1 Introduction

From the Greek work αὔξησις (auxesis), a noun meaning “increase”, comes the word αὐξητικός (auxetikos), which means “that which tends to increase”; the latter leads to the term “auxetics”, which has been coined for referring to materials and structures that exhibit negative Poisson’s ratio [1]. Due to the exponential increase in the research activities on auxetic materials and systems in recent years, it is no longer possible to provide a comprehensive list of auxetics literature; the reader is therefore referred to comprehensive reviews [2,3,4,5,6,7,8,9,10,11,12,13,14,15] and two monographs [16, 17] in the field of auxetics. While the introduction of auxetic materials paves a way for designing materials for structural applications in which it is advantageous to possess negative Poisson’s ratio, there are also other structural applications where it is advantageous to possess positive Poisson’s ratio. In cases where both conditions take place, it is prudent for such materials and structures to be able to possess both positive and negative Poisson’s ratio, depending on the loading condition. A preliminary design on such system has been attempted, in which the sign of Poisson’s ratio switches between positive and negative values in situ, in response to the change in stress direction, e.g. vxy > 0 when σxx > 0, but the Poisson’s ratio sign switches to vxy < 0 when σxx < 0, thereby always giving εyy < 0 regardless of whether σxx is tensile or compressive [18]. Following that, an attempt was made to explore the possibility in auto-locking of fibers onto their surrounding matrix materials, i.e. the radial dimension tends to increase regardless of whether the fiber is being pulled or being pushed, so as to prevent being displaced axially. By means of microstructural duality, each microstructure can be made to manifest two distinct effective microstructures such that vxy < 0 when σxx > 0, but the Poisson’s ratio sign toggles to vxy > 0 when σxx < 0, thereby always giving εyy > 0 regardless of whether σxx is tensile or compressive [19], i.e. the dimension that is lateral to the stress application tends to increase, thereby indicating auxetikos behavior transverse to the load line. Having shown the capability of materials to exhibit Poisson’s ratio sign switching upon stress direction reversal, it has recently been proven that the coefficient of thermal expansion (CTE) sign can be switched between positive and negative values in situ, in response to thermal fluctuation, specifically αeff > 0 when dT < 0, but the effective CTE sign flips to αeff < 0 when dT > 0, thereby always giving negative thermal strain ε < 0 regardless of whether the temperature increases or decreases [20], which is not auxetikos. Recently a class of metamaterial consisting of rectangular cells in triangular array has been conceptualized using alternating bimaterials that are joined to the ends of cross beams, whereby the direction parallel to the cross beams can be designed to be positive thermal expansion (PTE) or negative thermal expansion (NTE) while the direction parallel to the bimaterials always give negative thermal strain [21]—again, such a system is not auxetikos. A follow-up work on a 2D microstructure was then made by aligning the alternating bimaterial strips parallel to both axes to form rectangular cells. These cells are arranged in rectangular array and are interconnected by cross beams oriented along both axes to give in-plane PTE, NTE, thermal shearing or zero thermal expansion (ZTE) [22]. As before, such a system is not auxetikos. A graphical description that shows how the present auxetikos system relates with positive thermal expansion (PTE) and negative thermal expansion (NTE) systems is illustrated in Fig. 1. For a temperature change of dT, recall that the that the thermal strain is defined as [23,24,25]

$$\varepsilon = \alpha \,dT,$$
(1)

where the proportionality constant α is the CTE [23,24,25].

Fig. 1
figure 1

For a given a temperature variation with time, the responses for thermal strain with reference to time (left column), thermal strain with temperature (middle column) and CTE with temperature (right column) are illustrated for b positive thermal expansion, c negative thermal expansion, and d auxetikos systems

Full size image

It follows that under a thermal fluctuation with time t as shown in Fig. 1a, the conventional response in terms of thermal strain is shown in Fig. 1b in which the positive slope of the thermal strain versus temperature change plot indicates positive CTE α > 0, i.e.

