Strain energy release rates for an interface crack in orthotropic media--a finite element investigation

Strain energy release rate (SERR) components for an interface crack in two-dimensional orthotropic media were obtained using finite element (FE) analysis. The elastic analysis of interface cracks results in oscillatory singularity. This is prevalent over a very small zone near the crack-tip, where the traction free crack faces undergo unacceptable deformations resulting in the interpenetration of crack faces. The individual and total strain energy release rates are calculated using modified crack closure integral (MCCI) method. Although the total SERR converges, it is observed that the individual SERR components are dependent on the values of the smallest element size (Δa) at the crack-tip. It is observed that both the crack opening and sliding displacements are oscillatory when the interpenetration is allowed in the contact zone. The contact zone length (r_c) calculated using Suo's analytical expression [Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. Royal Soc. London, Ser A427 (1990) 331] is in good agreement with the results from FE analysis and MCCI calculations. However, for the chosen material properties, the estimated contact zone length based on the analytical expression proposed by Ni and Nemat-Nasser [J. Mech. Phys. Solids 39 (1991) 113] exhibits a large deviation from the present FE results. It is seen that the mode-II behavior dominates the crack growth, even under mode-I loading.     

Explicit expressions for energy release rates using virtual crack extensions

A new finite element technique for calculating energy release rates is presented. An explicit expression for energy changes due to virtual crack extensions is formulated based on a variation of isoparametric element mappings. Energy release rates are calculated directly from integral expressions evaluated over singular quarter-point isoparametric elements surrounding the crack tip. Since the energy release rates are expressed in variational form, there is no need for the analyst to select a small finite crack extension to simulate a virtual crack extension. The method is shown to produce very accurate solutions even with fairly coarse element meshes. A similar technique for mixed-mode fracture based on mutual potential energy release rates is described.     

Creep crack growth in an elastic-creeping material Part II: mode I

The quasi-static growth of a crack in an elastic-creeping material under mode I loading is investigated. The creep strain rate of the material is assumed to be governed by a power law involving the stress and creep strain. The major emphasis of this investigation is on elastic-primary creep response. The asymptotic crack tip fields for a quasi-statically extending crack under conditions of plane strain and plane stress are developed. The asymptotic fields are unambiguously determined in terms of the instantaneous crack speed and material parameters and are independent of the prior crack history, specimen geometry, and loading. A plane strain finite element analysis is performed to determine the complete stress and strain fields. These fields are compared with the asymptotic ones to establish the zone of dominance of the crack tip fields. The zone of dominance can be a very small fraction of the size of the creep zone attending the crack up.

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Résumé On étudie la croissance quasi-statique d'une fissure dans un matériau sujet à fluage élastique sous une sollicitation de Mode I. On suppose que la vitesse de déformation en fluage du matériau est régie par une loi parabolique comportant la contrainte et la déformation. L'accent est surtout mis sur la réponse au fluage élastique primaire.On établit les configurations asymptotiques règnant à l'extrémité de la fissure, dans le cas d'une fissure quasi-statique en extension sour état plan de dilatation ou sous état plan de tension. Ces champs sont déterminés sans ambiguïté par la vitesse instantanée de la fissuration et par les paramétres du matériau; ils sont indépendants de l'histoire primitive de la fissure, de la géométrie de l'éprouvette et de la sollicitation. On effectue une analyse par éléments finis en déformations planes pour déterminer complètement les champs de contrainte et de dilatation, que l'on compare aux champs asymptotiques aux fins d'établir la zone de prédominance des champs à l'extrémité de la fissure. Cette zone peut être une fraction très petite de la taille de la zone où s'effectue un fluage au voisinage de l'extrémité de la fissure.

