Elasticity of Auxetic Materials

Abstract

The auxeticity of elastic materials is described and explained by the use of the linear and nonlinear models of elastic deformations for a wide range of strain values up to moderate levels. This chapter consists of three parts – general information on auxetic materials, description of auxetics by the model of the linear theory of elasticity, description of auxetics by the models of the nonlinear theory of elasticity. The analytical expressions are offered 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 Murna-ghan 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.

1. Introduction

To begin with, let us recall the definition of elasticity of deformation of the material. So, the property of elasticity consists in that the body practically simultaneously takes the initial configuration after removing the deformation causes. In other words, if deformations are elastic, then they simultaneously vanish after removing the action of forces, caused the deformations.

This property, as also other properties, though, is displayed seldom in the pure form, that is, it is accompanied in real solid materials by several other properties. But in most cases, elasticity is the main and pre-vailing property.

It is worthy to note at beginning of this chapter that the analysis of auxetic materials as the deforming elastically materials is dominating over other types of deformation (thermoelastic, viscoelastic, elastoplastic, magnetoelastic, etc). Therefore, the theme “Elasticity of Auxetic Materials” is related to the main part of studies of auxetics.

At present, the auxetic materials are thought of as some subclass of nontraditional (nonconventional) ma-terials which are known as metamaterials. The metamaterials include the mechanical metamaterials, which in turn include the auxetic materials. At present, a sufficiently big group of scientists work in the area of auxetic materials. It includes mainly specialists from material science, to the lesser extent from statistical physics, and even to the lesser extent from experimental mechanics. The state-of-the-art in science on auxe tic materials is shown in the monographs [1, 2, 3] and the review articles [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

The auxetic materials were discovered and identified as a novel class of materials about forty years ago. Usually, two publications of Gibson L.J., Ashby M.F. et al [19, 20] are shown as the pioneer ones.

The term “auxetic material” was introduced by Evans in 1991 [21] for a new range of materials, which he defined them “the materials with negative Poisson ratio (NPR)”. This needs some scientific comments relative to the term and definition.

Comment 1 (to the term “auxetic material”). At present, Wikipedia and other sources propose for such materials the name “auxetics”. Both names come from the Greek word $\mathit{\text{αυξητικοζ}}$ (that which tends to increase). But this does not explain why just “auxetic”. The next comment on definition gives some clearness.

Comment 2 (to the definition of auxetic material). Auxetic materials are deformed elastically exhibiting the unconventional property of increasing the cross-section (growing swollen) of cylindrical or prismatic samp-le under uniaxial tension, whereas in the conventional materials this cross-section decreases (grows thin). Just this is reflected in the name “auxetic” and shown in Figure 1 [4].

Figure 1.

Test on uniaxial tension for conventional and non-conventional materials.

The point is that the property of the decrease is described in the linear theory of elasticity by the use of the Poisson ratio as the elastic constant. A change of the decrease of cross-section on the increase of one means a change of positive values of the Poisson ratio on the negative ones.

The presented short information on auxetics shows that their definition is based on the secondary fact – the negativity of the Poisson ratio, which corresponds to the model of the linearly elastic body. The primary fact consists in observation in the standard for mechanics of materials (which does not depend on the model of deformation) experiment of longitudinal tension of a prism when the transverse deformation of the prism is positive (a material as if swells) in contrast to the classical materials, where it is negative.

The adherence of researchers of auxetic materials to the foams can be seen in the often used (described verbally or by the picture) demonstration of auxeticity of the foam as increasing the volume of sample from the foam under tension. It is shown in Figure 2 [[10] (left), [20] (right)].

Figure 2.

Usually used test-demonstration of auxeticity.

These pictures are really very demonstrative because they show two basic features.

Feature 1. The sample length is possibly not sufficient to create the classical conditions of the test on the universal deformation of uniaxial tension-compression.

Feature 2. The longitudinal and transverse strains are seemingly not sufficiently small in this test to be described by the linear theory of elasticity.

Starting with the first works on auxetics, the discussed real materials were the different kinds of foams. It is considered that the first observed auxetic materials were the foams which are characterized by the small value of density and the porous internal structure (see Lakes [24] and Wojciechowski [25]). In the next studies, the new auxetics were revealed, the density of which was also small and which have a porous structure. But it was shown later that small density is not the defining property of auxetics, because the significant part of foams has not the property of auxeticity. The defining characteristics of auxetics are new three mechanical phenomena which will be described below.

The common concept was adopted almost at the initial part of studies that the auxeticity of materials is caused by the internal structure of these materials. This corresponds to the general concept of mechanics (which is clearly shown in mechanics of composite materials) that the specificities of deformation of materi-als can be explained by the existence of some specific internal structure. Only the answer should be found which concrete specificity is characteristic for the auxetic materials. Therefore, an essential part of studies of auxetics consists in the finding of diverse variants of internal structure that are further studied by methods of molecular physics and computational simulations. The most popular is a so-called hexagonal system (it is shown in Figure 3 [12]; left – before stretching, right – after stretching). Just this structure shows the swelling of the sample and is given by different authors to illustrate the auxeticity.

Figure 3.

The most known interpretation of the internal structure of auxetic material.

It should be noted that the mechanics of materials works with the continuum models. This means that any discrete models of the internal structure must be transformed into the continuum one (here the different ways of averaging are usually applied). In mechanics, the internal structure of materials can appear on two different stages of modeling the materials. First, on the stage of changing the discrete structure of a material by the continuous one (that is when the notion of the continuum is introduced according to the principle of continualization). Second, on the stage of modeling the piece-wise inhomogeneous continuum by the homo-geneous continuum (that is when the principle of homogenization is applied). The first stage is usually asso-ciated with methods of molecular physics, whereas the second stage is a standard one in mechanics of composite materials. This is peculiar to all the materials that are studied in mechanics and refers also to the theory of elasticity within the framework of which the elastic deformation of aux-etic materials is studied.

For the presence in the material property of auxeticity, its internal structure has to change under defor mation by the special way exhibiting the unusual (nontraditional) mechanical effects. Note that mechanics of materials studied traditionally first the elastic deformation and this concern both traditional (non-auxe-tic), and nontraditional (auxetic) materials.

As far as the number of known nonauxetic materials exceeds the number of auxetic ones on many or-ders, then the term “unusual effect” is looking appropriate. In contrast to the traditional effects that count tens, the effects of auxeticity are observed as now in the identical mechanical problems in three types of such problems that are realized experimentally and described theoretically. An identity consists in that the samples from material must be compared when the internal structure of a material in cases “auxetic non-auxetic” is differing by the only geometrical shape of pores. This case is shown in Figure 4 for the sample from the polyurethane foam (left – traditional structure, right – auxetic structure) [2].

Figure 4.

Sample from the polyurethane foam (left – traditional structure, right – auxetic structure).

Now, some facts from this theory should be shown concerning the phenomenon of auxeticity. But first three specific appearances of auxeticity must be described and commented on.

2. Three specific properties of auxetic materials

Only one of these specificities is well known – the swelling under the tension of the standard sample (standard mechanical test). This test is described above and shown in Figure 1.

But the fact is known that the auxeticity is generated by the special kind of internal structure of material and appears in three basic mechanical tests on deformation of material

1. Swelling under tension.

2. Hardening under indentation (impact).

3. Synclastic and anticlastic deformation of thin flexible plate.

Test 2 on indentation (statical Hertz problem, problem on hardness by Rockwell-Brinell-Wikkers) and impact (dynamical Hertz problem) shows the effect of hardness of auxetics in the contact zone. Within the framework of the theory of elasticity, this problem is solving numerically 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 [12].

Figure 5.

Test for hardness material (left – traditional structure, right – auxetic structure).

Test 3 on synclastic and anticlastic 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 a plate and directed upward, whereas two other identical forces are applied at the centers of two opposite ends of the plate and directed downward. Within the framework of the theory of flexible elastic plates, this problem is solving numerically with the given exactness. The simple experiment that exhibits the essential difference in deformation of the plate from the traditional and auxetic materials is shown in Figure 6 [4] (left traditional material, right – auxetic material).

Figure 6.

Test on synclastic (left) and anticlastic (right) deformation.

Note that these basic phenomena of deformation of auxetics can be described only in the terms of the theory of elasticity.

3. Some facts from the linear theory of elasticity necessary for describing the auxetic materials

Because the elastic deformation is described in mechanics only by the theory of elasticity, then some facts from this theory should be recalled before the discussion of the specificities of elastic deformation of auxetics. At that, the division of the theory of elasticity on the linear and nonlinear theories should be taken into account. It is important to remember that the linear theory is based on the one (Hookean) model, whereas the nonlinear uses many different models.

