3R , in each point of which the density of mass is given (the area
occupied by the material continuum). In this way, a real body, the shape of which coincides
with ,V is changed on a fictitious body. This fictitious body is the basic notion of mechanics.
The Lagrangian kx or Eulerian kX coordinate systems can be given in
3R . In the theory
of deformation of a body as a change of its initial shape, the notions are utilized that are as-
sociated with a geometry of body (kinematic notions) and with the forces acting on body
from outside and inside (kinetic notions). The notions of the configuration , the vector of
dis- placement ku u
, the principal extensions k , the strain tensor ik are referred to the
notions of kinematics. The external and internal forces as well as the tensors of internal
stresses refer to the notions of kinetics,
The configuration of body at a moment t is called the actual one, whereas the configu-
ration of body at arbitrarily chosen initial moment t is called the reference one. The co
ordinates of the body point before deformation is denoted by kx . It is assumed that after
deformation this point is displaced on 1 2 3( , , , ).ku x x x t Then the vector with components ku
is called the displacement vector and the coordinates of the point after deformation are pre-
sented in the form 1 2 3( , , , ).k k kx u x x x t The frequently used Cauchy-Green strain tensor
is given by the known displacement vector of ( , )ku x t
in the Lagrangian coordinates kx
and in the reference configuration
, , , ,, 1 2 .nm k n m m n n i m ix t u u u u (1)
As a result, the deformation of body is given by nine components of displacement
gradients ,i ku . Such a description of deformation is used in the most part of models of the
theory of elasticity. But the process of deformation can be described also by other parame-
ters of the geometry change of the body. It seems meaningful to use the first three algebraic
invariants of tensor (1) 1 mn mnA , 2
2 (1 2) ( )mn mn ik ikA
2( )mn mn ik ik ,
3 det mnA , which can be rewritten through the principal values of tensor (1) k by the
formulas 1 1 2 3,A 2 1 2 1 3 2 3,A 3 1 2 3.A The often used invariants
1 2 3, ,I I I of tensor ik are linked with the algebraic invariants of the same tensor by relations
2
1 1 2 1 1 2
2 3
3 1 1 2 3 2 1 1
3 2 3 2 ; 3 4 2( ) 3 4 2( );
det 2 1 2 2( ) (4 / 3)(2 3 ).
nn nn nn mm nm mn
pq pq
I A I A A A
I A A A A A A A
120
In a number of models of nonlinear deformation of materials, the elongation coefficients
(principal extensions) defined as a change of length of the conditional linear elements (the in-
finitesimal segments that are directed arbitrarily) are used
1 2k k .
A simpler formula 1k k is valid for the case of linear theory. Additionally to three
parameters above, in threes parameters should be introduced that characterize a change of
the angles between linear elements and areas of elements of coordinate surfaces.
It seems to be necessary to show the very often used notation of the displacements gradient
F
1,1 1,2 1,3
2,1 2,2 2,3
3,1 3,2 3,3
1
1
1
u u u
u u u
u u u
and notation of the left Cauchy-Green strain tensor B = F FT associated with it. The most
used are two tensors of internal stresses: the symmetric Cauchy-Lagrange tensor ,ik that is
measured on the unit of area of deformed body, and the nonsymmetric Kirchhoff tensor ikt ,
that is measured on the unit area of un-deformed body.
2.2. Universal deformation of simple shear. The experiments on simple shear are reali-
zed on the sufficiently long beam of quadratic cross-section, in which the uniform defor-
mation is created on some distance from the ends. The lower side of beam is fixed rigidly
and the surface tangential constant load 2T is applied to the upper side. The deformation of
beam can be described by one component of the deformation gradient 1,2 1 2 .u u x The
component 1,2u and the shear angle are linked as follows
1,2 tan 0.u
In the linear theory, the shear angle is assumed to be small and then tan [15,
28, 37].
