A mode I fracture analysis of a center-cracked infinite shape memory alloy plate under plane stress

The problem of a center plane crack in an infinite, thin, pseudoelastic Shape Memory Alloy (SMA) plate subjected to an in-plane uniform tensile stress at infinity is analyzed. The analysis follows closely the Dugdale?CBarenblatt model developed for conventional metals. It is found for low remote stress values??less than a critical value??that the SMA is not fully transformed in the vicinity of a crack tip. Closed form expressions for the size of the partial transformation zone, crack opening displacement and J-integral are given for this case. For remote stress levels above the critical value, the fully-transformed material near a crack tip is assumed to yield plastically. The sizes of the transformed (both partially and fully) and plastic regions are numerically evaluated by solving a system of integral equations and their sensitivity to the transformation characteristics (i.e., maximum transformation strain and temperature) is determined. Moreover, a relationship between the J-integral and the crack-tip opening displacement is derived. The results obtained are important in understanding the effect of stress-induced phase transformation in the fracture behavior of SMAs in the presence of static cracks, and subsequently in formulating conditions for initiation of crack propagation.     

Two cracks with coalesced interior plastic zones — The generalised Dugdale model approach

The problem investigated is of an elastic-perfectly plastic infinite plate containing two equal collinear and symmetrically situated straight cracks. The plate is subjected to loads at infinity inducing mode I type deformations at the rims of the cracks. Consequently, plastic zones are formed ahead of the tips of the cracks. The loads at infinity are increased to a limit such that the plastic zones formed at the neighbouring interior tips of the cracks get coalesced. The plastic zones developed at the tips of the cracks are closed by applying normal cohesive quadratically varying stress distribution over their rims. The opening of the cracks is consequently arrested. Complex variable technique is used in conjugation with Dugdale’s hypothesis to obtain analytical solutions. Closed form analytical expressions are derived for calculating plastic zone size and crack opening displacement. An illustrative numerical example is discussed to study the qualitative behaviour of the loads required to arrest the cracks from opening with respect to parameters viz. crack length, plastic zone length and inter-crack distance. Crack opening displacement at the tip of the crack is also studied against these parameters.     

CRACK CLOSURE AND PLASTIC ZONE SIZES IN FATIGUE

Abstract— An elastic-plastic finite element simulation of growing fatigue cracks which accounts for plasticity-induced crack closure is used to study the size of the forward and reversed plastic zones at the crack tip. Forward plastic zone widths for fatigue cracks and stationary, monotonically loaded cracks are compared and found to be similar. The width of the forward plastic zone at the tip of a fatigue crack is not significantly influenced by closure. The traditional Irwin-Rice estimate for crack tip plastic zone size in plane stress is found to be generally consistent with the finite element results. The width of the reversed plastic zone at the tip of a growing fatigue crack in plane stress is found to be considerably less than one-fourth the size of the forward plastic zone, the traditional Rice estimate. This decrease appears to be due to fatigue crack closure. A simple model is developed which permits estimation of the reversed plastic zone size for any stress ratio from information about maximum and minimum stresses and the closure stress. The predictions of this model agree closely with plastic zone sizes calculated by the finite element analysis. These observations appear to be consistent with experimental measurements of forward and reversed plastic zones sizes reported in the literature.     

Plastic intensity factors for cracked plates

An elastic-plastic analysis is performed for two problems relevant to fracture mechanics: a semiinfinite body with an edge crack in a far out-of-plane shearing field and an infinite plate under plane stress conditions containing a finite line crack in a remote tensile field. Amplitudes of the dominant singularity in the plastic region at the crack tip, the plastic stress and strain intensity factors, are calculated for applied stress levels approaching the yield stress. A technique is developed for using the dominant singular solution in conjunction with the finite element method to make accurate calculations for the near-tip fields. Additionally, a comparative study of deformation theory with flow theory is performed for cracks in an anti-plane shear field. Elastic fracture mechanics is extended to high levels of applied stress for which the plastic zone is no longer small compared to the crack length by relating the critical stress for fracture initiation to the plastic intensity factors.     