3.1 Universal deformations

Universal deformations (uniform deformations, universal states) occupy a special place in the theory of elasticity just owing to their universality [26]. This universality consists in that the theoretically and experi-mentally determining elastic constants of material in samples, in which the universal deformation is created purposely, are valid also for all other deformed states both samples and any different products made of this material. It is considered therefore that the particular importance of universal deformation (their fundamen tality) consists in the possibility to use them in the determination of properties of materials from tests [26, 27, 28, 29, 30, 31]. 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 more of-ten and in detail: simple shear, simple (uniaxial) tension-compression, uniform volume (omniaxial) tension-compression. In the linear theory of elasticity, the experiment with a sample, in which the simple shear is realized, allows determining the elastic shear modulus $\mu$. The experiment with a sample, in which the uniaxial tension is realized, allows determining Young elastic modulus $E$ and Poisson ratio $\nu$. 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, which is valid for only the very small deformations to the mo-dels of non-small (moderate or large) ones, that is, from the linear mechanics of materials to nonlinear me-chanics of materials, the universal states permit to describe theoretically and experimentally many nonlinear phenomena. The history of mechanics testifies to the experimental observation in the XIX century of the non linear effects that arose under the simple shear and were named later by the names of Poynting and Kelvin [27, 28, 29, 30, 31]. After about a hundred years in the XX century, these effects were described theoretically within the framework of the nonlinear Mooney-Rivlin model [31, 32, 33, 34, 35].

The mechanics of composite materials is one more area of application of universal deformations. The mo-del of averaged (effective, reduced) moduli is in this case the simplest and most used model. In the theory of effective moduli, the composite materials of the complex internal structure with internal links are treated usually as homogeneous elastic media. A possibility to create in such media the states with universal deformations was used in the evaluation of effective moduli by different authors and different methods. It was found that it is sufficient for isotropic (granular) composites to study the energy stored in the elementary volumes of composites under only two kinds of universal deformations: simple shear and omniaxial compression. In the case of transversely isotropic (fibrous or layered) composites, the different directions need analysis of universal deformations for each direction separately.

3.2 Classical procedures of estimating the values of elastic moduli in the linear theory of elasticity

Perhaps, the eldest and exhausting procedures are shown in the classical Love’s book [36]. Let us save the Love’s notations and write according to [36] the internal energy of deformation of the linearly elastic isotropic body $W$ in the form

$$W=\lambda {\left({\epsilon }_{\mathit{xx}}+{\epsilon }_{\mathit{yy}}+{\epsilon }_{\mathit{zz}}\right)}^2+2\mu \left({\epsilon }_{\mathit{xx}}^2+{\epsilon }_{\mathit{yy}}^2+{\epsilon }_{\mathit{zz}}^2\right)+\mu \left({\epsilon }_{\mathit{xy}}^2+{\epsilon }_{\mathit{xz}}^2+{\epsilon }_{\mathit{yz}}^2\right),$$E1

where $\lambda ,\mu$ are the Lame moduli, ${\epsilon }_{\mathit{xx}},\dots ,{\epsilon }_{\mathit{yz}}$ are the components of the strain tensor.

The Hooke law has the form

$$\begin{array}{l}{X}_x=\lambda \mathrm{\Delta }+2{\mathit{\mu \epsilon }}_{\mathit{xx}},Y_y=\lambda \mathrm{\Delta }+2{\mathit{\mu \epsilon }}_{\mathit{yy}},Z_z=\lambda \mathrm{\Delta }+2{\mathit{\mu \epsilon }}_{\mathit{zz}},\\ X_y=2{\mathit{\mu \epsilon }}_{\mathit{xy}},Z_x=2{\mathit{\mu \epsilon }}_{\mathit{zx}},Y_z=2{\mathit{\mu \epsilon }}_{\mathit{yz}}.\end{array}$$E2

Here $\mathrm{\Delta }={\epsilon }_{\mathit{xx}}+{\epsilon }_{\mathit{yy}}+{\epsilon }_{\mathit{zz}}$ is the dilatation.

The classical procedure of introducing the Young modulus and Poisson ratio is as follows: the cylinder or prism of any shape is considered, then the axis of the cylinder is chosen in direction $\mathit{Ox}$ and the prism is stretched at the ends by a uniform tension $T$. Because the lateral surface of the prism is assumed to be free of stresses, then the stress state of a prism is uniform and is characterized by only one stress $X_x=T$. In this case, the Hooke law becomes simpler

$$T=\lambda \mathrm{\Delta }+2{\mathit{\mu \epsilon }}_{\mathit{xx}},0=\lambda \mathrm{\Delta }+2{\mathit{\mu \epsilon }}_{\mathit{yy}},0=\lambda \mathrm{\Delta }+2{\mathit{\mu \epsilon }}_{\mathit{zz}}.$$E3

An expression for dilatation follows from equalities (3)$T=\left(3\lambda +2\mu \right)\mathrm{\Delta }\to \mathrm{\Delta }=T/\left(3\lambda +2\mu \right)$.

The substitution of the last expression for dilatation into the first equality (2) gives relations

$$T=\frac{\lambda }{3\lambda +2\mu }T+2{\mathit{\mu \epsilon }}_{\mathit{xx}}\to T=\frac{\mu \left(3\lambda +2\mu \right)}{\lambda +\mu }{\epsilon }_{\mathit{xx}}.$$E4

The expression (4) represents the elementary law $T=E{\epsilon }_{\mathit{xx}}$ of link between tension and deformation of the prism, in which the Young modulus $E$ is used

$$E=\frac{\mu \left(3\lambda +2\mu \right)}{\lambda +\mu }.$$E5

The substitution of expression for dilatation into the second and third equalities (2) gives relations

$$-{\epsilon }_{\mathit{yy}}=-{\epsilon }_{\mathit{zz}}=\frac{\lambda }{2\left(\lambda +\mu \right)}{\epsilon }_{\mathit{xx}},$$E6

which express the classical Poisson law on the transverse compression under the longitudinal extension and permit to introduce of the Poisson ratio

$$\sigma =\frac{-{\epsilon }_{\mathit{yy}}}{{\epsilon }_{\mathit{xx}}}=\frac{-{\epsilon }_{\mathit{zz}}}{{\epsilon }_{\mathit{xx}}}=\frac{\lambda }{2\left(\lambda +\mu \right)}.$$E7

Let us repeat now the procedure associated with introducing the universal (uniform) deformation – the uniform compression. Thus, the body of arbitrary shape is considered, to all points of which the constant pressure $-p$ is applied. In this body, the uniform stress state arises which is characterized by stresses $X_x=Y_y=Z_z=-p,X_y=Y_z=Z_x=0$. The Hooke law becomes simpler

$$-p=\lambda \mathrm{\Delta }+2{\mathit{\mu \epsilon }}_{\mathit{xx}},-p=\lambda \mathrm{\Delta }+2{\mathit{\mu \epsilon }}_{\mathit{yy}},-p=\lambda \mathrm{\Delta }+2{\mathit{\mu \epsilon }}_{\mathit{zz}},{\epsilon }_{\mathit{xy}}={\epsilon }_{\mathit{yz}}={\epsilon }_{\mathit{zx}}=0.$$E8

The relations (8) can be transformed to $-3p=\left(3\lambda +2\mu \right)\left({\epsilon }_{\mathit{xx}}+{\epsilon }_{\mathit{yy}}+{\epsilon }_{\mathit{zz}}\right)\to -p=\left[\lambda +\left(2/3\right)\mu \right]\mathrm{\Delta }$.

In this way, the modulus of compression $k$ is defined

$$k=\lambda +\left(2/3\right)\mu .$$E9

The classical Love’s reasoning, which is repeated in most books on the linear theory of elasticity, is based on the representation of moduli $\lambda ,\mu ,k$ through moduli $E,\sigma$

$$\lambda =\frac{\mathit{E\sigma }}{\left(1+\sigma \right)\left(1-2\sigma \right)},\mu =\frac{E}{2\left(1+\sigma \right)},k=\frac{E}{3\left(1-2\sigma \right)}.$$E10

The formulas (10) are commented in ([36], p. 104) as follows: “If $\sigma$ were $>1/2$, $k$ would be negative, or the material expands under pressure. If $\sigma$ were $<-1$, $\mu$ 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 $\sigma$ are not excluded by the condition of stability, but such values have not been found for any isotro-pic material.”

Because the comments of negativity of Poisson ratio is found in the books on the theory of elasticity very seldom, therefore a few sentences from Lurie’s book ([29], p. 117) are worthy to be cited: “A tension of the rod with negative $\nu$ (but the more than $-1$) would be accompanied by increasing of transverse sizes. Ener-getically, the existence of such elastic materials is not excluded.” “In hypothetic material with $\nu <-1$, the hy-drostatic compression of the cube would be accompanied by increasing its volume.”

Note that the Poisson ratio is denoted in the theory of elasticity by $\sigma$ “sigma” and $\nu$ “nu”. Love uses $\sigma$, whereas Lurie uses $\nu$.

It should be also noted that not all authors of books on the linear isotropic theory of elasticity discuss the restrictions on changing the Poisson ratio (for example, Germain, Nowacki, Hahn do not made this in their well-known books [37, 38, 39]). The constitutive relations and classical restrictions on elastic constants are discussed in the most comprehensive and modern treatment of the theory of elasticity [28] (Subsection 3.3 “Constitutive relations”).

But in some books, the discussion is presented and all authors start with one and the same postulate: in the procedure of restrictions in changing the Poisson ratio, the primary requirement is a positiveness of internal energy $W$(1). The representation of energy can be different for different elastic moduli. For example, Leibensohn [40], Love [36], Lurie [29] choose the pair $\lambda ,\mu$ and use the representation (1). Landau and Lifshits [41] use the pair $k,\mu$. In all the cases, $W$ has a form of a quadratic function with coefficients composed of elastic moduli.

