Modeling the brittle–ductile transition in ferritic steels: dislocation simulations

We present a model for the brittle–ductile transition in ferritic steels based on two dimensional discrete dislocation simulations of crack-tip plasticity. The sum of elastic fields of the crack and the emitted dislocations defines an elasto–plastic crack field. Effects of crack-tip blunting of the macrocrack are included in the simulations. The plastic zone characteristics are found to be in agreement with continuum models, with the added advantage that the hardening behavior comes out naturally in our model. The present model is composed of a macrocrack with microcracks ahead of it in its crack-plane. These microcracks represent potential fracture sites at internal inhomogeneities, such as brittle precipitates. Dislocations that are emitted from the crack-tip account for plasticity. When the tensile stress along the crack plane attains a critical value σ _F over a distance fracture is assumed to take place. The brittle–ductile transition curve is obtained by determining the fracture toughness at various temperatures. Factors that contribute to the sharp upturn in fracture toughness with increasing temperature are found to be: the increase in dislocations mobility, and the decrease in tensile stress ahead of the macrocrack tip due to increase in blunting, and the slight increase in fracture stress of microcracks due to increase in plasticity at the microcrack. The model not only predicts the sharp increase in fracture toughness near the brittle–ductile transition temperature but also predicts the limiting temperature above which valid fracture toughness values cannot be estimated; which should correspond to the ductile regime. The obtained results are in reasonable agreement when compared with the existing experimental data.     

Three-dimensional numerical simulations of dynamic fracture in silicon carbide reinforced aluminum

The three-point bending test by Kolsky-bar apparatus is a convenient technique to test the dynamic fracture properties of materials. This paper presents detailed three-dimensional finite element simulations of a silicon particle reinforced aluminum (SiC_p/Al) experiment (Li et al., [Proceedings of the US Army Symposium on Solid Mechanics]. In the simulations, the interaction between the input bar and the specimen is modeled by coupled boundary conditions. The material model includes large plastic deformations, strain-hardening and strain-rate hardening mechanisms. Furthermore, crack initiation and propagation processes are simulated by a cohesive element model. The simulation results quantitatively agree with the experimental measurements on three fronts: (1) the structural response of the specimen, (2) the time of unstable crack propagation, and (3) the local deformations at the crack-tip zone. The simulations reveal crack propagation characteristics, including crack-tip plastic deformation, crack front curving, and crack velocity profile. The effectiveness of Kolsky-bar type fracture tests is verified. It is shown that a rate-independent cohesive model can describe the complicated dynamic elastic–plastic fracture process in the SiC_p/Al material.     

Effect of loading rate on dynamic fracture initiation toughness of brittle materials

Numerical simulation is carried out to investigate the effect of loading rate on dynamic fracture initiation toughness including the crack-tip constraint. Finite element analyses are performed for a single edge cracked plate whose crack surface is subjected to uniform pressure with various loading rate. The first three terms in the Williams’ asymptotic series solution is utilized to characterize the crack-tip stress field under dynamic loads. The coefficient of the third term in Williams’ solution, A ₃, was utilized as a crack tip constraint parameter. Numerical results demonstrate that (a) the dynamic crack tip opening stress field is well represented by the three term solution at various loading rate, (b) the loading rate can be reflected by the constraint, and (c) the constraint A ₃ decreases with increasing loading rate. To predict the dynamic fracture initiation toughness, a failure criterion based on the attainment of a critical opening stress at a critical distance ahead of the crack tip is assumed. Using this failure criterion with the constraint parameter, A ₃, fracture initiation toughness is determined and in agreement with available experimental data for Homalite-100 material at various loading rate.     

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.     

A plastic flow-induced fracture theory for fatigue crack growth

A plastic flow-induced fracture theory for fatigue crack growth is presented. A new formulae for the fatigue stress intensity threshold and the fatigue crack growth rate law are derived by applying the principle of energy conservation in considering the fatigue crack growth process in the presence of local plastic flow ahead of the crack-tip. The present theory predicts not only the fatigue crack growth rate being just proportional to the rate of creation of dislocation at the crack-tip, but also the fatigue stress intensity threshold, which can be determined according to the applied fatigue stress amplitude and the characteristic size of microstructural fracture process ahead of the crack-tip, and can account for the fatigue crack growth characteristics at both low and high levels of applied fatigue stress intensity amplitude. All the results are universal and agree with the existing empirical results and experimental observations.     

