Strain-induced passivity breakdown in corrosion crack initiation

Corrosion crack nucleation and growth are modelled as a moving boundary problem. The model incorporates three physical processes––dissolution, passivation and straining––into a continuum mechanical framework. The dissolution triggers surface advance; the passivation restrains the access of the environment to bare metal; the deformation causes for passivity breakdown. Plane cracks nucleating from surface pits in an elastic–plastic material body under fatigue load are considered. The problem is solved using a FEM program and a moving boundary tracking procedure. The model simulates how cracks form and grow in a single continuous course. The geometry of the developed cracks is found independent of the initial pit size. Plasticity is found to influence the curvature at the tip of the nucleated corrosion cracks. The most important evolution length parameter, the width of the corrosion crack, is found to depend on the size constraints of the tracking procedure. It is concluded that the model is deficient for determining all length scales observed in reality. Physical processes to be considered in an advanced model are proposed and discussed.     

Analysis of electric boundary condition effects on crack propagation in piezoelectric ceramics

There are three types of cracks: impermeable crack, permeable crack and conducting crack, with different electric boundary conditions on faces of cracks in piezoelectric ceramics, which poses difficulties in the analysis of piezoelectric fracture problems. In this paper, in contrast to our previous FEM formulation, the numerical analysis is based on the used of exact electric boundary conditions at the crack faces, thus the common assumption of electric impermeability in the FEM analysis is avoided. The crack behavior and elasto-electric fields near a crack tip in a PZT-5 piezoelectric ceramic under mechanical, electrical and coupled mechanical-electrical loads with different electric boundary conditions on crack faces are investigated. It is found that the dielectric medium between the crack faces will reduce the singularity of stress and electric displacement. Furthermore, when the permittivity of the dielectric medium in the crack gap is of the same order as that of the piezoelectric ceramic, the crack becomes a conducting crack, the applied electric field has no effect on the crack propagation. The project supported by the National Natural Science Foundation of China (19672026, 19891180)     

Retardation of a crack with connections between the faces using an induced thermoelastic stress field

This paper considers local temperature variations near the tip of a crack in the presence of regions in which the crack faces interact. It is assumed that these regions are adjacent to the crack tip and are comparable in size to the crack size. The problem of local temperature variations consists of delay or retardation of crack growth. For a crack with connections between the crack faces subjected to external tensile loads, an induced thermoelastic stress field, and the stresses at the connections preventing crack opening, the boundary-value problem of the equilibrium of the crack reduces to a system of nonlinear singular integrodifferential equations with a Cauchy kernel. The normal and tangential stresses at the connections are found by solving this system of equations. The stress intensity factors are calculated. The energy characteristics of cracks with tip regions are considered. The limiting equilibrium condition for cracks with tip regions is formulated using the criterion of limiting stretching of the connections.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 133–143, January–February, 2005     

Contact Interaction of the Faces of a Rectangular Crack under Normally Incident Tension–Compression Waves

A three-dimensional problem on the contact interaction between the faces of a rectangular crack under a normally incident harmonic tension–compression wave is considered. The problem is solved by using the method of boundary integral equations and an iterative algorithm. The contact forces and the discontinuity in the displacement of the crack faces are studied. The results obtained are compared with those for a finite plane crack.     

Effect of contact interaction of the crack faces for a crack under harmonic loading

This paper is concerned with dynamic problems in fracture mechanics for elastic solids having cracks with contacting faces. The contact problem for a penny-shaped crack with a nonzero initial opening under normally incident harmonic wave is solved by the method of boundary integral equations. The solutions are compared with those that neglect the contact interaction of the crack faces. Results are presented for different values of the initial crack opening Presented at the 6th International Conference on Modern Practice in Stress and Vibration Analysis (Bath, United Kingdom, September 5–7, 2006). Published in Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 125–131, July 2007.     

Finite boundary effects in problem of a crack perpendicular to and terminating at a bimaterial interface

In this paper, the problem of a crack perpendicular to and terminating at an interface in bimaterial structure with finite boundaries is investigated. The dislocation simulation method and boundary collocation approach are used to derive and solve the basic equations. Two kinds of loading form are considered when the crack lies in a softer or a stiffer material, one is an ideal loading and the other one fits to the practical experiment loading. Complete solutions of the stress field including the T stress are obtained as well as the stress intensity factors. Influences of T stress on the stress field ahead of the crack tip are studied. Finite boundary effects on the stress intensity factors are emphasized. Comparisons with the problem presented by Chen et al. (Int. J. Solids and Structure, 2003, 40, 2731–2755) are discussed also.The project supported by the National Natural Science Foundation of China (10202023 and 10272103), and the Key Project of CAS (KJCX2-SW-L2).     

