On the undamped free vibration of a mass interacting with a Hertzian contact stiffness

Three methods are used to determine the natural frequency of undamped free vibration of a mass interacting with a Hertzian contact stiffness. The exact value is determined using the first integral of motion. The harmonic balance method is used on a transformed equation for an approximate solution, and the multiple scales method is used on an approximate equation. The maximum initial displacement avoiding contact loss is also determined, and the corresponding exact natural frequency is also obtained analytically. The methods are evaluated by studying the free vibration of an elastic sphere on a flat rigid surface. © 2011 Elsevier Ltd © 2011 Elsevier Ltd. All rights reserved.

On the contact stiffness and nonlinear vibration of an elastic body with a rough surface in contact with a rigid flat surface

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Reliability and Failure Modes of Implant-Supported Y-TZP and MCR Three-Unit Bridges

Purpose: Chipping within veneering porcelain has resulted in high clinical failure rates for implant-supported zirconia (yttria-tetragonal zirconia polycrystals [Y-TZP]) bridges. This study evaluated the reliability and failure modes of mouth-motion step-stress fatigued implant-supported Y-TZP versus palladium-silver alloy (PdAg) three-unit bridges. Materials and Methods: Implant-abutment replicas were embedded in polymethylmethacrylate resin. Y-TZP and PdAg frameworks, of similar design (n = 21 each), were fabricated, veneered, cemented (n = 3 each), and Hertzian contact-tested to obtain ultimate failure load. In each framework group, 18 specimens were distributed across three step-stress profiles and mouth-motion cyclically loaded according to the profile on the lingual slope of the buccal cusp of the pontic. Results: PdAg failures included competing flexural cracking at abutment and/or connector area and chipping, whereas Y-TZP presented predominantly cohesive failure within veneering porcelain. Including all failure modes, the reliability (two-sided at 90% confidence intervals) for a ""mission"" of 50,000 and 100,000 cycles at 300 N load was determined (Alta Pro, Reliasoft, Tucson, AZ, USA). No difference in reliability was observed between groups for a mission of 50,000. Reliability remained unchanged for a mission of 100,000 for PdAg, but significantly decreased for Y-TZP. Conclusions: Higher reliability was found for PdAg for a mission of 100,000 cycles at 300 N. Failure modes differed between materials.

Dynamic Analysis Model of Spherical Roller Bearings with Defects

Rolling element bearings are essential components of rotating machinery. The spherical roller bearing (SRB) is one variant seeing increasing use, because it is self-aligning and can support high loads. It is becoming increasingly important to understand how the SRB responds dynamically under a variety of conditions. This doctoral dissertation introduces a computationally efficient, three-degree-of-freedom, SRB model that was developed to predict the transient dynamic behaviors of a rotor-SRB system. In the model, bearing forces and deflections were calculated as a function of contact deformation and bearing geometry parameters according to nonlinear Hertzian contact theory. The results reveal how some of the more important parameters; such as diametral clearance, the number of rollers, and osculation number; influence ultimate bearing performance. Distributed defects, such as the waviness of the inner and outer ring, and localized defects, such as inner and outer ring defects, are taken into consideration in the proposed model. Simulation results were verified with results obtained by applying the formula for the spherical roller bearing radial deflection and the commercial bearing analysis software. Following model verification, a numerical simulation was carried out successfully for a full rotor-bearing system to demonstrate the application of this newly developed SRB model in a typical real world analysis. Accuracy of the model was verified by comparing measured to predicted behaviors for equivalent systems.