The Cauchy-Green strain tensor is characterized by only three nonzero components
2
11 1,1 1,1 1, 1, 1,2 1,2 1,3 1,31 2 1 2k ku u u u u u u u ;
12 21 1,2 2,1 1, 2,1 2 1 2 .k ku u u u
The principal extensions are written through the shear angle by formulas 1 1,
2 3 .
2.3. Universal deformation of uniaxial tension. A rod in the form of a straight long (of
circular or quadratic cross-section) cylinder with the axis in direction of axis 1Ox is conside-
red when the lateral surface of rod is free of stresses. The rod is stretched in the axial direc-
tion. Then the uniform stress-strain state is formed in the rod except for the area near the ends.
It is characterized by only one nonzero component 11 of the stress tensor and two nonzero
components 11 22 33, of the strain tensor (or two principal extensions 1 2 3, ).
On the Young modulus and Poisson’s ratio. Perhaps, the most old and exhausting proce-
dures are shown in the classical Love’s book [27]. Let us use the adopted at that time nota-
tions and write according to [9] the standard representation of the Hooke law through the
Lame moduli ,
2 ; 2 ; 2 ;x xx y yy z zzX Y Z
2 ; 2 ; 2 ,y xy x zx z yzX Z Y (2)
where the notation of dilatation is used xx yy zz .
121
Let us repeat the classical procedure of introducing the Young modulus and the Poisson’s
ratio. Toward this end, the universal deformation of uniaxial tension is considered, when the
axis is chosen in direction Ox and the prism is stretched at the ends by an uniform tension
T . The stress state of prism is uniform and is characterized by only the stress xX T (other
stresses are zero ones). In this case, the Hooke law becomes simpler 2 ,xxT
0 2 yy , 0 2 .zz The expression for dilatation is obtained by adding all three
equalities above (3 2 ) (3 2 ).T T
The substitution of the last expression for dilatation into the first equality (2) gives rela-
tions
3 2 2 3 2xx xxT T T .
The last expression represents the elementary law xxT E of link between tension and
deformation of prism, in which the Young modulus E is used. Comparison of this law with
relation (3) gives the classical expression for the Young modulus through the Lame moduli
3 2 .E (3)
The substitution of expression for dilatation into the second and third equalities (2) gives
relations 2yy zz xx , which express the classical Poisson’s law on the
transverse compression under the longitudinal extension and permit to introduce the Pois-
son’s ratio
2yy xx zz xx . (4)
Thus, the Poisson’s ratio is one of characteristics of linear deformation of elastic materi-
al and is considered as the basic notion of linear elasticity. But the ratio of transverse strain
to the longitudinal one can be used in any model of nonlinear elasticity (and not only elastici-
ty). In this case, this ratio will have its own representation in each model and possibly will
not be constant quantity for any level of strains.
2.4. Universal deformation of uniform (omniaxial) compression-tension. A sample has
the shape of a cube, to sides of which the uniform surface load (hydrostatic compression) is
applied. Then the uniform stress state is formed in the cube. The normal stresses are equal
with each other 11 22 33 , and the shear stresses ik i k are absent. This type of uni-
versal deformation is defined by the following components of displacement gradients
1,1 2,2 3,3 0;u u u 1,1 2,2 3,3 3 ;u u u e , 0 .k m k mu u x k m (5)
The Cauchy-Green strain tensor is as follows
2
11 22 33 1 2 ; 0 ,ik i k (6)
and the algebraic invariants are written in the form
1 11 22 33 ;I e 2 2 2
2 11 22 33 ;I 3 3 3
3 11 22 33 .I (7)
The principal extensions are equal with each other
1 2 3 . (8)
3. Three nonlinear models of hyperelastic deformation.
These models are related to the models of hyperelastic materials. This class of materials
is characterized by the way of introduction of constitutive equations. First the function of
kinematic parameters (elastic potential, internal energy) is defined, from which later the
constitutive equations are derived mathematically and substantiated physically.
