A comprehensive review on in-plane and through-the-thickness auxeticity in composite laminates for structural applications

Abstract

Auxetic laminates are part of a fascinating branch of materials denominated as auxetics, which display a NPR (Negative Poisson’s Ratio), an uncommon property in the engineering world. Auxeticity is at the root of enhancements in shear and indentation resistance, fracture toughness and energy absorption, making the design of NPR a desirable feat in structural engineering. In composite laminates made of conventional, i.e. positive Poissons ratio, materials, auxeticity is a result of the combination between particular angular configurations and anisotropic materials. Such laminates can be used in a wide array of engineering applications, especially those which require high energy absorption capacity, including aerospace, personal defense and sports industries. This review focuses on particular property enhancements reported in the scientific world brought about by the design and application of IP (In-plane) and TTT (Through-the-thickness) NPR fibre-reinforced polymer laminates under QSI (Quasi-static indentation), LVI (Low-velocity impact) and fatigue solicitations. Furthermore, some insight is given on some possible future paths for further investigation of this topic.

Introduction

Auxetic materials have been of increasing importance in the engineering world. Their unique mechanical behavior, due to a NPR (Negative Poissons Ratio) i.e., a synchronous orthogonal compression or dilation, can generate structural property enhancements on shear stiffness, fracture toughness, indentation and impact resistance. ¹ Generally, auxetics have foam or composite origin. The former takes advantage of the variation of internal voiding with load application. Regarding the latter, the current focus of the area falls on lattice structures with specific stiff material patterns, incorporated in a ductile matrix, which induce auxeticity under loading. The inclusion of an auxetic element in composite laminates - either the fibre or matrix - has also been studied, with reports of a four-fold increase in fibre pull-out resistance with auxetic fibres against conventional ones.^2,3

A further method concerns prompting auxetic behaviour in fibre-reinforced composite laminates made of conventional, i.e. positive Poissons ratio, materials, namely C/E (Carbon/epoxy), G/E (Glass/epoxy), K/E (Kevlar/epoxy) and C/P (Carbon/thermoplastic). Auxeticity in such laminates is a function of material (anisotropy) and structural (ply orientation) properties. UD (Unidirectional) fibre laminas are applied in this type of laminates. A distinction on the auxeticity plane must be defined based on the orthotropic behavior of laminates: IP (In-plane) auxeticity refers to NPR on the face plane of the laminate (ν _xy , ν _yx ); TTT (Through-the-thickness) auxeticity implies NPR along the thickness plane (ν _xz , ν _zx , ν _yz , ν _zy ). The adopted designation for local lamina (x₁-x₂-x₃, or 1-2-3, with x₁ as the fibre direction) and global laminate axis system for this text is depicted in Figure 1. The angle θ between a given UD lamina and the global axis system quantifies ply orientation. Important to note that the results discussed in this review concern major Poisson’s ratios, i.e. ν _xy , ν _xz and ν _yz .

Figure 1.

Adopted direction notation.

Furthermore, regarding the angular notation subsequently represented, the subscripts n, S and T are referent to the number of repetitions of a given set of angles, symmetric and total laminate angle stacking sequences or configurations, respectively.

Auxetic laminates display desirable properties for structural applications, as in the case of the aerospace industry. It is believed they have a beneficial effect on the poor inherent laminate’s LVI (Low-velocity impact) resistance, a deficiency that is very damaging to the industry as a simple tool drop during installation can induce internal delamination. ⁴ Moreover, these laminates can be exploited in mechanical joints to replace typical connectors. ⁵

This review focuses on the enhancements in shear, fracture, QSI (Quasi-static indentation) and LVI resistance prompted by auxeticity in conventional base-material composite laminates, and on the prior work of design and optimization of NPR in both the IP and TTT planes. The literature on the topic, presented in the following sections, focused, on an initial stage, on the design of the best combination of structural, i.e. largest NPR ply orientations, and material properties, chasing the highest anisotropy - inferred by IP moduli E _x /E _y and E _x /G _xy ratios (with E and G as the Young’s and shear modulus, respectively) - possible. Upon the achievement of such combinations, the scientific community turned their attention into the isolation of the auxetic effect and validation of their inherent enhancements. Hence, this review is subdivided into IP and TTT auxeticity design studies, and subsequent work performed to validate indentation, impact and fatigue resistance enhancements.

