Modeling of surface and subsurface crack behavior under contact load in the presence of lubricant

A new mathematical model for lubricated elastic solids weakened by cracks is proposed. Surface and subsurface cracks are taken into account, and the interaction of lubricant with elastic solids within cavities of surface cracks is regarded as the most interesting aspect of the problem. The boundary conditions characterizing the behavior of lubricant within crack cavities such as pressure rise in crack cavities fully filled with lubricant as well as other boundary and additional conditions are derived. The problem is reduced to a system of integro-differential equations with nonlinear boundary conditions in the form of alternating equations and inequalities. A new iterative numerical method is developed for solution of the proposed problem. The method guarantees conservation of lubricant volumes trapped within closed crack cavities and allows for all three functions (normal and tangential displacement jumps and normal stress applied to crack faces) characterizing the problem solution to be determined simultaneously. Examples of numerical results for surface and subsurface cracks are presented and numerical and asymptotic results for small subsurface cracks are compared to each other. The numerical analysis indicates that depending on a surface crack orientation its normal stress intensity factor may be two or more orders of magnitude higher than the one for a similar subsurface one.     

Application of the Geometrical Theory of Diffraction to closed cracks

The purpose of this paper is to compare the scattering of ultrasound by means of different types of smooth planar cracks in elastic solids. The boundary conditions on the flaws are of a type that incorporates interfacial forces. A crack that is partly closed by a static background pressure can thus be modeled. The Geometrical Theory of Diffraction (GTD) is used to predict the pulse-echo response from the crack. Previously obtained diffraction and reflection coefficients for cracks with interfacial forces are reviewed and some numerical difficulties are discussed. The backscattered echo amplitude is numerically calculated and compared for some different crack types and crack shapes. The presence of interfacial forces, due to background pressure, significantly influences the scattering behavior of the cracks. For a background pressure of 250 MPa, the backscattered amplitude, using our model of the interfacial forces, is typically reduced by 5–20 dB.     

Hypersingular integral formulation of elastic wave scattering

A hypersingular boundary integral formulation for calculating two dimensional elastic wave scattering from thin bodies and cracks is described. The boundary integral equation for surface displacement is combined with the hypersingular equation for surface traction. The difficult part in employing the traction equation, the derivation of analytical formulas for the hypersingular integral by means of a limit to the boundary, is easily handled by means of symbolic computation. In addition, the terms containing an integrable logarithmic singularity are treated by a straightforward numerical method, bypassing the use of Taylor series expansions. Example wave scattering calculations for cracks and thin ellipses are presented.     

Stress intensity in the vicinity of a surface flaw in the linear part of a pipeline

Real sharp-edted surface and subsurface flaws detected in a gas pipeline body are modeled by surface semi-elliptical mathematical cracks (cuts) in a closed cylindrical shell. A relationship is proposed that relates the geometrical dimensions of the flaws to the crack aspect ratio. Based on the line spring model, the problem of stress state and boundary equilibrium conditions of a closed cylindrical shell with a surface semi-elliptical crack is reduced to a system of singular integral equations. An algorithm was developed for computational solution of the problem, and numerical analysis was made for the dependence of stress intensity factors on loading conditions and geometrical parameters of shell and crack. For a shell subjected to internal pressure and weakened by a surface longitudinal semi-elliptical crack, a closed approximation formula is proposed that interrelates pressure level, shell/crack dimensions, and material mechanical properties in boundary equilibrium conditions. The maximal error value is indicated for the results obtained using this formula. Lvov Polytechnic State University, Lvov, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 38–47, July–August, 1999.     

Spring boundary model for a partially closed crack

The spring boundary conditions have been widely employed to describe the acoustic properties of rough surfaces in partial contact. Central to this model is the role of the interfacial stiffness. Recently, a singular stiffness model has been proposed to remove inconsistencies which arise when these boundary conditions are used to describe the acoustic response of partially closed cracks with a position-independent and finite stiffness. Here, an alternative solution to this problem is discussed which is based on the micromechanics of rough surfaces in contact. For cracks that are partially closed, it prescribes a finite crack stiffness that varies along the crack faces and, rather than diverging, becomes null within a small but finite neighborhood of the crack tip. The conditions under which the local stiffness of a non-planar crack can be evaluated are also reviewed.     

