Ugo Andreaus

At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola

Gabrio Piola’s scientific papers have been underestimated in mathematical physics literature. Indeed, a careful reading of them proves that they are original, deep and far-reaching. Actually, even if his contribution to the mechanical sciences is not completely ignored, one can undoubtedly say that the greatest part of his novel contributions to mechanics, although having provided a great impetus to and substantial influence on the work of many preeminent mechanicians, is in fact generally ignored. It has to be remarked that authors Capecchi and Ruta dedicated many efforts to the aim of unveiling the true value of Gabrio Piola as a scientist; however, some deep parts of his scientific results remain not yet sufficiently illustrated. Our aim is to prove that non-local and higher-gradient continuum mechanics were conceived already in Piola’s works and to try to explain the reasons for the unfortunate circumstance which caused the erasure of the memory of this aspect of Piola’s contribution. Some relevant differential relationships obtained in Piola (Memoria intorno alle equazioni fondamentali del movimento di corpi qualsivogliono considerati secondo la naturale loro forma e costituzione, 1846) are carefully discussed, as they are still too often ignored in the continuum mechanics literature and can be considered the starting point of Levi-Civita’s theory of connection for Riemannian manifolds.

Piezoelectric passive distributed controllers for beam flexural vibrations

Recent technological developments have made available efficient bender transducers based on the piezoelectric effect. In this paper an electrical circuit analog to the Timoshenko beam is synthesized using a Lagrangian method and by paralleling capacitive flux linkages to rotation and transverse displacement. A Piezo-ElectroMechanical (PEM) beam is conceived by uniformly distributing piezoelectric transducers on a beam and interconnecting their electric terminals via the found analog circuit, completed with suitable resistors. The high performance features of the synthesized novel circuit include the following. (i) The circuit topology is extremely reduced, the used components are all but one two-terminal elements, and the only two-port network needed is an ideal transformer. (ii) One and the same dissipative circuit ensures a multiresonance coupling with the vibrating beam and the optimal electrical dissipation of mechanical vibrations energy. (iii) For a prototype of a PEM beam, the design of the analog circuit is possible and the obtained nominal values of the circuital elements ensure that can be technically realized without any external feeding. The insertion of resistors in the analog circuit is determined according to two optimality criteria (namely minimization of strain energy time envelope and maximization of vibration time rate decay), based on specific engineering needs. The former seems to be suitable for applications in fatigue phenomena and the latter when the amplitude of vibrations must be rapidly decreased, independently of the initial conditions.

Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle

The dynamics of non-smooth oscillators has not yet sufficiently been investigated, when damping is simultaneously due to friction and impact. Because of the theoretical and practical interest of this type of systems, an effort is made in this paper to lighten the behaviour of a single-degree-of-freedom oscillator colliding with an obstacle and excited by a moving base, which transfers energy to the system via friction. The different nature of discontinuities arising in the combined problem of friction and impact has been recognized and discussed. Closed-form solutions are presented for both transient and steady-state response, assuming Coulombs friction law and a rigid stop-limiting motion. Furthermore, a deformable (hysteretic) obstacle has been considered, and its influence on the response has been investigated.

On the rocking-uplifting motion of a rigid block in free and forced motion : influence of sliding and bouncing

SummaryThe problem of rocking response of a rigid block in free and forced motion has been studied for a number of technical reasons. Apart from the technical interests, the problem of rigid block rocking is intrinsically of interest from a theoretical point of view. In fact, the problem is highly nonlinear in nature.Aim of this paper is to study the “contact-impact” problem of a rigid block colliding on a frictional base, by means of a numerical simulation, and to compare numerical results with analytical responses known from the literature.The influence of sliding and bouncing on the orbit type and stability is analyzed by a 3-DOF model of the system and by a new refined model of the contact forces between block and base. Furthermore, attention has been paid to two-dimensional free motion of the block with three degrees of freedom. Refined analytical stress-strain relations in either normal and tangential directions with respect to the contact surfaces are formulated which allow to account for (i) up-lifting and hysteretic damping in normal direction, (ii) coupling between shear strength and compression force, friction dissipation and cumulating damage in tangential direction.

A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials

In this paper, the phenomena of resorption and growth of bone tissue and resorption of the biomaterial inside a bicomponent system are studied by means of a numerical method based on finite elements. The material behavior is described by a poro-viscoelastic model with infiltrated voids. The mechanical stimulus that drives these processes is a linear combination of density of strain energy and viscous dissipation. The external excitation is represented by a bending load slowly variable with sinusoidal law characterized by different frequencies. Investigated aspects are the influence of the load frequency, of type of the stimulus and of the effective porosity on the time evolution of the mass densities of considered system.

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...

Coupling image processing and stress analysis for damage identification in a human premolar tooth

Non-carious cervical lesions are characterized by the loss of dental hard tissue at the cement-enamel junction (CEJ). Exceeding stresses are therefore generated in the cervical region of the tooth that cause disruption of the bonds between the hydroxyapatite crystals, leading to crack formation and eventual loss of enamel and the underlying dentine. Damage identification was performed by image analysis techniques and allowed to quantitatively assess changes in teeth. A computerized two-step procedure was generated and applied to the first left maxillary human premolar. In the first step, dental images were digitally processed by a segmentation method in order to identify the damage. The considered morphological properties were the enamel thickness and total area, the number of fragments in which the enamel is chipped. The information retrieved by the data processing of the section images allowed to orient the stress investigation toward selected portions of the tooth. In the second step, a three-dimensional finite element model based on CT images of both the tooth and the periodontal ligament was employed to compare the changes occurring in the stress distributions in normal occlusion and malocclusion. The stress states were analyzed exclusively in the critical zones designated in the first step. The risk of failure at the CEJ and of crack initiation at the dentin-enamel junction through the quantification of first and third principal stresses, von Mises stress, and normal and tangential stresses, were also estimated.

The complete works of Gabrio Piola: Volume I

Gabrio Piola works had an enormous impact on the development of applied mathematics and continuum mechanics. An excellent scientific committee who took it upon themselves to translate his complete works. In a second step, they commentedPiolas work and compared it to modern theories in mechanics in order to stress Piolas impact on modern science and proofs that he has set milestones in applied mathematics.This book presents Piolas original Italian text together with ist translations and their comments. It shows impressively that Gabrio Piolas work must still be regarded as a modern theory.

Optimal bone density distributions

In this paper a control and optimization procedure for bone remodeling simulations was adopted to study the effect of the osteocyte influence range on the predicted density distribution. In order to reach this goal, the osteocyte network regulating bone remodeling process in a 2-D bone sample was numerically simulated. The assumed proportional-integral-derivative (PID) bone remodeling rule was related to the error signal between the strain energy density and a selected target. Furthermore the control parameters and the target were optimally determined minimizing a suitable cost index: the goal was to minimize the final mass and the energy thus maximizing the stiffness. The continuum model results show that the developed and adapted trabecular structure was consistent with the applied loads and only depended on the external forces, the value of the cost index, the maximum attainable elastic modulus value (hence, the maximum density value) and the value of the energy target. The remodeling phenomenon determined the number and thickness of the trabeculae which are formed from a uniform distribution of mass density in the considered domain; this number and these thicknesses are controlled by the values assigned to the parameters of the model. In particular, the osteocyte decay distance (D) of the influence range affected the trabecular patterns formation, showing an important effect in the adaptive capacity of the optimization numerical model.