Rigid Half-space Research Articles

Axisymmetric contact problems for an elastic layer resting on a rigid base with a Winkler type excavitation

Axisymmetric contact problems for an elastic layer pressed by a rigid sphere or by a rigid flat cylinder are considered. It is assumed that the layer rests on the rigid half-space with a near-boundary cylindrical excavitation which is filled with a deformable material. This material is modelled by a Winkler medium. The Hankel integral transforms are applied and the problems are reduced to the system of integral equations. The numerical analysis is performed to investigate the contact parameters and the deflexion in the excavitation zone. Results are presented in diagrams.

Interface crack extension at any constant speed in orthotropic or transversely isotropic bimaterials––II. Two important examples

Abstract Part I [Int. J. Solids Struct., 39, 1165–1182] gives exact general solutions for steady dynamic extension by a semi-infinite crack along the interface of dissimilar orthotropic/transversely isotropic half-spaces under in-plane loading. Crack speed is any constant value, and results are valid for all six possible relations between the four body wave speeds. These results are applied here to sub-critical interface crack extension, and to debonding from a rigid half-space at any constant speed. The former case demonstrates well-known phenomena––the minimum Rayleigh speed is critical, and complex conjugate eigenvalues cause the crack edge oscillatory/square-root singular behavior that implies interpenetration. Calculations for representative materials show that eigenvalue and stress intensity factor amplitude variation with crack speed is small except near the critical value. Results for the latter case complement those of other recent studies: Singular behavior vanishes for super-critical/sub-sonic crack speeds, although oscillations remain. For trans-sonic crack speeds, eigenvalues are real, singular behavior is no longer square-root, and interpenetration can still occur below a critical speed. Lines of displacement gradient discontinuity radiate from the crack edge, but can vanish at a critical speed. Calculations show that eigenvalue variation with crack speed is more pronounced than in the first case.

The problem of a viscoelastic cylinder rolling on a rigid half-space

The problem of a viscoelastic cylinder rolling on a rigid base, propelled by a line force acting at its centre, is solved in the noninertial approximation. The method used is based on a decomposition of hereditary integrals developed by the authors in previous work, and on the viscoelastic Kolosov-Muskhelishvili equations which are used to generate a Hilbert problem. In this formulation, the problem reduces to a nonsingular integral equation in space and time, which simplifies under steady-state conditions and for exponential decay materials, to algebraic form. There are also two subsidiary conditions. In the case of a standard linear model, explicit analytic results and numerical examples are given for the pressure function, for surface displacements, and also for hysteretic friction.

Linear elastic contact of the Weierstrass profile

A contact problem is considered in which an elastic half–plane is pressed against a rigid fractally rough surface, whose profile is defined by a Weierstrass series. It is shown that no applied mean pressure is sufficiently large to ensure full contact and indeed there are not even any contact areas of finite dimension — the contact area consists of a set of fractal character for all values of the geometric and loading parameters. A solution for the partial contact of a sinusoidal surface is used to develop a relation between the contact pressure distribution at scale n − 1 and that at scale n . Recursive numerical integration of this relation yields the contact area as a function of scale. An analytical solution to the same problem appropriate at large n is constructed following a technique due to Archard. This is found to give a very good approximation to the numerical results even at small n , except for cases where the dimensionless applied load is large. The contact area is found to decrease continuously with n , tending to a power–law behaviour at large n which corresponds to a limiting fractal dimension of (2 − D ), where D is the fractal dimension of the surface profile. However, it is not a ‘simple’ fractal, in the sense that it deviates from the power–law form at low n , at which there is also a dependence on the applied load. Contact segment lengths become smaller at small scales, but an appropriately normalized size distribution tends to a limiting function at large n . † The authors dedicate this paper to the memory of Dr J. F. Archard, 1918–1989.

Exponential neutral stability of a floating ice layer

Linear stability of two-dimensional monochromatic waves, i.e., normal modes, in a homogeneous elastic ice layer of finite thickness and infinite horizontal extension floating on the surface of a water layer of finite depth is treated analytically. The water is assumed to be compressible but the viscous effects are neglected in the model. The treatment is an extension of the two-step analysis in Brevdo [4] for a homogeneous waveguide overlaying a rigid half space. First, we apply an energy-type method and show that, for real wavenumbers \(k, \omega^2\) is real, where \(\omega\) is a frequency. Further, to exclude purely imaginary frequencies for real k, we use the dispersion relation function of the problem \(D(k, \omega )\) and show that for \(\omega = is, s \in {\Bbb R}^+ ,\) the equation D(k, is) = 0 does not have real roots in k. Hence, due to the symmetry, all normal modes in this model are neutrally stable. This result on the one hand provides a theoretical support for the physical relevance of the model and on the other hand points to a possibility of resonant algebraically growing responses to localized harmonic in time perturbations. It is also shown that an unstable vertically stratified ice layer is always absolutely unstable. Based on this result, a conjecture is made concerning a possible mechanism of spontaneous ice breaking as a consequence of emergence of absolute instability, which is caused by a weather induced appearance of an unstable ice stratification.