122
Two-constant Neo-Hookean model (model 1). The elastic potential of Neo-Hookean mo
del is defined as follows [16, 22, 34, 37]
2 2/3
1 1 1 1 1 ,3 1 ; ; det ;i kW C I D J I J I J u
2/3 22 2 2
1 2 3 1 1 2 3 1 2 3 1 1 2 3, , 3 1 .W C D (9)
where the elastic constants of model are linked with the classical elastic constants by relations
1 12 ; 2C D k .
The constitutive equations have the form
5/3
1 1 12 1/ 3 2 1 ;nm nm nm nmC J B I D J (10)
5/3
1 1 12 1/ 3 2 1nn nC J I D J .
It is considered that this model describes well the deformation of rubber under the prin-
cipal extensions up to 20% from the initial state. Since these extensions are linked with the
principal values of the strain tensor by relation (2) 1 2k k , then it is assumed
1k kk approximately with exactness to 1% in the cases of universal deformations for
Neo-Hookean model, what is true in the case of linear theory too. Because the extensions in
the linear theory are two orders less, then this observation testifies the fact that the Neo-
Hookean model extends essentially the area of allowable values of strains as compared with
the Hookean model.
3.2. Three-constant Mooney-Rivlin model (model 2). The elastic potential of the Mo-
oney -Rivlin model is defined as follows [16, 22, 31, 35, 37]
2 4/3
10 1 01 2 1 2 2
2/3 2 2 2
1 2 3 10 1 2 3 1 2 3
4/3 22 2 2 2 2 2
01 1 2 3 1 2 1 2 2 3 1 1 2 3
3 3 1 ; ;
, , 3
3 1 ,
W C I C I D J I J I
W C
C D
(11)
where the elastic constants of the model are linked with the classical constants by relations
10 01 12( ) ; 2 .C C D k .
The stresses are determined by formulas
5/3 2/3 7/3
10 01 1 012 2J C C J I B J C BB
5/3 2/3
1 10 1 01 22 1 2 3 2 1;D J J C I C J I (12)
5/3 7/32 2 2 2 2 2 2
10 1 2 3 1 2 3 01 1 2 32 1 3 2kk k k k n m
k
W
C C
2 2 2 2 2 2
1 2 1 2 2 3 1 1 2 32 / 3 1 .k D (13)
Here the indexes knm form the cyclic permutation from numbers 123.
The Mooney-Rivlin model is classical one. This can be seen from the next historical in-
formation.
Information. An effect of nonlinear dependence of decreasing the shear stresses when
the torsion angle (deformation) to the level of nonsmall values is called “the Poynting ef-
fect” owing to his publication of 1909, where this effect was described. At that, Poynting
does not mentioned the results of Coloumb (1784), Wertheim (1857), Kelvin (1865), Bau-
schinger (1881), Tomlinson (1883), where this effect was also described in one way or an other.
123
But only within the framework of finite elastic deformations, that was developed in 20 cen-
tury, this effect was satisfactorily explained by Rivlin in 1951. He used the model of nonlin-
ear deformation which now is termed “the Mooney-Rivlin model”.
Five-constant Murnaghan model (model 3). The elastic potential in the Murnaghan model
has the form [14, 22, 32, 33, 37]
2 2 2 3( ) 1 2 ( ) ( ) 1 3 ( ) 1 3 ( ) ;ik mm ik ik im km ik mm mmW A B C (14)
2 3
1 2 3 1 2 3 1 2 1( , , ) 1 2 1 3 1 3W I I I I I AI BI I I .
The Cauchy-Green strain tensor ik and five elastic constants (two Lame elastic const-
ants , and three Murnaghan elastic constants , ,A B C ) are used in this potential.
The Murnaghan model can be considered as the classical one in the nonlinear theory of
hyperelastic materials. It takes into account all the quadratic and cubic summands from ex-
pansion of the internal energy and describes the deformation of big class of engineering and
other materials. If to unite the data on the constants of Murnaghan model, shown in books
[14, 22, 28], then the sufficiently full information can be obtained on many tens of materials.
