Localized Dissipative Mechanisms Research Articles

Thermal imaging of the structural damping induced by an acoustic black hole

An Acoustic Black Hole (ABH) is a passive and efficient way to control the flexural vibrations of beams or plates. In its classical form, an ABH consists of a local reduction of the thickness of a structure according to a power law profile, associated with a thin viscoelastic coating placed in the thinnest region. A focalization and a wave trap effect occur, leading to a localized energy dissipation, which induces a local temperature increase. The objective of this paper is twofold. First, the goal is to develop an adequate experimental methodology capable of accurately mapping the small temperature variations induced by the local dissipation mechanism. Second, from the thermal standpoint, the goal is to provide experimental evidence of a local temperature increase associated with a damping effect in the case of an ABH beam. This paper thus describes a new kind of experimental methodology able to provide original data, bringing some new insight into the ABH physical understanding and the analysis of structural damping.

Multi-surface plasticity model for concrete with 3D hardening/softening failure modes for tension, compression and shear

In this work we develop multi-surface plasticity model which can reproduce the inelastic behavior and failure modes of concrete in tension, compression and shear. The main novelty of the proposed concrete model can also capture all different phases of localized failure for massive structures, where the elastic behavior is followed by the creation of the fracture process zone with a large number of micro-cracks and subsequent final failure mode with micro-cracks coalescence into the macro-crack. The fracture process zone is represented by homogenized plasticity criterion with hardening (in particular the non-associated Drucker-Prager) since the number of micro-cracks is considered sufficiently large and their orientation random. The macro-crack is represented with a surface of displacement discontinuity, which is typical of all localized dissipative mechanisms due to the apparition and development of localization zones. The main novelty of proposed model is to provide the full set of 3D localization modes for tension, for compression and for shear, with each mode using corresponding fracture energy.

Damping Oriented Design of Thin-Walled Mechanical Components by Means of Multi-Layer Coating Technology

The damping behaviour of multi-layer composite mechanical components, shown by recent research and application papers, is analyzed. A local dissipation mechanism, acting at the interface between any two different layers of the composite component, is taken into account, and a beam model, to be used for validating the known experimental results, is proposed. Multi-layer prismatic beams, consisting of a metal substrate and of some thin coated layers exhibiting variable stiffness and adherence properties, are considered in order to make it possible to study and validate this assumption. A dynamical model, based on a simple beam geometry but taking into account the previously introduced local dissipation mechanism and distributed visco-elastic constraints, is proposed. Some different application examples of specific multi-layer beams are considered, and some numerical examples concerning the beam free and forced response are described. The influence of the multilayer system parameters on the damping behaviour of the free and forced response of the composite beam is investigated by means of the definition of some damping estimators. Some effective multi-coating configurations, giving a relevant increase of the damping estimators of the coated structure with respect to the same uncoated structure, are obtained from the model simulation, and the results are critically discussed.

Three-dimensional finite elements with embedded strong discontinuities for the analysis of solids at failure in the finite deformation range

This paper presents the development of new three-dimensional finite elements in the finite deformation range for the numerical modeling of material failure characterized by propagating crack-type surfaces in solids, commonly referred to as strong discontinuities. The singular fields associated with such discontinuous solutions are captured by an element-based enhancement, locally within a multi-scale framework. The complex kinematics arising from the general propagating discontinuity surfaces, especially in the considered finite deformation setting, is captured by the direct identification of the element strain fields associated to the separation modes of the discontinuity, rather than attempting to interpolate directly the corresponding discontinuous deformation fields. To avoid the over-stiff response known as stress-locking, by which the discontinuities cannot open as needed by the material response, the new brick elements consider full linear interpolations of the displacement jumps. This requires a total of nine local parameters for each localized finite element, parameters that are condensed out at the element level. The resulting formulation does not change then neither the number nor the connectivity of the global degrees of freedom employed in the finite element resolution of the underlying global mechanical problem, thus leading to a computationally efficient numerical technique that captures sharply these discontinuous solutions. In particular, the localized dissipative mechanism associated to the material failure is captured objectively through the consideration of the proper cohesive law along the discontinuity surface. Special care is taken to assure the material frame indifference of the final formulation. Complete details of the implementation of the new elements, including the consistent linearization of the governing equations, are also presented. The results obtained in numerical simulations of representative benchmark problems illustrate the performance of the new finite elements with embedded strong discontinuities.

