The strain-hardening effect in HRR plane fields according to T-criterion

The strain-hardening effect on fracture is investigated with the aid of the T-criterion using HRR stress fields [1–3] around a crack tip in a power hardening material. Using the appropriate components of strain energy density for the elastic-plastic as well as a nearly elastic expression of the T-criterion, we find the fracture angles, as well as fracture stresses in materials possessing an elastic extended up to a perfectly plastic behavior, by considering plane mixed-mode deformation at the crack tip.Significant influence of the strain hardening coefficient, n on the fracture stress, as well as the hardening parameter mainly appeared in plane strain conditions. This phenomenon was observed almost independently of the solution applied, which provides a nearly elastic or an elastic-plastic expression of the T-criterion describing the fracture conditions.     

HRR fields for damaged materials

The paper presents an investigation of the interaction between a macroscopic crack and distributed damage in an elastic-plastic material based on the HRR field model for virgin materials. This is achieved by describing the mechanical effects of the distributed micro-cracks in terms of the damage variable D on the HRR fields. Damage evolution equation and the constitutive equations coupled with damage are formulated and the resulting boundary value problems are solved numerically. Material constants , n and m ₀are varied to examine their effects on the resulting stress distributions. It is found that the HRR fields for damaged and virgin materials are surprisingly similar although the severity of damage equivalent stress is of several orders of magnitude higher than the conventional plastic equivalent stress without damage consideration. Furthermore, it is shown theoretically and justified numerically that the J-integral loses its path independency for damaged materials, causing the amplitude of the singularity K _D to remain an unknown variable in the asymptotic analysis.     

Persistence of the HRR fields for general yielding in mode II

The finite element program BERSAFE is used to analyse an edge-cracked square plate under mode II loading in the elastic-plastic regime. Plane strain constraint and power law hardening withn = 3 are assumed. Crack tip stress and strain fields are investigated and shown to equal the HRR fields at all loads from small scale yielding through into general yielding.

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Résumé Le programme d'éléments finis BERSAFE est utilisé pour l'analyse d'une plaque carrée fissurée à ses bords et soumise en régime élastoplastique à une sollicitation de mode II. On suppose un état plan de déformation et un écrouissage régi par une loi de puissancen = 3. On étudie les champs de contraintes et de déformations à l'éxtrémité des fissures et on montre que ces champs sont équivalents à ceux décrits par Hutchinson, Rice et Rosengren (HRR) pour toute la gamme de mise en charges allant de la plastification locale de faible envergure à la plastification généralisée.

     

The T-criterion for engineering materials based on strength anisotropy

Summary In most engineering materials the yield limits in tension and compression are in general different. Their ratio,R= _0C / _0T , characterizing the strength anisotropy of the material, was found to influence significantly the modes of both plane stress and plane strain fractures. The theoretical analysis presented in this paper introduced the necessary modifications of theT-criterion of fracture in order to cover the effect of the strength anisotropy, called the strength-differential effect (SDE). According to the statement of the proposed fracture criterion, the maximum value of the ratio of strain energy density components,T _R =T _V /T _D , when calculated along the elastic-plastic boundary around the tip of a crack indicates the angle of initial crack path. Crack onset is characterized by a critical value of the SED. For the determination of the elastic-plastic boundary the most general form of a failure criterion, that is the paraboloid failure condition was used.With 12 Figures     

Crack tip fields in ductile crystals

Results on the asymptotic analysis of crack tip fields in elastic-plastic single crystals are presented and some preliminary results of finite element solutions for cracked solids of this type are summarized. In the cases studied, involving plane strain tensile and anti-plane shear cracks in ideally plastic f c c and b c c crystals, analyzed within conventional small displacement gradient assumptions, the asymptotic analyses reveal striking discontinuous fields at the crack tip.For the stationary crack the stress state is found to be locally uniform in each of a family of angular sectors at the crack tip, but to jump discontinuously at sector boundaries, which are also the surfaces of shear discontinuities in the displacement field. For the quasi-statically growing crack the stress state is fully continuous from one near-tip angular sector to the next, but now some of the sectors involve elastic unloading from, and reloading to, a yielded state, and shear discontinuities of the velocity field develop at sector boundaries. In an anti-plane case studied, inclusion of inertial terms for (dynamically) growing cracks restores a discontinuous stress field at the tip which moves through the material as an elastic-plastic shock wave. For high symmetry crack orientations relative to the crystal, the discontinuity surfaces are sometimes coincident with the active crystal slip planes, but as often lie perpendicular to the family of active slip planes so that the discontinuities correspond to a kinking mode of shear.The finite element studies so far attempted, simulating the ideally plastic material model in a small displacement gradient type program, appear to be consistent with the asymptotic analyses. Small scale yielding solutions confirm the expected discontinuities, within limits of mesh resolution, of displacement for a stationary crack and of velocity for quasi-static growth. Further, the discontinuities apparently extend well into the near-tip plastic zone. A finite element formulation suitable for arbitrary deformation has been used to solve for the plane strain tension of a Taylor-hardening crystal panel containing, a center crack with an initially rounded tip. This shows effects due to lattice rotation, which distinguishes the regular versus kinking shear modes of crack tip relaxation. and holds promise for exploring the mechanics of crack opening at the tip.     

