Zener Stroh Crack Research Articles

An interfacial arc-shaped Zener–Stroh crack due to inclusion–matrix debonding in composites

Stress analysis has been carried out for a curved interfacial Zener–Stroh crack between a circular inclusion and an infinite matrix due to interface debonding in a composite. Using the distributed dislocation technique, the physical problem is formulated into singular integral equations of the 2nd kind. With the Jacobi polynomials and Gauss-Legendre integration methods, the integral equations are discretized and solved numerically. The stress intensity factors and energy release rates of the curved crack are evaluated accordingly. In the numerical examples, the effects of half debonding angle, the Dundurs’ constants, and the loading ratio \({{b_{x}^{T}}/{b_{y}^{T}}}\) on the stress intensity factors and the energy release rates are analyzed in detail. It is found that the stress intensity factors are greatly affected by the half debonding angle, the Dundurs’ constant β and the loading ratio \({{b_{x}^{T}}/{b_{y}^{T}}}\), while the influence of Dundurs’ constant α is relatively small especially when the loading ratio is close to zero.

A simple model for the study of the tolerance of interfacial crack under thermal load

Thermal stresses due to a mismatch between material properties may induce failure of a bonded interface. A simple model for linear dissimilar elastic bonded solids containing an interfacial Zener–Stroh crack under a uniform temperature shift is proposed. Its solution is derived. This result may provide some information about the interface defect tolerant size, which is mainly responsible for triggering interface failures under thermal load. Thus, it can be used to assess the interface integrity and reliability under thermal load. On the other hand, the practical interface crack problem in this study provides another background (mechanism) for the Zener–Stroh crack model.

On the fracture behavior of a Zener–Stroh crack with plastic zone correction in three-phase cylindrical composite material

With crack tip plastic zone correction, stress investigation on the fracture behavior of a Zener–Stroh crack in three-phase composite was carried out. A Zener–Stroh crack (in the matrix phase) is near a circular inclusion, with the three-phase cylindrical composite model used to represent the composite material. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. The Dugdale model of small scale yielding is used to introduce a thin strip of yielded plastic zone each crack tip. The physical problem is formulated into a set of singular integral equations, using the solution for a three-phase model with a single dislocation in the matrix phase as the Green’s function. The singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacements using Erdogan and Gupta’s method with some iterative numerical procedures.

On a finite crack partially penetrating two circular inhomogeneities and some related problems

We investigate a Mode-III finite slit crack partially penetrating two circular inhomogeneities embedded in an unbounded matrix. In order to obtain analytical solutions, it is assumed that the two circular inhomogeneity–matrix interfaces are Apollonius circles with respect to the two crack tips (or equivalently the two crack tips are just mutually image points with respect to each one of the two circular interfaces). Particularly closed-form expressions of the stress intensity factors at the two crack tips are obtained even though only series form solutions to the original boundary value problem can be derived. The loadings considered in this research include: (i) remote uniform anti-plane shearing; (ii) a straight screw dislocation at any position of the three-phase composite system; (iii) a Zener-Stroh crack. The results are verified by comparison with existing solutions. The related problem of a circular hole partially merged in two circular inhomogeneities is also addressed, with closed-form expressions of the stress concentration factors derived.

ZENER-STROH CRACK PROBLEM IN A FINITE PLATE

In this paper, a Zener-Stroh crack problem in a finite plate is studied. By using the principle of superposition, the original problem is converted into a Zener-Stroh crack problem in an infinite plate and a usual Griffith crack problem in a finite plate. The former problem has a solution in a closed form. The latter problem can be solved by using EEVM (eigenfunction expansion variational method). Finally, numerical examples for evaluating the stress intensity factors at crack tips are presented.

