Manuscript Topics

The central topic of the proposed Special Issue is the mathematical modeling of microstructured materials. Specifically, the Special Issue will focus on those theoretical analytical and computational aspects that are developed to respond to the increasing needs emerging from the design of innovative materials.

This Special Issue will indeed be the opportunity to establish a link between the multi-physics and multi-scale approaches, that are nowadays established as appropriate means for studying microstructured materials, and the large corpus of advancements in the related fields of applied mathematics that have taken place in the last decades.

To make this synergy effective, the Special Issue will put special emphasis on approaches that use and develop techniques and theoretical results of Mathematical Analysis, which can be transferred into the specific language of the Mechanics of Materials, hence creating space for concrete multi-disciplinary interaction in research.

Having this in mind, the contributions to this volume will present original results in different fields of Nonlinear Analysis and Calculus of Variations, examples and proposals of applications of these results to real design problems, and computational analyses adequate to test and improve their effectiveness, suggesting new directions of theoretical developments.

Mathematical topics of great interest for this issue are in particular the following.
Homogenization techniques in periodic and non-periodic environments, with also the possibility of including random effects, due to the recent advances in quantitative homogenization (see e.g. the recent body of work by Otto, Gloria, Armstrong, and others) and homogenization in non-periodic environments [3]. Homogenization and relaxation techniques are particularly efficient e.g. for studying problems of damage in elastic materials, to treat problems of optimal design, and to quantitatively describe structural properties of composites [12].

Nonlinear evolution techniques: the large body of theoretical works on the variational evolution in a non-smooth setting such as [2] and [6] provides a sound basis for future applied analyses of the evolution of complex mechanical microstuctured materials (see [5] for a first insight).

Techniques for free-discontinuity problems, which allow both the statement and analysis of problems in Fracture Mechanics, also from the standpoint of quasistatic evolution [4], for which we envisage the application to brittle materials. Spaces of functions of bounded variation and of bounded deformation, which can be used to describe mechanical phenomena involving plasticity and in the modeling of structures such as masonry.

These analyses are often coupled with a  discrete-to-continuum description via Γ-convergence, which allows on one hand the validation of macroscopic continuum models from microstructured geometries, and on the other hand, allows the introduction of novel microscopic effects (see e.g. the recent book [1]).

Topics in Mechanics of Materials of special interest for this issue are the following ones.

Discrete and semi-discrete models can be used to guide the development, optimization, and experimentation of new hierarchical metamaterials, like complex beam lattices, and also to validate their continuum description. To address the static and dynamic regimes, stepwise solution strategies can be developed based on different approaches.

Generalized continuum theories are a powerful tool to design, optimize, and analyze the performances of microstructured materials in the form of beams, plates, and solids, whose description requires an extension of standard mathematical models employed in Structural and Continuum Mechanics. These theories can dramatically improve the description, as an instance, of wave propagation in the bulk as well as surface waves and reflection/transmission across interfaces and defects(cracks, notches, and voids), damping properties, and material instabilities. The computational analysis of generalized continua requires the use of enhanced computational tools, like isogeometric analysis.

Mechanical models of granular materials can elucidate the complex behavior of those materials that, at some spatial scale, are recognizable as composed of discrete sub-bodies or structural elements, being them naturally or artificially placed, of sizes that can range from sub-nanometer to several meters. combined effects produced through the interaction of the microstructure and micromechanics can be termed as micro-mechano-morphology. For many materials utilized in engineering activities, their granular nature (or micro-mechano-morphology) has a profound effect on the behavior that emerges at the so-called macro-scale which represents the combined behavior of a large number of elements.

Damage detection revolves around the analytical exploration of material behavior and structural response to identify potential flaws or weaknesses. Theoretical models play a pivotal role in predicting and understanding the impact of damage on the mechanical performance of structures. The most studied approaches seek to correlate, through physical, statistical or artificial intelligence models any alteration in eigenfrequencies to the presence of damage. Indeed, localized damage often alters the stiffness or mass distribution of a structure, causing a shift in its natural frequencies.

This special issue aims at gathering together contributions from experts, in particular, those presented at the homonymous conference in honor of Emilio Turco on his 60th birthday, from different—but complementary—fields relevant to the modeling of microstructured materials, with the objective of providing a rich state-of-the-art overview, and research perspectives, on this highly interdisciplinary topic.

Selected references

[1] R. Alicandro, A. Braides, M. Cicalese and M. Solci. Discrete Variatonal Problems with Interfaces. Cambridge University Press, Cambridge (2024).
[2] L. Ambrosio, N. Gigli and G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich,          Birkhäuser, Basel, 2008
[3] X. Blanc and C. Le Bris. Homogenization Theory for Multiscale Problems. Springer Verlag, Berlin, 2023.
[4] B. Bourdin, G. A. Francfort, J. J.Marigo, The variational approach to fracture. J. Elast., 91 (2008), 5–148.
[5] A. Braides, Local Minimization, Variational Evolution and Γ-convergence. Lecture Notes in Mathematics 2094, Springer Verlag, Berlin, 2013.
[6] A. Mielke and T. Roubiček. Rate-Independent Systems. Applied Mathematical Science 193, Springer Verlag, Berlin, 2015
[7] L. Placidi, E. Barchiesi, E. Turco, and N.L. Rizzi. A review on 2D models for the description of pantographic fabrics. Zeitschrift für angewandte Mathematik und Physik 67, 2016.
[8] F. dell'Isola, I. Giorgio, M. Pawlikowski, and N.L. Rizzi. Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenization, experimental and numerical examples of equilibrium. Proc. R. Soc. A, 472, 2016
[9] D. J. Steigmann. Two-dimensional models for the combined bending and stretching of plates and shells based on three-dimensional linear elasticity.       International Journal of Engineering Science, 46, 2008.
[10] A. Misra and P. Poorsolhjouy. Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Mathematics and Mechanics of Complex Systems, 3, 2015.
[11] A. Cazzani, M. Malagù, and E. Turco. Isogeometric analysis: A powerful numerical tool for the elastic analysis of historical masonry arches. Continuum      Mechanics and Thermodynamics, 21, 2014.
[12] G. Milton. The Theory of Composites. Cambridge University Press, Cambridge (2002).

Keywords:
Discrete variational models; Periodic and non-periodic homogenization; Quantitative random homogenization; Variational evolution of microstructures; Isogeometric analysis; Load identification; Damage detection; Metamaterials mechanics; Second gradient media; Granular materials

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