Steady-state Crack Research Articles

Bridging crack propagation at the atomistic and mesoscopic scale for BCC-Fe with hybrid multiscale methods

The hybrid multiscale method bridging the atomistic and mesoscopic scale is proposed in the current study, which combines the concurrent generalized particle (GP) dynamics method and the bottom-up hierarchical cohesive zone model (CZM) with embedded traction–separation law. The primary purpose is to transfer the GP-obtained physical parameters to the upper mesoscale finite element method (FEM) to investigate the meso- and microscopic crack propagation. The local fracture energy, stress and opening relationships under tension loading during steady-state crack propagation are extracted from the three-scale GP model. In this procedure, the scale duality technique is conducted using the GP analog, which can allow material to exist as particles via a lumping process and allows them to decompose into atoms at crack tips and interfaces. Crack extension resistance is detected by coordination vector (CV) snapshots, and energy release rates in the subdomain are used to evaluate the material behavior against the crack propagation when the current crack tip grows. Using the unique cohesive element length, four-scale tension specimen FE models are designed to reveal the accuracy of the intrinsic correlation between the atomistic and mesoscopic scale under a stress intensity factor to study the brittle body-centered cubic (BCC)-Fe fracture behavior. The result appears reasonable and encouraging.

A Yoffe crack/cohesive zone model for a steady state moving crack

A Yoffe crack/cohesive zone model is presented to investigate steady state crack propagation in an infinite medium. In the current model, the rear crack tip closes and a cohesive zone exists only at the front crack tip during crack propagation, which eliminates the dilemma in the standard Yoffe crack/Dugdale zone model that the crack length remains constant but the rear crack tip opens up. The cohesive zone length and the crack tip opening displacement (CTOD, the opening at the physical crack tip) are determined in closed forms using a Fourier transform/integral equation approach. It is found that the cohesive zone length and CTOD differ significantly from those in the standard Yoffe crack/Dugdale model in which two cohesive zones exist at both crack tips during crack propagation.

Traction–separation relationships for hydrogen induced grain boundary embrittlement in nickel via molecular dynamics simulations

A statistical approach combined with molecular dynamics simulations is used to study the influence of hydrogen on intergranular decohesion. This methodology is applied to a Ni Σ3(112)[11¯0] symmetric tilt grain boundary. Hydrogenated grain boundaries with different H concentrations are constructed using an energy minimization technique with initial H atom positions guided by Monte Carlo simulation results. Decohesion behavior is assessed through extraction of a traction–separation relationship during steady-state crack propagation in a statistically meaningful approach, building upon prior work employing atomistic cohesive zone volume elements (CZVEs). A sensitivity analysis is performed on the numerical approach used to extract the traction–separation relationships, clarifying the role of CZVE size, threshold parameters necessary to differentiate elastic and decohesion responses, and the numerical averaging technique. Results show that increasing H coverage at the Ni Σ3(112)[11¯0] grain boundary asymmetrically influences the crack tip velocity during propagation, leads to a general decrease in the work of separation required for crack propagation, and provides a reduction in the peak stress in the extracted traction–separation relationship. The present framework offers a meaningful vehicle to pass atomistically derived interfacial behavior to higher length scale formulations for intergranular fracture.

Asymptotic solutions of mode I steady growth crack in materials under creep conditions

An asymptotic analysis is made on problems with a steady-state crack growth coupled with a creep law model under tensile loads. Asymptotic equations of crack tip fields in creep materials are derived and solved numerically under small scale conditions. Stress and strain functions are adopted under a polar coordinate system. The governing equations of asymptotic fields are obtained by inserting the stress field and strain field into the material law. The crack growth rate rather than fracture criterion plays an important role in the crack tip fields of materials with creep behavior.

Virtual Boundary Meshless with Trefftz Method for the Steady-State Heat Conduction Crack Problem

This article presents a virtual boundary meshless with Trefftz method for calculation of the two-dimensional steady-state heat conduction crack problem, which is based on the idea of multidomain combination. The virtual source function is constructed by the radial basis functions approximation in the subdomain not containing the crack. The temperature or heat flux is calculated by crack special solution of Trefftz function in the crack subdomain. Thus, the proposed method has the advantages of both the boundary-type meshless method and the Trefftz method. No nodes or elements are distributed on the crack, because the boundary condition of the crack is automatically satisfied by employing the crack special solution of the Trefftz function. The heat stress intensity factor is simply and directly calculated.

