Giuseppe Rosi

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.

Analytical continuum mechanics à la Hamilton-Piola: least action principle for second gradient continua and capillary fluids

In this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments. We remark that these fluids are sometimes also called Korteweg–de Vries or Cahn–Allen fluids. In general, continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola–Toupin, Mindlin, Green–Rivlin, Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler–Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and ∇C or on C−1 and ∇C−1, where C is the Cauchy–Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions are recovered. A version of Bernoulli’s law valid for capillary fluids is found and useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola’s contribution to analytical continuum mechanics are also presented.

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 ).

Optimization of piezoelectric patch positioning for passive sound radiation control of plates

The purpose of this article is the optimization of piezoelectric patch positioning for reducing the radiated sound power of thin plates. To this end, an aluminium plate equipped with a set of piezoelectric patches connected to a passive circuit is considered. The difficulties in designing a smart structure are not only related to the conception of the electric circuit used as controller, but also to the choice of how the circuit itself is coupled with the structure. The selection of the number of transducers to be used, and their positioning, is a crucial step in the designing process. In this work the entire design process of a smart structure is proposed, with the goal of obtaining the best efficiency in terms of reduction of radiated sound power, while using a lightweight optimization procedure in terms of computational costs. To this end, classical instruments of vibrations mechanics are used together with acoustic concepts, such as the modal radiation efficiency. The introduction of this acoustic characterization of the structure in the optimization process is done by using a new utility function, the acoustic controllability, which will be used for the optimization of transducer positioning. The proposed optimization process is applied to the case study of an aluminium plate with nonstandard boundary conditions, and experimental results confirm the validity of this novel procedure.

Reflection and transmission of elastic waves at interfaces embedded in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model

Abstract In this paper we propose to study wave propagation, transmission and reflection in band-gap mechanical metamaterials via the relaxed micromorphic model. To do so, guided by a suitable variational procedure, we start deriving the jump duality conditions to be imposed at surfaces of discontinuity of the material properties in non-dissipative, linear-elastic, isotropic, relaxed micromorphic media. Jump conditions to be imposed at surfaces of discontinuity embedded in Cauchy and Mindlin continua are also presented as a result of the application of a similar variational procedure. The introduced theoretical framework subsequently allows the transparent set-up of different types of micro-macro connections granting the description of both (i) internal connexions at material discontinuity surfaces embedded in the considered continua and, as a particular case, (ii) possible connections between different (Cauchy, Mindlin or relaxed micromorphic) continua. The established theoretical framework is general enough to be used for the description of a wealth of different physical situations and can be used as reference for further studies involving the need of suitably connecting different continua in view of (meta-)structural design. In the second part of the paper, we focus our attention on the case of an interface between a classical Cauchy continuum on one side and a relaxed micromorphic one on the other side in order to perform explicit numerical simulations of wave reflection and transmission. This particular choice is descriptive of a specific physical situation in which a classical material is connected to a phononic crystal. The reflective properties of this particular interface are numerically investigated for different types of possible micro-macro connections, so explicitly showing the effect of different boundary conditions on the phenomena of reflection and transmission. Finally, the case of the connection between a Cauchy continuum and a Mindlin one is presented as a numerical study, so showing that band-gap description is not possible for such continua, in strong contrast with the relaxed micromorphic case.

On the validity range of strain-gradient elasticity: a mixed static-dynamic identification procedure

Wave propagation in architectured materials, or materials with microstructure, is known to be dependent on the ratio between the wavelength and a characteristic size of the microstructure. Indeed, when this ratio decreases (i.e. when the wavelength approaches this characteristic size) important quantities, such as phase and group velocity, deviate considerably from their low frequency/long wavelength values. This well-known phenomenon is called dispersion of waves. Objective of this work is to show that strain-gradient elasticity can be used to quantitatively describe the behaviour of a microstructured solid, and that the validity domain (in terms of frequency and wavelength) of this model is sufficiently large to be useful in practical applications. To this end, the parameters of the overall continuum are identified for a periodic architectured material, and the results of a transient problem are compared to those obtained from a finite element full field computation on the real geometry. The quality of the overall description using a strain-gradient elastic continuum is compared to the classical homogenization procedure that uses Cauchy continuum. The extended model of elasticity is shown to provide a good approximation of the real solution over a wider frequency range.

Numerical investigations of ultrasound wave propagating in long bones using a poroelastic model

Ultrasonic responses probed from an axial transmission test (ATT) may provide useful information about material and structural properties of cortical bone. For the mathematical modeling of ultrasonic wave propagation in long bones, most of studies assumed an (visco-)elastic behavior for cortical bone tissue by neglecting the interstitial pressure in the pores presented within this material. Here, a functionally graded anisotropic poroelastic model is proposed for describing the behavior of long bones in the ultrasonic frequency range. The simulation of time-domain wave propagation can efficiently be carried out by using a semi-analytical finite element method. The proposed model allows us investigate the influence of the presence of the pores, as well as their distribution in a bone layer on the speed of sound propagated in a cortical bone layer coupled with the marrow and the soft tissue. The effects of emitted signal’s frequency will also be examined.

Influence of anisotropic bone properties on the biomechanical behavior of the acetabular cup implant: a multiscale finite element study

Abstract Although the biomechanical behavior of the acetabular cup (AC) implant is determinant for the surgical success, it remains difficult to be assessed due to the multiscale and anisotropic nature of bone tissue. The aim of the present study was to investigate the influence of the anisotropic properties of peri-implant trabecular bone tissue on the biomechanical behavior of the AC implant at the macroscopic scale. Thirteen bovine trabecular bone samples were imaged using micro-computed tomography (μCT) with a resolution of 18 μm. The anisotropic biomechanical properties of each sample were determined at the scale of the centimeter based on a dedicated method using asymptotic homogenization. The material properties obtained with this multiscale approach were used as input data in a 3D finite element model to simulate the macroscopic mechanical behavior of the AC implant under different loading conditions. The largest stress and strain magnitudes were found around the equatorial rim and in the polar area of the AC implant. All macroscopic stiffness quantities were significantly correlated (R2 > 0.85, p < 6.5 e-6) with BV/TV (bone volume/total volume). Moreover, the maximum value of the von Mises stress field was significantly correlated with BV/TV (R2 > 0.61, p < 1.6 e-3) and was always found at the bone-implant interface. However, the mean value of the microscopic stress (at the scale of the trabeculae) decrease as a function of BV/TV for vertical and torsional loading and do not depend on BV/TV for horizontal loading. These results highlight the importance of the anisotropic properties of bone tissue.