Sharp Crack Research Articles

Embrittlement: A Crack Tip View

Normally tough structural materials may fail catastrophically when contaminated with even minuscule quantities of ubiquitous elements such as sulfur, phosphorus or hydrogen. These so called embrittling elements exert their influence at the tip of atomically sharp cracks where applied stress is concentrated to a degree given by the stress concentration factor. When the concentrated stress exceeds the strength of the bonds across the crack tip, the crack will run. Hence, an element may embrittle a host either by decreasing bond strength and/or increasing the stress concentration. While the effects of embrittling elements on the earlier have been extensively studied, less attention has been directed to their effects on the latter. Here we exploit the nearsightedness of electronic matter principle to provide a measure of the stress concentration factor and study its changes in the presence of a known embrittling element. We find that for the well studied system of copper embrittled by dilute quantities of bismuth, bismuth atoms increase the crack tip stress concentration by more than 60%

Experimental and XIGA-CZM based Mode-II and mixed-mode interlaminar fracture model for unidirectional aerospace-grade composites

This article presents a fracture model for evaluation of Mode-II and Mixed-Mode (Mode-I + II) interlaminar fracture properties of unidirectional laminated composites. Mode-II and Mixed-Mode (Mode-I + II) interlaminar fracture tests have been carried out on End-Notched Flexure and Mixed-Mode bending specimens, respectively. The non pre-craked specimens are tested under pure Mode-II and Mixed-Mode loading conditions for the estimation of energy required to initiate crack growth from the natural crack tip. Pre-cracked specimens have been tested under pure Mode-II loading conditions to investigate the nature of delamination propagation from the sharp crack tip. A new set of interfacial fracture properties has been evaluated for AS4/914 laminates. An expression has been developed to correlate the experimentally obtained energy release rate (ERR) and equivalent crack growth (Δa) using a compliance based beam method. The curve-fitted expression between ERR and Δa is further introduced to the cohesive zone model for numerical modelling. Fractured surfaces at the final fracture load point in non pre-cracked specimens during Mode-II fracture tests are examined using scanning electron microscopy technique. Numerical simulations of crack initiation and propagation for all fracture tests have been performed using a combined formulation of fracture mechanics and damage mechanics. The introduction of experimentally obtained interlaminar interfacial properties into a combined formulation of extended isogeometric analysis and cohesive zone model shows a good agreement with experimentally obtained results.

A phase-field modeling method for the mixed-mode fracture of brittle materials based on spectral decomposition

The phase-field method is artful to handle sharp crack discontinuities by regularizing the sharp crack surface topology with a function. In the phase-field theory, the phase field characterizing the crack is driven by the strain energy. The strain energy of the crack-driving part is the key to simulate the realistic crack surface. In this study, a new framework of the phase-field method is proposed based on the unified tensile fracture criterion, in which the crack-driving strain energy is evaluated by the spectral representation with a thermodynamic consistent projection operator to ensure the thermodynamic consistency. The mixed-mode fracture of brittle materials can be well simulated. Furthermore, a new degradation function is proposed to simulate the brittle materials with different degradation rates. The proposed phase-field modeling method is implemented by a staggered algorithm, in which the phase field and displacement field are successively solved and updated by the subroutines UEL and UMAT of the commercial finite element software ABAQUS. The comparison with experimental data shows that the proposed phase-field modeling method can quantitatively and qualitatively simulate both the cracking and mechanical behavior of mixed-mode fracture.

