Linear micropolar elastic crack-tip fields under mixed mode loading conditions

Micropolar elasticity laws provide a possibility to describe constitutive properties of materials for which internal length scales may become important. They are characterized by the presence of couple stresses and nonsymmetric Cauchy stress tensor. Beyond the classical displacement field, the kinematical variables are augmented by a so-called microrotation field and its gradient, the latter introducing an internal length scale in the theory. For an isotropic, linear micropolar elastic material, the near-tip asymptotic field solutions for mode I and mode II cracks are derived. It is shown that these solutions behave similar to those according to the so-called couple stress theory, which has been investigated by Huang et al. (1997a), or similar to those derived for cellular materials by Chen et al. (1998). In particular, the singular fields have an order of singularity r ^–1/2 and are governed by some amplitude factors, having the meaning of stress intensity factors as in the classical linear elastic theory. The effect of material parameters on the stress intensity factors is studied by applying the finite element method to calculate the values of the stress intensity factors for an edge-cracked specimen of finite width.     

Quasi-statically growing crack-tip fields in elastic perfectly plastic pressure-sensitive materials under plane strain conditions

Quasi-statically growing crack-tip fields in elastic perfectly plastic pressure-sensitive materials under plane strain conditions are investigated in this paper. The materials are assumed to follow the Drucker-Prager yield criterion and the normality flow rule. The asymptotic mode I crack-tip fields are assumed to follow the five-sector assembly of Drugan et al. (1982) for Mises materials. The crack-tip sectors, in turns, from the front of the crack tip are a constant stress sector, a centered fan sector, a non-singular plastic sector, an elastic sector and finally a trailing non-singular plastic sector bordering the crack face. The results of the asymptotic analysis show that as the pressure sensitivity increases, the plastic deformation shifts to the front of the tip, the angular span of the elastic unloading sector increases, and the angular span of the trailing non-singular plastic sector bordering the crack surface decreases. As the pressure sensitivity increases to about 0.6, the angular span of the trailing non-singular plastic sector almost vanishes. The effects of the border conditions between the centered fan sector and the first non-singular plastic sector on the solutions of the crack-tip fields for both Mises and pressure-sensitive materials are investigated in details. This revised version was published online in August 2006 with corrections to the Cover Date.     

Higher-order crack-tip creep stress fields in materials under biaxial loading

We have developed and implemented a method for calculating the fields of parameters of the crack-tip creep stress-strain state by taking direct account of the higher-order terms. The paper presents some calculated data on the fields of crack-tip stresses, creep strain rates, and amplitude ratios in the creep case. The influence of loading biaxiality on redistribution of stresses between the creep stages and on the constraint parameters in failure is assessed. Translated from Problemy Prochnosti, No. 6, pp. 25–43, November–December, 2008.     

Asymptotic crack-tip fields in anisotropic power law material under antiplane shear

A hodograph transformation in conjunction with an appropriate affine transformation are both used to investigate the strain and stress fields near the crack tip in an anisotropic power law material under antiplane shear. Stress and strain exponents as well as angular distributions for the asymptotic stress and strain fields are obtained analytically. All the stress strain exponents are independent of material anisotropy, and the effect of material anisotropy on the asymptotic stress and strain field is discussed including higher order terms.     

Non-singular stress and velocity fields for crack-tip plastic zones and conditions for uniqueness

Non-singular plastic stress and velocity fields, for the tip of a crack of finite thickness and root radius, are developed as an elastic-plastic crack model that is likely to be more physically realistic than the classical infinitesimal crack with a plastic crack-tip singularity. With a non-singular plastic zone the velocity-field equations are not uniquely determined by the boundary conditions, under large geometrical changes, and they must therefore have the form of a wide set of kinematically-admissible velocity fields. These virtual velocity fields are used to establish the critical work-hardening rate to give a sufficient condition for uniqueness of the crack-tip velocity field in elastic-plastic fracture; it is shown that proof of uniqueness of the velocity field is likely to be an essential requirement for the valid application of elastic-plastic fracture mechanics.The elastic infinitesimal-crack model is shown to give an inadequate representation of the circumferential T-stress distribution at the surface of a crack of finite root radius, and this requires the adoption of a finite-thickness elliptical crack model to give approximate consistency between the elastic stress field and the non-singular plastic stress field at the crack tip.     

Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear

Analytical solutions of higher order fields in a fully plastic power-law hardening material are presented. By the use of hodograph transformation and asymptotic analysis the stress and strain exponents, angular distributions of shear stresses and strains are analytically determined. Special cases, such as linearly elastic, perfectly plastic materials are discussed. Similar characteristics between mode III and mode I plane strain, and mode II plane stress are examined. Comparison of four-term asymptotic solutions with exact and leading term solutions in an infinite strip with a semi-infinite crack under constant displacements along its edges is provided.     

