The influence of elastic mismatch on the size of the plastic zone of a crack terminating at a brittle-ductile interface

This paper presents the results of an investigation of the effects of elastic mismatch on the size of the plastic zone at the tip of cracks terminating at a bimaterial interface. Using the Williams technique, an asymptotic solution is obtained for the magnitude of the crack tip stress singularity and for the stress field associated with a semi-infinite crack impinging on an interface. This solution, together with the Von Mises yield criterion, is used to estimate the location of the plastic zone boundary r ₀ for various levels of the elastic mismatch, which are expressed in terms of the Dundurs constants. Results are expressed in terms of the non-dimensional quantity% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa% aaleaacaaIWaaabeaakmaadmaabaWaaSaaaeaacaaIXaaabaWaaOaa% aeaacaaIYaGaeqiWdahaleqaaaaakmaalaaabaGaam4AamaaBaaale% aacaqGjbaabeaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaaaa% kiaawUfacaGLDbaadaahaaWcbeqaaiabgkHiTiaaigdacaGGVaGaeq% 4UdWgaaaaa!4832!\[r_0 \left[ {\frac{1}{{\sqrt {2\pi } }}\frac{{k_{\text{I}} }}{{\sigma _0 }}} \right]^{ - 1/\lambda }\]where k _i is the stress intensity factor and ₀ is the yield stress. These results, together with an integral equation solution for k _i , are used to calculate the size of the plastic zone of a crack of length 2a loaded by uniform pressure. It is shown that the location of the boundary of the plastic zone depends strongly on the elastic mismatch.     

On the size of plastic zone ahead of crack tip

A method for the crack tip analysis of a tensile loaded crack (mode I) due to yielding of the material is developed. The stress/strain distribution within the plastic zone, as well as size of the plastic zone are presented. The development is based on the energy interpretation of the strain hardening exponent, and an analogy between mode III and mode I for the case of small scale yielding. Predictions of the proposed method are compared with the experimental results, and a fairly good agreement is observed. A number of proposed methods to estimate the plastic zone size for ductile materials are also discussed.     

A note on the plastic zone size ahead of the propagating fatigue crack

Crack length and plastic zone size measurements were made on a limited number of mild steel tensile specimens of ASTMGS No.3.5 and No. 7.0 grain size for various stress amplitudes and various mean stresses. The data are in agreement with the predictions of Bilby, Cottrell and Swinden concerning the static yielding, and indicate that maximum shearing stress rather than shear stress amplitude should be used for best correlation.

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Zusammenfassung An einer beschränkten Anzahl Zugprüflingen aus weichem Stahl der Korngrösse ASTMGS N 3,5 und No 7,0 wurden Risslänge und Grösse der plastischen Zone an der Risspitze bei verschiedenen Spannungsamplituden und verschiedenen mittleren Spannungen gemessen.Die erhaltenen Daten stimmen mit den von Bilby, Cottrell und Swinden für den Fall des statischen Versagens vorhergesagten Daten überein und zeigen an, dass im Hinblick auf eine gute Uebereinstimmung mit der Theorie eher die maximale Schubspannung als die Schubspannungsamplitude verwendet werden soll.

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Résumé Sur un nombre limité d'éprouvettes de traction d'un acier doux, dont la grosseur du grain ASTM était de 3,5 et de 7,0, on a mesuré la longueur de la fissure de fatigue, et l'étendue de la zone de déformation plastique précédant celle-ci, en fonction de diverses amplitudes de la tension alternée, et de divers niveaux de la tension moyenne.Les résultats sont en accord avec les prédictions de Bilby, Cottrell et Swinden concernant l'élongation statique et montrent que pour obtenir une corrélation optimale il convient de considérer la contrainte maximale de cisaillement plutôt que l'amplitude de la tension appliquée.

     

Influence of specimen size and configuration on the plastic zone size, toughness and crack growth

The main purpose of this investigation is to evaluate the effect of the thickness, width, aspect ratio and geometry of the fracture toughness specimen on the resultant displacement due to growth of plastic zone and crack.