4. Simple shear.
The following materials are used in the numerical evaluations below (elastic constants
are shown): 1. Rubber – 20 МPа, 2,0k GPа. 2. Foam – 9 90,58 10 , 0,39 10 ,
90,84 10 .k 3. Foam – 9 90,58 10 , 0,39 10 , 101,0 10 ,A 100,9 10 ,B C
101,1 10 . 4. Polystyrene – 93,7 10 , 91,14 10 , 101,1 10 ,A 100,79 10 ,B
100,98 10 .C
Description by model 1. In this case 2(1 ) ,J 2
1 1 2 .I Then expressions for
displacement gradients F and components of tensor В are simplified
F
1
0 1 0 ;
0 0 1
В
21 2
1 0
0 1
.
As a result, the components of stress tensor have the form
10/3
12 21 13 31 1 32 23
10/3
11 1 1
10/3
22 33 1 1
2 1 ; 0;
8 / 3 1 1 2 2 ;
4 / 3 1 1 2 2 2 .
C
C D
C D
(15)
The formulas (15) show that the Poynting
effect (when the values of shear angle in-
crease from the sufficiently small values
to the moderate ones, then the shear
stress depends nonlinearly on the shear
angle) is described by the Neo-Hookean
model, because equation (15) demon-
strates just this nonlinear dependence for
the moderate values of shear angle. The
Figure 7 shows the dependence of the
shear stress on the shear angle 12
for the silicon rubber (in all the plots, a
stress corresponds to 1 MPa).
Fig. 7
124
Description by model 2. The expressions for displacement gradient F and components of
tensor В are the same as for the Neo-Hookean model. As a result, the expressions from for-
mula (12) are simplified
1 2 31, 1 ; 2
1 ;J 2
1 1 2 ;I 2 2
2 1 2 1I
and components of the stress tensor have the form
10/3 14/3
12 21 10 012 1 2 1 1 4 ;C C (16)
14/3 2
23 32 012 1 ;C (17)
10/3 2
11 10 1
14/3 2 3 4
01
2 1 4 / 3 1 2 2 1 2
2 1 4 / 3 3 5 5 4 2 ;
C D
C
(18)
2 4/3 4/32
22 33 1 12 1 1 1 2 1 1 1 2 1 2 .C D (19)
Thus, the Mooney-Rivlin model (that is more complicate as compared with the Neo-
Hookean model) describes the more complicate stress state, which is characterized by six
components of stress tensor. This model describes well-known nonlinear effects. The Poyn-
ting effect follows from representation of the shear stresses by formula (16). The Kelvin
effect follows from formulas (18), (19).
Also, formula (17) describes one more
nonlinear effect: an initiation of shear
stresses 23 32 .The Figure 8 shows
the nonlinear dependence of shear
stress 12 on the shear strain , that is
built for the silicon rubber. Compari-
son with Figure 7, which corresponds
to the Neo-Hookean model, shows that
the Mooney-Rivlin model describes
the more essential deviation from the
linear Hookean description of simple
shear.
4.3. Description by model 3. The Cauchy-Green strain tensor is characterized by three
components
2
22 2,2 2,2 ,2 ,21 2 1 2 ;k ku u u u (20)
12 21 1,2 2,1 ,1 ,21 2 1 2k ku u u u . (21)
To calculate the stresses, it is necessary to write the potential (14) with respect to the
formulas (20), (21)
2 2 2 2
22 22 12 211 2ikW (22)
3 2 2 2 3
22 12 12 21 21 12 21 22 22 12 21 22 221 3 1 3 ;A B C
2 4 6( ) 1 2 1 8 2 1 24 3 .W A B A B C (23)
Fig. 8
125
The Lagrange stress tensor is determined by the formula ,ik n ikx t W and has
two nonlinear components
2
22 22 22 12 12 21 21 12 21
2 2 2 2
22 12 21 22
2 4
2 1 3
3 ( )
1 4 2 2 2 1 4 3 ;
A
B C
A B A B C
(24)
12 12 12 21 12 222 1 3 2 ;A B (25)
3
12 21 1 6 3 .A B (26)
The shear stress contains the linear and nonlinear summands and describes the simple
shear. The normal stress describes the change of volume under deformation and testifies the
break of the state of simple shear under nonlinear description of deformation. To build the
plots of dependence (24) choose two nonstandard for the Murnaghan mo del materials –
foam and polystyrene – which can experience not only the small by values strains, but also
the moderate ones.
