Experimental and finite element studies on mode I and mixed mode (I and II) stable crack growth—I. Experimental

Experimental results on mode I and mixed mode stable crack growth under static loadings through an aluminium alloy (D16AT) are presented. The compact tension type of geometry was employed for both the sets of tests. Data pertaining to load-deflection diagrams, crack opening displacements, crack front geometry, etc., are included. There is a greater spurt of crack growth at the initiation stage in a mixed mode than in mode I. The crack opening angle (COA) remained nearly constant during the whole stable growth. There is a substantial tunneling, the extent of which increases as the extension progresses in both mode I and mixed mode. The tunneling reduces as the ratio a₀/W increases. Because of this tunneling, the COD at a point finite distance behind the crack tip and on the specimen surface is much more than expected. At the maximum load the tunneling is 2 to 3.5 mm in the case of mode I. The crack extends initially almost along a straight line at an angle with the initial crack in a mixed mode. The maximum to initiation load ratio varied in the range 1.50 to 1.75 for the whole range of tests.     

A mixed mode crack tip finite element

A special finite element for plane analysis of elastic structures with through the thickness cracks is presented. Generalized displacements are used in developing the element, and it contains the proper singularities. The opening (K _I), in plane shearing (K _II), and combined (K _I+K _II) modes of deformation are present. The stiffness matrix is given explicitly and its eigenvalues are shown. Numerical results are presented and compared with other solutions.

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Résumé On présente un système spécial d'éléments finis pour l'analyse en état plan de structures élastiques comportant des fissures traversant l'épaisseur du produit. Les déplacements généralisés sont utilisés pour le développement de ces éléments, ceux-ci comportant leurs propres singularités. On présente l'ouverture (K _I) en cisaillement plan (K _II) et en modes combinés de déformation (K _I+K _II). La matrice de rigidité est donnée explicitement et on montre qu'elle correspond à eigenvalues. Les résultats numériques sont présentés et comparés avec d'autres solutions existantes.

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This work was sponsored by Martin-Marietta's Independent Research and Development Program.     

Determination of T-stress from finite element analysis for mode I and mixed mode I/II loading

The elastic T-stress has been recognised as a measure of constraint around the tip of a crack in contained yielding problems. A review of the literature indicates that most methods for obtaining T are confined to simple geometry and loading configurations. This paper explores direct use of finite element analysis for calculating T. It is shown that for mode I more reliable results with less mesh refinement can be achieved if crack flank nodal displacements are used. Methods are also suggested for calculating T for any mixed mode I/II loading without having to calculate stress intensity factors. There is good agreement between the results from the proposed methods and analytical results. T-stress is determined for a test configuration designed to investigate brittle and ductile fracture in mixed mode loading. It is shown that in shear loading of a cracked specimen T vanishes only when a truly antisymmetric field of deformation is provided. However this rarely happens in practice and the presence of T in shear is often inevitable. It is shown that for some cases the magnitude of T in shear is much more than that for tension. The effect of crack length is also investigated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Combination of finite element analysis and analytical crack tip solution for mixed mode fracture

A finite element program was developed which combines the analytical crack tip solution with a conventional finite element analysis and evaluates various crack tip parameters as part of the solution. This program was used to analyze cracked specimens subjected to mixed mode loading. The importance of retaining the second term of the series expansion for local stress, a contribution which is independent of the distance from the crack tip, was demonstrated. It was first shown analytically that the presence of a load applied parallel to the crack reveals itself only through this constant second term, which vanishes only for specific loading conditions. The results of the numerical analysis demonstrate that the stress intensity factor K_I is independent of the load applied parallel to the crack only when this term is included in the analytical crack tip solutions. Failure to include the constant term has the effect that K_I varies with the horizontal load. The parameter K₁₁ is independent of this load in both cases. This indicatesonce again that it is this constant term which accounts solely and entirely for the presence of a load applied parallel to the crack.     