Requirements for a one parameter characterization of crack tip fields by the HRR singularity

Detailed crack tip stress and strain fields are generated for the edge crack bar subjected to bending and the center cracked panel and single cracked panel subjected to tensile loads. These fields are compared with the singular fields, due to Hutchinson and Rice and Rosengren, at the same level of applied J. The comparisons show that the size R of the region at the crack tip dominated by the HRR singularity is substantially larger in the bend specimen than in the center-cracked panel for contained and large scale plasticity. Hardening properties also have a strong influence on R. The implications of these results on the minimum size requirement essential to a one parameter fracture criterion based on the J-integral or crack tip opening displacement are discussed.

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Résumé Lorsqu'un barreau fissuré sur ses bords est soumis à flexion, ou lorsqu'un panneau fissuré en son centre ou présentant une fissure simple est soumis à des contraintes de traction, des champs complexes de contrainte et de déformation aux extrémités des fissures considérées apparaissent. Ces champs sont comparés aux champs singuliers étudiés par Hutchinson et Rice et Rosengren pour le même niveau d'un J déterminé. Les comparaisons montrent que la dimension R de la région à l'extrémité d'une fissure sur laquelle agit de manière déterminante une singularité HRR se trouve être substantiellement plus grande dans une éprouvette de flexion que dans un panneau présentant une fissure centrale, et ce pour une plasticité contenue ou une plasticité à large échelle. Les propriétés de durcissement ont également une influence déterminante sur R. On discute les implications de ces résultats sur l'exigence de dimension minimum qui est essentielle dans le cas du critère de rupture à un paramètre basé sur une intégrale J ou dans le cas du COD.

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Appeared in General Electric Technical Information Series publication, October 1978.     

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.

     

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.     

Experimental study of the T-criterion in ductile fractures

In the present paper, the newly introduced T-criterion was applied in the case of the fracture of a ductile material. Polycarbonate of Bisphenol A (PCBA) specimens containing slant internal cracks were tested and the experimentally obtained angles of crack propagation, as well as the respective fracture loads were compared to corresponding predictions of the T-criterion. It was found out that the predicted and measured quantities are in good agreement, especially in cases where existing other fracture criteria fail. The prediction of the T-criterion, that for small angles of crack inclination there must be angles of propagation absolutely larger than 90°. was clearly verified. Also, the fracture load measured, for each of the same small angles of inclination, was more than three times lower, as compared to the values given by other criteria in use, and in excellent agreement with T-criterion predictions.     

A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems

This paper presents a novel numerical method for effectively simulating the singular stress field for mode-I fracture problems based on the edge-based smoothed finite element method (ES-FEM). Using the unique feature of the ES-FEM formulation, we need only the assumed displacement values (not the derivatives) on the boundary of the smoothing domains, and hence a new technique to construct singular shape functions is devised for the crack tip elements. Some examples have demonstrated that results of the present singular ES-FEM in terms of strain energy, displacement and J-integral are much more accurate than the finite element method using the same mesh.     

Invariability of singular stress fields in adhesive bonded joints

A numerical study of the influence of adherend stiffness on adhesive bonded joints provided an insight into some of their fundamental properties. The stress yield in the vicinity of singular locations, in stiff and flexible adherend joints, was investigated. Linear elastic finite element analysis was used to compare the different systems. A special gradual refinement technique in the vicinity of stress singularities was used and proved to be very efficient. Parametric finite element analysis allowed for development of closed form expressions for the stress distribution in the vicinity of the singular points The singular stress distribution zone is limited by the adhesive strip thickness. Thus the stress intensity factors are attenuated in comparison with homogeneous bodies. The stress field in the singular region normalized to the developed stress concentration factor, appears to be invariant, and independent of joint configuration. The subsequent invariability leads to a general relationship between the Stress Concentration Factor SCF, unique for adhesively bonded joints.     

A model for crack initiation in elastic/plastic indentation fields

A model is proposed for the initiation of microfracture beneath sharp indenters. Using a simple approximation for the tensile stress distribution in the elastic/plastic indentation field, in conjunction with the principle of geometrical similarity, fracture mechanics procedures are applied to determine critical conditions for the growth of penny-like median cracks from sub-surface flaws. The analysis provides a functional relationship between the size of the critical flaw and the indentation load necessary to make this flaw extend. Initiation is well defined (unstable) only if the critical flaw lies within a certain size range; outside this range, large flaws can extend stably but small flaws can not extend at all. No flaws can extend below a characteristic minimum load, values of the indentation variables at this load accordingly providing useful threshold parameters. These quantities involve the intrinsic deformation/fracture parameters, hardness and toughness, in a fundamental way, thereby establishing a basis for materials selection in fracture-sensitive applications.     

