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## Über dieses Buch

This book offers valuable insights and provides effective tools useful for imagining, creating, and promoting novel and challenging developments in structural mechanics. It addresses a wide range of topics, such as mechanics and geotechnics, vibration and damping, damage and friction, experimental methods, and advanced structural materials. It also discusses analytical, experimental and numerical findings, focusing on theoretical and practical issues and innovations in the field. Collecting some of the latest results from the Lagrange Laboratory, a European scientific research group, mainly consisting of Italian and French engineers, mechanicians and mathematicians, the book presents the most recent example of the long-term scientific cooperation between well-established French and Italian Mechanics, Mathematics and Engineering Schools. It is a valuable resource for postgraduate students, researchers and practitioners dealing with theoretical and practical issues in structural engineering.

## Inhaltsverzeichnis

### 5-Dimensional Thermodynamics of Dissipative Continua

Present work aims to develop a geometrization of thermodynamics of continua within the classical approximation where the velocity of the light is considered infinite, but nevertheless in the spirit of relativity. The connection on the manifold represents the gravitation. The temperature has the status of a vector and its gradient, called friction, merges the temperature gradient and strain velocity. We claim that the energy-momentum-mass tensor is covariant divergence free. It is a geometrized version of the first principle. The modeling of the dissipative continua is based on an additive decomposition of the momentum tensor into reversible and irreversible parts. The second principle is based on a tensorial expression of the local production of entropy which provides its Galilean invariance. On this ground, we propose a relativistic version of the second principle compatible with Poincare’s group.
Géry de Saxcé

### Quinze Ans Après...

We aim to present mathematical models of smart devices and smart structures. Smart devices are made of materials which present significant multiphysical couplings. They are integrated in smart structures which take technological advantages of some multiphysical effects. We first propose simplified but accurate models of thin plates or slender rods made of piezoelectric or electromagneto-elastic materials in both static and dynamic cases. Then we focus on smart structures such as piezoelectric patches bonded on a linearly elastic body and piezoelectric junctions between two linearly piezoelectric or elastic bodies.
Christian Licht, Thibaut Weller

### Multiplane Cohesive Zone Models Combining Damage, Friction and Interlocking

The present work describes a number of cohesive zone models (CZMs) developed over the last decade; the models are derived from a simplified approach to the micro-mechanics of the fracture process. The models are able to separately consider damage and frictional dissipation; moreover, the most recent proposed models account also for intelocking and dilatancy. Initially, the model developed by Alfano and Sacco [6], coupling together damage and friction, is reviewed. A damage variable is introduced, evolving from zero for no damage to one when cohesion is lost. The main idea is to assume that friction only acts on the damaged part of the interface. The evolution of damage is governed by a mixed-mode criterion widely used in composite materials. Then, some thermodynamical consideration is presented, which leads in a simplified context to the result that the value of the fracture energy in mode I and II has to be the same [44]. A microstructured interface model is presented, obtained as combination of more inclined planes; this model is named as Representative Multiplane Element (RME) and it shows different fracture energies in mode I and II as result of the interplay between residual adhesion and the frictional slips on the inclined elementary planes, which determines significant frictional dissipation also in pure more II. The RME model is also able to account for interlocking and dilatancy [43]. Numerical applications illustrating the capacity of the proposed models are presented.
Elio Sacco, Roberto Serpieri, Giulio Alfano

### On Alternative Approaches for Graded Damage Modelling

To prevent problem of spurious localization in damage mechanics, the introduction of a quadratic terms in gradient of damage has been used to govern the amplitude of the spatial gradient of damage. In more recent papers based on definition of Thick-Level-Set, the regularization is obtained considering that damage is function of a level-set, then the gradient of damage is bounded in zone where the level-set becomes a signed distance function. Here we consider classical damage modelling with introduction of an internal constraint on the spatial gradient of damage. This point of view produces a new regularization without the introduction of a level-set and signed distance.
Michel Frémond, Claude Stolz

