Dimpled elastic sheets: a new class of non-porous negative Poisson’s ratio materials

In this study, we report a novel periodic material with negative Poisson’s ratio (also called auxetic materials) fabricated by denting spherical dimples in an elastic flat sheet. While previously reported auxetic materials are either porous or comprise at least two phases, the material proposed here is non-porous and made of a homogeneous elastic sheet. Importantly, the auxetic behavior is induced by a novel mechanism which exploits the out-of-plane deformation of the spherical dimples. Through a combination of experiments and numerical analyses, we demonstrate the robustness of the proposed concept, paving the way for developing a new class of auxetic materials that significantly expand their design space and possible applications.


Additional Experimental Results
Mechanical properties of VeroClear material. All samples tested for this study are fabricated out of VeroClear material using a 3D printer (Connex 500 available from Objet, Ltd.). Since the glass transition temperature of VeroClear is T GT = 52 − 54 • C, the material is in the glassy phase when tested at room temperature. This is confirmed by a uniaxial tensile test conducted on a flat (non-dimpled) dog-bone shape sample with a testing section of height × width × thickness = 87.5×50×0.5 mm. During the test a maximum tensile displacement of u grip = 1.53 mm (resulting 1 in a nominal strain of ε nominal = u grip /height = 0.017) is applied to the sample using an Instron uniaxial testing machine and the displacement field is visualized using a digital image correlation (DIC) technique (see the Methods section of the main text for details).
In Figs. S1a and b we report the experimental contour maps for the horizontal (u x ) and vertical (u y ) components of the displacement fields, which are used to estimate the Poisson's ratio of the bulk material, ν. In particular, to minimize the boundary effects, we focus on a square region in the center of the sample (highlighted with black dashed lines in Fig. S1a and b) and use the the average displacement components along each of its four boundaries to calculate ν (see

Additional Numerical Results
Size effects. In Fig. 1 of the main text we report numerical results for a dimpled elastic sheet comprising a square array of 20×20 dimples with h/r = 0.5 and ψ = 75%. We show that, if all dimples dented on one side, the applied uniaxial stretch causes out-of-plane bending and results in a positive value of the macroscopic Poisson's ratio (see Fig. 1b). Differently, if the dimples are dented on both sides of the flat sheet to form a checkerboard pattern (see Fig. 1c), all dimples flatten toward the structure mid-plane under an applied uniaxial tension, resulting in a lateral expansion of the system and, therefore, an auxetic response.
While the results reported in the text are for an array of 20×20 dimples, in Fig. S4 we show numerical results for dimpled sheets comprising arrays of 10×10 (Figs. S4a and b) and 30×30 (Figs. S4c and d) dimples. These structures behave identically to that of the sheet with an array of 20×20 dimples presented in the main text, indicating that their response is not affected by the size of the system. Furthermore, we investigate the role played by boundary effects on the macroscopic Poisson's ratio,ν, when the size of the dimple array in a dog-bone shape sample is decreased. In particular, we use FE simulations to model the response of a sheet characterized by ψ = 75%, h/r = 0.5, and t/r = 0.08 and comprising an array of N × N dimples (with N = 9, 7, 5, 3).
From each simulation, we calculate ν using the same procedure introduced to postprocess the experimental results (see Experiments in the Results section 2 of the main text for more details).
The results reported in Fig. S5 indicate that by decreasing the size of the array the negative Pois-7 Figure S4: Numerical results showing the deformation under uniaxial tension of elastic sheets with square arrays of (a and b) 10×10 dimples and (c and d) 30×30 dimples. In a and c, all dimples are dented on one side while, in b and d, they are dented on both sides to form a checkerboard pattern. 8 Figure S5: Numerical results showing the Poisson's ratio, ν, of elastic sheets with square arrays of N × N dimples (N = 9, 7, 5, 3). The dimpled sheet is characterized by ψ = 75%, h/r = 0.5, and t/r = 0.08.
son's ratio of the structure becomes more pronounced. This is because the effect of the free lateral boundaries, at which dimples can freely expand, progressively increases.