In-plane auxeticity design

Table 1 summarizes the state of the art of IP NPR detection and validation in composite laminates. Tsai and Hahn studied auxetic bidirectional T300/5208 C/E laminates in 1980, reporting that auxeticity was driven by extension-shear coupling. ⁶ Furthermore, it was observed that distinct ply ratios, p, i.e. the quotient between the number of plies with a given fibre angle and the total number of plies of a laminate (p _i = N _i /N _L for θ _i in a laminate L), ranging from 0.1 to 0.5, have a negligible influence on ν _xy , although larger NPR values were verified for unbalanced, i.e. p₁ ≠ p₂, laminates. Additionally, the authors noted that bidirectional laminates may offer the best IP NPR properties, as an increase in the number of ply orientations complicates analytical study and drives the laminate closer to a quasi-isotropic state. The authors also reported a reduction of E _x in bidirectional auxetic laminates that may taint their performance.

Table 1.

Relevant bibliographic publications regarding IP NPR detection in laminates (Types: Analytical (A); Numerical (N); Experimental (E)).

Year	Lead author	Plane	Material	Type
1980	S. W. Tsai 6	IP	C/E	A
1989	M. Miki 5	IP	C/E	A+E
1998	R. Zhang 7	IP	G/E	A+E
1999	R. Zhang 8	IP	C/E	A
1999	H. Yeh 9	IP	—	A
2011	M. Shokrieh 10	IP	C/E	A

A subsequent study conducted in 1989 by Miki and Murotsu further solidified the idea of superiority of bidirectional configurations, by creating an iterative numerical method based on multidimensional nonlinear programming techniques. ⁵ The authors defined a set of initial angles for 6, 10 and 20-ply T300/5208 C/E laminates, and the solution converged towards unbalanced bidirectional laminates, in which final ply ratios and extreme values of ν _xy depend on the initial conditions. Consequently, by adopting an heuristic approach restricting the number of orientations to two, the minimization of ν _xy became independent from initial conditions, yielding a value of -0.380 for unbalanced ${\left[{\left(14/62\right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$ laminates (for p₁=0.681), albeit confirming that balanced laminates are sufficient to design IP NPR (Figure 2). The study for the maximization of ν _xy resulted in a value of 1.46 for ${\left[{\left(\pm 25\right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$ balanced laminates, a configuration of maximum TTT NPR, as later discussed. Large IP NPR configurations induce large positive TTT Poisson’s ratios, and vice-versa.

Figure 2.

ν _xy vs ply ratio (p₁ = N _n /N _m ) in bidirectional C/E laminates. ⁵

Zhang, H.-Y. Yeh and H.-L. Yeh further analysed the minimization of IP NPR in bidirectional balanced laminates. Experiments on G/E ${\left[{\left(22/68\right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$ laminates with a fibre volume fraction, v _f , of 0.92, yielded a small IP NPR of -0.05⁷. The same group of authors conducted additional analysis on the matter by analytically investigating the influence of lamina properties (E _f and E _m , subscripts f and m for fibre and matrix, respectively) on T300/5208 C/E laminate constants (E _x , E _y and G _xy ), concluding that increasing E _f and decreasing E _m lead to an increase of E _x and decrease of E _y , respectively, increasing anisotropy and auxeticity. ⁸ Different parameters were used to evaluate the maximum IP NPR configurations (Table 2), including A_NPR, a condition to produce NPR calculated by terms of the laminate’s extensional stiffness matrix A (A₁₆A₂₆ − A₁₂A₆₆), a parameter that is positive for IP NPR laminates and that is higher with larger auxeticity. The fluctuation of results can be explained due to a relative variation of ɛ _x and ɛ _y with stress - hence a larger ɛ _y might not equate to a larger NPR - and by the independence of A_NPR from the strain and stress fields.

Table 2.

