In this paper, we show one solution of the following synthesis problem: to find a planar, periodic, structure made up of straight bars linked by (perfect) hinges, which, once homogenized, can be modelled as a planar third-gradient one-dimensional (1D) continuum. One possible-solution structure is obtained by considering a suitably Modified Hart's Antiparallelograms Mechanism (MHAM). Such a mechanism has been conceived in order to get, once suitable kinematical constraints are added, what we call an MHAS, i.e., a Modified Hart's Antiparallelograms Structure. Each of these structures has a stress-free configuration, i.e., a configuration having vanishing deformation energy, which coincides with a circumference. By limiting to the case when all the mechanism bars are rigid and by replacing specific MHAM elements with two bars interconnected by elastic rotational joints, we get a truss structure which, once homogenized into an elastic 1 D continuum, can assume deformed shapes whose deformation energy is not vanishing only when its curvature is not constant. Choosing for the homogenized 1 D continuum, as a deformation measure, the derivative of curvature, and imposing a suitable rescaling of the rotational elastic moduli, we establish the asymptotic micro-macro relationship for its macro-deformation energy.

In this paper, we show, via an asymptotic expansion, that a 1D continuum model, with an internal variable, is suitable to describe the mechanical behavior of the Inverted Hart's micro-architectures, generalizing the treatment presented in Rizzi et al. (2026). An internal variable is needed to describe the kinematics of the 1D continuum: we find an expression for deformation energy, which depends on the gradient of curvature of a 1D continuum and on the introduced internal variable. We find, as a particular case, the results already presented in Rizzi et al. (2026): when the condensation of kinematical parameters is possible, the energy density of the 1D model depends only on the curvature gradient.

The aim of rationally designed composites called metamaterials or metasurfaces is to achieve effective properties that go beyond those of their constituent parts. For periodic architectures, the design can draw on concepts from solid-state physics, such as crystal symmetries, reciprocal space, band structures and Floquet–Bloch eigenfunctions. Recently, nonlocality has emerged as a design paradigm, enabling both static and dynamic properties that are unattainable with a local design. In principle, all material properties described by linear response functions can be nonlocal, but for ordinary solids, local descriptions are mostly good approximations, leaving nonlocal effects as corrections. However, metamaterials and metasurfaces can be designed to go far beyond local behaviour. This Review covers these anomalous behaviours in elasticity, acoustics, electromagnetism, optics and diffusion. In the dynamic regime, nonlocal interactions enable versatile band structure and refraction engineering. In the static regime, they result in large decay lengths of ‘frozen’ evanescent Bloch modes, leading to strong size effects. For zero modes, the decay length diverges. Nonlocality has gained increasing attention in metamaterial and metasurface design. This Review discusses recent advances, focusing on the physical mechanisms of nonlocality that lead to intriguing properties and functions.

Truss structures composed of members that work exclusively in tension or in compression appear in several problems of science and engineering, e.g., in the study of the resisting mechanisms of masonry structures, as well as in the design of spider web-inspired web structures. This work generalizes previous results on the existence of cable webs that are able to support assigned sets of nodal forces under tension. We extend such a problem to the limit analysis of compression-only 'strut nets' subjected to fixed and variable nodal loads. These systems provide discrete element models of masonry bodies, which lie inside the polygon/polyhedron with vertices at the points of application of the given forces ('underlying masonry structures'). It is assumed that fixed nodal forces are combined with variable forces growing proportionally to a scalar multiplier (load multiplier), and that the supporting strut net is subjected to kinematic constraints at given nodal positions.

We study, from a variational viewpoint, the asymptotic behavior of a planar beam with a periodic wavy shape when the amplitude and the wavelength of the shape tend to zero. We assume that the beam behaves, at the microscopic level, as a compressible Euler–Bernoulli beam and that the material properties have the same period as the geometry. We allow for distributed or concentrated bending compliance and for a non-quadratic extensional energy. The macroscopic Γ-limit that we obtain corresponds to a non-linear model of Timoshenko type.

Duoskelion structures have been recently introduced by Barchiesi et al. (2021) as a proof-of-concept motif for a new class of metamaterials. The properties of these periodic beam-like chiral structural elements have been investigated, up to now, by means of a discrete model formulation whose predictions are obtained by numerical methods. In this paper we select a specific scaling law for micro stiffnesses aimed at deriving, via asymptotic homogenization, an internally-constrained Cosserat one-dimensional planar continuum model as the limit of a duoskelion structure. We analyze qualitatively and quantitatively the family of equilibrium configurations of the homogenized continuum when subjected to axial loading and compare the results of the analysis with those obtained by means of the discrete model formulation.

We consider the problem of finding a net that supports prescribed point forces, yet avoids certain obstacles, with all the elements of the net being under compression (or all being under tension), and being confined within a suitable bounding box. In the case of masonry structures, when described through the simple, no-tension constitutive model, this consists, for instance, in finding a strut net that supports the forces, is contained within the physical structure, and avoids regions that may be not accessible. We solve such a problem in the two-dimensional case, where the prescribed forces are applied at the vertices of a convex polygon, and we treat the cases of both single and multiple obstacles. By approximating the obstacles by polygonal regions, the task reduces to identifying the feasible domain in a linear programming problem. For a single obstacle we show how the region Γ available to the obstacle can be enlarged as much as possible in the sense that there is no other strut net, having a region Γ ′ available to the obstacle with Γ ⊂ Γ ′ . The case where some of the forces are reactive is also treated.

