Closed-form solutions of the frictional sliding contact problem for a magneto-electro-elastic half-plane indented by a rigid conducting punch

This paper focuses on the study of a frictional sliding contact problem between a homogeneous magneto-electro-elastic material (MEEM) and a perfectly conducting rigid flat punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the main unknowns are the normal contact stress, the electric displacement and the magnetic induction. An analytical closed-form solution is obtained for the normal contact stress, electric displacement and magnetic induction distributions. The main objective of this paper is to study the effect of the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat stamp profile.     

Frictionless contact of a functionally graded magneto-electro-elastic layered half-plane under a conducting punch

This paper investigates the two-dimensional frictionless contact problem of a functionally graded magneto-electro-elastic materials (FGMEEMs) layered half-plane under a rigid flat or a cylindrical punch. It is assumed that the punch is a perfect electro-magnetic conductor with a constant electric potential and a constant magnetic potential. The magneto-electro-elastic (MEE) properties of the FGMEEM layer vary exponentially along the thickness direction. Using the Fourier transform technique, the contact problem can be reduced to Cauchy singular integral equations, which are then solved numerically to determine the normal contact stress, electric displacement and magnetic induction on the contact surface. Numerical results show that the gradient index, punch geometry and magneto-electro-mechanical loads have a significant effect on the contact behavior of FGMEEMs.     

Inclined circular flat punch on a transversely isotropic piezoelectric half-space

Summary Utilizing the general solution of transversely isotropic piezoelectricity, the paper analyzes the problem of an inclined rigid circular flat punch indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to take into account the effect of the electric field in piezoelectric materials. Assuming that the punch is maintained at a constant electric potential, exact expressions for the elastoelectric field are derived in terms of elementary functions. It is noted that the solution corresponding to a flat circular punch centrally loaded by a concentrated force can be obtained as a special case. Received 15 December 1998; accepted for publication 9 March 1999     

Torsion of rough elastic half-space by rigid punch

Torsion of an elastic half-space by a rigid punch is investigated. The boundary of the half-space is assumed to be rough. Two geometries of the punch-parabolic and flat end are considered. It is shown that the contact area consists of stick and slip zones. This fact, which is well-known in the classical torsional contact of the elastic half-space with the smooth surface and the parabolic punch, also holds true for the flat-ended punch if the boundary roughness is involved. The partial slip problems are reduced to the integral equations, which are solved numerically. The presented results show the effects of boundary roughness on the shear stresses, size of the stick area and the relation between the twisting moment and the angle of twist.     

On the half-infinite crack problem in thermo-electro-elasticity

This paper concerns itself with the fundamental solutions of the thermo-electro-elastic field in an infinite medium, weakened by a half-infinite plane crack with two identical point thermal loads applied on the crack surfaces. The corresponding mixed boundary value problem is solved by virtue of the potential theory method conjugated with the general solutions. The boundary governing equations are solved by using the results available in literature. Exact and complete three-dimensional (3D) fundamental solutions are presented in terms of elementary functions. The singularity at the crack tip is analyzed explicitly. The obtained solutions will be of high significance to the related BEM analysis.     

3D coupled field in a transversely isotropic magneto-electro-elastic half space punched by an elliptic indenter

The present paper is concerned with three-dimensional (3D) coupled field in a transversely isotropic magneto-electro-elastic half space punched by a rigid flat-ended elliptic indenter. Closed form solutions and corresponding numerical results are presented in this work, in a systematic manner. The material in question is transversely isotropic with the axis of symmetry normal to the surface of the half space. The indenter is assumed to be either electrically and magnetically conducting or insulating. Corresponding boundary integral equations (BIEs), to indenter with different magneto-electric properties, are solved by virtue of the method of generalized potential theory. For all four physical cases, corresponding coupled magneto-electro-elastic fields in the half space are obtained. The present analytical solutions indicate that the indentation forces and stiffness may be written as intrinsic combinations of a physical factor and a geometrical factor. The present study is an extension of the previous work on circular punch, and may find applications in guiding future indentation experiments.     

The contact problem of a rubber half-space dented by a rigid cone apex

Summary  The smooth contact of a rubber half-space dented by a rigid cone apex is analyzed based on the large deformation theory. The problem is treated as an axially symmetric case, and the material is assumed to be incompressible. The asymptotic equations for the domain near the apex are derived. They are solved analytically for the shrinking domain, while a numerical solution is found for the expanding domain in the vicinity of the stress singularity. The purpose of this paper is not only to solve a typical problem but also to provide an analytical method to solve a large-strain problem with a singular point. Received 10 July 2001; accepted for publication 24 January 2002     

Mechanics of adhesive contact on a power-law graded elastic half-space

We consider adhesive contact between a rigid sphere of radius R and a graded elastic half-space with Young's modulus varying with depth according to a power law E=E₀(z/c₀)^k (0<k<1) while Poisson's ratio ν remaining a constant. Closed-form analytical solutions are established for the critical force, the critical radius of contact area and the critical interfacial stress at pull-off. We highlight that the pull-off force has a simple solution of P_cr=−(k+3)πRΔγ/2 where Δγ is the work of adhesion and make further discussions with respect to three interesting limits: the classical JKR solution when k=0, the Gibson solid when k→1 and ν=0.5, and the strength limit in which the interfacial stress reaches the theoretical strength of adhesion at pull-off.     