Fig. 9
The Figure 9 shows a dependence of the shear stress on the shear angle.
Note to dependence 12 for models 1 – 3. These models describe the nonlinear
Poynting effect. At the same time many scientists working with auxetic materials report the
experimental dependences that coincide quantitatively with the shown here theoretical de-
pendences (for example, [4, 19]).
Conclusion to dependence 12 for models 1 – 3. The developed in mechanics of
materials nonlinear models of deformation of elastic materials can be recommended for de-
scription of auxetic materials.
5. Uniaxial tension.
5.1. Description by model 1. The formulas 2 3 , 2
1 2J , 2 2
1 1 22I are valid
and the normal stresses are given by the formulas
5/32 2 2 2
11 1 2 1 2 1 22 3 1 ;k
(27)
5/32 2 2 2
22 33 1 2 1 2 1 21 3 1 .k
(28)
If to assume that the normal stresses on outside of the sample are absent
5/32 2 2 2
1 2 1 2 1 21 3 1 0;k
(29)
then 2
11 1 23 1k . (30)
126
It follows from Eq. (30) that the Poynting-type effect (when the principal extensions
increase from the sufficiently small values to the moderate ones, then the normal stress in
the direction of tension depends nonlinearly on these extensions) is described by the Neo-
Hookean model.
It should be noted that the stretching in the longitudinal direction stress depends in the
model 1 on two principal extensions – longitudinal and transverse.
Fig. 10 Fig. 11
The Figure 10 shows a dependence of the longitudinal stress on principal extensions and
is built for the rubber with allowance for that the value ( 3 ) 0.00334k is very small com-
pared to the unit. Then Eq. (29) is simplified to the form
2
22 111 / 2 1 2 1/ 2 . (31)
The Figure 11 corresponds to formula (29) and shows a dependence of the longitudinal
principal extension on the transverse principal extension. Note that the silicon rubber is
characterized by the big difference between values of shear and bulk moduli that can reach
hundred times. Therefore, the new material is chosen further for the numerical analysis – the
foam, which values of elastic constants is characterized by about equal by the order.
It looks in this case to be illogical to neglect the first summand in Eq. (29). Note that the
ratio 2 1/ corresponds in the linear theory to the Poisson’s ratio.
The Figure 6 shows that with an increase of extension 1 the increase of extension 2 slows.
5.2. Description by model 2. The uniaxial tension in direction of abscissa axis is character-
ized by parameters: 2 3 , 2
1 2J , 2 2
1 1 22I , 4 2 2
2 2 1 22I , 2
11 1B ,
4
111
BB . The normal stresses are given by the formulas
5/32 2 2
11 10 1 2 1 22 2 3C
7/32 4 2 2 4 2
01 1 2 1 1 2 2 1 1 22 2 3 5 3 2 1 ;C D
(32)
5/32 2 2
22 33 10 1 2 2 12 3 C
7/32 2 2 2 2
01 1 2 2 2 1 1 1 22 1 3 2 1C D
. (33)
Assume that the normal stresses over the sample outer surface are absent. Then equation
(33) is simplified to the form
7/32 4 4 2
11 01 1 2 1 2 1 1 22 6 1C D
. (34)
The last formula testifies: the Mooney-Rivlin model describes the Poynting-type effect. Two
elastic constants are presented in Eq. (34) in contrast to the Neo-Hookean model, where the
shear modulus was absent.