On the dynamic crack propagation in an interface with spatially varying elastic properties

This article provides a comprehensive theoretical treatment of a finite crack propagating in an interfacial layer with spatially varying elastic properties under antiplane loading condition. The theoretical formulations governing the steady state solution are based upon the use of an integral transform technique. The resulting dynamic stress intensity factor of the propagating cracks is obtained by solving the appropriate singular integral equations, using Chebyshev polynomials, for different inhomogeneous materials. Numerical examples are provided to verify the technique and to show the effect of the thickness of the interfacial layer and the material properties upon the dynamic stress intensity factor of the crack and the associated singularity transition.     

Analysis of a crack-tip dual zone model

An analytic model is proposed for an opening mode of crack face displacement with crack-tip dual zones, i.e. an elastic core zone plus a plastic strip zone. A presence of such dual zones in the vicinity of the crack tip was experimentally observed in a recent study based on the generation of dislocations. A Papkovich-Neuber formulation of the resulting four-part mixed boundary value problem leads to a set of quadruple integral equations which are solved with an application of finite Hilbert transform technique. With conditions of boundedness on the stresses in the plastic strip zone, the results show an inverse square root of the distance type singularity at the base of the crack tip and a relaxation of stresses in the crack-tip elastic core zone is realized. The stress intensity factors and the crack-tip opening displacements are presented in exact forms involving elliptic integrals and Heuman's lambda function and are shown to depend upon the crack size, the applied loading and the crack-tip dual zone lengths. The analytic and graphical solutions are compared with the Dugdale model to which they reduce as a limiting case of vanishing elastic core zone.

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Résumé On propose un modèle analytique pour décrire la déformation d'ouverture selon le mode I, en utilisant une approche de mécanique des milieux continus pour décrire la double zone-élastique et plastique-située à la pointe d'une fissure.

     

The improved D-M model solutions for single edge cracked plates by considering the yielding at the back side

The method of calculating the stress intensity factor (SIF) and the crack opening displacement (COD) for double edge cracks in plates under arbitrary loadings that results in solving a system of Cauchy-type singular integral equations is presented. The improved D-M model is then constructed for edge cracked plates by considering the yielding at the back side. For the cases of tension and bending, the plastic zone sizes and the crack opening displacements are calculated from the improved model solution, and the envelopes for the beginning of backside yielding and ligament yielding are obtained. The numerical results are compared with known solutions which take no account of the yielding at the back side and with experimental results.     

Boundary element analysis of structures with bridged interfacial cracks

The direct boundary integral equations method has been applied to analyze stresses in a fracture process zone (a crack bridged zone) and to calculate stress intensity factors module for structures with bridged interfacial cracks under mechanical loading. Bridged zones at interfacial cracks are considered as parts of these cracks with assumption that surfaces of interfacial cracks are connected by distributed spring-like bonds with given bond deformation law. For numerical analysis of piecewise structures with bridged interfacial cracks the multi-domain formulation of the boundary elements method is used. The stress intensity factors module evaluation is performed on the basis of displacements and stresses computed at nodal points of special quadratic boundary elements adjoined to a crack tip. The comparative study between the results obtained by the boundary elements method and the results obtained previously by the singular integral–differential equations method is performed and the validity of the presented numerical formulation is demonstrated. The new problem for a bridged circumferential crack between a cylindrical inclusion and a matrix in plate of finite size is also solved. Stresses distributions along the bridged zone and the stress intensity factors modulus dependencies versus the bridged zone length and bonds stiffness are presented and discussed for this problem.     