Studies of rotor dynamics using a multibody simulation approach

The non-idealities in a rotor-bearing system may cause undesirable subcritical superharmonic resonances that occur when the rotating speed of the rotor is a fraction of the natural frequency of the system. These resonances arise partly from the non-idealities of the bearings. This study introduces a novel simulation approach that can be used to study the superharmonic vibrations of rotor-bearing systems. The superharmonic vibrations of complex rotor-bearing systems can be studied in an accurate manner by combining a detailed rotor and bearing model in a multibody simulation approach. The research looks at the theoretical background of multibody formulations that can be used in the dynamic analysis of flexible rotors. The multibody formulations currently in use are suitable for linear deformation analysis only. However, nonlinear formulation may arise in high-speed rotor dynamics applications due to the cenrrifugal stiffening effect. For this reason, finite element formulations that can describe nonlinear deformation are also introduced in this work. The description of the elastic forces in the absolute nodal coordinate formulation is studied and improved. A ball bearing model that includes localized and distributed defects is developed in this study. This bearing model could be used in rotor dynamics or multibody code as an interface elements between the rotor and the supporting structure. The model includes descriptions of the nonlinear Hertzian contact deformation and the elastohydrodynamic fluid film. The simulation approaches and models developed here are applied in the analysis of two example rotor-bearing systems. The first example is an electric motor supported by two ball bearings and the second is a roller test rig that consists of the tube roll of a paper machine supported by a hard-bearing-type balanceing machine. The simulation results are compared to the results available in literature as well as to those obtained by measuring the existing structure. In both practical examples, the comparison shows that the simulation model is capable of predicting the realistic responses of a rotor system. The simulation approaches developed in this work can be used in the analysis of the superharmonic vibrations of general rotor-bearing systems.

Mechanical microcontacts of powder particles studied with the quartz crystal microbalance

Within this work, a particle-polymer surface system is studied with respect to the particle-surface interactions. The latter are governed by micromechanics and are an important aspect for a wide range of industrial applications. Here, a new methodology is developed for understanding the adhesion process and measure the relevant forces, based on the quartz crystal microbalance, QCM. rnThe potential of the QCM technique for studying particle-surface interactions and reflect the adhesion process is evaluated by carrying out experiments with a custom-made setup, consisting of the QCM with a 160 nm thick film of polystyrene (PS) spin-coated onto the quartz and of glass particles, of different diameters (5-20µm), deposited onto the polymer surface. Shifts in the QCM resonance frequency are monitored as a function of the oscillation amplitude. The induced frequency shifts of the 3rd overtone are found to decrease or increase, depending on the particle-surface coupling type and the applied oscillation (frequency and amplitude). For strong coupling the 3rd harmonic decreased, corresponding to an “added mass” on the quartz surface. However, positive frequency shifts are observed in some cases and are attributed to weak-coupling between particle and surface. Higher overtones, i.e. the 5th and 7th, were utilized in order to derive additional information about the interactions taking place. For small particles, the shift for specific overtones can increase after annealing, while for large particle diameters annealing causes a negative frequency shift. The lower overtones correspond to a generally strong-coupling regime with mainly negative frequency shifts observed, while the 7th appears to be sensitive to the contact break-down and the recorded shifts are positive.rnDuring oscillation, the motion of the particles and the induced frequency shift of the QCM are governed by a balance between inertial forces and contact forces. The adherence of the particles can be increased by annealing the PS film at 150°C, which led to the formation of a PS meniscus. For the interpretation, the Hertz, Johnson-Kendall-Roberts, Derjaguin-Müller-Toporov and the Mindlin theory of partial slip are considered. The Mindlin approach is utilized to describe partial slip. When partial slip takes place induced by an oscillating load, a part of the contact ruptures. This results in a decrease of the effective contact stiffness. Additionally, there are long-term memory effects due to the consolidation which along with the QCM vibrations induce a coupling increase. However, the latter can also break the contact, lead to detachment and even surface damage and deformation due to inertia. For strong coupling the particles appear to move with the vibrations and simply act as added effective mass leading to a decrease of the resonance frequency, in agreement with the Sauerbrey equation that is commonly used to calculate the added mass on a QCM). When the system enters the weak-coupling regime the particles are not able to follow the fast movement of the QCM surface. Hence, they effectively act as adding a “spring” with an additional coupling constant and increase the resonance frequency. The frequency shift, however, is not a unique function of the coupling constant. Furthermore, the critical oscillation amplitude is determined, above which particle detach. No movement is detected at much lower amplitudes, while for intermediate values, lateral particle displacement is observed. rnIn order to validate the QCM results and study the particle effects on the surface, atomic force microscopy, AFM, is additionally utilized, to image surfaces and measure surface forces. By studying the surface of the polymer film after excitation and particle removal, AFM imaging helped in detecting three different meniscus types for the contact area: the “full contact”, the “asymmetrical” and a third one including a “homocentric smaller meniscus”. The different meniscus forms result in varying bond intensity between particles and polymer film, which could explain the deviation between number of particles per surface area measured by imaging and the values provided by the QCM - frequency shift analysis. The asymmetric and the homocentric contact types are suggested to be responsible for the positive frequency shifts observed for all three measured overtones, i.e. for the weak-coupling regime, while the “full contact” type resulted in a negative frequency shift, by effectively contributing to the mass increase of the quartz..rnThe interplay between inertia and contact forces for the particle-surface system leads to strong- or weak-coupling, with the particle affecting in three mentioned ways the polymer surface. This is manifested in the frequency shifts of the QCM system harmonics which are used to differentiate between the two interaction types and reflect the overall state of adhesion for particles of different size.rn

