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.     

A cumulative model of fatigue crack growth

A model of fatigue crack growth based on an analysis of elastic/plastic stress and strain at the crack tip is presented. It is shown that the fatigue crack growth rate can be calculated by means of the local stress/strain at the crack tip. The local stress and strain calculations are based on the general solutions given by Hutchinson, Rice and Rosengren. It is assumed that a small highly strained area existing at the crack tip is responsible for the fatigue crack growth. It is also assumed that the fatigue crack growth rate depends mainly on the width, x¹, of the highly strained zone and on the strain range, $\text{Δ?}{}^1$, within the zone. A relationship between stress intensity factor K and the local strain and stress has been developed. It is possible to calculate the local strain for a variety of crack problems. Then, the number of cycles N¹ required for material failure inside the highly strained zone is calculated. The fatigue crack growth rate is calculated as the ratio $\text{x}{}^1\text{N}{}^1$.The calculated fatigue crack growth rates were compared to the experimental ones. Two alloys steels and two aluminium alloys were analyzed. Good agreement between experimental and theoretical results is obtained.     

Elastodynamic near-tip fields for a crack propagating along the interface of two orthotropic solids

The elastodynamic stress field near a crack tip rapidly propagating along the interface between two dissimilar orthotropic elastic solids is solved numerically, for in-plane motion. The cartesian displacements are sought in the separated forms, $\text{r}{}^{\mathrm{p}}\text{U(θ)}$ and $\text{r}{}^{\mathrm{p}}\text{V(θ)}$, $\text{r}$ and θ being polar coordinates centered at the moving tip. This reduces the mathematical statement of the problem to two complex second-order linear ordinary differential equations for complex functions $\text{U(θ)}$ and $\text{V(θ)}$. By means of the finite difference method, a matrix eigenvalue problem of the type $\text{ΣA}{}_{\mathrm{ij}}\text{(p)X}{}_{\mathrm{j}}\text{= 0}$, is obtained where $\text{A}{}_{\mathrm{ij}}\text{(p)}$ are polynomials of the complex variable $\text{p}$ and $\text{X}{}_{\mathrm{j}}$, are complex unknowns. An iterative numerical scheme for determining $\text{Im(p)}$ is developed and the roots $\text{p}$ as well as angular stress and displacement distributions are calculated and plotted for various material combinations. Comparison with exact solutions for the case of dissimilar Isotropie solids indicates good accuracy of the numerical solution. The orthotropic nature of the materials is shown to have a significant effect on stress maximums.     

A cumulative model of fatigue crack growth and the crack closure effect

A model of fatigue crack growth based on an analysis of elastic/plastic stress and strain at the crack tip is presented. It is shown that the fatigue crack growth rate can be calculated using the local stress/strain at the crack tip by assuming that a small highly strained area $\text{x}{}^1$, existing at the crack tip, is responsible for the fatigue crack growth, and that the fatigue crack growth may be regarded as the cumulation of successive crack re-initiations over a distance $\text{x}{}^1$. It is shown that crack closure can be modelled using the effective contact zone g behind the crack tip. The model allows the fatigue crack growth rate over the near threshold and linear ranges of the general da/dN versus ΔK curve to be calculated. The fatigue crack growth retardation due to overload and fatigue crack arrest can also be analysed in terms of g and $\text{x}{}^1$.Calculated fatigue crack growth rates are compared with experimental ones for low and high strength steel.     

Dynamic stress distribution around the tip of a running crack

The stress distribution is obtained around the tip of a crack running in a brittle material. The stresses are written as the sum of the associated static solution and the wave-effect terms which depend upon the crack speed. The results obtained clearly reduce to the associated static solutions if the crack speed vanishes.Near the tip of the crack, the dynamic stress-intensity factor for the circumferential stress, $\text{σ}{}_{\mathrm{\theta \theta }}$, is written as the product of the associated static stress-intensity factor and the dynamic correction factor which is a nondimensional function of the crack speed, $\text{V}$, the angle from the crack plane, θ, and Poisson's ratio, ν. The value of the correction factor is computed for various values of $\text{V}$ and θ at $\text{ν = 0.25}$. It is shown that the maximum tensile value of $\text{σ}{}_{\mathrm{\theta \theta }}$, occurs on the crack plane for $\text{V}$ less than 0.7 time shear wave speed, $\text{c}{}_2$, and suddenly shifts to the plane of $\text{θ = 55°}$ for $\text{V}$ slightly larger than 0.7 $\text{c}{}_2$. For $\text{V > 0.7}$$\text{c}{}_2$, the angle θ for the maximum $\text{σ}{}_{\mathrm{\theta \theta }}$, θ being larger than 55°, varies continuously with the crack speed, $\text{V}$. The results obtained are used to discuss the growth of branching crack.     