J-T characterized stress fields of surface-cracked metallic liners bonded to a structural backing - I. Uniaxial tension

Surface crack-tip stress fields in a tensile loaded metallic liner bonded to a structural backing are developed using a two-parameter J-T characterization and elastic-plastic modified boundary layer (MBL) finite element solutions. The Ramberg-Osgood power law hardening material model with deformation plasticity theory is implemented for the metallic liner. In addition to an elastic plate backed surface crack liner model, elastic-plastic homogeneous surface crack models of various thicknesses were tested. The constraint effects that arise from the elastic backing on the thin metallic liner and the extent to which J-T two parameter solutions characterize the crack-tip fields are explored in detail. The increased elastic constraint imposed by the backing on the liner results in an enhanced range of validity of J-T characterization. The higher accuracy of MBL solutions in predicting the surface crack-tip fields in the bonded model is partially attributed to an increase in crack-tip triaxiality and a consequent increase in the effective liner thickness from a fracture standpoint. After isolating the effects of thickness, the constraint imposed by the continued elastic linearity of the backing significantly enhanced stress field characterization. In fact, J and T along with MBL solutions predicted stresses with remarkable accuracy for loads beyond full yielding. The effects of backing stiffness variation were also investigated and results indicate that the backing to liner modulus ratio does not significantly influence the crack tip constraint. Indeed, the most significant effect of the backing is its ability to impose an elastic constraint on the liner. Results from this study will facilitate the implementation of geometric limits in testing standards for surface cracked tension specimens bonded to a structural backing.     

A FATIGUE CRACK GROWTH MODEL FOR FIBER-REINFORCED METAL–MATRIX COMPOSITES

A fracture-mechanics based model is proposed for fatigue crack growth in fiber-reinforced metal-matrix composites (MMCs). The model incorporates most of the fracture micromechanisms commonly observed in fiber-reinforced MMCs, including (1) formation of microcracks ahead of the crack tip by either fiber fracture or interface decohesion, (2) interactions of the main crack tip with fibers and microcracks, (3) linkage of the main crack with microcracks, and (4) crack deflection by fibers. Statistical variations of fiber or interface strength are also considered. The essential feature of the model is to compute the changes in the local stress intensity due to various fracture mechanisms; the local stress intensity is then utilized to predict crack growth rate in MMCs via an elastic modulus normalization procedure. Application of the model to predicting crack growth in an alumina fiber Mg-alloy composite is presented.     

Measuring overload effects during fatigue crack growth in bainitic steel by synchrotron X-ray diffraction

In this work we present the results of in situ synchrotron X-ray diffraction measurements of fatigue crack-tip strain fields following a 100% overload (OL) under plane strain conditions. The study is made on a bainitic steel with a high toughness and fine microstructure. This allowed a very high (60 μm) spatial resolution to be achieved so that fine-scale changes occurring around the crack-tip were captured along the crack plane at the mid-thickness of the specimen. We have followed the crack as it grew through the plastic/residually stressed zone associated with the OL crack location. We observed two effects; one when the enhanced plastic zone is ahead of the crack and one after it has been passed. Regarding the former it was found that the compressive stress at the crack-tip initially falls sharply, presumably due to the increased plastic stretch caused by the OL. This is associated with a concomitant fall in peak tensile stress at K_max, the elastic excursion between K_min and K_max remaining essentially unchanged from before OL. Subsequently discontinuous closure as seen previously for plane stress caused by crack face contact at the OL location limits the elastic strain range experienced by the crack tip and thereby retards crack growth.     

An Investigation of the Effects of Crack Front Curvature on the Crack-Tip Opening Displacement of A707 Steel

This paper examines the effects of crack front curvature on the fracture toughness (crack-tip opening displacement) of A707 steel. Fracture mechanics specimens, in which the radii of curvature of the crack fronts are controlled in an effort to simulate potential variations in crack front profiles in fracture experiments, were produced by machining and fatigue pre-cracking. Three-point bend crack-tip opening displacements (CTOD_c) were measured in accordance with the ASTM E-1290 code. The results show that the critical CTOD_c increases with increasing crack front curvatures between 0.05 and 0.17 mm^–1. In all cases, stable crack growth and final catastrophic failure of the specimens are found to occur by transgranular ductile dimpled fracture, in which the ductile dimples are nucleated around MnS or Al₂(Mg)O₃ inclusions. The implications of the results are discussed for the measurement of critical CTOD_c in specimens with varying levels of crack front curvature.     

