Shakedown in elastic contact problems with Coulomb friction

Elastic systems with frictional interfaces subjected to periodic loading are sometimes predicted to ‘shake down’ in the sense that frictional slip ceases after the first few loading cycles. The similarities in behaviour between such systems and monolithic bodies with elastic–plastic constitutive behaviour have prompted various authors to speculate that Melan’s theorem might apply to them – i.e., that the existence of a state of residual stress sufficient to prevent further slip is a sufficient condition for the system to shake down.In this paper, we prove this result for ‘complete’ contact problems in the discrete formulation (i) for systems with no coupling between relative tangential displacements at the interface and the corresponding normal contact tractions and (ii) for certain two-dimensional problems in which the friction coefficient at each node is less than a certain critical value. We also present counter-examples for all systems that do not fall into these categories, thus giving a definitive statement of the conditions under which Melan’s theorem can be used to predict whether such a system will shake down.     

Non-destructive evaluation of the interface between silicon dies and copper leadframes in integrated circuit packaging

The silicon die and copper leadframe in integrated circuit (IC) packaging are bonded together by die attach adhesive, and the quality of the interface is a critical issue in the reliability of IC packaging. Common defects such as cracks and delaminations can be detected using the C-scan ultrasonic microscopy method with sufficient confidence. However, weak interfaces due to weak adhesion and poor cohesion have often gone undetected, to subsequently become potential defective areas. In this paper we present experimental work to evaluate the quality of the interfaces that typically exist in IC packages using longitudinal ultrasonic wave propagation with contact transducers. Three different conditioning processes, varying curing, moisture exposure and pre-curing moisture contamination, are used to degrade the interface bonding the silicon die and copper leadframe. Ultrasonic reflection coefficients from the interface are then measured. The results show that the reflection coefficient depends strongly on the interface quality, and can be used as a quantitative indicator to characterize the bond quality.     

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.     

Experimental contact pattern analysis for a gear-rack system

Gears perform their main task, namely load transmission, by means of very small contact areas originated by tooth interaction and thus, the analysis of phenomena occurring at the interface between mating teeth represents a critical issue in ensuring the optimal functioning of such devices. Nevertheless, while literature proposes a huge amount of numerical tooth contact analyses (TCA), a lack of experimental validation of such approaches is to be noted, since it is extremely difficult to inspect a contact interface which is, by its own nature, closed towards the outside world. One of the most promising techniques employed in investigating contact in metallic interfaces is based on the use of high frequency ultrasonic waves; their reflection from the interface (which is known to be related to contact conditions) can be graphically processed to build maps from which it is possible to assess geometrical features of the nominal contact area and, after a suitable calibration procedure, contact pressure distribution.     

The application of asymptotic solutions to contact problems characterised by logarithmic singularities

We give the contact pressure distribution near a contacting wedge having a slightly rounded form adjacent to a discontinuity in surface profile. It is shown that, well away from the rounding the pressure is logarithmic in form, just as it is near the apex of a sharp wedge. This pair of solutions may then be used to ‘patch in’ a roundness correction relevant to any punch having a discontinuous gradient. Further, it is noted that the multiplier on the logarithm term is pre-determined by the change in gradient. This process is applied to a finite, slightly blunt wedge, where the exact answer is known, and to a wheel having a worn flat. The agreement with the exact solution in the former case is seen to be very good.     

NDE of metal damage: ultrasonics with a damage mechanics model

This research describes a nondestructive method for the quantitative estimation of property variations due to damage in metal materials. The method employs a damage mechanics model, which accounts for stiffness degradation and damage evolution of a metal medium with a measurement of ultrasonic velocity. In order to describe the progressive deterioration of materials prior to the initiation of macrocracks, we have developed a new damage mechanics model. Thereafter, a finite element model valid for numerically describing such damage process has been developed by ABAQUS/Standard code, and correlations between damage state, elastic stiffness and plastic strain could be found by the results of the finite element simulation. The property variations due to damage evolution are calculated based on the Mori–Tanaka theory, and then the ultrasonic velocity can be predicted by Christoffel’s equation. When the measured velocity is coupled with the theoretically predicted velocity, the unknown damage variable is solved, from which other residual properties are determined by the predictions of damage model. The proposed technique is performed on type 304 stainless steel bars. The numerical results obtained by the simulation were compared with experimental ones in order to verify the validity of the proposed finite element model and good agreement was found. It is shown that the damaged properties of metals can be estimated accurately by the proposed method.     