Thus, in most cases, the expression (1) is analyzed. It is assumed that the sufficient and being in line with experimental observations condition is the condition of positiveness of Lame moduli

$$\lambda >0,\mu >0.$$E11

Further, the formulas (10) are considered, in which without controversy the Young modulus is assumed positive $E>0$. Then positiveness of expressions $1+\sigma >0,1-2\sigma >0$ provides validity of formula (11), from which the well-known restriction on the Poisson ratio follows

$$-1<\sigma <1/2.$$E12

Let us recall that all the elastic moduli in the classical linear isotropic theory of elasticity are always posi-tive. The obvious contradiction between the assumption of negativity of the Poisson ratio and the primary statement on the positivity of Lame moduli (11) in condition when the Poisson ratio is defined by formula (7) is commented in the classical theory of elasticity anybody. To all appearances, this situation is occurred owing to the incredibility of negative values if only one of the elastic moduli $\lambda ,\mu ,E,k$.

Note finally that two experimental approaches to determine the value of Poisson ratio for concrete material are used at present time ([27], subsections 2.18, 3.27, 3.28). The first approach is the older one. It is based on the experimental determination of Young, shear, and compression moduli and subsequent calculation of Poisson ratio by formulas (10)$\sigma =\left(E/2\mu \right)-1,\sigma =\left(1/2\right)\left[\left(E/3k\right)-1\right]$. Here, the problem of the exactness of calculation arises. Let us cite Bell’s book ([27], subsection 3.28): “Remind of the Grüneisen’s conclusion that the errors of $\pm 1\%$ in values $E$ and $\mu$ result in the error of 10% in the value of Poisson ratio.” Therefore, the second approach seems to be more preferable. It is associated with Kirchhoff’s experiments (1859), in which the Poisson ratio is determined from the direct experiment on simultaneous bending and torsion.

Let us recall that the primary phenomenon in the determination of the Poisson ratio is the contraction of a sample (transverse deformation of a sample) under its elongation (its longitudinal deformation).

3.3 Refinement of procedures of estimating the values of elastic moduli

Let us save the initial postulate that the primary requirement is the positivity of internal energy $W$(1) and reject the sufficient (and not necessary) condition of positivity of $W$ when the positivity of Lame moduli $\lambda ,\mu$ is assumed and suppose the general condition of positivity of $W$.

Because the Lame modulus $\mu$ has a physical sense of the shear modulus and until now the facts of observation its negativity (a shear in the direction opposite to the direction of shear force) are not reported, then we can agree to its positivity. This condition of positivity can be also substantiated theoretically based on ana-lysis of universal deformation of simple shear. To describe the simple shear, the coordinate plane (for example, $\mathit{xOy}$) should be chosen and only one non-zero component $u_{x,y}$ of the displacement gradient should be given. This can be commented geometrically as deformation of the elementary rectangle $\mathit{\text{ABCD}}$ with sides $\mathit{dx},\mathit{dy}$ parallel to the coordinate axes into the parallelogram $A{B}^{\prime }{C}^{\prime }D$, which results from the longitudinal shift of the rectangle side $\mathit{BC}$. Then the shear angle $\measuredangle \mathit{BA}{B}^{\prime }=\gamma$ is linked with $u_{x,y}$ in a next way $u_{x,y}=tan\gamma =\tau$ and ${\epsilon }_{\mathit{xy}}=\left(1/2\right)\tau$. The Hooke law becomes the simplest form ${\sigma }_{\mathit{xy}}=2{\mathit{\mu \epsilon }}_{\mathit{xy}}$ and the corresponding representation of internal energy is as follows $W=\left(1/2\right){\mathit{\mu \tau }}^2$. Then the positivity of shear modulus (14) follows from the positivity of energy $W$.

Now, the next refinement can be formulated.

Refinement 1. The Lame modulus $\lambda$ can be negative if the Poisson ratio $\sigma =\lambda /\left[2\left(\lambda +\mu \right)\right]$ can be assumed possible negative.

Refinement 2. If the Poisson ratio $\sigma$ is assumed to be possible negative and the shear modulus $\mu$ is positive, then according to definition (7) the negative Lame modulus $\lambda$ can not exceed by its absolute value the shear modulus

$$\left|\lambda \right|<\mu .$$E13

Let us return to the primary definition of the Poisson ratio (7), which is found from the solution of the problem of unilateral tension. In this case, the internal energy has the form

$$\begin{array}{l}W=\lambda {\left({\epsilon }_{11}+{\epsilon }_{22}+{\epsilon }_{33}\right)}^2+2\mu \left({\epsilon }_{11}^2+{\epsilon }_{22}^2+{\epsilon }_{33}^2\right)=\lambda {\left({\epsilon }_{11}+2{\mathit{\sigma \epsilon }}_{11}\right)}^2+2\mu \left({\epsilon }_{11}^2+{\left({\mathit{\sigma \epsilon }}_{11}\right)}^2+{\left({\mathit{\sigma \epsilon }}_{11}\right)}^2\right)\\ =\left[\lambda {\left(1+2\sigma \right)}^2+2\mu \left(1+2{\sigma }^2\right)\right]{{\epsilon }_{11}}^2>0.\end{array}$$E14

Then $\lambda +\left[2\left(1+2{\sigma }^2\right)/{\left(1+2\sigma \right)}^2\right]\mu >0.$ permits to the formulation of some new refinements.

Refinement 3. If the Poisson ratio $\sigma$ is assumed to be possible negative and the shear modulus $\mu$ is positive, then the condition of positivity of internal energy admits arbitrary negative values of the Poisson ratio.

(because the coefficient ahead of $\mu$ is always positive). The case $1+2\sigma =0\to \sigma =-0,5$ is the peculiar one – the value of modulus $\lambda$ is practically not restricted at its neighborhood.

Refinement 4. The Lame modulus $\lambda$ is already restricted from below according to (14), but also the additional condition (16) exists

$$\left|\lambda \right|<\left[2\left(1+2{\sigma }^2\right)/{\left(1+2\sigma \right)}^2\right]\mu .$$E15

The condition (15) is less strong: the coefficient ahead of $\mu$ exceeds 1 for all negative $\sigma$ (in condition (13), the coefficient ahead of $\mu$ is equal to 1). Therefore, the condition (13) remains.

Let us turn to formula (9), which expresses the compression modulus $k$ through the Lame moduli $\lambda ,\mu$. It follows from (9) that the modulus $k$ will be negative if only the negative Lame modulus $\lambda$ exceeds $\left(2/3\right)\mu$ by absolute value

$$k=\lambda +\left(2/3\right)\mu <0\to \left|\lambda \right|>\left(2/3\right)\mu =0,667\mu .$$E16

Comparison with restrictions (13) and (15) on the absolute values of negative Lame modulus $\lambda$ in the case of negative values of Poisson ratio $\sigma$ shows that (16) does not conflict with (13) and (15).

Refinement 5. If the Poisson ratio $\sigma$ is assumed to be possible negative and the shear modulus $\mu$ is positive, then the compression modulus $k$ can be negative.

The situation with refinements becomes clearer if the moduli $\lambda ,E$ and $k$ are written through $\mu$ and $\sigma$

$$\lambda =\frac{2\sigma }{1-2\sigma }\mu ,E=2\left(1+\sigma \right)\mu ,k=\frac{2}{3}\frac{1+\sigma }{1-2\sigma }\mu .$$E17

A few statements can be formulated at the end of this subchapter.

Statement 1. The classical restrictions of positivity of the elastic moduli in the isotropic theory of elasticity should be refined for auxetic materials: most elastic moduli can be negative.

Statement 2. Seemingly, the auxetics should be defined by the primary physical phenomenon of positivity of transverse deformation of a prism, which is observed in the standard in mechanics of materials experiment of longitudinal tension of a prism. In this case, the auxetics will be associated not only with the isotropic elastic materials.

Statement 3. In the case of auxetic materials, the Lame modulus $\lambda$ is always negative and the Young $E$ and compression $k$ moduli are negative when the negative Poisson ratio is less than −1: $\sigma <-1$.

Statement 4. When the problems of the linear isotropic theory of elasticity being studied for auxetic materials, then at least two elastic moduli for these materials should be determined from the direct experiments (unilateral tension, omnilateral compression, simple shear, torsion).

4. Specificities of describing the auxetic materials by the nonlinear theory of elasticity

4.1 Essentials of nonlinear theory of elasticity

While being studied the auxetics from the position of the nonlinear theory of elasticity, some essential differences between the linear and nonlinear descriptions should be taken into account. Therefore, the basic notions of the nonlinear approach seem to be worthy to show here very shortly [31, 34, 35, 42, 43].

