Critical study of existing solutions for a penny-shaped interface crack, comparing with a new boundary element solution allowing for frictionless contact

A numerical study of the fundamental problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces is carried out allowing for the possibility of frictionless contact between crack faces. A new, highly accurate axi-symmetric formulation of the boundary element method (BEM) for the solution of elastic contact problems is employed. The correctness and accuracy of available predictions of different kinds for several key characteristics of the solution of this problem are checked. First, comparison of the BEM results for the near-tip contact length shows a very good agreement with some existing predictions. Second, the global solution obtained by BEM is compared with existing asymptotic solutions, obtained with both the open and the frictionless contact models. BEM results show that at the closest neighborhood to the crack tip the global solution of the problem is governed by the first term of the asymptotic solution of the frictionless contact model (up to a distance of the order of a fraction of the near-tip contact length). After a small transition region, in an adjacent surrounding zone whose extent is almost independent of the near-tip contact length, the global solution of the problem is governed by the first term of the asymptotic solution of the open model. As a result of the comparison presented, the regions in which the classical fracture parameters, stress intensity factor (SIF) and energy release rate, can be accurately obtained from the global numerical solution of a crack of this kind have been determined. Third, BEM results and previous estimations show certain discrepancies with a recently published closed form solution of the near-tip contact length and the mode II SIF of the frictionless contact model. A new closed form expression of this mode II SIF, derived from the asymptotic solution of the open model, is proposed in this paper.     

Three-dimensional asymptotic stress field in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity

An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity, e.g., crack, anticrack (infinitely rigid lamella), etc., located at the center, edge or corner, and subjected to the far-field torsion (mode III), extension/bending (mode I), and sliding shear/twisting (mode II) loadings. Five different discontinuity-surface boundary conditions are considered: (1) bimaterial penny-shaped interface anticrack or perfectly bonded thin rigid inclusion, (2) bimaterial penny-shaped interfacial jammed contact, (3) bimaterial penny-shaped interface crack, (4) bimaterial penny-shaped interface crack with partial axisymmetric frictionless slip, and (5) bimaterial penny-shaped interface thin rigid inclusion alongside penny-shaped crack. Solutions to these cases except (3) are hitherto unavailable in the literature. Closed-form expressions for stress intensity factors subjected to various far-field loadings are also presented. Numerical results presented include the effect of the ratio of the shear moduli of the layer materials, and also Poisson’s ratios on the computed lowest real parts of eigenvalues for the case (5). Interesting and physically meaningful conclusions are also presented, especially with regard to cases (1) and (2).     

Finite element analysis of frictionless contact between a sinusoidal asperity and a rigid plane: Elastic and initially plastic deformations

A series of finite element simulations of frictionless contact deformations between a sinusoidal asperity and a rigid flat are presented. Explicit expressions of critical variables at plastic inception including interference, contact radius, depth of first yielding, and pressures are obtained from curve fitting of simulation results as a function of material and geometrical parameters. It is found Hertz solution is not applicable to the critical contact variables at plastic inception for sinusoidal contact, although contact responses of initially plastic deformation follow the same trend as that of purely elastic deformation. The contact pressure at incipient plasticity, which is defined as yield strength, is dependent on Poisson’s ratio, yield stress, and geometrical parameters, but independent of elastic modulus. It is not yield stress, but yield strength that correlates with indentation hardness. The results yield the insight into the specification of material properties to realize elastic contact. A larger ratio of yield stress to elastic modulus is beneficial to sustain a larger load before plastic deformation.     

Fracture mechanics analysis of cracking in plain and reinforced concrete using the boundary element method

Boundary element formulations for modelling the nonlinear behaviour of concrete are reviewed. The analysis based on the dual boundary element method (BEM) to represent the cracks in concrete is presented. The fictitious crack is adopted to represent the fracture process zone in concrete. The influence of reinforcements on the concrete is considered as a distribution of forces over the region of attachment. The yielding of reinforcement is considered when the total force at any section of the reinforcement is greater than the yielding force and is assumed to be broken when the strain reaches the maximum strain. In using the BEM to simulate cracks, the crack path need not be known in advance since it can be calculated during the iteration process and as such remeshing becomes obsolete. The numerical results obtained are compared to the FEM analysis.     

A study of the interaction between a propagating crack and an uncoated/coated elastic inclusion using the BE technique

The purpose of the present work is a parametrical study of the interaction between a propagating edge-crack and an embedded elastic fibre using the Boundary Element (BE) technique. Uniaxial fibre reinforced composites generally have very good properties in the direction of the fibres, but in conventional multi-layer crossply laminates it is cracking in the transverse direction which effectively limits the strength of a stressed body. Therefore, in this study the propagation of a crack in the transverse direction is considered, i.e., in a plane containing the fibre axes, rather than perpendicular to the fibres. Crack deflection/attraction mechanisms and their associated energy release rate variations are investigated for a range of Young's moduli and Poisson's ratio mismatches, and crack offsets with respect to the inclusion centreline. Furthermore, the effects of a third-phase, i.e., coating, applied to the fibre's surface are analysed, and results have been obtained for different coating thicknesses and elastic moduli ratios. From this investigation it was found that the Poisson's ratio of the different phases could have a significant effect on the crack trajectory, and hence the energetics involved in the process of crack deflection are also dramatically altered. This opens up the possibility of enhancing the fracture toughness of fibre reinforced composite materials by considering the Poisson's ratio of the individual phases when selecting the particular material combination.     

Determination of the opening load for a growing fatigue crack: evaluation of experimental data reduction techniques and analytical models

In metallic materials, growing cracks will remain closed or partially closed for a portion of the applied cyclic load as a consequence of plastically deformed material left in the wake of a growing crack, surface roughness along the crack surfaces, or corrosion debris. Proper characterization of this crack closure and the subsequent opening load is required for accurate prediction of crack growth. In the laboratory, global load–displacement data are commonly used in conjunction with a data reduction technique to estimate the opening load for a growing crack. Different data reduction techniques will be compared, and the influence of data smoothing will be demonstrated, using AA 7075-T651 specimens tested under constant amplitude cyclic loading with load ratios R = 0.1, 0.2, and 0.3. The ratio of maximum stress intensity factor to plane strain fracture toughness was approximately K _max / K _Ic = 0.5. The measured crack opening loads are also compared with predictions obtained from two different strip-yield models and three-dimensional elastic–plastic finite element analyses. Results show the necessity of using smoothed data, and the poor behaviour of the compliance offset data reduction technique, when analysing high load ratio data. A modification to this technique is proposed which improves crack opening load estimates. Overall, the analytical model predictions compare well with the experimental results; especially those results generated using the modified compliance offset technique.