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Design and testing of 3D-printed micro-architectured polymer materials exhibiting a negative Poisson’s ratio

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Abstract

This work proposes the complete design cycle for several auxetic materials where the cycle consists of three steps (i) the design of the micro-architecture, (ii) the manufacturing of the material and (iii) the testing of the material. We use topology optimization via a level-set method and asymptotic homogenization to obtain periodic micro-architectured materials with a prescribed effective elasticity tensor and Poisson’s ratio. The space of admissible micro-architectural shapes that carries orthotropic material symmetry allows to attain shapes with an effective Poisson’s ratio below \(-\,1\). Moreover, the specimens were manufactured using a commercial stereolithography Ember printer and are mechanically tested. The observed displacement and strain fields during tensile testing obtained by digital image correlation match the predictions from the finite element simulations and demonstrate the efficiency of the design cycle.

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Notes

  1. Characteristics of this material can be found in the manufacturers data sheet (see https://dl.airtable.com).

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Acknowledgements

This work is financed by the french-swiss ANR-SNF project MechNanoTruss (ANR-15-CE29-0024-01). The authors would like to express their gratitude to Chiara Daraio for fruitful discussion on the design of lattice structures and to Gregoire Allaire and Georgios Michailidis for lending their expertise on the numerical and algorithmic issues of the optimization.

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Correspondence to Andrei Constantinescu.

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Communicated by Johlitz, Laiarinandrasana and Marco.

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A computation of the effective Poisson’s ratio

A computation of the effective Poisson’s ratio

This appendix reviews the mathematical approach that was used to measure/compute the effective Poisson’s ratio of a unit cell in both measurements by digital image correlation and numerical estimation using a finite element method. For the following computation, we place ourselves in the case of small strain assumption (Fig. 12).

Fig. 12
figure 12

Representation of a unit cell

The effective material is supposed to carry a natural orthotropic material behaviour. The effective Poisson’s ratio \(\nu _{12}\), characterizing the transverse strain of the structure in the direction \((O,\mathbf {e_2})\) axis when stretched in the direction \((O,\mathbf {e_1})\), is defined as:

$$\begin{aligned} \nu _{12}^*=\frac{C^H_{1122}}{C^H_{2222}} \end{aligned}$$
(A.1)

We remind that \(C^H_{1122}\) and \(C^H_{2222}\) are coefficients of the effective elastic stiffness tensor. In general \(\nu _{12} \ne \nu _{21}\). During a uniaxial tensile test in the direction \((O,\mathbf {e_1})\), Eq. (A.1) yields to the negative of the ratio of macroscopic transverse strain to macroscopic axial strain:

$$\begin{aligned} \nu _{12}^* = -\, \frac{\varepsilon ^H_{22}}{\varepsilon ^H_{11}} \end{aligned}$$
(A.2)

In the small strain assumption, the strain field can be linearized as:

$$\begin{aligned} \mathbf {\varepsilon ^H} = \left\langle \mathbf {\varepsilon } \right\rangle _{\varOmega } = \frac{1}{2} \left( \left\langle {\mathbb {F}} \right\rangle _{\varOmega }^T + \left\langle {\mathbb {F}} \right\rangle _{\varOmega } \right) - {\mathbb {I}} \end{aligned}$$
(A.3)

where \({\mathbb {F}}\) is the average transformation gradient. Considering the small strain assumption:

$$\begin{aligned} \left\langle {\mathbb {F}} \right\rangle _{\varOmega } = \frac{1}{V_{\varOmega }} \int _{\varOmega } ({\mathbb {I}} +\mathbb {\nabla } \mathbf {u}) \hbox {d} \varOmega \end{aligned}$$
(A.4)

Using Ostrogradsky’s theorem, we can express the transformation gradient at the boundary \(\partial \varOmega \):

$$\begin{aligned} \left\langle {\mathbb {F}} \right\rangle _{\varOmega } = \frac{1}{V_{\varOmega }} \left( \int _{\varOmega } {\mathbb {I}} d \varOmega + \oint _{\varGamma } \mathbf {u} \otimes \mathbf {n} \hbox {d}\varGamma \right) \end{aligned}$$
(A.5)