On the plastic zone size and CTOD study for a Zener–Stroh crack interacting with a circular inclusion

An analytical solution is given for plastic yielding of a Zener–Stroh crack near a circular inclusion embedded in an infinite matrix. The crack is orientated along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. Using the Dugdale model of small-scale yielding, plastic zones are introduced at both crack tips. Using the solution of a circular inclusion, interacting with a single dislocation as the Green’s function, the physical problem is formulated into a set of singular integral equations. With the aid of Erdogan’s method and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement. The results obtained in the current work are verified by reduction to simpler cases of the Dugdale model. Various parameters such as the distance, shear modulus ratio, Poisson’s ratio, and loading condition are studied.

A Zener-Stroh crack interacting with an edge dislocation

A Zener-Stroh crack interacting with an edge dislocation is studied. The crack faces are assumed to be traction free. The applied ‘generalized loading’ for crack is the initial displacement jump and the intervention of the edge dislocation. Through decomposing the problem into two sub-simple problems, using the superposition principle, its solution is obtained. To demonstrate both the validity of the solution and its potential application, two simple examples related to the crack stress intensity factors are presented on the basis of the solution. The application of the solution to model crack initiations arising from dislocation-pileup is discussed.

Perturbation method for the solution of a Zener–Stroh crack with a slightly curved configuration

This paper investigates the Zener–Stroh crack with curved configuration in plane elasticity. A singular integral equation is suggested to solve the problem. Formulae for evaluating the SIFs and T-stress at the crack tip are suggested. If the curve configuration is a product of a small parameter and a quadratic function, a perturbation method based on the singular integral equation is suggested. In the method, the singular integral equation can be expanded into a series with respect to the small parameter. Therefore, many singular integral equations can be separated from the same power order for the small parameter. These singular integral equations can be solved successively. The solution of the successive singular integral equations will provide results for stress intensity factors and T-stress at the crack tip. It is found that the behaviors for the solution of SIFs and T-stress in the Zener–Stroh crack and the Griffith crack are quite different. This can be seen from the presented comparison results.

Distributed Zener-Stroh cracks give a Griffith crack, and their dipole is the Bueckner weight function

Distributed Zener-Stroh cracks give a Griffith crack, and their dipole is the Bueckner weight function

Closed-form solutions for a mode III radial matrix crack penetrating a circular inhomogeneity

In this research we address in detail a mode III radial matrix crack penetrating a circular inhomogeneity. One tip of the radial crack lies in the matrix, while the other tip of the radial crack lies in the circular inhomogeneity. In addition the two tips of the crack are mutually image points (or inverse points) with respect to the circular inhomogeneity-matrix interface. First we conformally map the crack onto a unit circle C a in the new ζ -plane. Meanwhile the inhomogeneity-matrix interface is mapped onto C b , a part of another circle in the ζ -plane. In addition C a and C b intersect at a vertex angle π /2. By using the method of image in the ζ -plane, closed-form solutions in terms of elementary functions are derived for three loading cases: (1) remote uniform antiplane shearing; (2) a screw dislocation located in the unbounded matrix; and (3) a radial Zener–Stroh crack.

Collinear Zener–Stroh crack problem in plane elasticity

This paper investigates the collinear Zener–Stroh crack problem in plane elasticity. Two cracks in series are chosen as an example in analysis. Different to the Griffith crack problem, an initial displacement discontinuity exists in the Zener–Stroh crack problem, which in turn is the increment of displacement when a moving point goes around the crack in a closed loop. From the traction free condition along the cracks, the dislocation distribution function as well as the complex potential with undetermined coefficients is suggested. The involved undetermined coefficients can be evaluated from the condition of the assumed initial displacement discontinuity. Finally, a closed form solution for the problem is obtained and the calculated stress intensity factors at crack tips are presented. A problem for two collinear Zener–Stroh cracks with different lengths is also studied.

Singular integral equation method for multiple Zener–Stroh crack problems in antiplane elasticity

In this paper, the multiple Zener–Stroh crack problems in anti-plane elasticity are studied. The crack faces are assumed to be traction free, and dislocation distributions on the cracks are chosen as the unknown functions in the solution. The singular integral equations for the problem are obtained. The constraint equations are also derived from the condition of the accumulation of dislocation on the cracks. After solving the integral equations, the stress intensity factors at crack tips can be evaluated immediately. Numerical examples are given. It is found that interactions between the Zener–Stroh cracks are quite different from those for the Griffith cracks, in qualitative and quantitative aspects.