A body is termed some area $V$ of 3D space ${\mathrm{R}}^3,$ in each point of which the density of mass $\rho$ 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 $\left\{x_k\right\}$ or Eulerian $\left\{X_k\right\}$ coordinate systems can be given in ${\mathrm{R}}^3$. In the theory of deformation of a body as a change of its initial shape, the notions are utilized that are associated with a geometry of body (kinematic notions) and with the forces acting on the body from outside and inside (kinetic notions). The notions of the configuration $\chi$, the vector of displacement $\overrightarrow{u}=\left\{u_k\right\}$, the principal extensions ${\lambda }_k$, the strain tensor ${\epsilon }_{\mathit{ik}}$ are referred to as 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 the body at a moment $t$ is called the actual one, whereas the configuration of the body at arbitrarily chosen initial moment $t^o$ is called the reference one. The coordinates of the body point before deformation are denoted by $x_k$. It is assumed that after deformation this point is displaced on the va-lue $u_k\left(x_1,x_2,x_3,t\right).$ Then the vector with components $u_k$ is called the displacement vector and the coordinates of the point after deformation are presented in the form ${\xi }_k=x_k+u_k\left(x_1,x_2,x_3,t\right)$. The frequently used Cauchy-Green strain tensor is given by the known displacement vector $\overrightarrow{u}\left(x_k,t\right)$ in the Lagrangian coordinates $\left\{x_k\right\}$ and the reference configuration

$${\epsilon }_{\mathit{nm}}\left(x_k,t\right)=\left(1/2\right)\left(u_{n,m}+u_{m,n}+u_{n,i}{u}_{m,i}\right).$$E18

As a result, the deformation of the body is given by nine components of displacement gradients $u_{i,k}$. Such a description of deformation is used in most models of the nonlinear theory of elasticity. But the process of deformation can be described also by other parameters of the geometry change of the body. It seems meaning ful to use often the first three algebraic invariants of tensor (18)$A_1={\epsilon }_{\mathit{mn}}{\delta }_{\mathit{mn}}$, $A_2=\left(1/2\right)\left[{\left({\epsilon }_{\mathit{mn}}{\delta }_{\mathit{mn}}\right)}^2-{\epsilon }_{\mathit{ik}}{\epsilon }_{\mathit{ik}}\right]$, $A_3=det{\epsilon }_{\mathit{mn}}$, which can be rewritten through the principal values of tensor (18)${\epsilon }_k$ by the formulas $A_1={\epsilon }_1+{\epsilon }_2+{\epsilon }_3$, $A_2={\epsilon }_1{\epsilon }_2+{\epsilon }_1{\epsilon }_3+{\epsilon }_2{\epsilon }_3$, $A_3={\epsilon }_1{\epsilon }_2{\epsilon }_3.$ The often used invariants $I_1,I_2,I_3$ of tensor ${\epsilon }_{\mathit{ik}}$ are linked with the algebraic invariants of the same tensor by relations

$$\begin{array}{c}{I}_1=3+2{\epsilon }_{\mathit{nn}}=3+2A_1,I_2=3+4{\epsilon }_{\mathit{nn}}+2\left({\epsilon }_{\mathit{nn}}{\epsilon }_{\mathit{mm}}-{\epsilon }_{\mathit{nm}}{\epsilon }_{\mathit{mn}}\right)=3+4A_1+2\left(A_1^2-A_2\right),\\ I_3=det‖{\delta }_{\mathit{pq}}+2{\epsilon }_{\mathit{pq}}‖=1+2A_1+2\left(A_1^2-A_2\right)+\left(4/3\right)\left(2A_3-3A_2{A}_1+A_1^3\right).\end{array}$$

In several models of nonlinear deformation of materials, the elongation coefficients (principal extensions) defined as a change of length of the conditional linear elements (the infinitesimal segments that are directed arbitrarily) are used

$${\lambda }_k=\sqrt{1+2{\epsilon }_k}.$$E19

A simpler formula ${\lambda }_k-1\approx {\epsilon }_k$ is valid for the case of linear theory. Additionally to three parameters (19), three 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 displacement gradient

$$\mathbf{F}=\left[\begin{array}{ccc}1+u_{1,1}& u_{1,2}& u_{1,3}\\ u_{2,1}& 1+u_{2,2}& u_{2,3}\\ u_{3,1}& u_{3,2}& 1+u_{3,3}\end{array}\right]$$

and notation of the left Cauchy-Green strain tensor B = F F^T associated with it. The most used are two ten-sors of internal stresses: the symmetric Cauchy-Lagrange tensor ${\sigma }_{\mathit{ik}},$ which is measured on the unit of area of the deformed body, and the nonsymmetric Kirchhoff tensor $t_{\mathit{ik}}$, which is measured on the unit area of the undeformed body.

4.2 Universal deformation of simple shear in the nonlinear approach

The simple shear is described in subsubsection 3.3, where the basic formula $u_{1,2}=tan\gamma =\tau >0$ is shown.

In the linear theory, the shear angle is assumed to be small and then $\gamma \approx tan\gamma =\tau$. The nonlinear app-roach introduces some complications. The Cauchy-Green strain tensor is characterized by only three non-zero components

$$\begin{array}{l}{\epsilon }_{11}=\left(1/2\right)\left(u_{1,1}+u_{1,1}+u_{1,k}{u}_{1,k}\right)=\left(1/2\right)\left(u_{1,2}{u}_{1,2}+u_{1,3}{u}_{1,3}\right)={\tau }^2;{\epsilon }_{12}={\epsilon }_{21}\\ =\left(1/2\right)\left(u_{1,2}+u_{2,1}+u_{1,k}{u}_{2,k}\right)=\left(1/2\right)\tau ,\end{array}$$E20

The principal extensions are written through the shear angle by formulas ${\lambda }_1=1,{\lambda }_2={\lambda }_3=\tau$.

4.3 Universal deformation of uniaxial tension in the nonlinear approach

This kind of deformation is also described above. It is characterized in the nonlinear approach by only one nonzero component ${\sigma }_{11}$ of the stress tensor and two nonzero components ${\epsilon }_{11},{\epsilon }_{22}={\epsilon }_{33}$ of the strain tensor (or two principal extensions ${\lambda }_1,{\lambda }_2={\lambda }_3$).

4.4 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 to each other ${\sigma }_{11}={\sigma }_{22}={\sigma }_{33}$, and the shear stresses ${\sigma }_{\mathit{ik}}\left(i\ne k\right)$ are absent. This type of deformation is defined as follows

$$u_{1,1}=u_{2,2}=u_{3,3}=\epsilon >0,u_{1,1}+u_{2,2}+u_{3,3}=3\epsilon =e,u_{k,m}=\left(\partial u_k/\partial x_m\right)=0\left(k\ne m\right).$$E21

The Cauchy-Green strain tensor is simplified ${\epsilon }_{11}={\epsilon }_{22}={\epsilon }_{33}=\epsilon +\left(1/2\right){\epsilon }^2,{\epsilon }_{\mathit{ik}}=0\left(i\ne k\right)$ and the algebraic invariants are written in the form

$$I_1={\epsilon }_{11}+{\epsilon }_{22}+{\epsilon }_{33}=e,I_2={\left({\epsilon }_{11}\right)}^2+{\left({\epsilon }_{22}\right)}^2+{\left({\epsilon }_{33}\right)}^2,I_3={\left({\epsilon }_{11}\right)}^3+{\left({\epsilon }_{22}\right)}^3+{\left({\epsilon }_{33}\right)}^3.$$E22

The principal extensions are equal to each other

$${\lambda }_1={\lambda }_2={\lambda }_3.$$E23

4.5 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 sub-stantiated physically. Model 1 is chosen as the simplest one. Model 2 is well-working for the not-small (large or finite) deformations. Model 3 belongs to the most used in the nonlinear mechanics of materials.

4.5.1 Two-constant Neo-Hookean model (model 1)

The elastic potential of this model is defined as follows [31, 34, 35, 42, 43]

$$W=C_1\left({\overline{I}}_1-3\right)+D_1{\left(J-1\right)}^2,{\overline{I}}_1=J^{-2/3}{I}_1,J=det{u}_{i,k},$$

$$W\left({\lambda }_1,{\lambda }_2,{\lambda }_3\right)=C_1\left[{\left({\lambda }_1{\lambda }_2{\lambda }_3\right)}^{-2/3}\left({\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2\right)-3\right]+D_1{\left({\lambda }_1{\lambda }_2{\lambda }_3-1\right)}^2.$$E24

Here the elastic constants of the model are linked with the classical elastic constants by relation $2C_1=\mu ;2D_1=k$.

The constitutive equations have the form

$${\sigma }_{\mathit{nm}}=2C_1{J}^{-5/3}\left[B_{\mathit{nm}}-\left(1/3\right)I_1{\delta }_{\mathit{nm}}\right]+2D_1\left(J-1\right){\delta }_{\mathit{nm}}$$E25

$${\sigma }_{\mathit{nn}}=2C_1{J}^{-5/3}\left({\lambda }_n-\left(1/3\right)I_1\right)+2D_1\left(J-1\right).$$

It is considered that this model describes well the deformation of rubber under the principal extensions up to 20% from the initial state. Since these extensions are linked with the principal values of the strain ten-sor by relation ${\lambda }_k=\sqrt{1+2{\epsilon }_k}$, then it is assumed ${\lambda }_k-1\approx {\epsilon }_{\mathit{kk}}$ approximately with exactness to $\le 1\%$ 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.