Fully plastic crack-tip fields for plane strain shallow-cracked specimens under pure bending

Detailed finite element (PE) analyses are performed to study the effect of crack depth on crack-tip constraint at full yielding for pure bending of plane strain single-edge-cracked specimens. Analyses are based on small-strain formulations and perfect plasticity. The crack depth a/W ranges from 0.1 to 0.7, and the deformation is applied up to the limiting state of full plasticity where crack-tip stresses reach steady-state limiting values.At load levels smaller than the limit load (contained yielding), the crack-tip constraint (stress triaxiality) gradually decreases as a/W decreases, but, at load levels close to the limit load (or at the limit load), it decreases very sharply. In terms of a/W, tractable closed-form approximations for fully plastic crack-tip stress and strain fields are proposed, and fully plastic values of crack-tip stresses are re-phrased in terms of the Q-parameter [1, 2]. The role of crack-tip strains on fracture of shallow-cracked bending specimens is briefly discussed.     

Effect of decohesion and sliding on bimaterial crack-tip fields

This work is concerned with the analytical characterization of the effect of bond decohesion and sliding on the fields surrounding the tip of an interface crack. We consider the two-dimensional problem of an interface crack along the bond between a pair of linearly elastic materials. The interface itself has a nonlinear constitutive property: it has maximum load carrying capacities in both tension normal to the bond and in shear. The interface therefore has the ability to slide and separate inelastically without loss of integrity. The effects of these physically motivated assumptions are deduced and discussed. Further impetus for this study stems from the recent resurgence of interest in interfacial fracture mechanics. This interest is partly driven by the desire to understand and alleviate the pathological difficulties associated with the crack-tip fields predicted by the linear theory of elasticity. By accounting for possible interfacial nonlinear behavior, we are able to find that near-tip fields are free of the offensive properties alluded to above.

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Résumé On considère le problème à deux dimensions d'une fissure d'interface suivant la liaison entre deux matériaux élastiques linéaires, et la caractérisation analytique de l'effet d'une décohésion et d'un glissement de cette liaison sur le champ entourant l'extrémité de la fissure d'interface.Ce dernier possède lui-même une propriété constitutive non linéaire et présente une capacité de chargement maximum à la fois suivant une traction normale par rapport au joint et en cisaillement. Dès lors, l'interface peut glisser ou se séparer de manière inélastique sans perte de l'intégrité de la liaison. Ces hypothèses physiques ont des effets qui sont discutés. Une telle étude est stimulée par le regain récent d'intérêt pour la mécanique de rupture des interfaces, en partie dû au désir de comprendre et d'alléger les difficultés auxquelles conduit une prédiction des champs à l'extrémité d'une fissure en se basant sur la théorie linéaire de l'élasticité. En prenant en compte la possibilité d'un comportement non linéaire de l'interface, on est à même de trouver que les champs au voisinage de l'extrémité de la fissure ne présentent pas de propriétés adverses.

     

Influence of damage in the vicinity of a crack-tip in embossed low-basis-weight paper

It is experimentally observed that the fracture process developed in an embossed paper sample having a periodic matrix organized pattern is distributed over several indentation rows in the region near a macroscopic crack. This observation suggests that the stresses at the tip of the crack is shielded by damage in neighboring rows of indentation: energy dissipation may occur not only at the tip of the main crack, but in several indentation rows parallel to the main crack.In this investigation, a model describing the in-plane fracture behavior of embossed low-basis-weight paper is developed. It is found that the model is capable of capturing the development of damage in rows parallel to the main crack and compares well with experimental results.     

Plane-stress crack-tip fields for pressure-sensitive dilatant materials

In this paper we present plane-stress crack-tip stress and strain fields for pressure-sensitive dilatant materials. A hydrostatic stress-dependent yield criterion and the normality flow rule are used to account for pressure-sensitive yielding and plastic dilatancy. The material hardening response is specified by a power-law relation. The plane-stress mode I singular fields are found in a separable form similar to the HRR fields (Hutchinson, J. Mech. Phys. Solids 16, 13–31 and 337–347, 1968; Rice and Rosengren, J. Mech. Phys. Solids 16, 1–12, 1968). The angular variations of the fields depend on the material hardening exponent and the pressure sensitivity parameter. Our low-hardening solutions for different degrees of pressure sensitivity agree well with the corresponding perfectly plastic solutions. An important aspect of the effects of pressure-sensitive yielding and plastic dilatancy on crack-tip fields is the lowering of the opening stress and the hydrostatic stress directly ahead of the crack tip. This effect, similar to that under plane-strain conditions (Li and Pan, to appear in J. Appl. Mech. 1989), has implications in the material toughening observed in some ceramic and polymeric composites.     