Zener–Stroh crack at the interface of multi-layered structures

A Zener–Stroh (Z–S) crack can be nucleated on the interface of a multi-layered structure when a dislocation pileup is stopped by the interface which works as an obstacle. During the entire fracture of a crack, Z–S crack mechanism controls the initial stage, or the first phase of crack initiation and propagation. In our current research, investigation on a Z–S crack at the interface of a multi-layered structure is carried out. The problem is formulated into a set of singular integral equations by applying the distributed dislocation based fracture mechanics. The obtained integral equations are then solved with numerical method after the singularities at crack tips are carefully checked. In the solution procedure, the contact zone model is adopted to cease the oscillation behavior. The contact zone length, the stress field near the crack tips and the stress intensity factors (SIFs) of the crack are discussed based on the numerical results of two typical structures. It was found that the contact zone length could be very large and was determined by the properties of all the three materials and loading conditions. Our analysis also shows that the thickness of the middle thin layer plays a critical role for the fracture behavior of the crack when it is comparable to the crack length.

A Zener?Stroh crack at the interface of a thin film bonded to a substrate

A Zener–Stroh crack can nucleate at the interface of a multi-layered structure when a dislocation pileup is stopped by the interface which works as an obstacle. During the entire fracture procedure of a crack, Zener–Stroh crack mechanism controls the initial stage, or the first phase of crack initiation and propagation. In our current research, stress investigation on a Zener–Stroh crack initiated at the interface of a thin film bonded to a half plane substrate has been carried out. With the application of dislocation-based fracture mechanics, the micro crack is simulated by the distributed dislocations along the crack line. To eliminate the contradictory oscillation phenomenon for the stress field near the interfacial crack tip, a contact zone behind the crack tip is introduced. The physical problem is thus formulated into a set of non-linear singular integral equations. Through careful examination of the crack singularities at the crack tips for different configurations, the formulated integral equations are solved with numerical methods developed in our research. The contact zone length, the stress fields near the crack tip and the stress intensity factors of the crack are evaluated accordingly. Numerical examples based on practical engineering structures are provided to discuss the influence of the key parameters, such as the thickness of the film, and the Dundurs constants, on the fracture behaviour of the crack.

Electro-elastic stress analysis for a Zener-Stroh crack interacting with a coated inclusion in a piezoelectric solid

The problem of a Zener-Stroh crack initiated near a coated circular inclusion in a piezoelectric medium is investigated in this paper. By using the solution of a single piezoelectric screw dislocation near a coating inclusion as the Green’s function, a Mode III displacement loaded crack is investigated. The proposed problem is formulated as a set of singular integral equations which are solved by numerical techniques. The influence of various parameters, such as the material constants of the inclusion, the coating, the matrix, the coating layer thickness, etc., on the crack behavior is studied. The stress and electric displacement intensity factors of the crack are derived. Several numerical examples are given and the results obtained are discussed in detail.

An inclined Zener–Stroh crack near a free surface

Machining of a material surface often produces large shear stress beneath the free surface, which causes dislocations piled up along a slip plane near the surface until those dislocations are stopped by an obstacle and nucleates a microcrack at the subsurface. The nucleated microcrack due to dislocation pile-up is called a Zener–Stroh crack which has many properties different to a conventional Griffith crack. In this paper, the physical mechanism of such microcrack initiation has been discussed. Stress investigation on the problem of a subsurface Zener–Stroh crack inclined to the free surface of a material has been carried out. By using the stress solution of a single dislocation in a half plane as the Green's function, the microcrack is simulated with distributed dislocations along the crack line. A set of singular integral equations are then formulated and solved with numerical method. Results show that the influence of the free surface on the stress intensity factors (SIFs) of the crack and the critical crack length greatly depends on the slant angle. The free surface also brings intrinsic coupling phenomenon for the Mode I and Mode II SIFs. As a result this coupling effect changes the fracture toughness of the material.

