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# Anisotropic auxetic composite laminates: a polar approach

Paolo Vannucci
LMV - Laboratoire de Mathématiques de Versailles, UMR8100
Université de Versailles et Saint Quentin - 45, Avenue des Etats-Unis, 78035 -
France
<EMAIL_ADDRESS>

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###### Abstract

The problem of obtaining anisotropic auxetic composite laminates, i.e. having
a negative Poisson’s ratio for at least some directions, is examined in this
paper. In particular, the possibility of obtaining auxeticity stacking uni-
directional identical plies is considered. It is shown that if the ply is
composed by isotropic matrix and fibers, then it is impossible to obtain
totally auxetic orthotropic laminates, i.e. auxeticity for each direction,
unless at least one among matrix and fibers is auxetic itself. Moreover, it is
shown what are the conditions, in terms of the mechanical properties of the
constituents and of the volume fraction of the fibers, to fabricate uni-
directional plies with which to realize laminates having a negative Poisson’s
ratio for some directions. Several existing materials are also examined. All
the analysis is done using the polar formalism, very effective for the study
of plane anisotropic problems.

Key words:  Poisson’s ratio, auxeticity, anisotropy, composite laminates,
polar formalism

##  1 Introduction

The property of having a negative Poisson’s ratio is known in the literature
as auxeticity[1]. Theoretically physically possible for isotropic materials,
auxeticity is normally obtained if some peculiar material microstructure is
present in an isotropic (at least at a sufficiently large scale) continuum.
Several studies have been done the last three decades on the possibility of
obtaining auxetic materials by the use of some peculiar microstructures, also
at the molecular level, from the pioneer works of Almgren[2], Evans[3, 4],
Lakes[5, 6, 7, 8], Milton[9]. Literature on the topic is very huge, it is not
the objective here to give a complete overview of the matter and the reader is
addressed to specific review articles on auxetic materials, like the papers of
Prawoto[10] or of Shukla & Behera[11].

For an anisotropic body, also auxeticity is an anisotropic property, i.e. the
Poisson’s coefficient can be negative for some directions, positive for some
others, and this without the need of any peculiar microstructure: the
directional variations typical of anisotropy make the auxeticity of classical
elastic materials à la Cauchy possible. For example, the Poisson’s ratio
$\nu_{12}$ for a board of pine wood in a plane-stress state is negative for
almost all the directions, Fig. 1.

Figure 1: The directional diagram of $\nu_{12}$ for pine wood in a plane-
stress state (material data in Tab. 1, source Lekhnitskii[12]). The thin
circle marks the zero value: inside it, $\nu_{12}$ is negative.

Because anisotropic composite materials are mainly used in the fabrication of
laminated structures, it has hence a certain interest the possibility of
obtaining auxetic laminates. A recent state of the art on auxetic composite
laminates for structural uses has been published by Veloso et al[13]. The
first studies on auxetic anisotropic laminates date back to the eighties[14,
15]. Clarke at al[16] and Hine et al[17] investigated experimentally the
auxeticity of balanced angle-ply laminates, a type of laminates frequently
considered for the simplicity of the stack. The possibility of obtaining
auxetic laminates is considered by Zhang et al[18, 19] and by Alderson et
al[20], while the determination of the highest negative Poisson’s ratio is
studied in[21, 22].

However, in all the existing literature on composite laminates, the problem of
the auxeticity of the elastic response is treated only partially, with regard
to some specific cases, i.e. angle-ply stacks, or concerning some special
situations or aspects, like for instance the constituent materials or the
maximization of the negative value of the Poisson’s ratio. The aim of this
paper is precisely to give some generality to the matter and namely to try to
respond as largely as possible to the following fundamental question: when is
it possible to fabricate an auxetic laminate stacking identical anisotropic
plies? In other terms, given an anisotropic ply, will it exist at least one
stacking sequence giving rise to a laminate that will have an auxetic
response? This leads to investigate the relations existing between the
material properties of the ply and the geometry of the stack. We want to
stress that we look for an auxetic behavior generated uniquely by anisotropy,
i.e. the component layers as well as their constituent materials are classical
elastic bodies à la Cauchy, with no microstructures able to generate
auxeticity, so just the classical anisotropic elasticity theory is used for
this study.

The study proposed here concerns only the in-plane properties of the laminate,
not the transversal ones. In addition, we consider laminates composed by
identical unidirectional layers (the case of plies reinforced by fabrics will
be the subject of a future study). Moreover, only orthotropic laminates are
considered, as this is the case for usual structures. The results are
presented for the extension behavior, but actually they can be rephrased
verbatim for the bending one (the auxeticity of the bending response is less
interesting for applications).

The problem is rather complicate and it reveals interesting, mathematically
speaking, to formulate it in an appropriate space of material and geometrical
parameters. In particular, for representing the material properties this study
makes use of dimensionless moduli deduced from the polar formalism for plane
anisotropy, introduced by G. Verchery in 1979[23, 24, 25] and of the, also
dimensionless, lamination parameters proposed in 1968 by Tsai and Pagano[26]
for the geometry of the stack.

In the paper, four different theoretically possible cases of in-plane
auxeticity for anisotropic laminates are introduced, all of them are
considered and analyzed. Several existing materials are also analyzed and the
question whether or not a type of auxeticity can be obtained for the laminate
using classical anisotropic plies, i.e. plies made of non auxetic materials,
is considered. Some examples of auxetic laminates are also given.

##  2 Fundamental relations

###  2.1 The in-plane Poisson’s ratio $\nu_{12}$

Let us consider an anisotropic layer in a plane-stress state, with
$\mathbb{S}$ and $\mathbb{Q}=\mathbb{S}^{-1}$ respectively the compliance and
the reduced stiffness tensors. The in-plane Poisson’s ratio $\nu_{12}(\theta)$
at the direction inclined of $\theta$ on the $x_{1}-$axis is defined as[12,
27, 28, 29] (we adopt in this study the Kelvin’s notation[30, 31] for the
elasticity tensors)

$\nu_{12}(\theta):=-\frac{\mathbb{S}_{12}(\theta)}{\mathbb{S}_{11}(\theta)}.$
(1)

Because $\mathbb{S}_{11}(\theta)>0\ \forall\theta$,

$\nu_{12}(\theta)<0\iff\mathbb{S}_{12}(\theta)>0.$ (2)

###  2.2 Polar formulation of the auxeticity condition

In the polar formalism, the above condition for an orthotropic layer
becomes[25]

$\mathbb{S}_{12}(\theta)=-t_{0}+2t_{1}-r_{0}\cos 4(\varphi_{0}-\theta)>0,$ (3)

with $t_{0},t_{1},r_{0}$ non-negative tensor invariants depending upon the
components of $\mathbb{S}$ and $\varphi_{0}$ a polar angle. In view of the
forthcoming developments, it is worth to transform the previous relation. To
this end, we express the polar parameters of the compliance $\mathbb{S}$ by
those of the stiffness $\mathbb{Q}$:

$\begin{array}[]{l}t_{0}=2\dfrac{T_{0}T_{1}-R_{1}^{2}}{\varDelta},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ t_{1}=\dfrac{T_{0}^{2}-R_{0}^{2}}{2\varDelta},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
r_{0}\mathrm{e}^{4\mathrm{i}\varphi_{0}}=\dfrac{2}{\varDelta}(R_{1}^{2}\mathrm{e}^{4\mathrm{i}\varPhi_{1}}-T_{1}R_{0}\mathrm{e}^{4\mathrm{i}\varPhi_{0}}),\end{array}$
(4)

where $T_{0},T_{1},R_{0},R_{1}$ are the polar invariant (positive) moduli of
$\mathbb{Q}$ and $\varPhi_{0},\varPhi_{1}$ the two polar angles of
$\mathbb{Q}$, whose difference is the fifth invariant. $\varDelta$ is the
invariant given by

$\varDelta=4T_{1}(T_{0}^{2}-R_{0}^{2})-8R_{1}^{2}[T_{0}-R_{0}\cos
4(\varPhi_{0}-\varPhi_{1})].$ (5)

Some simple passages give

$\begin{split}r_{0}\cos
4(\varphi_{0}-\theta)&=\frac{2}{\varDelta}[R_{1}^{2}\cos
4(\varPhi_{1}-\theta)-\\\ &-T_{1}R_{0}\cos 4(\varPhi_{0}-\theta)].\end{split}$
(6)

As, actually, $\varDelta=\det\mathbb{Q}>0$, it can be ignored in the
following, because it does not change the sign of $\mathbb{S}_{12}$. So, in
terms of the polar parameters of $\mathbb{Q}$ the auxeticity condition is