Estimation of the plastic zone by finite element method under mixed mode (I and II) loading

Material fracture by opening (mode I) is not lonely responsible for fracture propagation. Many industrial examples show the presence of mode II and mixed mode I + II. The present work consists in the elaboration of a code to estimate the size of the plastic zone at the crack tip under mode I, mode II and mixed mode I + II loading. The computations are made according to Von Mises and Tresca criteria. The results obtained are compared to those measured by experiments.     

Fully automated mixed mode crack propagation analyses based on tetrahedral finite element and VCCM (virtual crack closure-integral method)

The authors have been developing a crack propagation analysis system that can deal with arbitrary shaped cracks in three-dimensional solids. The system is consisting of mesh generation software, a large-scale finite element analysis program and a fracture mechanics module. To evaluate the stress intensity factors, virtual crack closure-integral method (VCCM) for the quadratic tetrahedral finite element is adopted and is included in the fracture mechanics module. The rate and direction of crack propagation are predicted by using appropriate formulae based on the stress intensity factors. In this paper, the crack propagation system is briefly described and some numerical results are presented.     

Experimental and finite element study on stable crack growth in three point bending

Experimental and finite element results are presented on mode I and mixed mode (involving I and II only) stable crack growth under static loading through an aircraft grade aluminium alloy (D16AT) in three point bending. The results include load-displacement diagrams, J-integrals, plastic zones, tunneling (or crack front curving), etc. During experiment a substantial amount of tunneling is observed, the extent of which increases as the extension progresses in both mode I and mixed mode. The tunneling reduces as a_o/w increases. The crack extends initially almost along a straight line at an angle with the initial crack in a mixed mode. The maximum load is observed to be as high as 1.6 times the initiation load in the whole range examined. From the finite element study it is seen that, in a mixed mode, the J-integral at the onset of extension is the lowest compared with the values at the later stages. The plastic zone size grows as the stable extension progresses; the growth is approximately the maximum along the crack extension line. The direction of initial crack extension in a mixed mode can be predicted through an elastic finite element analysis and using the criterion of maximum tangential principal stress. The study also indicates that the load-displacement diagram associated with a mixed mode stable crack growth can be predicted reasonably accurately using the criterion of crack opening angle.     

Coupled meshfree and fractal finite element method for mixed mode two‐dimensional crack problems

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 John Wiley & Sons, Ltd.     

A mixed hybrid finite element for three-dimensional elastic crack analysis

A new three-dimensional crack tip element is proposed, which is based on a mixed hybrid stress/displacement model. A truncated series expansion of eigenfunctions for the straight semi-infinite crack is deduced and assumed for the internal stress and displacement fields in the element. The basic approach of constructing these hybrid elements is outlined. Their good capability, efficiency and accuracy for analyzing three-dimensional elastic crack problems are demonstrated by first numerical examples.

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Résumé On propose un nouveau type d'élément tridimensionnel pour l'extrémité d'une fissure, basé sur un modèle mixte contraintes hybrides/déplacements. On en tire un développement en séries tronquées des eigenfonctions relatives à une fissure droite semi-infinie, et on suppose qu'elle est représentative des champs de contraintes internes et de déplacements dans l'élément. L'approche de base utilisée pour construire ces éléments hybrides est soulignée. On démontre par de premiers exemples numériques qu'ils ont la capacité, l'efficacité et la précision nécessaires à l'analyse des problèmes élastiques et tridimensionnels de fissuration.

     

An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi‐brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well‐known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.     

Calculation of mixed mode (I/II) stress intensities at sharp V‐notches using finite element notch opening and sliding displacements

In this paper, a simple, robust, and an efficient technique has been proposed for accurate estimation of mixed mode (I/II) notch stress intensity factors (NSIFs) of sharp V‐notched configurations using finite element notch opening and sliding displacements at the selected number of nodes along the notch flanks. Unlike the crack problems, displacement field is rarely employed in the notch problems due to complexities introduced by the presence of rigid body displacements. One of the main emphasis of the present work is to neatly bypass these rigid body displacements and develop a simple approach for accurate computation of the NSIFs so that it can be easily incorporated in the existing code. Several benchmark problems have been analyzed. The results obtained using the present method show excellent agreement with the solutions available in the literature. Some new results have also been reported in the present work.     