Дослідження особливостей руйнування шаруватої кераміки в її мікрооб’ємах методами індентування

Technology and modes of ZrB2―SiC layered ceramic composites manufacturing have been developed. The structures, elastic characteristics and strength properties of the materials under investigation have been studied. Effect of internal stress fields on fracture processes in the indentation area and mechanical properties of the ceramics in its microvolumes has been investigated both in layers and at their interfaces. Using values of contact tensile strength along different directions in layers of the composites, effective residual thermal stresses have been calculated (≈180 MPa). The obtained data on contact tensile strength and effective crack resistance, taking into consideration the contributions of residual stresses to their values, have been used for estimations of contact strength and crack resistance of the layer materials themselves. The fracture toughness measured by the three-point bending method is 3,3—3,7 MPa · m1/2. Analysis of the data obtained indicates that the spark notch provides a greater sharpness of the crack tip and better conditions for measuring K1c, while processing with a blade picks up a sharp thermal crack in the notch tip. The elastic properties of the multilayer system (SiC—15% ZrB2) + (SiC—30% ZrB2) were studied using ultrasonic research methods. The values of the velocities of sound and elastic characteristics are sufficiently large and close to those expected from the models of the composite, which does not contain noticeable porosity and microcracks in the layers themselves and in the region of their boundaries. For directions along and across the plane of the layers, the values of Young's moduli differ by about 6%. For the directions of propagation of an ultrasonic wave along and across the layers, anisotropy of ultrasonic velocities of ~5% and elastic moduli of ~10—12% is observed, which may be due to the texture that develops in the structure of the layers during hot pressing. Keywords: layered ceramics, indentation, strength properties, thermal stresses.

Applications of phase field fracture in modelling hydrogen assisted failures

The phase field fracture method has emerged as a promising computational tool for modelling a variety of problems including, since recently, hydrogen embrittlement and stress corrosion cracking. In this work, we demonstrate the potential of phase field-based multi-physics models in transforming the engineering assessment and design of structural components in hydrogen-containing environments. First, we present a theoretical and numerical framework coupling deformation, diffusion and fracture, which accounts for inertia effects. Several constitutive choices are considered for the crack density function, including choices with and without an elastic phase in the damage response. The material toughness is defined as a function of the hydrogen content using an atomistically-informed hydrogen degradation law. The model is numerically implemented in 2D and 3D using the finite element method. The resulting computational framework is used to address a number of case studies of particular engineering interest. These are intended to showcase the model capabilities in: (i) capturing complex fracture phenomena, such as dynamic crack branching or void-crack interactions, (ii) simulating standardised tests for critical components, such as bolts, and (iii) enabling simulation-based paradigms such as Virtual Testing or Digital Twins by coupling model predictions with inspection data of large-scale engineering components. The evolution of defects under in-service conditions can be predicted, up to the ultimate failure. By reproducing the precise geometry of the defects, as opposed to re-characterising them as sharp cracks, phase field modelling enables more realistic and effective structural integrity assessments.

Consolidated derivation of fracture mechanics parameters and fatigue theoretical evolution models: basic review

Presence of cracks lead to structural steels failure below critical yield strength. The primary aim of the present article is to simplify and consolidate mathematical derivation of stress concentration, fracture stress, stress intensity factor, crack tip opening displacement and J-integral parameters from the first principle as well as application to fatigue. The review explains the mathematical derivation of fracture mechanics parameters from the theoretical concept, including alternatives to fatigue life prediction with strain-based approach method. The stress concentration around a notch can only be performed if the radius of the notch is far greater than zero and the stress field at sharp crack shows singularity when the crack tip radius is equal to zero. Furthermore, blunted crack tip violates stress singularity, while the crack tip opening displacement and J-integral parameters show the solution of a crack extending beyond zero crack tip radius, thus are used to characterize material stress fields with blunted crack tip. The review highlights benefit of characterizing fatigue crack growth with J-integral and crack tip opening displacement parameters over stress intensity factor. This paper would benefit majorly engineers and specialists in nuclear, aviation, oil and gas industries.

