Transient shear waves in a wedge subjected to rapid tearing

The surface of an elastic wedge is subjected to sudden antiplane surface tractions and displacements sufficient to cause tearing. The subsequent crack instability is investigated. The wedge faces subtend an angle κπ with the line of antisymmetry, along which the crack propagates with a constant velocity v. For the externally applied disturbances that are considered here, and for constant crack tip velocities, the particle velocity and $\text{?t}{}_{\mathrm{\theta z}}$ are functions of $\text{r}\text{t}$ and θ only, which allows Chaplygin's transformation and conformai mapping to be used. The theory of analytic functions is then used. For various values of the crack propagation velocity, the dependence of the elastodynamic stress intensity factor, and energy flux into the crack tip, on the wedge angle 2κπ is investigated.     

Annular punch on an elastic layer overlying an elastic foundation

The problem solved here is the axisymmetric mixed boundary value problem of the isotropic homogeneous theory of elasticity, in which the normal displacement is specified inside an annular area $\text{a ≤ r ≤ b}$, the normal stress is zero in $\text{r < a, r \# b}$ and the shearing stress is zero on the whole face $\text{z = ?h}$, the upper face of the elastic layer; the continuity of the normal and radial displacements and the normal and shearing stresses is assumed at the interface $\text{z = 0}$ between the elastic layer and the elastic foundation having different elastic constants. The problem is reduced to the solution of a Fredholm integral equation of the first kind. The Fredholm integral equation is further put in terms of four simultaneous Fredholm integral equations of the second kind in four unknown functions. The iterative solution of these integral equations has been obtained for $\text{epsi = b/h ? 1}$, and $\text{λ = a/b ? 1}$ for the case of an annular cylindrical punch. The expressions for the normal stress $\text{σ}{}_{\mathrm{zz}}\text{(r, ?h)}$ for $\text{a ≤ r ≤ b}$ and the total load $\text{P}$ on the punch have been obtained.     

Growth of fatigue cracks in metals and polymers

Empirical data on the propagation of tensile fatigue cracks in metals and thermoplastics have been examined. It was found that a cyclic crack propagation relationship, based on the stress intensity factor concept, exists which can be successfully utilised for both types of materials.The proposed equation has a form $\text{/.a}{}_{\mathrm{x}}\text{= MA}{}^{\mathrm{n}}$ where $\text{A}$ is a function of$\text{ΔK}$ and mean $\text{K}$. The analysis of results suggests that this equation incorporating the influence of mean stress intensity factor provides an excellent fit to the investigated data. The possible modified forms of such a relationship in terms of strain energy release rate, the crack tip yielding and the crack opening displacement concepts are also indicated.     

Impact response of a finite crack in a finite strip under anti-plane shear

The response of a through-the thickness crack with finite dimensions to impact in a finite elastic strip is investigated in this study. The elastic strip is assumed to be subjected to anti-plane shear deformation. Laplace and Fourier transform were used to formulate the mixed boundary value problem. The dynamic stress intensity factor and crack opening displacement are obtained as a function of time and the strip width to crack length ratio, h/a. The results indicate that the intensity of the crack-tip stress field reaches a peak very quickly and then decreases in magnitude oscillating about the static value. In general, the dynamic stress intensity factor is higher for small $\text{h/a}$. Similar behavior has also been found for the crack surface displacement.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

A fracture mechanics approach to high temperature fatigue crack growth in Udimet 700

Crack growth behavior under high temperature fatigue in Udimet 700 has been analyzed using both linear and non-linear elastic fracture mechanics concepts. It is shown that crack growth data for various loads in a compact tension specimen correlate well with the stress intensity factor, even at temperatures as high as 850°C. Using these results, a self consistent procedure has been developed for the determination of the $\text{J-}\text{integral}$ parameter under load-controlled fatigue and is shown to be compatible with data based on the stress intensity factor. The spread in the crack growth data is smaller in terms of $\text{J-}\text{integral}$ as compared to stress intensity or crack opening displacement parameters. Also based on a detailed fractographic analysis, it is suggested that the micromechanism of crack growth in Stages I and II is the environmentally assisted cleavage process, whereas in Stage III creep assisted crack growth processes are superimposed on the cleavage mode of crack growth. Effects of stress and temperature on the fatigue crack growth behavior of the Udimet alloy are discussed in detail.     