Numerical simulation of contact problems in vane machinery by a parametric quadratic programming method

Interference fits are widely used for connecting impeller and shaft assembly that are forced together slowly by pressing. The interference fit design ensures stable balance behavior and allows for positive contact between the impeller and shaft assembly throughout the range of operating speeds. In addition to maintaining radial contact, sufficient net radial interface pressure must remain in order to transmit torque when the rotational speed is very high. Therefore, the interference fit between the impeller and the shaft assembly is one of the most important factors influencing the performance of the turbo unit in the design of turbocharger compressor. A suitable fit tolerance needs to be considered in the structural design. A locomotive-type turbocharger compressor with 24 blades under combined centrifugal and interference fit loading is used for the analysis. The finite-element (FE) parametric quadratic programming (PQP) method developed based on the parametric variational principle (PVP) is used for the analysis of the stress distribution in the three-dimensional (3D) contact problem of impeller. The advantages of the parametric programming method compared with conventional approaches are that the penalty factors can be canceled and that solutions can be obtained directly without tedious iterative procedures such as the general incremental iterative method. To save time in the computation, a~multi-substructure technique is adopted for structural modeling. This not only simplifies the calculation, but also provides a convenient service for process computer-aided design (CAD) by means of FE simulation. The effects of the fit tolerance, coefficient of friction and rotational speed (centrifugal force), wall thickness of the shaft sleeve and the contact stress on the interference-fitting surfaces are studied in detail in the numerical computation. It is found that a nonuniform initial amount of interference in the structural design avoids the relative displacement generated and ensures uniformity of the contact stress. To assure quality of press-fitting, the amount of interference between the shaft sleeve and shaft should be strictly controlled to avoid the rapid increase of the contact stress. The numerical results demonstrate the high accuracy and good convergence of the algorithm presented here, which provides an effective approach that achieves more-reliable interference-fitted connections and more-precise assembly accuracy with lower manufacturing cost in the structural design.     

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.     

Indentation of a transversely isotropic piezoelectric half-space by a rigid sphere

This paper exactly analyzes the problem of a rigid sphere indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to include the piezoelectric effect. By using the previous results of elasticity, an exact solution is derived. It is found that all the elastoelectric variables can be expressed in terms of elementary functions. The work was partly supported by the National Natural Science Foundation of China (No. 19872060).     

On the relationship between the solutions of static contact problems of elasticity and electroelasticity for a half-space

The paper establishes the relationship between the static contact problems of elasticity and electroelasticity (in the absence of friction) for a transversely isotropic half-space whose surface is the isotropy plane. This makes it possible to avoid solving the electroelastic problem by finding all the characteristics of electroelastic contact from known cases of purely elastic interaction. Moreover, the electroelastic state of the half-space can be fully described using a known harmonic function, which is a solution of the purely elastic problem. The approach is exemplified by solving contact problems of electroelasticity for flat, elliptic, two circular, conical, and paraboloidal (circular and elliptic in plan) punches __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 69–84, November 2006.     

A general solution of the axisymmetric contact problem for biphasic cartilage layers

A unilateral axisymmetric contact problem for articular cartilage layers is considered. The articular cartilages bonded to subchondral bones are modeled as biphasic materials consisting of a solid phase and a fluid phase. It is assumed that the subchondral bones are rigid and shaped like bodies of revolution with arbitrary convex profiles. The obtained closed-form analytical solution is valid over time periods compared with the typical diffusion time and can be used for increasing loading.     

Three-dimensional analytical solution for a rotating disc of functionally graded materials with transverse isotropy

Based on the basic equations for axisymmetric problems of transversely isotropic elastic materials, the displacement components are expressed in terms of polynomials of the radial coordinate with the five involved coefficients, named as displacement functions in this paper, being undetermined functions of the axial (thickness) coordinate. Five equations governing the displacement functions are then derived. It is shown that the displacement functions can be found through progressive integration by incorporating the boundary conditions. Thus a three-dimensional analytical solution is obtained for a transversely isotropic functionally graded disc rotating at a constant angular velocity.The solution can be degenerated into that for an isotropic functionally graded rotating disc. A prominent feature of this solution is that the material properties can be arbitrary functions of the axial coordinate. Thus, the solution for a homogeneous transversely isotropic rotating disc is just a special case that can be easily derived. An example is finally considered for a special functionally graded material, and numerical results shows that the material inhomogeneity has a remarkable effect on the elastic field.     

