Showing papers on "Stress concentration published in 1977"
TL;DR: In this paper, an integral equation for the displacement and stress fields due to propagation of two-dimensional shear cracks in an infinite, homogeneous medium which is linearly elastic everywhere off the crack plane is found.
Abstract: Summary
We present a numerical technique to determine the displacement and stress fields due to propagation of two-dimensional shear cracks in an infinite, homogeneous medium which is linearly elastic everywhere off the crack plane. Starting from the representation theorem, an integral equation for the displacements inside the crack is found. This integral equation is solved by a method proposed by Hamano for various initial and boundary conditions on the crack surface. We verified the accuracy of our numerical method by comparing it with the analytical solution of Kostrov, and the numerical solution of Madariaga. A critical stress jump across the tip of a crack (between a grid-point inside the crack and a neighbouring point out-side the crack) is used as our fracture criterion. We find that our critical stress jump is the finite difference approximation to the critical stress-intensity factor used in Irwin's fracture criterion.
For an in-plane shear crack starting from the Griffin critical length and controlled by the above fracture criterion, the propagation velocity of the crack-tip is found to be sub-Rayleigh or super-shear depending on the strength of the material (given by the critical stress jump) and the instantaneous length of the crack. In fact, the crack-tip velocity may even reach the P-wave velocity for low-strength materials. Additionally we find that once the crack starts propagating, it accelerates rapidly to its terminal velocity, and that the average rupture velocity over an entire length of fault cannot be much smaller than the terminal velocity, for smooth rupture propagation.
381 citations
TL;DR: In this paper, the authors present a theory for the radiation of high-frequency waves by earthquake faults, where they model the fault as a planar region in which the stress drops to the kinematic friction during slip and show that the high frequencies originate from the stress and slip velocity concentrations in the vicinity of the fault's edges.
Abstract: Summary. We present a theory for the radiation of high-frequency waves by earthquake faults. We model the fault as a planar region in which the stress drops to the kinematic friction during slip. This model is entirely equivalent to a shear crack. For twodimensional fault models we show that the high frequencies originate from the stress and slip velocity concentrations in the vicinity of the fault’s edges. These stress concentrations radiate when the crack expands with accelerated motion. The most efficient generation of high-frequency waves occurs when the rupture velocity changes abruptly. In this case, the displacement spectrum has an u-’ behaviour at high frequencies. The excitation is proportional to the intensity of the stress concentration near the crack tips and to the change in the focusing factor due to rupture velocity. We extend these two-dimensional results to more general three-dimensional fault models in the case when the rupture velocity changes simultaneously on the rupture front. Results are similar to those described for twodimensional faults. We apply the theory to the case of a circular fault that grows at constant velocity and stops suddenly. The present theory is in excellent agreement with a numerical solution of the same problem. Our results provide upper bounds to the high-frequency radiation from more realistic models in which rupture velocity does not change suddenly. The u-’ is the minimum possible decay at high frequencies for any crack model of the source.
339 citations
TL;DR: In this paper, the displacements and stresses are determined in an elastic plate, weakened by a sharpedged line crack, and the plate is loaded by a uniform tension perpendicular to the line of the crack at infinity.
Abstract: S olutions are presented for the one- and two-dimensional Griffith crack problems in non-local elasticity. The displacements and stresses are determined in an elastic plate, weakened by a sharpedged line crack. The plate is loaded by a uniform tension perpendicular to the line of the crack at infinity. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis.
325 citations
Book•
01 Jul 1977
TL;DR: In this paper, the authors discuss the concepts of force, stress, strain, and displacement in the theory of elasticity, and propose a finite element model for stress and strain analysis.
Abstract: 1 Basic Concepts of Force, Stress, Strain, and Displacement 2 Stress and Strain. Transformations, Equilibrium, and Compatibility3 Fundamental Formulations of Stress, Strain, and Deflection4 Concepts from the Theory of Elasticity5 Topics from Advanced Mechanics of Materials6 Energy Techniques in Stress Analysis7 Strength Theories and Design Methods8 Experimental Stress Analysis9 Introduction to the Finite Element Method10 Finite Element Modeling TechniquesAppendicesA SI and USCU ConversionsB Properties of Cross SectionsC Beams in BendingD Singularity FunctionsE Principal Second-area MomentsF Stress Concentration FactorsG Strain Gage Rosette EquationsH Corrections for Transverse Sensitivity of Strain GagesI Matrix Algebra and Cartesian Tensors
287 citations
TL;DR: In this paper, the results of experiments on three isotropic and one anisotropic materials are discussed and compared with the criteria proposed up to the present, and a new criterion is proposed, which coincides with the Gough's empirical formula for the brittle materials under combined stress.