$$\left. {\begin{array}{*{20}c} {\varepsilon > 0 \Leftarrow dT > 0} \\ {\varepsilon < 0 \Leftarrow dT < 0} \\ \end{array} } \right\} \Rightarrow \alpha = \frac{\varepsilon }{dT} > 0$$
(2)

If, however, the material responds in the manner shown in Fig. 1c, i.e. the sign of the thermal strain always opposes the sign of the temperature change, then the corresponding negative slope of the thermal strain versus temperature change plot indicates negative CTE α < 0, i.e.

$$\left. {\begin{array}{*{20}c} {\varepsilon < 0 \Leftarrow dT > 0} \\ {\varepsilon > 0 \Leftarrow dT < 0} \\ \end{array} } \right\} \Rightarrow \alpha = \frac{\varepsilon }{dT} < 0$$
(3)

Suppose the material behaves in the manner described in Fig. 1d, whereby the thermal strain is non-negative, then the CTE sign toggles between positive and negative depending on how the temperature changes; specifically,

$$\varepsilon > 0 \Leftarrow \left\{ {\begin{array}{*{20}c} {dT > 0 \Rightarrow \alpha = \varepsilon /dT > 0} \\ {dT < 0 \Rightarrow \alpha = \varepsilon /dT < 0} \\ \end{array} } \right.$$
(4)

Note that the thermal strain for auxetikos system is always positive to indicate it having the tendency to increase in dimension. This characteristic translates into two distinct slopes of the plot of thermal strain versus temperature, and the corresponding abrupt change in CTE at the original state. There has been no precedence whereby any material system expands based on temperature change magnitude, and therefore the auxetikos system considered herein is being presented for the first time.

In this paper, we attempt the development of a 2D auxetikos material system that tends to increase in dimension under both heating and cooling effect, i.e. αx = αy > 0 when dT > 0, but the effective CTE sign changes to αx = αy < 0 when dT < 0. The implementation of alternating CTE sign with thermal fluctuation is attained herein by incorporating a set of rotating rods that are redundant during heating but functional during cooling. These are indicated as inclined linkages that are parallel to the shuriken sides in the original state illustrated in Fig. 2a, while each shuriken is connected to its four closest neighbors by the horizontally and vertically aligned connecting rods. Upon cooling, all rods contract such that the hinges at the corners move away from the shuriken to a greater extent than the shortening of the connecting rods (Fig. 2, left), thereby creating an overall distancing between the shurikens. This translates to effective NTE behavior during cooling. Upon heating, only the connecting rods elongate (Fig. 2, right); the rotating rods encounter build-up of compressive thermal stress as the hinges are locked at the shuriken corners. For this reason, the next section on analysis consists of two parts for catering to two different overall expansion mechanisms.

Fig. 2
figure 2

Schematics of 3-by-3 interconnected shuriken network under cooling (left) and heating (right) at a original state, b 5% rod strain magnitudes and c 10% rod strain magnitudes. Green squares indicate original size of 3-by-3 unit cells for comparison

Full size image

2 Analysis

We consider a network of interconnected shuriken whereby every shuriken possesses four axes of symmetry, i.e. two axes of symmetry with each axis passing through the opposing sharp edges, and two axes of symmetry with each axis of symmetry passing through the opposing corners. Figure 3a shows a unit cell of the interconnected shuriken network, where the rotating rods and connecting rods in the original state are indicated by purple color, forming an angle of θ between them. The rotating rod is of length lr while the entire connecting rod is of length 2lc for connecting two shurikens. The half-length of each connecting rod lc is therefore assigned to each connecting rod within the boundary of the unit cell. The horizontal and vertical distances between neighboring hinges on the sharp edges within each shuriken is 2ls, so that its half-length

$$l_{s} = l_{r} \sin \theta$$
(5)

can be expressed by the rotating rod length and its angle formed with the connecting rod. Upon an increase in temperature of dT, the half-length of each connecting rod increases by

$$dl_{c} = l_{c} \alpha_{c} dT$$
(6)