2D SH modelling of ultrasonic testing for cracks near a non-planar surface

A model of 2D SH ultrasonic nondestructive testing for interior strip-like cracks near a non-planar back surface in a thick-walled elastic solid is presented. The model employs a Green's function to reformulate the 2D antiplane wave scattering problem as two coupled boundary integral equations (BIE): a displacement BIE for the back surface displacement and a hypersingular traction BIE for the crack opening displacement (COD). The integral equations are solved by performing a boundary element discretization of the back surface and expanding the COD in a series of Chebyshev functions which incorporate the correct behaviour at the crack edges. The transmitting ultrasonic probe is modelled by prescribing the traction underneath it, enabling the consequent calculation of the incident field. An electromechanical reciprocity relation is used to model the action of the receiving probe. A few numerical examples which illustrate the influence of the non-planar back surface are given.     

TEM investigations of intergranular stress corrosion cracking in austenitic alloys in PWR environmental conditions

Abstract

Analytical transmission microscopy has been used to investigate the initiation of stress corrosion cracking in Inconel 600 subjected to constant load testing under simulated pressured water reactor primary water conditions. The observations revealed that intergranular attack proceeded by the development of a zone of polycrystalline chromia along the boundary plane intersecting either the free surface or a blunted, open crack in contact with the free surface. Ni-rich metal particles were interspersed within the chromia. Conversely, open cracks were filled with nanocrystalline NiO and large compound particles of spinel and NiO, indicating a difference in potential between closed, attacked boundaries and open cracks. Open cracks appeared to have initiated by fracture of the chromia zones, such fracture being strongly dependent on boundary geometry with respect to loading direction. The observations suggest that stress corrosion crack initiation and propagation is dependent on diffusion of oxygen through the porous oxides. Dislocations and stress could enhance diffusion as chromia was observed along slip planes at the arrested tips of blunt cracks.     

Elastic wave scattering by a rectangular crack near a non-planar back surface

A 3D model of non-destructive ultrasonic testing for cracks near a non-planar back surface is presented. The scattering by an interior rectangular crack in a thick-walled component with a back surface of general geometry is considered. The 3D wave scattering problem is solved using boundary integral equation methods (BIEMs): the boundary element method (BEM) for the back surface displacement is combined with an analytical technique for the hypersingular traction boundary integral equation for the crack opening displacement. The solution method generates many unknowns, but by applying a threshold criterion a sparse approximation of the system matrix is obtained such that a fast sparse solver may be used. The computations are accelerated further using the stationary phase approximation for the computation of probe field integrals. The action of ultrasonic probes in transmission and reception, calibration by side-drilled holes and effects of material damping are taken into account in the model, and a few numerical examples illustrate the influence of the back surface geometry.     

On the Exact Solution of Boundary Value Problems with Non-separable Geometries,Like a Rough Surface

A Dirichlet problem is considered for the Helmholtz equation and for a class of geometries for which the Helmholts operator does not separate (e.g. a rough surface). It is shown that, contrary to the widely held view, it is possible to obtain the solution of this problem in a closed form which resembles closely the solutions obtained for separable geometries—expansions generated by Sturm-Liouville theory. As with separable geometries, we show in particular that the expansion coefficients can be written explicitly as integrals containing a priori known functions—that matrix inversion is not required for the determination of the expansion coefficients. The problem includes as a special case the scattering of electromagnetic waves from a rough cylindrical surface which is the boundary of a perfect conductor. The method is very general and can be used for much more complicated boundary value problems, such as scattering by dielectric interface.     

Coupled boundary element and finite element model for fluid-filled membrane in gravity waves

This paper describes a three-dimensional, coupled boundary element and finite element model for dynamic analysis of a fluid-filled membrane in gravity waves. The model consists of three components, describing respectively, the membrane deflection and the motions of fluids inside and outside the membrane. Small amplitude assumptions of the surface waves and membrane deflection lead to linearization of the mathematical problem and an efficient solution in the frequency domain. A finite element model, based on the membrane theory of shells, relates the membrane deflection to the internal and external fluid pressure. Two boundary element models, which describe the potential flows inside and outside the membrane, are coupled to the finite element model through the kinematic and dynamic boundary conditions on the membrane. As a demonstration, the resulting model is applied to evaluate the dynamic response of a bottom-mounted fluid-filled membrane in a wave flume. Previous two-dimensional numerical model results and three-dimensional laboratory data verify and validate the present three-dimensional model. Analysis of the computed membrane response and surface wave pattern reveals intricate resonance characteristics that explain the discrepancies between the numerical model results and the laboratory data.