Abstract: Fatigue crack initiates in the slip band and exists also in it near fatigue limit; many slip bands are apt to appear in the direction of the maximum shearing stress; crack propagates by the normal tensile stress; the maximum shearing stress on a plane at fatigue limit is reduced by the effect of the normal stress on the same plane. From these results of the experiment, a new criterion is proposed, which coincides with the Gough's empirical formula for the brittle materials under combined stress. As the plane of the maximum shearing stress is varied by the various combination of torsion and bending, the isotropic material should be used in the combined stress experiment. In this paper, the results of experiments on three isotropic and one anisotropic materials are discussed and compared with the criteria proposed up to the present.
287 citations
TL;DR: The brittle fracture propensity was shown to occur during the decompression of large compacts made by nonisostatic compression and seems to be related to the ability to relieve stresses by plastic deformation.
242 citations
TL;DR: In this article, small axially loaded smooth specimens of A533B steel subjected to strain cycling fatigue at large plastic strains were reported for the first time, and the growth rates for different strain levels were correlated in fracture mechanics fashion by the J integral concept.
Abstract: Crack growth data are reported for small axially loaded smooth specimens of A533B steel subjected to strain cycling fatigue at large plastic strains. Surface crack lengths were monitored using cellulose acetate replicas, and occasional specimens were broken open to determine crack depth. Experimental crack growth rates for different strain levels are correlated in fracture mechanics fashion by the J integral concept, with J values being estimated from stress-strain hysteresis loops. The crack growth rate data of this investigation are compared with previous data for the same material obtained from linear elastic fracture mechanics tests. It is suggested that research on the behavior of small cracks is fundamental to a better understanding of the fatigue process.
216 citations
TL;DR: In this paper, the authors defined equivalent cracks in notched and unnotched situations as cracks with equal growth rates, and the concept of notch contribution to crack length was introduced.
175 citations
TL;DR: In this article, a method for computing stress intensity factors for cracks embedded in structural details is described, which consists of adding to accepted solutions for cracks in finite plates and bodies of uniform contour a geometry correction factor which accounts for the stress gradient produced by the geometric discontinuity of the detail.
Abstract: A method for computing stress intensity factors for cracks embedded in structural details is described. It consists of adding to accepted solutions for cracks in finite plates and bodies of uniform contour a geometry correction factor which accounts for the stress gradient produced by the geometric discontinuity of the detail. This correction factor is determined by integrating away the stresses normal to the line where the crack is to be inserted. The method is applied to the case of a crack emanating from a circular hole in a plate, and the results are found to be in good agreement with Bowie's numerical solution. Values of the stress intensity factor for cracks emanating from spherical weld porosities, and part-through cracks at stiffeners and cover plate ends are computed.
172 citations
TL;DR: In this paper, a quasi-continuous evaluation of the load-deflection curve recorded during controlled fracture is obtained as a function of the achieved crack length, which may be regarded as the crack extension resistance of the material because the balance between the energy release rateg1 andR is maintained throughout the experiment.
Abstract: Sub-critical crack extension can readily be observed in controlled fracture tests in fourpoint bending. A natural crack of any desired lengthc which exceeds the notch depthc0 by the amount Δc =c −c0 can be introduced into bend specimens by stable crack propagation. The stress intensity factor to achieve Δc increases considerably with increasing Δc. In pre-cracked specimens the stress intensity factorKI0 to start the crack and the critical valueKIC strongly depend on the natural crack length Δc whereasKI0 andKIC are independent ofc0 in solely notched specimens. From a quasi-continuous evaluation of the load-deflection curve recorded during controlled fracture, the “differential work of fracture” can be obtained as a function of the achieved crack length. It may be regarded as the crack extension resistanceR of the material because the balance between the energy release rateg1 andR is maintained throughout the experiment. By that, a formal analogy to theR-curve concept of fracture mechanics is given. The steady increase ofR is explained by multiple crack formation and by the interference of the fracture surfaces due to the angular development of the crack front.
166 citations
Book•
01 Jan 1977
TL;DR: In this paper, a stress analysis of non-circular and thin-walled sections of a diaphragm is presented, along with contact stress, residual stress and stress concentrations.
