Luca Placidi

A unifying perspective: the relaxed linear micromorphic continuum

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict nonpolar size effects. It is intended for the homogenized description of highly heterogeneous, but nonpolar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models, our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models.

Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials

In this paper reflection and transmission of compression and shear waves at structured interfaces between second-gradient continua is investigated. Two semi-infinite spaces filled with the same second-gradient material are connected through an interface which is assumed to have its own material properties (mass density, elasticity and inertia). Using a variational principle, general balance equations are deduced for the bulk system, as well as jump duality conditions for the considered structured interfaces. The obtained equations include the effect of surface inertial and elastic properties on the motion of the overall system. In the first part of the paper general 3D equations accounting for all surface deformation modes (including bending) are introduced. The application to wave propagation presented in the second part of the paper, on the other hand, is based on a simplified 1D version of these equations, which we call “axial symmetric” case.

The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics

We study well-posedness for the relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. In contrast to classical micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. Another interesting feature concerns the prescription of boundary values for the micro-distortion field: only tangential traces may be determined which are weaker than the usual strong anchoring boundary condition. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch, and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes.

Towards the Design of Metamaterials with Enhanced Damage Sensitivity: Second Gradient Porous Materials

Numerous computational and conceptual difficulties are often encountered when conceiving techniques which are effective in detecting damage intensity, localization, and onset. Actually, also when the semi-inverse or the material characterization problems (which are commonly formulated in this context) can be recognized to be well posed, the numerical and computational obstacles which need to be overcome can render useless the conceived methodology. In the present paper we propose to change the paradigm used up to now when addressing the problem of damage assessment in engineering materials. In fact, we propose to conceive a metamaterial the properties of which make more expedite and effective the detection of cracks onset and damage evolution via the study of reflection and transmission of waves. More particularly, porous materials with underlying heterogeneous micro-structure may magnify the effects of reflection and transmission of waves at damaged sites depending on the considered boundary conditions. Materials of this type would make easier the structural health monitoring via nondestructive evaluation of local damage and would permit to detect incipient structural failure in a more efficient way. By analyzing the characteristic patterns of the reflection and transmission properties of surfaces where damage is concentrated, we show that, in the considered metamaterials, slow incident waves can be used to detect the onset and evolution of first gradient macroscopic damage (δ e ), while fast incident waves can be used to reveal loss of contact at the microscopic level, i.e. to detect the onset of second gradient macroscopic damage (δ r ).

Soft impact dynamics of a cantilever beam: equivalent SDOF model versus infinite-dimensional system

Non-smooth dynamics of a cantilever beam subjected to a transverse harmonic force and impacting onto a soft obstacle is studied. Upon formulating the equations of motion of the beam, proper attention is paid to identifying the mechanical properties of an equivalent single-degree-of-freedom (SDOF) piecewise linear impacting model. A multi-degree-of-freedom (MDOF) model of the impacting beam is also derived via standard finite elements. An ‘optimal’ identification curve of the obstacle spring rigidities in the two models is obtained by comparing the relevant pseudo-resonance frequencies. The identification is then exploited in the non-linear dynamic regime to get hints on some main, mostly regular, features of non-linear dynamic response of the impacting beam by the actual investigation of the behaviour of the sole equivalent SDOF model, with a definitely lower computational effort. Sample regular and non-regular responses of the MDOF model are also presented where the identification does not work. Overall, useful points are made as regards the possibility and the limitations of referring to an SDOF impacting model to investigate the non-linear response of the underlying infinite-dimensional system.

Microcantilever dynamics in tapping mode atomic force microscopy via higher eigenmodes analysis

Microcantilever dynamics in tapping mode atomic force microscopy (AFM) is addressed via a multimode approximation, which allows to consider external excitation at primary or secondary resonance and to highlight the effect of higher order eigenmodes. Upon presenting the AFM model and its multimode discretization, the dynamic response is investigated via numerical simulation of single- and three-mode models by considering different bifurcation parameters. Typical features of tapping mode AFM response as nonlinear hysteresis, bistability, higher harmonics contribution, impact velocity, and contact force are addressed. The analysis is conducted by evaluating damping of higher modes according to the Rayleigh criterion, which basically accounts for structural damping representative of the behavior of AFMs in air. Nominal damping situations more typical of AFMs in liquids are also investigated, by considering sets of modal Q-factors with different patterns and ranges of values. Variable attractive-repulsive effe...