where αc is the coefficient of thermal expansion (CTE) of the connecting rod. However, the there is no increase in the length of the rotating rods, as they are locked in place by the corners, thereby resulting in a build-up of compressive thermal stress. Suppose the origin of the coordinate system is placed at the center of the shuriken, the half-lengths of the unit cells as measured along the x-axis in the original state is

$$x_{0} = l_{s} - l_{r} \cos \theta + l_{c}$$
(7)

while the same half-length upon heating is

$$x = l_{s} - l_{r} \cos \theta + l_{c} (1 + \alpha_{c} dT)$$
(8)

upon considering Eq. (6). If, in addition to the four axes of symmetry imposed on the shuriken we let y0 = x0 to give square array, then the in-plane strain is equibiaxial. Substituting Eq. (5) into Eqs. (7) and (8) gives the thermal strain in x-direction εx = (x − x0)/x0 as

$$\varepsilon_{x} = \alpha_{c} dT\frac{{\frac{{l_{c} }}{{l_{r} }}\;}}{{\sin \theta - \cos \theta + \frac{{l_{c} }}{{l_{r} }}}}$$
(9)

upon heating.

Fig. 3
figure 3

A unit of the shuriken network a in original state indicated by purple rods and extension of connecting rods by dlc indicated in red upon heating, and b contraction of all rods upon cooling indicated in blue, where dlr and dlc take on negative values

Full size image

Upon a decrease in temperature, the half-length of the connecting rod changes by the amount indicated in Eq. (6), but this value is negative because dT < 0. Likewise, the change in the length of the rotating rod

$$dl_{r} = l_{r} \alpha_{r} dT,$$
(10)

where αr is the CTE of the rotating rod, takes on a negative value for the same reason. Due to the contraction of the rotating rod, its angle formed with the connecting rod increases to θ + dθ, as shown in Fig. 3b. For these changes in rod lengths and angles, we have the updated half-length of the unit cell in x-direction

$$x = l_{s} - l_{r} (1 + \alpha_{r} dT)\cos (\theta + d\theta ) + l_{c} (1 + \alpha_{c} dT).$$
(11)

Substituting cos dθ = 1 and sin dθ = dθ as dθ → 0 for infinitesimal deformation, we have

$$x = l_{s} - l_{r} (1 + \alpha_{r} dT)[\cos \theta - d\theta \sin \theta ] + l_{c} (1 + \alpha_{c} dT).$$
(12)

The angular change can be eliminated by equating the vertical components of the rotating rod before and after cooling

$$l_{r} \sin \theta = l_{r} (1 + \alpha_{r} dT)\sin (\theta + d\theta )$$
(13)

to give

$$\frac{1}{{1 + \alpha_{r} dT}} = \cos d\theta + \frac{\sin d\theta }{\tan \theta }.$$
(14)

Considering infinitesimal deformation again, we have

$$d\theta = \left( {\frac{1}{{1 + \alpha_{r} dT}} - 1} \right)\tan \theta$$
(15)

which, upon substitution into Eq. (12) gives rise to

$$x = l_{s} - l_{r} (1 + \alpha_{r} dT)\left[ {\cos \theta - \sin \theta \tan \theta \left( {\frac{1}{{1 + \alpha_{r} dT}} - 1} \right)} \right] + l_{c} (1 + \alpha_{c} dT).$$
(16)

Using Eq. (5) and recalling the original dimension described by Eq. (7), one obtains the thermal strain

$$\varepsilon_{x} = - \alpha_{r} dT\frac{{\sin \theta \tan \theta + \cos \theta - \frac{{l_{c} }}{{l_{r} }}\frac{{\alpha_{c} }}{{\alpha_{r} }}}}{{\sin \theta - \cos \theta + \frac{{l_{c} }}{{l_{r} }}}}$$
(17)

upon cooling.