End-shaped copper fibers in an epoxy matrix—predicted versus actual fracture toughening

End-shaped copper fibers are placed in a brittle thermoset epoxy matrix at 10 vol% and tested in four-point bending to determine the fracture toughness of the composite. Results from four-point bend tests agree well with the theoretical predictions of the fracture toughness increment ‘ΔG’ of a metal fiber/brittle thermoset matrix composite based on single fiber pullout (SFP) tests. This close agreement demonstrates that SFP testing, along with the theoretical model, can be used as an effective end-shape screening tool for ductile fibers before full scale composite testing. The model predicts that the composite’s fracture toughness will be 46% higher with flat end-impacted fibers and 4% lower with rippled fibers compared to straight fibers at a 0° orientation. Four-point bend results show the actual composite’s fracture toughness is 49% higher with flat end-impacted fibers and 5% lower with rippled fibers compared to straight fibers. Further, four-point bend results show that end-shaped copper fibers improve both the flexural strength and modulus of the composite, demonstrating that end-shaped ductile fibers provide a good stress transfer to the fibers by anchoring the fibers into the matrix. Lastly, experimental validation of the model also indicates that at low fiber volume fractions, fiber–fiber interaction has only a minor influence on the fracture toughness for the tested ductile fiber/brittle matrix composite.     

Intermittency and Regularity Issues in 3<Emphasis Type="Italic">D</Emphasis> Navier-Stokes Turbulence

Two related open problems in the theory of 3D Navier-Stokes turbulence are discussed in this paper. The first is the phenomenon of intermittency in the dissipation field. Dissipation-range intermittency was first discovered experimentally by Batchelor and Townsend over fifty years ago. It is characterized by spatio-temporal binary behaviour in which long, quiescent periods in the velocity signal are interrupted by short, active ‘events’ during which there are violent fluctuations away from the average. The second and related problem is whether solutions of the 3D Navier-Stokes equations develop finite time singularities during these events. This paper shows that Leray’s weak solutions of the three-dimensional incompressible Navier-Stokes equations can have a binary character in time. The time-axis is split into ‘good’ and ‘bad’ intervals: on the ‘good’ intervals solutions are bounded and regular, whereas singularities are still possible within the ‘bad’ intervals. An estimate for the width of the latter is very small and decreases with increasing Reynolds number. It also decreases relative to the lengths of the good intervals as the Reynolds number increases. Within these ‘bad’ intervals, lower bounds on the local energy dissipation rate and other quantities, such as ||u(·, t)||_∞ and ||∇u(·, t)||_∞, are very large, resulting in strong dynamics at sub-Kolmogorov scales. Intersections of bad intervals for n≧1 are related to the potentially singular set in time. It is also proved that the Navier-Stokes equations are conditionally regular provided, in a given ‘bad’ interval, the energy has a lower bound that is decaying exponentially in time.Final version 17 March 05. Original version November 03.     

A fretting fatigue crack detection feasibility study using shear wave non-destructive inspection

A study was conducted to assess the viability of using ultrasonic shear wave non-destructive inspection (NDI) methods to detect fatigue cracks nucleating in the vicinity of a contact regionin situ. Use of this method is hampered by the presence of electronic and acoustic noise in the laboratory environment and by the contact in the experimental configuration. A previously established fretting fatigue test fixture was selected, in which nominally flat pads are held in contact against a thin, flat specimen and gross sliding between the pad and specimen is eliminated. Experiments were performed with Ti-6Al-4V at 300 Hz andR=0.5 for average clamping stresses of 200 and 620 MPa, and applied fatigue stresses of 330 and 250 MPa. The shear wave response was monitored during each test, and the test was interrupted when changes in the waveform were thought to indicate a crack. Also, the effect of the contact load and the sensitivity of the technique under the contact conditions were assessed. For the lower clamping stress, a sizeable portion of life was spent nucleating cracks, and the propagation life was too short to allow interruption of the tests. At the higher clamping stress, cracks with surface lengths of ∼2.5 mm were detected on, 10 mm wide specimens in tests conducted using the shear wave NDI technique. While the presence of the contact produced changes in the ultrasonic waveform, additional changes occurred as the crack propagated that permitted crack detection. A simple waveform correlation was used post-test to quantify the waveform changes, and thereby validate the viability of this NDI method for use in contact regions. In the configuration used for this study, the shear wave NDI technique was insensitive to small cracks. Some refinements that could dramatically improve crack detection capability are discussed. The original color figures found in this article can be seen in the online verson of this article, or can be obtained from A. Hutson.     

Non-uniform, axisymmetric misfit strain： in thin films bonded on plate substrates/substrate systems： the relation between non-uniform film stresses and system curvatures

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to non-uniform, but axisymmetric misfit strain distributions in the thin film, we derived relations between the film stresses and the misfit strain, and between the plate system’s curvatures and the misfit strain. These relations feature a ‘‘local’’ part which involves a direct dependence of the stress or curvature components on the misfit strain at the same point, and a ‘‘non-local’’ part which reflects the effect of misfit strain of other points on the location of scrutiny. Most notably, we also derived relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary radial non-uniformities. These relations also feature a ‘‘non-local’’ dependence on curvatures making a full-field measurement a necessity. Finally, it is shown that the interfacial shear tractions between the film and the substrate are proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.     

