Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories

It is known that the plasticity models affect characterization of the crack tip fields. To predict failure one has to understand the crack tip stress field and control the crack. In the present work the element-free Galerkin methods for gradient plasticity theories have been developed and implemented into the commercial finite element code ABAQUS and used to analyze crack tip fields. Based on the modified boundary layer formulation it is confirmed that the stress singularity in the gradient plasticity theories is significantly higher than the known HRR solution and seems numerically to equal to 0.78, independently of the strain-hardening exponent. The strain singularity is much lower than the known HRR one. The crack field in gradient plasticity under small-scale yielding condition consists of three zones: The elastic K-field, the plastic HRR-field dominated by the J-integral and the hyper-singular stress field. Even under gradient plasticity there exists an HRR-zone described by the known J-integral, whereas the hyper-singular zone cannot be characterized by J. The hyper-singular zone is very small (r ? J/σ₀) and contained by the HRR zone in the infinitesimal deformation framework. The finite strains under the gradient plasticity will not eliminate the stress singularity as r → 0, in contrast to the known finite strain results under the Mises plasticity. Numerically no significant changes in characterization of the stress field were found in comparison with the infinitesimal deformation theory. Since the hyper-singular stress field is much smaller than the HRR zone and in the same size as the fracture process zone, one may still use the known J concept to control the crack in the gradient plasticities. In this sense the gradient plasticity will not change characterization of the crack.     

Extensions of access structures and their cryptographic applications

In secret sharing schemes a secret is distributed among a set of users ${\mathcal{P}}In secret sharing schemes a secret is distributed among a set of users P{\mathcal{P}} in such a way that only some sets, the authorized sets, can recover it. The family Γ of authorized sets is called the access structure. To design new cryptographic protocols, we introduce in this work the concept of extension of an access structure: given a monotone family G ì 2^P{{\it \Gamma} \subset 2^\mathcal{P}} and a larger set P^￠ = P è[(P)\tilde]{\mathcal{P}^{\prime} = \mathcal{P} \cup \tilde{\mathcal{P}}}, a monotone access structure G^￠ ì 2^P￠{{\it \Gamma}^{\prime}\subset 2^{\mathcal{P}^{\prime}}} is an extension of Γ if the following two conditions are satisfied: (1) The set P{\mathcal{P}} is a minimal subset of Γ′, i.e. P ? G^￠{\mathcal{P} \in {\it \Gamma}^{\prime}} and P - {R_i} ? G^￠{\mathcal{P} - \{R_i\}\notin {\it \Gamma}^{\prime}} for every R_i ? P{R_i \in \mathcal{P}}, (2) A subset A ì P{A \subset \mathcal{P}} is in Γ if and only if the subset A è[(P)\tilde]{A \cup \tilde{\mathcal{P}}} is in Γ′. As our first contribution, we give an explicit construction of an extension Γ′ of a vector space access structure Γ, and we prove that Γ′ is also a vector space access structure. Although the definition may seem a bit artificial at first, it is well motivated from a cryptographic point of view. Indeed, our second contribution is to show that the concept of extension of an access structure can be used to design encryption schemes with access structures that are chosen ad-hoc at the time of encryption. Specifically, we design and analyze a dynamic distributed encryption scheme and a ciphertext-policy attribute-based encryption scheme. In some cases, the new schemes enjoy better properties than existing ones.     

Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration

For R a Galois ring and m ₁, . . . , m _l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ?_i=1^lm_i{\sum_{i=1}^lm_i} is an R[x]-submodule of R[x]/(x^m₁-1)×?×R[x]/(x^m_l-1){R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}. Suppose m ₁, . . . , m _l are all coprime to the characteristic of R and let {g ₁, . . . , g _t } be the set of all monic basic irreducible polynomials in the factorizations of x^m_i-1{x^{m_i}-1} (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ?_i=1^lm_i{\sum_{i=1}^lm_i} are identified with G₁×?×G_t{{\mathcal G}_1\times\cdots\times {\mathcal G}_t}, where G_j{{\mathcal G}_j} is an R[x]/(g _j )-submodule of (R[x]/(g_j))^n_j{(R[x]/(g_j))^{n_j}}, where n _j is the number of i for which g _j appears in the factorization of x^m_i-1{x^{m_i}-1} into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.     