$\begin{split}2(T_{0}T_{1}-R_{1}^{2})-T_{0}^{2}+R_{0}^{2}+2[R_{1}^{2}\cos
4(\varPhi_{1}-\theta)-&\\\ -T_{1}R_{0}\cos
4(\varPhi_{0}-\theta)]&<0.\end{split}$ (7)

A rotation $\alpha$ of the frame corresponds to subtract $\alpha$ from the two
polar angles. If $\alpha=\varPhi_{1}$ or, equivalently, if the reference frame
is chosen in such a way that $\varPhi_{1}=0$, which corresponds, for
unidirectional (UD) plies, to put the $x_{1}-$axis aligned with the fibres,
the above relation becomes

$\begin{split}2(T_{0}T_{1}-R_{1}^{2})-T_{0}^{2}+R_{0}^{2}+2[R_{1}^{2}\cos
4\theta-&\\\ -T_{1}R_{0}\cos 4(\varPhi-\theta)]&<0,\end{split}$ (8)

with $\varPhi=\varPhi_{0}-\varPhi_{1}$, the angular invariant of $\mathbb{Q}$.
Common orthotropy corresponds to the condition[25]

$\varPhi=K\frac{\pi}{4},\ \ K\in\\{0,1\\},$ (9)

the value of $K$ determining two different types of orthotropic materials
sharing the same polar moduli of $\mathbb{Q}$. As said in the Introduction, we
consider in this paper only UD plies (i.e. layers reinforced by fabrics are
not considered). It can be shown that such plies can be only ordinary
orthotropic hence $R_{0}\neq 0,R_{1}\neq 0$.

So finally, for an UD layer, the auxeticity condition resumes to

$\begin{split}\lambda(\theta):&=2(T_{0}T_{1}-R_{1}^{2})-T_{0}^{2}+R_{0}^{2}+\\\
&+2[R_{1}^{2}-(-1)^{K}T_{1}R_{0}]\cos 4\theta<0,\end{split}$ (10)

###  2.3 Auxeticity condition for an orthotropic laminate

Let us now consider a laminate composed of identical UD plies all sharing the
same reduced stiffness tensor $\mathbb{Q}$. We focus on the extension
response, described by tensor[28, 25]

$\mathbb{A}=\frac{1}{h}\sum_{j=1}^{n}({z_{j}}-{z_{j-1}})\mathbb{Q}_{(}\delta_{j}),\\\
$ (11)

with $\delta_{j}$ the orientation of the $j-$th layer among the $n$ composing
the laminate and $h$ the plate’s thickness. We further assume that the
laminate is extension-bending uncoupled. This assumption is necessary, because
otherwise the Poisson’s ratio for the extension, or also for the bending,
behavior, should be practically impossible to be analyzed (in fact, for
coupled laminates the compliances, in extension and in bending, depend in a
very complicate manner upon $\mathbb{A},\mathbb{B}$ and $\mathbb{D}$,
respectively the stiffness tensors in extension, coupling and bending[32, 33,
34]). It is worth to recall that uncoupling ($\mathbb{B}=\mathbb{O}$) can be
obtained by suitable stacking sequences, but, contrarily to what commonly
believed, not necessarily symmetric. Actually, asymmetric uncoupled laminates
are much more numerous than the symmetric ones[35, 36]. So, uncoupling can be
obtained rather easily and to assume it does not constitute a true limitation.

The auxeticity condition for an orthotropic tensor $\mathbb{A}$ is the same of
that for $\mathbb{Q}$, provided that the polar parameters of $\mathbb{A}$
(denoted in the following by a superscript $A$) replace those of the layer:

$\begin{split}\lambda^{A}(\theta):&=2(T_{0}^{A}T_{1}^{A}-{R_{1}^{A}}^{2})-{T_{0}^{A}}^{2}+{R_{0}^{A}}^{2}+\\\
&+2[{R_{1}^{A}}^{2}-(-1)^{K^{A}}T_{1}^{A}R_{0}^{A}]\cos 4\theta<0.\end{split}$
(12)

Because the plies are identical, the polar parameters of the extension
stiffness tensor $\mathbb{A}$ can be put in the form[25]

$\begin{array}[]{l}T_{0}^{A}=T_{0},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
T_{1}^{A}=T_{1},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
R_{0}^{A}\mathrm{e}^{4\mathrm{i}\varPhi_{0}^{A}}={R_{0}\mathrm{e}^{4\mathrm{i}\varPhi_{0}}}{\left(\xi_{1}+\mathrm{i}\xi_{2}\right)},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
R_{1}^{A}\mathrm{e}^{2\mathrm{i}\varPhi_{1}^{A}}={R_{1}\mathrm{e}^{2\mathrm{i}\varPhi_{1}}}{\left(\xi_{3}+\mathrm{i}\xi_{4}\right)}.\end{array}$
(13)

The quantities $\xi_{i},i=1,...,4$ are the lamination parameters[26] for
$\mathbb{A}$:

$\xi_{1}+\mathrm{i}\xi_{2}=\frac{1}{n}\sum_{j=1}^{n}\mathrm{e}^{4\mathrm{i}\delta_{j}},\
\
\xi_{3}+\mathrm{i}\xi_{4}=\frac{1}{n}\sum_{j=1}^{n}\mathrm{e}^{2\mathrm{i}\delta_{j}},$
(14)

Eq. (13) shows, on the one hand, that the isotropic part of $\mathbb{A}$ is
identical to that of the layer, $T_{0}$ and $T_{1}$, and, on the other hand,
because the layer is UD, if we consider, as said above, an orthotropic
extension behavior, choosing $\varPhi^{A}_{1}=0$ to fix the reference frame
for the laminate, we get easily

$\begin{array}[]{c}\xi_{2}=\xi_{4}=0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
(-1)^{K^{A}}R_{0}^{A}=(-1)^{K}R_{0}\xi_{1},\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ R_{1}^{A}=R_{1}\xi_{3}.\end{array}$ (15)

So, the polar auxeticity condition for $\mathbb{A}$ becomes

$\begin{split}\lambda^{A}(\theta)&=2(T_{0}T_{1}-R_{1}^{2}\xi_{3}^{2})-T_{0}^{2}+R_{0}^{2}\xi_{1}^{2}+\\\
&+2[R_{1}^{2}\xi_{3}^{2}-(-1)^{K}T_{1}R_{0}\xi_{1}]\cos 4\theta<0.\end{split}$
(16)

###  2.4 Dimensionless auxeticity condition

In the expression of $\lambda^{A}(\theta)$, there are three kind of
variables/parameters: the independent variable $\theta$, determining the
direction, the two lamination parameters $\xi_{1}$ and $\xi_{3}$, accounting
for the geometry of the stack (i.e., the sequence of the orientation angles
$\delta_{j}$), and the polar parameters $T_{0},T_{1},(-1)^{K}R_{0}$ and
$R_{1}$ of the layer, representing the material part of $\lambda^{A}(\theta)$.
It is worth to introduce new parameters for the material part, dimensionless
like $\xi_{1}$ and $\xi_{3}$. This will reduce the number of independent
material parameters and allow an easier study of the relation between geometry
and material. To this end, we introduce the following ratios[37]:

$\tau_{0}=\frac{T_{0}}{R_{1}},\ \ \tau_{1}=\frac{T_{1}}{R_{1}},\ \
\rho=\frac{R_{0}}{R_{1}}.$ (17)

It is worth noting that these ratios can be introduced because, for a UD ply,
$R_{1}\neq 0$. The auxeticity condition (16) can be rewritten in the
equivalent form

$\begin{split}\lambda^{A}(\theta)&=2(\tau_{0}\tau_{1}-\xi_{3}^{2})-\tau_{0}^{2}+\rho^{2}\xi_{1}^{2}+\\\
&+2[\xi_{3}^{2}-(-1)^{K}\tau_{1}\rho\ \xi_{1}]\cos 4\theta<0,\end{split}$ (18)

depending on five dimensionless quantities: three for the ply, material part,
and two for the stack, geometric part.

We present in Tab. 1 the characteristics of twenty different UD plies commonly
used for the fabrication of composite laminates.

Table 1: Some examples of UD plies. Modules are in GPa,
$\varPhi_{0}=\varPhi_{1}=0\Rightarrow K=0$ for all the plies.