Stresses in a Thin Piezoelectric Element Bonded to a Half-Space

In this paper, a thin piezoceramic element is considered which is bonded to an elastic or a rigid half-space. Such a model may be an approximation of the interaction between piezoceramic elements and elastic structures like beams and plates. For an elastic half-space, the determination of the shear stress in the bonding layer leads to a singular integral equation. A half-space which is very stiff may be modeled as a rigid substrate. For this case, displacement functions are introduced. Hamilton’s principle for electromechanical systems allows the use of Lagrange multipliers to incorporate the condition of a stress free upper surface of the piezoceramic element. The stresses in the bonding layer and in the piezoceramic element are estimated by this method and compared with Finite Element results. Though the singularity near the ends of the piezoceramic element cannot be modeled by both methods, stress concentrations can clearly be seen for the shear stress as well as for the normal stress. As infinite stresses due to the singularity do not occur in reality, the results allow an estimation of the bonding stresses except in the near vicinity of the edges. The knowledge of these stresses is important to prevent failure due to delamination.

Adhesion of elastic spheres

Bradley (1932) showed that if two rigid spheres of radii R 1 and R 2 are placed in contact, they will adhere with a force 2πΔ R γ, where R is the equivalent radius R 1 R 1 /( R 1 + R 2 ) and Δγ is the surface energy or ‘work of adhesion’ (equal to γ1+γ2-γ12). Subsequently Johnson et al. (1971) (JKR theory) showed by a Griffith energy argument (assuming that contact over a circle of radius a introduces a surface energy -π a 2 Δγ) how the Hertz equations for the contact of elastic spheres are modifed by surface energy, and showed that the force needed to separate the spheres is equal to (3/2)πΔ R γ, which is independent of the elastic modulus and so appears to be universally applicable and therefore to conflict with Bradley9s answer. The discrepancy was explained by Tabor (1977), who identified a parameter 3 Δγ 2 / 3 / E * 2 / 3 \e governing the transition from the Bradley pull-off force 2π R Δ|γ to the JKR value (3/2)π R Δγ. Subsequently Muller et al. (1980) performed a complete numerical solution in terms of surface forces rather than surface energy, (combining the Lennard–Jones law of force between surfaces with the elastic equations for a half-space), and confirmed that Tabor9s parameter does indeed govern the transition. The numerical solution is repeated more accurately and in greater detail, confirming the results, but showing also that the load–;approach curves become S-shaped for values of μ greater than one, leading to jumps into and out of contact. The JKR equations describe the behaviour well for values of μ of 3 or more, but for low values of μ the simple Bradley equation better describes the behaviour under negative loads.

Contact Pressures as an Elastic Roller Crosses a Scratch

The Westergaard method of plane elastic analysis is used to obtain an exact solution to the problem of an elastic roller crossing a gap, intended to represent a scratch, on a rigid half-space.

GROUND VIBRATION IN THE VICINITY OF A RECTANGULAR LOAD ACTING ON A VISCOELASTIC LAYER OVER A RIGID FOUNDATION

The transmission of vibrations in the vicinity of a rectangular harmonic vertical load, acting on a viscoelastic layer overlaying a rigid half-space, is investigated theoretically, using a semi-analytic approach. The solution involves a double Fourier transform with respect to two of the space variables of Navier's elastodynamic equations. The inverse double Fourier transform is achieved with the FFT algorithm. Results presented include transformed displacements in the wavenumber domain, actual displacements in the near-field of the load, and the direct receptance at the load.

Analytical approaches to the determination of pressure distribution under a plantar prominence

INTRODUCTION:: The purpose of the current research was to develop an analytical model for the interface between the metatarsal head, surrounding soft tissue, and the shoe. Results of 'peak plantar pressures' computed from this model were compared to previous results obtained from clinical data and from a finite element (FE) model. METHODS:: An analytical model for determining the pressure distribution at the interface between the foot and shoe insole was developed based on the solution[Figure: see text] for a rigid cylinder and a single elastic layer mounted on a rigid half-space originally developed by Meijers (see Figure 1). The rigid cylinder was used to represent the metatarsal head region (including bone and soft tissue). Material properties and loading conditions were chosen to reflect those used by Lemmon et al. for 'Normal' and 'Reduced' soft tissue thickness cases. RESULTS:: Peak pressures computed from the analytical model are plotted against insole thickness in Figure 1, alongside experimental and FE model data from Lemmon et al. Correlations between analytical, FE, and experimental results for both the Normal and Reduced soft tissue cases were all highly significant (r(2) = 0.86, p < 0.01). DISCUSSION:: The analytical model produced unrealistically high peak pressures compared to the FE and experimental results, with differences of approximately one order of magnitude. This was not surprising, given that the entire metatarsal head region (soft tissue included) was modelled as a rigid cylinder. The soft tissue on the bottom of the foot, and the much greater surface area over which the loads are carried in the real case, contribute greatly to reducing these peak pressure values. However, the downward trend in peak pressure with increased insole thickness was retained. CONCLUSION:: It is anticipated that the use of more accurate and detailed models will greatly improve the initial results.

