GENERALIZED CONTINUUM THEORIES

In the framework of continuum theories, the systematic use of Cauchy theories may sometimes represent a too drastic simplification of reality, especially when dealing with metamaterials, since some essential characteristics related to the heterogeneity of microstructures are implicitly neglected in such models. Every material is actually heterogeneous if one considers sufficiently small scales: it suffices to go down to the molecular or atomic level to be aware of such heterogeneity. Nevertheless, very often, the effect of microstructure cannot be detected at the engineering scale. In such cases, continuum Cauchy theory is a suitable choice for modeling the mechanical behavior of considered materials in the simplest and more effective way. However, there are some cases in which the considered materials are heterogeneous even at relatively large scales and, as a consequence, the effect of microstructure on the overall mechanical behavior of the medium cannot be neglected. In such situations, Cauchy continuum theory may not be sufficient to fully describe the mechanical behavior of considered materials especially when considering particular loading and/or boundary conditions.

Generalized continuum theories (micromorphic, second gradient etc.) may be good candidates to model such micro-structured materials in a more appropriate way (both in the static and dynamic regime) since they are able to account for the description of some macroscopic manifestations of the presence of microstructure in a rather simplified way.

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WAVE PROPAGATION IN MICROSTRUCTURED MATERIALS

A fascinating field of application of generalized continuum theories which may have a deep impact in engineering and technology is that of the study of wave propagation in metamaterials. In fact, classical Cauchy models are not sufficient to describe the dynamical behavior of some metamaterials with complex microstructures and which show exotic responses such as, for example, the inhibition of wave propagation (frequency band gaps). Such peculiar dynamical behavior can be related to two main different phenomena occurring at the micro-level:

• local resonance phenomena (Mie resonance): the micro-structural components, excited at particular frequencies, start oscillating independently of the matrix so capturing the energy of the propagating wave which remains confined at the level of the microstructure. Macroscopic wave propagation thus results to be inhibited.

• diffusion phenomena (Bragg scattering): when the propagating wave has wavelengths which are small enough to start interacting with the microstructure of the material, reflection and transmission phenomena occur at the level of the microstructure that globally result in an inhibited macroscopic wave propagation.

Independently of the triggering mechanism, such inhibition of wave propagation typically intervenes for precise frequency ranges which are known as “frequency band gaps”. A relaxed micromorphic theory is the correct choice to account for the existence of such band gaps always remaining in the framework of a continuum theory.

One of the main advantages of using a continuum theory to describe this kind of phenomena may be found in the fact that a continuum model introduces only few elastic parameters so providing a reasonable compromise between the complexity of the model to be used and the detail at which microstructures can be described.

Such simplified continuum framework allows the possibility of performant numerical implementations of otherwise extremely complex systems.

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FIBROUS COMPOSITE REINFORCEMENTS

One of the most promising fields of application of generalized continuum theories is that of the modeling of the mechanical behavior of woven fibrous composite reinforcements. Such metamaterials are constituted by two order of fibers which have very high elongation stiffness, but very low shear stiffness. This strong contrast in the mechanical properties of the mesostructure is such that the homogenized material must necessarily be described at least in the framework of second gradient theories. As a matter of fact, classical Cauchy theories are not sufficient for the description of specific deformation patterns usually observed in fibrous composite reinforcements such as concentration of high gradients of strains in thin transition layers which can be seen to be related to flexural strains of the fibers. It is worth to stress the fact that a classical Cauchy continuum theory is not able in any case to take into account the effect of flexural bending stiffness of the yarns on the overall mechanical behavior of fibrous composite reinforcements. On the other hand, it is easy to understand that such a mesoscopic deformation mechanism may have an important macroscopic effect on the overall deformation of the considered material, at least for particular boundary conditions and/or applied external loads.

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BONE/BIO-MATERIAL REMODELING

Another field of application of generalized continuum theories may be found in the study of the bio-mechanical behavior of bone tissues in presence or absence of bioresorbable artificial materials. It is indeed well established, both theoretically and experimentally, that the application of externally applied loads may favor the regeneration and remodeling of bone. It is clear that the application of suitable mechanical loads may be beneficial for the remodeling of bone also in presence of bioresorbable materials. Among all the possible generalized continuum models, an internal variable model is the most suitable one which can be used in order to describe the remodeling of a bone-biomaterial mixture in the framework of a continuum theory. Indeed, a model of this type can be suitable to:

• account for the effect of the application of mechanical loads on the macroscopic process of remodeling of both bone and biomaterial

• include in the modeling the existence of an underlying activity of cells at the lower scales: even if the used model is intrinsically macroscopic, the evolution laws for the introduced internal variables allow to account for the effect of cells activity on the density variation of both bone and biomaterial at the macroscopic scale.

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MECHANICS OF POROUS MEDIA AND CONCRETE

Another possible application of generalized continuum theories is the description of the mechanical behavior of concrete, with particular reference to energy dissipation in the dynamical regime. In fact, it is well known that concrete is an engineering material which can be considered to be almost homogeneous at the scale of the structure, but which indeed presents strong heterogeneities at lower scales due to the very different mechanical properties of its basic constituents. In particular, it is well known that the heterogeneous microstructure of concrete can be schematized by considering a set of micro-cracks immersed in a quasi-brittle matrix.

The most natural way of approaching the problem of modeling the heterogeneity of concrete in the framework of a generalized continuum theory is that of considering an enriched kinematics which is able to simultaneously account for:

• the standard macro-displacement of the material points

• the microscopic relative displacement of the crack lips.

The microscopic relative displacement can be seen as an internal variable whose evolution is able to macroscopically influence the behavior of the overall material.

In particular, friction phenomena associated to the microscopic motions of cracks can be seen to directly produce macroscopic dissipative effects in concrete specimens subjected to dynamical loading conditions.

Concrete can also be modeled in the framework of the mechanics of porous media.

Porous media can be considered to belong to the class of generalized continua in the sense that an extended kinematics (fluid placement in addition to the standard solid one) is needed in order to describe their motion. A rigorous development of the mechanics of porous media may be of use to master the behavior of other complex systems such as saturated and unsaturated soils, petroleum and gas reservoirs, etc.

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