Two equal symmetric circular arc cracks in an elastic medium – the Dugdale approach

The Dugdale model for two equal, symmetrically situated coplanar circular arc cracks contained in an infinite elastic perfectly-plastic plate is proposed. Biaxial loads are applied at the infinite boundary of the plate. Consequently, the rims of the cracks open in Mode I and develop a plastic zone ahead of each of the cracks. These plastic zones are then closed by the distribution of uniform normal closing stresses over the rims of the plastic zones. Based on the complex-variable technique and the superposition principle, the solution for the above problem is obtained. Closed-form analytic expressions are obtained for the determination of the sizes of the plastic zones and the crack-opening displacement (COD) at the tip of the crack. Numerical studies are carried out to calculate the load ratio (load applied at infinity/yield point stress applied at the rims of the plastic zones) required for the closure of the plastic zones, for various radii of arc cracks and for various angles subtended by them at the centre. The crack-opening displacement is also investigated with respect to these parameters.     

Crack arrest model for a piezoelectric plate — A generalised Dugdale model

The constant search for new materials has provided impetus to research in piezoelectric materials. An anti-plane problem for a cracked unbounded two-dimensional poled piezoelectric plate has been investigated. The crack rims open on account of shear mechanical forces applied at the remote boundary and in-plane electric displacement field prescribed at the infinite boundary. Thus the crack yields both mechanically and electrically. Consequently, a plastic zone and a saturation zone protrude ahead of each tip of the crack. These developed zones are in turn closed by applying yield point shear stress at the rims of plastic zone and normal closing saturation limit displacement on the rims of saturation zones. Two cases are investigated when (i) the developed saturation zone length exceeds that of the developed plastic zone, and (ii) saturation zone length is smaller than that of the plastic zone. Fourier integral transform method is used in each case to obtain the length of plastic zone and saturation zone. Closed form analytic expressions are obtained in each case. Crack opening displacement and potential drop across the rims of the crack are also obtained. The effect of mechanical loads on crack closure in the presence of electric field is investigated and vice-versa. Also energy release rate expressions are obtained for both the cases.     

Crack–surface displacements for cracks emanating from a circular hole under various loading conditions

The purpose of this paper is to calculate and develop equations for crack–surface displacements for two‐symmetric cracks emanating from a circular hole in an infinite plate for use in strip‐yield crack‐closure models. In particular, the displacements were determined under two loading conditions: (1) remote applied stress and (2) uniform stress applied to a segment of the crack surface (partially loaded crack). The displacements were calculated by an integral‐equation method based on accurate stress–intensity factor equations for concentrated forces applied to the crack surfaces and those for remote applied stress or for a partially loaded crack surface. A boundary‐element code was also used to calculate crack–surface displacements for some selected cases. Comparisons made with crack–surface displacement equations previously developed for the same crack configuration and loading showed significant differences near the location where the crack intersected the hole surface. However, the previous equations were fairly accurate near the crack‐tip location. Herein an improved crack–surface displacement equation was developed for the case of remote applied stress. For the partially loaded crack case, only numerical comparisons were made between the previous equations and numerical integration. A rapid algorithm, based on the integral‐equation method, was developed to calculate these displacements. Because cracks emanating from a hole are quite common in the aerospace industry, accurate displacement solutions are crucial for improving life‐prediction methods based on the strip‐yield crack‐closure models.     

On the plastic zone size and CTOD study for a Zener�CStroh crack interacting with a circular inclusion

An analytical solution is given for plastic yielding of a Zener?CStroh crack near a circular inclusion embedded in an infinite matrix. The crack is orientated along the radial direction of the inclusion. In the solution procedure, the crack is simulated as a continuous distribution of edge dislocations. Using the Dugdale model of small-scale yielding, plastic zones are introduced at both crack tips. Using the solution of a circular inclusion, interacting with a single dislocation as the Green??s function, the physical problem is formulated into a set of singular integral equations. With the aid of Erdogan??s method and iterative numerical procedures, the singular integral equations are solved numerically for the plastic zone sizes and crack tip opening displacement. The results obtained in the current work are verified by reduction to simpler cases of the Dugdale model. Various parameters such as the distance, shear modulus ratio, Poisson??s ratio, and loading condition are studied.     

