Pantographic metamaterials: an example of mathematically driven design and of its technological challenges Abstract In this paper, we account for the research efforts that have been started, for some among us, already since 2003, and aimed to the design of a class of exotic architectured, optimized (meta) materials. At the first stage of these efforts, as it often happens, the research was based on the results of mathematical investiga-tions. The problem to be solved was stated as follows: determine the material (micro)structure governed by those equations that specify a desired behavior. Addressing this problem has led to the synthesis of second gradient materials. In the second stage, it has been necessary to develop numerical integration schemes andDipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma "La Sapienza.", Via Eudossiana 18, 00184 Rome, Italy E-mail: barchiesiemilio@gmail.com the corresponding codes for solving, in physically relevant cases, the chosen equations. Finally, it has been necessary to physically construct the theoretically synthesized microstructures. This has been possible by means of the recent developments in rapid prototyping technologies, which allow for the fabrication of some complex (micro)structures considered, up to now, to be simply some mathematical dreams. We show here a panorama of the results of our efforts (1) in designing pantographic metamaterials, (2) technology of rapid prototyping, and (3) in the mechanical testing of many real prototypes. Among the key findings that have been obtained, there are the following ones: pantographic metamaterials (1) undergo very large deformations while remaining in the elastic regime, (2) are very tough in resisting to damage phenomena, (3) exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties, (4) have superior strength to weight ratio, (5) have predictable damage behavior, and (6) possess physical properties that are critically dictated by their geometry at the microlevel.
The multiform bio-mechanical phenomena occurring in bones grafted with the addition of artificial materials urge for the formulation of models which are sophisticated enough to describe their complexity. In the present paper we present a continuum poro-elastic mixture model in which two apparent mass densities are introduced to describe, at a macroscopic length scale, situations in which bone tissues and artificial materials coexist and interact. We focus on the final healing stage process when the bone remodelling becomes the dominant phenomenon. Artificial materials used are obviously to be biocompatible and must resist to externally applied mechanical loads. More recently in order to favour bone tissue re-growth in grafts, which improves the long term performances of grafted bones, it has been conceived to use substitute materials which may be, similarly to bone tissue, bio-resorbed by osteoclasts and eventually replaced by newly synthesised living tissue. To account for resorption and synthesis phenomena suitable evolution equations are introduced for Lagrangian mass densities of the mixture constituents in which an integrodifferential operator defined on deformation fields appears. This operator is chosen to model some features of the coupling between mechanical compliance and biological bone tissue activity. The obtained system of integrodifferential equations is not trivial also when one considers one dimensional cases. Treating this simplified situations will allow us to individuate more easily some important remodelling scenarios. The numerical simulations which we present here show that the introduced model is promising and deserves to be developed to give previsions in more realistic applications.
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