The transition from subsonic to supersonic cracks.

We present the full analytical solution for steady-state in-plane crack motion in a brittle triangular lattice. This allows quick numerical evaluation of solutions for very large systems, facilitating comparisons with continuum fracture theory. Cracks that propagate faster than the Rayleigh wave speed have been thought to be forbidden in the continuum theory, but clearly exist in lattice systems. Using our analytical methods, we examine in detail the motion of atoms around a crack tip as crack speed changes from subsonic to supersonic. Subsonic cracks feature displacement fields consistent with a stress intensity factor. For supersonic cracks, the stress intensity factor disappears. Subsonic cracks are characterized by small-amplitude, high-frequency oscillations in the vertical displacement of an atom along the crack line, while supersonic cracks have large-amplitude, low-frequency oscillations. Thus, while supersonic cracks are no less physical than subsonic cracks, the connection between microscopic and macroscopic behaviour must be made in a different way. This is one reason supersonic cracks in tension had been thought not to exist.

Creep relaxation in the presence of residual stress

In this paper, the elastic and elastic–plastic creep behaviour of cracked structures in the presence of a residual stress field have been studied, numerically and analytically. A self-equilibrating residual stress field is introduced by prior mechanical loading. The magnitude of the residual stress field is varied and the corresponding transient crack tip parameter, C(t), is evaluated in single edge notched specimens, using the finite element (FE) method. Since the steady state crack tip parameter, C∗, and redistribution time, tred, are no longer valid for the case of residual stress only (as for this case C∗→0 and thus tred→∞), new normalising factors of Cnorm and tnorm are introduced and used to normalise C(t) and time, respectively. Existing analytical solutions are modified based on these normalising factors, to describe creep relaxation in the presence of a residual stress field. The values of C(t) obtained from FE analysis are compared to those from the proposed equations. It has been found that the transient C(t) value provides an accurate characterisation of the crack tip fields under residual stress only.

Failure Analysis of High Strength Steel Bar Used in a Wind Turbine Foundation

Screw-thread steel bars made of high strength steel fractured after exposure to a constant pre-stress of 650 MPa for more than 10 h. Visual examination, chemical analysis, mechanical tests, metallographic analysis, and fractographic examination were carried out in order to determine the causes of this failure. The fracture surfaces consisted of three regions: the crack source or fracture origin region, steady-state crack growth, and unstable or final fracture region. The crack initiated at the center of the cross section and then propagated to the periphery. The fracture exhibited a brittle appearance, accompanied with secondary cracks, tear ridges, and micropores on the grain boundary. The results revealed that this failure was due to hydrogen embrittlement of the steel.

Strain rate effects on dynamic fracture of pipeline steels: Finite element simulation

The present work develops a strain rate-dependent cohesive zone model and related finite element model to analyze speed-dependent dynamic fracture of pipeline steels observed in recent drop-weight tear tests. Different than most of existing cohesive zone models, the traction-separation law of the present model considers the rate of separation in the cohesive zone, and a rate-dependent elastic-viscoplastic constitutive model is employed for the bulk material. The speed-dependences of CTOA and energy dissipation observed experimentally are reproduced in our simulations for moderate steady-state crack speed (up to 150 m/s). The present model gives detailed stress-strain fields surrounding the moving crack tip, which offer plausible explanation why the rate-effects in the bulk material and the cohesive zone could be largely responsible for all observed speed-dependent dynamic fracture phenomena of pipeline steels.

Modeling of steady-state crack growth in shape memory alloys using a stationary method

Shape memory alloys experience phase transformation from austenite to martensite around crack tip. When the crack advances, martensitic transformation occurs at the tip and the energy that goes into this transformation results in stable crack growth like in the case of plastic deformation. In literature, steady-state crack growth in elasto-plastic materials with small scale yielding at the crack tip has been successfully modeled using stationary methods. In this work, Mode I steady-state crack growth in an edge-cracked Nitinol plate is modeled using a non-local stationary method (Moumni, 1995). The Zaki–Moumni (ZM) constitutive model is utilized for this purpose. The model is implemented in ABAQUS by means of a user-defined material subroutine (UMAT) to determine transformation zones around the crack tip. Steady-state crack growth is first simulated without considering reverse transformation to calculate the effect of transformation on stress distribution in the wake region, then reverse transformation is taken into account. The effect of reorientation of martensite near the crack tip as a result of non-proportional loading is also studied. The stress distribution and the phase transformation region are compared to results obtained for the case of a static crack.