The form of crack tip plastic zones

The “radius” of the plastic zone at a crack tip is a parameter that has numerous applications in fracture mechanics. However, attention is drawn here to the confusion that is apparent, even in text-books, concerning the calculation of the plastic zone “radius” under plane strain conditions. The aim of this work has been to resolve this point, to determine the actual shape and size of the zone and to investigate the influence of stress state and other factors.The plastic zone dimensions have been simply calculated, over a range of values of Poisson's ratio, for isotropic materials subjected to loading under plane stress and plane strain conditions; the analysis has been further extended to cover some effects of anisotropy. It has been demonstrated that, for isotropic materials, the maximum extent of the plastic zone directly ahead of, and in the plane of, a crack is ($\text{K}{}_{\mathrm{I}}\text{/Y)}{}^2\text{18π}$ under plane stress loading and is ($\text{K}{}_{\mathrm{I}}\text{/Y)}{}^2\text{18π}$ under plane strain loading. This latter result is smaller, by a factor of $\text{1}\text{3}$ than the plastic zone “radius” under plane strain conditions that is widely quoted in fracture mechanics texts. That “radius”, ($\text{K}{}_{\mathrm{I}}\text{/Y)}{}^2\text{6π}$ is, in fact, the maximum size of the zone parallel to, but not in, the plane of the crack, if Poisson's ratio is taken to be $\text{1}\text{3}$.A lower value of Poisson's ratio or an increased material anisotropy can lead to an enlarged plastic zone; this latter conclusion suggests that test-pieces for valid fracture toughness measurements on anisotropic materials could be required to be larger than defined in the relevant British Standard.     

Incipient fracture angle,fracture loci and critical stress for mixed mode loading

A prediction of the direction of incipient crack growth in brittle-like materials and the associated fracture loci under mixed mode loading is proposed. It is postulated that the direction of unstable crack propagation is determined by the “weakest” near-tip element defined as the one which would relax maximum potential energy upon prospective crack extension. Starting from the energy rate principle of crack extension (Eshelby energy-momentum tensor and Rice $\text{J-}\text{internal}$ vector) it is deduced that a crack will extent in the direction along which the following stress criterion is satisfied, $\text{(δ}{}_{\mathrm{\theta \theta }}{}^2\text{? δ}{}_{\mathrm{rr}}{}^2\text{) →}\text{maximum (for}\text{δ}{}_{\mathrm{\theta \theta \; >\; 0)}}$ The fracture angle in pure Mode II (70.4° away from the original straight path) is shown to be unstable in the sense that any slight tension along the crack (non-singular at the crack tip) affects considerably (up to 22%) the directionality of crack extension. It appears to be sensitive to the extent of the near-tip zone ($\text{r}{}_0$) in which linear elasticity does not hold and the non-singular stress term (squared).The fracture loci in mixed mode loading (generated by projecting the $\text{J-}\text{integral}$ vector along the prospective fracture path and letting this scalar function attain a critical value) is quadratic in $\text{K}{}_1$ and $\text{K}{}_2$ with an interactive cross product term $\text{K}{}_1\text{× K}{}_2$.The suggested criterion with its implication in predicting critical fracture load, exhibits behavior which is consistent with experimental observations collected from several sources. The common and uncommon features with respect to other known criteria are compared and discussed.     