From the definition of CTE α = ε/dT, one can obtain the effective CTE of the interconnected shuriken network from Eqs. (9) and (17) for heating and cooling, respectively. When normalized against the CTE of the rotating rods, we have the dimensionless effective CTE

$$\frac{{\alpha_{x} }}{{\alpha_{r} }} = \frac{{\frac{{l_{c} }}{{l_{r} }}\;\frac{{\alpha_{c} }}{{\alpha_{r} }}}}{{\sin \theta - \cos \theta + \frac{{l_{c} }}{{l_{r} }}}};\quad dT > 0$$
(18)

for heating, and

$$\frac{{\alpha_{x} }}{{\alpha_{r} }} = - \frac{{\sin \theta \tan \theta + \cos \theta - \frac{{l_{c} }}{{l_{r} }}\frac{{\alpha_{c} }}{{\alpha_{r} }}}}{{\sin \theta - \cos \theta + \frac{{l_{c} }}{{l_{r} }}}};\quad dT < 0$$
(19)

for cooling.

3 Results and discussion

Effects from the various geometrical and mechanical properties of the interconnected shuriken network are investigated for rotating rod CTE of αr = 20 × 10−6 K−1, connecting-to-rotating rod length ratio of lc/lr = 1 and the angle of θ = 60° between these two rods. This choice of αr is representative of typical metals with high CTEs, such as aluminum (αAl = 23.1 × 10−6 K−1), brass (αBr = 19 × 10−6 K−1), and copper (αCu = 17 × 10−6 K−1), which gives an average of (αAl + αBr + αCu)/3 = 19.7 × 10−6 K−1≈20 × 10−6 K−1. The effective thermal strains in x-direction εx are plotted in Fig. 4 (left side) while the corresponding effective CTEs in the same direction are furnished in Fig. 4 (right side) with reference to the change in temperature dT. Here, effects of the connecting rod CTE αc is shown in Fig. 4a using rod CTE ratio of αc/αr = 0, 0.25, 0.5, 0.75 and 1. In deciding the range of αc/αr ratio, we begin with the premise that the both types of rods possess non-negative CTEs due to the rarity of intrinsically NTE materials. While the attainment of positive thermal strain during heating is easily achieved solely by the expansion of the connecting rod arising from the prevention of rotating rod expansion due to the locking effect, as indicated in Fig. 3a, we note from Fig. 3b that the displacement of the rotating-to-connecting rod hinge must be greater than the shortening of the connecting rod in order to achieve positive thermal strain upon cooling. As such, we identify the range 0 ≤ αc/αr ≤ 1 as being reasonable with increments of 0.25 to produce five sets of data. Influence from the connecting-to-rotating rod subtending angle is furnished in Fig. 4b for θ = 50°, 55°, 60°, 65° and 70°. In determining the range of θ ratio, we note that from Fig. 3 that 45° < θ < 90° by geometrical reason. A central value of θ = 60° was chosen as it would give convenient trigonometric expressions. Hence a range of 50° ≤ θ ≤ 70° with an increment of 5° would also give five sets of data. The effect from the connecting-to-rotating rod length ratio is plotted in Fig. 4c for lc/lr = 0.5, 0.75, 1, 1.25 and 1.5. In deliberating the range of lc/lr ratio, we note that under the central value of θ = 60°, it is required that lc > 0.5lr to prevent overlapping of the shurikens’ pointed edges, hence lc/lr = 0.5 sets the lower bound, while there is no upper bound for lc/lr. Perusal to Eqs. (18) and (19) indicates that both expressions are simplified when lc/lr = 1. If this rod ratio is selected as the central value, then the choice of 0.5 ≤ lc/lr ≤ 1.5 would be reasonable, and an increment of 0.25 would again give five sets of data.

Fig. 4
figure 4

Plots of thermal strain (left) and effective CTE (right) against thermal change for various a connecting-to-rotating rod CTE ratio, b rotating-to-connecting rod angles, and c connecting-to-rotating rod length ratio