Stresses near Crack Tips on the Boundary of Two Media in the Presence of Plastic Strips

Under the conditions of plane deformation, we study stresses in a piecewise-homogeneous isotropic body near the tip of a mode I crack appearing at the angular point on the boundary of two media. It is assumed that plastic strips (modeled by plastic slip lines) are formed on the boundary of the media. To determine the stresses, we use the Mellin integral transformation and the Wiener–Hopf method. The angular point is a stress concentrator with power singularity and, therefore, immediately after the appearance of lateral plastic strips (zones), a new plastic zone begins to develop from this point. We study the dependence of the power of singularity of stresses on the angle made by the boundary and the elastic characteristics of the media.     

Interaction between cracks and rigid lines in an infinite plate

Summary Interaction between cracks and rigid lines in an infinite plate is investigated in this paper. The rigid lines are assumed in an equilibrium condition and may have some rotation in the deformation process of the adjacent material. After placing some distributed dislocations along the cracks and some distributed body forces along the rigid lines, a system of singular integral equations is obtained. The obtained system of the singular integral equations is reduced to a system of Fredholm integral equations by appropriate substitution of the unknown functions. The regularized integral equations are solved numerically. Stress intensity factors at the crack tips and stress singularity coefficients are investigated in the numerical examples.     

On the effect of the residual stresses on the crack opening displacement in a cracked sheet

The residual stress and displacement fields caused by localized plastic flow near a mode I crack tip in a sheet under plane stress conditions are investigated. The present study founds on the classical Dugdale scheme of the plastic flow localization. The residual stress field is considered to be induced by reversed plastic flow near the crack tip caused by an unloading. As it is found the residual stresses around the crack compress the crack tip, while the residual tensile stresses in a distant from the crack tip zone occur. It is shown the maximum residual tensile stresses can reach the significant value of the one third of the yield limit. The length of the compressed plastic zone and the residual displacement distributions are obtained. The exact formula for the residual crack opening displacement to estimate the crack closure is found. Then the next loading of the cracked plate is considered. It is shown that the second loading causes the origin of two plastic zones localized near the crack tip and at the point, where the maximum residual tensile stresses are concentrated. Again, according to the Dugdale scheme of the plastic localization, both the plastic flow zones are modelled as narrow stripes on the line extending the crack. To determine three non-dimensional parameters, characterizing the position of the segment-like plastic flow zones, a non-linear system of equations is obtained and analyzed. The exact formula for the crack opening displacement after a loading–unloading cycle is obtained. An asymptotic analysis (as the linear dimension of the distant plastic flow zone compared with the actual crack length is small) is given. It shows that the effect of the distant plastic flow zone appears as some complementary crack closure.     

A unification of small- and large-crack growth laws

A generalized model enhancement is proposed to link small- and large-crack growth laws. The enhancement is based on crack growth rate laws with crack tip plastic zone size formulations. Transition functions are used to transform small-crack plastic zone sizes and crack growth law exponents to those predicted by linear-elastic fracture mechanics. In doing so, influences on crack growth, e.g. constraint, crack aspect ratio and specimen geometry are accounted for. The applicability of the enhancement is directed toward instances where small cracks start from geometric features and grow through stress gradients to eventually become large cracks under nominal LEFM conditions. The enhancement is applied to the Wang model, and crack growth rate and fatigue lifetime predictions are made. The enhancement is shown to provide a good correlation to experimental results for Ti–6Al–4V under various maximum stresses at a stress ratio of R = 0.4.     

Crack closure in friction stir weldment using non‐linear model for fatigue crack propagation

The fatigue crack propagation in a friction stir‐welded sample has been simulated herein by means of two 3‐dimensional finite element method (FEM)‐based analyses. Numerical simulations of the fatigue crack propagation have been carried out by assuming a residual stress field as a starting condition. Two initial cracks, observed in the real specimen, have been assessed experimentally by performing fatigue tests on the welded sample. Hence, the same cracks have been placed in the corresponding FE model, and then a remote load with boundary conditions has been applied on the welded specimen. The material behaviour of the welded joint has been modelled by means of the Ramberg‐Osgood equation, while the non‐linear Kujawski‐Ellyin (KE) model has been adopted for the fatigue crack propagation under small‐scale yielding (SSY) conditions. Owing to the compressive nature of the residual stress field that acts on a part of the cracked regions, the crack closure phenomenon has also been considered. Then, the original version of the KE law has been modified to fully include the closure effect in the analysis. Later, the crack closure effect has also been assessed in the simulation of fatigue propagation of three cracks. Finally, an investigation of the fracture process zone (FPZ) extension as well as the cyclic plastic zone (CPZ) and monotonic plastic zone (MPZ) extensions have been assessed.     