Decay of Solutions to Damped Korteweg–de Vries Type Equation

In the present paper we establish results concerning the decay of the energy related to the damped Korteweg–de Vries equation posed on infinite domains. We prove the exponential decay rates of the energy when a initial value problem and a localized dissipative mechanism are in place. If this mechanism is effective in the whole line, we get a similar result in H k -level, k∈ℕ. In addition, the decay of the energy regarding a initial boundary value problem posed on the right half-line, is obtained considering convenient a smallness condition on the initial data but a more general dissipative effect.

Numerical modeling of softening hinges in thin Euler–Bernoulli beams

This paper presents new finite elements for thin Euler–Bernoulli beams that incorporate the softening hinges observed at failure. The proposed methods rely crucially on the identification of the classical notion of inelastic hinge with strong discontinuities of the generalized displacements describing the beam’s deformation. The development of a multi-scale framework that effectively incorporates the localized dissipative mechanisms associated with these discontinuous solutions into the large-scale problem of the beam, or general frame system, defines a crucial step undertaken here. This framework defines the setting of its numerical implementation by finite elements enhanced with the singular strains corresponding to the discontinuities. A general procedure is presented that leads, in particular, to finite elements free of stress-locking and that resolve exactly the kinematics of the hinge. Several numerical simulations are presented to illustrate the performance of the proposed methods.

An analysis of strain localization and wave propagation in plastic models of beams at failure

This paper presents an analysis of localized failures in Timoshenko beams/rods. Standard elastoplastic models are considered first. A spectral analysis of the linearized problem shows the instability of the viscous problem and the ill-posedness of the inviscid rate-independent problem in the presence of strain softening. The consequences of the ill-posedness are illustrated by obtaining the exact solution of wave propagation in a simply supported beam and observing its lack of physical significance: the failure of the beam occurs with no energy dissipation. Finite element solutions of the problem are presented, exhibiting the characteristic pathological mesh-size dependence in these conditions. The lack of a localized dissipative mechanism in these models is identified as the main cause of these inconsistencies, regardless of the length scale present in the problem (namely, the thickness of the beam). To solve these difficulties, we formulate the so-called localized models, incorporating these localized dissipative mechanisms in the form of strong discontinuities in the generalized displacements, leading naturally to the classical notion of plastic hinges. These models consider a localized softening law between the bending moment and the rotation jump. The model problem of wave propagation in a simply supported beam is then reexamined in this context, arriving again at the exact modal solution. This solution recovers its physical significance (the breaking of the beam is modeled with a non-zero energy dissipation), illustrating the well-posedness of the problem and the regularizing effects introduced by the localized models. The comparison of this exact solution with finite element solutions of the problem allows to evaluate the approximation properties of the numerical techniques, showing now the required objective energy dissipation with respect to the mesh size.

Analysis and numerical simulation of strong discontinuities in finite strain poroplasticity

This paper presents an analysis of strong discontinuities in coupled poroplastic media in the finite deformation range. A multi-scale framework is developed for the characterization of these solutions involving a discontinuous deformation (or displacement) field in this coupled setting. The strong discontinuities are used as a tool for the modeling of the localized dissipative effects characteristic of the localized failures of typical poroplastic systems. This is accomplished through the inclusion of a cohesive-frictional law relating the resolved stresses on the discontinuity and the accumulated fluid content on it with the displacement and fluid flow jumps across the discontinuity surface. The formulation considers the limit of vanishing small scales, hence recovering a problem in the large scale involving the usual regular displacement and pore pressure variables, while capturing correctly these localized dissipative mechanisms. All the couplings between the mechanical and fluid problems, from the modeling of the solid's response through effective stresses and tractions to the geometric coupling consequence of the assumed finite deformation setting, are taken into account in these considerations. The multi-scale structure of the theoretical formulation is fully employed in the development of new enhanced strain finite elements to capture these discontinuous solutions with no regularization of the singular fields appearing in the formulation. Several numerical simulations are presented showing the properties and performance of the proposed localized models and the enhanced finite elements used in their numerical implementation.

Bone tissue ultrastructural response to elastic deformation probed by Raman spectroscopy.