Influence of microcracking on crack-tip toughness of alumina

Al₂O₃-samples with different grain sizes were produced and the crack-tip toughness K_I0, also called intrinsic fracture toughness, was determined using two different measurement techniques. It appeared that K_I0 depends strongly on specimen grain size with the mechanistic link provided by the increase of microcrack density with grain size. Results obtained using measurements of crack opening displacements (COD) lie considerably lower than values based on a method using bending tests of pre-notched bend bars. It is suggested that the latter method relies on the use of a notch with a microcrack, which in fact is different than the behavior of a long crack alone.     

Plane-stress crack-tip fields for perfectly plastic orthotropic materials

Plane stress mode I crack-tip fields for perfectly plastic orthotropic materials are studied. Plastic orthotropy is described by Hill's quadratic yield function. The construction of crack-tip fields is based on the general crack-tip field analysis for elastic perfectly plastic materials given by Rice [1] and guided by the corresponding low-hardening power-law solutions. Two very different types of plane-stress crack-tip fields emerge as plastic orthotropy is varied. The first one consists of a centered fan sector in front of the crack tip and two neighboring constant stress sectors. The second one consists of a constant stress sector in front of the crack tip, a constant stress sector bordering the crack face, and a centered fan sector between the two constant stress sectors. All the perfectly plastic crack-tip solutons are verified by the corresponding low-hardening power-law solutions. General trends of crack-tip stress solutions as functions of plastic orthotropy and implications of these solutions to the design of ductile composite materials are discussed.

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Résumé On étudie les champs de contraintes planes de mode I à l'extrémité d'une fissure, dans les matériaux orthotropiques parfaitement plastique. L'orthotropie plastique est décrite par la fonction quadratique de plasticité de Hill. On base les constructions des champs de constraintes sur l'analyse générale des constraintes à l'extrémité d'une fissure fournie par Rice pour les matériaux élastiques parfaitement plastiques, que l'on règle par les lois paraboliques caractérisant un faible écrouissage. Lorsque l'on modifie l'orthotropie plastique, il apparaît deux types de champs de contraintes à l'extrémité de la fissure très différents. Le premier comporte un secteur en éventail centré sur le front de fissure, et deux secteurs voisins à contraintes constantes. Le second consiste en une secteur à contrainte au bord de la surface de la fissure, et un secteur en éventail centré sur les deux secteurs à contraintes constantes. Toutes les solutions relatives à une extrémité de fissures parfaitement plastique sont vérifiées par les fonctions paraboliques d'écrouissage faible correspondantes. On discute des tendances générales que suivent les solutions pour les contraintes en extrémité de fissure selon l'orthrotropie plastique, et des implications que comportent ces solutions dans la conception de matériaux composites ductiles.

     

Plane-strain crack-tip fields for power-law hardening orthotropic materials

Near-tip stress and strain fields for power-law hardening orthotropic materials under plane-strain conditions are presented. Plastic orthotropy is described by Hill's quadratic yield function. The angular variations of these HRR-type fields depend on a single parameter which specifies the state of plastic orthotropy. Near-tip fields for highly orthotropic materials differ substantially from the fields for isotropic materials. Mode I (symmetric) and mode II (anti-symmetric) solutions for different degrees of plastic orthotropy are given. The angular stress distributions for the low-hardening material agree remarkably well with the plane-strain slip-line fields. Based on the singularity fields, effective stress contours are constructed. The applicability of these fields in the context of a fiber-reinforced composite containing a macroscopic flaw is discussed.     

Numerical analysis of crack-tip fields using the hodograph transformation

A numerical method is discussed for analysis of anti-plane shear crack-tip fields in linear and non-linear materials, including possible anisotropy. The method is based on the hodograph transformation. It allows efficient computation of singular stress fields because the transformed partial differential equation to be solved numerically is linear. The inversion of the transformation is also done numerically. The method is promising for the study of localizations, since regions with the largest stress gradients are automatically discretized with the finest mesh owing to the nature of the transformation.     