Mat. | $E_{1}$ | $E_{2}$ | $G_{12}$ | $\nu_{12}$ | $T_{0}$ | $T_{1}$ | $R_{0}$ | $R_{1}$ | $\tau_{0}$ | $\tau_{1}$ | $\rho$
---|---|---|---|---|---|---|---|---|---|---|---
1 | 10.00 | 0.42 | 0.75 | 0.24 | 1.66 | 1.34 | 0.91 | 1.20 | 1.383 | 1.116 | 0.758
2 | 181.00 | 10.30 | 7.17 | 0.28 | 26.88 | 24.74 | 19.71 | 21.43 | 1.254 | 1.154 | 0.919
3 | 205.00 | 18.50 | 5.59 | 0.23 | 29.80 | 29.14 | 24.21 | 23.42 | 1.272 | 1.244 | 1.033
4 | 38.6 | 8.27 | 4.14 | 0.26 | 7.47 | 6.49 | 3.33 | 3.85 | 1.940 | 1.686 | 0.865
5 | 86.90 | 5.52 | 2.14 | 0.34 | 12.23 | 12.11 | 10.09 | 10.25 | 1.193 | 1.181 | 0.984
6 | 47.66 | 13.31 | 4.75 | 0.27 | 9.24 | 8.70 | 4.49 | 4.38 | 2.108 | 1.984 | 1.023
7 | 54.00 | 18.00 | 9.00 | 0.25 | 12.54 | 10.34 | 3.54 | 4.59 | 2.729 | 2.250 | 0.771
8 | 207.00 | 5.00 | 2.60 | 0.25 | 27.52 | 26.85 | 24.93 | 25.29 | 1.088 | 1.062 | 0.986
9 | 76.00 | 5.50 | 2.10 | 0.34 | 10.85 | 10.74 | 8.75 | 8.88 | 1.221 | 1.209 | 0.984
10 | 207.00 | 21.00 | 7.00 | 0.30 | 30.67 | 30.35 | 23.67 | 23.46 | 1.307 | 1.293 | 1.009
11 | 45.00 | 12.00 | 4.50 | 0.30 | 8.62 | 8.22 | 4.13 | 4.22 | 2.041 | 1.945 | 0.976
12 | 134.00 | 7.00 | 4.20 | 0.25 | 19.34 | 18.12 | 15.14 | 15.93 | 1.214 | 1.138 | 0.951
13 | 85.00 | 5.60 | 2.10 | 0.34 | 11.98 | 11.89 | 9.88 | 10.00 | 1.198 | 1.189 | 0.988
14 | 220.00 | 140.00 | 7.50 | 0.25 | 41.50 | 55.98 | 34.00 | 10.41 | 3.985 | 5.375 | 3.265
15 | 294.50 | 6.34 | 4.90 | 0.23 | 39.73 | 38.01 | 34.83 | 36.06 | 1.102 | 1.054 | 0.966
16 | 109.70 | 8.55 | 5.31 | 0.30 | 16.89 | 15.53 | 11.58 | 12.73 | 1.327 | 1.220 | 0.910
17 | 131.70 | 8.76 | 5.03 | 0.28 | 19.55 | 18.26 | 14.52 | 15.45 | 1.265 | 1.182 | 0.940
18 | 133.10 | 9.31 | 3.74 | 0.34 | 19.02 | 18.74 | 15.28 | 15.60 | 1.219 | 1.201 | 0.979
19 | 135.00 | 9.24 | 6.28 | 0.32 | 20.55 | 18.89 | 14.27 | 15.83 | 1.298 | 1.193 | 0.901
20 | 128.00 | 13.00 | 6.40 | 0.30 | 20.00 | 18.77 | 13.60 | 14.51 | 1.378 | 1.293 | 0.937

1: Pine wood[12] |  11: Glass-epoxy[38]
---|---
2: Carbon-epoxy T300/5208[27] |  12: Carbon-epoxy[38]
3: Boron-epoxy B(4)-55054[27] |  13: Kevlar-epoxy[38]
4: Glass-epoxy s-ply1002 Vf=0.45[27] |  14: Boron-aluminium[38]
5: Kevlar-epoxy 149[39] |  15: Carbon-epoxy GY70/34[40]
6: S-glass-epoxy S2-449/SP381[40] |  16: Carbon-bismaleide AS4/5250-3[40]
7: Glass-epoxy[28] |  17: Carbon-peek AS4/APC2[40]
8: Carbon-epoxy[28] |  18: Carbon-epoxy AS4/3502[40]
9: Kevlar-epoxy[28] |  19: Carbon-epoxy T300/976[40]
10: Boron-epoxy[28] |  20: Carbon-epoxy 3 MXP251S[41]

###  2.5 Thermodynamic necessary conditions

The polar moduli, like any other set of parameters representing elasticity,
must satisfy some thermodynamic conditions, stating the positivity of the
elastic potential[42, 29, 25, 43]. In[37] it has been shown that using the
dimensionless parameters (17), for an orthotropic ply these conditions are

$\begin{split}&\tau_{0}-\rho>0,\\\
&\tau_{1}\left[\tau_{0}+(-1)^{K}\rho\right]-2>0.\end{split}$ (19)

They are necessary for establishing the set, in the space of the polar
parameters, of materials giving rise to auxetic laminates, see below.

###  2.6 Theoretical considerations

An interesting interpretation can be given to the auxeticity condition: Eq.
(3) can equivalently be rewritten as

$\frac{1}{4[t_{0}+r_{0}\cos 4(\varphi_{0}-\theta)]}>\frac{1}{8t_{1}}.$ (20)

The two members of the above inequality have a direct mechanical
interpretation[25]: the shear modulus $G_{12}(\theta)$ is

$\begin{split}G_{12}(\theta)&=\frac{1}{4[t_{0}-r_{0}\cos
4(\varphi_{0}-\theta)]}=\\\ &=\frac{1}{4[t_{0}+r_{0}\cos
4(\varphi_{0}-\theta-\frac{\pi}{4})]}.\end{split}$ (21)

The bulk modulus $\kappa$ is

$\kappa=\frac{1}{8t_{1}}.$ (22)

So, the auxeticity condition for the direction $\theta+\frac{\pi}{4}$ actually
corresponds to state that

$G_{12}(\theta)>\kappa.$ (23)

This condition also includes and generalizes the already discussed auxeticity
condition[9, 10] for isotropy ($r_{0}=0$), that in 2D elasticity reads like:

$\frac{1}{4t_{0}}>\frac{1}{8t_{1}}\Rightarrow G>\kappa.$ (24)

###  2.7 The lamination domain

Miki[44, 45, 28] has shown that the lamination domain of an orthotropic
$\mathbb{A}$, i.e. the set of points of the plane $(\xi_{3},\xi_{1})$, that
can correspond to an orthotropic behavior in extension, is the sector of
parabola $\Omega$ in Fig. 2 defined by the bounds

$2\xi_{3}^{2}-1\leq\xi_{1}\leq 1,\ \ -1\leq\xi_{1}\leq 1.$ (25)

Each lamination point $(\xi_{3},\xi_{1})$ corresponds to an extension
response, i.e. to a tensor $\mathbb{A}$, that, in general, can be realized by
more that one stacking sequence.

Figure 2: The lamination domain of an orthotropic $\mathbb{A}$.

We just recall that the lamination point P${}_{1}=(1,1)$ corresponds to a
unidirectional laminate with all the plies with $\delta_{j}=0\ \forall j$,
P${}_{2}=(-1,1)$ to a unidirectional laminate with
$\delta_{j}=\frac{\pi}{2}\forall j$, P${}_{3}=(0,-1)$ to a balanced angle ply
with $\delta_{j}=\pm\frac{\pi}{4}$, P${}_{4}=(0,0)$ to an isotropic laminate.
Moreover, any angle-ply laminate is represented by a point of the parabolic
boundary and any cross-ply laminate by a point of the line P${}_{1}-$P2.

Miki[46] has also shown that a tensor $\mathbb{D}$ describing the bending
behavior has exactly the same lamination domain, although the definition of
the relevant lamination parameters is different. That is why all the results
of this paper, concerning extension, can be exported identically to bending,
the stacking sequence apart.