Large strain response in acceptor- and donor-doped Bi<Subscript>0.5</Subscript>Na<Subscript>0.5</Subscript>TiO<Subscript>3</Subscript>-based lead-free ceramics

Effects of Fe and La addition on the dielectric, ferroelectric, and piezoelectric properties of Bi_0.5Na_0.5TiO₃–Bi_0.5Li_0.5TiO₃–BaTiO₃–Mn ceramics were investigated. Similar to the doping effect in lead-based piezoelectric materials, here the Fe-doped ceramic created a hard effect with an improved mechanical quality factor (Q _m) ~ 160, coercive field (E _c) ~ 2.9 kV/mm, decreased dielectric constant ( e₃₃^T /e₀ ) ~ 80 3, \left( {\varepsilon_{33}^{T} /\varepsilon_{0} } \right)\sim 80 3, and loss (tanδ) ~ 0.024 while the La-doped one indicated a soft feature with improved piezoelectric constant (d ₃₃) ~ 184 pC/N, e₃₃^T /e₀   ~ 983, \varepsilon_{33}^{T} /\varepsilon_{0} \,\sim { 983}, tanδ ~ 0.033, and decreased E _c ~ 2.46 kV/mm. In addition, the temperature dependence of the ferroelectric hysteresis loops and strain response under unipolar electric field was also studied. Around the depolarization temperature T _d, large strain value was obtained with the normalized d₃₃^* d_{33}^{*} up to ~1,000 pC/N, which was suggested originated from the development of the short-range order or non-polar phases in the ferroelectric matrix. All these would provide a new way to realize high piezoelectric response for practical application in different temperature scale.     

Gc versus crack velocity in single groove double cantilever beam specimens of polycarbonate

The dependence of G _c on crack velocity in single-groove double cantilever beams (SG DCB) is negligible over the range of crack velocities from 0.2 to 1100 in. min^–1. The constancy of G _c appears to reflect a counterbalancing of the rise in yield stress by a decrease in crack opening displacement. The latter occurs through a decrease in the gauge length of material engulfed by yielding rather than by a decrease in the ultimate plastic strain in the crack tip plastic zone. The energy of shear lip formation at any crack velocity appears to be accurately estimated from G _c's for SG DCB specimens fractured at low velocity.     

Subcritical interlaminar crack growth in fibre composites exhibiting a risingR-curve

Subcritical crack growth behaviour has been evaluated in composite laminates based on uniaxial carbon fibres in poly(ether-ether ketone) matrices. Double cantilever beam (DCB) specimens have been employed to give mode I loading and it is first shown that the materials exhibit a risingR-curve, i.e. the value of the interlaminar fracture energy,G _IC, increases as the crack propagates through the specimens. Secondly, when a DCB specimen is held at a constant displacement, subcritical crack growth is found to occur. The velocity of the subcritical crack growth,v, has been measured using a load-relaxation technique. Hence, values of the crack velocity,v, have been obtained as a function of the strain-energy release rate,G _I applied during subcritical crack growth. Owing to the presence of theR-curve, these data have been measured at various stages during the development of theR-curve. The relationships betweenv andG _I are modelled using power-law expressions. Finally, it is considered that theR-curve behaviour is most likely caused by the fibre bridging which develops behind the crack tip as the delamination propagates through the specimen. Fibre bridging allows stress to be transferred across the crack faces, behind the advancing crack tip, and so results in a shielding of the stress field at the crack tip from the applied stress. Therefore, the expression ascertained for the relationship between the velocity,v, of subcritical crack growth and the corresponding value ofG _I has been further refined and modelled to account for the presence of fibre bridging.     