Effects of partial crack–face contact for the bending of thin shell structures

The physical occurrence that crack surfaces are in contact at the compressive edges when a flat or a shell is subjected to a bending load has been recognized. This article presents a theoretical analysis of crack–face contact effect on the stress intensity factor of various shell structures such as spherical shell, cylindrical shell containing an axial crack, cylindrical shell containing a circumferential crack and shell with two non-zero curvatures, under a bending load. The formulation of the problem is based on the shear deformation theory, incorporating crack–face contact by introducing distributed force at the compressive edge. Material orthotropy is concerned in this analysis. Three-dimensional finite element analysis (FEA) is conduced to compare with the theoretical solution. It is found that due to curvature effect crack–face contact behavior in shells differs from that in flat plates, in that partial contact of crack surfaces may occur in shells, depending on the shell curvature and the nature of the bending load. Crack–face contact has significant influence on the stress intensity factor and it increases the membrane component but decreases the bending component.     

Residual strength of a fiber reinforced metal matrix composite with a crack

The residual strength of a cracked unidirectional fiver reinforced metal matrix composite is studied. We propose a bridging model based on the Dugdale strip yielding zones in the matrix ahead of the crack tips that accounts for ductile deformations of the matrix and fiber debonding and pull-out in the strip yielding zone. The bridging model is used to study the fracture of an anisotropic material and its residual strength is calculated numerically. The predicted results for a SiC/titanium composite agree well with the existing experimental data. It is found that a higher fiber bridging stress and a larger fiber pull-out length significantly contribute to the composite's residual strength. The composite's strength may be more notch-insensitive than the corresponding matrix material's strength depending on several factors such as fiber-matrix interface properties and the ratio of the matrix modulus to an ‘effective modulus’ of the composite.     

A damage mechanics approach to stress softening and its application to rubber

We analyze stress softening phenomena within the framework of the ‘generalized standard material’ based on the notion of a ‘normal dissipative mechanism’. We prove that the monotonicity properties of the ‘yield function’ governing such mechanism lead to local and global uniqueness of the response. Applications to oscillators with a single degree of freedom, whose anharmonic spring exhibits stress softening, are also presented.     

Flow patterns and interfacial velocities near a moving contact line

Experimental data on velocity fields and flow patterns near a moving contact line is shown to be at variance with existing hydrodynamic theories. The discrepancy points to a new hydrodynamic paradox and suggests that the hydrodynamic approach may be incomplete and further parameters or forces affecting the surfaces may have to be included. A contact line is the line of intersection of three phases: (1) a solid, (2) a liquid, and (3) a fluid (liquid or gas) phase. A moving contact line develops when the contact line moves along the solid surface. A flat plate moved up and down, inside and out of a liquid pool defines a simple, reliable experimental model to characterize dynamic contact lines. Highlighted are three important conclusions from the experimental results that should be prominent in the development of new theoretical models for this flow. First, the velocity along the streamline configuring the liquid–fluid interface is remarkably constant within a distance of a couple of millimeters from the contact line. Second, the relative velocity of the liquid–fluid interface, defined as the ratio of the velocity along the interface to the velocity of the solid surface, is independent of the solid surface velocity. Third, the relative interface velocity is a function of the dynamic contact angle.     

A Comparison of Contact Stiffness Measurements Obtained by the Digital Image Correlation and Ultrasound Techniques

The digital image correlation (DIC) and ultrasound techniques have both previously been employed to measure the contact stiffness of real engineering interfaces, but a comprehensive comparison of these techniques has not previously been carried out. Such a comparison is addressed in the present paper. The principal novelty in this work is that DIC and ultrasound are used to simultaneously measure contact stiffness in the same tests and on the same contact interface. The results show that ultrasound measures somewhat higher contact stiffness magnitudes than DIC: at an average normal contact pressure of 70 MPa, ultrasound was around three times stiffer. Given that the techniques are vastly different in their measurement approach (DIC measures micron-scale relative displacements from external side-on images of the interface, while ultrasound uses the reflection of an Ångstrom scale ultrasonic perturbation from the interior of the interface itself), this level of agreement is thought to be encouraging. The difference in results can partly be explained by consideration of inherent physical differences between the techniques which have previously received little attention. Ultrasound measurement will always give the local elastic ‘unloading stiffness’ (even at a plastically deforming contact); whereas, a load-deflection technique like DIC, will give the ‘loading stiffness’. The reason for this difference is discussed in the paper and tests carried out under increasing tangential load in the pre-sliding regime illustrate this difference experimentally. Under normal loading, the increase in real contact area obscures the effect to some extent as both DIC and ultrasound stiffnesses increase with normal load. The results suggest that rough interfaces may be satisfactorily modelled as a variable stiffness spring whose stiffness increases with contact pressure as the smooth contact case is approached.     