The effect of elastic modulus and equilibrium vapor adsorption on capillary forces

Die Kapillarkraft entsteht durch die Bildung eines Meniskus zwischen zwei Festkörpen. In dieser Doktorarbeit wurden die Auswirkungen von elastischer Verformung und Flϋssigkeitadsorption auf die Kapillarkraft sowohl theoretisch als auch experimentell untersucht. Unter Verwendung eines Rasterkraftmikroskops wurde die Kapillarkraft zwischen eines Siliziumoxid Kolloids von 2 µm Radius und eine weiche Oberfläche wie n.a. Polydimethylsiloxan oder Polyisopren, unter normalen Umgebungsbedingungen sowie in variierende Ethanoldampfdrϋcken gemessen. Diese Ergebnisse wurden mit den Kapillarkräften verglichen, die auf einem harten Substrat (Silizium-Wafer) unter denselben Bedingungen gemessen wurden. Wir beobachteten eine monotone Abnahme der Kapillarkraft mit zunehmendem Ethanoldampfdruck (P) fϋr P/Psat > 0,2, wobei Psat der Sättigungsdampfdruck ist.rnUm die experimentellen Ergebnisse zu erklären, wurde ein zuvor entwickeltes analytisches Modell (Soft Matter 2010, 6, 3930) erweitert, um die Ethanoladsorption zu berϋcksichtigen. Dieses neue analytische Modell zeigte zwei verschiedene Abhängigkeiten der Kapillarkraft von P/Psat auf harten und weichen Oberflächen. Fϋr die harte Oberfläche des Siliziumwafers wird die Abhängigkeit der Kapillarkraft vom Dampfdruck vom Verhältnis der Dicke der adsorbierten Ethanolschicht zum Meniskusradius bestimmt. Auf weichen Polymeroberflächen hingegen hängt die Kapillarkraft von der Oberflächenverformung und des Laplace-Drucks innerhalb des Meniskus ab. Eine Abnahme der Kapillarkraft mit zunehmendem Ethanoldampfdruck hat demnach eine Abnahme des Laplace-Drucks mit zunehmendem Meniskusradius zur folge. rnDie analytischen Berechnungen, fϋr die eine Hertzsche Kontakt-deformation angenommen wurde, wurden mit Finit Element Methode Simulationen verglichen, welche die reale Deformation des elastischen Substrats in der Nähe des Meniskuses explizit berϋcksichtigen. Diese zusätzliche nach oben gerichtete oberflächenverformung im Bereich des Meniskus fϋhrt zu einer weiteren Erhöhung der Kapillarkraft, insbesondere fϋr weiche Oberflächen mit Elastizitätsmodulen < 100 MPa.rn

Determining the elastic-plastic properties of metallic materials through instrumented indentation