On the influence of electric boundary conditions on dynamic SIFs in piezoelectric materials

Dynamic stress intensity factors (SIFs) for a straight crack in a piezoelectric material under time-harmonic L- and SH-wave loading are determined for different electric boundary conditions. Impermeable, permeable and limited permeable cracks are compared. The problem is formulated and numerically solved using a nonhypersingular traction-based boundary integral equation method where the fundamental solution is obtained by Radon transform. A parametric study in the frequency domain shows the dependence of the SIFs on the choice of the electrical boundary conditions at the crack faces.     

Interaction of a growing crack with the boundary in a bicrystal

Dynamic photoelasticity has been used in conjunction with selective etching on lithium fluoride bicrystals to examine the interaction of a growing crack with inclined boundaries; it is found that the stresses at the head of the crack alter as the boundary is approached. The speed of the crack is related to the angle of incidence on the boundary and the angular disorientation of the latter. The change in crack speed is related to the change in state of stress at the vertex. Analytical and experimental distributions are presented for the stresses ahead of a growing crack.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 138–143, July–August, 1973.     

A mixed electric boundary value problem for a two-dimensional piezoelectric crack

In this paper, a mixed electric boundary value problem for a two-dimensional piezoelectric crack problem is presented, in the sense that the crack face is partly conducting and partly impermeable. By the analytical continuation method, the unknown electric charge distributions on the upper and lower conducting crack faces are reduced to two decoupled singular integral equations and then these two equations are converted into algebraic equations to find the full field solution. Though the results suggest that the stress intensity factors at the crack tip are identical to those of conventional piezoelectric materials, but the electric field and electric displacement are related to the electric boundary conditions on the crack faces. The electric field and electric displacement are singular not only at crack tips but also at the junctures between the impermeable part and conducting parts. Numerical results for the variations of the electric field, electric displacement field and J-integral with respect to the normalized impermeable crack length are shown. Some discussions for the energy release rate and the J-integral are made.     

Opening-Function Simulation of the Three-Dimensional Nonstationary Interaction of Cracks in an Elastic Body

Integral relations between three-dimensional dynamic displacements (stresses) in an infinite elastic body with arbitrarily located plane cracks and discontinuities in the displacements of the opposite crack faces are presented. The influence of opening cracks on each other is considered in the problem on crack faces loaded by pulse forces. This problem is reduced to a system of boundary integral equations of the wave-potential type in a time domain. The dynamic mode I stress intensity factors are determined for two coplanar elliptic cracks under forces in the form of the Heaviside function     

NEW BOUNDARY ELEMENT METHOD FOR TORSION PROBLEMS OF CYLINDER WITH CURVILINEAR CRACKS

The Saint-Venant torsion problems of a cylinder with curvilinear cracks were considered and reduced to solving the boundary integral equations only on cracks. Using the interpolation models for both singular crack tip elements and other crack linear elements, the boundary element formulas of the torsion rigidity and stress intensity factors were given. Some typical torsion problems of a cylinder involving a straight, kinked or curvilinear crack were calculated. The obtained results for the case of straight crack agree well with those given by using the Gauss-Chebyshev integration formulas, which demonstrates the validity and applicability of the present boundary element method.     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

Closed form solution for two unequal collinear semi-permeable straight cracks in a piezoelectric media

The problem of two unequal collinear straight cracks weakening a poled transversely isotropic piezoelectric ceramic is addressed under semi-permeable electric boundary conditions on the crack faces. The plate has been subjected to combined in-plane normal(to the faces of the cracks) mechanical and electric loads. Problem is formulated employing Stroh formalism and solved using complex variable technique. The elastic field, electric field and energy release rate are obtained in closed analytic form. A case study is presented for poled PZT-5H cracked plate to study the effect of prescribed mechanical load, electric load, inter-crack distance and crack lengths on crack arrest parameters stress intensity factor (SIF), electric displacement intensity factor (EDIF) and mechanical and total energy release rates (ERR). Moreover a comparative study is done of impermeable and semi-permeable crack face boundary conditions on SIF, EDIF and ERR, and results obtained is presented graphically. It is observed that the effect of dielectric medium in the crack gap cannot be ignored.     