The effect of pre-stress cycles on fatigue crack growth—an analysis of crack growth mechanism

Cyclic pre-stress increases subsequent fatigue crack growth rate in 2024-T351 aluminum alloy. This increase in growth rate, caused by the pre-stress, and the increased rate, caused by temper embrittlement as observed by Ritchie and Knott, cannot be explained by the crack tip blunting model alone. Each fatigue crack increment consists of two components, a brittle and a ductile component. They are respectively controlled by the ductility of the material and its cyclic yield strength.     

A mode II fracture specimen—finite element analysis

A series of studies which demonstrate the potential of a proposed mode II fracture specimen are reported. First, previously obtained photoelastic results[1–3] are reviewed so that this paper will be self-contained. Then, a finite element analysis is presented in order not only to corroborate the experimental results but also to calibrate the specimen for $\text{K}{}_{\mathrm{IIC}}$ measurements. Finally, a preliminary $\text{K}{}_{\mathrm{IIC}}$ test performed on a brittle steel specimen is described.     

A finite element analysis of stable crack growth in an aluminium alloy

Extensive stable cracking has been observed in large test pieces of 25 mm thick weldable AlMgZn alloy which is used in the construction of a portable bridge. Standard fracture specimens produced valid K_IC values, with short cracks exhibiting unstable fracture. Finite element analysis of the large specimens determined a valid J_R-curve that can increase the effective K_C by several times the K_IC value. The R-curve has an unusual ‘concave’ shape that is associated with the change from initially flat fracture to fully slant fracture. The early stages of the R-curve are affected by in-plane constraint that can be indexed by the T-stress. The R-curve can be used to explain the stability of long cracks in full-scale tests on a bridge prototype, compared with the instability of short cracks in small, standard test pieces.     

Adaptive finite element analysis of mixed‐mode crack problems with automatic mesh generator

Fully automatic advancing front type mesh generator to take care of crack and fracture problems has been presented. It is coupled with Zienkiewicz and Zhu error estimator, the refinement methodology depends on the concept of strain energy concentration for adaptive analysis of mixed‐mode crack problems. No investigation is reported in this direction so far. It has been found that the above combination proved to be very powerful for adaptive finite element analysis of mixed‐mode crack problems in two‐dimensional isotropic solids. Very accurate stress intensity factors have been obtained for a target error of 10 per cent with a minimum number of steps. Copyright © 2000 John Wiley & Sons, Ltd.     

Experimental study of I + III mixed mode fatigue crack transformation propagation

In this paper, compact tension specimens with tilted cracks under monotonic fatigue loading were tested to investigate I + III mixed mode fatigue crack propagation in the material of No. 45 steel with the emphasis on the mode transformation process. It is found that with the crack growth, I + III mixed mode changes to Mode I. Crack mode transformation is governed by the Mode III component and the transformation rate is a function of the relative magnitude of the Mode III stress intensity factor. However, even in the process of the crack mode transformation the fatigue crack propagation is controlled by the Mode I deformation.     

Solute distribution in crack tips under mixed mode I/II conditions

Summary Concentration solutions of crack problems under steady state conditions are expressed in terms of the Westergaard stress function. The problem of a central crack in an infinite plate subjected to a biaxial stress field at any angle of inclination with respect to the crack axis is considered in detail. The concentration distribution in the vicinity of the crack tip is obtained and is expressed in terms of the opening-mode and sliding-mode stress intensity factors. Constant terms, usually omitted, are incorporated into the concentration solution. The crack growth criterion based on the maximum concentration of diffusing species in front of the crack tip is reformulated by incorporating the constant terms of the concentration solution. It is shown that the omission of the constant terms may result in a significant error in the prediction of the critical quantities for crack growth.     