En los últimos años, y asociado al desarrollo de la tecnología MEMS, la técnica de indentación instrumentada se ha convertido en un método de ensayo no destructivo ampliamente utilizado para hallar las características elástico-plásticas de recubrimientos y capas delgadas, desde la escala macroscópica a la microscópica. Sin embargo, debido al complejo mecanismo de contacto debajo de la indentación, es urgente proponer un método más simple y conveniente para obtener unos resultados comparables con otras mediciones tradicionales. En este estudio, el objetivo es mejorar el procedimiento analítico para extraer las propiedades elástico-plásticas del material mediante la técnica de indentación instrumentada. La primera parte se centra en la metodología llevada a cabo para medir las propiedades elásticas de los materiales elásticos, presentándose una nueva metodología de indentación, basada en la evolución de la rigidez de contacto y en la curva fuerza-desplazamiento del ensayo de indentación. El método propuesto permite discriminar los valores de indentación experimental que pudieran estar afectados por el redondeo de la punta del indentador. Además, esta técnica parece ser robusta y permite obtener valores fiables del modulo elástico. La segunda parte se centra en el proceso analítico para determinar la curva tensión-deformación a partir del ensayo de indentación, empleando un indentador esférico. Para poder asemejar la curva tension-deformación de indentación con la que se obtendría de un ensayo de tracción, Tabor determinó empíricamente un factor de constricción de la tensión () y un factor de constricción de la deformación (). Sin embargo, la elección del valor de y  necesitan una derivación analítica. Se describió analíticamente una nueva visión de la relación entre los factores de constricción de tensión y la deformación basado en la deducción de la ecuación de Tabor. Un modelo de elementos finitos y un diseño experimental se realizan para evaluar estos factores de constricción. A partir de los resultados obtenidos, las curvas tension-deformación extraidas de los ensayos de indentación esférica, afectadas por los correspondientes factores de constricción de tension y deformación, se ajustaron a la curva nominal tensión-deformación obtenida de ensayos de tracción convencionales. En la última parte, se estudian las propiedades del revestimiento de cermet Inconel 625-Cr3C2 que es depositado en el medio de una aleación de acero mediante un láser. Las propiedades mecánicas de la matriz de cermet son estudiadas mediante la técnica de indentación instrumentada, haciendo uso de las metodologías propuestas en el presente trabajo. In recent years, along with the development of MEMS technology, instrumented indentation, as one type of a non-destructive measurement technique, is widely used to characterize the elastic and plastic properties of metallic materials from the macro to the micro scale. However, due to the complex contact mechanisms under the indentation tip, it is necessary to propose a more convenient and simple method of instrumented indention to obtain comparable results from other conventional measurements. In this study, the aim is to improve the analytical procedure for extracting the elastic plastic properties of metallic materials by instrumented indentation. The first part focuses on the methodology for measuring the elastic properties of metallic materials. An alternative instrumented indentation methodology is presented. Based on the evolution of the contact stiffness and indentation load versus the depth of penetration, the possibility of obtaining the actual elastic modulus of an elastic-plastic bulk material through instrumented sharp indentation tests has been explored. The proposed methodology allows correcting the effect of the rounding of the indenter tip on the experimental indentation data. Additionally, this technique does not seem too sensitive to the pile-up phenomenon and allows obtaining convincing values of the elastic modulus. In the second part, an analytical procedure is proposed to determine the representative stress-strain curve from the spherical indentation. Tabor has determined the stress constraint factor (stress CF), and strain constraint factor (strain CF), empirically but the choice of a value for and is debatable and lacks analytical derivation. A new insight into the relationship between stress and strain constraint factors is analytically described based on the formulation of Tabor’s equation. Finite element model and experimental tests have been carried out to evaluate these constraint factors. From the results, representative stress-strain curves using the proposed strain constraint factor fit better with the nominal stress-strain curve than those using Tabor’s constraint factors. In the last part, the mechanical properties of an Inconel 625-Cr3C2 cermet coating which is deposited onto a medium alloy steel by laser cladding has been studied. The elastic and plastic mechanical properties of the cermet matrix are studied using depth-sensing indentation (DSI) on the micro scale.