Stress in the vicinity of a griffith crack at the interface of a layer bonded to a half plane—an approximate method

Approximations to the stress field in the vicinity of a Griffith crack located at the interface of a layer bonded to a dissimilar half plane are determined. A systematic use of Fourier transforms reduces the problem to that of solving a set of simultaneous dual integral equations with trigonometric kernels and weighting functions. This latter problem is reduced to the solution of an uncoupled pair of singular integral equations. An approximate technique using Legendre polynomial expansions is discussed. The analysis shows that when a constant pressure is applied to the faces of the crack, the stress components have the distinctive oscillatory singularities at the crack tip. Expressions up to the order of $\text{h}{}^{\mathrm{?4}}$, where $\text{h}$ is the thickness of the layer and is much greater than 1, are derived for the stress components.     

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.     

Torsion of a nonhomogeneous transversely isotropic half space

The statical Reissner Sagoci problem for a transversely isotropic, nonhomogeneous elastic solid is investigated. The modulus of rigidity of the medium is assumed to be variable as a power of the radial coordinate in the form $\text{r}{}^{\mathrm{\beta }}\text{(β ? 0)}$. The expressions for stresses, displacement and torque are given.     

Cyclic analysis of a propagating crack and its correlation with fatigue crack growth

Stress and strain field of a propagating fatigue crack and the resulting crack opening and closing behavior were analysed. It was found that a propagating fatigue crack was closed at tensile external loads due to the cyclically induced residual stresses. Strain range value $\text{Δ?}{}_{\mathrm{y}}$ in the vicinity of the crack tip was found to be closely related with the effective stress intensity factor range $\text{ΔK}{}_{\mathrm{eff}}$ which was determined on the basts of the analytical crack opening and closing behavior at its tip. Application of this analysis to the non-propagating fatigue crack problem and the fatigue crack propagation problems under variable stress amplitude conditions revealed that both $\text{Δ?}{}_{\mathrm{y}}$ and $\text{ΔK}{}_{\mathrm{eff}}$ were essential parameters governing fatigue crack growth rate.     

Fatigue crack propagation from a crack inclined to the cyclic tensile axis

Cyclic stresses with stress ratio R = 0.65 were applied to sheet specimens of aluminium which have an initial crack inclined to the tensile axis at angles of 30°, 45°, 72° or 90°. The threshold condition for the non-propagation of the initial crack was found to be given by a quadratic form of the ranges of the stress intensity factors of modes I and II. The direction of fatigue crack extension from the inclined crack was roughly perpendicular to the tensile axis at stress ranges just above the threshold value for non-propagation. On the other hand, at stress ranges 1.6 times higher than the threshold values the crack grew in the direction of the initial crack. The rate of crack growth in the initial crack direction was found to be expressed by the following function of stress intensity factor ranges of mode I, K₁, and mode II, K₂: $\text{dc}\text{dN}\text{= C(K}{}_{\mathrm{eff}}\text{)sum}$, where $\text{K}{}_{\mathrm{eff}}\text{= [K}{}_1{}^4\text{+ 8K}{}_2{}^4\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. This law was derived on the basis of the fatigue crack propagation model proposed by Weertman.     

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.     

A multilayer hybrid-stress finite element analysis of a through-thickness edge crack in A ± 45° laminate

A solution is given for the three-dimensional stress field near a through-thickness edge crack in a thin ± 45° laminate having elastic ply moduli typical of graphite/epoxy. The stress distribution was obtained by a three-dimensional multilayer finite element analysis based on the hybrid stress model, formulated through the minimum complementary energy principle. The results indicate that the in-plane stresses of each individual ply follow the classical $\text{1}\text{√r}$ stress singularity, but that the shape of isostress contours in the crack tip region is strongly distorted from predictions based on two-dimensional anisotropic fracture mechanics theory. The interlaminar shear stresses increase rapidly as the crack tip is approached, but are restricted to a local region around the crack tip and flanks. The interlaminar normal stress is assumed to be negligible in the formulation of the analysis.     

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 $\text{ρ}{}_0\text{(}\text{DK}\text{kT}\text{)}{}^{\mathrm{<mtext>4</mtext><mtext>5</mtext>}}$.     