A framework to correlate a/W ratio effects on elastic-plastic fracture toughness (J c )

Single edge-notched bend (SENB) specimens containing shallow cracks (a/W < 0.2) are commonly employed for fracture testing of ferritic materials in the lower-transition region where extensive plasticity (but no significant ductile crack growth) precedes unstable fracture. Critical J-values J _c ) for shallow crack specimens are significantly larger (factor of 2–3) than the J _c )-values for corresponding deep crack specimens at identical temperatures. The increase of fracture toughness arises from the loss of constraint that occurs when the gross plastic zones of bending impinge on the otherwise autonomous crack-tip plastic zones. Consequently, SENB specimens with small and large a/W ratios loaded to the same J-value have markedly different crack-tip stresses under large-scale plasticity. Detailed, plane-strain finite-element analyses and a local stress-based criterion for cleavage fracture are combined to establish specimen size requirements (deformation limits) for testing in the transition region which assure a single parameter characterization of the crack-tip stress field. Moreover, these analyses provide a framework to correlate J _c )-values with a/W ratio once the deformation limits are exceeded. The correlation procedure is shown to remove the geometry dependence of fracture toughness values for an A36 steel in the transition region across a/W ratios and to reduce the scatter of toughness values for nominally identical specimens.     

Multiscale analysis of stiffness and fracture of nanoparticle-reinforced composites using micromechanics and global–local finite element models

A combined micromechanics analysis and global–local finite element method is proposed to study the interaction of particles and matrix at the nano-scale near a crack tip. An analytical model is used to obtain the effective elastic modulus of nanoparticle-reinforced composites, then a global–local multi-scale finite element model with effective homogeneous material properties is used to study the fracture of a compact tension sample. For SiO₂ particle-reinforced epoxy composites with various volume fractions, the simulation results for effective elastic modulus, fracture toughness, and critical strain energy release rate show good agreement with previously published experimental data. It is demonstrated that the proposed parametric multi-scale model can be used to efficiently study the toughness mechanisms at both the macro and nano-scale.     

Fracture of single crystal MgAl2O4

The fracture of single crystal, stoichiometric MgAl₂O₄ spinel, was investigated for the (1 0 0), (1 1 0), and (1 1 1) from room temperature to 1500° C. Two regions of fracture behaviour were observed; a low temperature elastic region where K _lc decreased with increasing temperature, and an elevated temperature region where K _lc increased rapidly with increasing temperature. The elastic region is explained primarily by the decrease of elastic modulus with increasing temperature, whereas the rapid increase of K _lc at elevated temperature is attributed to plastic flow in the vicinity of the crack tip     

Deformation and fracture of Al-8Fe rapidly solidified alloys

The effects of the rapid solidification on the deformation and fracture of Al-8Fe alloys, from TEM fracture specimens, have been studied. The most general conclusion which can be drawn in this study is clearly in agreement with a plastic deformation mechanism. Crack propagation occurs by localized plastic rupture mechanisms which result from enhanced slip along {111} planes. Crack propagation occurs within the deformed zone either by the nucleation, growth and coalescence of holes ahead of the crack-tip, or through the emission of dislocation from the crack-tip. The resulting fracture is along the active {111} slip planes. The principal effect of secondary phases (Al₁₃Fe₄) on the fracture propagation in Al-8Fe alloys was that the secondary phases increased the stress level at which plastic deformation occurs at the crack-tip and increased the stress level at which the crack propagates. This work clearly shows that in order to obtain coarse intermetallic precipitates in the specimens after ageing heat treatments the crack propagation and deformation processes occur at lower stresses compared to as-received rapidly solidified samples.     

A local modulus analysis of rapid crack propagation in polymers

This work is concerned with the analysis of rapid crack propagation (RCP) in Polymethylmethacrylate (PMMA), Polycarbonate (PC) and two-layer PMMA/PC systems. Remarkably constant crack speeds were observed, and higher crack speeds corresponded to the higher preloads. Uniform fracture surfaces were associated with these constant speed RCPs. An indirect method was used to characterise dynamic fracture properties of the materials. The method relies on the recorded crack length histories and boundary conditions which are incorporated in a dynamic Finite Element (FE) code to generate the crack resistance (G _ID). The numerical simulation of the constant speed RCPs generated highly scattered G _ID data. Very large variations of the computed G _ID with the crack length did not correspond to fracture surface appearances. Geometry dependent and multivalued crack resistance results with respect to the crack speed cast doubt on the uniqueness of G _ID. In this work, attempts were made to overcome these difficulties by exploring the concept that the anomalies arise from large local strains around the rapidly moving crack tip, resulting in the crack seeing a low local modulus. It is demonstrated that the critical source of error on the analysis of RCP, is the improper linear elastic representation of the material behaviour around the propagating crack tip. Since the parameters describing the behaviour of the materials near the propagating crack tip were unknown, local non-linear effects were approximated by a local low modulus strip along the prospective crack path. The choice of the local modulus was justified by measurements of the strain histories along the crack path during RCP. The local strip low modulus model generated a larger amount of the kinetic energy in the sample and the crack resistance was reduced compared to results from the single constant modulus approach. Most importantly, G _ID data were nearly independent of the crack length, crack speed and the specimen size. This local modulus concept was also successfully applied to the analysis of RCP in the duplex specimen configuration.     

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.     