Shear band characterization of mixed mode I and II fully plastic crack growth

Fully plastic crack growth in singly-grooved plane strain tensile specimens is here characterized by the directions and amounts of fracture and slip on three planes. This model gives the crack growth ductility, defined as the axial displacement per unit ligament reduction (of practical importance in determining the stiffness of the surrounding structure that is needed to prevent unstable fracture) in terms of the fracture surface lengths and directions, as well as the deformation of the back surface. It also gives the directions and magnitudes of slip and fracture.Applied to six different structural alloys with strain-hardening exponents from 0.1 to 0.2, the model gave crack growth ductilities within 10 percent of the observed ones for the symmetrical configurations, where the values ranged from 0.25 to 0.40 and were unrelated to the strain-hardening exponent. For the asymmetrical configurations (that could occur near welds or shoulders), the crack growth ductility for the low hardening materials drops from 0.07 to 0.11. The predicted values (larger for the higher hardening alloys) were within 30 percent of the observed ones. Thus this slip plane model of fully plastic crack growth provides a useful correlation between macroscopic measurements made on the specimens after fracture, and the important loss of crack growth ductility that occurs in fully plastic asymmetric configurations with low strain-hardening materials.

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Résumé Dans ce travail, on caractérise la croissance complètement plastique d'une fissure dans des éprouvettes de traction à rainure simple en état plan de déformations par les directions et l'intensité de la rupture et des glissements selon les trois plans de référence. Ce modèle fournit la ductilité vis-à-vis de la croissance d'une fissure, définie comme le déplacement axial par unité de réduction de ligament, en fonction des longeurs et directions de la surface de rupture, ainsi que la déformation de la surface arrière. Cette ductilité présente une importance pratique pour la détermination de la raideur de la structure d'environnement nécessaire pour éviter une rupture instable. Le modèle fournit également les directions et amplitudes des glissements et de la rupture.Appliqué à six alliages de construction aux modules d'écrouissage compris entre 0.1 et 0.2, le modèle fournit les ductilités vis-à-vis dé la croissance d'une fissure avec un écart de 10% par rapport à celles observées dans des configurations symétriques où les valeurs, non liées aux modules d'écrouissage, s'étaient entre 0.25 et 0.40. Pour des configurations asymétriques, telles qu'on les rencontre près des soudures ou dans les épaulements, la ductilité vis-à-vis de la croissance des fissures tombe à des valeurs de 0.07 à 0.11, dans le cas de matériaux à faible écrouissage. Les valeurs prédites, plus élevées dans les alliages fortement sensibles au vieillissement, s'écartent de 30% des valeurs observées. Ainsi, le modèle à plans de glissement d'une croissance complètement plastique d'une fissure fournit une corrélation utile entre des mesures macroscopiques sur éprouvettes après rupture et l'importante perte de ductilité vis-à-vis de la croissance d'une fissure, rencontrée dans des configurations asymétriques totalement plastiques avec des matériaux à faible sensibilité à l'écrouissage.

     

Determination of crack tip parameters for ASCB specimen under mixed mode loading using finite element method

In this paper, a new asymmetric semicircular bend specimen (ASCB) is presented. Having more geometric parameters in asymmetric bend elements gives the opportunity of covering a wider range of K_?, K_?? and T-stress in comparison with classical SCB specimens. Finite element method is used to obtain these parameters from pure mode I to pure mode II. Extensive numerical calculations are made to get a wide range data for crack tip parameters of this specimen. It is observed that for ASCB specimens with specified geometries under pure mode II loading, one of the bottom supports can move horizontally without significant variation in Y_I. The complete sets of numerical results are obtained and can be used for verification and interpretation of future experimental results.     

The plastic stress ahead of a mixed mode I and II plane stress crack

The small scale yielding for mixed mode I and II plane stress crack problems in elastic perfectly-plastic solids is analysed by considering the stress field near the crack line. By expanding the stresses near the crack line and matching the stress field in the plastic zone with the elastic dominant field for a blunt crack near the crack line at the elastic-plastic boundary, the problem is reduced to solving a system of nonlinear algebraic equations. The relationship between the near-field mixity parameter M^p and the far-field mixity parameter M^e is detennined by solving the system of equations numerically. Analogous to Shih's calculation by the finite element method for the small scale yielding of mixed mode plane strain crack problems, the numerical results indicate that the shift from a mixed mode to a pure mode may not be a smooth one.