History effect in fatigue crack growth under mixed-mode loading conditions

Plastic deformation within the crack tip region introduces internal stresses that modify subsequent behaviour of the crack and are at the origin of history effects in fatigue crack growth. Consequently, fatigue crack growth models should include plasticity-induced history effects. A model was developed and validated for mode I fatigue crack growth under variable amplitude loading conditions. The purpose of this study was to extend this model to mixed-mode loading conditions. Finite element analyses are commonly employed to model crack tip plasticity and were shown to give very satisfactory results. However, if millions of cycles need to be modelled to predict the fatigue behaviour of an industrial component, the finite element method becomes computationally too expensive. By employing a multiscale approach, the local results of FE computations can be brought to the global scale. This approach consists of partitioning the velocity field at the crack tip into plastic and elastic parts. Each part is partitioned into mode I and mode II components, and finally each component is the product of a reference spatial field and an intensity factor. The intensity factor of the mode I and mode II plastic parts of the velocity fields, denoted by dρ_I/dt and dρ_II/dt, allow measuring mixed-mode plasticity in the crack tip region at the global scale. Evolutions of dρ_I/dt and dρ_II/dt, generated using the FE method for various loading histories, enable the identification of an empirical cyclic elastic–plastic constitutive model for the crack tip region at the global scale. Once identified, this empirical model can be employed, with no need of additional FE computations, resulting in faster computations. With the additional hypothesis that the fatigue crack growth rate and direction can be determined from mixed-mode crack tip plasticity (dρ_I/dt and dρ_II/dt), it becomes possible to predict fatigue crack growth under I/II mixed-mode and variable amplitude loading conditions. To compare the predictions of this model with experiments, an asymmetric four point bend test system was setup. It allows applying any mixed-mode loading case from a pure mode I condition to a pure mode II. Initial experimental results showed an increase of the mode I fatigue crack growth rate after the application of a set of mode II overload cycles.     

Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements

Triangular and prismatic quadratic isoparametric elements, formed by collapsing one side and placing the mid-side node near the crack tip at the quarter point, are shown to embody the (1/√r) singularity of elastic fracture mechanics and the (1/r) singularity of perfect plasticity. The procedure of performing the fracture analysis for the case of small scale yielding is discussed, and the finite element results are compared with theoretical results. The proposed elements have wide application in the fracture analysis of structures where ductile fracture is investigated. They permit a determination of the relationship between crack tip field parameters, loading, and geometry. And for a given fracture criterion can be applied to the prediction of fracture in structures such as pressure vessels under in service conditions.     

Plastic,viscoplastic and creep fracture problems with the boundary element method

The present paper shows the applicability of the dual boundary element method to analyse plastic, viscoplastic and creep behaviours in fracture mechanics problems. Several models with a crack, including a square plate, a holed plate and a notched plate, are analysed. Special attention is taken when the discretization of the domain is performed. In fact, for the plasticity and viscoplasticity cases, only the region susceptible to yielding was discretized, whereas the creep case required the discretization of the whole domain. The proposed formulation is presented as an alternative technique to study these kinds of nonlinear problems. Results from the present formulation are compared to those of the well‐established finite element technique, and they are in good agreement. Important fracture mechanic parameters like K_I, K_II, J‐integrals and C‐integrals are also included. In general, the results, for the plastic, viscoplastic and creep cases, exhibit that the highest stress concentrations are in the vicinity of the crack tip and they decrease as the distance from the crack tip is increased.     