Matrix cracking of fiber-reinforced ceramic composites in shear

The mechanics of cracking in fiber-reinforced ceramic matrix composites (CMCs) under general loadings remains incomplete. The present paper addresses one outstanding aspect of this problem: the development of matrix cracks in unidirectional plies under shear loading. To this end, we develop a model based on potential energy differences upstream and downstream of a fully bridged steady-state matrix crack. Through a combination of analytical solutions and finite element simulations of the constituent stresses before and after cracking, we identify the dominant stress components that drive crack growth. We show that, when the axial slip lengths are much larger than the fiber diameter and when interfacial slip precedes cracking, the shear stresses in the constituents are largely unaffected by the presence of the crack; the changes that do occur are confined to a ‘core’ region within a distance of about one fiber diameter from the crack plane. Instead, the driving force for crack growth derives mainly from the axial stresses—tensile in the fibers and compressive in the matrix—that arise upon cracking. These stresses are well-approximated by solutions based on shear-lag analysis. Combining these solutions with the governing equation for crack growth yields an analytical estimate of the critical shear stress for matrix cracking. An analogous approach is used in deriving the critical stresses needed for matrix cracking under arbitrary in-plane loadings. The applicability of these results to cross-ply CMC laminates is briefly discussed.

On the Effect of Latent Heat on the Fracture Toughness of Pseudoelastic Shape Memory Alloys

A finite element analysis of steady-state crack growth in pseudoelastic shape memory alloys under the assumption of adiabatic conditions is carried out for plane strain, mode I loading. The crack is assumed to propagate at a critical level of the crack-tip energy release rate and the fracture toughness is obtained as the ratio of the far-field applied energy release rate to the crack-tip critical value. Results related to the influence of latent heat on the near-tip stress field and fracture toughness are presented for a range of parameters related to thermomechanical coupling. The levels of fracture toughness enhancement, associated with the energy dissipated by the transformed material in the wake of the growing crack, are found to be lower under adiabatic conditions than under isothermal conditions [Baxevanis et al., 2014, J. Appl. Mech., 81, 041005]. Given that in real applications of shape memory alloy (SMA) components the processes are usually not adiabatic, which is the case with the lowest energy dissipation during a cyclic loading–unloading process (hysteresis), it is expected that the actual level of transformation toughening would be higher than the one corresponding to the adiabatic case.

Nonuniform Ferro-elastic Domain Switching for the Interfacial Crack

Existing theoretical researches about interfacial crack of bimaterial piezoelectric composites are confined to linear piezoelectric constitutive relation, neglecting the nonlinear part driven by domain switching. This paper attempts to extend the research of mode III interfacial crack from linear piezoelectricity to nonlinear ferroelectricity. The analysis focuses on the variation of stress intensity factor caused by domain switching, with an arbitrary three-dimensional initial poling orientation. Due to the mismatch of material properties, an asymmetric nonuniform domain switching zone is achieved around the interfacial crack under the nonuniform domain switching criterion. By employing the weight function method, analytic forms of mono-domain and multi- domain toughening effects about stationary and quasi-static steady-state crack are obtained, respectively. The conclusions come to that domain switching of mode III quasi-static steady-state growing interfacial crack has a positive effect to toughen the material. The research can provide some guidance for material selection to maximize the effect of ferro-elastic switch toughening.

Crack propagation in pre-strained single layer graphene sheets

We use molecular dynamics simulations to delineate crack propagation speed as a function of the crack length and the axial prestrain in a single layer graphene sheet. A covalent bond between two carbon atoms is assumed to break when the bond length has been stretched by 100%. For a pristine single layer graphene sheet the maximum axial force is attained at a nominal axial strain of 15.5%. A pristine graphene sheet is first deformed in tension in the armchair direction to the desired value of the axial strain, and then one or two cracks are simultaneously inserted in it at central locations by breaking the bonds. Five such problems have been studied with four different values of the axial prestrain up to 15.3%. For each problem, crack-tip speeds are found to reach steady state values as the crack elongates. The steady state crack speed increases with an increase in the axial nominal prestrain. The crack propagation is found to be stable in the sense that the value of the J-integral increases with an increase in the crack length. For the same normalized crack length the value of the J-integral increases with an increase in the nominal axial prestrain.