### Edge Debonding Prediction in Beams Strengthened by FRP Composite Plates

Edge debonding initiation and propagation in beams strengthened with fiber-reinforced composite plates is here studied. The structural system is composed by three physical components, namely the beam, the adhesive layer and the bonded plate, which are modeled by one or several first-order shear deformable layers according to a multi-layer formulation, wherein both strong and weak interface constitutive relations are introduced to model interfaces. Debonding onset is predicted with the aid of a mixed mode coupled stress and energetic criterion, and propagation in different locations across the adhesive thickness is studied by using a mixed mode fracture criterion. The proposed models are implemented according to the 1D finite element technique, allowing accurate evaluation of interlaminar stresses and fracture energies.
Domenico Bruno, Fabrizio Greco, Stefania Lo Feudo, Paolo Nevone Blasi

### A Concurrent Multiscale Model for Crack Propagation Analysis in Composite Materials

An innovative concurrent multiscale model is proposed for simulating transverse crack propagation in fiber-reinforced composite materials, based on a domain decomposition technique equipped with an adaptive zooming-in strategy. Under general loading, the crack path is not a priori known, as both fiber/matrix interface debonding and matrix cracking are involved. Therefore, a suitable crack path tracking strategy is proposed, based a moving mesh approach coupled with a shape optimization method. A number of numerical experiments have been carried out for assessing the validity of the proposed model, with reference to the complete failure analysis of a single notched fiber-reinforced composite beam subjected to both mode-I and mixed-mode crack propagation conditions.
Domenico Bruno, Fabrizio Greco, Lorenzo Leonetti, Paolo Lonetti

### Quasi-static Evolution, Variational Principles and Implicit Scheme in Gradient Plasticity

This paper is devoted to the theory of gradient plasticity. Our attention is focussed on the description of the constitutive equations, on the formulation of the governing equations in terms of the energy potential and the dissipation potential of the solid. The evolution equation is discussed for quasi-static responses. A time-discretization by the implicit scheme of the evolution equation leads to the study of the incremental problem which is different from the rate problem. The incremental problem and associated incremental variational principles are discussed in relation with some existing results of the literature.
Quoc-Son Nguyen

### Deviatoric Strength of Nanoporous Materials: A Limit Analysis Approach

In this paper, deviatoric strength properties of nanoporous materials are investigated by addressing the limit state of a hollow sphere undergoing axisymmetric deviatoric strain-rate based loading conditions. The hollow sphere is assumed to be comprised of a rigid ideal-plastic matrix obeying to a von Mises strength criterion. Void-size effects are consistently described by introducing a coherent-imperfect homogeneous interface at the cavity boundary. In the framework of a kinematic approach, the limit-analysis problem on the hollow sphere is solved by referring to a particular trial velocity field, expressed in terms of some free model parameters, chosen as a result of an optimization strategy. A closed-form expression for estimating the macroscopic deviatoric strength is obtained and successfully compared with available benchmarking data.
Stella Brach, Luc Dormieux, Djimédo Kondo, Giuseppe Vairo

### On Melan’s Theorem in Temperature-Dependent Viscoplasticity

In plasticity, Melan’s theorem is a well-known result that is both of theoretical and practical importance. That theorem applies to elastic-plastic structures under time-dependent loading histories, and gives a sufficient condition for the plastic dissipation to remain bounded in time. That situation is classically referred to as shakedown. Regarding fatigue, shakedown corresponds to the most favorable case of high-cycle fatigue. The original Melan’s theorem rests on the assumption that the material properties remain constant in time, independently on the applied loading. Extending Melan’s theorem to time fluctuating elastic moduli is a long standing issue. The main motivation is to extend the range of applications of Melan’s theorem to thermomechanical loading histories with large temperature fluctuations: In such case, the variation of the elastic properties with the temperature cannot be neglected. In this contribution, an extension of Melan’s theorem to elastic-viscoplastic materials with time-periodic elastic moduli is presented. Such a time-dependence may for instance result from time-periodic temperature variations. An illustrative example is presented and supported by numerical results obtained from incremental analysis.
Michaël Peigney