4.5.2 Three-constant Mooney-Rivlin model (model 2)

The elastic potential of the Mooney -Rivlin model is defined as follows [31, 32, 33, 34, 35, 42, 43]

$$W=C_{10}\left({\overline{I}}_1-3\right)+C_{01}\left({\overline{I}}_2-3\right)+D_1{\left(J-1\right)}^2,{\overline{I}}_2=J^{-4/3}{I}_2,$$E26

$$\begin{array}{l}W\left({\lambda }_1,{\lambda }_2,{\lambda }_3\right)=C_{10}\left[{\left({\lambda }_1{\lambda }_2{\lambda }_3\right)}^{-2/3}\left({\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2\right)-3\right]\\ +C_{01}\left[{\left({\lambda }_1{\lambda }_2{\lambda }_3\right)}^{-4/3}\left({\lambda }_1^2{\lambda }_2^2+{\lambda }_1^2{\lambda }_2^2+{\lambda }_2^2{\lambda }_3^2\right)-3\right]+D_1{\left({\lambda }_1{\lambda }_2{\lambda }_3-1\right)}^2,\end{array}$$

where the elastic constants are linked with the classical constants by relations $2\left(C_{10}+C_{01}\right)=\mu ;2D_1=k$.

The stresses are determined by formulas

$$\begin{array}{c}\sigma =2J^{-5/3}\left(C_{10}+C_{01}{J}^{-2/3}{I}_1\right)B-2J^{-7/3}{C}_{01}\mathit{BB}\\ +\left[2D_1\left(J-1\right)-\left(2/3\right)J^{-5/3}\left(C_{10}{I}_1+2C_{01}{J}^{-2/3}{I}_2\right)\right]1,\end{array}$$E27

$$\begin{array}{l}{\sigma }_{\mathit{kk}}={\lambda }_k\frac{\partial W}{\partial {\lambda }_k}\\ =2C_{10}{\left({\lambda }_1{\lambda }_2{\lambda }_3\right)}^{-5/3}\left[{\lambda }_k^2-\left(1/3\right)\left({\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2\right)\right]\\ +2C_{01}{\left({\lambda }_1{\lambda }_2{\lambda }_3\right)}^{-7/3}\left[{\lambda }_k^2\left({\lambda }_n^2+{\lambda }_m^2\right)\left.-\left(2/3\right){\lambda }_k\left({\lambda }_1^2{\lambda }_2^2+{\lambda }_1^2{\lambda }_2^2+{\lambda }_2^2{\lambda }_3^2\right)\right]+D_1\left({\lambda }_1{\lambda }_2{\lambda }_3-1\right).\right.\end{array}$$E28

Here the indexes $\mathit{knm}$ form the cyclic permutation from numbers 123.

The Mooney-Rivlin model is the classical one. This can be seen from the next historical information.

Information. An effect of nonlinear dependence of decreasing the shear stresses when the torsion angle (de-formation) to the level of nonsmall values is called “the Poynting effect” owing to his publication of 1909, where this effect was described. At that, Poynting does not mention the results of Coloumb (1784), Wert-heim (1857), Kelvin (1865), Bauschinger (1881), Tomlinson (1883), where this effect was also described in one way or another. But only within the framework of finite elastic deformations, which was developed in 20 century, this effect was satisfactorily explained by Rivlin in 1951. He used the model of nonlinear defor-mation which now is termed “the Mooney-Rivlin model”.

4.5.3 Five-constant Murnaghan model (model 3)

The elastic potential in the Murnaghan model has the form [31, 34, 35, 42, 43, 44, 45]

$$W\left({\epsilon }_{\mathit{ik}}\right)=\left(1/2\right)\lambda {\left({\epsilon }_{\mathit{mm}}\right)}^2+\mu {\left({\epsilon }_{\mathit{ik}}\right)}^2+\left(1/3\right)A{\epsilon }_{\mathit{ik}}{\epsilon }_{\mathit{im}}{\epsilon }_{\mathit{km}}+B{\left({\epsilon }_{\mathit{ik}}\right)}^2{\epsilon }_{\mathit{mm}}+\left(1/3\right)C{\left({\epsilon }_{\mathit{mm}}\right)}^3,$$E29

$$W\left(I_1,I_2,I_3\right)=\left(1/2\right)\lambda I_1^2+\mu I_2+\left(1/3\right){\mathit{AI}}_3+{\mathit{BI}}_1{I}_2+\left(1/3\right)I_1^3.$$

The Cauchy-Green strain tensor ${\epsilon }_{\mathit{ik}}$ and five elastic constants (two Lame elastic constants $\lambda ,\mu$ 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 ma-terials. It takes into account all the quadratic and cubic summands from the expansion of the internal energy and describes the deformation of a big class of engineering and other materials. If to unite the data on the constants of Murnaghan model, shown in books [21, 42, 44], then the sufficiently full information can be ob-tained on many tens of materials.

5. Description of deformations of the auxetic materials by the models 1–3

5.1 Universal deformation of simple shear

This kind of deformation of the auxetics needs some preliminary discussion. First, mechanics distinguishes the simple and pure shears. The state of such deformations is standard in the test for the determination of the shear modulus. Second, it is a common position in mechanics that this modulus is always positive. This means that new effects relative to auxetic materials will most likely not be found. Third, owing to the written above comments, the one only positive result can be reached: the degree of the description of the classical nonlinear effects the Poynting and Kelvin effects – can be considered for the chosen three nonlinear models.

The following materials are used in the numerical evaluations below (elastic constants are shown): 1. Rubber - $\mu =20\mathrm{MPa}$, $k=2.0\mathrm{GPa}$. 2. Foam - $\lambda =0.58\cdot {10}^9,\mu =0.39\cdot {10}^9,$$k=0.84\cdot {10}^9$. 3. Foam - $\lambda =0.58\cdot {10}^9,\mu =0.39\cdot {10}^9$, $A=-1.0\cdot {10}^{10}$, $B=-0.9\cdot {10}^{10}$, $C=-1.1\cdot {10}^{10}$. 4. Polystyrene - $\lambda =3.7\cdot {10}^9$, $\mu =1.14\cdot {10}^9$, $A=-1.1\cdot {10}^{10}$, $B=-0.79\cdot {10}^{10}$, $C=-0.98\cdot {10}^{10}$.

5.1.1 Simple shear in model 1

In this case $J={\left(1+\tau \right)}^2$, $I_1=1+2{\tau }^2$. Then expressions for displacement gradients F and components of tensor В are simplified

$$\mathbf{F}=\left[\begin{array}{ccc}1& \tau & \tau \\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\mathsf{В}=\left[\begin{array}{ccc}1+2{\tau }^2& \tau & \tau \\ \tau & 1& 0\\ \tau & 0& 1\end{array}\right].$$

As a result, the components of stress tensor have the form

$$\begin{array}{l}{\sigma }_{12}={\sigma }_{21}={\sigma }_{13}={\sigma }_{31}=2C_1{\left(1+\tau \right)}^{-10/3}\tau ,{\sigma }_{32}={\sigma }_{23}=0,\\ {\sigma }_{11}=\left(8/3\right)C_1{\left(1+\tau \right)}^{-10/3}\left(\tau -1\right)\tau +2D_1\tau \left(\tau +2\right),\\ {\sigma }_{22}={\sigma }_{33}=-\left(4/3\right)C_1{\left(1+\tau \right)}^{-10/3}\left(1+2\tau \right)\tau +2D_1\tau \left(\tau +2\right).\end{array}$$E30

The formulas (30) show that the Poynting effect (when the values of shear angle increase 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 Eq. (30) demonstrates just this nonlinear dependence for the moderate values of shear angle.

Figure 7 shows the dependence of the shear stress on the shear angle ${\sigma }_{12}\sim \tau$ for the silicon rubber (here and in all next plots, stress is measured by MPa).

Figure 7.

Dependence of the shear stress on the shear angle ${\sigma }_{12}\sim \tau$.

5.1.2 Simple shear in model 2

The expressions for gradient F and components of tensor В are the same as for the Neo-Hookean model. As a result, the expressions from formula (30) are simplified ${\lambda }_1=1,{\lambda }_2={\lambda }_3=1+\tau$, $J={\left(1+\tau \right)}^2$, $I_1=1+2{\tau }^2$, $I_2={\left(1+\tau \right)}^2\left[2+{\left(1+\tau \right)}^2\right]$ and components of the stress tensor have the form

$${\sigma }_{12}={\sigma }_{21}=2C_{10}{\left(1+\tau \right)}^{-10/3}\tau -2C_{01}{\left(1+\tau \right)}^{-14/3}\left(1+4\tau \right)\tau ,$$E31

$${\sigma }_{23}={\sigma }_{32}=-2C_{01}{\left(1+\tau \right)}^{-14/3}{\tau }^2,$$E32

$$\begin{array}{c}{\sigma }_{11}=2C_{10}{\left(1+\tau \right)}^{-10/3}\left(4/3\right)\left(1+\tau +2{\tau }^2\right)+2D_1\tau \left(1+2\tau \right)+\\ +2C_{01}{\left(1+\tau \right)}^{-14/3}\left(4/3\right)\left(3+5\tau +5{\tau }^2+4{\tau }^3-2{\tau }^4\right),\end{array}$$E33

$$\begin{array}{l}{\sigma }_{22}={\sigma }_{33}\\ =2C_1{\left(1+\tau \right)}^{-2}\left[{\left(1+\tau \right)}^{-4/3}+\right.\left.\left(1+2{\tau }^2\right)\left(1-{\left(1+\tau \right)}^{-4/3}\right)-1\right]+2D_1\tau \left(1+2\tau \right).\end{array}$$E34

Thus, the Mooney-Rivlin model (that is, more complicated as compared with the Neo-Hookean model) describes the more complicated stress state, which is characterized by six components of the stress tensor. This model describes well-known nonlinear effects. The Poynting effect follows from the representation of the shear stresses by formula (31). The Kelvin effect follows from formulas (33) and (34).