The analytical part evaluates the effect of the width, aspect ratio, geometry and flow properties on the displacement due to the growth of plastic zone as well as the crack. The experimental part evaluates the effect of thickness and width of a compact tension specimen on the displacement and on the thickness direction contraction due to the growth of the plastic zone.

The main result of the investigation is that the plastic zone size decreases and the constraint to yielding increases as the width of a CT specimen increases. Based on this and other analytical and experimental result, a new procedure for the determination of K_IC has been proposed. The procedure is verified by experimental data obtained by other workers. The procedure overcomes the limitation of ASTM E399 for the determination of K_IC.     

Effect of thickness on plastic zone size in BCS theory of fracture

In the fracture theory of Bilby, Cottrell and Swinden, the plastic zone in an infinitely thick specimen was represented as a distribution of dislocations. For a specimen of finite thickness, Eshelby and Stroh have shown that the interaction force between two screw dislocations can be expressed by a modified Bessel function which decays much faster than the inverse first power law as the distance between the dislocations is increased. By assuming a continuous distribution of dislocations, an integral equation based on the Eshelby and Stroh interaction is formulated to examine the effect of thickness on the plastic zone size as well as the relative displacement along the crack surfaces. The distribution function of dislocations is obtained numerically from this integral equation. Numerical results indicate that for a given crack opening displacement the length of the plastic zone decreases with decreasing specimen thickness. Energy release rate is discussed from the dislocation point of view. A previous result that the energy release rate is equal to the total force on the dislocations in the plastic zone is generalized to the case of a plate with finite thickness.     

Closed-form solution for the size of plastic zone in an edge-cracked strip

This paper is concerned with the problem of plastic zone at the tip of an edge crack in an isotropic elastoplastic strip under anti-plane deformations. By means of complex potential and Dugdale model, the stress intensity factor and the size of plastic zone are obtained in closed-form. Furthermore, the analytic solutions for an edge crack at the free boundary of a half-space and a semi-infinite crack heading towards a free surface are determined as the limiting cases of the strip geometries.     

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.     

Elastic-plastic analysis of stress state and spread of plastic zone around single edge-cracked plate by finite element method

A two-dimensional finite element model for estimating elastic-plastic displacements and stresses near a crack for the elastic-plastic situation of loadings is presented. Singularity effects near the crack tip are solved by introducing 12-node cubic isoparametric elements in the crack tip region and collapsing them into triangular elements. The proposed finite element model is used to determine the spread of the plastic zone and mouth opening displacements of an edge-cracked structural steel plate. The spread of the plastic zone is obtained by successive increments of applied nominal tensile stress transverse to the crack length. The mouth opening displacement values obtained by this model are also compared with those measured experimentally.     

The shielding effects of the crack-tip plastic zone

In this paper we demonstrate that a plastically deformed zone around a stressed crack tip can be, mechanically, identified with an inclusion of transformation strain by means of Eshelby equivalent inclusion method. Thus, the shielding effect of the plastic zone can be quantitatively evaluated by the present transformation toughening theory. A closed-form solution to determine the change in the stress intensity factor induced by the plastic zone is given both for plane stress and plane strain mode I cracks under small-scale yielding conditions. By using the present solution, the effects of the strain-hardening behavior of the material, the plane stress and plane strain states and the T-stress on the crack-tip shielding effects are identified.     

A new method of plastic zone size determined based on maximum crack opening displacement

This paper presents a new method to determine the crack-tip plastic zone size (R_y) in the center cracked plate in tension. The maximum crack opening displacement (MCOD) is used to estimate R_y in this method. Based on the series of calculation results by finite element analysis, the explicit expression for the crack-tip plastic zone size versus MCOD is fitted by least square method. The expression enables to eliminate the influences of the yield stress of material, crack geometry, plate sizes and transverse stress. Therefore, the presented method of R_y determined by MCOD is suitable to any center crack finite plate of any material under uniaxial or biaxial tension.