Bounding theorems for stress intensity factors

Bounding theorems for the changes in elastic energy due to a crack in a finite body and crack tip Stress Intensity Factors are established by use of the variational principles of the theory of elasticity. It is shown that the change in elastic energy due to a crack in a finite body is larger than the change in energy due to the same crack in an infinite body with the same boundary conditions when stresses are prescribed on the external boundary, and smaller when displacements are prescribed. The energy changes can be expressed as functions of the crack tip Stress Intensity Factors and for special loadings, bounds for single mode Stress Intensity Factors are obtained. The obtained inequalities are in agreement with known numerical solutions of finite cracked bodies.     

Size and density influence on overall moduli of finite media with cracks

This paper deals with the numerical calculations of overall moduli of cracked solid. The variational approach of the displacement discontinuity method is used. Both crack size influence and crack density influence are studied. It is found that for a given crack density, increasing crack size reduce the overall moduli when the overall stresses are assumed to be prescribed. Numerical results are compared with classical estimation schemes such as self-consistent method and differential scheme.     

Instability of a complex-shaped circumferential crack in a pipe subjected to bending loads

The paper examines the instability of a complex-shaped circumferential crack in a Type 304 stainless steel pipe that is subjected to rotations at the pipe ends. A through-wall crack extends over a prescribed fraction of the pipe circumference, while there is also an internal circumferential crack extending around the remainder of the pipe section. Instability of circumferential growth of the through-wall crack segment is examined using a crack tip opening angle growth criterion, and an assessment is made of the effects of crack configurational parameters, pipe size and material properties, on the instability criterion. The paper highlights the undesirability, from a safety perspective, of having a complex-shaped crack in a region with adverse fracture resistance characteristics.     

The penny-shaped crack problem in micropolar thermoelasticity

The axisymmetry problem of a penny-shaped crack opened out by thermal loads is studied. The linear theory of micropolar elasticity is employed. Two types of thermal loads are considered—prescribed temperature on the crack faces and prescribed heat flux across the faces. It is shown that, in both the cases, the problem is equivalent to the isothermal problem of the crack opened out by suitable normal tractions on the crack faces. The stress intensity factors are found to depend on, in addition to the usual parameters, two parameters N and M; N is a number characterising the coupling of the displacement field with the microtation field and M is the ratio N/τ where τ is a non-dimensional material characteristic length. The classical values of the stress intensity factors are recovered as a limiting case. Numerical results are presented for the case of constant heat flux across the crack faces. These results show that the presence of couple stresses elevates the values of the stress intensity factors.     

Unsteady growth of mode III crack in elastic perfectly-plastic material

In this paper the assembly of the near-tip fields given by J. R. Rice is completed for the mode III crack growing quasi-statically and unsteadily in elastic perfectly-plastic material. The obtained results provide a particular example for the general theoretical relations between the steady state and unsteady state crack growth. Further, the general expression of the rate of crack opening displacement is obtained, which is similar to one by J.R. Rice and co-workers for mode I crack growing in elastic perfectly-plastic material. The fracture criterion of the critical opening displacement at a prescribed distance behind the crack tip is discussed. As a result, the theoretical J-resistance curves are given.     

Distribution of temperature necessary for maintaining a crack in a prescribed shape

The problem considered here is one dealing with the distribution of surface temperature required to maintain a crack in an elastic solid, in a prescribed shape.     

Stationary heat conduction in an anisotropic slab containing a crack

The stationary temperature field in an anisotropic slab containing an internal longitudinal crack is investigated. On each face of the slab the temperature is presumed to have an arbitrary variation in one direction, while on the crack surface either heat flux or temperature is prescribed. The problem is reduced to a pair of Fredholm integral equations which can be solved numerically. As an illustrative example, the temperature distribution is computed for a slab having a temperature rise on one face and containing a central crack across which no heat is transferred. Numerical results are given for orthotropic and isotropic slabs with various ratios of crack length to slab thickness.     

Inverse crack problem in elasticity

Summary Exact solution is given to the problem of a penny-shaped crack embedded in a transversely isotropic elastic half-space when arbitrary normal displacements are prescribed at its faces. A new integral representation of the kernel of the governing integral equation allowed to obtain closed form expressions for all the quantities of interest like, stresses inside and outside the crack, stress intensity factor, work done to open the crack, directly through the given displacements. Several illustrative examples are considered.     

