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# Design and optimization of metamaterials

### Optimization of auxetic materials

**People involved:** V. Zega, M. Bruggi, A. Corigliano

Complex inertial Micro Electro Mechanical Systems (MEMS) usually require the simultaneous motion of the masses in more than one direction and an overall linear behavior. Furthermore, since it is usually very difficult to actuate the device in all the required directions, complex spring configurations for the conversion of the motion are needed.

The goal of the work is to propose a new topology optimization procedure which can be used as a tool during the design phase of the optimal auxetic structure, that is completely compatible with the MEMS fabrication processes available so far and that allows the motion conversion in a MEMS device without entering the nonlinear regime. The most popular feature of auxetic materials or structures is the fact that they can expand in the direction perpendicular to an externally exerted tension, thus showing an equivalent negative Poisson’s ratio. Auxetic behavior is a scale-independent property which can be achieved at different structural levels from molecular to macroscopic levels.

First, auxetic configurations are found by performing the synthesis of compliant mechanisms, i.e. finding the distribution of material that maximizes a displacement component of the workpiece obtained as the output of the elastic response to an input represented by an imposed displacement applied elsewhere through an actuator. The main problem of this topology optimization procedure is the fact that the resultant auxetic structures have a truss-like shape and are, generally, too stiff for typical MEMS applications.

To solve this problem, a new formulation of topology optimization is implemented: it addresses the synthesis of compliant mechanisms using a Cosserat micro-polar material model instead of the Cauchy material model previously employed. The key point of the new procedure is that the constitutive law of a 2D micro-polar solid requires the introduction of a characteristic material length that governs the intrinsic material bending stiffness and that a suitable tuning of that characteristic length allows to generate optimal layouts whenever a bending-resistant design is preferred to a conventional truss-like solution.

**Bibliography:**

[1] M. Bruggi, V. Zega and A. Corigliano ‘Synthesis of auxetic structures using optimization of compliant mechanisms and a micropolar material model’ Structural and Multidisciplinary Optimization 55 (2017) 1-12

[2] V. Zega, M. Bruggi, A. Corigliano ‘Optimization of auxetic structures’ IV ECCOMAS young investigator conference, Milan, Italy, September 13-15, 2017

[3] M. Bruggi, V. Zega, A. Corigliano ‘Optimization of auxetic structures for MEMS applications’ Eurosime 2016, Montpellier, France, April 17-20, 2016

### Deformation of the optimized auxetic structures

Schematic view of the auxetic structure obtained through the first topology optimization procedure in which a Cauchy elastic material is considered.

Schematic view of the auxetic structure obtained through the new topology optimization procedure in which a micro-polar Cosserat elastic material is considered.

### Auxetic materials for vibrations insulation

**People involved:** V. Zega, L. D’Alessandro, R. Ardito, A. Corigliano

Auxeticity and phononic crystals bandgap properties are properly combined to obtain a single phase periodic structure with a tridimensional wide tunable bandgap.

When an external tensile load is applied to the structure, the auxetic unit cells change their configurations by exploiting the negative Poisson’s ratio and this results in the tuning, either hardening or softening, of the frequencies of the modes limiting the 3D bandgap.

Moreover, the expansion of the unit cell in all the directions, due to the auxeticity property, guarantees a fully 3D bandgap tunability of the proposed structure. Numerical simulations and analytical models are proposed to prove the claimed properties. The first experimental evidence of the tunability of a wide 3D bandgap is then shown thanks to the fabrication of a prototype by means of additive manufacturing.

Figure 1: Unit cell topology. (a) 3D representation of the unit cell. (b) 2D cross section with respect to one of the principal planes. c) Auxetic (v ≈- 0.6) deformed shape of the unit cell computed by means of the Solid Mechanics Module of COMSOL Multiphysics v5.3. The contour of the displacement magnitude field is shown in color.

Figure 2: Tuning of the first bandgap: in solid lines the experimental results, in dashed lines the numerical visco-elastic model results.

**Bibliography:
**[1] L. D’Alessandro, V. Zega, R. Ardito, A. Corigliano ‘3D auxetic single material periodic structure with ultrawide tunable bandgap’ Scientific Reports 8:2262 (2018)