Abstract: Introduction Notation Unsymmetrical bending Struts Strains beyond the elastic limit Rings, discs and cylinders subjected to rotation and thermal gradients Torsion of non-circular and thin-walled sections Experimental stress analysis Circular plates and diaphragms Introduction to advanced elasticity theory Introduction to the Finite Element Method Contact stress, residual stress and stress concentrations Fatigue, creep and fracture Miscellaneous topics Appendices Index.
TL;DR: In this article, the influence of mean stress and microstructure on fatigue crack growth near the threshold stress intensity for crack propagation, was investigated in humid air over a very wide spectrum of growth rates from 10/sup -8/ to 10/Sup -1/ mm/cycle.
Abstract: Fatigue crack propagation behavior of an ultra-high strength steel (300-M) has been investigated in humid air over a very wide spectrum of growth rates from 10/sup -8/ to 10/sup -1/ mm/cycle. Particular emphasis has been devoted to the influence of mean stress (or load ratio R = K/sub min//K/sub max/) and microstructure on fatigue crack growth near the threshold stress intensity for crack propagation, ..delta..K/sub 0/. Increasing the load ratio from R = 0.05 to 0.70 was found to lead to increased near-threshold growth rates, and a decrease in the threshold stress intensity. Similarly, increasing material strength, by varying the microstructure through quench and tempering and isothermal transformation, resulted in higher near-threshold growth rates, and a marked reduction of ..delta..K/sub 0/. These effects are contrasted with behavior at higher growth rates. The infuence of strength on ..delta..K/sub 0/ is rationalized in terms of the cyclic hardening or softening response of the material, and hence it is shown that cyclic softening can be beneficial to fatigue crack propagation resistance at very low growth rates. The results are discussed in the light of crack closure and environmental contributions to fatigue crack growth at low stress intensities.
TL;DR: In this paper, the influence of dynamic effects on the crack arrest process is investigated using a shadow optical technique in combination with a Cranz Schardin high-speed camera, and the results show that the dynamic stress intensity factor oscillates with decreasing amplitude around the static stress intensity at arrest.
Abstract: The influence of dynamic effects on the crack arrest process is investigated. For propagating and subsequently arresting cracks, actual dynamic stress intensity factors were measured applying a shadow optical technique incombination with a Cranz Schardin high-speed camera. The experiments were performed in wedge-loaded double-cantilever-beam (DCB) specimens machined from an epoxy resin (Araldite B). In the initial phase of crack propagation the measured dynamic stress intensity factors were found smaller; in the arresting phase, however, they were larger than the corresponding static values. After arrest the dynamic stress intensity factor oscillates with decreasing amplitude around the static stress intensity factor at arrest. Crack arrest toughness values determined according to a static analysis showed a dependence on the crack velocity prior to arrest, but the dynamic crack arrest toughness yielded a single value only, indicating that this quantity represents a true material property.
TL;DR: In this paper, the authors examined the stress field of a finite-width, symmetrical laminate when subjected to a uniform temperature change and employed an advanced sparse matrix solution technique to study the singular nature of stresses near the free-edge region.
Abstract: Thermally induced stresses in fiber composite laminates play an important role in the strength and failure of such laminates as load carrying members. When laminates have free edges, cutouts or holes, interlaminar stress concentration will usually develop near the free-edge region under service loads. An accurate evaluation of thermally induced edge stresses will further the understanding of the laminates' behavior. The present paper examines the stress field of a finite-width, symmetrical laminate when subjected to a uniform temperature change. A finite-element proce dure employing advanced sparse matrix solution technique is employed to study the singular nature of stresses near the free-edge region. Edge effects on the overall response of the laminate under thermal loading are discussed.
TL;DR: In this paper, the authors investigated the implications of subdividing the lithosphere into upper elastic and lower viscoelastic layers by finite-element analysis, and they showed that applying uniform horizontal boundary stresses at the ends of a lithospheric plate leads to amplification of the stress in the elastic layer by a factor about equal to the ratio of lithosphere to elastic layer thicknesses.
TL;DR: In this article, the fracture surface characteristics of five commercially available amorphous polymers [poly(methylmethacrylate) (PMMA), polycarbonate (PC), polyvinyl chloride (PVC), polystyrene (PS), and polysulphone (PSF) were studied to determine if common mechanisms of fatigue crack propagation prevail among these glassy polymers.