Atomic-Scale Studies of Overlapping Grain Boundaries between Parallel and Quasi-Parallel Grains in Low-Symmetry Monolayer ReS2

Summary Grain boundaries (GBs) are significant microstructures that dominate properties of polycrystalline two-dimensional (2D) materials. Low-symmetry rhenium disulfide monolayers provide an ideal platform to investigate diverse configurations of GBs and their orientation-dependent characteristics. Here, we utilize the preferential deposition of platinum-based nanoparticles on grain edges to rapidly locate nanoscale-wide overlapping GBs during microscopic-scale observation. Atomic-resolution investigations by aberration-corrected annular dark-field scanning transmission electron microscopy reveal diverse overlapping GBs constructed by parallel grains with transversal displacement. They prefer to align along several primary lattice directions with different interlayer atomic registries. Calculations show that the electronic band structures of overlapping GBs are strongly direction dependent, suggesting their potential in tunable electronic and photonic devices. Incommensurate overlapping GBs formed by quasi-parallel grains with ultrasmall twist angles are also observed. These results unveil the rich polymorphism of GBs and new opportunities to tailor properties of anisotropic 2D materials via GB engineering.

A combined XFEM phase-field computational model for crack growth without remeshing

This paper presents an adaptive strategy for phase-field simulations with transition to fracture. The phase-field equations are solved only in small subdomains around crack tips to determine propagation, while an extended finite element method (XFEM) discretization is used in the rest of the domain to represent sharp cracks, enabling to use a coarser discretization and therefore reducing the computational cost. Crack-tip subdomains move as cracks propagate in a fully automatic process. The same mesh is used during all the simulation, with an h-refined approximation in the elements in the crack-tip subdomains. Continuity of the displacement between the refined subdomains and the XFEM region is imposed in weak form via Nitsche’s method. The robustness of the strategy is shown for some numerical examples in 2D and 3D, including branching and coalescence tests.

Phase‐field modeling of brittle fracture in a 3D polycrystalline material via an adaptive isogeometric‐meshfreeapproach

SummaryPhase‐field modeling, which introduces the regularized representation of sharp crack topologies, provides a convenient strategy for tackling 3D fracture problems. In this work, an adaptive isogeometric‐meshfree approach is developed for the phase‐field modeling of brittle fracture in a 3D polycrystalline material. The isogeometric‐meshfree approach uses moving least‐squares approximations to construct the equivalence between isogeometric basis functions and meshfree shape functions, thus inheriting the flexible local mesh refinement scheme from a meshfree method. This refinement scheme is improved by introducing an error estimator that includes both the phase field and its gradient. With the present approach, numerical implementations of the adaptive phase‐field modeling that introduces the anisotropy of fracture resistance in polycrystals are proposed. In this way, propagating cracks can be dynamically tracked, and the mesh near cracks is refined in a meshfree manner without requiring a priori knowledge of crack paths. Furthermore, the intergranular and transgranular crack propagation patterns in polycrystalline materials can be simulated by the present approach. A series of numerical examples that deal with the isotropic and anisotropic fracture are investigated to demonstrate the robustness and effectiveness of the proposed approach.

Exact analysis of mode-III cohesive fracture in layered elastic composites

This paper examines the mechanics of mode-III defect initiation and quasi-static growth in a composite consisting of N layers separated by nonuniform nonlinear cohesive surfaces. The exact analysis is based on the elasticity solution to the problem of a single layer of finite width and thickness subjected to arbitrary, nonuniform but equilibrated shear tractions on top and bottom surfaces. The formulation leads to a system of integral equations governing rigid body layer translations and cohesive surface slip fields (tangential jump discontinuities across layer interfaces). Cohesive surfaces are modelled by traction-slip relations (exponential, modified exponential) characterized by a cohesive strength, a force length and possibly friction parameters. Symmetric center, edge and array defects are modeled by cohesive strength functions that vary with surface coordinate. Infinitesimal strain equilibrium solutions are sought by eigenfunction approximation of the solution of the governing integral equations. Solutions indicate that quasi-static defect initiation/propagation occur under increasing applied shear traction. For small values of force length, brittle behavior occurs corresponding to sharp crack growth. At larger values of force length, ductile response occurs similar to linear “spring” cohesive surfaces. Both behaviors ultimately cause abrupt failure of the surface. Results for the stiff, strong cohesive surface with a center defect under a small applied traction agree well with static crack solutions taken from the literature. Detailed results are obtained for (i) an array of symmetric collinear defects accounting for defect–defect interaction, (ii) a symmetric defect in a bilayer with nonuniform material properties/thicknesses, (iii) a center defect accounting for decohesion and friction.