Tensile cracks in polycrystalline ice under transient creep: Part I — Numerical modeling

The interaction between creep deformations and a stationary or growing crack is a fundamental problem in ice mechanics. Knowledge concerning the physical mechanisms governing this interaction is necessary: (1) to establish the conditions under which linear elastic fracture mechanics can be applied in problems ranging from ice-structure interaction to fracture toughness testing; and (2) to predict the ductile-to-brittle transition in the mechanical behavior of ice and, especially, the stability and growth of cracks subjected to crack-tip blunting by creep deformations. This requires a quantitative estimate of the creep zone surrounding a crack-tip, i.e., the zone within which creep strains are greater than the elastic strains.

The prediction of the creep zone in previous ice mechanics studies is based on the theory developed by Riedel and Rice (1980) for tensile cracks in creeping solids. This theory is valid for a stationary crack embedded in an isotropic material obeying an elastic, power-law creep model of deformation and for a suddenly applied uniform far-field tension load that is held constant with time. The deformation of ice at strain-rates ahead of a crack (i.e., 10⁻⁶ to 10⁻² s⁻¹) is dominated, however, by transient (not steady power-law) creep and the loading, in general, is not instantaneous and constant.

A numerical model is developed in this paper to investigate the role of transient creep and related physical mechanisms in predicting the size, shape and time evolution of the creep zone surrounding the tip of a static crack in polycrystalline ice. The model is based on the fully consistent tangent formulation derived in closed form (Shyam Sunder et al., 1993) and used in the solution of the physically-based constitutive theory developed by Shyam Sunder and Wu (1989a, b) for the multiaxial behavior of ice undergoing transient creep. The boundary value problem involving incompressible deformations ahead of a stationary, traction-free mode I crack in a semi-infinite medium is modeled and solved by a finite element analysis using the boundary layer approach of Rice (1968). This model is verified by comparing its predictions with (i) the known theoretical solutions for the elastic and HRR asymptotic stress and strain fields in an elastic-plastic material of the Ramberg-Osgood type, and (ii) the creep zone size for an isotropic material obeying the elastic power-law creep model of deformation.     

Micro- and macro-mechanical testing of transparent MgAl2O4 spinel

Advanced transparent ceramics with high chemical and thermal stability are gaining increasing interest as replacement of glass-based materials in technical window applications. The mechanical reliability and performance of transparent MgAl₂O₄ with a grain size of 5 μm has been characterized at ambient temperature using micro-mechanical indentation and macroscopic bending tests. The measurements focused on elastic modulus, fracture toughness, crack kinetics, and strength, the latter analyzed with Weibull statistics. The effect of slow crack growth is assessed using a strength–probability–time plot. Complementary fractography by optical, confocal and scanning electron microscopy provided a correlation between failure origin and fracture stress. The results and reliability aspects are discussed in terms of linear elastic fracture mechanics.     

PASSIVATION EFFECTS ON CRACK CLOSURE IN THE FATIGUE CRACK PROPAGATION OF A PEAK-AGED Al-Li-Zr ALLOY IN NaCl AND Na2SO4 SOLUTIONS

Abstract Fatigue cracking of a peak-aged Al-Li-Zr alloy was investigated by measuring crack closure as a function of applied anodic potential in 0.6 M NaCl and 0.5 M Na₂SO₄ solutions with an unloading elastic compliance technique, and by comparison with crack closure in dry air. The present work involves complementary anodic behaviour of the Al-Li-Zr alloy in both solutions by potentiodynamic polarization and potentiostatic current transient experiments. From the repassivation rates in the passivation potential range in both solutions, it is indicated that a more stable passive film is formed at lower applied anodic potential than at higher applied anodic potential. The intrinsic fatigue crack propagation (FCP) rates under unstable passivation potential in both solutions were significantly larger than those obtained in dry air. Under stable passivation potential in both solutions, however, the intrinsic FCP rates in the low ΔK_eff range were slightly lower than those obtained in dry air. The crack closure in the low ΔK_eff range increased under stable passivation potential, in dry air and under unstable passivation potential. The high crack closures appearing in the low ΔK_eff range were characterized by a tortuous fracture surface in dry air, and the occurrence of various crack paths such as rolling plane delamination under unstable passivation potential. The difference between environmental crack closures under stable and unstable passivation conditions is discussed in terms of environment-assisted crack-tip damage processes.     

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.     

The use of quarter-point crack-tip elements for T-stress determination in boundary element method analysis

A simple formula for obtaining the elastic T-stress in boundary element method fracture mechanics analysis is presented in this communication. This formula is obtained by comparing the variation of the displacements along the quarter-point crack-tip element with the classical field solution for the crack-tip. Its validity is tested with four example problems for a range of crack sizes and good agreement with solutions in the literature is generally obtained.