Also, formula (32) describes one more nonlinear effect: an initiation of shear stresses ${\sigma }_{23}={\sigma }_{32}$. Figure 8 shows the nonlinear dependence of shear stress ${\sigma }_{12}$ on the shear strain $\tau$, that is built for the silicon rubber. Comparison 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.

Figure 8.

Dependence of shear stress ${\sigma }_{12}$ on the shear strain $\tau$.

5.1.3 Simple shear in model 3

The Cauchy-Green strain tensor is characterized by three components

$${\epsilon }_{22}=\left(1/2\right)\left(u_{2,2}+u_{2,2}+u_{k,2}{u}_{k,2}\right)=\left(1/2\right){\tau }^2,$$E35

$${\epsilon }_{12}={\epsilon }_{21}=\left(1/2\right)\left(u_{1,2}+u_{2,1}+u_{k,1}{u}_{k,2}\right)=\left(1/2\right)\tau .$$E36

To calculate the stresses, it is necessary to write the potential (29) concerning the formulas (35) and (36)

$$\begin{array}{l}W\left({\epsilon }_{\mathit{ik}}\right)=\left(1/2\right)\lambda {\left({\epsilon }_{22}\right)}^2+\mu \left[{\left({\epsilon }_{22}\right)}^2+{\left({\epsilon }_{12}\right)}^2+{\left({\epsilon }_{21}\right)}^2\right]\\ +\left(1/3\right)A\left[{\epsilon }_{22}\left({\epsilon }_{12}{\epsilon }_{12}+{\epsilon }_{21}{\epsilon }_{21}+{\epsilon }_{12}{\epsilon }_{21}\right)+{\left({\epsilon }_{22}\right)}^3\right]\\ +B\left[{\left({\epsilon }_{22}\right)}^2+{\left({\epsilon }_{12}\right)}^2+{\left({\epsilon }_{21}\right)}^2\right]{\epsilon }_{22}+\left(1/3\right)C{\left({\epsilon }_{22}\right)}^3,\end{array}$$E37

$$W\left(\tau \right)=\left(1/2\right){\mathit{\mu \tau }}^2+\left(1/8\right)\left[\left(\lambda +2\mu \right)+A+B\right]{\tau }^4+\left(1/24\right)\left[A+3B+C\right]{\tau }^6$$E38

The Lagrange stress tensor is determined by the formula ${\sigma }_{\mathit{ik}}\left(x_n,t\right)=\partial W/\partial {\epsilon }_{\mathit{ik}}$ and has two nonlinear com-ponents

$$\begin{array}{l}{\sigma }_{22}=\left(\lambda +2\mu \right){\epsilon }_{22}+A\left[{\left({\epsilon }_{22}\right)}^2+\left(1/3\right)\left({\epsilon }_{12}{\epsilon }_{12}+{\epsilon }_{21}{\epsilon }_{21}+{\epsilon }_{12}{\epsilon }_{21}\right)\right]\\ +B\left[3{\left({\epsilon }_{22}\right)}^2+{\left({\epsilon }_{12}\right)}^2+{\left({\epsilon }_{21}\right)}^2\right]+C{\left({\epsilon }_{22}\right)}^2\\ =\left(1/4\right)\left[2\left(\lambda +2\mu \right)\right.+\left.\left(A+2B\right)\right]{\tau }^2+\left(1/4\right)\left(A+3B+C\right){\tau }^4,\end{array}$$E39

$${\sigma }_{12}={\sigma }_{21}=2{\mathit{\mu \epsilon }}_{12}+\left[\left(1/3\right)A\left({\epsilon }_{12}+{\epsilon }_{21}\right)+2B{\epsilon }_{12}\right]{\epsilon }_{22}=\mathit{\mu \tau }+\left(1/6\right)\left(A+3B\right){\tau }^3.$$E40

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 in the nonlinear description of deformation. To build the plots of dependence (40) choose two nonstandard for the Murnaghan model materials – foam and polystyrene – which can experience not only the small by values strains but also the moderate ones. Figures 9 and 10 show the dependence of the shear stress ${\sigma }_{12}$ on the shear angle $\tau$ for the foam and polystyrene.

Figure 9.

Dependence of the shear stress on the shear angle (foam).

Figure 10.

Dependence of the shear stress on the shear angle (polystyrene).

The dependences ${\sigma }_{12}\sim \tau$ for models 1–3 can be commented in the following way: these models describe well 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 (and based on them numerical) dependences (for example, [46, 47, 48]). Also, some conclusions to dependence ${\sigma }_{12}\sim \tau$ for models 1–3 can be formulated: the developed in mechanics of materials nonlinear models of deformation of elastic materials can be recommended for the description of auxetic materials.

5.2 Universal deformation of uniaxial tension

This kind of deformation is fundamental for the auxetics because just in tests on the uniaxial tension-compression the phenomenon of auxeticity was first observed.

5.2.1 Uniaxial tension in model 1

The formulas for the principal extensions ${\lambda }_2={\lambda }_3$, $J={\lambda }_1{\lambda }_2^2$, $I_1={\lambda }_1^2+2{\lambda }_2^2$ are valid in this model and the normal stresses (the shear stresses are absent in this state of deformation) are given by the formulas

$${\sigma }_{11}=\left(2/3\right)\mu {\left({\lambda }_1{\lambda }_2^2\right)}^{-5/3}\left({\lambda }_1^2-{\lambda }_2^2\right)+k\left({\lambda }_1{\lambda }_2^2-1\right),$$E41

$${\sigma }_{22}={\sigma }_{33}=-\left(1/3\right)\mu {\left({\lambda }_1{\lambda }_2^2\right)}^{-5/3}\left({\lambda }_1^2-{\lambda }_2^2\right)+k\left({\lambda }_1{\lambda }_2^2-1\right).$$E42

Note that the stresses are depending in model 1 on two principal extensions – longitudinal and transverse.

If to assume that all three normal stresses on the lateral surface of the sample are absent (the surface is free of stresses), then

$${\sigma }_{11}=3k\left({\lambda }_1{\lambda }_2^2-1\right).$$E43

It follows from (43) 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.

Figure 11 shows the dependence of the longitudinal stress on principal extensions and is built for the rubber with allowance for that the value $\left(\mu /3k\right)=0,00334$ is very small compared to the unit (the bulk mo-dulus is essentially more of the shear one). Then the dependence is valid

Figure 11.

Dependence of the longitudinal stress on the principal extensions.

$${\epsilon }_{22}=\left(1/2\right){\left(1+2{\epsilon }_{11}\right)}^{-2}-1/2.$$E44

Figure 12 corresponds to formulas (41) and (42). 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. Figure 12 shows also that with an increase of extension ${\lambda }_1$ the increase of extension ${\lambda }_2$ slows.

Figure 12.

Dependence of the longitudinal principal extension on the transverse principal extension.

It looks, in this case, to be illogical to neglect the first summand in (41) and (42). Note here that the ratio $\left({\lambda }_2/{\lambda }_1\right)$ corresponds in the linear theory to the Poisson’s ratio.

5.2.2 Uniaxial tension in model 2

The uniaxial tension in direction of the abscissa axis is characterized by parameters: ${\lambda }_2={\lambda }_3$, $J={\lambda }_1{\lambda }_2^2$, $I_1={\lambda }_1^2+2{\lambda }_2^2$, $I_2={\lambda }_2^4+2{\lambda }_1^2{\lambda }_2^2$, $B_{11}={\lambda }_1^2$, ${\left(\mathit{BB}\right)}_{11}={\lambda }_1^4$. The normal stresses are given by the formulas

$$\begin{array}{l}{\sigma }_{11}=2C_{10}\left(2/3\right){\left({\lambda }_1{\lambda }_2^2\right)}^{-5/3}\left({\lambda }_1^2-{\lambda }_2^2\right)+2C_{01}{\left({\lambda }_1{\lambda }_2^2\right)}^{-7/3}\left({\lambda }_1^4+\left(2/3\right){\lambda }_1^2{\lambda }_2^2-\left(5/3\right){\lambda }_2^4\right)\\ +2D_1\left({\lambda }_1{\lambda }_2^2-1\right),\end{array}$$E45

$${\sigma }_{22}={\sigma }_{33}=\left(2/3\right)C_{10}{\left({\lambda }_1{\lambda }_2^2\right)}^{-5/3}\left({\lambda }_2^2-{\lambda }_1^2\right)+2C_{01}{\left({\lambda }_1{\lambda }_2^2\right)}^{-7/3}\left(1/3\right){\lambda }_2^2\left({\lambda }_2^2-{\lambda }_1^2\right)+2D_1\left({\lambda }_1{\lambda }_2^2-1\right).$$E46

Assume that all three normal stresses over the sample lateral surface are absent. Then Eq. (45) is simplified to the form

$${\sigma }_{11}=2C_{01}{\left({\lambda }_1{\lambda }_2^2\right)}^{-7/3}\left({\lambda }_1^4-{\lambda }_2^4\right)+6D_1\left({\lambda }_1{\lambda }_2^2-1\right).$$E47

The last formula testifies: the Mooney-Rivlin model describes the Poynting-type effect.

