Orthotropic Laminated Open-cell Frameworks Retaining Strong Auxeticity under Large Uniaxial Loading

Anisotropic materials form inside living tissue and are widely applied in engineered structures, where sophisticated structural and functional design principles are essential to employing these materials. This paper presents a candidate laminated open-cell framework, which is an anisotropic material that shows remarkable mechanical performance. Using additive manufacturing, artificial frameworks are fabricated by lamination of in-plane orthotropic microstructures made of elbowed beam and column members; this fabricated structure features orthogonal anisotropy in three-dimensional space. Uniaxial loading tests reveal strong auxeticity (high negative Poisson’s ratios) in the out-of-plane direction, which is retained reproducibly up to the nonlinear elastic region, and is equal under tensile and compressive loading. Finite element simulations support the observed auxetic behaviors for a unit cell in the periodic framework, which preserve the theoretical elastic properties of an orthogonal solid. These findings open the possibility of conceptual materials design based on geometry.


Supplementary Discussion 1: Fabrication of ABS Specimens Immersed in Silicone Oil
We assessed improvements in the brittle ABS specimens after immersion in silicone oil in a prior loading test. Figure S1 summarizes our findings. We fabricated a hollow cylinder (pipe) using a 3D-printing machine (uPrint SE Plus, Stratasys Ltd.), having a cross-section with an outer-diameter of 45 mm and thickness of 1 mm. The pipes had a micro-patterned structure with a pair of rectangular pores alternately arranged in the transverse and vertical directions (see Fig. S1(a)). As shown in Fig. S1(b), we developed a loading test device for the cellular pipe using specially processed jigs, and examined the uniaxial tensile test at a displacement speed of 1 mm/min. Figure S1(c) shows the measured load-stain curves and Table S1 shows the experimental conditions and results. The ductility of structures immersed in silicone oil was enhanced compared with that of untreated structures. The maximum loads and fracture strains of the silicone-oil-treated structures were roughly twice as high as those of the untreated structures.
Thermoplastic polymer materials are in general soluble in various organic solvents; in this case, toluene containing silicone promoted the erosion of the ABS resin components.
Such erosion often induces chemical stress cracks inside the material, which result in reduced ductility. However, the fine structures fabricated by 3D-printing machines tend to be composed of low density solids, with properties determined by the in-plane resolution, stacking pitch, and path planning algorithm of the machine. In our experiments, the silicone elastomers might penetrate microscopic pores in the ABS resin specimens at the same time as toluene dissolves into the material. In this way the silicone oil reinforced the structures.
In the main experiments, we also prepared cellular structure specimens made of ABS resin, which were immersed in silicone oil. We have included information on the prepared cellular structures and their loading test results along with those of the structure made of nylon resin in Table S2.

Supplementary Discussion 2: Design of Jig for Tensile and Compressive Tests
For precise load testing of the proposed cellular structure, a pair of the specialized steel jigs was manufactured. The upper jig, illustrated in Fig. S2, consisted of one main platen and a pair of jig caps. In the case of tensile loading, the platen and two caps were joined with six bolts, between which ten steel pins were inserted. Each of the pins passed through the side of the specimen so that the tensile forces were transferred to the structure via the pins. Under x-axial uniform tensile displacement the structure was fixed in the z-direction at both the sides because frictional forces acted on the interfaces between the pins and specimen. However, when the perpendicular force driven by the out-of-plane deformation overcame the maximum static frictional force, the side bars were prone to slip in the z-direction (see Movie S1).
Specially-processed side bars were welded in-plane with the inside cells, which were fixed in the y-direction at the ends. In the case of compressive loading, a pair of the platens in direct contact with the specimen was used for pressing the ends of the cellular structure. The structure remained fixed to the surface of the two platens until the onset of buckling.

Fig. S2
Schematic of the upper jig attachment.