Raman spectroscopy is used as a probe of ultrastructural (molecular) changes in both the mineral and matrix (protein and glycoprotein, predominantly type I collagen) components in real time of murine cortical bone as it responds to elastic deformation. Because bone is ia composite material, its mechanical properties are dependent on the structure and composition at a variety of dimensional scales. At the ultrastructural level, crystal structure and protein secondary structure distort as the tissue is loaded. These structural changes are followed as perturbations to tissue spectra. We load murine femora in a custom-made mechanical tester that fits on the stage of a Raman microprobe and can accept hydrated tissue specimens. As the specimen is loaded in tension, the shifts in mineral P-O4 v1 are followed with the microprobe. Average load and strain are measured using a load cell. These devices ensure that specimens are not loaded to or beyond the yield point. Changes occur in the mineral component of bone as a response to loading in the elastic regime. We propose that the mineral apatitic crystal lattice is deformed by movement of calcium and other ions. Raman microspectroscopy shows that bone mineral is not a passive contributor to tissue strength. The mineral active response to loading may function as a local energy storage and dissipation mechanism, thus helping to protect tissue from catastrophic damage.

An analysis of strain localization in a shear layer under thermally coupled dynamic conditions. Part 2: Localized thermoplastic models

AbstractThe analyses presented in Part 1 of this work have shown the ill‐posedness of the governing equations for a shear layer governed by a coupled thermomechanical continuum model with strain softening. Finite element solutions of the problem have been shown to exhibit a pathological mesh‐size dependence in this case, as a consequence of the aforementioned ill‐posedness. The lack of a localized dissipative mechanism has been traced back to the origin of these difficulties. We consider in this paper the incorporation of such localized dissipative mechanism in the form of a cohesive law along a discontinuity of the displacements, the so‐called strong discontinuity. A coupled thermomechanical model of these discontinuities is formulated involving a continuous temperature field. The discontinuity of the heat flow (i.e. the derivative of the temperature) accommodates the presence of the localized dissipation associated to the thermomechanical cohesive law. In this context, we obtain the exact solution of a problem of wave propagation in the shear layer when the response of the bulk of the layer is elastic. The physical nature of the solution, in contrast with the solutions obtained in Part 1 for continuum models, illustrates the regularizing effect of these localized models, showing in particular the continuous dependence of the solution on the material parameters. Furthermore, we obtain the exact solution of the problem involving a regularized discontinuity in a domain of finite length. This analysis allows to gain a full understanding of the properties of finite element approximations of the problem. Several numerical simulations are presented in this context corroborating the conclusions drawn from the analysis. Copyright © 2003 John Wiley & Sons, Ltd.

On the characterization of localized solutions in inelastic solids: an analysis of wave propagation in a softening bar

This paper presents a study of the solutions characteristic of the localized failures in inelastic solids under general dynamic conditions. The paper is divided into two parts. In the first part, we present a general framework for the inclusion of localized dissipative mechanisms in a local continuum. This is accomplished by the consideration locally of discontinuities in the displacement field, the so-called strong discontinuities, as a tool for the modeling of these localized effects of the material response. We present in this context a thermodynamically based derivation of the resulting governing equations along these discontinuities. These developments are then incorporated in the local continuum framework characteristic of typical large-scale structural systems of interest. The general multi-dimensional case is assumed in this first part of the paper. In the second part, we present in the context furnished by the previous discussion a study of the wave propagation in the one-dimensional case of a localized softening bar. We obtain first the exact closed-form solution involving a strong discontinuity with a general localized softening law. We consider next the approximate problem involving the softening response of the material in a zone of finite length. Closed-form analytic solutions are obtained for the case of a linear softening law. This analysis reveals the properties of the approximation introduced by the spatial discretization in numerical solutions of the problem. Finally, we present finite element simulations that confirm the conclusions drawn from the previous analyses.