Abstract: Fatigue fracture surface characteristics of five commercially available amorphous polymers [poly(methylmethacrylate) (PMMA), polycarbonate (PC), poly(vinyl chloride) (PVC), polystyrene (PS), and polysulphone (PSF)] as well as bulk-polymerized PMMA prepared over a wide range of molecular weights were studied to determine if common mechanisms of fatigue crack propagation prevail among these glassy polymers. In those polymers with viscosity-average molecular weight ¯Mv≲2×105, the macroscopic appearance of the fracture surface showed the presence of a highly reflective mirror-like region which formed at low values of stress intensity and high cyclic test frequencies (∼100 Hz). The microscopic appearance of this region revealed that many parallel bands exist oriented perpendicular to the direction of crack growth and that the bands increase in size with ΔK. In all instances, the crack front advanced discontinuously in increments equal to the band width after remaining stationary for hundreds of fatigue cycles. Electron fractographic studies verified the discontinuous nature of crack extension through a craze which developed continuously with the load fluctuations. By equating the band size to the Dugdale plastic zone dimension ahead of the crack, a relatively constant yield strength was inferred which agreed well with reported craze stress values for each material. At higher stress intensity levels in all polymers and all values of ¯Mv, another series of parallel bands were observed. These were also oriented perpendicular to the direction of crack growth and likewise increased in size with the range in stress intensity factor, ΔK. Each band corresponded to the incremental advance of the crack during one load cycle, indicating these markings to be classical fatigue striations.
TL;DR: In this article, a method of calculating stress intensity factors for branched and bent cracks embedded in an infinite body has been developed, where the branches are always assumed to be sharp cracks and are modelled by dislocation distributions.
Abstract: A method of calculating stress intensity factors for branched and bent cracks embedded in an infinite body has been developed. The branches are always assumed to be sharp cracks and are modelled by dislocation distributions. The original crack may be either sharp or of elliptical cross-section with finite root radius. Hence, the method which has a precision ±2%, is also applicable to the study of crack branches emanating from elliptical holes and, approximately, also from notches. The following detailed calculations have been made assuming mode I loading: branched sharp crack with branches of equal and different length, bent sharp crack, and one and two crack branches emanating from the crack with a finite root radius. Bending of a sharp crack under mixed mode loading has also been studied. The criteria of maximum tensile stress and maximum energy release rate used in the study of direction of crack propagation are discussed.
TL;DR: In this paper, the authors investigated the propagation of fatigue cracks in notched and unnotched biaxially stressed plates and found that the rate of propagation is affected by both the stress field associated with the notch and the biaXial stress state of the bulk material.
Abstract: The propagation of fatigue cracks in notched and un-notched biaxially stressed plates is investigated. The rate of propagation is found to be affected by both the stress field associated with the notch and the biaxial stress state of the bulk material. It is found that the propagation rate of a crack from a notch may be predicted by the use of a theoretical notch contribution factor in conjunction with propagation data for a crack in un-notched material.
TL;DR: In this article, the authors investigated the effects of normal stress σx on the rate of propagation of a Stage II crack extending by a Mode I mechanism, which has different effects should it be tensile or compressive, cyclic or constant.
Abstract: Fatigue-crack propagation in biaxially stressed plates has been investigated. The normal stress σx, applied on a plane perpendicular to the crack front, affects the rate of propagation of a Stage II crack extending by a Mode I mechanism. This stress, which does not affect singularity conditions in a linear elastic fracture mechanics analysis, has different effects should it be tensile or compressive, cyclic or constant. The changes in crack-growth rate are related to changes in the size of the plastic shear ears at the tip of the crack. Fatigue-crack propagation is related to two parameters in the plane of the plate, the maximum shear stress range and the stress normal to the plane of maximum shear. Both these parameters will affect the crack-opening displacement and hence crack-growth rate.
TL;DR: In this paper, a computer simulation of the time dependent development of the plastic zone ahead of a crack loaded in uniform tension was performed, where the material was assumed to deform according to a creep law relating the local strain rate to the local stress.
Abstract: A computer simulation of the time dependent development of the plastic zone ahead of a crack loaded in uniform tension was performed. The material was assumed to deform according to a creep law relating the local strain rate to the local stress. The plastic zone was modelled by an array of edge dislocations coplanar with the crack. For a given time the stress was found to be uniform in a region ahead of the crack. This region increased and the local stress decreased with increasing time. The distribution of dislocations in the zone at a given time was found to be almost the same as that given by the Bilby, Cottrell and Swinden model (1963) if the friction stress in that model was replaced by an apparent friction stress equal to the uniform stress ahead of the crack. This apparent friction stress is dependent on both the applied stress and time. Assuming a critical crack opening displacement (COD) or a critical value of theJ integral,J
c, to be the criteria for the onset of the creep crack growth the initiation time can be calculated using the results of this study. A good agreement between the theory and experiment is obtained for two different CrMoV steels. This comparison with experiments suggests that the COD is an appropriate crack growth initiation parameter for both ductile and brittle materials whilstJ
cdoes not seem to be applicable in creep fracture.
