Axi-symmetric bodies of equal material in contact under torsion or shift

Summary Two axi-symmetric bodies are pressed together, so that their axes of symmetry coincide with the contact normal and the normal force is held constant. A small torque about the contact normal or a small tangential force is applied. For bodies of equal material, the normal and tangential stress states are uncoupled, and can solved separately. The surfaces of the bodies are thought as a superposition of infinitesimal rigid flat-ended punches. Consequently, the normal stress distribution can be calculated as a summation of differential flat punch solutions. A formula results, which is identical with the solution of Green and Collins. After application of a torque an annular sliding area forms at the border of the contact area. For reasons of symmetry, the common displacement of the inner stick area must be a rigid body rotation. Similarly to the normal problem, the solution can be thought as a superposition of rigid punch rotations. The tangential solution can be derived analogically, in form of a superposition of rigid punch displacements. The present method also solves the problem of simultanous normal and torsional or tangential loading with complete adhesion. As an example, Steuermann's problem for polynomial surfaces of the formA _2nr²ⁿis solved. The solutions for constant normal forces can be used as basic functions for loading histories with varying normal and tangential forces.     

The effect of compliant coatings on coherent structure in turbulent boundary layers

The velocity profile, turbulence intensity profile, streaky structure and bursting frequency in turbulent boundary layers over a flat plate with compliant coatings were investigated by Laser Doppler Anemometry and conditional sampling techniques. This experiment led to the conclusions that in boundary layer flows on a compliant wall, as compared with that on a rigid wall, the log law region was extended further away from the wall, and that the maximum value of each turbulence intensity profile in the near wall region was reduced and the bursting frequency obviously decreased with the compliant coatings. One point worthy of notice was that the above results were very much like those of polymer drag reduction experiments. The project is supported by the National Natural Science Foundation of China.     

Multiple bifurcations in wrinkling analysis of thin films on compliant substrates

Wrinkling phenomena of stiff thin films on compliant substrates are investigated based on a non-linear finite element model. The resulting non-linear equations are then solved by the Asymptotic Numerical Method (ANM) that gives interactive access to semi-analytical equilibrium branches, which offers considerable advantage of reliability compared with classical iterative algorithms. Bifurcation points are detected through computing bifurcation indicators well adapted to the ANM. The effect of boundary conditions and material properties of the substrate on the bifurcation portrait is carefully studied. The evolution of wrinkling patterns and post-bifurcation modes including period-doubling has been observed beyond the onset of the primary sinusoidal wrinkling mode in the post-buckling range.     

Mechanics of shaft-loaded blister test for thin film suspended on compliant substrate

Based on the von Kármán plate theory, the mechanics of a shaft-loaded blister test for thin film/substrate systems is studied by considering elastic substrate deformations and residual stresses in these films. In testing, films are attached to a substrate provided with a circular hole, through which loading is applied to the film by a flat-ended shaft of circular cross-section. The effect of substrate deformation on the deflection of the loaded film is taken into account by using a line spring model. For small deflections, an analytical solution is derived, while for large deflections a numerical solution is obtained using the shooting method. The resulting load-shaft displacement relation, which is essential in blister tests, compares favorably with finite element analysis.     

On the absolute instability in a boundary-layer flow with compliant coatings

The question of absolute instabilities occuring in a boundary-layer flow with compliant coatings is reassessed. Compliant coatings of the Kramer's type are considered. Performing a local, linear absolute/convective stability analysis, a family of spring-backed elastic plates with damping is shown to be absolutely unstable for sufficiently thin plates. The absolute instability arises from the coalescence between an upstream propagating evanescent mode and the Tollmien–Schlichting wave. To reinforce the local, linear stability results the global stability behaviour of the system is investigated, integrating numerically the full nonparallel and nonlinear two-dimensional Navier–Stokes system coupled to the dynamical model. Injecting Gaussian-type, spatially localized flow disturbances as initial conditions, the spatio-temporal evolution of wave packets is computed. The absolute stability behaviour is retrieved in the global system, for a compliant panel of finite length. It is demonstrated numerically that the global stability behaviour of the wall, triggered by finite-end-effects, may be independent of the disturbance propagation in the flow.     

Buckling of a stiff thin film on a compliant substrate in large deformation

A finite-deformation theory is developed to study the mechanics of thin buckled films on compliant substrates. Perturbation analysis is performed for this highly nonlinear system to obtain the analytical solution. The results agree well with experiments and finite element analysis in wavelength and amplitude. In particular, it is found that the wavelength depends on the strain. Based on the accurate wavelength and amplitude, the membrane and peak strains in thin films, and stretchability and compressibility of the system are also obtained analytically.     