It should be noted that in both models – Neo-Hookean and Mooney-Rivlin – the tension
in the longitudinal direction stress 11 depends already on two principal extensions.
127
The Figure 12 shows a dependence
of the longitudinal stress on principal
extensions is built for the silicon rub-
ber. It coincides practically with Figure
10 (Neo-Hookean model) and shows
that the constant 01C of the Mooney-
Rivlin model effects not essentially on
the stress 11 and the dependence (32)
rests the weakly nonlinear within the
accepted restrictions.
The equation (33) can be trans-
formed to the form
3 36 3 2 4 2 4 2
1 1 10 1 01 1/ 2 / 6 2 / 6 ( 1) 0;C D C D (35)
2 1 ;
3 4
10 13 2 2 4 2
1
3 2
01 1
2 / 6
1/ 2 1/ 2 1 ( 1).
2 / 6
C D
C D
The corresponding to the model 1 plot from Figure 11 is practically identical with the plot
corresponding to the model 2.
5.3. Description by model 3. The uniaxial tension is characterized by three nonzero
components of the strain tensor kk and one non-zero component of the stress tensor 11 .
Then the constitutive equations are somewhat simplified and have the form
2
11 11 22 33 1 11 11
2 2 2
11 22 33 11 1 22 11 33 11
2
2 2 2 .
I A
B I C
(36)
2
1 22 22 22 1 22 33 22 110 2 2 2 2 ;I A B I C (37)
2
1 33 33 33 1 22 11 22 330 2 2 2 2 .I A B I C (38)
Let us remind that in the linear theory of elasticity, corresponding to the Hookean
model, the constitutive equations are significantly simpler
11 1 112 ;I 1 220 2 ;I 1 330 2 .I (39)
Apply further to the nonlinear equations (36) – (38) the procedure of analysis of the state
of uniaxial tension that is used in the linear theory of elasticity as applied to equations (39).
Sub traction of equation (38) from equation (37) gives the formula
2 2
22 33 22 33 22 33 11 22 330 2 2 ,A B
from which follows the equality of components of transverse strains 22 33 .
Addition of formulas (36)-(38) results in the following formula
2 2 2
11 11 22 11 223 2 3 3 2 2 2 3 2 2A B C B
2
22 22 11 11 224 3 2 2 2 .C (40)
Substitution of formula (40) into the relation (36) gives new relation
Fig. 12
128
2
11 11 11
2 3
E A B C
(41)
2
22 11 22
2 24 2 2
.A B C B C
The last relation shows that the model 3, like the models 1 and 2, describes the
Poyntingtype effect.
The Figure 13 shows a dependence among the longitudinal stress 11 and strains 11 22,
11 11 11 22( , ) for the foam and polystyrene and the moderate values of strains. Both
plots demonstrate an essential nonlinearity under moderate strains.
Fig. 13
Write now the constitutive equation (36) with allowance for equality 22 33 and
transform it to the form of quadratic equation relative to the ratio 22 11/
2
11 1122 22
11 11
2 0
6 4 6 4
B C B C
A B C A B C
.
The solution of this equation has the form
22 11 11/ 6 4B C A B C
11
2
11
6 4 /
1 1
/
A B C B C
B C
. (42)
Thus, equation (42) shows that for the ratio 22 11( / ) is not constant in the Murnaghan
nonlinear model.
The plots in Fig. 14 shows a dependence of the ratio 22 11( / ) on the strain 11 .and are
built for the foam and polystyrene for the moderate strains. The plot main feature: the ratio
22 11( / ) is decreased essentially from the initial value, which corresponds to the Poisson
ratio for small strain, to the negative values under the moderate values of longitudinal strain.
