A strip yield model for two collinear cracks in plates of arbitrary thickness

As two cracks grow and approach each other under fatigue loading, a deleterious interaction between them can considerably affect the crack growth rate, making theoretical evaluations and experimental data from a single isolated crack case considerably inaccurate. The aim of the present study is to investigate the interaction between two collinear cracks of equal length, taking into account the plate??s thickness effect, which was demonstrated to have a large effect on fatigue crack growth in the case of a single crack. The obtained solution to the problem is based on the Dugdale strip yield model and the distributed dislocation technique. In addition, a fundamental solution for an edge dislocation in a finite thickness plate is utilised. The present solution shows a very good agreement with previously published results for some limiting cases. The obtained results confirm a significant dependence of the interaction and stress intensity factors on the plate thickness, which can dramatically affect the plastic collapse conditions as well as fatigue crack growth rates.     

Effective spring stiffness for a planar periodic array of collinear cracks at an interface between two dissimilar isotropic materials

Explicit analytical expressions are obtained for the longitudinal and transverse effective spring stiffnesses of a planar periodic array of collinear cracks at the interface between two dissimilar isotropic materials; they are shown to be identical in a general case of elastic dissimilarity (the well-known open interface crack model is employed for the solution). Since the interfacial spring stiffness can be experimentally determined from ultrasound reflection and transmission analysis, the proposed expressions can be useful in estimating the percentage of disbond area between two dissimilar materials, which is directly related to the residual strength of the interface. The effects of elastic dissimilarity, crack density and crack interaction on the effective spring stiffness are clearly represented in the solution. It is shown that in general the crack interaction weakly depends on material dissimilarity and, for most practical cases, the crack interaction is nearly the same as that for crack arrays between identical solids. This allows approximate factorization of the effective spring stiffness for an array of cracks between dissimilar materials in terms of an elastic dissimilarity factor and two factors obtained for cracks in a homogeneous material: the effective spring stiffness for non-interacting (independent) cracks and the crack interaction factor. In order to avoid the effect of the crack surface interpenetration zones on the effective spring stiffness, the range of the tensile to transverse load ratios is obtained under the assumption of small-scale contact conditions. Since real cracks are often slightly open (due to prior loading history and plastic deformation), it is demonstrated that for ultrasound applications the results obtained are valid for most practical cases of small interfacial cracks as long as the mid-crack opening normalized by the crack length is at least in the order of 10⁻⁵.     

Interaction of coplanar surface cracks in a half-space

Boundary integral equations are applied to the interaction of closely spaced surface-coplanar cracks with various geometrical dimensions in a half-space that is tensioned by forces perpendicular to the crack surfaces. The numerical results show that the crack depths and the distances between them greatly influence the stress intensity coefficients at the edges of the cracks, and the interaction between the cracks can be neglected only for shallow ones.Translated from Problemy Prochnosti, No. 10, pp. 12–16, October, 1994.     

Cracks propagation and interaction in an orthotropic elastic material: Analytical and numerical methods

An elastic orthotropic material containing a crack in Mode I is considered to formulate a new analytical model. The boundary conditions for the crack existence in the material lead to the solution of the homogeneous Riemann–Hilbert problems. The mathematical model was elaborated for a single and two collinear cracks of different lengths and distance for Mode I in order to investigate cracks interaction problem. Using the theory of Cauchy’s integral and the numerical analysis, the fields in the vicinity of the crack tips were determined.Finite Element Method was applied to compare the mathematical analytical solution and to determine the fields in the vicinity of the crack tips. The critical values of applied stress which caused cracks propagation were evaluated. The interaction of cracks in an orthotropic aramid-epoxy material was studied in details. Comparison of both approaches to crack propagation leads to the conclusion that the new analytical model is correct and can be applied to more complex cracks geometries, including inclined cracks.     

