A note on Orowan mechanism below the root of an yielding crack like notch in a strain-hardening metal

Based on Orowan's historic model, a local fracture initiation criterion is proposed for high strength strain-hardening metals. A crack like notch geometry is studied in the following Von Mises yielding materials: (a) an ideally plastic-elastic solid, and (b) a power law strain-hardening metal. Orowan mechanism is seen to be dominant below the root region of an yielding crack like notch, just before the onset of fracture. Close to the elastic-plastic interface, Orowan's Kink-band forms inside the plastic region, when a mismatch Kink-stress (in radial component only) reaches a critical value in this particular region. It is extremely important to note that, this narrow Kink-band zone, where the micro-cracks are likely to nucleate, lies at a distance approx. $\text{3}\text{4}\text{s}$, where s is the size of the plastic zone on the crack extension plane under a plane strain deformation. A reverse slip mechanism operates in this region in addition to the presence of a pure hydrostatic tension, just before the release of this critical Kink-stress. Due to this stress history, a large inhomogeneous strain-localization occurs in a narrow band, which could then interact with the free notch surface before the onset of final instability. Thus, at the onset of crack extension, (satisfying Griffith-Irwin criterion of fracture), the stress intensification at the notch tip root is directly proportional to the strength of this critical strain-localization and inversely proportional to the plastic zone size on the crack extension plane. Hence, it is concluded that: Orowan's mechanism and McClintock's criteria for critical strain-localizations should play the most important roles for predicting the local fracture behaviour of metals.     

Dynamic growth of anti-plane shear cracks in ideally plastic crystals

A near-tip asymptotic analysis is given for the stress and deformation field near the tip of crack propagating dynamically under anti-plane shear in an ideally plastic single crystal. A paricular class of orientations of the crack relative to the crystals is considered so that the yield is so simple diamond shape (relative to directions along perpendicular to the crack) in the plane of the anti-plane shear stresses. The near-tip solution is shown to consists of sectors which carry constant stresses, at yield levels, corresponding to adjacent vertices on the diamond-shaped yield locus, and which are joined along an elastic-plastic shock discontinuity. All plastic flow in the near-tip region occurs in the shock. Plastic strains and particle velocity are finite at the crack tip. The plastic strain is proportional to the elastic strain at onset of yielding and is inversely proportional to the elastic Mach number associated with the speed of crack growth.     

Energy density approach for calculation of three-dimensional inelastic strain and stress at the crack tip in compact tension specimens of polycarbonate

An energy-based method is utilized for calculating elastic-plastic strains and stresses near fatigue crack tip in specimens of Merlon polycarbonate. The stress redistribution caused by the plastic yielding around the crack tip is taken into account so that theoretical crack tip strain is improved. The estimated values of crack tip strain based on an energy density approach are compared with experimental results obtained from an embedded grid moire technique and embedded strain gages. Large-scale yielding seems to dominate near the crack tip. In fact, the measured strain is in agreement with the elastic solution, which means, in reality, only small-scale yielding takes place near the crack tip. The strain in the mid-plane (plane strain) is found to be higher than in the surface plane (plane stress). The experimental and theoretical results are in good agreement.     

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.     

An interface crack in a coating-substrate composite with mixed-mode Dugdale corrections

Considering both plane stress and plane strain conditions, the plastic zone size and the crack tip opening displacement of an interface crack between a coating and a semi-infinite substrate under a normal load on the crack surfaces are investigated by the mixed-mode Dugdale model. In the model, stresses applied in the plastic zones satisfy the Von Mises yield criterion. The plastic zone size can be calculated by satisfying the condition that the complex stress intensity factors vanish. After the plastic zone size is solved, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, a uniform load is considered, and the effects of the normalized elastic modulus (the ratio of the elastic modulus of the coating to the elastic modulus of the substrate) and the normalized crack depth (the ratio of the coating thickness to the interface crack length) on the normalized plastic zone size and the normalized crack tip opening displacement are examined. Numerical examples show in the case of thin coatings, the value of the normalized plastic zone size decreases with increasing the normalized elastic modulus.     