Efficient solution of multiple cracks in great number using eigen COD boundary integral equations with iteration procedure

A newly developed computational approach is proposed in the paper for the analysis of multiple crack problems based on the eigen crack opening displacement (COD) boundary integral equations. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix to determine all the unknown CODs step by step. To deal with the interactions among cracks for multiple crack problems, all cracks in the problem are divided into two groups, namely the adjacent group and the far-field group, according to the distance to the current crack in consideration. The adjacent group contains cracks with relatively small distances but strong effects to the current crack, while the others, the cracks of far-field group are composed of those with relatively large distances. Correspondingly, the eigen COD of the current crack is computed in two parts. The first part is computed by using the fictitious tractions of adjacent cracks via the local Eshelby matrix derived from the traction boundary integral equations in discretized form, while the second part is computed by using those of far-field cracks so that the high computational efficiency can be achieved in the proposed approach. The numerical results of the proposed approach are compared not only with those using the dual boundary integral equations (D-BIE) and the BIE with numerical Green's functions (NGF) but also with those of the analytical solutions in literature. The effectiveness and the efficiency of the proposed approach is verified. Numerical examples are provided for the stress intensity factors of cracks, up to several thousands in number, in both the finite and infinite plates.     

Complex fracture parameters for an interface crack between twohardening materials: a photoelastic study

In this paper, complex stress intensity factors (SIFs) at an interface crack are determined for a range of applied loads, crack lengths and remote mode mixes using automated photoelasticity. The specimen geometries comprise epoxy resin and aluminium alloy halves bonded together, and are loaded in either compact tension in mixed‐mode conditions or in three‐point bend under mode I conditions. In the experiments, full‐field isochromatic data were obtained from the epoxy half using an established phase‐stepping technique. A reworked approach to the determination of the SIFs was developed by combining a least‐squares over‐deterministic method for fitting crack‐tip stress equations to the data and a weighting factor that ensures that only data in the singularity zone are used. For comparison, some of the specimens were tested using a linear‐elastic finite element (FE) analysis and/or by experiment using homogeneous test specimens. Excellent agreement between the experimental and numerical SIF moduli was achieved for remote mode I loadings. However, for good agreement to be made between the phase angle results requires an additional phase term to be added to the FE solution at each load to account for the development of a crack‐tip plastic zone. Further, results for the SIFs from remote mixed‐mode loadings of the compact tension specimen only have a meaningful interpretation in light of small‐scale yielding conditions. It is shown, qualitatively, that the experiments verify some of the predictions made in the literature of asymptotic behaviour at interface crack tips from results of elasto‐plastic FE analyses.     

A set of interface cracks with contact zones in a combined tension-shear field

Summary. A set of cracks lying along the interface of two dissimilar isotropic materials under a mixed-mode loading is considered. The interface cracks are assumed to be fully open, partially closed with frictionless contact zones and fully closed. The problem is reduced to a homogeneous combined Dirichlet-Riemann boundary value problem, which is solved in closed form. A set of transcendental equations for the determination of the contact zone lengths for an arbitrary number of cracks and the closed-form expressions for the stresses and the displacement jumps on the material interface are obtained. A single crack with one and two contact zones has been considered in details. An explicit set of two transcendental equations for the relative contact zone length and closed-form expressions for the stress intensity factors at the crack tips are obtained for both cases. The contact zone lengths and the stress intensity factors are investigated numerically for different material pairs under different values of the loading, and a comparison of the results for a crack with one and two contact zones is carried out.