Multiple Zener-Stroh crack problem in an infinite plate

In this paper, the multiple Zener-Stroh crack problem is studied. We choose the distributed dislocations as unknown functions in the integral equation. The crack faces are assumed to be traction free. The applied “generalized loading” for cracks is the initial displacement jump (abbreviated as IDJ), which in turn is the increment of displacement when a moving point goes around the crack in a closed loop. A system of singular integral equations is obtained. After solving the integral equations, the stress intensity factors at crack tips can be evaluated immediately. Numerical examples are given. It is found that interactions between Zener-Stroh cracks are quite different from those for the Griffith cracks, in the qualitative and quantitative aspects.

Electro-elastic stress analysis for a Zener–Stroh crack at the metal/piezoelectric bi-material interface

A Zener–Stroh crack may nucleate at the interface of metal/piezoelectric bi-materials, when piled-up dislocations along a slip plane in metal material are stopped by the interface which works as an obstacle. As a kind of microcrack, Zener–Stroh crack controls the initial phase of crack nucleation. In the first step of our current research, the stress and electric displacement fields for a single interfacial dislocation in general piezoelectric/metal bimaterials are obtained in an explicit close form. With this solution as the Green function, the interfacial Zener–Stroh crack problem is formulated into a set of singular integral equations with distributed dislocation technique. These singular integral equations are then solved by numerical method based on the singularity nature at the crack tips. To deal with the oscillation behaviors near the interfacial crack tips, both open crack tip model and contact zone model are considered and compared each other in the paper. Solutions on the stress field, stress and electric displacement intensity factors and contact zone length are obtained. A numerical example of a Zener–Stroh crack at the Pt/PZT-5H bimaterial interface is given. Some useful electro-elastic characteristics related to the crack are found and discussed.

Micro-crack initiation at tip of a rigid line inhomogeneity in piezoelectric materials

For the square-root singularity shear stress found at the tip of a rigid line inhomogeneity (an anti-crack) in piezoelectric media, one possible way of releasing high strain energy is to initiate a micro-crack at the inhomogeneity tip. In our current study, a dislocation pileup model for micro-crack initiation at the inhomogeneity tip is proposed based on Zener–Stroh crack initiation mechanism. An interesting and important physical result that emerges from the analysis is that the critical stress intensity factor for the anti-crack (the line inhomogeneity) can be related to the fracture toughness of a conventional Griffith crack in the same material. Analytical results further show that under mechanical loading, the critical stress and electric displacement intensity factors of an anti-crack are only related to the corresponding intensity factors of stress and electric displacement of the crack, respectively. While if the anti-crack is under displacement loading (with net dislocation pile-up at the inhomogeneity tip), the critical stress and electric displacement intensity factors of an anti-crack depend on both of the total mechanical dislocations bT and electricity dislocations bD.

Stress analysis for a Zener–Stroh crack interacting with a coated inclusion

Abstract A microcrack can be initiated by coalescing dislocations piled up near an inhomogeneity. This mechanism was firstly proposed by Zener and later analyzed by Stroh. The microcrack thus is called a Zener–Stroh crack, a counterpart of the well-known Griffith crack in linear elastic fracture mechanics. In the current study we consider a Zener–Stroh crack initiated near a coated circular fiber in a solid material. The interaction between the crack and the coated inclusion (fiber) for both Mode I and II displacement loading is investigated using the solution of an edge dislocation interacting with a coated fiber derived by Xiao and Chen [International Journal of Solids and Structure, in press] as the Green’s function. The problem is formulated as a set of singular integral equations and is solved numerically by Erdogan–Gupta method. The influence of several parameters, such as the distance between the crack and the inclusion, the material constants combination of the fiber, the coating layer and the matrix, on the stress intensity factors of the crack is analyzed. Several numerical examples are given and the results obtained are discussed in detail.