L. M. Keer

A mathematical analysis for indentation tests of articular cartilage

Abstract A mathematical model is developed for indentation tests of articular cartilage. The cartilage, normally bonded to the subchondral bone, is modeled as an infinite elastic layer bonded to a rigid half space, and the indenter is assumed to be a rigid axisymmetric punch. The problem is formulated as a mixed boundary value problem of the theory of elasticity and solutions are obtained for the indentation of the layer by the plane end of a rigid circular cylinder and by a rigid sphere. Subject to detailed verification with independent tests, the present solutions are suggested as useful for the determination of the elastic shear modulus of intact cartilage.

Solution of Crack Problems: The Distributed Dislocation Technique

Preface. 1. Introduction to Fracture Mechanics. 2. Distributed Dislocation Fundamentals. 3. Further Topics in Plane Crack Problems. 4. Interface Cracks. 5. Solution of Axi-Symmetric Crack Problems. 6. Three-Dimensional Cracks: An Introduction. 7. Three-Dimensional Cracks: Further Concepts. 8. Concluding Remarks. A: Dislocation Influence Functions. B: Numerical Solution of SIEs with Cauchy Kernel. C: Plane and Ring Dipole Influence Functions. D: Contour Integral and Kernel Function. References. Index.

A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques

Abstract An alternative numerical method for solving contact problems for real rough surfaces is proposed. The real area of contact and the contact pressure distribution are determined using a single-loop iteration scheme based on the conjugate gradient method, which converges for arbitrary rough surfaces. The surface deflections and subsurface stresses are computed using an alternative two-dimensional multi-level multi-summation algorithm, which allows the summation error to be kept under the discretization error for any number of contact points. The proposed method is fast: rough contact problems for surface samples with 10 5 –10 6 data points are solved on a personal computer in a few hours. The numerical algorithms are described in full detail so that an interested reader can implement the new contact solver in a computer code. Numerical examples demonstrating the method advantages are presented. The method is compared with other fast contact solvers that have emerged in the last few years.

Unstable growth of thermally induced interacting cracks in brittle solids

Abstract The growth and stability of thermally induced equally spaced parallel cracks in a half-plane consisting of a homogeneous isotropic linearly elastic brittle material are studied. At the initial time, the uniform temperature of the half-plane is reduced at its surface by a large increment, T 0 , and then kept constant (at the surface). Because of heat conduction and possible heat convection due to fluid flow, a temperature gradient forms close to the surface and penetrates into the half-plane. Thermal contraction results in the formation of cracks perpendicular to the free surface. It is shown that if the cracks are initially parallel and equally spaced, and if the possibility of branching is excluded, then they grow in time until a critical state is reached. At this state alternate cracks stop growing, while the others begin to grow at a much faster rate. This process continues until another critical state is attained, where the cracks which had stopped growing (together with some other cracks, depending on the temperature profile), suddenly close, while the cracks which have continued growing, suddenly “snap” into a finitely longer length. At this state the crack spacing is doubled (or quadrupled, depending on the temperature profile). The whole process then repeats itself. Applications to geothermal energy extraction from hot dry rock masses is mentioned.

Vibration and stability of cracked rectangular plates

Abstract The present paper deals with eigenvalue problems of cracked rectangular plates. Vibration and buckling problems are solved for a plate with a crack emanating from one edge and for a plate with a centrally located internal crack. The problems are formulated as dual series equations and reduced to homogeneous Fredholm integral equations of the second kind. The singularity of the solution in each case is isolated and treated analytically. Numerical results for the natural frequencies and moment distributions are compared with the work of other investigators. Vibration and buckling mode shapes are also illustrated for a cracked plate.

Wear‐Contact Problems and Modeling of Chemical Mechanical Polishing

Wafer shape and contact pressure evolution during chemical mechanical polishing, and the characteristics of the steady-state regime are analyzed on the basis of approaches developed in contact mechanics. Nonplanarity caused by the geometrical nonuniformity (erosion) and by the presence of different material on the surface (recess) is considered. The possibility of the process optimization and the determination of system parameters based on the polished surface profiles is discussed.

Scattering of elastic waves by a surface-breaking crack

Abstract Scattering of incident surface waves and incident body waves by a surface-breaking crack is investigated in a two-dimensional geometry. By decomposing the scattered fields into symmetric and antisymmetric fields with respect to the plane of the crack, two boundary value problems for a quarter-plane have been obtained. The formulation of each boundary-value problem has been reduced to a singular integral equation which has been solved numerically. For incident surface waves the back-scattered and forward-scattered surface waves have been plotted versus the dimensionless frequency. Curves are also presented for the scattered displacement fields in the interior of the body generated by incident body waves, both versus the angle of incidence and versus the dimensionless frequency.

An Analytical Model of Joint Contact

The stress distribution in the region of contact between a layered elastic sphere and a layered elastic cavity is determined using an analytical model to stimulate contact of articulating joints. The purpose is to use the solution to analyze the effects of cartilage thickness and stiffness, bone stiffness and joint curvature on the resulting stress field, and investigate the possibility of cracking of the material due to tensile and shear stresses. Vertical cracking of cartilage as well as horizontal splitting at the cartilage-calcified cartilage interface has been observed in osteoarthritic joints. The current results indicate that for a given system (material properties mu and nu constant), the stress distribution is a function of the ratio of contact radius to layer thickness (a/h), and while tensile stresses are seen to occur only when a/h is small, tensile strain is observed for all a/h values. Significant shear stresses are observed at the cartilage-bone interface. Softening of cartilage results in an increase in a/h, and a decrease in maximum normal stress. Cartilage thinning increases a/h and the maximum contact stress, while thickening has the opposite effect. A reduction in the indenting radius reduces a/h and increases the maximum normal stress. Bone softening is seen to have negligible effect on the resulting contact parameters and stress distribution.

Micromechanics Modeling of Crack Initiation Under Contact Fatigue

Using dislocation pileup theory, a model is givenfor the prediction of crack initiation life under contact fatigue. Near surface crack initiation is investigated by introducing the sliding contact boundary condition. Crack initiation originated at the surface and substrate are treated as extreme cases. The new model physically explains how a surface crack can be initiated and shows that the surface crack initiation life should be shorter than the subsurface crack initiation life under the same stress amplitude conditions. A discussion is given about the influence of residual stress, hardness, temperature, irreversibility of the plastic deformation, as well as other parameters that affect the crack initiation life

A Fast and Accurate Method for Numerical Analysis of Elastic Layered Contacts

Solution of contact problems for layered elastic solids generally required the use of numerical methods. Recently, the fast Fourier transform (FFT) technique has been applied to such contacts. While very fast. FFT is strictly applicable only to periodic contact problems. When it is applied to essentially non-periodic contacts, an error is introduced in the numerical solution. A new method that overcomes the limitation of the straightforward FFT approach for solving non-periodic layered contact problems is introduced in the present article. A special correction procedure based on the multi-level multi-summation technique is used to compensate the FFT results for the periodicity error. The use of a robust iteration scheme based on the conjugate gradient method ensures that the new method is applicable to contact problems involving real rough surfaces. Numerical examples demonstrate that the new method is both accurate and fast.