Two-parameter characterization of the near-tip stress fields for a bi-material elastic-plastic interface crack

A particular case of interface cracks is considered. The materials at each side of the interface are assumed to have different yield strength and plastic strain hardening exponent, while elastic properties are identical. The problem is considered to be a relevant idealization of a crack at the fusion line in a weldment. A systematic investigation of the mismatch effect in this bi-material plane strain mode I dominating interface crack has been performed by finite strain finite element analyses. Results for loading causing small scale yielding at the crack tip are described. It is concluded that the near-tip stress field in the forward sector can be separated, at least approximately, into two parts. The first part is characterized by the homogeneous small scale yielding field controlled by J for one of the interface materials, the reference material. The second part which influences the absolute value of stresses at the crack tip and measures the deviation of the fields from the first part can be characterized by a mismatch constraint parameter M. Results have indicated that the second part is a very weak function of distance from the crack tip in the forward sector, and the angular distribution of the second part is only a function of the plastic hardening property of the reference material.     

Asymptotic deformation and stress fields at the tip of a sharp notch in an elastic-plastic material

The singular elastic-plastic stress, strain and the displacement fields at the tip of a sharp notch for both plane stress and plane strain conditions are investigated analytically. The material is assumed to be governed by the deformation theory of plasticity with linear strain hardening characteristic. Since the elastic strain is retained in the analysis, the final strain and displacement fields can be separated into the elastic and the plastic parts. In the case with zero notch angle, the results reduce to the classical crack problem. The relationship of the amplitude of the near crack tip elastic-plastic field to the elastic far field is obtained. Both mode I and mode II cases are investigated. The mixed mode case is also discussed.     

Strain localization below the root of an yielding notch

Prandtl-Orowan mechanism for plastic crack extension is discussed. A method of calculating strain localization below an yield notch in Prager's elastic-plastic material is described. Firstly, a boundary value problem is solved for an elastic notch under the combined loadings of normal and shear stresses acting on its flank faces. Secondly, the stresses and strains in the compressibility gradient are calculated on the crack extension plane assuming a forward slip. Thirdly, taking a sticking plastic friction on the flank faces, it is found that a large transverse strain localizes at the plastic incompressibility-compressibility gradient ahead of an yielding notch. Finally, it is suggested that Rayleigh-Lin type fluid mechanics instability criterion may help to understand such plastic flow localization problem.     

Interfacial crack-tip constraints and <Emphasis Type="Italic">J</Emphasis>-integral for bi-materials with plastic hardening mismatch

This paper investigates interfacial crack tip stress fields and the J-integral for bi-materials with plastic hardening mismatch via detailed elastic-plastic finite element analyses. For small scale yielding, the modified boundary layer formulation with the elastic T-stress is employed. For fully plastic yielding, plane strain single-edge- cracked specimens under pure bending are considered. Interfacial crack tip stress fields are explained by modified Prandtl slip-line fields. It is found that, for bi-materials consisting of two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and the J-contribution there. The implication of asymmetric J-integral in bi-materials is also discussed.     

Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials,Part 1: Plane strain case

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.     

THE INFLUENCE OF NOTCH PLASTICITY ON SHORT FATIGUE CRACK BEHAVIOUR

Abstract— A study has been made of fatigue crack formation and growth at the root of different notch profiles in a structural steel subjected to fully reversed tension-compression loading. The scale of stage I microstructural crack growth at notches decreased with increasing notch root strain and was comparable to the size of stage I cracks in shallow hourglass profile specimens at the same strain. Stage II crack growth rates were faster within the notch plastic field than in the elastic stress field of the bulk material.     

