A boundary layer framework considering material gradation effects

This paper describes the development and application of a novel modified boundary layer (MBL) model for graded nonhomogeneous materials, e.g. functionally graded materials (FGMs). The proposed model is based on a middle-crack tension, M(T), specimen with traction boundary conditions applied to the top and lateral edges of the model. Finite element analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates elements with graded elastic and plastic properties. Elastic crack-tip fields obtained from the proposed MBL model show excellent agreement with those obtained from full models of the cracked component for homogeneous and graded nonhomogeneous materials. The K-T dominance of FGMs is investigated by comparing the actual stress fields with the asymptotic stress fields (the Williams’ solution). The examples investigated in the present study consider a crack parallel to the material gradient. Results of the present study provide insight into the K-T dominance of FGMs and also show the range of applicability of the proposed MBL model. The MBL model is applied to analyze the elastic-plastic crack-tip response of a Ti/TiB FGM SE(T) specimen. The numerical results demonstrate that the proposed MBL model captures the effect of T-stress on plastic zone size and shape, constraint effects, etc. for such configurations.     

J–Q characterized stress fields of surface-cracked metallic liners – II. Composite overwrapped pressure vessels

Two-parameter J–Q elastic–plastic crack front fields are developed for surface-cracked metallic liners of composite overwrapped pressure vessels (COPV). Uniaxial tensile data from 6061-T6 aluminum coupon specimens for the metallic liner and anisotropic elastic material properties for the filament wound carbon fiber epoxy are used in three-dimensional finite element models. Modified boundary layer (MBL) finite element solutions are used to evaluate near tip dominance and parameterization limits. Semicircular surface cracks of varying depths inserted in the inner liner surface are investigated. J and Q crack front distributions, and the corresponding parameterization limits, and near tip triaxiality trends are obtained and the effects of elastic–plastic material discontinuity of the heterogeneous joint and the biaxiality of stresses are evaluated. J–Q predicted fields maintain accuracy for higher far-field loads and lower near tip deformations compared to the liner only models for angles near the free surface. However, for the critical crack growth region, Q does not maintain a radially independent measure of constraint for loads seen in a typical COPV; therefore, these fracture predictors may not be applicable. In the COPV, large-scale yielding marks a transition where triaxiality is higher as a function of constraint compared to the linear relationship common to homogeneous structures. Results from this study will facilitate the implementation of proof test logic and accurate fracture prediction of COPV liners with emphasis on geometric limits and fracture specimen applicability.     

Two-parameter characterization of elastic-plastic crack front fields: Surface cracked plates under tensile loading

In this paper the J-Q two-parameter characterization of elastic-plastic crack front fields is examined for surface cracked plates under uniaxial and biaxial tensile loadings. Extensive three-dimensional elastic-plastic finite element analyses were performed for semi-elliptical surface cracks in a finite thickness plate, under remote uniaxial and biaxial tension loading conditions. Surface cracks with aspect ratios a/c = 0.2, 1.0 and relative depths a/t = 0.2, 0.6 were investigated. The loading levels cover from small-scale to large-scale yielding. In topological planes perpendicular to the crack fronts, the crack stress fields were obtained. In order to facilitate the determination of Q-factors, modified boundary layer analyses were also conducted. The J-Q two-parameter approach was then used in characterizing the elastic-plastic crack front stress fields along these 3D crack fronts. Complete distributions of the J-integral and Q-factors for a wide range of loading conditions were obtained. It is found that the J-Q characterization provides good estimate for the constraint loss for crack front stress fields. It is also shown that for medium load levels, reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from finite element analysis. These results are suitable for elastic-plastic fracture mechanics analysis of surface cracked plates.     

J-integral evaluation for cases involving non-proportional stressing

Calculation of J for cases where the proportional stressing condition cannot be satisfied is investigated. A modified J definition is derived and implemented into an ABAQUS post-processing program for both 2-D and axisymmetric problems. The modified J-integral is path independent for cases of proportional and non-proportional stressing. For cases with proportional stressing, the modified integral gives the same value as does the standard ABAQUS J function. It is also found that the modified J is equivalent to the stress intensity factor for a linear elastic material and provides a measure of the intensity of the crack-tip fields for non-linear elastic and elastic-plastic materials. The modified J formulation is applied to the case of a cylinder with an external circumferential crack under various load conditions.     