Shape prediction and stability analysis of Mode-I planar cracks

This paper presents a numerical technique for simulating stable growth of Mode-I cracks in two and three dimensions, using energy release rate and its derivatives. The crack growth model used in the numerical simulation is based on the concept of maximizing potential energy of the system released as cracks evolve. Therefore, a series of quadratic programming (QP) problems with linear constraints and bounds are solved to simulate stable growth of Mode-I planar cracks. The derivative of energy release rate provides a stability condition for crack growth in structures and can be regarded as a discretized influence function that represents the strength of the interaction among crack extensions at different crack tips in 2-D and different locations along a crack front in 3-D. The energy release rate and its derivative are accurately calculated by the analytical virtual crack extension method [Engng. Fract. Mech. 59 (1998) 521; 68 (2001) 925] in a single analysis. Numerical examples are presented to demonstrate the capabilities of the proposed approach. Examples include a central crack subjected to wedge forces in a 2-D finite plate, a system of interacting thermally induced parallel cracks in a two-dimensional semi-infinite plane and a 3-D penny-shaped crack embedded in a large cylinder, pressurized in a central circular region.     

Effect of electromechanical coupling on the dynamic interaction of cracks in piezoelectric materials

Summary This article provides a comprehensive treatment of the dynamic interaction between two arbitrarily located and oriented cracks in a piezoelectric medium under steady-state inplane electrical and antiplane mechanical loads. Using an impermeable condition along the crack surfaces, a fundamental dynamic solution was developed for the single crack problem. In this fundamental solution, the single crack problem was treated using Fourier transform and the appropriate singular integral equations. The fundamental solution was then implemented into a pseudo-incident wave method to account for the interaction between the cracks. Numerical examples are provided to show the effect of the geometry of the cracks, the material constants, the frequency of the incident wave and the applied electrical field upon the dynamic stress intensity factors. The results show the significant effect of electromechanical coupling upon the stress intensity factor at the crack tip.     

Evaluation of C* integral for interacting cracks in plates under tension

A closed form solution for C* integral of two interacting cracks in plates under tension is developed on the basis of reference stress method. Comprehensive finite element (FE) creep analyses are carried out to provide the benchmark of the interaction evaluation of multiple cracks. Results indicate that more pronounced interaction is observed between the C* of double cracks and that of a single crack compared to that denoted by stress intensity factor (SIF). Overall good agreement is achieved between the proposed method for C* of multiple crack interaction and the FE results which provides confidence in practical application.     

Computation of stress intensity factors of interface cracks based on interaction energy release rates and BEM sensitivity analysis

Stress intensity factors of bimaterial interface cracks are evaluated based on the interaction energy release rates. The interaction energy release rate is defined based on the energy release rates of a cracked body, corresponding to two independent loading conditions, actual field and an auxiliary field, and is related to the sensitivities of the potential energies for crack extensions. The potential energy of a cracked body is expressed with a domain integral, which is converted to a boundary integral expression by applying the divergence theorem. By differentiating this expression with the crack length, a boundary integral expression for the interaction energy release rate is obtained. The boundary integral representation for the interaction energy release rate involves the displacement, the traction, and their sensitivity coefficients with respect to the crack length. The boundary element sensitivity analyses are used to calculate these quantities accurately. A regularized boundary integral equation relating the boundary displacement and traction is differentiated with respect to an arbitrary shape parameter to derive the regularized boundary integral equation for the sensitivity coefficients of the boundary displacement and traction. The proposed approach is applied to several cracks in dissimilar media and the results are compared with those obtained by the conventional approach based on the extrapolation method. The analytical displacement and stress solutions for an interface crack between two infinite dissimilar media subjected to uniform stresses at infinity are used to give the auxiliary field, in which the values of the stress intensity factors are known. It is demonstrated that the present method can give accurate results for the stress intensity factors of various bimaterial interface cracks under coarse mesh discretizations.     