. We formulate a linear programming approach to the limit analysis of discrete element models of masonry structures. Numerical results show the ability of the given procedure to predict the limit value of the multiplier of variable horizontal forces, which are applied to masonry walls in association with a fixed vertical loading

Second-gradient continua are defined as those continua whose internal virtual work functionals depend on the first and second-gradient of the virtual displacement. These functionals can be represented either in Lagrangian (referential) or Eulerian (spatial) description thus defining respectively the Piola–Lagrange as well as the Cauchy–Euler stress and double-stress. In this paper, we deduce the Piola transformation formulae, i.e., those relationships between all Lagrangian and Eulerian fields relevant for the formulation of the Principle of Virtual Work. In particular, we derive the Piola transformations of stress and double-stress as well as the Piola transformations for external virtual work functionals compatible with second-gradient internal work functionals. The latter transformations contain in fact the Piola transformations of the contact surface and line forces as well as the contact surface double-forces.

This paper presents an architectured material featuring significant strain-gradient effects and called pantographic material. It is easy to fabricate, being a plate made of a single and continuous linear elastic material containing voids. The pattern consists of triangles connected by thin junctions and arranged in such a way that two floppy strain modes are present. A homogenization scheme based on the two-scale asymptotic expansion is suggested, keeping only significant strain-gradient contributions in the homogenized energy through an adequate projection. The predictions from the homogenization scheme are validated against a full-scale simulation and yield very good L2 error estimates whereas the classical first-gradient homogenization fails. Furthermore, the relative position of the unit cell in the full-scale computation does not have a significant influence on the quality of the prediction. Finally, with an adequate choice of scalings between the scale separation and the junction thinness, it is possible to ensure that strains as well as displacements in compliant junctions remain bounded while preserving macroscopic strain-gradient effects.

In this paper, we represent second-gradient internal work functionals in Lagrangian (referential) and Eulerian (spatial) descriptions, and we deduce the corresponding expressions for the Piola transformations of stress and double-stress tensors and of external forces and double-forces. We also derive, in both the Eulerian and Lagrangian description, the expression of surface and edge contact interactions (which include forces and double-forces) for second-gradient continua in terms of the normal and the curvature of contact boundary surfaces and edge shapes.

We provide in this paper homogenization results for the L2-topology leading to complete strain-gradient models and generalized continua. Actually, we extend to the L2-topology the results obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5, 259–288) using a topology adapted to minimization problems set in varying domains. Contrary to (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5, 259–288) we consider elastic lattices embedded in a soft elastic matrix. Thus our study is placed in the usual framework of homogenization. The contrast between the elastic stiffnesses of the matrix and the reinforcement zone is assumed to be very large. We prove that a suitable choice of the stiffness on the weak part ensures the compactness of minimizing sequences while the energy contained in the matrix disappears at the limit: the Γ-limit energies we obtain are identical to those obtained in (Abdoul-Anziz & Seppecher, 2018 Homogenization of periodic graph-based elastic structures. Journal de l’Ecole polytechnique–Mathématiques 5, 259–288).

While homogenization of periodic linear elastic structures is now a well-known procedure when the stiffness of the material varies inside fixed bounds, no homogenization formula is known which enables to compute the effective properties of highly contrasted structures. Examples have been given in which the effective energy involves the strain-gradient but no general formula provides this strain-gradient dependence. Some formulas have been proposed which involve such terms and provide a small correction to the classical effective energy still when the stiffness of the material varies inside fixed bounds. The goal of this paper is to check the applicability of these formulas for highly contrasted structures. To that aim we focus on structures whose limit energy is already known and we compare the energies given by (i) the convergence results, (ii) the corrective formulas and (iii) by a direct numerical simulation of the complete structure.

Everybody has an intuitive interpretation of continuum mechanics in terms of microscopic forces: Many punctual forces applied to a material lead in a continuous approximation, that is, through a homogenization process, to a continuous surface density of forces applied on its boundary. Things are less clear when dealing with generalized continua or, worse, with strain-gradient models. For instance, interpretations of Cosserat variable are often proposed in terms of the rotation of some internal small substructures which are more rigid than the rest of the material; then, boundary actions are understood as a surface density of torques; however, it is seldom explained how these actions propagate through the body. In this chapter, we provide examples of lattices for which homogenization can be performed rigorously; they lead to generalized or strain-gradient continua; hence, the peculiarities of these models are enlightened by a possible microscopic interpretation.

The Lagrange multipliers method is used in Mathematical Analysis, in Mechanics, in Economics and in several other fields, to deal with the search of the global maximum or minimum of a function, in presence of a constraint. The usual technique, applied to the case of finite-dimensional systems, transforms the constrained optimization problem into an unconstrained one, by means of the introduction of one or more multipliers and of a suitable Lagrangian function, to be optimized. In Mechanics, several optimization problems can be applied to infinite-dimensional systems. Lagrange multipliers method can be applied also to these cases.

We determine in the framework of static linear elasticity the homogenized behavior of three-dimensional periodic structures made of welded elastic bars. It has been shown that such structures can be modeled as discrete systems of nodes linked by extensional, flexural/torsional interactions corresponding to frame lattices and that the corresponding homogenized models can be strain-gradient models, i.e., models whose effective elastic energy involves components of the first and the second gradients of the displacement field. However, in the existing models, there is no coupling between the classical strain and the strain-gradient terms in the expression of the effective energy. In the present article, under some assumptions on the positions of the nodes of the unit cell, we show that classical strain and strain-gradient strain terms can be coupled. In order to illustrate this coupling we compute the homogenized energy of a particular structure that we call asymmetrical pantographic structure.