On the singularity of temperature gradient near an inclined crack terminating at bimaterial interface

This paper deals with the singularity of temperature gradient near an inclined crack terminating at a bimaterial interface. The temperature field is solved by considering the continuity of temperature and heat flux at the interface and appropriate thermal boundary conditions on crack surfaces. The singularity of temperature gradient around the crack tip is then studied for the cases for which the temperature on crack surfaces is prescribed or crack surfaces are insulated. It is found that, unlike the oscillatory singularity of the stress field, no oscillatory character near the crack tip is observed for these problems. The dependence of the singularity of temperature gradient on the inclined angle of crack and thermal conductivity ratio of two dissimilar media is also shown.     

The extended finite element method in thermoelastic fracture mechanics

The extended finite element method (XFEM) is applied to the simulation of thermally stressed, cracked solids. Both thermal and mechanical fields are enriched in the XFEM way in order to represent discontinuous temperature, heat flux, displacement, and traction across the crack surface, as well as singular heat flux and stress at the crack front. Consequently, the cracked thermomechanical problem may be solved on a mesh that is independent of the crack. Either adiabatic or isothermal condition is considered on the crack surface. In the second case, the temperature field is enriched such that it is continuous across the crack but with a discontinuous derivative and the temperature is enforced to the prescribed value by a penalty method. The stress intensity factors are extracted from the XFEM solution by an interaction integral in domain form with no crack face integration. The method is illustrated on several numerical examples (including a curvilinear crack, a propagating crack, and a three‐dimensional crack) and is compared with existing solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

On the acoustic emission due to the fracture of brittle inclusions

The far-field characteristics of the emission from a theoretical model for the fracture of brittle inclusions are presented in detail. The model is a circular crack growing at constant speed from zero size until it attains a prescribed size. The far-field radiation pattern is the same as that of a simple combination of force doublets, and some qualitative similarities between force doublets and acoustic dipoles are noted. The initial shape of the far-field pulses due to the growing stage and the stopping is determined, but difficulties arise in accounting for the diffraction of a surface wave on the crack faces generated by the stopping of the crack.     

Analytical modelling of dynamic fracture and crack arrest in DCB specimens under fixed displacement conditions

An analytical solution via the beam theory considering shear deformation effects is developed to solve the static and dynamic fracture problem in a bounded double cantilever beam (DCB) specimen. Fixed displacement condition is prescribed at the pin location under which crack arrest occurs. In the static case, at first, the compliance function of a DCB specimen is obtained and shows good agreement with the experimental results cited in the literature. Afterward, the stress intensity factor is determined at the crack tip via the energy release rate formula. In the dynamic case, the obtained governing equations for the model are solved supposing quasi‐static treatment for unstable crack propagation. Finally, a closed form expression for the crack propagation velocity versus beam parameters and crack growth resistance of the material is found. It is shown that the reacceleration of crack growth appears as the crack tip approaches the finite boundary. Also, the predicted maximum crack propagation velocity is significantly lower than that obtained via the Euler–Bernoulli theory found in the literature.     

Inverse problems of material distributions for prescribed apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media

This study is concerned with the inverse problem of calculating material distributions intending to realize prescribed apparent fracture toughness in functionally graded material (FGM) coatings around a circular hole in infinite elastic media. The incompatible eigenstrain induced in the FGM coatings after cooling from the sintering temperature, due to mismatch in the coefficients of thermal expansion, is taken into consideration. An approximation method of determining stress intensity factors is introduced for a crack in the FGM coatings in which the FGM coatings are homogenized simulating the nonhomogeneous material properties by a distribution of equivalent eigenstrain. A radial edge crack emanating from the circular hole in the homogenized coatings is considered for the case of a uniform pressure applied to the surfaces of the hole and the crack. The stress intensity factors determined for the crack in the homogenized coatings represent the approximate values of the stress intensity factors for the same crack in the FGM coatings, and are used in the inverse problem of calculating material distributions in the FGM coatings intending to realize prescribed apparent fracture toughness in the coatings. Numerical results are obtained for a TiC/Al₂O₃ FGM coating, which reveal that the apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media can be controlled within possible limits by choosing an appropriate material distribution profile in the coatings.     

Solutions of multiple crack problems of a circular plate or an infinite plate containing a circular hole by using Fredholm integral equation approach

Presented is an elementary solution, which is a particular solution of the circular plate containing one crack. The solution consists of two parts and satisfies the following conditions: (i) the first part corresponds to a pair of normal and tangential concentrated forces acting at a prescribed point on both edges of a single crack; (ii) the second part corresponds to some distributed tractions along both edges of the crack; (iii) the obtained elementary solution, i.e. the sum of the first and second parts, satisfies a traction free condition on the circular boundary. Using this elementary solution and taking some undetermined density of the elementary solution along each crack, a system of Fredholm integral equations of multiple crack problems can always be obtained. The multiple crack problems of an infinite plate containing a circular hole can be solved in a similar way. Several numerical examples are given in this paper.     