Vertical crack inside a semi-infinite plane

In this paper we consider the problem of determining the stress-intensity factors and the crack energy in a semi-infinite plane containing an inside crack perpendicular to the straight boundary of the plane. By the use of Mellin transform, we reduce the problem to solving a single singular integral equation. Approximate solution of the integral equation is obtained as a series of Chebyshev polynomials of the first kind. The coefficients $\text{B}{}_{\mathrm{n}}$ of the series are determined from a system of linear algebraic equations. Expressions for the stress-intensity factors at the edges of the crack, the shape of the crack and the crack energy are derived in terms of the coefficients $\text{B}{}_{\mathrm{n}}$. The numerical values of these quantities have been displayed graphically for three particular cases.     

Interaction of shear waves with a griffith crack situated in an infinitely long elastic strip

This paper deals with the propagation of shear waves in a wave guide which is in the form of an infinite elastic strip with free lateral surfaces. This strip contains a Griffith crack. An integral transform method is used to find the solution of the equation of motion from the linear theory for a homogeneous, isotropic elastic material. This method reduces the problem into an integral equation. It has been observed that only shear waves with frequencies less than a parameter-value, depending on the width of the wave guide, can propagate. The integral equation is solved numerically for a range of values of wave frequency and the width of the strip. These solutions are used to calculate the dynamic stress intensity factor, displacement on the surface of the crack and crack energy. The results are shown graphically.     

On crack opening stress level in fatigue by Eddy current technique

Linear elastic fracture mechanics relates fatigue crack growth with the stress intensity factor at the crack tip. Presence of residual deformations at the tip of a fatigue crack reduces the crack tip stress intensification such that effective stress intensity range $\text{ΔK}{}_{\mathrm{e}}\text{= U · ΔK}$. In this paper use of eddy current technique is exhibited to find the values of test value of effective stress range factor $\text{U}{}_{\mathrm{test}}$. A reasonable comparison between computed and experimental results of $\text{U}{}_1$ and $\text{U}{}_{\mathrm{test}}$ on two Al alloys 6061-T6 and 6063-T6 has recommended the Eddy Current Technology for finding out the values of crack opening stress level under given loading conditions.     

Thermal stresses near a penny-shaped crack in an elastic sphere embedded in an infinite elastic space

This paper gives an analysis of the distribution of thermal stresses in a sphere which is bonded to an infinite elastic medium. The thermal and the elastic properties of the sphere and the elastic infinite medium are assumed to be different. The penny-shaped crack lies on the diametral plane of the sphere and the centre of the crack is the centre of the sphere. By making a suitable representation of the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Using suitable solution of the thermoelastic displacement differential equation, the problem is then reduced to the solution of a Fredholm integral equation, in which the solution of the earlier integral equation arising from heat conduction problem occurs as a known function. Numerical solutions of these two Fredholm integral equations are obtained. These solutions are used to evaluate numerical values for the stress intensity factors. These values are displayed graphically.     

A griffith crack problem for an inhomogeneous elastic material

Summary This paper examines the problem of a Mode I crack in a nonhomogeneous elastic medium. It is assumed that the shear modulus varies exponentially with the coordinate perpendicular to the plane of the crack. The problem is reduced to a Fredholm integral equation and in terms of its solution the normal components of stress and displacement are described. Expressions are also derived for the stress intensity factor and the crack energy. The effect of the inhomogeneity is examined and comparisons made with the corresponding results for the homogeneous material.     