The Equivalent Strain Energy Density approach re-formulated and applied to sharp V-shaped notches under localized and generalized plasticity

In the case of a rounded notch, the stress and strain at the notch tip can be determined by the traditional Neuber rule or by the Equivalent Strain Energy Density (ESED) approach, as formulated by Glinka and Molski. In the latter case the elastoplastic strain energy density at the notch tip is thought of as coincident with that determined under purely elastic conditions. For sharply V‐shaped notches this approach is not directly applicable, since the strain energy density at the notch tip tends toward infinity both for a material obeying an elastic law and a material obeying a power hardening law. By using the notch stress intensity factors, the present paper suggests a re‐formulation of the ESED approach which is applied no longer at the notch tip but to a finite size circular sector surrounding the notch tip. In particular we have adopted the hypothesis that, under plane strain conditions, the value of the energy concentration due to the notch is constant and independent of the two constitutive laws. When small scale yielding conditions are present, such a hypothesis immediately results in the constancy of the strain energy averaged over the process volume. As a consequence, plastic notch stress intensity factors valid for sharp V‐shaped notches can be predicted on the basis of the linear elastic stress distributions alone.     

Plastic yielding at the tip of a blunt notch during static and fatigue loading

Rice's analytical Mode III solution for the relationship between anti-plane stress and anti-plane strain was used to determine the small scale plastic yielding at the tip of a two-dimensional blunt notch. The results were applied to fatigue loading. The plastic zone size and crack opening displacement derived in the present analysis were determined as functions of applied stress, geometric factors (notch radius and length) and material properties (yield stress and the work hardening rate). The minimum stress intensity required for plastic yielding at a blunt notch tip was postulated to be the experimentally observed threshold stress intensity for fatigue crack initiation. The threshold stress intensity so determined depends not only on the notch geometry but also on material properties. There is good agreement with calculated and measured values of the threshold stress intensity for fatigue crack initiation.     

EFFECT OF ELASTIC MISMATCH ON THE GROWTH OF A CRACK INITIALLY TERMINATED AT AN INTERFACE IN ELASTIC PLASTIC BIMATERIALS

Abstract A crack perpendicular to, and initially with the tip on, a bimaterial interface is studied. An asymptotic analysis is performed and crack growth proceeds straight ahead at constant remote load. Mode I conditions and plane strain are assumed. The materials on both sides of the interface are elastic perfectly-plastic with different elastic properties and the same yield stress. A finite element analysis is made and crack growth is simulated by an element relaxation technique. Because of the interface, the crack-tip driving force is not constant, which is reflected in the near-tip state. The development of the plastic zone and the crack opening displacements is presented for different elastic mismatches. Small scale yielding like results are obtained after a crack extension of about the plastic zone size from the interface, i.e. long before a square-root singular stress field may be expected to embed the plastic zone. An important observation is that the development of the crack opening displacement at the initial stage of growth is reversed when plasticity is introduced, as compared to the prediction by an elastic model. A region of stable crack growth is identified at the initial phase of growth into a stiffer material, solely due to elastic mismatch.     

Precise elastic-plastic analysis of crack line field for mode II plane strain crack

The near crack line analysis method has been used to investigate the exact elastic-plastic solutions of a mode II crack under plane strain condition in an elastic-perfectly plastic solid. The significance of this paper is that the assumptions of the conventional small scale yielding theory have been completely abandoned. The inappropriateness of matching conditions formerly taken at the elastic-plastic boundary ths been corrected as well. By eatching the general solution of the plastic stress (but not the special solution that was adopted) with the exact elastic stresses (but not the crack tip K-dominant field) at the elastic-plastic boundary near the crack line, the plastic stresses, the length of the plastic zone and the unit normal vector of the elastic-plastic boundary, which are sufficiently precise near the crack line region, have been given. The solutions are suitable not only under the condition that the plastic region is sufficiently small but also under the condition that the plastic region is large.     

Strain energy-based evaluations of plastic notch stress intensity factors at pointed V-notches under tension

An analytical study is carried out on the existing link between elastic and plastic notch stress intensity factors at pointed V-notches in plates under tension.The frame is developed on the basis of the elastic and plastic energy concentration factors of the notch defined here as the ratio between the local and the nominal strain energy densities. The link varies under plane stress and plane strain conditions. The local strain energy density is evaluated over a control volume drawn by the energy contour lines ahead of the notch and allows plastic notch stress intensity factors to be predicted on the basis of an ideally linear elastic analysis, both under small and large scale yielding.     