On the possibility of elastic strain localisation in a fault

The phenomenon of strain localisation is often observed in shear deformation of particulate materials, e.g., fault gouge. This phenomenon is usually attributed to special types of plastic behaviour of the material (e.g., strain softening or mismatch between dilatancy and pressure sensitivity or both). Observations of strain localisation in situ or in experiments are usually based on displacement measurements and subsequent computation of the displacement gradient. While in conventional continua the symmetric part of the displacement gradient is equal to the strain, it is no longer the case in the more realistic descriptions within the framework of generalised continua. In such models the rotations of the gouge particles are considered as independent degrees of freedom the values of which usually differ from the rotation of an infinitesimal volume element of the continuum, the latter being described for infinitesimal deformations by the non-symmetric part of the displacement gradient. As a model for gouge material we propose a continuum description for an assembly of spherical particles of equal radius in which the particle rotation is treated as an independent degree of freedom. Based on this model we consider simple shear deformations of the fault gouge. We show that there exist values of the model parameters for which the displacement gradient exhibits a pronounced localisation at the mid-layers of the fault, even in the absence of inelasticity. Inelastic effects are neglected in order to highlight the role of the independent rotations and the associated additional parameters. The localisation-like behaviour occurs if (a) the particle rotations on the boundary of the shear layer are constrained (this type of boundary condition does not exist in a standard continuum) and (b) the contact moment-or bending stiffness is much smaller than the product of the effective shear modulus of the granulate and the square of the width of the gouge layer. It should be noted however that the virtual work functional is positive definite over the range of physically meaningful parameters (here: contact stiffnesses, solid volume fraction and coordination number) so that strictly speaking we are not dealing with a material instability.

Dynamics of Granular Crystals with Elastic-Plastic Contacts

We study the behavior of granular crystals subjected to impact loading that creates plastic deformation at the contacts between constituent particles. Granular crystals are highly periodic arrangements of spherical particles, arranged into densely packed structures resembling crystals. This special class of granular materials has been shown to have unique dynamics with suggested applications in impact protection. However, previous work has focused on very low amplitude impacts where every contact point can be described using the Hertzian contact law, valid only for purely elastic deformation. In this thesis, we extend previous investigation of the dynamics of granular crystals to significantly higher impact energies more suitable for the majority of applications. Additionally, we demonstrate new properties specific to elastic-plastic granular crystals and discuss their potential applications as well. We first develop a new contact law to describe the interaction between particles for large amplitude compression of elastic-plastic spherical particles including a formulation for strain-rate dependent plasticity. We numerically and experimentally demonstrate the applicability of this contact law to a variety of materials typically used in granular crystals. We then extend our investigation to one-dimensional chains of elastic-plastic particles, including chains of alternating dissimilar materials. We show that, using the new elastic-plastic contact law, we can predict the speed at which impact waves with plastic dissipation propagate based on the material properties of the constituent particles. Finally, we experimentally and numerically investigate the dynamics of two-dimensional and three-dimensional granular crystals with elastic-plastic contacts. We first show that the predicted wave speeds for 1D granular crystals can be extended to 2D and 3D materials. We then investigate the behavior of waves propagating across oblique interfaces of dissimilar particles. We show that the character of the refracted wave can be predicted using an analog to Snell's law for elastic-plastic granular crystals and ultimately show how it can be used to design impact guiding "lenses" for mitigation applications.