A study of crack closure in fatigue

Crack closure phenomenon in fatigue was studied by using a Ti-6Al-4V titanium alloy. The occurrence of crack closure was directly measured by an electrical potential method, and indirectly by load-strain measurement. The experimental results showed that the onset of crack closure depends on both the stress ratio, $\text{R}$, and the maximum stress intensity factor, $\text{K}{}_{\max }$. Crack closure was not observed for stress ratio, $\text{R}$, greater than 0.3 in this alloy.A two-dimensional elastic model was used to explain the behavior of the recorded load-strain curves. Closure force was estimated by using this model. Based on the estimated closure force, the crack opening displacement was calculated. This result showed that onset of crack closure detected by electrical-potential measurement and crack-opening-displacement measurement is the same.The implications of crack closure on fatigue crack are considered. The experimental results show that crack closure cannot fully account for the effect of stress ratio, $\text{R}$, on crack growth, and that it cannot be regarded as the sole cause for delay.     

Near-tip behavior of a crack in a plane anisotropic elastic body

The asymptotic form of the stress and displacement components near the tip of a straight crack in a generally rectilinear anisotropic plane elastic body are resolved. As in the isotropic analysis, the solutions for the stresses display a $\text{r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ dependence, where r is the distance from the tip, while the angular dependence depends upon the anisotropy in a complicated way. The effect of some special anisotropies upon these solutions is fully explored. Finally, these solutions are used to solve the problem of a finite length straight crack in an anisotropic elastic plane when uniform stresses are applied far from the crack. This solution includes obtaining the stress intensity factors, and the nature and magnitude of the crack face displacements.     

Mean stress effects on fatigue crack growth and failure in a rail steel

Over a limited range, the effect of mean stress has been studied on fatigue crack propagation and on the critical fatigue crack size associated with sudden fast fracture in centre-notched plate specimens of a rail steel under pulsating loading. The results have been presented in terms of the stress intensity factor range $\text{ΔK}$ and the ratio $\text{R}$ of the minimum to maximum stress. Increasing $\text{R}$ was found to both accelerate cracking and reduce the critical crack size at instability. The data have been correlated with three crack growth equations currently used in the literature and it was found that the equation of Forman et al. relating crack growth rate to $\text{ΔK}$ and $\text{R}$ gave the best fit. This equation was used to predict life in the finite range of the S-N curve. Fractographic examination revealed that the fracture surfaces were complex and a number of fracture modes contributed to cracking.     

Diffraction of SH waves by Griffith cracks or rigid strips lying in a laterally and vertically non-homogeneous infinite medium

The problem of diffraction of obliquely incident SH waves by one Griffith crack in an infinite non-homogeneous elastic medium is solved. The problems of diffraction of normally incident SH waves by two coplanar cracks or by two coplanar rigid strips in an infinite non-homogeneous elastic medium are also solved. Approximate expressions are derived for the displacement component and the stress intensity factors. The modulus of rigidity (μ) and the density (ρ) of the material are assumed to vary both in the horizontal (x) and vertical (z) directions, and the ratio $\text{μ}\text{ρ}$ is assumed to be constant. The numerical values of the stress-intensity factors are displayed graphically to show the effect of non-homogeneity of the material. The results for the corresponding problems for an isotropic, homogeneous, elastic medium are derived as particular cases.     

Stress intensity factor at a bilaterally-bent crack in the bending problem of thin plate

An infinite plate with an asymmetric bilaterally-bent crack is analyzed as a bending problem of a thin plate. Stress distributions and stress intensity factors are obtained for some angles and length of bent crack. These are obtained for the some Poisson's ratio. Influence of the initial crack width on the stress intensity factor are also investigated. Three loading conditions are taken into consideration: uniform out of plane bending at infinity in the $\text{x}$ and $\text{y}$ directions and uniform out of plane twist. The rational mapping function in the form of a sum of fractional expressions and the complex variable method are used for the analysis.     