On the nature of the mechanical resonance associated with the electrical resonance of a poled PZT65/35 disc

In this paper we present experimental results concerning the axial mechanical displacements of a partially and fully poled ferroelectric ceramic disc of PZT65/35 composition in the neighborhood of its lowest detectable electrical resonance. The measured displacements are resolved into their flexural and thickness components which are due, respectively, to a recently discovered new electric-to-mechanical coupling phenomenon and conventional piezoelectric coupling. Flexural resonances and a ‘thickness’ resonance are detected. The frequencies of the former increases with remanent polarization, and the frequency' of the latter decreases with remanent polarization. In particular, the nature of the ‘thickness’ resonance indicates that this resonance is not one-dimensional in character as postulated in many theoretical considerations.     

Small edge crack in a semi-infinite solid subjected to thermal stratification

The mode I stress intensity factor for a small edge crack in an elastic half-space is found when the space is in contact with two stratified fluids of different temperatures, the boundary between the fluids oscillating sinusoidally over the solid surface. The variation in the stress intensity factor, which may lead to thermal fatigue crack growth, is examined as a function of time, crack depth, amplitude and temporal frequency of oscillation, surface heat transfer coefficient and material properties of the half-space. It is shown how this ‘boundary layer’ solution may be applied to problems involving finite geometries.     

Effect of Membrane Preparation Method on Performance of Polyol Supported Membrane Used for Separation of Phenol

With increasing demands on the environment and on natural resources, there is a growing need to develop practical technologies that not only can remediate waste streams but also recover valuable components from these effluents. Membranes and membrane-based processes have attained technical and commercial importance with respect to their industrial and environmental applications. In the present paper, studies on stability of supported liquid membranes (SLMs) is reported. It has been shown that the method of preparation for SLM has an influence on the stability and lifetime of the SLM. Membranes prepared with ‘dry’ outer surfaces, free from organic wetting, were found to be more stable than the conventional SLM prepared with external surfaces wetted with a film of the organic membrane liquid phase. For phenol transport the ‘dry’ surface SLM had a similar initial flux to the ‘wet’ surface SLM, and about 2 times the flux after 50 h. Over a 50 h period the ‘dry’ SLM lost about 10% of its membrane liquid, whereas the ‘wet’ SLM lost about 45%. The difference is attributed to the loss of membrane liquid by emulsion formation at one of the aqueous-organic interfaces which would be greater for the ‘wet’ SLM with a continuous liquid film over the surface of the support.     

Ultrasonic assessment of rough surface contact between solids from elastoplastic loading-unloading hysteresis cycle

An ultrasonic method to characterize the elastoplastic contact between two rough surfaces is presented. Ultrasonic experiments are performed on three different interfaces formed by aluminum surfaces with different levels of roughness. The frequency-dependent ultrasonic reflection coefficient from the interface is measured during loading and unloading cycles as a function of pressure, from which the ultrasonic interfacial contact stiffness is reconstructed by the least-squares inversion procedure. It is shown that one should distinguish between the ultrasonic (dynamic) interfacial stiffness and static interfacial stiffness for rough surfaces in elastoplastic contact (they are identical for purely elastic contact). It is shown that ultrasonic stiffness is associated with local unloading stiffness. An elastoplastic micromechanical model is used to describe the plasticity-induced hysteresis in the ultrasonically measured interfacial stiffness during loading-unloading cycles. The topographic parameters of the interface contact are reconstructed by matching the model-predicted results with the experimentally determined ultrasonic stiffness. Using these parameters the real area of contact, which is not directly measurable, is predicted during loading-unloading cycles using the model.     

Measuring Contact Mechanics Deformations Using DIC through a Transparent Medium

This paper describes the experimental methodology used to study the contact mechanics of a rigid, rough surface and a compliant, nominally flat surface using digital image correlation (DIC). The rough surface was produced by 3-D printing PMMA and the flat surface was produced with transparent PDMS (silicone rubber). The deformation of the speckled top surface (contact) of the PDMS was measured via DIC viewed through the transparent media. Four different PDMS formulations with moduli ranging from 64 to 2120 kPa were used in the experiment program to cover a wide range of modulus normalized loads. The deformation of the contact surface and depth of penetration versus normalized load were measured. The results were overlaid with previous measurements of contact area and complemented them extremely well. Additionally, it was shown that scaling laws associated with such contact mechanics problems extend many length scales.