     

Prediction of energy release rates for crack growth using FEM-based energy derivative technique

This paper presents a new method, named energy derivative technique, to calculate energy release rate for a variety of crack growth scenarios. The new method is based on energy conservation principle for crack growth, and is applicable to crack development in any quasi-static condition in which dynamic energy for crack growth is negligible. The method has the advantage over existing finite element-based methods in that the former does not require an elaborate fine mesh in the vicinity of a crack tip, and is not limited to linear deformation behaviour. Several case studies are presented to demonstrate validity of the method, which are (i) growth of penny-shaped crack for linear elastic fracture behaviour, (ii) crack growth in rubber sheet under tension for nonlinear elastic fracture behaviour, (iii) delamination in end-notched flexure specimen with friction, and (iv) crack growth with plastic deformation in double-edge-notched plate under tension. Results from these case studies show excellent agreement with data available in the literature, which were determined using either analytical or other FEM-based techniques.     

Elastic-plastic analysis of an arbitrarily-oriented mode III crack touching an interface

In this paper we investigate a semi-infinite crack terminating at an arbitrarily oriented interface between two elastic-plastic materials under an anti-plane shear loading. An analytical solution is first developed for general power-law hardening materials under a mode III loading. If both materials have the same hardening exponent, the formulation results in a nonlinear eigenequation which can be solved numerically. The stress singularities are determined as a function of two material constants: the hardening exponent n and parameter G which represents the relative resistance of the two materials. In addition to the power of the singularity, the stress, strain and displacement asymptotic fields are also determined. If the hardening exponents are not the same, the leading order terms of an expansion model ensure the stress continuity across the interface. The results show that the stress singularity mainly depends upon the material having the larger hardening exponent, with the highest stresses in the material having the smaller hardening exponent. By taking the hardening exponent n , the perfectly plastic bimaterial problem is studied. It has been found that if the crack lies in the less stiff material, the entirely plastic asymptotic fields around the crack tip can be determined. On the other hand, if the crack lies in the stiffer material, the crack-tip fields are partially elastic and partially plastic. For both cases, unique asymptotic fields can be determined explicitly. For those cases when the materials present a strain hardening property, different mathematical models are established.     

The energy release rate for a quasi-static mode I crack in a nonhomogeneous linearly viscoelastic body

The Energy Release Rate (ERR) for the quasi-static problem of a semi-infinite mode I crack propagating through an inhomogeneous isotropic linearly viscoelastic body is examined. The shear modulus is assumed to have a power-law dependence on depth from the plane of the crack and a very general behavior in time. A Barenblatt type failure zone is introduced in order to cancel the singular stress and a formula for the ERR is derived which explicitly displays the combined influences of material viscoelasticity and inhomogeneity. The ERR is calculated for both power-law material and the standard linear solid and the qualitative features of the ERR are presented along with numerical illustrations.     

Energy release rate calculations for an interface mode III edge crack based on a conservation integral

The M-integral is applied to the calculation of energy release races for interface edge cracks of the Mode III type. Specifically, for an edge crack along the interface between two elastic wedges of different opening angles and dissimilar elastic properties, and that is subjected to point loads at the apex, a relation is derived among the length of the crack, the energy release race of the crack, the applied loads, the wedge angles and the material parameters.     

Energy release rates of bi-material interface crack including residual thermal stresses: Application of crack tip element method

Crack tip element method is applied to the formulation of the energy release rate associated with interfacial crack growth of laminates with residual thermal stresses using the Timochenko beam model. Special attention is paid to the energy release rates of double cantilever beam and mixed-mode bending tests of bi-material specimens, and mode-I and mode-II energy release rates are formulated including residual thermal stresses. The derived results are verified by the comparison to finite element analysis, and the effect of residual thermal stresses on the mode mixity of the double cantilever beam and mixed-mode bending tests is discussed.     

On the stress singularity at an interface crack

The plane elastostatic stress field near an interface crack is considered and a model developed for the interface crack tip which is free from the inter-penetration of the crack flanks present in some earlier investigations, yet permits propagation in a crack opening mode along or near the interface. The basis of the model lies in taking, as the configuration to which the loads are applied, a crack tip which subtends an opening angle. By means of an asymptotic eigen-analysis, the main features of this departure from the norm in LEFM are identified and some examples presented which demonstrates its potential usefulness.