On the evaluation of the J-integral by a plastic crack tip element

Two crack tip elements are formulated for a stationary, mode I plastic crack in planar structures using hybrid assumed stress approach, based on the secant modulus and the Newton-Raphson schemes, respectively. The stress distribution in the crack tip element is assumed to be the HRR field superimposed by the regular polynomial terms. The formulated (hybrid) crack tip elements are compatible with the isoparametric element so that they can be used conveniently along with the conventional displacement-based finite elements. The intensity of the HRR stress field, the J-integral, is determined directly from the finite element equations together with the nodal displacements. The dominance of the HRR stress field at the crack tip is pertinent to the present approach, which depends on geometry and loading conditions. Since the J-integral is globally path-independent for nonlinear elastic materials (deformation plasticity model), in order to assess the accuracy and efficiency of the methodology as compared to the contour integration approach, numerical studies of common plane-stress cracked configurations are performed for these materials. The results indicate that for a sufficiently small crack tip element size, J from the present approach correlates well, within 6 percent difference, with that from the contour integration for a wide range of material hardening coefficients if the HRR zone exists at the crack tip. These highly accurate results for J from the crack tip stresses could not be achieved without using (newly) modified variational principles and a refined numerical technique. It should be emphasized that the present methodology also can be applied to cracks in J ₂ flow materials under HRR dominance. In such case, the J integral may not be globally path independent, and hence it now must be determined from the stress and strain fields near the crack tip.     

Quasi-statically steady crack growth in rate dependent solids-one-parameter field encompassing different constraints

A finite element analysis is presented for quasi-statically steady crack growth in an elastic-viscoplastic material under Mode I, plane strain and small scale yielding conditions. The effects of material rate-sensitivity on the fields in the vicinity of the moving crack tip are examined. Our analysis employs a modified boundary layer formulation whereby the remote tractions are given by the first two-terms of elastic asymptotic stress field, characterized by K _Iand T. When the physical coordinates are scaled by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Hcaca% WFlbWaaSbaaSqaaGqaaiaa+fdaaeqaaOGaai4laiabeo8aZnaaBaaa% leaacaaIWaaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaaaa!3ECA!\[(K_1 /\sigma _0 )^2 \], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa% aaleaacaaIWaaabeaaaaa!3A07!\[\sigma _0 \] is the tensile yield stress, the near-tip fields over a wide range of stress triaxialities are members of a family of self-similar solutions parameterized by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-rfaca% WFVaGaeq4Wdm3aaSbaaSqaaiaa-bdaaeqaaaaa!3B8E!\[T/\sigma _0 \]. Members of this family are found to collapse into a single near-tip distribution when the physical coordinates are normalized by a characteristic length L _g, which is a significant fraction of the plastic zone length directly ahead of the crack tip. This distribution depends only on the relative crack speed given by the dimensionless number % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-zfaca% WFVaGaa8hkaiaa-XeadaWgaaWcbaGaa83zaaqabaacciGccuGFiiIZ% gaGaamaaBaaaleaacaaIWaaabeaakiaacMcaaaa!3EB2!\[V/(L_g \dot \in _0 )\] where V is the crack speed and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGGaciqb-HGioB% aacaWaaSbaaSqaaiaaicdaaeqaaaaa!39D6!\[\dot \in _0 \] is the material's viscoplastic strain rate at a reference stress. Near-tip field distributions are obtained for several values of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-zfaca% WFVaGaa8hkaiaa-XeadaWgaaWcbaGaa83zaaqabaacciGccuGFiiIZ% gaGaamaaBaaaleaacaaIWaaabeaakiaacMcaaaa!3EB2!\[V/(L_g \dot \in _0 )\] and material strain rate sensitivity, m. Our results show that strong material rate sensitivity and high crack speed elevate the stress level ahead of the moving crack tip.     

Fracture parameters for interfacial cracks: an experimental-finite element study of crack tip fields and crack initiation toughness

Crack tip measurements and analysis of interfacial parameters for PMMA-aluminum bimaterial system are presented. A variety of crack tip mode-mixities are obtained by subjecting asymmetric four-point-bend specimens to different boundary loads. The crack tip fields are mapped using the optical method of Coherent Gradient Sensing (CGS). The complex stress intensity factors and the associated crack tip mixities () are measured from CGS fringe patterns. The asymptotic expansion field for interface cracks is used for extracting fracture parameters by accounting for higher order contributions to the experimental data. The measurements are compared with complementary finite element computations. A linear relationship between crack tip mixity and the applied load mixity is experimentally demonstrated in this large elastic mismatch system. The fracture load and hence the energy release rate G _cr () at crack initiation is measured as applied load mixities are varied. Limited discussion on the influence of surface roughness prior to bonding on the fracture toughness is included. Positive and negative shear on the crack plane produce different failure responses in this bimaterial system and the observed asymmetry is akin to the one predicted by the T&H model that includes crack tip nonlinearty.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