Tractive rolling contact mechanics of graded coatings

In this study, the tractive rolling contact problem between a rigid cylinder and a graded coating is investigated. The main objective of this study is to investigate the effect of the stiffness ratio, the coefficient of friction and the coating thickness on the surface contact tractions, the surface in-plane stress, the stick zone length and the creep ratio parameter that may have a bearing on the fatigue life of the component. Assuming that the shear modulus varies exponentially through the thickness of the coating, the governing integral equations associated with the rolling contact problem are constructed. Furthermore, it is supposed that the contact patch is controlled by a central stick zone accompanied by two slip zones. The conventional Goodman approximation is employed in order to decouple the governing singular integral equations. Finally, the numerical solution of the integral equations is obtained by applying the Gauss–Chebyshev integration method.     

The effect of longitudinal high-frequency in-plane vibrations on a 1-DoF friction oscillator with compliant contact

Nonlinear Dynamics - The effect of vibrational smoothing of dry friction has been studied intensively in the past. High-frequency vibrations have been shown to reduce average friction forces and...     

Micromechanical analysis of failure in thin protective coatings

The integrity of protective coatings is the subject of this investigation. Specifically, we concentrate on the case of thermo-protective coatings, so-called thermal barrier coatings, aiming at development of methodology for service life estimation of the coated components under thermal loading. At the current phase of the development, we establish relationships between various thermal loading patterns and failure driving parameters at different failure stages by developing an analytical–computational model of the process. The analysis addresses a typical failure development pattern consisting of a system of multiple surface cracks leading to and branching along or near the interface between the coating and the base material. The process is driven by thermal stresses. The developed model is applicable to thin coatings and provides insight into the processes taking place during failure development and the effect of the details of the applied thermal loading.The methods developed in the course of this investigation may be applied to the analysis of environmental effects on protective coatings and to investigations of failure development in layered composite systems under general loading conditions.     

Crack propagation in glass coatings under expanding spherical contact

The growth of transverse cracks under expanding spherical contact in a model system consisted of soda-lime glass bonded to a polycarbonate substrate is observed in situ from below or from the polished edge of the bilayer. Abrasion or chemical etching is employed on the coating surfaces to control the initial fracture. In the limit case of monoliths, the crack mouth becomes fully engulfed by the expanding contact, which results in a much steeper crack angle compared to the classical Hertzian cone case. As the coating thickness is reduced, flexure stresses are set in the coating which drive the cone crack to well away from the contact circle and initiate semi-elliptical-like radial cracks at the subsurface, right under the contact. Common to all three fracture modes is an initial unstable propagation phase following by a stable growth, with detrimental failure associated with severe damage to the top surface and/or delamination at the coating/substrate interface taking place at loads several times the fracture initiation loads.LEFM in conjunction with a large-strain FEM contact code is used to study the post-initiation fracture, with the crack path controlled by the principal stress trajectory or zero-mode II S.I.F. The analysis exposes the leading geometric and material parameters in each fracture mode, which may be useful in the design of bilayer structures for optimal mechanical performance. The well-known Auerbach law governing the initial fracture of monoliths is found to apply also to the bilayer crack systems within a certain range of the problem parameters. The numerical prediction for the crack profiles and the fracture envelopes generally collaborate well with the tests.     

The electro-mechanical response of highly compliant substrates and thin stiff films with periodic cracks

We present results that describe the mechanical response of highly compliant substrates coated with ultra-thin stiff films, with thickness and elastic moduli differences spanning four orders of magnitude. Dimensional analysis based on shear-lag models of cracked films is used to identify key parameters that control the effective elastic properties of the cracked multi-layer, crack opening displacements, and the steady-state energy release rate for channeling crack formation. Analytical forms that describe multi-layer response in terms of film properties and crack spacing are presented and corroborated with numerical models for linear elastic materials. A key result is that the energy release rate scales with 1/(1 − α), where α is one of the Dundurs’ parameters describing elastic mismatch. The results can also be used to evaluate the performance of electrostrictive actuators comprised of cracked blanket electrodes and elastomer dielectrics. In this scenario, an interesting result is that ultra-thin cracked films can continue to distribute charge, since crack openings may be small enough to allow breakdown in air at typical operating voltages.     