Transient crack-tip fields for mixed-mode power law creep

Mixed-mode stationary crack-tip fields are obtained for an elastic-nonlinear viscous power law creeping solid under conditions of plane strain and small-scale creep. Power law exponents of 2 and 5 are considered which are representative of the creep response of a wide range of ceramics and metals. The imposed far-field mixity ranges from pure mode I to pure mode II. Crack tip fields are calculated during the transient regime using a detailed finite element analysis and are shown to be governed by a Hutchinson-Rice-Rosengren type singularity over the inner one fifth of the creep zone. Dominance of universal mixed-mode near-tip fields within the inner creep zone is found for several mixtures of far-field mode I and mode II. The pronounced effects of the amount of mixity on the size and shape of the creep zone as well as on the time required to reach extensive creep conditions are determined. For a creep exponent of 5, it is estimated that the creep zone grows about seven times faster in mode II than in mode I, with a corresponding decrease in the transition time from small-scale to extensive creep. For a creep exponent of 2, the creep zone grows about six times faster in mode II than in mode I. Finally, the mixed-mode creep fields are used to assess possible beneficial effects of crack deflection or branching in metals and ceramics at elevated temperatures.     

Singular crack-tip fields for pressure sensitive plastic solids

Plain strain mode-I singular plastic fields are examined for cracks embedded in pressure sensitive solids. Material response is described by a small strain deformation theory in conjunction with elliptic yield criterion and plastic potential. Non-associativity is accounted for and a pure power law is assumed to characterize strain hardening. The material does not admit a strain energy function hence it is not possible to deduce a-priori the J-integral motivated stress singularities. A standard separation of variables representation of near-tip eigenfunctions has been evaluated numerically, over a range of material parameters. It has been found that stress singularities may deviate from J-integral predictions, with increasing non-associativity, by up to nearly 20%. Sample illustrations are provided for singular field profiles and some aspects of pressure sensitive non- associated plasticity are discussed.     

Plane-stress crack-tip fields for power-law hardening orthotropic materials

Plane stress mode I near-tip fields in orthotropic materials are examined. Plastic orthotropy is described by Hill's quadratic yield function and the strain hardening behavior is given by an appropriate generalization of a uniaxial tensile power-law stress-strain relation. Pronounced changes in the pattern of the angular variations of crack-tip fields have been observed with the degree of plastic orthotropy and the amount of strain hardening. Possible shapes and sizes of plastic zones (as inferred from effective stress contours) are presented for high- and low-hardening materials and a wide range of plastic orthotropy. The shape of the plastic zone for a particular case of plastic orthotropy agreed remarkably well with the zone of intense straining induced by an appropriately orientated crack within a graphite/epoxy laminate.

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Résumé On examine les champs de contraintes planes selon un mode I au voisinage de l'extrémité d'une fissure dans des matériaux orthotropes. L'orthotrope plastique est décrite par la fonction quadratique de plastification de Hill, et le comportement à l'écrouissage est donné par une généralisation adéquate d'une relation tensioncilatation de forme parabolique, sous traction mono-axiale. On a observé des modifications profondes dans l'aspect des variations angulaires des champs d'extrémité de fissure, selon le degré d'orthotropie plastique et infensité de l'écrouissage. Pour des matériaux très sujets ou peu sujets à l'écrouissage, et pour une large gamme d'orthotropies plastiques, on présente les formes et dimensions possibles des zones plastiques, telles qu'elles se deduisent des contours effectifs de contraintes. La forme de la zone plastique correspondant au cas particulier d'une orthotropie plastique s'accorde remarquablement bien à la zone de dilatation importante créée par une fissure d'orientation appropriée, dans une plaque de graphite-epoxy.

     

Mixed-mode crack-tip stress fields for orthotropic functionally graded materials

Quasi-static mixed mode stress fields for a crack in orthotropic inhomogeneous medium are developed using asymptotic analysis coupled with Westergaard stress function approach. In the problem formulation, the elastic constants E ₁₁, E ₂₂, G ₁₂, ν ₁₂ are replaced by an effective stiffness ${E=\sqrt {E_{11} E_{22}}}$ , a stiffness ratio ${\delta =\left({{E_{11}}\mathord{\left/ {\vphantom {{E_{11}} {E_{22}}}}\right. \kern-0em} {E_{22}}} \right)}$ , an effective Poisson’s ratio ${\nu =\sqrt {\nu_{12}\nu _{21}} }$ and a shear parameter ${k=\left({E \mathord{\left/ {\vphantom {E {2G_{12}}}}\right. \kern-0em} {2G_{12}}}\right)-\nu }$ . An assumption is made to vary the effective stiffness exponentially along one of the principal axes of orthotropy. The mode-mixity due to the crack orientation with respect to the property gradient is accommodated in the analysis through superposition of opening and shear modes. The expansion of stress fields consisting of the first four terms are derived to explicitly bring out the influence of nonhomogeneity on the structure of the mixed-mode stress field equations. Using the derived mixed-mode stress field equations, the isochromatic fringe contours are developed to understand the variation of stress field around the crack tip as a function of both orthotropic stiffness ratio and non-homogeneous coefficient.