Self‐similar crack patterns induced by spatial stress fluctuations

Abstract A mechanism of the formation of multiscale self‐similar crack structures in brittle materials is proposed based on the action of spatial random stress fluctuations (random stress fields) with vanishing mathematical expectation. The presence of self‐equilibrating stress fluctuations is, in many cases, characteristic of heterogeneous brittle materials even in the absence of external loading. Some examples are given by residual stresses—stresses caused by phase transformations or internal microscopic heat sources. If the stress fluctuations are strong enough to cause cracking, the cracks will be originated in regions with and perpendicular to the highest local tension. As a legacy of this special location, additional local tractions opening the crack in its centre are developed even in self‐equilibrating stress fields. This mechanism is modelled by disklike cracks opened by a pair of concentrated forces applied at the centre. Two cases are considered: (i) the crack growth in the moving equilibrium conditions caused by the self‐equilibrating stress fluctuations of increasing amplitude; and (ii) the subcritical crack growth under the stress fluctuations of either sustained (creep) or oscillating (fatigue) amplitudes. As the cracks grow, the interaction between them leads to a separation of crack sizes. The effects of the interaction are modelled by the differential self‐consistent method that is shown to become asymptotically accurate as the ratio between the sizes of the interacting cracks increases. The interaction results in the crack distribution tending to a self‐similar one with the distribution function proportional to the inverse fourth power of the crack radius.     

Close interaction of coplanar circular cracks under shear loading

A new method is proposed for the stress analysis of an elastic space weakened by several arbitrarily located coplanar circular cracks subjected to an arbitrary shear loading. The method is based on a new type of integral equation and has definite advantages over the existing methods: equations are non-singular, the iteration procedure is rapidly convergent even for very close interactions. The method allows us to obtain a practically exact numerical solution to the problem of very close interactions. An accurate analytical solution is obtained for the case of two cracks, which are separated by a half of their radius or more. The stress intensity factors and the crack energy increase due to the interaction are computed for various distances between the cracks.The reported research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada     

T-shaped crack problem for bonded orthotropic layers

The plane elasticity problem of two perfectly bonded orthotropic layers containing cracks perpendicular to and along the interface is considered. Cracks are extended to intersect the boundaries and each other in such a way that a crack configuration suitable to study the T-shaped crack problem is obtained. The problem is reduced to the solution of a system of singular integral equations with Cauchy type singularities. Numerical results for stress intensity factors and energy release rates are presented for various loading conditions and for isotropic as well as orthotropic material pairs. These results indicate that elementary strength of material type calculations for energy release rates provide a good approximation to the actual elasticity solution even for relatively short cracks, as long as the layer thicknesses are not very different.     

Generic Overlapping Cracks in Polymers: Modeling of Interaction

This paper deals with modeling of the interaction in overlapping cracks that the authors have earlier identified to be generic to a wide range of polymeric systems (Ramasamy and Lesser, J Polym Sci B Phys, 2003). A complex stress function method is used for evaluating stress intensity factors for interacting cracks. The interaction between two parallel overlapping cracks is considered first. It is shown for this case that the stress intensity factor can fall below the threshold value when there is sufficient overlap, leading to arrest of crack growth at the overlapping tip. Then the interaction in a doubly periodic infinite array of cracks is considered. The interaction in the array is found to be non-linear. However, at a given stress level, the highest density of stable cracks is related to the threshold value for crack propagation K_th though a simple set of equations. It is also shown that in an infinite array of cracks, the energy release rate criterion for crack growth is different from the stress intensity factor criterion due to a reduced stiffness of the material.     

Periodic system of interface cracks with contact zones in the isotropic bimaterial in the fields of tension and shear

The methods of the theory of functions of complex variable are used to construct the solution of the plane problem of the theory of elasticity in the closed form for an infinite isotropic bimaterial plane (space) with a periodic system of interface cracks in the presence of the zones of smooth contact of crack lips of near the crack tips. As a result of the numerical analysis of the obtained solution, it becomes possible to study the interaction of the cracks and establish, in particular, the regularities of variation of the length of the contact zones and the stress intensity factors as functions of the distance between the cracks. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 4, pp. 101–107, July–August, 2006.     

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.     

Stress field interaction and strain energy distribution between a stationary main crack and its process zone

The damage process zone developed by brittle materials in front of a macrocrack is simulated by means of a distribution of microcracks. Crack mutual interactions are taken into account by means of a numerical technique, based on a displacement discontinuity boundary element method that is able of considering both the macrocrack–microcrack and microcrack–microcrack interactions inside the process zone. In the frame of linear elastic fracture mechanics the stress field at each crack tip and the related elastic strain energy are calculated. The main features of the interaction phenomena turn out to be almost independent of the microcrack density. Some considerations both on the shielding and amplification effects on the main crack and on the strain energy distribution between cracks give explanation to experimental evidence and prove that crack interaction is not such a short-range effect as sometimes expected.     