Exact solution of external tangential contact problem for a transversely isotropic elastic half-space

Summary  The following mixed boundary-value problem for a transversely isotropic elastic half-space is considered. Arbitrary tangential displacements are prescribed at the exterior of a circle, while the interior of the circle is free of tangential stress, and the normal stress vanishes all over the boundary. The governing integral equation is solved exactly, in closed form, and in terms of elementary functions. The method of continuation of solutions previously published by the author has been used here. Several examples are considered. No similar results has been reported before, even in the case of an isotropic body. Received 8 May 2000; accepted for publication 20 July 2000     

Applicability of the Hertz contact theory to rail-wheel contact problems

Summary Rail-wheel contact problems have been analyzed by applying three-dimensional finite element models. Based on these models, the applicability of the Hertz contact theory (HCT) to rail-wheel contact problems is verified in the present paper. Beside a standard rail, also a crane rail and a switching component are considered in the verification. In the case of a contact between the standard rail UIC60 and the standard wheel UICORE, different transverse contact positions are analyzed. Numerically calculated distributions of the contact pressure for different types of rails with respect to different initial contact positions agree with the results from the HCT only if either the contact zone does not spread into a region of changed surface curvature or if plastification does not occur. Finally, the convective part of the dissipation power due to friction is calculated, which cannot be provided directly by the HCT. Received 5 August 1998; accepted for publication 6 May 1999     

Further improvement on fundamental solutions of plane problems for orthotropic materials

On the basis of the ecisting fundamental solutions of displacements, further improvement is made, and then the general fundamental solutions of both plane elastic and plane plastic problems for orthotropic materials are obtained. Two parameters based on material constants α₁ = α₁ are used to derive the relevant expressions in a real variable form. Additionally an analytical method of solving the singular integral for the internal stresses is introduced, and the corresponding results are given. If α₁ = α₁ = 1, all the expressions obtained for orthotropy can be reduced to the corresponding ones for isotropy. Because all these expressions and results can be directly used for both isotropic problems and orthotropic problems, it is convenient to use them in engineering with the boundary element method (BEM).     

Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch

This paper studies the contact vibration problem of an elastic half-space coated with functionally graded materials (FGMs) subject to a rigid spherical punch. A static force superimposing a dynamic time-harmonic force acts on the rigid spherical punch. Firstly, we give the static contact problem of FGMs by a least-square fitting approach. Next, the dynamic contact pressure is solved by employing the perturbation method. Lastly, the dynamic contact stiffness with different dynamic contact displacement conditions is derived for the FGM coated half-space. The effects of the gradient index, coating thickness, internal friction, and punch radius on the dynamic contact stiffness factor are discussed in detail.     

A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method

A unified treatment of axisymmetric adhesive contact problems is provided using the harmonic potential function method for axisymmetric elasticity problems advanced by Green, Keer, Barber and others. The harmonic function adopted in the current analysis is the one that was introduced by Jin et al. (2008) to solve an external crack problem. It is demonstrated that the harmonic potential function method offers a simpler and more consistent way to treat non-adhesive and adhesive contact problems. By using this method and the principle of superposition, a general solution is derived for the adhesive contact problem involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile. This solution provides analytical expressions for all non-zero displacement and stress components on the contact surface, unlike existing ones. In addition, the newly derived solution is able to link existing solutions/models for axisymmetric non-adhesive and adhesive contact problems and to reveal the connections and differences among these solutions/models individually obtained using different methods at various times. Specifically, it is shown that Sneddon’s solution for the axisymmetric punch problem, Boussinesq’s solution for the flat-ended cylindrical punch problem, the Hertz solution for the spherical punch problem, the JKR model, the DMT model, the M-D model, and the M-D-n model can all be explicitly recovered by the current general solution.     

Scattering of SHPlease define abbreviation SH waves by arc-shaped interface cracks between a cylindrical magneto-electro-elastic inclusion and matrix: near fields

Summary In this paper, the scattering of SH waves by a magneto-electro-elastic cylindrical inclusion partially debonded from its surrounding magneto-electro-elastic material is investigated by using the wavefunction expansion method and a singular integral equation technique. The debonding regions are modeled as multiple arc-shaped interface cracks with non-contacting faces. The magneto-electric impermeable boundary conditions are adopted. By expressing the scattered fields as wavefunction expansions with unknown coefficients, the mixed boundary-value problem is firstly reduced to a set of simultaneous dual-series equations. Then, dislocation density functions are introduced as unknowns to transform these dual-series equations to Cauchy singular integral equations of the first type,which can be numerically solved easily. The solution is valid for arbitrary number and size of the arc-shaped interface cracks. Finally, numerical results of the dynamic stress intensity factors are presented for the cases of one debond. The effects of incident direction, crack configuration and various material parameters on the dynamic stress intensity factors are discussed. The solution of this problem is expected to have applications in the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.The work was supported by the National Natural Science Fund of China (Project No. 19772029) and the Research Fund for Doctors of Hebei Province, China (Project No. B2001213).