##  3 Auxeticity conditions for an orthotropic laminate

As mentioned above, for anisotropic bodies also auxeticity is an anisotropic
property. However, when an UD ply is used to fabricate a laminate, it is of
interest, at least theoretically, to examine the different situations that can
determine an auxetic behavior of the laminate. First of all, a natural
question is: because physics allows auxetic isotropic materials ($-1<\nu<0$),
one can wonder whether or not it is possible to realize, stacking identical UD
plies, a Totally Auxetic Anisotropic Laminate (TAAL), i.e. an anisotropic, or
also isotropic, laminate that is auxetic for each direction. Secondarily, one
could consider a relaxed requirement: an anisotropic laminate that should be
auxetic for some directions, situation that in short we will denote by the
acronym PAAL (Partially Auxetic Anisotropic Laminate). For a given UD
material, four different situations are, in principle, possible:

  1. 1.

the material allows to fabricate a TAAL for each possible lamination point
$(\xi_{3},\xi_{1})\in\Omega$;

  2. 2.

the material allows to fabricate a TAAL for some lamination points
$(\xi_{3},\xi_{1})\in\Omega$;

  3. 3.

the material allows to fabricate a PAAL for each possible lamination point
$(\xi_{3},\xi_{1})\in\Omega$;

  4. 4.

the material allows to fabricate a PAAL for some lamination points
$(\xi_{3},\xi_{1})\in\Omega$.

The objective of this paper is precisely to determine the conditions to be
satisfied by a material in order to obtain one of the four cases above. It is
a matter of fact that composite anisotropic auxetic laminates can exist[13],
but the general conditions allowing to obtain them and, more precisely, of
what type among the four cases above, has never been elucidated in the
literature, only some partial recommandations have been proposed. We aim at
giving a general response through the use of classical anisotropic elasticity
and the polar formalism.

###  3.1 The conditions for obtaining a TAAL

For an anisotropic laminate to be totally auxetic, equation (18) must be
satisfied $\forall\theta\in\left[0,\frac{\pi}{2}\right]$. But
$\lambda^{A}(\theta)\leq 0\ \forall\theta\iff\max\lambda^{A}(\theta)\leq 0$
which gives the condition

$\begin{split}\varPsi(\xi_{3},\xi_{1}):&=2(\tau_{0}\tau_{1}-\xi_{3}^{2})-\tau_{0}^{2}+\rho^{2}\xi_{1}^{2}+\\\
&+2|\xi_{3}^{2}-(-1)^{K}\tau_{1}\rho\ \xi_{1}|<0.\end{split}$ (26)

For a chosen ply, the function $\varPsi(\xi_{3},\xi_{1})$ is defined on
$\Omega$; it is shown in Fig. 3 for materials 2 and 4 in Tab. 1.

Figure 3: The function $\varPsi(\xi_{3},\xi_{1})$ on $\Omega$ for materials 2
and 4 of Tab. 1.

Then:

  * •

an UD ply can fabricate a TAAL $\forall(\xi_{3},\xi_{1})\in\Omega\iff\max\
\varPsi(\xi_{3},\xi_{1})<0$;

  * •

an UD ply can produce a TAAL for some lamination points
$(\xi_{3},\xi_{1})\in\Omega\iff\min\ \varPsi(\xi_{3},\xi_{1})<0$.

The maximum and minimum of $\varPsi(\xi_{3},\xi_{1})$ depend on the material
parameters $\tau_{0},\tau_{1}$ and $\rho$ and are rather cumbersome though not
difficult to be found. The maximum has the same expression regardless the type
of orthotropy, $K=0$ or $K=1$:

$\varPsi_{\max}=\varPsi(0,\pm 1)=(\tau_{0}+\rho)(2\tau_{1}-\tau_{0}+\rho).$
(27)

So, because the polar moduli are non-negative quantities, the condition for
obtaining a TAAL $\forall(\xi_{3},\xi_{1})$ is

$\varPsi_{0}:=2\tau_{1}-\tau_{0}+\rho<0.$ (28)

The minimum can take different values and depends also on the type of
orthotropy. If $K=0$,

$\begin{array}[]{l}\varPsi_{\min}=-(\tau_{0}-\tau_{1})^{2}:=\varPsi_{1}\ \ \ \
\mathrm{if}\ \ \tau_{1}<\rho<\dfrac{\tau_{1}}{2\tau_{1}^{2}-1},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\
\varPsi_{\min}=\left(\dfrac{\rho}{2\tau_{1}\rho-1}\right)^{2}-\dfrac{2\tau_{1}\rho}{2\tau_{1}\rho-1}-\tau_{0}^{2}+2\tau_{0}\tau_{1}:=\varPsi_{2}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \hskip 25.60747pt\mathrm{if}\ \ \
\rho>\dfrac{1}{\tau_{1}}\ \ \mathrm{and}\ \
\dfrac{\rho(2\tau_{1}^{2}-1)-\tau_{1}}{2\tau_{1}\rho-1}>0,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\varPsi_{\min}=(\tau_{0}-\rho)(2\tau_{1}-\tau_{0}-\rho):=\varPsi_{3}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \hskip 25.60747pt\mathrm{if}\ \ \
\rho<\tau_{1}\ \ \mathrm{and}\ \ \rho<\dfrac{1}{\tau_{1}}.\end{array}$ (29)

The condition for obtaining a TAAL for some points $(\xi_{3},\xi_{1})$ is
hence, for $K=0$ orthotropic materials,

$\varPsi_{\min}\in\\{\varPsi_{1},\varPsi_{2},\varPsi_{3}\\}<0.$ (30)

Moreover, referring to Fig. 4: $\varPsi_{1}$ is get on the segment
$q_{1}q_{2}$ and on its symmetric with respect to the $\xi_{1}-$axis;
$\varPsi_{2}$ is get on the point $q_{3}$ and on its symmetric; $\varPsi_{3}$
is get on the segment $q_{4}q_{5}$ and on its symmetric.

Figure 4: The points of minimum of the function $\varPsi(\xi_{3},\xi_{1})$.

The coordinates of the points $q_{i}$ are:

$\begin{array}[]{l}q_{1}=\left(\tau_{1},\dfrac{\tau_{1}}{\rho}\right),\
q_{2}=\left(\sqrt{\dfrac{\rho+\tau_{1}}{2\rho}},\dfrac{\tau_{1}}{\rho}\right),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
q_{3}=\left(\sqrt{\dfrac{\tau_{1}\rho}{2\tau_{1}\rho-1}},\dfrac{1}{2\tau_{1}\rho-1}\right),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ q_{4}=\left(\sqrt{\tau_{1}\rho},1\right),\
q_{5}=(1,1).\end{array}$ (31)

A similar analysis can be done for materials with $K=1$:

$\begin{array}[]{l}\varPsi_{\min}=\varPsi_{1}\ \ \ \ \mathrm{if}\ \ \ \
\rho(2\tau_{1}^{2}-1)+\tau_{1}<0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\varPsi_{\min}=\left(\dfrac{\rho}{2\tau_{1}\rho+1}\right)^{2}-\dfrac{2\tau_{1}\rho}{2\tau_{1}\rho+1}-\tau_{0}^{2}+2\tau_{0}\tau_{1}:=\varPsi_{4}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \hskip 25.60747pt\mathrm{if}\ \ \
\rho(2\tau_{1}^{2}-1)+\tau_{1}>0,\end{array}$ (32)

so the condition for obtaining a TAAL for some points $(\xi_{3},\xi_{1})$
using a $K=1$ orthotropic material is

$\varPsi_{\min}\in\\{\varPsi_{1},\varPsi_{4}\\}<0.$ (33)

Still referring to Fig. 4, $\varPsi_{1}$ is get on the segment $q_{6}q_{7}$
and on its symmetric with respect to the $\xi_{1}-$axis, while $\varPsi_{4}$
is get on the point $q_{8}$ and on its symmetric and the coordinates of these
points are:

$\begin{array}[]{l}q_{6}=\left(\tau_{1},-\dfrac{\tau_{1}}{\rho}\right),\
q_{7}=\left(\sqrt{\dfrac{\rho-\tau_{1}}{2\rho}},-\dfrac{\tau_{1}}{\rho}\right),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
q_{8}=\left(\sqrt{\dfrac{\tau_{1}\rho}{2\tau_{1}\rho+1}},-\dfrac{1}{2\tau_{1}\rho+1}\right).\end{array}$
(34)

It is interesting to remark that, while $\varPsi_{\min}$ depends on all the
material parameters, i.e., in the end, on the whole $\mathbb{Q}$, its position
on $\Omega$ as well as the formula for its calculation depend only on
$\tau_{1}$ and $\rho$. The zones of the different $\varPsi_{i},\ i=1,...,4,$
in the plane $(\tau_{1},\rho)$ are depicted in Fig. 5. On the same diagram,
the positions of the materials in Tab. 1 are indicated by a small circle. It
is apparent that all of them, of the type $K=0$, apart material 1, pine wood,
belong to the zone where $\varPsi_{\min}=\varPsi_{2}$. Hence, for all of them,
the minimum is get at the point $q_{3}$, on the boundary of $\Omega$. Such a
lamination point can always be obtained by an angle-ply sequence, but not
exclusively.