Interfacial crack-tip constraints and <Emphasis Type="Italic">J</Emphasis>-integral for bi-materials with plastic hardening mismatch

This paper investigates interfacial crack tip stress fields and the J-integral for bi-materials with plastic hardening mismatch via detailed elastic-plastic finite element analyses. For small scale yielding, the modified boundary layer formulation with the elastic T-stress is employed. For fully plastic yielding, plane strain single-edge- cracked specimens under pure bending are considered. Interfacial crack tip stress fields are explained by modified Prandtl slip-line fields. It is found that, for bi-materials consisting of two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and the J-contribution there. The implication of asymmetric J-integral in bi-materials is also discussed.     

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.     

Variation of elastic T-stresses along slightly wavy 3D crack fronts

The non-singular terms in the series expansion of the elastic crack-tip stress field, commonly referred to as the elastic T-stresses, play an important role in fracture mechanics in areas such as the stability of a crack path and the two-parameter characterization of elastic-plastic crack-tip deformation. In this paper, a first order perturbation analysis is performed to study some basic properties of the T-stress variation along a slightly wavy 3D crack front. The analysis employs important properties of Bueckner-Rice 3D weight function fields. Using the Boussinesq-Papkovitch potential representation for the mode I weight function field, it is shown that, for coplanar cracks in an infinite isotropic and homogeneous linear elastic body, the mean T-stress along an arbitrary crack front is independent of the shape and size of the crack. Further, a universal relation is discovered between the mean T-stress and the stress field at the same crack front location under the same loading but in the absence of a crack. First-order-accurate solutions are given for the T-stress variation along a slightly wavy crack front with nearly circular or straight confifurations. Specifically, cosine wave functions are adopted to describe smooth polygonal and slightly undulating planar crack shapes. The results indicate that T ₁₁, the 2D T-stress component acting normal to the crack front, increases with the curvature of the crack front as it bows out but T ₃₃, acting parallel to the crack front, decreases with the crack front curvature.     

Numerical modelling of plasticity-induced crack closure for interfacial cracks in bi-material specimens

This paper reports the results of a fairly detailed finite element study which modelled the plasticity-induced crack closure (PICC) behaviour of interfacial cracks in various bi-material specimens. In particular, the fatigue crack-opening stress (S_op) level and the crack-tip deformation fields (Modes I and II) have been assessed for a number of different material combinations, chosen so as to throw some light on the effects of modulus of rigidity and strength level of the alloy on PICC. The material combinations included specimens based on aluminium alloy steel, medium strength-high strength steel, and aluminium or steel specimens coated with a rigid ceramic. Results obtained indicate that stabilised values of closure, S_op, can be interpreted as supporting the hypothesis that it is the elastic constraint on, and deformability of, the plastic zone surrounding a crack that are the major contributors to PICC, rather than any permanent ‘stretch’ associated with crack growth. Positive Mode II slip of the upper crack face over the lower face (i.e. the upper surface moving over the lower surface towards the crack-tip) can elevate S_op level, while a negative slip (i.e. the upper surface moving over the lower surface away from the crack-tip) causes a reduction in its value.     

Characteristic crack-tip distances in fracture criteria: Is crack propagation discontinuous?

The present paper describes a possible mechanism for discontinuous crack advance in which surface separation occurs initially not at the crack-tip itself but within the crack-tip plastic zone of size r_p, at the mid-point of the crack-tip characteristic distance d (identified here with the finite growth step Δa), i.e., at the region of maximum opening tensile stress, spreading towards (and also away from) the crack-tip. The crack extension occurs when the crack-tip is reached and full opening over the distance d is completed.Finite element analyses show that this mechanism causes the formation of a rippled crack face surface in elastic-plastic materials in which irreversible plastic deformations take place during each growth step, in sharp contrast with the smooth surface created in ideal elastic materials in which all deformations are fully reversible. Some pictorial evidence of void formation ahead of the crack tip and of ripples during propagation, found in the literature, is presented.Although the present analysis is from a continuum standpoint it is acknowledged that micro structural features and mechanisms can condition the fracture events taking place in the process zone.The implication to the brittle-ductile transition of the dependence of the energy release rate, G^ΔΞ, on the ratio q (=Δa/r_p) is also discussed.     