RESIDUAL STRESS FIELDS DURING FATIGUE CRACK GROWTH

This study outlines the distinction between (1) residual stresses at an ideal crack tip, undergoing reversed deformation in the absence of crack closure, and (2) additional residual stresses generated due to plasticity induced closure upon fatigue crack growth. Residual stresses resulting from reversed deformation in plane strain were higher compared to the plane stress case, while residual stresses generated behind the crack tip were more significant in plane stress compared to plane strain. The origin of these residual stresses was studied for two specimen geometries over a wide range of loading conditions. We define a new crack tip parameter, S_tt as the applied stress level that corresponds to the development of tensile stresses immediately ahead of crack tips. The S_tt levels were significantly higher for a fatigue crack than for an ideal crack. We attribute the difference in S_tt levels between these two cases to plasticity induced closure. The results demonstrate the importance of the S_tt parameter, since the stresses ahead of crack tips could remain compressive even when the crack surfaces are open. Moreover, the study emphasizes the need, when describing fatigue crack growth, to incorporate both the closure concept and residual stress field ahead of a crack tip.     

A finite element analysis of plane strain dynamic crack growth at a ductile-brittle interface

In this work, steady, dynamic crack growth under plane strain, small-scale yielding conditions along a ductile-brittle interface is analysed using a finite element procedure. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behaviour. The objectives of this work are to establish the validity of an asymptotic solution for this problem which has been derived recently [12], and to examine the effect of changing the remote (elastic) mode-mixity on the near-tip fields. Also, the influence of crack speed on the stress fields and crack opening profiles near the propagating interface crack tip is assessed for various bi-material combinations. Finally, theoretical predictions are made for the variation of the dynamic fracture toughness with crack speed for crack growth under a predominantly tensile mode along ductile-brittle interfaces. Attention is focused on the effect of mismatch in stiffness and density of the constituent phases on the above aspects.     

Computation of X-ray powder diffractograms of cement components and its application to phase analysis and hydration performance of OPC cement

The importance of computed X-ray diffraction patterns of various polymorphs of alite (M₃, T₁, R{\bf \emph{M}_{3}, \emph{T}_{1}, \emph{R}}), belite (b\boldsymbol{\beta}, g\boldsymbol{\gamma}), aluminate (cubic, orthorhombic), aluminoferrite, gypsum and hemihydrate in the quantitative phase analysis of cement and its early stage hydration performance is highlighted in this work with three OPC samples. The analysis shows that the predominant silicate phases present in all the samples are M₃{\bf \emph{M}_{3}}-alite phase and b\boldsymbol{\beta}-belite phase, respectively. Both cubic and orthorhombic phases of C ₃ A, brownmillerite, gypsum and hemihydrates are present at different levels. Quantitative phase analysis of cement by Rietveld refinement method provides more accurate and comprehensive data of the phase composition compared to Bogue method. The comparative hydration performance of these samples was studied with w/c{\bf \emph{w/c}} ratio, 0·5 and the results are interpreted in the light of difference in phase compositions viz. b\boldsymbol{\beta}-C ₂ S/C ₃ S ratio, fraction of finer cement particles present in the samples and theoretical modeling of C ₃ S hydration.     