Bidirectional configurations for the optimal values of ${\nu }_{x{y}_{\min }}$ $\left({\left[{\left(62.5/15\right)}_{\mathrm{n}}\right]}_{\mathrm{S}}\right)$ , $A_{{\text{NPR}}_{\max }}$ $\left({\left[{\left(68.5/21.5\right)}_{\mathrm{n}}\right]}_{\mathrm{S}}\right)$ and ${\epsilon }_{y_{\max }}$ $\left({\left[{\left(65/25\right)}_{\mathrm{n}}\right]}_{\mathrm{S}}\right)$ . ⁸

Configuration	ν xy	A NPR	ɛ y (%)	ɛ x (%)
${\left[{\left(62.5/15\right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$	−0.326	26.58	0.0572	0.175
${\left[{\left(68.5/21.5\right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$	−0.277	31.34	0.0679	0.245
${\left[{\left(65/25\right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$	−0.257	29.00	0.0742	0.288

Additional work from these authors on auxeticity in randomly orientated quasi-isotropic composite laminates reported that the decrease of E _y /E _x , increase of G _xy /E _x and decrease of ν _xy produce the largest values of IP NPR. ⁹

Shokrieh and Assadi developed an optimization algorithm based on thickness ratios (design variable) and thickness normalization of T300/5208 C/E laminates to calculate the minimum major Poisson’s ratio in xy. A ν _xy of −0.3536 was reached in unbalanced ${\left[{15}_2,{60}_1\right]}_T$ laminates, reporting that the minimum number of laminas to produce auxeticity is 3. ¹⁰

Although the present work does not focus on auxeticity in laminates of elastomer-based matrices, it should be mentioned that these type of laminates can achieve very large IP NPR values. An honourable mention should be given to the work of Peel in the study of the behaviour of fibre-reinforced elastomer laminates, and later with the findings of improved damping on laminates with ν _xy in the region of −6.^11,12 Another possible application for this type of laminate was studied by Chen et al., who also explored the effect of IP NPR in carbon fibre-reinforced elastomer laminates on the design, manufacturing, characterization and parametric modelling of auxetic flexible skins for aircraft wings. ¹³

Through-the-thickness auxeticity design

Table 3 displays an overview on the research performed on the topic of TTT auxetic laminates.

Table 3.

Relevant bibliographic publications regarding TTT NPR detection in laminates.

Year	Lead author	Plane	Material	Type
1984	C. T. Herakovich 14	TTT	C/E	A
1994	J. F. Clarke 15	TTT	C/E	A+E
1997	P. J. Hine 16	TTT	C/E	A+E
2007	H. Harkati 17	TTT	C/E, G/E,	A
			K/E
2009	H. Harkati 18	TTT	C/E	A

The first analytical publication dates to 1984, when Herakovich reported the possibility of TTT auxeticity in ${\left[{\left(\pm \theta \right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$ T300/5208 C/E laminates due to a high degree of extension-shear coupling. ¹⁴ Concurrently, this phenomenon generates high IP shear stresses and a very high IP Poisson’s ratio, as previously mentioned. In the xz plane, the NPR domain occurs between 15^◦ and 40^◦, with a minimum of -0.21 for a θ of 25^◦. Results for the yz plane are similar, albeit phased out about the 45^◦ orientation (Figure 3). TTT NPR combinations were found not to be restricted to ${\left[{\left(\pm \theta \right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$ , with accounts of auxeticity in ${\left[{\left(0_2/\pm \theta \right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$ and ${\left[{\left(0/\pm \theta \right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$ laminates.

Figure 3.

ν _xz and ν _yz for ${\left[{\left(\pm \theta \right)}_n\right]}_{\mathrm{S}}$ T300/5208 C/E laminates. ¹⁴

Experiments by Clarke et al. in 1994 proved the existence of TTT NPR in ${\left[{\left(\pm \theta \right)}_{\mathrm{n}}\right]}_{\mathrm{S}}$ balanced laminates. An ultrasonic velocity technique was employed to measure all six Poisson’s ratios in 24-ply ${\left[{\left(\pm \theta \right)}_6\right]}_{\mathrm{S}}$ T300/913 pre-preg carbon/epoxy laminates with v _f =50%, and with θ = 0, 10, 15, 20, 25, 30 and 40°, together with analytical predictions for the remaining angle domain. A negative ν _xz was reported, for the experimental data, between 20° and 30°, with this range again coinciding with a maximum in-plane Poisson’s ratio as a constancy in material volume must be met. ¹⁵

This work was further developed by Hine et al. in 1997. ¹⁶ Maintaining the same operational conditions from the previous work, with the exception of the usage of a high-modulus Courtaulds HM370 carbon fibre (E₁= 350 GPa) in order to increase the material’s anisotropy, for ${\left[{\left(\pm \theta \right)}_6\right]}_{\mathrm{S}}$ laminates with θ = 0, 10, 20 and 40°. Analytical predictions showed that a ν _xz of −0.5 was possible for θ = 23°, with an IP Poisson’s ratio of +2, and that the xz TTT NPR domain ranged from 10° to approximately 40°, which was corroborated by the experimental trials conducted.