So, the ratio, that is treated as the Poisson’s ratio for small strain, in the case of moderate
strain becomes the characteristics of transition of the material from the category of conven-
tional materials into the category of nonconventional materials. This can be considered as
the new revealed theoretically nonlinear effect.
Thus, an analysis of universal deformation of uniaxial tension for the model 3 revealed
the new property: the material with conventional properties under small strains is transfor-
med under moderate strains into the nonconventional (auxetic) material. Uncommonness of
this observation consists in that usually the material is considered either the conventional or
the nonconventional during all the process of deformation.
129
Fig. 14
Let us compare the plots from Fig. 14 with the experimental data from [4, Fig.4] shown
here as Fig. 15 (dependence of the ratio 22 11( / ) on the strain 11 ), where the deformation
of the foams was studied for the finite strans with increasing the longitudinal strain 11 from
0,1 to 1,4. Note that the theoretical plots are constructed for the range from 11 0 to the
moderate values 0.23 (foam) and 0,33 (polystyrene). This comparison shows that 22 11/
increases within the range 11 0,0; 0,3 . Thus, the model 3 describes some experimental
observation of the foam.
Fig. 15
The plots in Figure 16 show dependence of longitudinal and transverse strains. Three
stages can be marked out: 1. A decrease of transverse strain becomes slower under transition
to the moderate strains. 2.The strain 22 reaches the local minimum and further increases.
3. When the strain 11 continues to increase, the strain 22 possesses zero value and fur-
ther increases possessing already the positive values.
The shown feature confirms once again the new mechanical effect – a transition of the
material under its deformation to the level of moderate values of the longitudinal stretching
from the class of conventional materials into the class of the auxetic materials. In other
words, the standard sample in conditions of universal deformation of uniaxial tension is de-
formed for small strains as if it is made of the conventional material (its cross-section is de-
creased) and with increasing the values of longitudinal stretching to the moderate values the
sample cross-section starts to increase, what is the characteristic just for auxetic materials.
The plots from Figure 11 can be compared with the plot, obtained experimentally in [41].
130
Fig. 16
This article reports that the new metamaterials were created from the soft silicon rubber.
The samples were deformed in conditions of uniaxial compression up to moderate values of
longitudinal strain 0,35. The shown in Figure 17 plot corresponds to Figure 2a in [41] and
shows a dependence of longitudinal and transverse strains. Comparison of plots from Fig-
ure16 (uniaxial stretching) and Figure 17 (uniaxial compression) demonstrates the common
property of forming the hump in the area of negative values of transverse strain, which is
transformed with the increasing values of longitudinal strain roughly into the straight line in
the area of positive values of transverse strain.
Fig. 17
Main conclusion to subsection 6. The nonlinear Murnaghan model describes within con-
ditions of uniaxial tension some nonlinear phenomena of deformation, which can be linked
with the properties of deformation of auxetic materials. Note that the shown feature is clear-
ly visible only within the framework of the Murnaghan model, but the Neo-Hookean and
Mooney -Rivlin models also describe the hump formation, as it can be seen in Fig. 11.
131
6. Multiaxial tension.
6.1. Description by model 1. In this case 1 2 3, 3 2
1 1 1, 3J I and the normal
stress is equal
3
11 1 12 ( 1).D (43)
The formula (43) describes the Poyn-
tingtype effect relative to the bulk modu-
lus (the dependence of 11 on the exten-
sion 1 is evidently nonlinear). The Fig-
ure 18 shows a dependence of longitudi-
nal stress on the longitudinal principal
extension and is built for the silicon rub-
ber. The plot testifies that the model 1 de-
scribes the nonlinear change of the sam-
ple volume, while being subjected the
universal deformation of uniform com-
pression-tension.
6.2. Description by model 2. In this case 1 2 3, 3 2
1 1 1, 3 ,J I 4
2 13I and
they are true for any nonlinear model. The formula for normal stress coincides with the
analogous formula for the model 1 (43) and verifies the nonlinear dependence of tension
stresses on the principal extension.