Full size image

The continuous effect of αc/αr, θ and lc/lr on the dimensionless effective CTE are displayed in Fig. 5. It shows that the negativity of the effective CTE upon cooling can be intensified (or the positivity of the effective CTE upon heating can be reduced) by decreasing αc/αr or lc/lr, or by increasing θ. It is of interest to note that while the material system fulfils the auxetikos criterion—as evident from the increasing thermal strain regardless of whether dT is positive or negative, and hence this system exhibits PTE under heating but reverses to NTE under cooling—the negativity of the CTE is of a greater extent in comparison to its positive counterpart. The existence of two curves, one each for heating and cooling, suggests that the effective CTE for the investigated material system herein is undefined when there is no change in temperature, for the CTE and its sign come into play only when one specifies the condition of temperature change—whether increasing or decreasing. In other words,

$$\mathop {\lim }\limits_{{dT \to 0^{ - } }} \alpha_{x} \ne \mathop {\lim }\limits_{{dT \to 0^{ + } }} \alpha_{x}$$
(20)
Fig. 5
figure 5

Variation of dimensionless effective CTE with reference to the a connecting-to-rotating rod CTE ratio, b rotating-to-connecting rod angles, and c connecting-to-rotating rod length ratio under heating and cooling

Full size image

It is therefore unsurprising that Fig. 5 exhibits two sets of CTE—one for increasing temperature and one for decreasing temperature—although the CTE is non-existent for no change in temperature.

Thus far the characteristic of the interconnected shuriken model has been demonstrated for αc/αr = 0, 0.25, 0.5, 0.75 & 1, θ = 50°, 55°, 60°, 65° & 70°, and lc/lr = 0.5, 0.75, 1, 1.25 & 1.5, with central values at αc/αr = 0.5, θ = 60° and lc/lr = 1. Outside the range of plotted results, the characteristics of the shuriken network can be approximated by means of extrapolated estimation. To obtain more accurate behavior of this network with different material and geometrical properties, one may perform calculations based on Eqs. (18) and (19) for heating and cooling, respectively, subjected to the following conditions to prevent the overlapping of neighboring shurikens or any other geometrical inconsistencies:

$$l_{c} \ge l_{r} \cos \theta$$
(21)

and

$$45^\circ < \theta = \sin^{ - 1} \left( {\frac{{l_{s} }}{{l_{r} }}} \right) < 90^\circ .$$
(22)

Finally, it is noted that the analysis is based on the assumption of straight lines with negligible width for the rods while the pin-joints are merely points. For practical consideration, a scaled-up model for a unit cell has been constructed to simulate the effects of cooling (both the rotating and connecting rods shorten) and heating (only the connecting rods are permitted to lengthen) so as to enable comparison between the theoretical and measured thermal strains. This is furnished in Fig. 6. It can be seen that the theoretical plots underestimate and overestimate the measured data points for cooling and heating, respectively. The discrepancies may well be attributed to the simplifications in the theoretical formulation wherein (a) the rods are of zero width and the pin joints are dimensionless points, and (b) the temperature change is infinitesimal.

Fig. 6
figure 6

Comparison between theoretical plot (line) and measured data (points) for a scaled-up unit cell model with αc/αr = 0.5, θ = 63° and lc/lr = 1 at αr = 0.002 K−1

Full size image

4 Conclusions and recommendation

An auxetikos material, i.e. one that tends to increase, has been defined herein as a material system that exhibits positive strain whether the stimulus is of positive or negative value. By adopting the interconnected shuriken network, it has been shown that the material system expands based on the magnitude of the thermal change. This has been made possible by the presence of two sets of effective microstructure. One set of microstructure, whereby the rotating rod is redundant, takes effect upon heating wherein only the connecting rods expand. The other set of microstructure, in which the rotating and connecting rods are permitted to contract simultaneously, takes effect upon cooling. The capability of demonstrating two effective microstructures from just a single microstructure has been made possible through the implementation of a jamming or locking mechanism to render the rotating rods redundant under heating but functional under cooling. Results reveal that the effective CTE can be made more negative (under cooling) or less positive (under heating) by decreasing the αc/αr or lc/lr ratios, or by increasing the θ angle. It is herein proposed that a refined model be developed to take into consideration the finite change in temperature and more realistic geometrical details be incorporated, such as rod width, so as to produce a more accurate predictive model. The capability for materials to exhibit similar response under opposing conditions—due to their ability to manifest opposing properties under opposing conditions—paves a way for engineers to design materials that are able to change their behavior to suit the environment.