The arbitrarily loaded single-edge cracked circular disc; accurate weight function solutions

A general solution for the stresses and displacements of collinear cracks in an infinite homogeneous anisotropic medium subjected to uniform loading at infinity has been given in this paper by using the Stroh's formulation. The solutions are valid not only for plane problems but also for antiplane problems and the problems whose inplane and antiplane deformations couple each other. Two special collinear crack problems are solved explicitly: (1) two collinear cracks, (2) an infinite row of evenly spaced collinear cracks. A closed form solution of the stresses and displacements in the entire domain is obtained. Through the use of identities developed in the literature, the stress intensity factors, crack opening displacements and energy release rate are expressed in real form, which are valid for any kind of anisotropic materials including the degenerate materials such as isotropic materials. The simple explicit form solutions for the crack opening displacements and energy release rate reveal that the effect of anisotropy is totally determined by the fundamental elasticity matrix L. The relation between the stress intensity factors and energy release rate is obtained in quadratic form and related to L.     

On the dynamic behaviour of interfacial cracks between a piezoelectric layer and an elastic substrate

Surface-bonded piezoelectric layers can be used as actuators/sensors for advanced structural applications. The current paper provides a theoretical study of the dynamic behaviour of interacting cracks between a piezoelectric layer and an elastic medium under antiplane mechanical loads. The electromechanical field of a single interfacial crack is determined first using Fourier transform technique and solving the resulting integral equations. This fundamental solution is then imple- mented into a pseudo-incident wave method to account for the interaction between different cracks. The dynamic behaviour of the resulting stress field is studied with special attention being paid to the stress intensity factors at the crack tips. Typical examples are provided to show the effect of the size and position of the cracks, the material combination and the loading frequency upon the stress intensity factors.     

Dynamic stresses around two cracks placed symmetrically to a large crack

Dynamic stresses around three cracks in an infinite elastic plate have been solved. Two cracks, which are small and equal, are situated ahead of a large crack so as to allow for geometrical symmetry. Time-harmonic normal traction acts on each surface of these cracks. To solve the problem, two solutions are combined. One of them is a solution for a crack in an infinite plate and another is that for two collinear cracks in an infinite plate. The Schmidt method is used to satisfy the boundary conditions on the cracks' surfaces with use of the combined solutions. Stress intensity factors are calculated numerically for some of these crack configurations.     

Cracks with bonding between the lips in bushings of friction couples

We study the problem of contact deformation of a bushing of friction couple. The friction surface of the bushing contains rectilinear cracks with end zones characterized by the presence of interaction between the crack lips. In the case of repeated reciprocal motion of a plunger, the material of the bushing finally fails as a result of contact interaction. The boundary-value problem of equilibrium of the bushing of friction couple with cracks whose lips are interacting is reduced to the solution of a system of nonlinear singular integro-differential equations with Cauchy-type kernel. The solution of this system is used to find the normal and tangential forces in the bonds. The condition of limiting equilibrium of a crack with end zone is formulated with regard for the criterion of ultimate length of the bonds in the material. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 2, pp. 53–61, March–April, 2006.     

Multiple penny-shaped cracks interaction in a finite body and their effect on stress intensity factor

In this paper we investigate the stress intensity factors (SIFs) of multiple penny-shaped cracks in an elastic solid cylinder under mode I (axial tension) loading. The cracks are located symmetrically and in parallel to one another in the isotropic cylinder. The fractal-like finite element method (FFEM) is employed to study the interaction of multiple cracks and to demonstrate the efficiency of the FFEM for multiple crack problems. The results show that the SIF values of the inner cracks, which are denoted as crack number 1,2,3,…,(n+1)/2 of a stack of n parallel cracks, are lower than the SIF values of a single crack by between 16% and 48%. Also, the outermost crack, that is the crack closest to the boundaries of a multiple cracked body, has the highest SIF values and is, therefore, likely to fail first.