Harkati et al. further developed the topic analytically, by modelling the influence of orientation and fibre reinforcement on TTT Poison’s ratio in C/E, G/E and K/E eight-ply laminates with a v _f of 70%. ¹⁷ With [±β/±θ]_S configurations, no TTT NPR was detected in G/E laminates, with maximum xz NPR of −0.243 for K/E ${\left[{\left(\pm 25\right)}_2\right]}_{\mathrm{S}}$ laminates, and −0.746 for C/E ${\left[{\left(\pm 20\right)}_2\right]}_{\mathrm{S}}$ laminates. Some further key observations remitted to the possibility of NPR design in β = 0° laminates but not with 90°, a minimum ν _xz did not lead to an extreme in IP moduli and the zeros of the 2^nd derivative of E _x and G _xy coincided with the location of the minimum ν _xz .

A subsequent publication by the same lead author focused on the influence of carbon fibre and resin type, and fibre volume fraction on the xz TTT Poisson’s ratio on ${\left[{\left(\pm \theta \right)}_4\right]}_{\mathrm{S}}$ C/E laminates. ¹⁸ UM (Ultra-high modulus) graphite fibres produced the most anisotropic and auxetic (ν _xz = −0.713, θ = 20°) laminates. Results for ν _xz were similar, in terms of behavior, for different epoxy resins (914 and 5208) and different fibres types versus θ (AS4 and IM6 carbon), respectively, with variations arising only from distinct anisotropy ratios. The study of the influence of fibre volume fraction in graphite/epoxy laminates showed that peak anisotropy is reached with a very reduced v _f (33.6%), and that increasing this fraction in domains usually applied in aerospace engineering (v _f ∈ [65;80] %) reduces auxeticity (Figure 4).

Figure 4.

Variation of E _x /Ey with v _f . ¹⁸

Quasi-static indentation and impact resistance enhancements

Several studies have been conducted on QSI and LVI enhancements of auxetic origin (Table 4). Most of the experimental work in the area focused on the isolation of the auxetic effect from other factors that could influence said enhancements, only achieved with the design of suitable control groups. For this purpose, the FORTRAN program developed by Wenchao and Evans in 1992 was key - unlike commercial software packages, in which it was necessary to conduct an approximate manual trial-and-error optimization study on an initially known lay-up configuration, this algorithm automatically calculates the optimal lay-up configuration based on input design requirements and on lamina mechanical properties and thickness, by iteratively minimizing the difference between calculated and required properties. ¹⁹

Table 4.

Relevant bibliographic publications regarding QSI and LVI resistance enhancements in auxetic laminates.

Year	Lead author	Plane	Material	Type
2004	K. Evans 20	IP, TTT	C/E	A+E
2005	K. Alderson 3	TTT	C/E	E
2008	K. Alderson 4	TTT	C/E	E
2011	V. Coenen 21	TTT	C/E	E
2019	C. Gonçalves 22	IP	C/E, C/P	E
2022	Y. Wang 23	TTT	C/E	A+N

By employing this algorithm, Evans et al. determined control configurations for auxetic IP [0/15/75/15]_S and xz TTT ${\left[{\left(\pm 30\right)}_6\right]}_{\mathrm{S}}$ laminates with matched elastic moduli but PPR (Positive Poisson’s ratio): [0/− 70/10/25]_S and [35/− 20/25/40/− 85/40/25/− 45/35/15/25/40]_S (Tables 5 and 6) with matched E _x and E _z , respectively, with constant volume fractions and overall laminate dimensions. ²⁰ Alderson et al. defined an extra control laminate for the TTT case with a ZPR (Near-zero Poisson’s ratio) ν _xz - ${\left({\left[0/-45/5/40\right]}_{\mathrm{S}}\right)}_3$ . ³ There was a noticeable enhancement of predicted shear modulus in auxetic TTT laminates when compared to its counterparts.