Two elastic constants are presented in (47) in contrast to the Neo-Hookean model, where the shear modulus was absent. It should be noted that in both models – Neo -Hookean and Moo-ney-Rivlin –the tension in the longitudinal direction stress ${\sigma }_{11}$ depends already on two principal extensions. Figure 13 shows a dependence of the longitudinal stress on principal extensions is built for the silicon rubber. It coincides practically with Figure 11 (Neo-Hookean model) and shows that the constant $C_{01}$ of the Mooney-Rivlin model effects not essentially on the stress ${\sigma }_{11}$ and the dependence (45) rests weakly nonlinear within the accepted restrictions.

Figure 13.

Dependence of the longitudinal stress on principal extensions.

The Eq. (46) can be transformed into the form

$$\begin{array}{c}{\lambda }_1^6-{\lambda }_1^3/{\sigma }^2+\left[\left(2C_{10}/6D_1\right)\sqrt[-3]{{\sigma }^4}+\left(2C_{01}/6D_1\right)\sqrt[-3]{{\sigma }^2}\right]{\sigma }^{-4}\left({\sigma }^2-1\right)=0,\\ \sigma =\left({\lambda }_2/{\lambda }_1\right),{\lambda }_1^3=1/2{\sigma }^2\pm 1/2{\sigma }^2\sqrt{1-\left[\begin{array}{l}\left(2C_{10}/6D_1\right)\sqrt[-3]{{\sigma }^4}\\ +\left(2C_{01}/6D_1\right)\sqrt[-3]{{\sigma }^2}\end{array}\right]{\sigma }^{-4}\left({\sigma }^2-1\right)}\end{array},$$E48

The corresponding to the model 1 plot from Figure 11 is practically identical with the plot from Figure 13 corresponding to model 2.

5.2.3 Uniaxial tension in model 3

The uniaxial tension in this model is characterized by three nonzero components of the strain tensor ${\epsilon }_{\mathit{kk}}$ and one non-zero component of the stress tensor ${\sigma }_{11}$. Then the constitutive equations are somewhat simp-lified.

$$\begin{array}{c}{\sigma }_{11}=\lambda I_1+2{\mathit{\mu \epsilon }}_{11}+A{\left({\epsilon }_{11}\right)}^2+B\left(\mathrm{E}+2{\epsilon }_{11}{I}_1\right)+C\left(\mathrm{E}+2{\epsilon }_{22}{\epsilon }_{11}+2{\epsilon }_{33}{\epsilon }_{11}\right)\\ I_1={\epsilon }_{11}+{\epsilon }_{22}+{\epsilon }_{33},\mathrm{E}={\left({\epsilon }_{11}\right)}^2+{\left({\epsilon }_{22}\right)}^2+{\left({\epsilon }_{33}\right)}^2.\end{array}$$E49

$$0=\lambda I_1+2{\mathit{\mu \epsilon }}_{22}+A{\left({\epsilon }_{22}\right)}^2+B\left(\mathrm{E}+2{\epsilon }_{22}{I}_1\right)+C\left(\mathrm{E}+2{\epsilon }_{22}{\epsilon }_{33}+2{\epsilon }_{22}{\epsilon }_{11}\right),$$E50

$$0=\lambda I_1+2{\mathit{\mu \epsilon }}_{33}+A{\left({\epsilon }_{33}\right)}^2+B\left(\mathrm{E}+2{\epsilon }_{33}{I}_1\right)+C\left(\mathrm{E}+2{\epsilon }_{22}{\epsilon }_{11}+2{\epsilon }_{22}{\epsilon }_{33}\right).$$E51

Let us remind that in the linear theory of elasticity, corresponding to the Hookean model, the constitutive equations are significantly simpler

$${\sigma }_{11}=\lambda I_1+2{\mathit{\mu \epsilon }}_{11},0=\lambda I_1+2{\mathit{\mu \epsilon }}_{22},0=\lambda I_1+2{\mathit{\mu \epsilon }}_{33}.$$E52

Apply further to the nonlinear Eqs. (49)–(51) the procedure of analysis of the state of uniaxial tension that is used in the linear theory of elasticity as applied to Eqs. (52). Subtraction of Eq. (51) from Eq. (50) gives the formula

$$0=2\mu \left({\epsilon }_{22}-{\epsilon }_{33}\right)+A\left({\left({\epsilon }_{22}\right)}^2-{\left({\epsilon }_{33}\right)}^2\right)+2B\left({\epsilon }_{22}-{\epsilon }_{33}\right)\left({\epsilon }_{11}+{\epsilon }_{22}+{\epsilon }_{33}\right),$$

from which the equality of components of transverse strains ${\epsilon }_{22}={\epsilon }_{33}$ follows.

The addition of formulas (36)–(38) results in the following formula

$$\begin{array}{l}{\sigma }_{11}/\left(3\lambda +2\mu \right)-\left[\left(A+3B+C\right)/3\lambda +2\mu \right]\left[{\left({\epsilon }_{11}\right)}^2+2{\left({\epsilon }_{22}\right)}^2\right]-\left[2B/\left(3\lambda +2\mu \right)\right]{\left({\epsilon }_{11}+2{\epsilon }_{22}\right)}^2\\ -\left[4C/\left(3\lambda +2\mu \right)\right]\left[{\left({\epsilon }_{22}\right)}^2+2{\epsilon }_{22}{\epsilon }_{11}\right]={\epsilon }_{11}+2{\epsilon }_{22}.\end{array}$$E53

Substitution of formula (53) into the relation (49) gives new relation

$$\begin{array}{l}{\sigma }_{11}=E{\epsilon }_{11}+\left(A+\frac{2\lambda +3\mu }{\lambda +\mu }B+C\right){\left({\epsilon }_{11}\right)}^2-\frac{\lambda }{\lambda +\mu }\left(A+\frac{4\lambda -2\mu }{\lambda }B-\frac{2\mu }{\lambda }C\right){\left({\epsilon }_{22}\right)}^2\\ +\frac{2\left(\lambda +2\mu \right)}{\lambda +\mu }\left(B+C\right){\epsilon }_{11}{\epsilon }_{22}.\end{array}$$E54

The relation (54) shows that model 3, like models 1 and 2, describes the Poynting-type effect.

Figures 14 and 15 show the dependence ${\sigma }_{11}={\sigma }_{11}\left({\epsilon }_{11},{\epsilon }_{22}\right)$ among the longitudinal stress ${\sigma }_{11}$ and strains ${\epsilon }_{11},{\epsilon }_{22}$ for the foam and polystyrene and the moderate values of strains. Both plots demonstrate an essential nonlinearity under moderate strains. This new nonlinear effect will be true for auxetic materials too.

Figure 14.

Dependence ${\sigma }_{11}={\sigma }_{11}\left({\epsilon }_{11},{\epsilon }_{22}\right)$ (foam).

Figure 15.

Dependence${\sigma }_{11}={\sigma }_{11}\left({\epsilon }_{11},{\epsilon }_{22}\right)$ (polystyrene).

Write now the constitutive Eq. (49) with allowance for equality ${\epsilon }_{22}={\epsilon }_{33}$ and transform it to the form of a quadratic equation relative to the ratio ${\epsilon }_{22}/{\epsilon }_{11}$

$${\left(\frac{{\epsilon }_{22}}{{\epsilon }_{11}}\right)}^2+2\frac{\left[\left(\left(\lambda +\mu \right)/{\epsilon }_{11}\right)+\left(B+C\right)\right]}{\left(A+6B+4C\right)}\frac{{\epsilon }_{22}}{{\epsilon }_{11}}+\frac{\left(\lambda /{\epsilon }_{11}\right)+\left(B+C\right)}{\left(A+6B+4C\right)}=0.$$

The solution of this equation has the form

$$\left({\epsilon }_{22}/{\epsilon }_{11}\right)=-\left\{\left[\left(\lambda +\mu \right)/{\epsilon }_{11}+\left(B+C\right)\right]/\left(A+6B+4C\right)\right\}\left[1\pm \sqrt{1-\frac{\left(A+6B+4C\right)\left[\lambda /{\epsilon }_{11}+\left(B+C\right)\right]}{{\left[\left(\lambda +\mu \right)/{\epsilon }_{11}+\left(B+C\right)\right]}^2}}\right].$$E55

Thus, Eq. (55) shows that the ratio $\left(-{\epsilon }_{22}/{\epsilon }_{11}\right)$ is not constant in the Murnaghan nonlinear model. This can be treated as the new mechanical nonlinear effect which is looking very promising for the auxetic materials.

Figures 16 and 17 show a dependence of the ratio $\left(-{\epsilon }_{22}/{\epsilon }_{11}\right)$ on the strain ${\epsilon }_{11}$ and are built for the foam and polystyrene for the moderate strains. The plot’s main features are as follows: the ratio $\left(-{\epsilon }_{22}/{\epsilon }_{11}\right)$ is de-creased essentially from the initial value, which corresponds to the Poisson ratio for small strain in the con-ventional materials, to the negative values under the moderate values of longitudinal strain that is observed in the auxetic materials. So, the ratio, that is, treated as the Poisson’s ratio for small strain, in the case of mo-derate strain becomes the characteristics of transition of the material from the category of conventional ma-terials into the category of nonconventional materials. This can be considered as the newly revealed theore-tically nonlinear effect.

