An analysis of the symmetries characterizing the micro-architecture topologies and the elastic material properties is performed. The goal is to elucidate a systematic procedure that facilitates the design of elastic metamaterial with a prescribed target elasticity tensor via inverse homogenization methodologies. This systematic procedure, which is defined through a set of rules, is based on the relationship established between the elasticity tensor symmetries and the symmetry displayed by the micro-architecture topology.Following this procedure, it can be guaranteed that the designed composites, with the attained micro-structures, have effective elasticity tensors that possess the same or higher symmetries than those shown by the target elasticity tensors. Furthermore, the micro-architectures designed through this technique display simple topologies.Both properties that are supplied by the procedure, i.e., the accomplishment of the required symmetry of the composite homogenized elasticity tensor combined with the topology simplicity, are assessed through numerical simulations of several micro-architecture design problems. They are designed by formulating the inverse homogenization problem as a topology optimization problem which is solved with two different standard algorithms. The proposed procedure and the conclusions here obtained do not depend on the algorithm adopted for solving this problem.

Summary New tools for the design of metamaterials with periodic microarchitectures are presented. Initially, a two‐scale material design approach is adopted. At the structure scale, the material effective properties and their spatial distribution are obtained through a Free Material Optimization technique. At the microstructure scale, the material microarchitecture is designed by appealing to a Topology Optimization Problem (TOP). The TOP is based on the topological derivative and the level set function. The new proposed tools are used to facilitate the search of the optimal microarchitecture configuration. They consist of the following: (i) a procedure to choose an adequate shape of the unit cell domain where the TOP is formulated and shapes of Voronoi cells associated with Bravais lattices are adopted and (ii) a procedure to choose an initial material distribution within the Voronoi cell being utilized as the initial configuration for the iterative TOP.

The objective of this paper is the design of three-dimensional elastic metamaterials with periodic microarchitectures. The microarchitectures of these materials are attained by following an inverse design technique jointly with an homogenization-based topology optimization algorithm.In this context, we have particularly studied the connection between the symmetry of the material layout at the microscale of 3D periodic composites and the symmetry of the effective elastic properties. We have analyzed some possible Bravais lattices and space groups, which are typically associated with crystallography, to study the way in which the symmetries of these geometrical objects can be usefully used for the microarchitecture design of 3D elastic metamaterial.Following a previous work of the authors for two-dimensional problems, we suggest adopting the design domain of the topology optimization problem coincident with the Wigner-Seitz cells of specific Bravais lattices having the same point group to that of the target elasticity tensor.The numerical assessment described in this papers aims at the design of an extreme material. The solutions obtained with this procedure show that different composite microarchitectures emerge depending on the cell shape selection.

Keywords: multi-scale cohesive models, computational homogenization, heterogeneous material failure, Embedded Finite Elements (EFEM). AbstractThe paper describes the computational aspects and numerical implementation of a two-scale cohesive surface methodology developed for analyzing fracture in heterogeneous materials with complex microstructures. This approach can be categorized as a semi-concurrent model using the Representative Volume Element (RVE) concept.A variational multi-scale formulation of the methodology has been previously presented by the authors. Subsequently, the formulation has been generalized and improved in two aspects: i) cohesive surfaces have been introduced at both scales of analysis, they are modeled with a strong discontinuity kinematics (new equations describing the insertion of the macro-scale strains, into the micro-scale and the posterior homogenization procedure have been considered); ii) the computational procedure and numerical implementation have been adapted for this formulation. The first point has been presented elsewhere, and it is summarized here. Instead, the main objective of this paper is to address a rather detailed presentation of the second point.Finite element techniques for modeling cohesive surfaces at both scales of analysis (FE 2 approach) are described: i) finite elements with embedded strong discontinuities (EFEM) are used for the macroscale simulation, and ii) continuum-type finite elements with high aspect ratios, mimicking cohesive surfaces, are adopted for simulating the failure mechanisms at the micro-scale.The methodology is validated through numerical simulation of a quasi-brittle concrete fracture problem. The proposed multi-scale model is capable of unveiling the mechanisms that lead from the material degradation phenomenon at the meso-structural level to the activation and propagation of cohesive surfaces at the structural scale.

SUMMARYA topology optimization technique based on the topological derivative and the level set function is utilized to design/synthesize the microstructure of a pentamode material for an acoustic cloaking device. The technique provides a microstructure consisting of a honeycomb lattice composed of needle-like and joint members. The resulting metamaterial shows a highly anisotropic elastic response with effective properties displaying a ratio between bulk and shear moduli of almost three orders of magnitude. Furthermore, in accordance with previous works in the literature, it can be asserted that this kind of microstructure can be realistically fabricated. The adoption of a topology optimization technique as a tool for the inverse design of metamaterials with applications to acoustic cloaking problems is one contribution of this paper. However, the most important achievement refers to the analysis and discussion revealing the key role of the external shape of the prescribed domain where the optimization problem is posed. The efficiency of the designed microstructure is measured by comparing the scattering wave fields generated by acoustic plane waves impinging on bare and cloaked bodies.

The present contribution describes an optimization-based design technique of elastic isotropic periodic microarchitectures with crystal symmetries aiming at the realization of composites with extreme properties. To achieve this goal, three consecutive procedures are followed: i) a series of inverse homogenization problems with symmetry constraints, ii) a correlation analysis between symmetries and effective elastic properties of the attained microarchitectures, and, iii) the pattern resemblance recognition of these topologies and their redesign, by adopting microstructures with two length-scales, through optimized parametric geometries. This paper is devoted to assessing the third procedure because the first two procedures have been evaluated in previous works of the authors, and here they are only summarized.By applying the methodology, two plane group symmetries are assessed to define two families of 2D periodic parameterized microarchitecture. Once the parameters have been optimized, the resulting composites achieve elastic isotropic properties close to the whole range of the theoretically estimated bounds. Particularly, an unprecedented microstructure attaining the theoretical maximum stiffness is reported. Starting from these parameterized topologies, simple, one-length scale and easily manufacturable geometries are defined. One of the so-designed microarchitectures has been manufactured and tested, displaying an effective Poisson's ratio of -0.90 simultaneously with a high shear modulus.

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