Table 5.

Predicted elastic constants for AS4/3501-6 C/E laminates with negative ([0/15/75/15]_S, NPR) and positive ([0/− 70/10/25]_S, PPR) ν _xy . ²⁴

Configuration	ν xy	E x (GPa)	E y (GPa)	G xy (GPa)
NPR	−0.134	63.4	27.2	6.29
PPR	0.466	60.2	25.6	8.57

Table 6.

Predicted elastic constants for AS4/3501-6 C/E laminates with negative ( ${\left[{\left(\pm 30\right)}_6\right]}_{\mathrm{S}}$ , NPR), near-zero ( ${\left({\left[0/-45/5/40\right]}_{\mathrm{S}}\right)}_3$ , ZPR) and positive ([35/− 20/25/40/− 85/40/25/− 45/35/− 15/25/40]_S, PPR) ν _xz . ³

Config	ν xz	E x (GPa)	E y (GPa)	E z (GPa)	G xy (GPa)
NPR	−0.156	49.6	11.9	9.5	26.5
ZPR	0.086	75.3	20.1	9.6	19.3
PPR	0.187	49.8	25.0	9.6	20.3

In the aftermath of the definition of aforementioned control groups, three papers from the authors K. L. Alderson, A. Alderson and V. L. Coenen, amongst others, were published on the topic, concerning experimental impact testing on 80×80×3 mm 24-ply vacuum bagged IM7/5882 C/E pre-preg panels, for the TTT case, subjected to flexural QSI and LVI tests under similar constraint conditions.^3,4,21 Results indicated that the level of auxetic enhancements is a function of the failure mechanism(s) and the level of impact energy.

Preliminary QSI tests conducted with a 12.7 mm diameter hemispherical nose indenter, with 1.76 kg, showed higher sustained load and energy absorption to first failure (Table 7) and peak load on auxetic specimens. A fractography analysis of the damaged samples showed very localized fibre breakage and few delaminations on the TTT auxetic plates, and larger delaminations on the control laminates. The enhancements verified were originated via two damage limitation mechanisms: ³

• Local densification: The contraction of an auxetic laminate forces the material to flow to the indented area due to a synchronized TTT orthogonal contraction, increasing local density and enhancing local and shear resistance, and decreasing the global span of delamination damage (not without severe fibre breakage that negatively affects residual properties, as discussed later ⁴ );

• Lamina mismatch: Auxetic bidirectional TTT configurations have an inherent balanced lamina mismatch (θ and −θ) that sequentially cancels out adjacent plies’ interlaminar stress and orderly distributes shear strain through the laminate’s thickness, limiting delamination. In auxetic bidirectional θ₁ and θ₂ IP laminates, albeit with a similar 50 to 60 absolute angle mismatch as the former, interlaminar stresses add up, magnifying delamination ²⁴ - unbalanced mismatch.

Table 7.

Energy absorbed by IM7/5882 C/E TTT plane laminates in QSI tests to first failure. ²¹

Indenter diameter	Energy absorbed (J)
	Negative ν xz	Near-zero ν xz	Positive ν xz
2 mm	0.98	0.75	0.97
12.7 mm	2.38	1.37	1.83
20 mm	1.61	1.50	1.27

A follow-up study assessed the damage mechanisms induced by QSI using three distinct indenter nose diameters: 2 mm (with full specimen penetration), 12.7 mm (5 mm penetration) and 20 mm (up to catastrophic failure). Fractography and C-scans revealed a reduction in internal damaged area (Table 8) and extension of damage with the 2 mm and 12.7 mm diameter indenters in the auxetic specimens. However, enhancements in load sustained and energy absorbed in first and full failure were only observed with the roundest, or largest, indenters (12.7 mm and 20 mm), and decrease with the increase of nose diameter. ²¹ Through the results of this TTT study it could be concluded that:

• Delamination is the dominant failure mechanism with rounder indenter noses - the delamination load onset threshold in auxetic specimens is increased due to their previously mentioned limitation mechanisms, hence when delamination is induced, its propagation will be truncated;

• Fibre breakage is the dominant failure mechanism with sharper indenter noses - once the fibre breakage onset threshold is reached, there is no suppression for it as in the former. Although delamination is still constrained, due to the aforementioned damage limitation mechanisms, there is a bias towards fibre breakage, coupled with a higher concentration of force (due to the reduced area of load application) that will target fibres in a more catastrophic way without a limitation mechanism, auxetic enhancements are reduced or nonexistent.