An estimation of the plastic zone size for a bimaterial interfacial crack

This note deals with the size of the plastic zone ahead of an interfacial crack between two dissimilar isotropic elastic materials. Dugdale's concept of finite stress at the tip of the crack is used in the study. The plastic zone size is determined for plane stress problems under the von Mises yield condition.     

Three-dimensional linear elastic distributions of stress and strain energy density ahead of V-shaped notches in plates of arbitrary thickness

This paper presents an analytical solution, substantiated by extensive finite element calculations, for the stress field at a notch root in a plate of arbitrary thickness. The present approach builds on two recently developed analysis methods for the in-plane stresses at notch root under plane-stress or plane strain conditions, and the out-of-plane stresses at a three-dimensional notch root. The former solution (Filippi et al., 2002) considered the plane problem and gave the in-plane stress distributions in the vicinity of a V-shaped notch with a circular tip. The latter solution by Kotousov and Wang (2002a), which extended the generalized plane-strain theory by Kane and Mindlin to notches, provided an expression for the out-of-plane constraint factor based on some modified Bessel functions. By combining these two solutions, both valid under linear elastic conditions, closed form expressions are obtained for stresses and strain energy density in the neighborhood of the V-notch tip. To demonstrate the accuracy of the newly developed solutions, a significant number of fully three-dimensional finite element analyses have been performed to determine the influences of plate thickness, notch tip radius, and opening angle on the variability of stress distributions, out-of-plane stress constraint factor and strain energy density. The results of the comprehensive finite element calculations confirmed that the in-plane stress concentration factor has only a very weak variability with plate thickness, and that the present analytical solutions provide very satisfactory correlation for the out-of-plane stress concentration factor and the strain constraint factor.     

Mathematical modeling of damage in unidirectional composites

Solutions are developed for the two-dimensional region containing unidirectional fibers embedded in an elastic matrix with an initial flaw in the form of a transverse notch, a rectangular cut-out, and a circular hole. Subsequent damage due to the presence of the flaw is generated by remote stresses acting parallel to the fibers. This work is an extension of the paper by Goree and Gross [1] in which the flaw was taken in the form of a notch (crack) and the subsequent damage, due to loading, consisted of longitudinal matrix yielding and splitting at the end of the notch. The present study accounts for longitudinal matrix damage as in [1] and, in addition, includes transverse matrix and fiber damage in the vicinity of the flaw for the above three initial shapes. The fibers are taken as linearly elastic, the matrix material as elasticperfectly plastic and the classical shear-lag stress displacement assumptions are used. An ultimate stress failure criterion is used for both the fibers and the matrix; simple tension for the fibers and shear failure for the matrix.

For ductile matrix composites (boron/aluminum) the present results indicate that both longitudinal matrix yielding and transverse notch extension must be included in order for the model to agree with experimental results. Interestingly, the extent of the transverse damage region at failure is shown to be approximately constant, independent of the initial flaw shape or length.

Very little difference is found between the results for the three types of initial damage, i.e. the notch, rectangular cut-out and circular hole. In all cases, the presence of additional damage changes the nature of the stress distribution in the unbroken fibers. For the original Hedgepeth[2] problem of a notched laminate the stresses decay as the square root of the distance from the notch tip. Inclusion of longitudinal or transverse damage significantly reduces the maximum stress concentration in the unbroken fibers and gives a much more uniform stress state. It is shown that this behavior cannot be accounted for by introducing an effective notch length or crack tip damage zone with a square root behavior.     

The mechanism of brittle fracture in notched impact tests on polycarbonate

The impact testing of notched polycarbonate bars that are thick enough to yield in plane strain has been investigated. Shear bands occur in the plastic zone that resemble the slip line field for yielding from a circular notch. Eventually, an internal craze nucleates at the tip of the plastic zone, where the stresses are highest, and a crack forms in the thickest part of the craze. Above –15 C the stress for the craze to nucleate is a nearly constant multiple of the yield stress. It is shown that previous observations that annealing polycarbonate causes a ductile to brittle transition is a consequence of testing bars of thickness less than 5 mm.