A length scale insensitive anisotropic phase field fracture model for hyperelastic composites

Fracture of composites consisting of isotropic matrix and anisotropic fibers is an essential problem in many engineering applications. The computational simulation of such a fracture is complicated, but the use of phase field models (PFMs) is promising. Indeed, in PFMs, sharp cracks are not treated as discontinuities; instead, they are approximated as thin damage bands. Thus, PFMs can seamlessly model complex crack patterns like branching, merging, and fragmentation. However, previous PFMs for anisotropic fracture, which are mostly based on a PFM using a simple quadratic degradation function without any user-defined parameters, provide solutions that are sensitive to a length scale (that controls the width of the damage band). Moreover, those PFMs have considered a same softening behavior for isotropic and anisotropic part of composite – which is not correct. This paper presents a length scale insensitive PFM for brittle fracture of anisotropic hyperelastic solids considering distinct softening behavior of isotropic matrix and anisotropic fibers. This model is an extension of the model of Wu [JMPS, 103 (2017)] with a rational degradation function dependent on elasticity and fracture related material parameters. Fracture of fiber-reinforced composites and biological tissues, simulated using the proposed model within the framework of the finite element method, is presented with predictions in good agreement with previous findings and experiments. Most importantly, the results are independent of the incorporated length scale parameter. Moreover, preliminary results show that the proposed model is as efficient as the previous models.

A length scale insensitive phase field model for brittle fracture of hyperelastic solids

Fracture of hyperelastic materials such as synthetic rubber, hydrogels, textile fabrics is an essential problem in many engineering fields. The computational simulation of such a fracture is complicated, but the use of phase field models (PFMs) is promising. Indeed, in PFMs, sharp cracks are not treated as discontinuities; instead, they are approximated as thin damage bands. Thus, PFMs can seamlessly model complex crack patterns like branching, merging, and fragmentation. However, previous PFMs for hyperelastic materials, which are mostly based on a PFM with a simple quadratic degradation function without any user-defined parameters, provide solutions that are sensitive to a length scale (that controls the width of the damage band). The current practice of considering this length scale as a material parameter suffers from two issues. First, such a calculated length scale might be too big (compared with the problem dimension) to provide meaningful crack patterns. Second, it might be too small, which results in undesirable computationally expensive simulations. This paper presents a length scale insensitive PFM for brittle fracture of hyperelastic materials. This model is an extension of the model of Wu (2017) with a material parameter dependent rational degradation function, which converges to Cohesive Zone Model (CZM) at least for 1D problems (Wu and Nguyen, 2018), and also can deal with crack nucleation and propagation simultaneously. Results of mode-I and mixed-mode fracture problems obtained with the method of finite elements are in good agreement with previous findings and independent of the discretization resolution. Most importantly, they are independent of the incorporated length scale parameter. Moreover, preliminary results show that the proposed model is as efficient as, if not more than the previous models.

On the Computational Derivation of Bond-Based Peridynamic Stress Tensor

The concept of ‘contact stress’, as introduced by Cauchy, is a special case of a nonlocal stress tensor. In this work, the nonlocal stress tensor is derived through implementation of the bond-based formulation of peridynamics that uses an idealised model of interaction between points as bonds. The method is sufficiently general and can be implemented to study stress states in problems containing stress concentration, singularity, or discontinuities. Two case studies are presented, to study stress concentration around a circular hole in a square plate and conventionally singular stress fields in the vicinity of a sharp crack tip. The peridynamic stress tensor is compared with finite element approximations and available analytical solutions. It is shown that peridynamics is capable of capturing both shear and direct stresses and the results obtained correlate well with those obtained using analytical solutions and finite element approximations. A built-in MATLAB code is developed and used to construct a 2D peridynamic grid and subsequently approximate the solution of the peridynamic equation of motion. The stress tensor is then obtained using the tensorial product of bond force projections for bonds that geometrically pass through the point. To evaluate the accuracy of the predicted stresses near a crack tip, the J-integral value is computed using both a direct contour approximation and the equivalent domain integral method. In the formulation of the contour approximation, bond forces are used directly while the proposed peridynamic stress tensor is used for the domain method. The J-integral values computed are compared with those obtained by the commercial finite element package Abaqus 2018. The comparison provides an indication on the accurate prediction of the state of stress near the crack tip.