Studies on crack growth rate under high temperature creep,fatigue and creep-fatigue interaction—II: On the role of fracture mechanics parameter aeffσg

For high temperature creep, fatigue and creep-fatigue interaction, several authors have recently attempted to express crack growth rate in terms of stress intensity factor $\text{K}{}_{\mathrm{I}}\text{= α}\text{aσ}{}_{\mathrm{g}}$, where a is the equivalent crack length as the sum of the initial notch length a₀ and the actual crack length $\text{a}{}^1$, that is, $\text{a = a}{}_0\text{+ a}{}^1$. On the other hand, it has been shown by Yokobori and Konosu that under the large scale yielding condition, the local stress distribution near the notch tip is given by the fracture mechanics parameter of $\text{aσ}{}_{\mathrm{g}}\text{?(σ}{}_{\mathrm{g}}$), where a is the cycloidal notch length, σ_g is the gross section stress and $\text{?(σ}{}_{\mathrm{g}}$) is a function of σ_g. Furthermore, when the crack growth from the initial notch is concerned, it is more reasonable to use the effective crack length a_eff taking into account of the effect of the initial notch instead of the equivalent crack length a. Thus we believe mathematical formula for the crack growth rate under high temperature creep, fatigue and creep-fatigue interaction conditions may be expressed at least in principle as function of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}\text{, σ}{}_{\mathrm{g}}$ and temperature.In the present paper, the geometrical change of notch shape from the instant of load application was continuously observed during the tests without interruption under high temperature creep, fatigue and creep-fatigue interaction conditions. Also, the effective crack length a_eff was calculated by the finite element method for the accurate estimation of local stress distribution near the tip of the crack initiated from the initial notch root. Furthermore, experimental data on crack growth rates previously obtained are analysed in terms of the parameter of $\text{a}{}_{\mathrm{eff}}\text{σ}{}_{\mathrm{g}}$ with gross section stresses and temperatures as parameters, respectively.     

On a hybrid method to determine strain fields around propagating cracks

The moving singularity of the crack tip in a plane-stress plate causes a highly dynamic stress field of varying intensity with time, throughout the period of the propagation of the crack. This dynamic stress field results in a considerable change of the mechanical and optical properties of a strain-rate dependent material. An analysis of this varying dynamic stress field was presented in this paper which contradicts assumptions and simplifications introduced in a previous paper [7], referring to the same problem. For the experimental determination of the $\text{K}{}^{\mathrm{d}}{}_{\mathrm{I}}\text{-}\text{factor}$ the optical method of the dynamic caustics was utilized in combination with a high-speed camera and a comparison was sketched between the possibilities of this method and the strain-gauge method used in Ref. [7].     

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.     

The stress field near the blunt crack tip and the fracture criterion

The asymptotic expansion solution containing two terms for the stress field near the blunt crack tip is obtained. It is proposed that the slit be divided into the ideal crack, blunt crack I, blunt crack II and the notch in accordance with the geometrical structure of the slit tip. Whether the blunt crack can be considered as the ideal crack will depend mainly on the following three factors: $\text{√}\text{2R}{}_0\text{C}$, $\text{√}\text{R}{}_0\text{r}{}_{\mathrm{c}}$ and the profile of the crack. In this paper, the influence of the crack tip radius on the fracture criterion is studied and it is shown that the classical strength theories belong to the unconditional extremum criteria while the S criterion, etc. in fracture mechanics belong to the conditional extremum criteria. A modified maximum tension stress theory is developed, in which the fracture theories of the crack and the notch can be roughly unified.     

Strain rate effects on viscoelastic crack bifurcation

The effects of applied strain rate on the viscoelastic crack bifurcation phenomenon in Polymethyl Methacrylate (PMMA) were investigated. It was still verified that the product $\text{σ}{}_{\mathrm{f}}\text{C}{}_{\mathrm{b}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ was constant, as was already observed by Congleton and Petch, and Anthony, Chubb and Congleton, for brittle elastic materials, for any strain rate, where σ_f = the gross fracture stress and C_b= the main crack length until the bifurcation starts. However, it was found that the higher strain rate increases the main crack length C_b resulting in the decrease in the gross fracture stress σ_f and vice versa. This might be interpreted that the higher stress concentration at the initiation crack tip, which is realized by becoming more brittle due to the higher strain rate owing to the predominance of the elastic element in the viscoelastic material, decreases the gross fracture stress leading to the longer main crack length.