Study of a unit cell

$$\begin{aligned} \left\langle {\mathbb {F}} \right\rangle _{\varOmega }= & {} {\mathbb {I}} + \frac{1}{V_{\varOmega }} \left( \int _{\partial \varOmega _T} \mathbf {u} \otimes \mathbf {e_2} \hbox {d}\,\varGamma + \int _{\partial \varOmega _B} \mathbf {u} \otimes (-\mathbf {e_2}) \hbox {d}\,\varGamma + \int _{\partial \varOmega _R} \mathbf {u} \otimes \mathbf {e_1} \hbox {d}\,\varGamma + \int _{\partial \varOmega _L} \mathbf {u} \otimes (-\mathbf {e_1}) \hbox {d}\,\varGamma \right) \nonumber \\\end{aligned}$$
(A.6)
$$\begin{aligned} \left\langle {\mathbb {F}} \right\rangle _{\varOmega }= & {} {\mathbb {I}} + \frac{1}{V_{\varOmega }} \begin{bmatrix} \int _{\partial \varOmega _R} u_1 d\,\varGamma - \int _{\partial \varOmega _L} u_1 \hbox {d}\,\varGamma&\int _{\partial \varOmega _T} u_1 \hbox {d}\,\varGamma - \int _{\partial \varOmega _B} u_1 \hbox {d}\,\varGamma \\&\\ \int _{\partial \varOmega _R} u_2 \hbox {d}\,\varGamma - \int _{\partial \varOmega _L} u_2 \hbox {d}\,\varGamma&\int _{\partial \varOmega _T} u_2 \hbox {d}\,\varGamma - \int _{\partial \varOmega _B} u_2 \hbox {d}\,\varGamma \\ \end{bmatrix} \end{aligned}$$
(A.7)

Thus, from Eq. (A.3):

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon _{11} = \frac{1}{V_{\varOmega }} \left( \int _{\partial \varOmega _R} u_1 \hbox {d}\,\varGamma - \int _{\partial \varOmega _L} u_1 \hbox {d}\,\varGamma \right) \\ \varepsilon _{22} = \frac{1}{V_{\varOmega }} \left( \int _{\partial \varOmega _T} u_2 \hbox {d}\,\varGamma - \int _{\partial \varOmega _B} u_2 \hbox {d}\,\varGamma \right) \end{array}\right. } \end{aligned}$$
(A.8)

For each edge of the square unit cell, the integral of the contour is computed by integrating the displacement of the material in contact with the edge. In other words, the void phase is not considered in the computation.

$$\begin{aligned} \nu _{12}^* = -\,\frac{\int _{\partial \varOmega _T} u_2 \hbox {d}\,\varGamma - \int _{\partial \varOmega _B} u_2 \hbox {d}\,\varGamma }{\int _{\partial \varOmega _R} u_1 \hbox {d}\,\varGamma - \int _{\partial \varOmega _L} u_1 \hbox {d}\,\varGamma } \end{aligned}$$
(A.9)

In practice, using a finite element method, Eq. (A.9) becomes:

$$\begin{aligned} \nu _{12}^* = -\,\frac{\frac{1}{N_T} \sum \nolimits _{i = 1}^{N_T} u_2^i - \frac{1}{N_B} \sum \nolimits _{b}^{N_B} u_2^i}{ \frac{1}{N_R} \sum \nolimits _{i = 1}^{N_R} u_1^i - \frac{1}{N_L} \sum \nolimits _{i = 1}^{N_L} u_1^i} \end{aligned}$$
(A.10)

where \(N_i, i \in \{ T,B,R,L\}\) are, respectively, the number of nodes on top, bottom, right and left edges.

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Agnelli, F., Constantinescu, A. & Nika, G. Design and testing of 3D-printed micro-architectured polymer materials exhibiting a negative Poisson’s ratio. Continuum Mech. Thermodyn. 32, 433–449 (2020). https://doi.org/10.1007/s00161-019-00851-6

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