Fracture mechanics analysis of thin coatings under plane-strain indentation

A combined experimental/analytical work is carried out to elucidate the fracture resistance of a thin, hard coating bonded to a semi-infinite substrate due to indentation by a cylindrical surface. The bending of the coating under the softer substrate induces concentrated tensile stress regions at the lower and upper surfaces of the coating, from which cracks may ensue. The evolution of such damage in a model transparent system (glass/polycarbonate) is viewed in situ from below and from the side of the specimen. The critical load needed to initiate a crack on the lower coating surface generally increase proportionally to the coatings thickness, d. An interesting departure from this trend occurs for thin coatings, where the fracture load, although marred by a large scatter, increases somewhat with decreasing d. The fracture data for the upper coating surface are limited to relatively thick coatings due to the recurrence of premature failure from the coating edges. The behavior in this range is similar to that for the lower surface crack, albeit with an order of magnitude greater fracture resistance.A fracture mechanics analysis in conjunction with FEM is performed to elucidate the stress intensity factors responsible for crack propagation. A crack normal to the coating surface is assumed to emanate either from the lower or upper surface of the coating. A major feature of the solution is the occurrence of a bending-induced compression stress field over a region ahead of the crack tip. This effect, which become more dominant as the ratio between the contact length and the coating thickness is increased, tends to delay the onset of crack propagation, especially for the lower surface crack. Consequently, in applications associated with large indenters, thin and/or tough coatings and stiff substrates, cracking from the upper coating surface may precede that from the lower surface. An interesting feature of this crack shielding mechanism is that when the coating surface contains a distribution of flaws rather than a single crack, small flaws in this population may be more detrimental than large ones. Incorporation of these aspects into the analysis leads to a good correlation with the test results. In the special case of line loading, which constitutes a lower bound for the critical loads, a closed-form, approximate solution for the stress intensity factors or the critical loads are obtained.Plane-strain indentation, although less common than spherical indentation, allows for characterizing the fracture resistance of opaque films through observation from the specimen edge. This approach is not easily implemented to thin films (i.e., less than about a hundred microns), however.     

On a multiplication technique applied to very thin photoelastic coatings

Procedures of multiplication of relative and absolute retardations, leading to the difference and the sum of principal strains, are studied in the case of very thin photoelastic coatings. Firstly, for multiplication of the birefringence effect simultaneously with an interference measurement, different wavelengths of laser light and a special mirror system capable of eliminating the influence of model displacement are used; secondly, for multiplication of the interference effect, properties of light frequency doubling in nonlinear optics are investigated.     

Semi-analytical analysis of thermally induced damage in thin ceramic coatings

An efficient procedure to analyze damage evolution in brittle coatings under influence of thermal loads is suggested. The approach is based on a general computational scheme to determine damage evolution parameters, which incorporates an analytical solution of the appropriate interim boundary-value thermoelasticity problem. For thin inhomogeneous coatings, the simplification in the analysis is achieved by application of the mathematical model with generalized boundary conditions of thermomechanical conjugation of the substrate with environment via the coating. Efficiency of the suggested approach is illustrated by an example of damage evolution in the alumina coating on the titanium-alloy and tungsten substrates under uniform heating.     

Fretting contact of two elastic solids with graded coatings under torsion

In this paper, the fretting contact problem for two elastic solids with graded coatings is investigated. We assume a conventional axisymmetric Hertzian contact takes place between two elastic solids under the action of the normal pressure. The application of the torque produces an annulus of slip. It is assumed that the surface shear traction within the contact area is limited by Coulomb’s friction law and the torsion angel was produced within the central adhesion zone as a rigid body. The linear multi-layer model is used to model the functionally graded coating with arbitrarily varying shear modulus. This model divides the coating into a series of sub-layers with the elastic modulus varying linearly in each sub-layer and continuous on the sub-interfaces. By using the transfer matrix method and Hankel integral transform technique, this problem is formulated as the solution of the Cauchy singular integral equations. The contact tractions are calculated by solving the equations numerically. The results show that the appropriate gradual variation of the shear modulus can significantly alter the contact tractions. Therefore, graded coatings may have potential applications in improving the resistance to fretting contact damage at the contact surfaces.     

Generalized impedance boundary conditions for thin dielectric coatings with variable thickness

We derive the so-called Generalized Impedance Boundary Conditions (GIBC) that model thin dielectric coatings with variable width located on a perfect electric or magnetic conducting surface. We treat the 2-D electromagnetic problem for both TM and TE polarizations. The expressions of the GIBCs are obtained up to the third order (with respect to the coating width). The order of convergence is numerically validated through various numerical examples using a finite element type method. Particular attention is given to the cases where the inner boundary has corner singularities.     

Boundary layer receptivity measurements on compliant surfaces

This paper describes receptivity measurements in a pre-transitional boundary layer flowing over either a rigid or a compliant surface. Fluctuating velocities and frequency spectra were determined on one rigid and nine compliant surfaces. The results showed that the near wall receptivity grows linearly with Re_θ. An empirical correlation of the gain frequency spectrum for a rigid wall was also established. For the compliant surfaces, the near wall gain is increased markedly near the leading edge of the plate due to the amplification of high and mid-frequencies. These frequencies are dissipated though as the flow progresses over the compliant surface such that the receptivity is lower on all the compliant surfaces than on the rigid surface at the trailing edge. An empirical correlation for the ratio of the gains on compliant and rigid surfaces in terms of the compliant surface coefficient ζ²/Eρ_CSL² and Re_θ was established. This correlation indicates that compliant surfaces can suppress receptivity by up to 25% for a Re_θ = 400.