Effect of the rough surface on the transient frictional temperature and thermal stresses near a single contact area

The contact between a rough elastic sphere and a smooth rigid half-space is considered. It is assumed that the sphere slides with time-dependent speed. Sliding is accompanied by heat generation over the contact interface in the form of a heat flux with intensity proportional to the contact pressure distribution. The transient temperature behaviour and the thermal stresses in the vicinity of the heating region are evaluated by Hankel-Laplace integral transforms. The effects of the variation in roughness parameter, duration of the heating and speed are analysed.

Scattering of elastic waves by a rigid cylindrical inclusion partially debonded from its surrounding matrix—I. SH case

In Part I of this two-part paper, the scattering of SH waves by a rigid cylindrical inclusion partially debonded from its surrounding matrix is investigated by using the wave function expansion method and singular integral equation technique. The debonding regions are modeled as multiple arc-shaped interface cracks with non-contacting faces. Expressing the scattered fields as the wave function expansions with unknown coefficients and considering the mixed boundary conditions, we reduce the problem to a set of simultaneous dual series equations. Then dislocation density functions are introduced as unknowns to transform these dual series equations to a set of singular integral equations of the first type which can be easily solved numerically by using the quadrature method of Erdogan and Gupta [Int. J. Solids Structures7, 1089–1107 (1972)]. The solution is valid for arbitrary values of KT0r0 (where KT0 is the wave number and r0 the inclusion radius) and arbitrary numbers and sizes of the debonds. Explicit solutions are obtained in two limiting situations: (i) the long wavelength limit (KT0r0 ≪ 1). In this case, the solution reduces to the quasistatic solution; (ii) the small debond limit with KT0r0 = O(1). This means the wavelength greatly exceeds the debond size and the solution is identical to that of a flat interface crack between a rigid half space and an elastic one subjected to static loading at infinity. If the debond is small and KT0r0 ≪ 1, the solution will give the results of a flat interface crack subjected to an incident SH wave. Finally, the numerical results of the dynamic stress intensity factors, the rigid body translations of the inclusion and the scattering cross-sections are presented for an inclusion with one or two debonds. The phenomenon of low frequency resonance discovered by Yang and Norris [J. Mech. Phys. Solids39, 273–294 (1991)] for an elastic inclusion with one debond is shown and its dependence upon the various parameters is discussed. The solution of this problem is relevant to ultrasonic nondestructive detection of debonding and is expected to have applications to the question of how dynamic loading can lead to growth of debonds [Norris and Yang, Mech. Mater.11, 163–175 (1991)].

On elastic joints containing interfacial cracks

We study quasistatic deformations of a massive elastic body bonded to an inifinite elastic layer of finite thickness whose other surface is glued to a half space. The adhesive is modeled as a nonlinear elastic material. It is assumed that interfacial cracks are present at the body/adhesive interface and at the interface between the elastic layer and adhesive material closer to the half space. Variational formulations of the problems when the material of the half space is modeled as isotropic linear elastic or as rigid are given and are shown to be equivalent to the corresponding boundary-value problems. Analytical solutions of two example problems are given; one involves a long rigid cylinder bonded to a rigid half space and the other a thin infinite linear plate bonded to a rigid half space. The results depicted graphically include the dependence of the crack closure length on the applied loads and the dependence of the gradient of the potential energy on the crack length for the first problem, and distributions of the normal stress and normal displacement on the crack surface for the second problem.

A NOTE ON HYPERBOLIC HEAT CONDUCTION IN A RIGID SOLID

Two hyperbolic heat conduction theories which predict a finite speed of propagation of thermal effects are considered. Numerical results obtained from these theories are presented for a one spatial dimension boundary-initial value problem, which involves transient heat conduction in a rigid half space. These results are compared with results from the classical theory, which predicts an infinite speed of propagation of thermal effects. It is shown that, for the problem considered, results from the three theories are very similar at times which are large compared to the relaxation time of the hyperbolic theories.