### Incompressibility and Large Deformations

We present a new point of view on the motion of an incompressible solid with large deformations. The description of the shape changes of the solid involves the stretch matrix $$\mathbf {W}$$ of the classical polar decomposition. The incompressibility condition is $$\det \mathbf {W}\,\ge \,1$$, accounting for possible cavitation or phase change. The reaction to the incompressibility condition is a pressure which is positive. There is cavitation or phase change when the pressure is null. The motion of a three-dimensional solid is investigated between time 0 and a final time $$T>0$$. It is possible to prove that the model is coherent in terms of mechanics and mathematics. Let us note that the pressure is a measure allowing possible internal collisions due to cavitation.
Elena Bonetti, Michel Frémond

### Crowd-Structure Interaction in Laterally Vibrating Footbridges: Comparison of Two Fully Coupled Approaches

Two models that deal with the crowd-structure interaction have been developed. The first is a 1D continuous model and the other is a 2D discrete one. In this paper, a summary of the formulation of these two models is presented. Both approaches used to represent the pedestrian-structure coupling phenomenon are detailed and compared. We start by introducing the partial and ordinary differential equations that govern the dynamics of both the continuous and the discrete models. First, the equation of dynamics of the footbridge for the case of lateral vibrations is recalled. Then, the Kuramoto phase equation is implemented for describing the coupling between the pedestrians and the laterally moving deck of a footbridge. Results obtained from numerical simulations are presented and compared with available experimental data.
Bachar Kabalan, Pierre Argoul, Silvano Erlicher

### Computational Modeling of Backward Erosion Piping

This work presents a short account of some recent advances in the numerical simulation of backward erosion piping, accomplished in the framework of a collaborative research between Italian and French research groups. After a brief review of the state of the art in engineering practice and research, we outline the key points of a novel approach and show some of the results obtained in an extensive validation work.
Andrea Francesco Rotunno, Carlo Callari, Francesco Froiio

### Experimental Analysis of a Tuned Mass Damper with Eddy Currents Damping Effect

A Tuned Mass Damper (TMD) is a structural passive control device fixed on a structure and composed of a linear oscillator which natural frequency is tuned to that of the structure, or to the dominant resonance frequency. In this paper, an experimental TMD with adjustable stiffness and eddy current damping is proposed. The first step is to check if the dynamical properties of the proposed TMD are constant during the dynamic test and for different values of stiffness and damping. Therefore, the instantaneous modal parameters are evaluated by applying the continuous wavelet transform on the experimental data. Then, the TMD is set with optimal parameters and used to control vibrations of a frame scale model. The structure response with and without the TMD is evaluated from the experimental measurements in case of a shock applied to the top floor.
Stefania Lo Feudo, Anissa Allani, Gwendal Cumunel, Pierre Argoul, Franco Maceri, Domenico Bruno

### Experimental Study on a Scaled Model of Offshore Wind Turbine on Monopile Foundation

Offshore wind turbines are slender structures with sensitive dynamics, strongly influenced by soil-structure interaction. The structure is subjected to cyclic and dynamic loads with frequencies close to the first natural frequency of the offshore wind turbine. To avoid resonance phenomenon due to external excitations, it is essential to precisely evaluate the initial first natural frequency of the wind turbine and its long term evolution. The present work deals with the design and analysis of a scaled model of an offshore wind turbine with monopile foundation. This study is aimed at experimental evaluation of the initial first natural frequency of this scaled model, followed by the comparison of experimental results with those obtained from the existing analytical models.
Laura Kerner, Jean-Claude Dupla, Gwendal Cumunel, Pierre Argoul, Jean Canou, Jean-Michel Pereira
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