Curvature Induced by Deflection in Thick Meta-Plates.

Mohammad J Mirzaali,Mehdi Habibi,Mahdiyeh Nouri‐Goushki,Angelo Accardo,José Bico,Aref Ghorbani,Shahram Janbaz,Amir A Zadpoor,Sebastien J P Callens,Nazli Tümer,Kenichi Nakatani

doi:10.1002/adma.202008082

Abstract

The design of advanced functional devices often requires the use of intrinsically curved geometries that belong to the realm of non-Euclidean geometry and remain a challenge for traditional engineering approaches. Here, it is shown how the simple deflection of thick meta-plates based on hexagonal cellular mesostructures can be used to achieve a wide range of intrinsic (i.e., Gaussian) curvatures, including dome-like and saddle-like shapes. Depending on the unit cell structure, non-auxetic (i.e., positive Poisson ratio) or auxetic (i.e., negative Poisson ratio) plates can be obtained, leading to a negative or positive value of the Gaussian curvature upon bending, respectively. It is found that bending such meta-plates along their longitudinal direction induces a curvature along their transverse direction. Experimentally and numerically, it is shown how the amplitude of this induced curvature is related to the longitudinal bending and the geometry of the meta-plate. The approach proposed here constitutes a general route for the rational design of advanced functional devices with intrinsically curved geometries. To demonstrate the merits of this approach, a scaling relationship is presented, and its validity is demonstrated by applying it to 3D-printed microscale meta-plates. Several applications for adaptive optical devices with adjustable focal length and soft wearable robotics are presented.

Highlights

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not
An initially flat panel can be reasonably bent in a single direction to adapt the shape of an arch, plates can be obtained, leading to a negative transforming the panel into a shell or positive value of the Gaussian curvature upon bending, respectively
Bending a wide plate generally results in a single curvature, except for a fraction of the width of the plate, which is of the order (h/κ1 )1/2, where h is the thickness of the plate.[76]

Summary

Results and Discussion

Bending a rubber eraser along its length results in a curvature of opposite sign along the transverse direction (Figure 1a). To study the effects of the boundary conditions on the second principal curvature, we created computational models of meta-plates with smaller and larger widths (i.e., W /4, W /2, 2W , and 4W ) than that of our reference models (i.e., W ) for both the maximum and minimum values of the Poisson’s ratio For those cases, we kept the out-of-plane thickness of the metaplates (i.e., h = 5 mm) constant. There is a linear relationship between the thickness of the meta-plate, h, and its second principal curvature, κ2, (i.e., κ2 ∝ h) obtained using our computational models for extremely negative (Figure 4a) and extremely positive (Figure 4b) values of the Poisson’s ratio. Regardless of the value of the Poisson’s ratio, the second principal curvature linearly decreases as the in-plane thickness of the struts of the meta-plates increases (Figure S2, Supporting Information). With a simple tuning of the geometrical parameters of the elementary cells, a wide range of curvatures could be exploited to create adaptive optical devices (e.g., mirrors) whose focal length is dependent on the level of the applied compressive load (Figures 5e–h)

Conclusion

Experimental Section

Conflict of Interest

Data Availability Statement