Supplementary Discussion 3: Image Processing and Definition of Poisson's Ratios
To measure the cell deformation from the photograph sequence during the loading tests, we used conventional image processing, as follows: in the first step, the RGB values of the pixel data in the original image were converted to YCbCr color space. The third and second data groups of the color space, which correspond to red-and blue-difference chroma components, respectively, were selectively extracted (see Fig. S2(i) and (i)'). In the second step, for each of the data sets, a binary image was generated by threshold adjustment after reducing the image noise with Gaussian filtering (see Fig. S2(ii) and (ii)'). In the final step, the centroids of the characteristic points were calculated. We estimated the height and width of the inner and outer cells from the average positions of all the centroids on each cell edge.
Note that, for the inner cells of the x-y surface, the cell lengths were calculated using the center unit cell length that we captured. The four nodal positions were connected with straight lines, each of which was the centroid of the four-point-groups that the sixteen red makers were equally divided into (see Fig. 1(c)).

Fig. S3
Example showing the image processing steps for x-z surface image of ABS structure.

Fig. S4
Schematic of the camera parameters used to correct the dimensions of the specimen.
Assuming a point light source, the measured dimensions obtained from the image data were corrected to the real dimensions of the deformed cells with high accuracy. We ignored errors with respect to lens aberration. Let  and d be the angle and distance from the object, respectively (see Fig. S4). The dimension Xi * on a camera image can be described as: where the subscripts i = 1, 2 and 3 correspond to the coordinates x, y, z in the real space, respectively. From Eq. (S1), the real dimension Xi can be expressed as, Using Eq. (S2), we converted all the cell dimensions from the images during deformation to their real dimensions.
Poisson's ratio is commonly calculated by designating engineering (nominal) strain as ij  , where the first index refers to axial strain and the second to transverse strain. However, the Poisson's ratio defined by engineering strain is sometime misleading when the targeted structure deforms greatly. Thus, we considered other types of Poisson's ratio [S1].
We defined the specimen dimensions of the n-th image as Xi (n) , where Xi (0) indicates the initial configuration. The engineering strain was calculated as follows: The true strain is then obtained by, Accounting for the current deformation state, the ratio of the extension to the instantaneous length could then be approximated as The strain described in Eq. (S5) is referred to as the instantaneous true strain [S1]. From Eqs.
(S3)-(S5), the three types of Poisson's ratios are described as follows:  Figure S5 indicates the data processing for the int ij  -curves measured by the tensile tests for the nylon structure (see Fig. 2c). Note that the four-point median processing was skipped when calculating int xy  for the outer cell because the raw data had satisfactory accuracy. The raw data of the Poisson's function out-of-plane (x-z) showed a wider variation of errors than those in-plane (x-y). One of the main reasons is the different number of pixels between the images taken by the two types of the cameras.

Supplementary Discussion 4: Finite Element Modeling of a Periodic Unit Cell
To obtain the elastic properties of the proposed periodic framework, we performed the finite element analyses using Abaqus (ver. 6.14-1, Dassault Systèms) [S2]. Figure S6 shows an overview of the finite element model. The unit cell consisted of beam and column components of the nylon material. The Young's modulus and Poisson's ratio were input as 1700 MPa and 0.40, respectively. All the boundary surfaces of the cell framework were connected with rigid plates, and were set with a Young's modulus a million times as large as that of nylon to impose multi-point constraints in a periodic fashion; all the displacements of the plates A and C at one side, not including ones in the major axes, were consistent with those of the plates at the other side. One side of the structural unit in each of the Xi-directions was fixed on the Xj-Xk plane and the four forces were applied to opposite sides in either direction (X1-, X2-or X3-axis). In the mesh processing, the body was divided with triangular quadratic elements (Abaqus element code CPE6H), as shown in Fig. S6. To account for geometric nonlinearity the nonlinear finite-element code ABAQUS/STANDARD was selected to analyze the uniaxial deformation of the structure. This was necessary for representing the out-of-plane behaviors.  Figure S1:

Captions for Supplementary Figures:
Uniaxial tensile tests for the ABS microstructure pipes with or without silicone oil; (a) the CAD-data, (b) the uniaxial tensile testing, and (c) load-stain curves under uniaxial extension.

Figure S2:
Schematic of the upper jig attachment.

Figure S3:
Example showing the image processing steps for x-z surface image of ABS structure.

Figure S4:
Schematic of the camera parameters used to correct the dimensions of the specimen.   Table S1:

Captions for Supplementary Tables:
Information on as-prepared pipe specimens and test results. Table S2: Information on as-prepared cellular structure specimens and test results.