Point temperature solution for a penny-shaped crack in an infinite transversely isotropic thermo-piezo-elastic medium

The solution of an impermeable penny-shaped crack subjected to a concentrated thermal load (prescribed point temperature) applied arbitrarily at the crack surfaces is derived using the generalized potential theory method. The integral equation governing the temperature field is found to have the same structure as that for the elastic punch problem and the integro-differential equations related to the electroelastic field are similar to that reported for the elastic crack problem. Significant solutions to these integro-differential equations are obtained by generalizing the previous results available in literature. Exact three-dimensional expressions for the full-space thermo-electro-elastic field are finally obtained by simple differentiation, all in terms of elementary functions. The exact analysis for a permeable crack is also presented and discussed. The obtained point temperature solutions play an important role in the related BEM analysis.     

Superposition method for calculating singular stress fields at kinks, branches and tips in multiple crack arrays

A method is developed for calculating stresses and displacements around arrays of kinked and branched cracks having straight segments in a linearly elastic solid loaded in plane stress or plain strain. The key idea is to decompose the cracks into straight material cuts we call `cracklets', and to model the overall opening displacements of the cracks using a weighted superposition of special basis functions, describing cracklet opening displacement profiles. These basis functions are specifically tailored to induce the proper singular stresses and local deformation in wedges at crack kinks and branches, an aspect that has been neglected in the literature. The basis functions are expressed in terms of dislocation density distributions that are treatable analytically in the Cauchy singular integrals, yielding classical functions for their induced stress fields; that is, no numerical integration is involved. After superposition, nonphysical singularities cancel out leaving net tractions along the crack faces that are very smooth, yet retaining the appropriate singular stresses in the material at crack tips, kinks and branches. The weighting coefficients are calculated from a least squares fit of the net tractions to those prescribed from the applied loading, allowing accuracy assessment in terms of the root-mean-square error. Convergence is very rapid in the number of basis terms used. The method yields the full stress and displacement fields expressed as weighted sums of the basis fields. Stress intensity factors for the crack tips and generalized stress intensity factors for the wedges at kinks and branches are easily retrieved from the weighting coefficients. As examples we treat cracks with one and two kinks and a star-shaped crack with equal arms. The method can be extended to problems of finite domain such as polygon-shaped plates with prescribed tractions around the boundary.     

Annular crack surrounding an elastic fibre in transversely isotropic elasticity

The axisymmetric problem of an infinitely long transversely isotropic elastic fibre perfectly bonded to a dissimilar transversely isotropic elastic matrix containing an annular crack is considered. The annular crack, surrounding the fibre, is subjected to prescribed longitudinal tension. A potential function approach is used to find the solution of the basic equations. The mixed boundary value problem is reduced to the solution of a singular integral equation, which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations.     

Cruciform crack in an orthotropic elastic plane

The problems of determining the stress and displacement fields in an infinite orthotropic plane containing a cruciform crack 387-1, y=0 and 387-2, x=0 when (I) the shape of the crack is prescribed and (II) the cracks are opened by given normal pressures, are reduced to mixed boundary value problems for the quarter plane. Using integral transform techniques, a closed form solution is obtained for problem I, whereas the solution of problem II has been reduced to solving a Fredholm integral equation of second kind with non-singular kernel. Numerical calculation of the stress intensity factor and crack energy in the case of a linear loading function for various crack lengths are presented for problem II, using the values of material constants for a Boron-Epoxy composite.     

Stress intensity factors for an elliptical crack approaching the surface of a semi-infinite solid

Stress intensity factors for an embedded elliptical crack approaching the free surface of the semi-infinite solid that is subjected to uniform tension perpendicular to the plane of crack are presented in a nondimensional form for various crack aspect ratios and crack distances from the free surface. Stress intensity factors are determined numerically using an alternating technique with two solutions. The first solution involves an elliptical crack in a solid and subjected to normal loading expressible in a polynomial of x and y. The second solution involves stresses in the half space due to prescribed normal and shear stresses on the surface. Effect of the Poisson's ratio on these stress intensity factors is also investigated. Stress intensity factors for a semi-elliptical surface crack in a tinite thickness plate are then estimated in a nondimensional form for various crack aspect ratios and crack depth to plate thickness ratios.Specialist Engineer, Aerospace Group, The Boeing Company, Seattle, Washington.Professor, Department of Mechanical Engineering, University of Washington, Seattle, Washington, and also Aerospace Group, The Boeing Company, Seattle.