Table 8.

Damaged area as a function of the total plate area in IM7/5882 C/E TTT plane laminates in QSI tests. ²¹

Indenter diameter	Damage area (%)
	Negative ν xz	Near-zero ν xz	Positive ν xz
2 mm	7	11	10
12.7 mm	10	14	15

LVI studies were conducted with a 12.7 mm diameter indenter nose using different energy levels, defined in accordance with static load-displacement curves: 2 J or 4 J (before first failure), 7 J (just after first failure), 12 J (at peak failure), and 18 J (to induce catastrophic failure, i.e. full back surface failure).^3,4 No damage was reported at the first level, as expected. Enhancements were reported at 7 and 12 J, with C-scan and fractography damage assessment revealing analogous patterns with large delaminations near the front face in positive and near-null ν _xz specimens, and only one small delamination towards the back face of auxetic samples. At an impact of 18 J, several damage mechanisms were activated, including matrix crushing under the impact site, severe fibre breakage leading to back surface failure, large delaminations and fibre splitting. The auxetic specimens showed localized damage around the impact area with smaller delaminations, however enhancements between configurations were almost negligible (Table 9). It is noteworthy that the first failure load in auxetic samples was the highest load recorded during impact, making it the most significant failure event in auxetic specimens.

Table 9.

Energy absorbed by IM7/5882 C/E TTT (xz plane) laminates in LVI tests at first failure.^3,4

Impact energy	Energy absorbed (J)
	Negative ν xz	Near-zero ν xz	Positive ν xz
7 J	3.8 ± 0.2	2.6 ± 0.2	2.7 ± 0.4
12 J	3.4 ± 0.1	2.6	2.8 ± 0.2
18 J	3.3 ± 0.2	2.8	3.1 ± 0.1

These findings highlight an energetic rate sensitivity of auxetic laminates, as energy absorption enhancements degrade with the increase of loading, specially after first failure - a trend only verified for auxetic samples. The significance of this impact event, allied with a more localized damaged area, is further denounced by the lesser residual properties in these specimens in response to a follow-up impact on the damaged area of 7 and 12 J impacted specimens. ⁴ Yet, this localization of damage is a plus for aerospace applications, as damaged areas are smaller, which makes them easier and cheaper to repair, simplifying maintenance operations. ²⁵

A numerical FEM (Finite Element Method) LVI impact study by Wang was executed on ten-ply IM7/977-3 C/E laminates with TTT NPR ([25₂/− 25₂/25₂/− 25₂/25₂]), and positive ν _xz : one configuration with matched E _x and E _z to low tolerance (≤0.7%), and other with all Young’s moduli matched to higher tolerances (Configurations 1 and 2 of Table 10, respectively). Three levels of energy (3, 5 and 8 J, without penetration) were applied using a 2 kg impactor with a 16 mm diameter indenter nose. The auxetic lay-up registered higher impact forces, shorter impact times and less energy dissipation, effects which intensified with higher impact energy. A lower maximum displacement in the TTT NPR laminate was observed for lower energy levels dominated by the NPR dominant factor, but at 8 J an increased E _y in Configuration 1 was dominant. ²³

Table 10.

Laminate configurations for xz auxetic ([25₂/− 25₂/25₂/− 25₂/25₂]) and matched-moduli positive ν _xz laminates (Configuration 1 (C1): [50₂/0₂/50₂/0₂/50₂]; Configuration 2 (C2): [20₂/10₂/5₂/10₂/20₂]). ²³

Configuration	ν xz	E x (GPa)	E y (GPa)	E z (GPa)
Negative ν xz	−0.327	70.8	9.45	9.95
C1	0.264	70.3	14.2	9.95
C2	0.260	71.9	9.46	9.24

Important to note that the maximum energy level on this study (8 J) is close to the first failure energy level from Alderson and Coenen’s study (7 J), where a reduction in enhancements with increasing energy levels was verified specially beyond 7 J⁴. As this numerical study did not use larger energy levels, it should not be expected that the trend of enhancement escalation with increasing impact energy would persist as it would then oppose experimental observations, with consideration for the distinct indentation, configuration and material conditions.