Investigating the mesh dependency and upscaling of 3D grain-based models for the simulation of brittle fracture processes in low-porosity crystalline rock

The application of 3D grain-based modelling techniques is investigated in both small and large scale 3DEC models, in order to simulate brittle fracture processes in low-porosity crystalline rock. Mesh dependency in 3D grain-based models (GBMs) is examined through a number of cases to compare Voronoi and tetrahedral grain assemblages. Various methods are used in the generation of tessellations, each with a number of issues and advantages. A number of comparative UCS test simulations capture the distinct failure mechanisms, strength profiles, and progressive damage development using various Voronoi and tetrahedral GBMs. Relative calibration requirements are outlined to generate similar macro-strength and damage profiles for all the models. The results confirmed a number of inherent model behaviors that arise due to mesh dependency. In Voronoi models, inherent tensile failure mechanisms are produced by internal wedging and rotation of Voronoi grains. This results in a combined dependence on frictional and cohesive strength. In tetrahedral models, increased kinematic freedom of grains and an abundance of straight, connected failure pathways causes a preference for shear failure. This results in an inability to develop significant normal stresses causing cohesional strength dependence. In general, Voronoi models require high relative contact tensile strength values, with lower contact stiffness and contact cohesional strength compared to tetrahedral tessellations. Upscaling of 3D GBMs is investigated for both Voronoi and tetrahedral tessellations using a case study from the AECL’s Mine-by-Experiment at the Underground Research Laboratory. An upscaled tetrahedral model was able to reasonably simulate damage development in the roof forming a notch geometry by adjusting the cohesive strength. An upscaled Voronoi model underestimated the damage development in the roof and floor, and overestimated the damage in the side-walls. This was attributed to the discretization resolution limitations.

Dissipative contact force models

"Series: Solid mechanics and its applications, vol. 226"

Repeated sprinting on natural grass impairs vertical stiffness but does not alter plantar loading in soccer players.

This study aimed to determine changes in spring-mass model (SMM) characteristics, plantar pressures, and muscle activity induced by the repetition of sprints in soccer-specific conditions; i.e., on natural grass with soccer shoes. Thirteen soccer players performed 6 × 20 m sprints interspersed with 20 s of passive recovery. Plantar pressure distribution was recorded via an insole pressure recorder device divided into nine areas for analysis. Stride temporal parameters allowed to estimate SMM characteristics. Surface electromyographic activity was monitored for vastus lateralis, rectus femoris, and biceps femoris muscles. Sprint time, contact time, and total stride duration lengthened from the first to the last repetition (+6.7, +12.9, and +9.3%; all P < 0.05), while flight time, swing time, and stride length remained constant. Stride frequency decrease across repetitions approached significance (-6.8%; P = 0.07). No main effect of the sprint number or any significant interaction between sprint number and foot region was found for maximal force, mean force, peak pressure and mean pressure (all P > 0.05). Center of mass vertical displacement increased (P < 0.01) with time, together with unchanged (both P > 0.05) peak vertical force and leg compression. Vertical stiffness decreased (-15.9%; P < 0.05) across trials, whereas leg stiffness changes were not significant (-5.9%; P > 0.05). Changes in root mean square activity of the three tested muscles over sprint repetitions were not significant. Although repeated sprinting on natural grass with players wearing soccer boots impairs their leg-spring behavior (vertical stiffness), there is no substantial concomitant alterations in muscle activation levels or plantar pressure patterns.

Importance of footing-soil separation on dynamic stiffness of piled embedded footings

[EN]Different phenomena such a soil consolidation, erosion, and scour beneath an embedded footing supported on piles may lead to loss of contact between soil and the pile cap underside. The importance of this separation on the dynamic stiffness and damping of the foundation is assessed in this work.

Non contact inspection of the fatigue damage state of carbon fiber reinforced polymer by optical surface roughness measurements

This work presents the evaluation of a new non-contact technique to assess the fatigue damage state of CFRP structures by measuring surface roughness parameters. Surface roughness and stiffness degradation have been measured in CFRP coupons cycled with constant amplitude loads, and a Pearson?s correlation of 0.79 was obtained between both variables. Results suggest that changes on the surface roughness measured in strategic zones of components made of the evaluated CFRP, could be indicative of the level of damage due to fatigue loads. This methodology could be useful for other FRP due to similarities in the fatigue damage process.