Figure 5: The zones of the minimum of $\varPsi(\xi_{3},\xi_{1})$ for the case
$K=0$, in black, and $K=1$, in grey. Each circle corresponds to a material of
Tab. 1.

###  3.2 The conditions for obtaining a PAAL

An anisotropic laminate is partially auxetic if Eq. (18) has at least one
solution in $\Omega$. Hence, a PAAL can be fabricated with a UD material if
and only if $\min\lambda^{A}(\theta)\leq 0$ which gives the condition

$\begin{split}\eta(\xi_{3},\xi_{1}):&=2(\tau_{0}\tau_{1}-\xi_{3}^{2})-\tau_{0}^{2}+\rho^{2}\xi_{1}^{2}-\\\
&-2|\xi_{3}^{2}-(-1)^{K}\tau_{1}\rho\ \xi_{1}|<0.\end{split}$ (35)

This case is rather similar to that of TAALs and can be treated in the same
way. The function $\eta(\xi_{3},\xi_{1})$, defined on $\Omega$, is shown in
Fig. 6 for the same materials 2 and 4 in Tab. 1.

Figure 6: The function $\eta(\xi_{3},\xi_{1})$ on $\Omega$ for materials 2 and
4 of Tab. 1.

It is evident that:

  * •

an UD ply can fabricate a PAAL $\forall(\xi_{3},\xi_{1})\in\Omega\iff\max\
\eta(\xi_{3},\xi_{1})<0$;

  * •

an UD ply can produce a PAAL for some lamination points
$(\xi_{3},\xi_{1})\in\Omega\iff\min\ \eta(\xi_{3},\xi_{1})<0$.

Like for function $\varPsi(\xi_{3},\xi_{1})$, the maximum and minimum of
$\eta(\xi_{3},\xi_{1})$ depend on the material parameters $\tau_{0},\tau_{1}$
and $\rho$ and can be found by standard, though slightly articulated,
differential calculus. Like for $\varPsi_{\max}$, also $\eta_{\max}$ is the
same for $K=0$ and $K=1$ orthotropy:

$\begin{array}[]{l}\eta_{\max}=(\tau_{0}-\rho)(2\tau_{1}-\tau_{0}-\rho):=\eta_{1}\
\mathrm{if}\ \rho\geq 2\tau_{1},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\eta_{\max}=\tau_{0}(2\tau_{1}-\tau_{0}):=\eta_{2}\ \mathrm{if}\
\rho<2\tau_{1}.\end{array}$ (36)

To remark that $\eta_{1}=\varPsi_{3}$; $\eta_{1}$ is get at $(0,-1)$ if $K=0$,
at $(0,1)$ if $K=1$ i.e., in the two cases, for cross-ply balanced laminates,
while $\eta_{2}$ is get at $(0,0)$ in both the cases, i.e. for isotropic
laminates. The zones of $\eta_{1},\eta_{2}$ in the plane $(\tau_{1},\rho)$ are
shown in Fig. 7, where the positions of the materials in Tab. 1 are again
denoted by a small circle.

Figure 7: The zones of the maximum of $\eta(\xi_{3},\xi_{1})$. Each circle
corresponds to a material of Tab. 1.

The condition for obtaining a PAAL $\forall(\xi_{3},\xi_{1})\in\Omega$ is
hence

$\eta_{\max}\in\\{\eta_{1},\eta_{2}\\}<0.$ (37)

Also $\eta_{\min}$ can take different values and depends on the type of
orthotropy. For the case $K=0$,

$\begin{array}[]{l}\eta_{\min}=\eta_{1}\ \ \mathrm{if}\ \
\rho^{2}-\tau_{1}\rho+1<0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\eta_{\min}=(\tau_{0}+\rho)(2\tau_{1}-\tau_{0}+\rho)-4:=\eta_{3}\\\ \hskip
82.51299pt\mathrm{if}\ \ \rho^{2}+\tau_{1}\rho-1<0,\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\eta_{\min}=-\left(\dfrac{\tau_{1}\rho-1}{\rho}\right)^{2}-\tau_{0}^{2}-2+2\tau_{0}\tau_{1}:=\eta_{4}\\\
\hskip 19.91692pt\mathrm{if}\rho^{2}-\tau_{1}\rho+1>0\ \mathrm{and}\
\rho^{2}+\tau_{1}\rho-1>0.\end{array}$ (38)

The condition for obtaining a PAAL for some points $(\xi_{3},\xi_{1})$ is
hence, for $K=0$ orthotropic materials,

$\eta_{\min}\in\\{\eta_{1},\eta_{3},\eta_{4}\\}<0.$ (39)

The minimum $\eta_{1}$ is get in $\Omega$ at the point $(0,-1)$ and on the
line $([-1,1],1)$, i.e. for any cross-ply combination and also for a laminate
with all the layer orientations $\delta_{j}=0$ or $\delta_{j}=\dfrac{\pi}{2}$;
$\eta_{3}$ is given by points $(\pm 1,1)$, which is the case of a laminate
with the orientations of all the plies at the angle $\delta_{j}=0$ or
$\delta_{j}=\dfrac{\pi}{2}$ and $\eta_{4}$ by the two symmetric points, on the
boundary of $\Omega$,
$\left(\pm\sqrt{\dfrac{\rho^{2}-\tau_{1}\rho+1}{2\rho^{2}}},\dfrac{1-\tau_{1}\rho}{\rho^{2}}\right)$,
that can be obtained by angle-ply sequences, but not only.

For $K=1$ orthotropic materials, it is

$\begin{array}[]{l}\eta_{\min}=(\tau_{0}-\rho)(2\tau_{1}-\tau_{0}-\rho)-4:=\eta_{5}\\\
\hskip 82.51299pt\mathrm{if}\ \ \rho^{2}-\tau_{1}\rho-1<0,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\eta_{\min}=-\left(\dfrac{\tau_{1}\rho+1}{\rho}\right)^{2}-\tau_{0}^{2}-2+2\tau_{0}\tau_{1}:=\eta_{6}\\\
\hskip 108.12047pt\mathrm{if}\ \ \rho^{2}-\tau_{1}\rho-1>0.\end{array}$ (40)

The condition for obtaining a PAAL for some points $(\xi_{3},\xi_{1})$ is
hence, for $K=1$ orthotropic materials,

$\eta_{\min}\in\\{\eta_{5},\eta_{6}\\}<0.$ (41)

The minimum $\eta_{5}$ is get at the points $(\pm 1,1)$ i.e. by laminates with
all the layer orientations $\delta_{j}=0$ or $\delta_{j}=\dfrac{\pi}{2}$;
$\eta_{6}$ is given by the two symmetric points, on the boundary of $\Omega$,
$\left(\pm\sqrt{\dfrac{\rho^{2}+\tau_{1}\rho+1}{2\rho^{2}}},\dfrac{1+\tau_{1}\rho}{\rho^{2}}\right)$,
also these lamination points can be obtained, but not only, by angle-ply
sequences. Just like $\varPsi_{\min}$, also $\eta_{\min}$ depends on all the
material parameters, but its position on $\Omega$ as well as the formula for
its calculation depend only on $\tau_{1}$ and $\rho$. The zones of the
different $\eta_{i},\ i=1,...,4,$ in the plane $(\tau_{1},\rho)$ are shown in
Fig. 8, where, once more, the positions of the materials in Tab. 1 are also
indicated by a small circle. Apart materials 7, a glass-epoxy ply, and 14, a
composite boron-aluminium, all the layers belong to the zone where
$\eta_{\min}=\eta_{4}$. For all of them, the minimum is get at a lamination
point on the boundary of $\Omega$, that can always be obtained by an angle-ply
sequence, but not exclusively.

Figure 8: The zones of the minimum of $\eta(\xi_{3},\xi_{1})$ for the case
$K=0$, in black, and $K=1$, in grey. Each circle corresponds to a material of
Tab. 1.

##  4 Materials suitable for the fabrication of auxetic laminates

Looking at Figs. 3 and 6, we can see that it is not guaranteed that functions
$\varPsi(\xi_{3},\xi_{1})$ and $\eta(\xi_{3},\xi_{1})$ be negative on $\Omega$
or just on a subset of it. Physically, this means that it is not guaranteed
that a TAAL or also a PAAL can always be realized, for any given anisotropic
ply. For the examples in Figs. 3 and 6, the only condition that is satisfied,
on a subset $\Xi\subset\Omega$, is that on $\eta_{\min}$ for material 2. So,
for the cases at hand, only a PAAL can be realized, by laminates whose
lamination point $(\xi_{3},\xi_{1})\in\Xi$ and made of material 2. The subset
$\Xi$ for material 2 is shown in Fig. 9.