TL;DR: In this article, a model of fatigue crack growth is proposed based on a line plastic zone analysis, which is assumed that the effect of cycling is to reduce the craze stress to some proportion of the original value depending on the degree of unloading.
Abstract: A model of fatigue crack growth is proposed based on a line plastic zone analysis. It is assumed that the effect of cycling is to reduce the craze stress to some proportion of the original value depending on the degree of unloading. Successive loadings result in growth of the craze with a corresponding increase in crack opening displacement. At some critical value of this displacement, crack growth occurs and the rate of growth is related to the applied stress intensity factor and the critical static value. The results of the model are applied to data on several polymers and a good description of growth rate, mean stress and frequency effects is given. Finally, some fatigue lives are predicted.
TL;DR: In this article, it has been shown that the threshold stress intensity depends more on the elastic modulus rather than on any other material property in agreement with many experimental results, and that the stress intensity can be related to the stress necessary to nucleate a dislocation at the crack tip.
Abstract: It has been shown that the ratio of threshold stress intensity for fatigue crack growth to the shear modulus is nearly a constant for many materials This implies that fatigue crack growth is related to some fundamental phenomenon occurring at the crack tip In the following a dislocation model has been developed to predict the threshold stress intensity It is shown that the stress intensity can be related to the stress necessary to nucleate a dislocation at the crack tip The most important outcome of the present analysis is that the threshold stress intensity depends more on the elastic modulus rather than on any other material property in agreement with many experimental results
TL;DR: Theoretically and empirically a fatigue crack propagation rate of the form d c d N ∞ (ΔK) 4 μσ 2 U is indicated where μ is the shear modulus, σ is an appropriate measure of the alloy's strength, and U is the energy to make a unit area of fatigue crack as discussed by the authors.
TL;DR: In this article, the Paris equation for fatigue crack growth is explained in relation to the crack closure concept suggested by Elber, and it is concluded that the exponent parameter m reflects mainly the dependency of crack closure behavior on ΔK.
Abstract: The two parameters, C and m, which characterize the Paris equation for fatigue crack growth are explained in relation to the crack closure concept suggested by Elber. It is proposed that the range of effective incremental change in stress intensity factor (ΔK) needed for crack growth should have a second power correlation with the growth rate. The crack growth is essentially determined by cumulative damage to the material in cycled plastic zone near the crack tip, and is relatively insensitive to the applied ΔK-values and the mechanical properties of material. However, the crack closure behavior is expected to depend on both the stress range and the material properties. Thus it is concluded that the exponent parameter m reflects mainly the dependency of crack closure behavior on ΔK. For example, in the case of m=4 the crack opening level increases linearly with increase in ΔK, while in the case of m=2 it remains constant. It is suggested that the cyclic straining at the crack tip possibly varies with ΔK, thus changing primarily the crack closure behavior rather than the damage accumulation process in the plastic zone.
TL;DR: In this article, the effects of specimen thickness, stress ratio and maximum stress intensity factor (K max ) on crack closure or opening were studied using a 2219-T851 aluminum alloy.
TL;DR: In this article, the authors measured the stress corrosion crack velocity and stress corrosion threshold stress intensity for Type 304L austenitic stainless steel in water with 42% MgCl2 at 130 C.
Abstract: Stress corrosion crack velocity and stress corrosion threshold stress intensity have been measured for Type 304L austenitic stainless steel in water with 42% MgCl2 at 130 C. The stress corrosion crack velocity vs stress intensity curve has a shape similar to the one for other austenitic steels and many other metallic and nonmetallic materials. The threshold stress intensity (KISCC is 8 ± 1 MN·m−3/2, and the plateau crack velocity is 3 to 6 × 10−8 m/s. Linear elastic fracture mechanics techniques can be applied to SCC testing of soft and tough materials, provided the KISCC is low enough, and crack branching can be avoided. Criteria for the formation of single, branched, and circular stress corrosion cracks are indicated.
TL;DR: In this article, the authors investigated the possibility of a point of a crack front propagating in a direction not in the normal plane of the crack front, and the results were used in conjunction with the strain energy density fracture criterion for the problem of an elliptical crack.
TL;DR: In this paper, it is proved that holes with equi-stressed boundaries are optimal, and the problem of finding the optimal hole shapes in elastic bodies, which cause minimal stress concentration, results in minimax optimization problems with a local criterion.