6.3. Description by model 3. The components of displacement gradients and Cauchy-
Green strain tensor are as follows
1,1 2,2 3,3 0;u u u 1,1 2,2 3,3 3 ;u u u e , 0k m k mu u x k m ;
2
11 22 33 1 2 ; 0 .ik i k (44)
The corresponding algebraic invariants of the Cauchy-Green tensor are written in the
form
1 11 22 33 ;I e 2 2 2 2
2 11 22 33 1 / 3 ;I e
3 3 3 3
3 11 22 33 1 / 9I e . (45)
The formulas for invariants (45) allow to write the potential in the simpler form
2 3
4
( ) 3 2 3 2 9 2 3 9 9
3 2 4 3 2 9 9
W A B C
A B C
5 63 4 9 9 1 8 9 9 .A B C A B C (46)
The stresses are evaluated by the formulas
2
11 22 33 3 2 3 2 3 2 9 7A B C
3 4
12 23 319 7 (1 4) ; 0.A B C
Thus, the normal stresses only are nonzero and they contain the linear and nonlinear
summands.
The interdependence between the first invariant of the stress tensor kk and the parame-
ter of the multiaxial tension e has the form
Fig. 18
132
3 2 1 2 3 2kk e
2 3 41/ 3 9 7 9 7 1/ 9 1 108 .A B C e A B C e e (47)
The plots in Figure 19 shows a dependence ( )kk e for the foam and polystyrene and are
evaluated by formula (47). It follows from them that they are similar to the parabola with a
vertex in a positive half of the plane kkOe . The parabola right branch of then passes into
the negative half of the plane. Both plots have “the hump” in the positive branch of the plane.
Fig. 19
A presence of “the hump” testifies that the nonlinear Murnaghan model describes the transi-
tion of the material of the sample-cube from the class of conventional materials into the
class of auxetic materials. The fact is that the sample is compressed for the small values of
uniform tension and in the following increase of the tension strain the sample swells. But this
phenomenon is characteristic for only auxetic materials.
Conclusions.
Conclusion 1. Three nonlinear models are used in the analysis. They describe the
nonlinear Poynting-type effects in conditions of three used above universal deformations and
the moderate strains. This agrees quantitatively with experimental observations of nonlinear
dependences in auxetic materials for the moderate strains.
Conclusion 2. In the case of uniaxial and omniaxial tension, the nonlinear Murnaghan
model describes a transition of the material from the class of conventional materials into the
class of the auxetic materials. This occurs when the material is deformed to the level of
moderate values of the longitudinal stretching. In other words, the shown experiments and
proposed theoretical analysis testify that the standard sample in conditions of the mentioned
universal deformation of uniaxial tension is deformed for small strains as if it is made of the
conventional material (its cross-section is decreased) and with increasing the values of longi-
tudinal stretching to the moderate values the sample cross-section starts to increase, what is
the characteristic just for auxetic materials.
133
РЕЗЮМЕ. Запропоновано теоретичну спробу пояснення ауксетичності матеріалів за допомо-
гою нелінійних моделей пружного деформування у широкому діапазоні значень деформацій від ма-
лого до помірного рівня. Отримано аналітичні вирази, які відповідають трьом видам універсальної
деформації (простий зсув, односторонній розтяг, всесторонній розтяг) і рамках трьох відомих в кла-
сичній теорії нелінійної пружності моделей – двоконстантної неогукової моделі, триконстантної
моделі Муні-Рівліна, п‘ятиконстантної моделі Мурнагана. Найбільш цікавий новий результат полягає
у тому, що як в показаних експериментах, так і в запропонованих теоретичних розрахунках зразок з
пружного матеріалу деформується як традиційний матеріал для малих значень деформацій, тоді як
при збільшеннi деформацій до помірних він деформується як нетрадиційний (ауксетичний) матеріал.
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Поступила 19.11.2018 Утверждена в печать 04.06.2019
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