Mixed mode fatigue crack growth predictions

This work is aimed at developing a predictive capability for the quantitative assessment of crack growth under fatigue loadings. The crack growth rate relation, $\text{Δa}\text{ΔN}$, may involve all three stress intensity factors k₁-k₃ such that the direction of crack growth may not be known in advance and must be predicted from a preassumed criterion. In principle, both the stress amplitude and the mean stress level should be included in the original expression for $\text{Δa}\text{ΔN}$.The strain energy density factor range, ΔS, is found to be a convenient parameter for predicting fatigue crack growth and can be applied expediently to examine the combined influence of crack geometry, complex loadings and material properties. Assumed is the accumulation of energy, $\text{ΔW}\text{ΔV}$, stored in an element ahead of the crack which triggers subcritical crack growth upon reaching a number of loading cycle, say ΔN. The proposed $\text{δa}\text{ΔN}$ relationship includes both the stress amplitude and mean stress effects.     

The elastic interaction between solute atoms and stress field at the tip of cracks and notches

As for a mechanical interaction between a general stress singularity and an-environmental factor (e.g. hydrogen) which has influence on delayed failure characteristics of materials, it has not hitherto been analysed, although it is an important problem. In the present article an analytical basis is given for the problem of elastic interaction between solute atom as an environmental factor accelerating delayed failure and stress field around a crack and further a general V-shaped notch.The result of analysis shows that the number of solute atoms which accumulate to crack tip due to elastic interaction is proportional to a parameter. $\text{ρ}{}_0\text{(}\text{DKt}\text{kT}\text{)}{}^{\mathrm{<mtext>4</mtext><mtext>5</mtext>}}$ where K is stress intensity factor,ρ₀ initial uniform density of the solute atom, D diffusion constant, t time, k Boltzman's constant and T absolute temperature. It is further shown that the abovementioned parameter is generalized to $\text{ρ}{}_0\text{(}\text{Dkt}\text{kT}\text{)}{}^{\mathrm{<mtext>2</mtext><mtext>(2+q)</mtext>}}$ for the number of what accumulate to the apex of a general V-shaped notch due to elastic interaction where k is a factor representing a stress singularity of the notch apex and q is a constant related to apex angle a.The above analysis indicates that it is possible to describe development of delayed failure in terms of a parametre of fracture mechanics K, when viewed from a standpoint inclusive of diffusion and mechanical processes, since it is shown that the rate of concentration to crack tip of solute atoms as an environmental factor is uniquely determined by a factor $\text{ρ}{}_0\text{(}\text{DK}\text{kT}\text{)}{}^{\mathrm{<mtext>4</mtext><mtext>5</mtext>}}$.     

Elastodynamic near-tip fields for a rapidly propagating interface crack

The elastodynamic stress field near a crack tip rapidly propagating along the interface between two dissimilar isotropic elastic solids is investigated. Both anti-plane and in-plane motions are considered. The anti-plane displacements and the in-plane displacement potentials are sought in the separated forms $\text{r}{}^{\mathrm{q}}\text{F(θ)}$, $\text{r}$ and θ being polar coordinates centered at the moving tip. The mathematical statement of the problem reduces to a second-order linear ordinary differential equation in θ, which can be solved analytically. Formulation of the boundary and interface conditions leads to an eigenvalue problem for the singularity exponent $\text{q}$. For the in-plane problem, root $\text{q}$ is found to be complex. Thus, the stresses exhibit violent oscillations within a small region around the crack tip, and the solutions have physical significance only outside this region. The angular stress distributions are plotted for various crack speeds, and it is found that at a high enough speeds the direction θ of maximum stress moves out of the interface. This result indicates that a running interface crack may move into one of the adjoining materials.     

The inclined crack under biaxial load

The problem of a sheet with an inclined crack subject to a system of biaxial loads is examined in a manner similar to that used in our previous publications. The combined effect of load biaxiality and crack orientation on $\text{K}{}_{\mathrm{l}}$, $\text{K}{}_{\mathrm{II}}$, maximum shear, anile of initial crack extension, elastic strain energy density, as well as on the local strain energy rate is made explicit. Once again it is shown that use of the so-called “singular solution” for expression of the local crack-tip elastic stress and displacements is not sufficient to give adequate account of the biaxial load effect.