Longitudinal Impact of a Shaped Projectile on a Hopkinson Bar

A computational method is developed to calculate numerically the normal force generated by the impact of a nonuniform linearly elastic projectile on a Hopkinson bar. One-dimensional relations are used. In order to assess the validity of the method, it is compared with an analytical solution in the case of the impact of a truncated cone on a rigid half-space. Then some practical applications are performed to predict the force generated by the impact of different truncated cones on a Hopkinson bar. We show that a perfect ramp load is only generated by the impact of a truncated cone on a rigid half-space.

Analytical and numerical solutions for wave propagation in water-saturated porous layered half-space

A fluid-saturated one-layer continuum underlain by a rigid half-space is considered. An exact solution is developed in frequency domain for analyzing disturbance induced by a strip footing located at the surface with vertical harmonic excitation. Since the analytical solution can be used only for very simple conditions, a finite element model has been developed also and compared with the exact solution. It is shown that finite element results are in close agreement with the results which have been obtained by a transformation technique. The proposed finite element scheme can take into account the complex geometry and inhomogeneity for practical problems. Besides this, the analytical results exhibit the overall characteristic of wave propagation in porous media and will provide a representative test problem which can be used for a quantitative evaluation of the accuracy of various numerical solution methods.

Unsteady shear vibrations of a die for a nonuniform base

A dynamic shear contact problem is examined for uniform and nonuniform bases. A flexible layer on a rigid half-space is chosen as the nonuniform base. The problem is solved on the basis of applied equations for strain. By using this approach, it is possible to reduce the problem to a system of integrodifferential equations. A numerical algorithm is proposed for its solution. Attention is focused on the effect of the flexible layer on the displacement and reaction of the die. The numerical results are analyzed.

Analysis of recorded earthquake response and identification of a multi-story structure accounting for foundation interaction effects

Analysis of recorded earthquake response and identification, of the 14-story, reinforced concrete Hollywood Storage building shaken during the 1987 Whittier Narrows earthquake, are presented. Since only the translational components of motion of the foundation were recorded and considered as input motions in the identification analysis, the inferred parameters reflect not only the characteristics of the superstructure but also the characteristics of the foundation and of the underlying soil, and as such are referred to as apparent system parameters. Further processing of the apparent system parameters is performed in order to extract the modal parameters of the superstructure. The pronounced interaction of the superstructure with the soil in the longitudinal direction reduced considerably (25%) the base shear which the building would have experienced if the superstructure were supported in a rigid half-space. On the other hand, the weak soil-structure interaction in the transverse direction reduced the base shear only by a very small amount ( ∼1.2%).

Indentation stiffness of young canine knee articular cartilage—Influence of strenuous joint loading

The indentation stiffness of knee articular cartilage subjected to strenuous physical training (SPT: treadmill running 20 km day −1 for 15 weeks, n = 6) of young Beagles was tested and compared to that obtained from age-matched (55 weeks, n = 9) controls. The mathematical solution for the shear modulus, as determined from indentation of an elastic layer bonded to a rigid half space, was extended to small Poisson's ratios and applied to the analysis of cartilage response after a step stress (0.39 MPa) application. In these measurements with an impervious, plane-ended indenter, the equilibrium deformation was systematically greater than values predicted from the instant response by the linear biphasic theory. Therefore, the accurate determination of Poisson's ratio from the creep curves was not possible. The mean shear modulus (calculated by using the deformation at 900 s after load application and assuming a constant Poisson's ratio of 0.40 for the matrix) of canine knee articular cartilage was 0.37 MPa. While the cartilage thickness was not affected by SPT, the cartilage of the lateral tibial plateau was stiffer (13.3%, p<0.05) than that in controls. However, in the femoral condyles, the stiffness was at the control level or even below. Our results on cartilage structure and properties suggest that SPT, in contrast to our previous findings with moderate training, does not necessarily improve the biological properties of articular cartilage in young animals.

Response of a Compliant Slab to Viscous Incompressible Fluid Flow

The interaction of a compliant slab and the flow of a viscous fluid is explored theoretically. Both laminar and turbulent boundary-layer flow cases are considered. The slab is treated as an infinite, elastic or viscoelastic solid of finite thickness, bonded to a rigid half-space. The two media are coupled through stresses and velocities, in both tangential and normal directions, on the deformed surface of the slab. The proposed mathematical model is used to predict the appearance of unstable surface waves of the coating. According to the model, instabilities of a highly viscoelastic slab originate in the form of waves with phase speeds at about 2.3 percent of the flow velocity while the wave speed for a slab with zero to moderate viscosity is in the range of 50–60 percent of the flow velocity, which confirms previous experimental results. The predicted onset velocities of slow propagating waves for turbulent flow over plastisol gel are within 10 percent of those observed experimentally.