On Preparation of Advance Ceramic for Single-edge V-Notch Beam Fracture Toughness Test of ISO/FDIS 23146:2008(E) Standards

With establishment of fracture toughness test for advance ceramic according ISO/FDIS 23146:2008(E) with the sample test of Single-edge V-Notch Beam/SEVNB, then the researcher related this activity should understand and able to realize how to prepare the sample of advance ceramic in order the requirements of the standard are fulfilled. In contrary with metals and polymers that usually easy to cut as the standard size for testing, the advance ceramics are found much more difficult to prepare since it is very hard and brittles. More even, specifically for fracture toughness test, the level of difficulty is added since the sharp initial crack should be introduced in the edge of the sample. This article introduce method on preparation of advance ceramic for the purpose of fracture toughness test completed with method to introduce the initial sharp V-notch crack which is valid and possible to be measured its length and diameter of the tip that yield valid result of fracture toughness test.

Fracture toughness of zirconia with a nanometer size notch fabricated using focused ion beam milling.

ObjectivesZirconia with 3 mol% yttria (3Y‐TZP) has been used for dental crowns and bridges due to its excellent mechanical behavior. Performing fracture toughness testing on this nanograin material, however, can be a challenge. For reliable results, fracture toughness testing requires an extremely sharp notch in the test specimen that closely approximates a very sharp crack. This study was to investigate an alternative method to produce nanometer‐sized notches, which are less than the average grain size of 3Y‐TZP, during the preparation of single‐edge V‐notched beam specimens and report the resulting fracture toughness value.MethodsWe present a method using focused ion beam (FIB) milling to fabricate nanometer‐sized notches in 3Y‐TZP. The notch tip is <100 nm wide, which is smaller than the grain size, and is consistent throughout the thickness of the specimen.ResultsThe FIB‐notched specimens show a much reduced average fracture toughness of 5.64 ± 1.14 MPa√m compared to 8.90 ± 0.23 MPa√m for the specimens without FIB‐notches. The FIB‐milling did not appear to create any monoclinic phase prior to fracture toughness testing. Fractures originated at the FIB‐notches, and the notch size can be readily identified post‐mortem using a microscope. A considerable amount of tetragonal‐to‐monoclinic phase transformation was observed throughout the fracture surfaces.SignificanceFIB milling provides an alternative method to fabricate nanometer‐sized notches that are smaller than the grain size of tetragonal zirconia polycrystal. The fracture toughness determined using FIB‐notches was ~5.64 MPa√m, smaller than the specimens with V‐notches fabricated using saw blades.

A phase field model for cohesive fracture in micropolar continua

While crack nucleation and propagation in the brittle or quasi-brittle regime can be predicted via variational or material-force-based phase field fracture models, these models often assume that the underlying elastic response of the material is not size-dependent. Yet, a length scale parameter must be introduced to these models to enable sharp cracks properly represented by a regularized implicit function. However, many materials with internal microstructures that contain surface tension, micro-cracks, micro-fracture, inclusion, cavity or those of particulate nature often exhibit size-dependent behaviors in both the path-independent and path-dependent regimes. This paper is intended to introduce a unified treatment that captures the size effect of the materials in both elastic and damaged states. By introducing a cohesive micropolar phase field fracture theory, along with the computational model and validation exercises, we explore the interacting size-dependent elastic deformation and fracture mechanisms exhibits in materials of complex microstructures. To achieve this goal, we introduce the distinctive degradation functions of the force-stress–strain and couple-stress-micro-rotation energy-conjugated pairs for a given regularization profile such that the macroscopic size-dependent responses of the micropolar continua are insensitive to the length scale parameter of the regularized interface. Then, we apply the variational principle to derive governing equations from the micropolar stored energy and dissipative functionals. Numerical examples are introduced to demonstrate the proper way to identify material parameters and the capacity of the new formulation to simulate complex crack patterns in the quasi-static regime.