On One Contact Problem in Fracture Mechanics for a Normally Incident Tension–Compression Wave

The contact interaction of the faces of a penny-shaped crack in a three-dimensional space is studied for the case of normal incidence of a harmonic tension–compression wave. The problem is solved by the method of boundary integral equations. The dependence of the mode I stress intensity factor on the wave number is studied. The solution is compared with the results obtained for a penny-shaped crack when the contact interaction is neglected.     

Some new aspects of boundary conditions at cracks in piezoelectrics

Fracture mechanical investigations of piezoelectric materials as components of smart structures have become popular in the last 30?years. In the early years of research, boundary conditions at crack faces have been adopted from pure mechanical systems under the assumption that boundaries were traction free. From the electrostatic point of view, cracks have been assumed to be either free of charge or fully permeable. Later, limitedly permeable crack boundary conditions have become popular among the community, nevertheless still assuming traction-free crack faces. Recently, the theoretical framework has been extended to include electrostatically induced mechanical tractions in crack models yielding a significant crack closure effect. However, these models are still simple, neglecting, e.g., the piezoelectric field coupling. In this work, we present an extended model for crack surface tractions yielding some interesting effects. In particular, the orientation of the electrical field with respect to the poling axis becomes important. Furthermore, applying a collinear stress parallel to the crack faces influences the Mode-I stress intensity factor and a Mode-II shear loading couples to the Mode-I SIF.     

Elastic–plastic and elastic anti-plane shear of two parallel cracks

Two infinite interacting parallel cracks in an elastic–plastic and in an elastic body under anti-plane strain (mode III) loading conditions are considered. The body is subjected to vanishing remote loading and the cracks are traction free. Closed-form solution is found for the elastic–plastic problem in terms of elementary functions, where the shape of the plastic boundary is obtained. The complete stress distribution is obtained in an inverse form i.e. physical coordinates are functions of stresses.     

Shape control of thin rigid inclusions and cracks in elastic bodies

The paper is concerned with a control of thin rigid inclusion and crack shapes in elastic bodies. It is assumed that rigid inclusions are delaminated; thus, cracks are located on the boundary of inclusions as well as outside of inclusions. We provide the problem formulations and analyze the shape sensitivity with respect to geometrical perturbations in the frame of free boundary models. Inequality type boundary conditions are considered at the crack faces to guarantee a mutual non-penetration between crack faces. Inclusion and crack shapes are considered as control functions. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry of the structure. We prove an existence of optimal shapes in the problems considered.     

Viscous boundary layers in flows through a domain with permeable boundary

The effect of small viscosity on nearly inviscid flows of an incompressible fluid through a given domain with permeable boundary is studied. The Vishik–Lyusternik method is applied to construct a boundary layer asymptotic at the outlet in the limit of vanishing viscosity. Mathematical problems with both consistent and inconsistent initial and boundary conditions at the outlet are considered. It is shown that in the former case, the viscosity leads to a boundary layer only at the outlet. In the latter case, in the leading term of the expansion there is a boundary layer at the outlet and there is no boundary layer at the inlet, but in higher order terms another boundary layer appears at the inlet. To verify the validity of the expansion, a number of simple examples are presented. The examples demonstrate that asymptotic solutions are in quite good agreement with exact or numerical solutions.     

The generalised weight function method for crack problems with mixed boundary conditions

The generalised weight function method for determination of stress intensity factors for crack problems with mixed boundary conditions is presented. The method is based on the application of Betti's reciprocal theorem to equivalent crack problems with special boundary conditions, which are converted from the original problems by the principle of superposition. Expressions of stress intensity factors for center cracks subjected to arbitrary stresses in finite plates with various boundary conditions are derived. Examples of practical interest are given. The results reveal the important roles of the boundary displacement constraint and finite dimensions in the crack parameter evaluation.     

Study into the Interaction of Cracks in an Elastic Half-Space under a Shock Load by Means of Boundary Integral Equations

The effect of a shock load on the interaction of circular cracks in an elastic half-space is studied. In the space of Fourier time transforms, the problem is reduced to a system of two-dimensional boundary integral equations in the form of the Helmholtz potential with unknown densities characterizing the discontinuities in the displacements of the opposite crack faces. Discrete analogs of those equations are constructed. As an example, two cracks are considered whose faces are under the action of shock tensile loads varying in time as the Heaviside function. The time dependences of the dynamic stress intensity factors are obtained. Their dependence on the relative position of the cracks in the half-space is analyzed.