Elastoplastic crack analysis for pressure-sensitive dilatant materials-Part II: Interface cracks

The problem of a plane strain crack lying along an interface between a rigid substrate and an elastic-plastic material has been studied. The elastic-plastic material exhibits pressure-sensitive yielding and plastic volumetric deformation. Two-term expansions of the asymptotic solutions for both closed frictionless and open crack-tip models have been obtained. The Mises effective stress in the interfacial crack-tip fields is a decreasing function of the pressure-sensitivity in both open and closed-crack tip models. The variable-separable solution exists for most pressure-sensitive materials and the limit values for existence of the variable-separable solution vary with the strain-hardening exponents. The governing equations become singular as the pressure-sensitivity limit is approached. Strength of the leading stress singularity is equal 1/(n+1) for both crack-tip models, regardless of the pressure-sensitivity. The second-order fields have been solved as an additional eigenvalue problem and the elasticity terms do not enter the second-order solutions as n2. The second exponents for the closed crack model are negative for the weak strain hardening, whereas the negative second-order eigenvalue in the open crack model slightly grows with the pressure-sensitivity. The second-order solutions are of significance in characterising the crack-tip fields. The leading-order solution contains the dominant mode I components for both open and closed crack-tip models when the materials do not have substantial strain hardening. The second-order solutions are generally mode-mixed and depend significantly on the pressure-sensitivity.     

Effect of Plate Thickness on Crack-Tip Plasticity

This paper presents an analytical method for determining the three-dimensional stress fields in plates with a through-the-thickness crack, especially under elastic-plastic conditions. Using the generalised plane strain theory in conjunction with the deformation theories of plasticity, exact solutions are obtained for the effects of plate thickness on the crack-tip plastic zone size and a plastic constraint factor, which is shown to correlate well with published finite element solutions.     

The jump-like crack growth model, the estimation of fracture energy and JR curve

In this paper the jump-like crack growth model for monotonic loading is applied to re-examine both the onset of crack growth and process of stable crack growth. In the former case the fracture energy associated with a new surface creation is estimated and the in-plane constraint influence on this quantity is examined using the J-A₂ approach. In the later case the formula to compute the J-resistance curve is re-examined and compared with the one known from the standards. In the analysis the plane strain model of a structural element made of elastic-plastic material is assumed.     

The T-stress solutions for through-wall circumferential cracks in cylinders subjected to general loading conditions

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stress solutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a circumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered; defined by the mean radius of the cylinder (R) to wall thickness (t) ratios: R/t = 5, 10, and 20. The T-stress was obtained at eight crack lengths (θ/π = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, θ is the crack half angle). Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubic crack face pressures and remote tension and bending. The results for constant and linear crack face pressure were used to derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validated against several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculations for circumferential cracks in cylinders under complex stress fields.     

Mode I plane stress crack-tip constraint for elastic-plastic materials with different strain hardening

Finite element analyses and simulations have been undertaken to investigate the triaxial constraint in the crack-tip regions of a stationary crack and a steady-state growing crack under mode I plane stress for elastic-plastic materials with different strain hardening. The results show that the triaxial constraint in the crack-tip region is independent of specimen geometry, and material strain hardening, both for a stationary and an extending crack quasi-statically. The triaxial constraints for the various configurations examined are in better accordance with those required by the HRR solution for a stationary crack, which defines the low and similar constraints in crack-tip regions for different material strain hardening in the plane stress case. Along the entire ligament ahead of a crack tip, the constraint level transites gradually from that defined by the HRR solution within the near tip zone to that characterized by the stress intensity factor K _I in the far field.     

T-Stress solutions for a semi-elliptical axial surface crack in a cylinder subjected to mode-I non-uniform stress distributions

This paper presents the T-stress solutions (T₁₁ and T₃₃) for semi-elliptical axial surface cracks in a cylinder subjected to mode-I non-uniform stress on the crack surface. Two cylindrical geometries with inner radius (R_i) to wall thickness (t) ratios R_i/t = 5 and 10 were considered. The T-stresses were applied along the crack front for normalized crack depth values a/t of 0.2, 0.4 and 0.5 and aspect ratios a/c of 0.2, 0.4, 0.6 and 1.0. Three stress distribution; uniform, linear and parabolic were applied to the crack face. In addition to these solutions, concrete formulation of the superposition principle is given for the T₃₃-stress, which is known as an elastic parameter that describes the out-of-plane crack tip constraint effect. Then, the validity of the formulation was shown through application of our T-stress solutions to the problem of an axial semi-elliptical surface crack in a cylinder subjected to internal pressure, and checking that the principle of superposition holds for the problem.