Figure 16.

Dependence of the ratio $\left(-{\epsilon }_{22}/{\epsilon }_{11}\right)$ on the strain ${\epsilon }_{11}$ (foam).

Figure 17.

Dependence of the ratio $\left(-{\epsilon }_{22}/{\epsilon }_{11}\right)$ on the strain ${\epsilon }_{11}$(polystyrene).

Thus, an analysis of universal deformation of uniaxial tension for model 3 revealed the new property: the material with conventional properties under small strains is transformed under moderate strains into the nonconventional (auxetic) material. The uncommonness of this observation consists in that usually the material is considered either the conventional or the nonconventional during all the processes of deformation.

Let us compare the plots from Figures 16 and 17 with the experimental data from ([49], Figure 4) shown here as Figure 18 (dependence of the ratio $\left(-{\epsilon }_{22}/{\epsilon }_{11}\right)$ on the strain ${\epsilon }_{11}$), where the deformation of the foams was studied for the finite strains with increasing the longitudinal strain ${\epsilon }_{11}$ from 0.1 to 1.4. Note that the theoretical plots are constructed for the range from ${\epsilon }_{11}=0$ to the moderate values 0.23 (foam) and 0.33 (polystyrene). This comparison shows that $\left({\epsilon }_{22}/{\epsilon }_{11}\right)$ increases within the range ${\epsilon }_{11}\in \left(0,0;0,3\right).$ Thus, model 3 describes some experimental observations of the foam.

Figure 18.

Experimental dependence of the ratio $\left(-{\epsilon }_{22}/{\epsilon }_{11}\right)$ on the strain ${\epsilon }_{11}$.

Figures 19 and 20 show the 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 ${\epsilon }_{22}$ reaches the local minimum and further increases. 3. When the strain ${\epsilon }_{11}$ continues to increase, the strain ${\epsilon }_{22}$ possesses zero value and further increases possessing already positive values.

Figure 19.

Dependence of longitudinal and transverse strains (foam).

Figure 20.

Dependence of longitudinal and transverse strains (polystyrene).

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 ma-terials into the class of the auxetic materials. In other words, the standard sample in conditions of 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 longitudinal stretching to the moderate values the sample cross-section starts to increase, what is the characteristic just for auxetic materials.

The plots from Figures 19 and 20 can be compared with the plot, obtained experimentally in [48]. This article reports that the new metamaterials were created from soft silicon rubber. The samples we-re deformed in conditions of uniaxial compression up to moderate values of longitudinal strain 0,35. The shown in the Figure 21 plot corresponds to Figure 2a in [48] and shows a dependence of longitudinal and transverse strains. Comparison of plots from Figure 11 (uniaxial stretching) and Figure 12 (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.

Figure 21.

Experimental dependence of longitudinal and transverse strains.

Thus, the nonlinear Murnaghan model describes within conditions 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 clearly visible only within the framework of the Murnaghan model, but the Neo-Hookean and Mooney -Rivlin models also describe the hump formation, as can be seen in Figure 6.

5.3 Universal deformation of omniaxial tension

5.3.1 Omniaxial tension in model 1

In this case ${\lambda }_1={\lambda }_2={\lambda }_3$, $J={\lambda }_1^3,I_1=3{\lambda }_1^2$ and the normal stress is equal

$${\sigma }_{11}=2D_1\left({\lambda }_1^3-1\right).$$E56

The formula (56) describes the Poynting-type effect relative to the bulk modulus (the dependence ${\sigma }_{11}$ on the extension ${\lambda }_1$ is evidently nonlinear).

Figure 22 shows a dependence of the stress on the principal extension and is built for the silicon rubber. The plot testifies that model 1 describes the nonlinear change of the sample volume while being subjected to the universal deformation of uniform compression-tension.

Figure 22.

Dependence of stress on principal extension.

5.3.2 Omniaxial tension in model 2

In this case ${\lambda }_1={\lambda }_2={\lambda }_3$, $J={\lambda }_1^3,I_1=3{\lambda }_1^2$, $I_2=3{\lambda }_1^4$ (note that they are identical for any nonlinear model). The formula for normal stress coincides with the analogous formula for model 1 (56) and verifies the non-linear dependence of the tension stress on the principal extension.

5.3.3 Omniaxial tension in model 3

The components of displacement gradients and Cauchy-Green strain tensor are as follows

$$\begin{array}{c}{u}_{1,1}=u_{2,2}=u_{3,3}=\epsilon >0,u_{1,1}+u_{2,2}+u_{3,3}=3\epsilon =e,u_{k,m}=\left(\partial u_k/\partial x_m\right)\\ =0\left(k\ne m\right);{\epsilon }_{11}={\epsilon }_{22}={\epsilon }_{33}=\epsilon +\left(1/2\right){\epsilon }^2,{\epsilon }_{\mathit{ik}}=0\left(i\ne k\right)\end{array}$$E57

The corresponding algebraic invariants of the Cauchy-Green tensor are written in the form

$$\begin{array}{l}{I}_1={\epsilon }_{11}+{\epsilon }_{22}+{\epsilon }_{33}=e,I_2={\left({\epsilon }_{11}\right)}^2+{\left({\epsilon }_{22}\right)}^2+{\left({\epsilon }_{33}\right)}^2=\left(1/3\right)e^2,I_3\\ ={\left({\epsilon }_{11}\right)}^3+{\left({\epsilon }_{22}\right)}^3+{\left({\epsilon }_{33}\right)}^3=\left(1/9\right)e^3.\end{array}$$E58

The formulas for invariants (58) allow writing the potential in the simpler form

$$\begin{array}{c}W\left(\epsilon \right)=\left(3/2\right)\left(3\lambda +2\mu \right){\epsilon }^2+\left(\left(9/2\right)\lambda +3\mu +A+9B+9C\right){\epsilon }^3+\\ +\left(3/2\right)\left(4\left(3\lambda +2\mu \right)+\left(A+9B+9C\right)\right){\epsilon }^4+\left(3/4\right)\left(A+9B+9C\right){\epsilon }^5+\left(1/8\right)\left(A+9B+9C\right){\epsilon }^6.\end{array}$$E59

The stresses are evaluated by the formulas (the normal stresses only are nonzero)

$$\begin{array}{l}{\sigma }_{11}={\sigma }_{22}={\sigma }_{33}=\left(3\lambda +2\mu \right)\epsilon +\left[\left(3/2\right)\left(3\lambda +2\mu \right)+\left(A+9B+7C\right)\right]{\epsilon }^2\\ +\left(A+9B+7C\right)\left({\epsilon }^3+\left(1/4\right){\epsilon }^4\right),{\sigma }_{12}={\sigma }_{23}={\sigma }_{31}=0.\end{array}$$

Thus, the stresses contain linear and nonlinear summands.

The interdependence between the first invariant of the stress tensor ${\sigma }_{\mathit{kk}}$ and the parameter of the omni-axial tension $e$ has the form

$$\begin{array}{l}{\sigma }_{\mathit{kk}}=\left(3\lambda +2\mu \right)e+\left[\left(1/2\right)\left(3\lambda +2\mu \right)+\right.\left.\left(1/3\right)\left(A+9B+7C\right)\right]e^2\\ +\left(A+9B+7C\right)\left[\left(1/9\right)e^3+\left(1/108\right)e^4\right].\end{array}$$E60

The plots in Figures 23 and 24 show a dependence ${\sigma }_{\mathit{kk}}\left(e\right)$ for the foam and polystyrene evaluated formula (60). It follows from them that they are similar to the parabola with a vertex in a positive half of the plane ${\sigma }_{\mathit{kk}}\mathit{Oe}$. The parabola’s right branch then passes into the negative half of the plane. Both plots have “the hump” in the positive branch of the plane.

Figure 23.

Dependence ${\sigma }_{\mathit{kk}}\left(e\right)$ (foam).

Figure 24.

Dependence ${\sigma }_{\mathit{kk}}\left(e\right)$ (polystyrene).

A presence of “the hump” testifies that the nonlinear Murnaghan model describes the transition 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 the strain the sample swells. But this phenomenon is characteristic of only auxetic materials.

Thus, three nonlinear models which are used in the analysis 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 $\sigma \sim \epsilon$ (stress versus strain) in auxetic materials for the moderate strains.

The main new effects are revealed: the nonlinear Murnaghan model describes in the case of uniaxial and omniaxial tension the 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 stan-dard 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 longitudinal stretching to the moderate values the sample cross-section starts to increase, what is the characteristic just for auxetic materials.

6. Final conclusions

The elasticity is the property of auxetic materials, which is especially characteristic and most studied for these materials. Historically, the auxetics were treated from the point of view of the linear theory of elasticity what was not quite adequate in some cases.

As the part of classical mechanics of elastic materials, the mechanics of auxetic materials needs at present more and more experimental studies (the level of such studies as compared with the classical ones can be seen from the famous Bell’s book [27]).

The nonlinear theory of elasticity is seemingly quite prospective for a description of elastic deformation of the auxetic materials but it is essentially more complicated in the mathematical apparatus and concrete investigations.