Figure 9: Subset $\Xi\subset\Omega$ where $\eta_{\min}<0$, material 2 in Tab.
1.

Its existence, form and dimensions depend on the material properties
$\tau_{0},\tau_{1}$ and $\rho$.

The possibility of realizing a TAAL or a PAAL with a given layer is
determined, in the space of the dimensionless polar moduli of the ply, by the
conditions on the maximum or on the minimum of $\varPsi(\xi_{3},\xi_{1})$ for
the TAALs or of $\eta(\xi_{3},\xi_{1})$ for the PAALs, jointly to eq. (19). In
the following, we rearrange such conditions for each one of the four cases
treated above. So, for a given UD material with parameters
$\tau_{0},\tau_{1},\rho$:

  * •

it is possible to obtain a TAAL $\forall(\xi_{3},\xi_{1})\in\Omega\iff$

$\left\\{\begin{array}[]{l}\tau_{0}>\rho+2\tau_{1},\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
\tau_{1}\left[\tau_{0}+(-1)^{K}\rho\right]-2>0.\end{array}\right.$ (42)

The corresponding domain in the space $\\{\tau_{0},\tau_{1},\rho\\}$ is shown
in Fig. 10.

Figure 10: Domain of the space $\\{\tau_{0},\tau_{1},\rho\\}$ where TAALs are
possible $\forall(\xi_{3},\xi_{1})\in\Omega$. The exterior, light and
transparent, domain corresponds to materials with $K=0$ and surrounds the,
darker, part of materials with $K=1$.

  * •

a TAAL can be obtained for some $(\xi_{3},\xi_{1})\in\Omega\iff$

$\left\\{\begin{array}[]{l}\tau_{0}-\rho>0,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \tau_{1}\left[\tau_{0}+(-1)^{K}\rho\right]-2>0,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\varPsi_{\min}\in\\{\varPsi_{1},\varPsi_{2},\varPsi_{3},\varPsi_{4}\\}<0.\end{array}\right.$
(43)

The domain in the space $\\{\tau_{0},\tau_{1},\rho\\}$ where TAALs are
possible for some $(\xi_{3},\xi_{1})\in\Omega$ is shown in Fig. 11.

Figure 11: Domain of the space $\\{\tau_{0},\tau_{1},\rho\\}$ where TAALs are
possible for some $(\xi_{3},\xi_{1})\in\Omega$. The exterior, light and
transparent, domain corresponds to materials with $K=0$ and surrounds the,
darker, part of materials with $K=1$.

  * •

a PAAL can be fabricated $\forall(\xi_{3},\xi_{1})\in\Omega\iff$

$\left\\{\begin{array}[]{l}\tau_{0}-\rho>0,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \tau_{1}\left[\tau_{0}+(-1)^{K}\rho\right]-2>0,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\eta_{\max}\in\\{\eta_{1},\eta_{2}\\}<0.\end{array}\right.$ (44)

The domain in the space $\\{\tau_{0},\tau_{1},\rho\\}$ where PAALs are
possible $\forall(\xi_{3},\xi_{1})\in\Omega$ is shown in Fig. 12. The part of
the domain concerning materials with $K=0$ surrounds completely that for
materials with $K=1$

Figure 12: Domain of the space $\\{\tau_{0},\tau_{1},\rho\\}$ where PAALs are
possible $\forall(\xi_{3},\xi_{1})\in\Omega$. The darker, upper part,
$\rho>2\tau_{1}$, corresponds to $\eta_{\max}=\eta_{1}$, the lighter, lower
part, $\rho<2\tau_{1}$, to $\eta_{\max}=\eta_{2}$.

  * •

a PAAL can be fabricated for some $(\xi_{3},\xi_{1})\in\Xi\subset\Omega\iff$

$\left\\{\begin{array}[]{l}\tau_{0}-\rho>0,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \tau_{1}\left[\tau_{0}+(-1)^{K}\rho\right]-2>0,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\eta_{\min}\in\\{\eta_{1},\eta_{3},\eta_{4},\eta_{5},\eta_{6}\\}<0.\end{array}\right.$
(45)

The domain in the space $\\{\tau_{0},\tau_{1},\rho\\}$ where PAALs are
possible for some $(\xi_{3},\xi_{1})\in\Xi\subset\Omega$ is shown in Fig. 13.

Figure 13: Domain of the space $\\{\tau_{0},\tau_{1},\rho\\}$ where PAALs are
possible for some $(\xi_{3},\xi_{1})\in\Xi\subset\Omega$. The darker part
corresponds to materials with $K=0$, the lighter one to $K=1$.

The above results, represented in Figs. 10 to 13, show that the four different
types of auxetic laminates can theoretically exist: just like thermodynamics
allows the existence of isotropic materials with a negative Poisson’s ratio,
in the same way it is physically possible to fabricate the four types of
auxetic orthotropic laminates introduced above.

##  5 Auxetic laminates made of non auxetic materials

Although the results of the previous Section show that the four types of
auxetic laminates can exist, a question still remains: a layer whose
representative point in the space $\\{\tau_{0},\tau_{1},\rho\\}$ belongs to
one of the domains in Figs. 10 to 13 can be fabricated using classical, i.e.
non auxetic, isotropic components (fibres and matrix)? In other words, is it
possible to obtain an auxetic laminate of one of the four types above using
non auxetic materials? To give a response to this question we need to express
the polar moduli of $\mathbb{Q}$, i.e. $\tau_{0},\tau_{1},\rho$ and $K$, by
the five parameters $E_{f},E_{m},\nu_{f},\nu_{m}$ and $v_{f}$, respectively
the Young’s moduli of the fibres and of the matrix, the Poisson’s ratio of the
fibres and of the matrix, and the volume fraction of the fibres. Through a
homogenization criterion, we first calculate the equivalent parameters of the
layer (reference frame of the layer $\\{x_{1},x_{2}\\}$, with $x_{1}$ aligned
with the fibres): $E_{1},E_{2},G_{12}$ and $\nu_{12}$.

In this study, the matrix and the fibres are isotropic non auxetic materials
and the classical rule of mixtures[28, 38] for UD layers is used to evaluate
$E_{1},E_{2},G_{12}$ and $\nu_{12}$. In order to reduce the size of the
problem, also for the technical moduli we introduce dimensionless parameters:

$E:=\frac{E_{f}}{E_{m}},\ \ \ \nu:=\frac{\nu_{f}}{\nu_{m}},\ \ \
G:=\frac{G_{f}}{G_{m}}$ (46)

with (we suppose that the fibres are stiffer than the matrix)

$E>1,\ \ \ -\frac{1}{\nu_{m}}<\nu<\frac{1}{2\nu_{m}}.$ (47)

In the above equations, $G_{f}$ and $G_{m}$ are, respectively, the shear
moduli of the fibres and of the matrix, both assumed to be isotropic:

$G_{f}=\frac{E_{f}}{2(1+\nu_{f})},\ \ \ G_{m}=\frac{E_{m}}{2(1+\nu_{m})}.$
(48)

Through the rule of mixtures we get the following dimensionless constants for
the layer:

$\begin{split}&\overline{E}_{1}:=\frac{E_{1}}{E_{m}}=1+v_{f}(E-1),\\\
&\overline{E}_{2}:=\frac{E_{2}}{E_{m}}=\frac{E}{v_{f}+(1-v_{f})E},\\\
&\overline{\nu}_{12}:=\frac{\nu_{12}}{\nu_{m}}=1+v_{f}(\nu-1),\\\
&\overline{G}_{12}:=\frac{G_{12}}{G_{m}}=\frac{G}{v_{f}+(1-v_{f})G}=\\\
&\hskip 19.91692pt=\frac{(1+\nu_{m})E}{v_{f}(1+\nu\
\nu_{m})+(1-v_{f})(1+\nu_{m})E}.\end{split}$ (49)

We can now calculate the components of $\mathbb{S}$ and, by inversion, those
of $\mathbb{Q}$, then used to calculate $T_{0},T_{1},(-1)^{K}R_{0}$ and
$R_{1}$[25] and finally $\tau_{0},\tau_{1},\rho$ and $K$. The mathematical
details are given in the Appendix, where it is also shown that an UD material
cannot have $K=1$; actually, all the UD plies in Tab. 1 have $K=0$. The final
expressions of $\tau_{0},\tau_{1},\rho$ as functions of $E,\nu,v_{f}$ for a
given $\nu_{m}$ are:

$\begin{split}\tau_{0}&=\frac{\left(\begin{array}[]{c}2E\\{(v_{f}-1)v_{f}+E^{2}(v_{f}-1)v_{f}+\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
E[2v_{f}-2v_{f}^{2}-1+(\nu_{m}+v_{f}(\nu-1)\nu_{m})^{2}]\\}\end{array}\right)}{\left(\begin{array}[]{c}(E-1)^{2}(v_{f}-1)v_{f}[1+(E-1)v_{f}]\times\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\times[v_{f}+v_{f}\nu\nu_{m}-E(v_{f}-1)(1+\nu_{m})]\end{array}\right)}+\\\
&+\frac{2E[1-\nu_{m}+v_{f}(v_{f}-1+\nu_{m}-\nu\nu_{m})]}{v_{f}(1-v_{f})(E-1)^{2}}+\frac{1+E^{2}}{(E-1)^{2}},\end{split}$
(50)
$\tau_{1}=\frac{\left(\begin{array}[]{c}(v_{f}-1)v_{f}+E^{2}(v_{f}-1)v_{f}-\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-2E\\{1+\nu_{m}+v_{f}[v_{f}-1+(\nu-1)\nu_{m}]\\}\end{array}\right)}{v_{f}(v_{f}-1)(E-1)^{2}},$
(51)
$\rho=\frac{\left(\begin{array}[]{c}(E-1)[E(v_{f}-1)-v_{f}][1+(E-1)v_{f}]+\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ +(1+(E-1)v_{f})[E^{2}(v_{f}-1)+v_{f}\nu-\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ -E(1+v_{f}+(v_{f}-2)\nu)]\nu_{m}+\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\
+2E[1+(\nu-1)v_{f}](E-\nu)\nu_{m}^{2}\end{array}\right)}{\left(\begin{array}[]{c}(E-1)[1+(E-1)v_{f}]\times\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\times[E(v_{f}-1)(1+\nu_{m})-v_{f}(1+\nu\nu_{m})]\end{array}\right)}.$ (52)

If the above expressions of $\tau_{0},\tau_{1},\rho$ are used in eqs. (42) to
(45), the domain of existence of the four cases of auxetic orthotropic
laminates can be traced in the space of the variables $E,\nu,v_{f}$, i.e. the
corresponding of Figs. 10 to 13 in the space $\\{E,\nu,v_{f}\\}$ can be found.
They are shown in Figs. 14 to 17 for a matrix having $\nu_{m}=0.3$ (the
influence of this parameter is rather low and in any case does not vary much
from one matrix to another).

Figure 14: Domain of the space $\\{E,\nu,v_{f}\\}$ where TAALs are possible
$\forall(\xi_{3},\xi_{1})\in\Omega$. Figure 15: Domain of the space
$\\{E,\nu,v_{f}\\}$ where TAALs are possible for some
$(\xi_{3},\xi_{1})\in\Omega$. Figure 16: Domain of the space
$\\{E,\nu,v_{f}\\}$ where PAALs are possible
$\forall(\xi_{3},\xi_{1})\in\Omega$. Figure 17: Domain of the space
$\\{E,\nu,v_{f}\\}$ where PAALs are possible for some
$(\xi_{3},\xi_{1})\in\Xi\subset\Omega$.

In all these figures, the plane $\nu=0$ is shown: the parts of the domain
belonging to the half-space $\nu<0$ corresponds to solutions that can be
obtained only if one of the two constituent materials of the layer, the fibres
or the matrix, is itself auxetic, that is, when one of the two constituents is
not a classical, non auxetic elastic material. Looking at Figs. 14-16, we see
that the admissible domains are completely in the half-space $\nu<0$. This
means that it is impossible to fabricate auxetic orthotropic laminates that
are completely auxetic or also partially auxetic for any possible orthotropic
stacking sequence. Only in Fig. 17 the feasible domain is for a part of it in
the subspace $\nu>0$. Physically, this means that only PAALs for some
$(\xi_{3},\xi_{1})\in\Xi\subset\Omega$ can be fabricated using non auxetic
layers. It is worth to notice that this check could be done, theoretically,
directly, adding the condition $\nu>0$ to the inequalities defining the
domains for the four cases; unfortunately, the expressions are so complicate
that this is impossible, in practice. Also, when tracing the domains in the
space $\\{E,\nu,v_{f}\\}$ bounded by eqs. (47), conditions (19) are
automatically satisfied and can be discarded.

##  6 Orthotropic PAALs

Finally, two facts are crucial for obtaining auxetic orthotropic laminates:
the material of the layers and the stacking sequence. An UD layer is suitable
for the realization of a PAAL only if condition (39) is satisfied. In fact,
for a physical, existing, layer, the thermodynamic conditions (19) are
necessarily satisfied, and because UD layers cannot have $K=1$, only condition
(39) must be satisfied in order to obtain a PAAL for some $(\xi_{3},\xi_{1})$
in a subset $\Xi\subset\Omega$, i.e. for some stacking sequences. It is rather
easy to check whether or not condition (39) is satisfied by a layer, and hence
whether or not it is possible to realize with it auxetic laminates: it is
sufficient to calculate its dimensionless polar parameters $\tau_{0},\tau_{1}$
and $\rho$ and then determine $\eta_{\min}$ and check if it is negative. In
particular, for the materials in Tab. 1 it is always $\eta_{\min}=\eta_{4}$,
apart materials 7 and 14, that have $\eta_{\min}=\eta_{1}$. The value of
$\eta_{\min}$ for all these materials is given in Tab. 2.

Table 2: Value of $\eta_{\min}$ for the materials of Tab. 1. Mat. | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
$\eta_{\min}$ | -0.86 | -0.68 | -0.52 | 0.49 | -0.63
Mat. | 6 | 7 | 8 | 9 | 10
$\eta_{\min}$ | 0.91 | 1.96 | -0.87 | -0.57 | -0.42
Mat. | 11 | 12 | 13 | 14 | 15
$\eta_{\min}$ | 0.92 | -0.72 | -0.62 | 2.52 | -0.89
Mat. | 16 | 17 | 18 | 19 | 20
$\eta_{\min}$ | -0.54 | -0.62 | -0.59 | -0.59 | -0.38

Looking at the values of $\eta_{\min}$, we can see that it is not possible to
obtain a PAAL with all the layers; namely, $\eta_{\min}>0$ for materials 4, 6,
7, 11 (glass-epoxy layers) and 14 (boron-aluminium). All of them are
characterized by a low ratio ${E_{1}}/{E_{2}}$, whose value varies between
1.57 to 4.67. For all the other materials, having $\eta_{\min}<0$, this ratio
varies from 9.85, materials 10 and 20, to 46.45, material 15. All these
materials, apart material 1, pine wood, are composites obtained reinforcing a
matrix with stiff fibres (carbon, kevlar, boron). It seems hence that
auxeticity by anisotropy can be get only when anisotropy is sufficiently
strong, practically, reinforcing a matrix with fibres that are much stiffer
than it and with an adequate volume fraction. This is confirmed by Fig. 17,
where the part of the feasible domain with $\nu>0$ has high values of the
ratio $E$.

##  7 Naturally auxetic anisotropic plies

The case of material 1, pine wood, is rather peculiar. Unlike all the other
materials in Tab. 1, it is a naturally auxetic anisotropic ply, in the sense
that, as already observed in the Introduction, it is, by itself, already
auxetic along some directions, see Fig. 1. This happens whenever the point
$(1,1)\in\Xi$, see Fig. 18.

Figure 18: Domain $\Xi$ for material 1 of Tab. 1: pine wood.

The condition for an anisotropic ply to be naturally auxetic can be get from
equation (10): once divided by $R_{1}^{2}$ to obtain its dimensionless form,
the condition is

$\begin{array}[]{c}\mu:=\min\lambda(\theta)<0\Rightarrow\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\mu=2(\tau_{0}\tau_{1})-\tau_{0}^{2}+\rho^{2}-2|1-(-1)^{K}\tau_{1}\rho|<0,\end{array}$
(53)

which gives

$\begin{array}[]{l}\mu_{\min}=(\tau_{0}+\rho)(2\tau_{1}-\tau_{0}+\rho)-4=\eta_{3}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \mathrm{if}\ \ K=0,\
\rho<\dfrac{1}{\tau_{1}},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\mu_{\min}=(\tau_{0}-\rho)(2\tau_{1}-\tau_{0}-\rho)=\eta_{1}\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \mathrm{if}\ \ K=0,\ \rho>\dfrac{1}{\tau_{1}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\mu_{\min}=(\tau_{0}-\rho)(2\tau_{1}-\tau_{0}-\rho)-4=\eta_{5}\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \mathrm{if}\ \ K=1.\end{array}$ (54)

In the case of material 1,
$\rho=0.758<\dfrac{1}{\tau_{1}}=0.896\rightarrow\mu_{\min}=\eta_{3}=-0.559<0.$
In Fig. 19 the admissible domains in the space $\\{\tau_{0},\tau_{1},\rho\\}$
and in the space $\\{E,\nu,v_{f}\\}$ are represented for the case $K=0$. It
can be noticed that also in this case the part of the domain in the half space
$\nu>0$ is null. This means that a UD cannot be naturally auxetic. The fact
that material 1, pine wood, is naturally auxetic just means that its behavior
cannot be reduced to that of a UD ply whose characteristics are evaluated
through the rule of mixtures.