An analysis of strong discontinuities in a saturated poro-plastic solid

We present in this paper an analysis of strong discontinuities in fully saturated porous media in the infinitesimal range. In particular, we describe the incorporation of the local effects of surfaces of discontinuity in the displacement field, and thus the singular distributions of the associated strains, from a local constitutive level to the large-scale problem characterizing the quasi-static equilibrium of the solid. The characterization of the flow of the fluid through the porous space is accomplished in this context by means of a localized (singular) distribution of the fluid content, that is, involving a regular fluid mass distribution per unit volume and a fluid mass per unit area of the discontinuity surfaces in the small scale of the material. This framework is shown to be consistent with a local continuum model of coupled poro-plasticity, with the localized fluid content arising from the dilatancy associated with the strong discontinuities. More generally, complete stress–displacement–fluid content relations are obtained along the discontinuities, thus identifying the localized dissipative mechanisms characteristic of localized failures of porous materials. The proposed framework also involves the coupled equation of conservation of fluid mass and seepage through the porous solid via Darcy's law, and considers a continuous pressure field with discontinuous gradients, thus leading to discontinuous fluid flow vectors across the strong discontinuities. All these developments are then examined in detail for the model problem of a saturated shear layer of a dilatant material. Enhanced finite element methods are developed in this framework for this particular problem. The finite elements accommodate the different localized fields described above at the element level. Several representative numerical simulations are presented illustrating the performance of the proposed numerical methods. Copyright © 1999 John Wiley & Sons, Ltd.

Theory of noncontact dissipation force microscopy

The interaction between the tip of the noncontact atomic force microscope (nc-AFM) and a surface is analyzed using the Langevin equation approach. The existence of an intrinsically local energy dissipation mechanism is demonstrated. This mechanism fundamentally differs from that related to adhesion hysteresis between the tip and the surface. A scheme referred to as noncontact dissipation force microscopy is proposed for producing surface images in ultrahigh vacuum. Moreover, a prospect for scanning probe microscopy is raised, namely, the possibility of measuring nanomechanical surface properties. It is also shown how nc-AFM operated at large amplitude could be used in principle to measure profiles of both the tip-surface interaction force and its gradient.

Large‐scale modeling of localized dissipative mechanisms in a local continuum: applications to the numerical simulation of strain localization in rate‐dependent inelastic solids

This paper presents a general framework for the formulation of constitutive models that incorporate a localized dissipative mechanism. The formalism of strong discontinuities is employed, allowing for the decoupling of the constitutive characterization of the continuum and localized responses of the material. A procedure for incorporating the localized small-scale effects of the material response in the large-scale problem characterized by the standard local continuum is described in detail. The resulting large-scale model is able to capture objectively the localized dissipation observed in localized failures of solids and structures. A localized viscous slip model is presented as a model example. The finite element implementation of the proposed formulation arises naturally as a local element enhancement of the finite element interpolations, with no regularization of the discontinuities. The above considerations are formulated first in the infinitesimal range, and then extended to the finite strain regime. Furthermore, it is shown that the proposed framework allows for the development of effective finite element methods capturing in the large scale the localized dissipation observed in the failure of rate-dependent materials, avoiding the resolution of small length scales associated to the localization bands in these regularized models. Several representative numerical simulations are presented to illustrate these ideas. Copyright © 1999 John Wiley & Sons, Ltd.

Large‐scale modeling of localized dissipative mechanisms in a local continuum: applications to the numerical simulation of strain localization in rate‐dependent inelastic solids

Large‐scale modeling of localized dissipative mechanisms in a local continuum: applications to the numerical simulation of strain localization in rate‐dependent inelastic solids

On the dissipation due to wave ringing in nonelliptic elastic materials

Initial-boundary value problems describing the mechanics of nonelliptic elastic materials give rise to solutions that involve phase boundaries, the motion of which can dissipate mechanical energy. We investigate whether this dissipation, acting alone, can drive such a system toward equilibrium. Moving phase boundaries are regarded as a localized dissipative mechanism, and we consider a model which specifically excludes dissipation away from a phase boundary (such as that due to viscoelastic damping). In the problem under consideration, wave packets reverberate between the fixed external boundary and a single internal phase boundary. The phase boundary remains stationary unless it is acted upon by one of these wave packets, and each such interaction dissipates a finite amount of energy while causing the initiating wave packet to split into a reflected wave packet and a transmitted wave packet. Consequently, the number of wave packets increases in a geometric fashion. Each individual interaction of a wave packet with the phase boundary is, in a certain sense, mechanically underdetermined, and we augment the mechanical theory with two alternative energy criteria, each of which determines a different interaction dynamics. These alternative energy criteria are motivated by considerations of maximizing the energy dissipation in the system. We treat a system that is perturbed out of an initial minimum energy equilibrium state by a disturbance at the external boundary. A framework is developed for treating the resulting wave reverberations and calculating the energy dissipation for large time. Numerical computation indicates that the total energy dissipated in both versions of the dynamical problem is that which is necessary to settle into a new energy-minimal equilibrium state. We then establish the same result analytically for a meaningful limit involving a vanishingly small dynamical perturbation.