J-Dominance in Mixed Mode Ductile Fracture Specimens

The asymptotic mixed mode crack tip fields in elastic-plastic solids are scaled by the J-integral and parameterized by a near-tip mixity parameter, M _{_p} . In this paper, the validity and range of dominance of these fields are investigated. To this end, small strain elastic-plastic finite element analyses of mixed mode fracture are first performed using a modified boundary layer formulation. Here, a two term expansion of the elastic crack tip field involving the stress intensity factor |K| the elastic mixity parameter M _{_e} as well as the T-stress is prescribed as remote boundary conditions. The analyses are conducted for different values of M _{_e} and the T-stress. Next, several commonly used mixed mode fracture specimens such as Compact Tension Shear (CTS), Four Point Bend (4PB), and modified Compact Tension specimen are considered. Here, the complete range of loading from contained yielding to large scale yielding is analyzed. Further, different crack to width ratios and strain hardening exponents are considered. The results obtained establish that the mixed mode asymptotic fields dominate over physically relevant length scales in the above geometries, except for predominantly mode I loading and under large scale yielding conditions. This revised version was published online in August 2006 with corrections to the Cover Date.     

An engineering methodology to assess effects of weld strength mismatch on cleavage fracture toughness using the Weibull stress approach

This work describes the development of an engineering approach based upon a toughness scaling methodology incorporating the effects of weld strength mismatch on crack-tip driving forces. The approach adopts a nondimensional Weibull stress, [`(s)]_w{\bar{{\sigma}}_w}, as a the near-tip driving force to correlate cleavage fracture across cracked weld configurations with different mismatch conditions even though the loading parameter (measured by J) may vary widely due to mismatch and constraint variations. Application of the procedure to predict the failure strain for an overmatch girth weld made of an API X80 pipeline steel demonstrates the effectiveness of the micromechanics approach. Overall, the results lend strong support to use a Weibull stress based procedure in defect assessments of structural welds.     

J-A 2 Characterization of Crack-Tip Fields: Extent of J-A 2 Dominance and Size Requirements

The requirements for J-dominance, limits of the single-parameter criterion to characterize the fracture of engineering structures, and two-parameter fracture analyses are first reviewed. Through comparison, it is argued that the two-parameter fracture methodology based on the J-A ₂ theory is a reasonable extension of the single parameter (J-integral) fracture methodology. Consequently the extent of J-A ₂ dominance in various specimens under either tension or bending is investigated in detail in this paper. Using the J ₂ flow theory of plasticity and within the small-strain framework, full field finite element solutions are obtained for both deep and shallow crack geometries of single edge notch bar under pure bending [SEN(B)] and central cracked panel in uniform tension [CC(T)]. These crack-tip stresses are compared with those in the HRR singularity fields and the J-A ₂ asymptotic fields at the same level of applied J. The comparison indicates that the size R of the region dominated by the J-A ₂ field is much larger than that of the HRR field around the crack tip. Except for deeply-cracked SEN(B) in low hardening material (n=10) under fully plastic conditions, the numerical results near the crack tip in both SEN(B) and CC(T) match very well with the J-A ₂ asymptotic solutions in the area of interest 1<r/(J/σ₀)<5 from well-contained to large scale plasticity. The implications of these results on the minimal specimen size requirements essential to a two-parameter fracture criterion based on the J-A ₂ asymptotic solution are then discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Quasi-static crack tip fields in rate-sensitive FCC single crystals

In this work, the effects of loading rate, material rate sensitivity and constraint level on quasi-static crack tip fields in a FCC single crystal are studied. Finite element simulations are performed within a mode I, plane strain modified boundary layer framework by prescribing the two term (K − T) elastic crack tip field as remote boundary conditions. The material is assumed to obey a rate-dependent crystal plasticity theory. The orientation of the single crystal is chosen so that the crack surface coincides with the crystallographic (010) plane and the crack front lies along [10[`1]][10\overline 1] direction. Solutions corresponding to different stress intensity rates [(K)\dot]\dot{{K}}, T-stress values and strain rate exponents m are obtained. The results show that the stress levels ahead of the crack tip increase with [(K)\dot]\dot{{K}} which is accompanied by gradual shrinking of the plastic zone size. However, the nature of the shear band patterns around the crack tip is not affected by the loading rate. Further, it is found that while positive T-stress enhances the opening and hydrostatic stress levels ahead of crack tip, they are considerably reduced with imposition of negative T-stress. Also, negative T-stress promotes formation of shear bands in the forward sector ahead of the crack tip and suppresses them behind the tip.