Tensile crack tip fields in elastic-ideally plastic crystals

Crack tip stress and deformation fields are analyzed for tensile-loaded ideally plastic crystals. The specific cases of (0 1 0) cracks growing in the [1 0 1] direction, and (1 0 1) cracks in the [0, 1, 0] direction, are considered for both fcc and bcc crystals which flow according to the critical resolved shear stress criterion. Stationary and quasistatically growing crack fields are considered. The analysis is asymptotic in character; complete elastic-plastic solutions have not been determined. The near-tip stress state is shown to be locally constant within angular sectors that are stressed to yield levels at a stationary crack tip, and to change discontinuously from sector to sector. Near tip deformations are not uniquely determined but fields involving shear displacement discontinuities at sector boundaries are required by the derived stress state. For the growing crack both stress and displacement must be fully continuous near the tip. An asymptotic solution is given that involves angular sectors at the tip that elastically unload from, and then reload to, a plastic state. The associated near-tip velocity field then has discontinuities of slip type at borders of the elastic sectors. The rays, emanating from the crack tip, on which discontinuities occur in the two types of solutions are found to lie either parallel or perpendicular to the family of slip plane traces that are stressed to yield levels by the local stresses. In the latter case the mode of concentrated shear along a ray of discontinuity is of kink type. Some consequences of this are discussed in terms of the dislocation generation and motion necessary to allow the flow predicted macroscopically.     

On the effect of plastic constraint on ductile tearing in a structural steel

The effect of plastic constraint on the initiation of ductile tears in a structural steel has been studied by measuring the crack opening displacement at initiation in 3-point bend specimens with deep and shallow notches. It is shown that δ_i for shallow notches where plastic flow reaches the surface is about twice that for deep notches. The crack opening displacement at maximum load is shown to be proportional to the specimen size. It is also shown that sidegrooves reduce this crack opening displacement.     

Development of plastic nonlinear waves in one-dimensional ductile granular chains under impact loading

A modified split Hopkinson pressure bar (SHPB) was used to load one-dimensional granular chains of metallic spheres under impact loading rates. These homogeneous chains, comprised of brass spherical beads ranging from a single sphere to a chain of sixteen, are of interest because of their unique wave propagation characteristics. In the elastic range, for loads around 10 s of N, nonlinear elastic solitary waves have been observed to form. In this work, loading magnitudes spanning from 9 kN to 40 kN – considerably higher than most previous works on these systems which have been conducted in the elastic regime – cause the granular chains to severely deform plastically. The aim of this study is to identify whether a nonlinear solitary-type wave will be generated under such high load levels, and if so, under what conditions (e.g., chain length, load level, etc.) it will do so. The propagating pulse was found to assume a distinctive shape after travelling through five beads, similar to the elastic case where solitary waves are realized with a traveling wavelength of five bead diameters. The wave speed of the plastic pulses observed here was seen to depend on maximum force, indicating that indeed it is a nonlinear wave in nature and is comparable to the elastic solitary wave. Locally, the plastic dissipation at every contact point through the chains was studied by measuring the residual plastic contact area. It was found that after the formation of the plastic nonlinear solitary wave had occurred there is also decreasing plastic deformation along the chain length except at the end beads in contact with the SHPB, which rebound into the SHPB bar causing larger plastic dissipation locally. To our knowledge this research is the first effort to investigate in detail the development and evolution of solitary-like waves in the plastic regime and will form the basis of future work in this area.     

Performance of variable order singular elements in analyzing interfacial fractures

This work concerns the development of singular boundary elements and the investigation of their numerical performance in analyzing interfacial cracks. In the vicinity of such cracks arise singular stress fields with variable order of singularity depending on the material characterizing parameters. The development of these elements which approximate displacement and traction functions is accomplished through controlled relocation of the mid-side node determined by compatibility and continuity requirements which must obey shape functions. These elements were applied to simulate the elastic behavior of cracks which are perpendicular and terminate on the interface of a bimaterial structure. Their efficiency in conjunction to the boundary only element method, are demonstrated in crack opening displacement diagrams and crack tip stress tabulated results.     

Asymptotic tensile crack-tip stress fields in elastic-perfectly plastic crystals

For a crack in an elastic-perfectly plastic crystal, there exist more than one asymptotic stress field around the crack tip. The condition governing which field occurs cannot be determined by the asymptotic fields, but depends on external loading conditions. For a plane-strain tensile crack in the (0 1 0) plane and growing in the [1 0 1] direction in a face-centered-cubic (fcc) crystal, there exist three asymptotic stress fields around the crack tip, including a previously established four-sector field and two new families of three-sector fields. The four-sector field gives a large mean stress ahead of the crack, 6.12, where is the critical resolved shear stress for slip systems {1 1 1}1 1 0. The first family of three-sector fields is parameterized by a single parameter p ranging from 0 to 1. In the limit p=0, the field degenerates to uniform tension in the direction parallel to the crack surface. In the other limit p=1, the corresponding field gives the maximum mean stress, 4.90 , in the family. The other family of three-sector fields also has two limits: one corresponds to uniform compression parallel to the crack, and the other provides the maximum mean stress in the family, 2.45 much less than the four-sector field. The stress distributions obtained by the finite element method confirm not only the four-sector field, but also two families of fields.