Intorno alle equazioni fondamentali del movimento di corpi qualsivogliono, considerati secondo la naturale loro forma e costituzione

Avviene non di rado che i nuovi ritrovamenti mediante i quali fu accresciuto qualche ramo delle Matematiche applicate, non appajano subito nel concetto e nella esposizione sgombri da superfluita o lungaggini. La complicazione de’ procedimenti analitici puo giungere anche a tale da non parer piu possibile l’andare innanzi: ed e invece allora che talvolta si scopre un punto di vista piu generale, si concentrano molte particolarita e si forma una teorica compendiosa e cosi bene assicurata da infondere lena per ulteriori progressi. Sarebbe desiderabile che questo avvenisse anche per le ultime aggiunte fatte da moderni Geometri alla Meccanica razionale: e quanto a me direi che il modo di riuscirvi l’abbiamo nelle nostre mani: resta da vedere se altri vorranno essere del mio avviso.

First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model

In this paper, we propose the first estimate of some elastic parameters of the relaxed micromorphic model on the basis of real experiments of transmission of longitudinal plane waves across an interface separating a classical Cauchy material (steel plate) and a phononic crystal (steel plate with fluid-filled holes). A procedure is set up in order to identify the parameters of the relaxed micromorphic model by superimposing the experimentally based profile of the reflection coefficient (plotted as function of the wave-frequency) with the analogous profile obtained via numerical simulations. We determine five out of six constitutive parameters which are featured by the relaxed micromorphic model in the isotropic case, plus the determination of the micro-inertia parameter. The sixth elastic parameter, namely the Cosserat couple modulus μc, still remains undetermined, since experiments on transverse incident waves are not yet available. A fundamental result of this paper is the estimate of the non-locality intrinsically associated with the underlying microstructure of the metamaterial. We show that the characteristic length Lc measuring the non-locality of the phononic crystal is of the order of 13 of the diameter of its fluid-filled holes.

Energy approach to brittle fracture in strain-gradient modelling

In this paper, we exploit some results in the theory of irreversible phenomena to address the study of quasi-static brittle fracture propagation in a two-dimensional isotropic continuum. The elastic strain energy density of the body has been assumed to be geometrically nonlinear and to depend on the strain gradient. Such generalized continua often arise in the description of microstructured media. These materials possess an intrinsic length scale, which determines the size of internal boundary layers. In particular, the non-locality conferred by this internal length scale avoids the concentration of deformations, which is usually observed when dealing with local models and which leads to mesh dependency. A scalar Lagrangian damage field, ranging from zero to one, is introduced to describe the internal state of structural degradation of the material. Standard Lamé and second-gradient elastic coefficients are all assumed to decrease as damage increases and to be locally zero if the value attained by damage is one. This last situation is associated with crack formation and/or propagation. Numerical solutions of the model are provided in the case of an obliquely notched rectangular specimen subjected to monotonous tensile and shear loading tests, and brittle fracture propagation is discussed.

3D modelling of reinforced concrete slab with yielding supports subject to impact load

Reinforced concrete slabs are the common structural elements in almost all reinforced concrete structures. An accidental impact load in such an element can be caused by mishaps in industry as well as accidents provided from transportation and artificial disasters or natural hazard as rock fall. The behaviour of reinforced concrete slabs under impact loads is not always respected or correctly taken into account. There are a number of ways to predict how an impact load will affect a reinforced concrete slab, some of which may be impractical or expensive, but because there have been significant developments in technology, numerical techniques rather than experimental approaches have become popular methods for developing detailed responses. The intent of this study is not to give an original development to any of the specific aspects (e.g. numerical cracking localisation, mesh discretisation, steel bar and concrete interactions and/or contact law between the hammer and concrete) involved in the problem. The intent is to give a finite element model to investigate the dynamic response and behaviour of reinforced concrete slabs subject to low velocity impact loading. Taking advantage of a commercial finite element package ABAQUS/Explicit code, three-dimensional numerical simulations have been performed, addressing the parameters associated with the impact load and reinforced concrete slab. The obtained results show a good correlation with the experiments.