A cutting mechanism map for machining metallic sheets using electric current pulses

We describe a tool-less and free-form manufacturing process for cutting metallic sheets by extending an initial cut by passing a series of electric current pulses. As the electric current reverses its direction across a sharp cut, an electromagnetic force is generated that tries to open the initial cut. The reversal of electric current around the cut also results in current crowding at the cut-tip, thereby concentrating Joule heating in the vicinity of the cut-tip. The self-induced electromagnetic force works in tandem with the concentrated Joule heating at the cut-tip, resulting in extension of the initial cut by a finite amount upon passing an electric pulse. The cut grows parallel to the initial orientation via one of the three mechanisms, namely sharp crack extension, blow-hole formation and formation of elongated hole with buckled edges, depending on the current density and the temperature. A detailed cutting mechanism map has been developed that can be used for selecting proper parameters for electro-pulse induced cutting for obtaining desired cut profile. This work paves path for realization of a unique methodology for machining hard-to-cut conducting materials.

Interaction of a screw dislocation with a parabolic cavity and a semi-infinite crack

We derive an analytic solution to the problem of a screw dislocation interacting with a parabolic cavity and a semi-infinite sharp crack using conformal mapping techniques and the method of images. Closed-form expressions for the image force acting on the screw dislocation, the mode III stress intensity factor at the crack tip and the generalized mode III stress intensity factor for the parabolic cavity are obtained. The correctness of the solution is validated by comparison with existing solutions in the literature.

A Reconsideration of the Log-Secant Model for Failure Pressure Prediction of Surface-Breaking Axial Cracks in Pipelines

Abstract In the late 1960s and early 1970s, the researchers of the NG-18 Committee at the Battelle Institute in Columbus Ohio completed a seminal study on the failure pressures of axial flaws in oil and gas pipelines. Key developments included the “ASME B31G” equations for assessment of blunt metal loss flaws, the log-secant model for sharp through-wall cracks, and the log-secant model for sharp surface-breaking cracks. These equations are well-established and feature in various industry standards, recommended practices, and federal regulatory requirements. This work is a reconsideration of the log-secant model for axial surface-breaking cracks. The original equations were derived based on a through-wall crack, for which the crack length is the driving force for crack extension. However, for a surface crack, the crack depth is the correct driving force for crack extension. This work rederives the log-secant model starting with an infinitely long surface crack, and then empirically corrects for a finite length. The result is a new failure pressure model of similar form to the original log-secant model, but with a few key differences. Preliminary validation work using the original NG-18 data shows promising results.

Scattering on a square lattice from a crack with a damage zone.

A semi-infinite crack in an infinite square lattice is subjected to a wave coming from infinity, thereby leading to its scattering by the crack surfaces. A partially damaged zone ahead of the crack tip is modelled by an arbitrarily distributed stiffness of the damaged links. While an open crack, with an atomically sharp crack tip, in the lattice has been solved in closed form with the help of the scalar Wiener–Hopf formulation (Sharma 2015 SIAM J. Appl. Math., 75, 1171–1192 (doi:10.1137/140985093); Sharma 2015 SIAM J. Appl. Math. 75, 1915–1940. (doi:10.1137/15M1010646)), the problem considered here becomes very intricate depending on the nature of the damaged links. For instance, in the case of a partially bridged finite zone it involves a 2 × 2 matrix kernel of formidable class. But using an original technique, the problem, including the general case of arbitrarily damaged links, is reduced to a scalar one with the exception that it involves solving an auxiliary linear system of N × N equations, where N defines the length of the damage zone. The proposed method does allow, effectively, the construction of an exact solution. Numerical examples and the asymptotic approximation of the scattered field far away from the crack tip are also presented.