Damaged area evaluation showed a two-fold increase of delamination in auxetic samples compared to Configuration 2 - a problem that worsens with increasing impact energy - with potentiated longitudinal growth and restricted transverse growth. Matrix compressive damage (often negligible) was also larger. The energy enhancements emerged from significant reductions in matrix (40%) and fibre (42% on average) tensile damaged areas, which are more sensitive to TTT NPR than to higher moduli: the synchronized orthogonal contraction during impact contracts the laminate in the x direction, reducing tensile damage. ²³ It should be noted that an experimental recreation of this study could prove difficult due to the singularization of each one of the addressed damage mechanisms, although this numerical analysis provides a relevant insight into the problem at hand.

LVI influence on IP NPR laminates was analysed by Gonçalves et al. with eight-ply [0/15/75/15]_S and ${\left[{\left(0\right)}_8\right]}_{\mathrm{S}}$ Grafil 34-700/CR83 C/E and Grafil 34-700/2426 H (LDPE, Low-Density Polyethylene) C/P laminates manufactured by vacuum infusion moulding and film stacking compression, respectively, with similar thickness, areal density and fibre volume fraction. Tensile tests on the C/E indicated worse strength, modulus and elongation at break for the IP auxetic laminates, with a NPR of −0.12 against a 0.33 ν _xy for the unidirectional control sequence. Auxetic specimens fared better in LVI tests performed with a 20 mm diameter hemispheric steel striker of 10.044 kg, at an energy level of 49 J, with superior energy absorption, a phenomenon more expressive in C/P samples due to the the high plastic deformation of LDPE. ²² This is an interesting result within a topic (C/P auxetic laminates) that should be examined further, specially as big manufacturers replace thermoset for thermoplastic matrices that can be recycled and reused.

Fatigue and fracture enhancements

Table 11 exhibits research performed on fatigue, and subsequent fracture, of auxetic laminates, critical parameters for mechanical performance – e.g., in the aerospace industry, auxetics laminates can reduce or simplify maintenance operations. ²⁵

Table 11.

Relevant bibliographic publications regarding enhancements in fatigue and fracture toughness of auxetic laminates.

Year	Lead author	Plane	Material	Type
2009	J. Donoghue 24	IP, TTT	C/E	E
2009	A. Bezazi 26	TTT	C/E	E

Donoghue et al. explored fracture behaviour on 16-ply AS4/3501-6 C/E pre-preg vacuum bagged laminates manufactured with double edge notched DEN specimens. The configurations shown in Tables 5 and 6 were used except for a near-zero ${\nu }_{xz}{\left({\left[-10/40/-40/40\right]}_2\right)}_3$ lay-up. In the IP plane, due to the aforementioned damage limitation mechanisms, auxetic samples showed larger crack opening displacement, i.e. displacement for the onset of a crack, as those specimens absorbed higher strains to first ply (crack propagation) and ultimate failure, specially for notch lengths above 8 mm. The critical stress intensity factor K _Ic fell faster with notch length for positive ν _xy samples (Figure 5), indicating less notch sensitivity of auxetic laminates that can have relative benefits in inhibiting the growth of large cracks. The authors refer that, as K _Ic could not be matched (it is higher for the control lay-up due to its higher tensile strength and due to the negative effect of unbalanced angular mismatch of IP NPR lay-ups), fracture toughness with respect to strength measurements is perhaps not the most appropriate manner to assess NPR enhancements. ²⁴

Larger values of compliance and a steeper rise of its value with increasing notch length were reported in TTT auxetic samples, indicating higher strain energy release rate and fracture toughness: the increased thickness and reduced width of auxetic samples, under tensile loading, distributes stress concentration at the crack tip over a wider area, reducing stress concentration and the probability of crack propagation onset. Furthermore, their balanced angular mismatch limits delamination (Table 12). ²⁴

Figure 5.

Variation of critical stress intensity parameter (K _Ic ) with notch length for AS4/3501-6 C/E [0/15/75/15]_S and [0/− 70/10/25]_S laminates. ²⁴

Table 12.