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Résumé Le champ de contrainte plane élastostatique au voisinage d'une fissure d'interface est prise en considération, et on développe un modéle pour l'extrémité de la fissure d'interface qui ne dépend pas de l'interpénétration des bords de la fissure telle qu'on l'avait envisagée dans des études précédentes, mais qui permet une propagation selon un mode d'ouverture de fissure le long ou au voisinage de cet interface. La base du modèle consiste à prendre comme configuration à laquelle les charges sont appliquées une extémité de fissure sous-tendant un angle d'ouverture. Au moyen d'une eigen-analysis asymptotique, les caractéristiques principales de cette déviation des normes de la mécanique de rupture linéaire élastique sont identifiées, et quelques exemples sont présentés qui démontrent son utilité potentielle.

     

Fatigue crack growth rates under sequential mixed-mode I and II loading cycles

One common mode of failure that occurs in rolling bodies such as gears, bearings and rails is due to the fatigue process. Several research workers suggest that rolling contact fatigue cracks are subjected to mixed mode I and II loading cycles. It is believed that the correct modelling of loading cycles can help us to study the mechanics of crack growth because fatigue comprises a major safety consideration in the design process. Experiments have been performed under nonproportional mixed-mode I and II loading cycles with fixed degrees of overlap, so that coplanar cracks were produced. Three empirical crack propagation laws have been established which are related to both mode I and mode II effective stress intensity factor ranges.     

Roughness of a mode I in-plane crack front propagating along a heterogeneous cohesive interface

The roughness of the crack front of an interfacial crack propagating along a weak plane in a heterogeneous disordered medium has been repeatedly studied both experimentally and numerically. For an interfacial toughness varying randomly on the interface, the front is self-affine. Quite often, however, the calculated roughness exponent differs from the experimental estimate. Several theoretical models have been employed up to now in the numerical simulations (elastic line depinning, random fuse and spring or beam models). In this paper we present finite element simulations (FEA) of the macroscopic mode-I static propagation of a crack front along a planar interface of an elastic or elastic-plastic coating adhered to a rigid substrate. The interfacial elements separate obeying a cohesive law, their toughness spatially fluctuating at random. The cohesive elements here employed allow for taking into account local I + II + III mixed-opening mode, i.e., allow for mode mixity at the local level. Our results indicate that for a given macroscopic toughness the crack front roughness is strongly sensitive to both the local cohesive law and the local fracture criterion.     

Intersonic mode II crack acceleration

An analytic solution is given for intersonic mode II crack acceleration. The solution is obtained by first considering a self-similar problem, motion of a semi-infinite crack at constant velocity, under the action of a constant crack face load, appearing behind the crack edge. This solution is then repeatedly used in a superposition scheme, which finally yields the solution for an accelerating crack.     

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.

     

On the calculation of energy release rates for cracked laminates

A general method is given for calculating the energy release rate G from the local values of bending moments and loads in a cracked laminate. This total value is then partitioned into mode I and II components. Examples are given of the analysis of several test geometries including both variable and constant ratio mixed mode tests. Solutions for compression failures with buckling are also given. Finally there is some discussion of specimen compliances and stability criteria for fixed load and fixed displacement.

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Résumé On fournit une méthode générale de calcul de la vitesse G de relaxation de l'énergie, à partir des valeurs locales des moments de flexion et des charges agissant sur un composite lamellaire fissurée. On divise selon les composantes de Mode I et de Mode II la valeur totale obtenue. Des exemples d'analyse appliquée à diverses géométries d'essai sont fournis, qui sont relatifs à des essais de mode mixte à ratios variable ou constant. On fournit également les solutions relatives à de la ruine par compression avec flambage. Enfin, on discute de la compliance des éprouvettes et des critères de stabilité dans le cas de charge imposées et de déplacements imposés.