Figure 19: The feasible domains of naturally anisotropic auxetic plies.

##  8 Some examples of PAALs

We consider now some examples of PAALs made using the materials in Tab. 1. In
particular, as criterion for the choice, we have taken for each material the
angle-ply corresponding to the point of the boundary of $\Xi$ where
$\eta_{\min}$ is get. The angle, $\pm\delta$, of the plies is hence

$\delta=\frac{1}{2}\arccos\xi_{3}=\frac{1}{4}\arccos\xi_{1}.$ (55)

In Tab. 3 we list the cases considered, indicating for each of them
$\eta_{3},\eta_{\min}$, the coordinates $\xi_{3}^{\min},\xi_{1}^{\min}$ of
$\eta_{\min}$, the orientation angle $\delta$ and the maximum,
$\nu_{12}^{\max}$, and minimum, $\nu_{12}^{\min}$, of the Poisson’s ratio.

The cases of materials 2, 5 and 15 show clearly that (partially) auxetic
laminates can be obtained stacking UD layers that are not naturally auxetic,
as a simple result of anisotropy.

Table 3: Characteristics of some PAALs; the material numbers refer to Tab. 1. Mat. | $\eta_{3}$ | $\eta_{\min}$ | $\xi_{3}^{\min}$ | $\xi_{1}^{\min}$ |  $\delta$ (∘) | $\nu_{12}^{\max}$ | $\nu_{12}^{\min}$
---|---|---|---|---|---|---|---
1 | -0.56 | -0.86 |  $\pm$0.796 | 0.266 | 18.6 | 1.03 |  $-$0.40
2 | 0.29 | -0.68 |  $\pm$0.681 |  $-$0.073 | 21.5 | 1.42 |  $-$0.32
5 | 0.69 | -0.63 |  $\pm$0.645 |  $-$0.168 | 24.9 | 1.64 |  $-$0.34
15 | 0.08 | -0.89 |  $\pm$0.700 |  $-$0.020 | 22.8 | 2.62 |  $-$0.93
Figure 20: Directional diagrams of $\nu_{12}$ for the four materials in Tab.
3, respectively material 1, 2, 5 and 15 from top to bottom. The dashed line is
$\nu_{12}$ of the ply. The thin circle marks the zero value: inside it,
$\nu_{12}$ is negative.

##  9 Conclusion

The question of the possible realization of auxetic orthotropic laminates
using layers made of non auxetic materials has been considered in this paper.
The use of dimensionless polar parameters to define the material properties of
the layer and of the lamination parameters for the description of the stack
has allowed to define the admissible sets for auxetic laminates. Using the
rule of mixture for the homogenization of the layer’s properties allows us to
show that actually only partially auxetic laminates can be fabricated, for
some admissible stacking sequences, using classical materials. This proves
also that auxeticity can actually be produced simply by anisotropy.

##  10 Appendix

In the Kelvin’s notation[30, 31], the relations between the polar parameters
and the Cartesian components of $\mathbb{Q}$ for a UD ply are[25]

$\begin{array}[]{l}T_{0}=\dfrac{1}{8}(\mathbb{Q}_{11}-2\mathbb{Q}_{12}+2\mathbb{Q}_{66}+\mathbb{Q}_{22}),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
T_{1}=\dfrac{1}{8}(\mathbb{Q}_{11}+2\mathbb{Q}_{12}+\mathbb{Q}_{22}),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
(-1)^{K}R_{0}=\dfrac{1}{8}(\mathbb{Q}_{11}-2\mathbb{Q}_{12}-2\mathbb{Q}_{66}+\mathbb{Q}_{22}),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
R_{1}=\dfrac{1}{8}(\mathbb{Q}_{11}-\mathbb{Q}_{22}),\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ \end{array}$ (56)

Because $R_{0}$ is intrinsically non negative, if $(-1)^{K}R_{0}<0$, then
$K=1$, otherwise $K=0$. The formulae for the polar parameters as functions of
the engineering moduli $E_{1},E_{2},G_{12},\nu_{12}$ of the layer can be
obtained expressing the $\mathbb{Q}_{ij}$s in terms of these last; for an UD
layer

$\begin{array}[]{l}\mathbb{Q}_{11}=\dfrac{E_{1}}{1-\nu_{12}\nu_{21}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\mathbb{Q}_{12}=\dfrac{\nu_{12}E_{2}}{1-\nu_{12}\nu_{21}},\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \mathbb{Q}_{22}=\dfrac{E_{2}}{1-\nu_{12}\nu_{21}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \mathbb{Q}_{66}=2G_{12},\end{array}$ (57)

with $\nu_{21}$ given by the reciprocity relation
$\nu_{21}=\nu_{12}\dfrac{E_{2}}{E_{1}}.$ Replacing these quantities into
equation (56) gives

$\begin{array}[]{l}T_{0}=\dfrac{1}{8}\left[\dfrac{E_{1}(E_{1}+E_{2}-2E_{2}\nu_{12})}{E_{1}-E_{2}\nu_{12}^{2}}+4G_{12}\right],\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
T_{1}=\dfrac{1}{8}\dfrac{E_{1}(E_{1}+E_{2}+2E_{2}\nu_{12})}{E_{1}-E_{2}\nu_{12}^{2}},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
(-1)^{K}R_{0}=\dfrac{1}{8}\left[\dfrac{E_{1}(E_{1}+E_{2}-2E_{2}\nu_{12})}{E_{1}-E_{2}\nu_{12}^{2}}-4G_{12}\right],\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
R_{1}=\dfrac{1}{8}\dfrac{E_{1}(E_{1}-E_{2})}{E_{1}-E_{2}\nu_{12}^{2}}.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \end{array}$ (58)

The dimensionless parameters $\tau_{0},\tau_{1}$ and $(-1)^{K}\rho$ can now be
obtained:

$\begin{array}[]{l}\tau_{0}=\dfrac{E_{1}(E_{1}+E_{2}+4G_{12})-2E_{1}E_{2}\nu_{12}-4E_{2}G_{12}\nu_{12}^{2}}{E_{1}(E_{1}-E_{2})},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\tau_{1}=\dfrac{E_{1}+E_{2}+2E_{2}\nu_{12}}{E1-E2},\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\
(-1)^{K}\rho=\dfrac{E_{1}(E_{1}+E_{2}-4G_{12})-2E_{1}E_{2}\nu_{12}+4E_{2}G_{12}\nu_{12}^{2}}{E_{1}(E_{1}-E_{2})}.\end{array}$
(59)

If in these expressions we set

$\begin{array}[]{c}E_{1}=\overline{E}_{1}E_{m},\ \
E_{2}=\overline{E}_{2}E_{m},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\nu_{12}=\overline{\nu}_{12}\nu_{m},\ \
G_{12}=\overline{G}_{12}G_{m},\end{array}$ (60)

with $\overline{E}_{1},\overline{E}_{2},\overline{\nu}_{12},\overline{G}_{12}$
and $G_{m}$ given by equations (48) and (49), then equations (50) to (52) are
get, giving the polar dimensionless parameters $\tau_{0},\tau_{1}$ and $\rho$
as functions of the dimensionless constants $E,\nu,v_{f}$ and of the Poisson’s
ratio of the matrix, $\nu_{m}$.

In Fig. 21 the parameter $(-1)^{K}\rho$ is shown for three different ratios
$\nu$ and for a matrix with $\nu_{m}=0.3$. It is apparent that in all the
cases $(-1)^{K}\rho>0\ \ \forall(E,v_{f})$. Because $\rho$ is intrinsically a
non negative quantity, $\rho>0\Rightarrow K=0$. Physically, this means that a
UD ply cannot have $K=1$.

Figure 21: The parameter $(-1)^{K}\rho$ for three different ratios $\nu$ and
for $\nu_{m}=0.3$.

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