Strain energy release rate (G _I ) and plane strain fracture toughness (K _I ) for AS4/3501-6 C/E negative ([±30]_6s), near-zero ${(\left({\left[-10/40/-40/40\right]}_2\right)}_3)$ and positive ([35/− 20/25/40/− 85/40/25/− 45/35/15/25/40]_S) ν _xz laminates with matched-moduli. ²⁴

Configuration	ν xz	G I (×104Nm−1)	K I (×107Nm−2√m)
Negative ν xz	−0.156	5.80	5.45
Near-zero ν xz	0.017	2.84	3.75
Positive ν xz	0.187	3.06	3.06

Bezazi et al. analysed eight-ply T300/914 C/E laminates with TTT auxetic configurations ( ${\left[{\left(\pm 20\right)}_2\right]}_{\mathrm{S}}$ , ${\left[{\left(\pm 25\right)}_2\right]}_{\mathrm{S}}$ and ${\left[{\left(\pm 30\right)}_2\right]}_{\mathrm{S}}$ ), with a fibre volume fraction of 60%, under cyclic fatigue. Static bending tests displayed higher stiffness, strength, lower strain at failure and more delamination with lower θ values. Fatigue tests for ${\left[{\left(\pm 20\right)}_2\right]}_{\mathrm{S}}$ , ${\left[{\left(\pm 25\right)}_2\right]}_{\mathrm{S}}$ under a N₅ criteria, i.e. a 5% decrease of the load in relation to the initial value, showed more resistance from the former, with the latter dissipating less energy for a given loading level, specially with its increase, and exhibiting in its S-N curve (i.e. the number of cycles, N, a material can withstand under a repeated stress range S before failure) a faster decay of fatigue resistance over a number of cycles. The fracture topography of the samples revealed that the main damage mechanism was delamination. ²⁶

Discussion

Auxetic laminates occupy an interesting position in the scientific and industrial world, as their mechanical enhancements seem to display structural enhancements, specially in shear, indentation resistance and fatigue and fracture, with consequential improved energy absorption, that can be positively exploited in mechanical design of composite laminate parts subjected to low-velocity impact and cyclic loading.

More studies are necessary in the area to provide more knowledge regarding the limitations of auxetic laminates, i.e. up to what threshold of impact conditions is it beneficial to apply an auxetic configuration. It is proven that in laminates with similar IP stiffness, auxeticity leads to improved performance,^3,4 however a reduction of elastic moduli is to be expected in such stack-up sequences which can negatively affect the property enhancements achieved by auxeticity. Furthermore, there seems to be an impact energy limit from which impact resistance and damage extension in auxetic laminates becomes close to the one of its positive counterpart in configurations with similar IP stiffness. ⁴ Further studies in this area are necessary to delineate the full range of impact capabilities of auxetic laminates.

Moreover, the efficiency of auxetic enhancements varies with the geometry of a given impactor - that can prompt a dominance of a certain damage mechanism. ²¹ Auxetic laminates proved to be effective on delamination-dominant cases, but not under penetration impacts that generate large fibre-breakage phenomena. An experimental approach to this question is rather complex, hence more numerical studies - such as the one by Wang ²³ - could be developed, specially in order to better analyze and correlate different impactor geometries and induced damage mechanisms, and to explore larger levels of energy in order to make a bridge with the already performed experimental work, particularly in order to numerically validate the evolution of the delamination suppression mechanism.

With regards to the analysis of fracture enhancements, as mentioned by Donoghue et al., ²⁴ new approaches of property matching are necessary, as investigating fracture toughness with respect to strength measurements method is perhaps not the most appropriate manner to assess NPR enhancements.

The effectiveness of conventional auxetic laminates should also be explored in more areas of structural mechanics, for example, for dynamic purposes, such as in the case of vibration damping, for which auxetic fibre-reinforced elastomers have already yielded positive results. ¹²

Nevertheless, the field of auxetic laminates has a lot of potential. Although it might not have the reach of lattice auxetic structures, it still does tap into the domain of composite laminates, vastly used in the industrial world. Overall, the study of auxetic materials is still relatively recent, and the development of this domain in the upcoming years is certainly a space to watch closely within the scientific community.