Investigation of thermal effects on fatigue crack closure using multiscale digital image correlation experiments

Hastelloy X, a nickel-based superalloy, has been extensively used for high temperature applications. In this work, Hastelloy X notched samples were used to investigate fatigue crack growth and crack closure at elevated temperatures. Isothermal, thermal jump, and thermal overload experiments at varying temperatures (up to 650°C), were performed. Macroscale (2μm/pixel) digital image correlation was performed on images taken at various stages of crack growth and microscale (0.4μm/pixel) digital image correlation was used on images obtained directly behind the crack tip to quantify the local effects of crack closure. Experiments focused on the effects of isothermal conditions and thermal overloads on measured crack closure levels. Each isothermal experiment showed steady state crack closure levels of 0.30 while thermal jumps and thermal overloads created significant decreases (or, in some cases, complete elimination) in closure levels immediately following the temperature change. Similar to the case of mechanical overloads, as crack growth was continued beyond the plastic zone enlargement created by the thermal spike, closure levels were reestablished near the original steady state values. Competing mechanisms, including crack tip blunting, crack bifurcation, change in temperature, yield stress, elastic modulus, and plastic zone size, thought to be responsible for the changes in closure levels following the thermal jumps and during the thermal overload, were investigated.

Experimental approach for mixed-mode fatigue delamination crack growth with large-scale bridging in polymer composites

An experimental apparatus utilizing double cantilever beam specimens loaded with uneven bending moments was developed to study the mixed-mode fatigue crack growth in composites. The approach is suitable when large-scale bridging of cracks is present. To illustrate the testing method, cyclic growth of delaminations in a typical fibre-reinforced polymer composite was investigated under a constant cyclic loading amplitude. Pure mode I, mode II and mixed-mode crack growth conditions were examined. The results, analysed using a J-integral approach, show that the double cantilever beam loaded with uneven bending moments configuration provides a robust approach to investigate the fatigue crack growth of composites for pure mode and mixed-mode cracking. A steady-state crack growth regime was observed for mode I and mixed-mode loading. For mode II loading, steady-state was absent, and a progressively decreasing crack growth rate observed. In addition to details concerning the equipment, a general discussion of the development of cyclic bridging laws for delamination growth in the presence of large-scale bridging is provided.

On the Fracture Toughness of Pseudoelastic Shape Memory Alloys

A finite element analysis of quasi-static, steady-state crack growth in pseudoelastic shape memory alloys is carried out for plane strain, mode I loading. The crack is assumed to propagate at a critical level of the crack-tip energy release rate. Results pertaining to the influence of forward and reverse phase transformation on the near-tip mechanical fields and fracture toughness are presented for a range of thermomechanical parameters and temperature. The fracture toughness is obtained as the ratio of the far-field applied energy release rate to the crack-tip energy release rate. A substantial fracture toughening is observed, in accordance with experimental observations, associated with the energy dissipated by the transformed material in the wake of the growing crack. Reverse phase transformation, being a dissipative process itself, is found to increase the levels of toughness enhancement. However, higher nominal temperatures tend to reduce the toughening of an SMA alloy—although the material's tendency to reverse transform in the wake of the advancing crack tip increases—due to the higher stress levels required for initiation of forward transformation.

Microbranching in mode-I fracture in a randomly perturbed lattice

We study mode-I fracture in lattices using atomistic simulations with randomly distributed bond lengths. By using a small parameter that measures the variation of the bond length between the atoms in perfect lattices and using a three-body force law, simulations reproduce the qualitative behavior of the beyond-steady-state cracks in the high-velocity regime, including reasonable microbranching. In particular, the effect of the lattice structure on the crack appears minimal, even though in terms of the physical properties such as the structure factor g(r) and the radial or angular distributions, these lattices share the physical properties of perfect lattices rather than those of an amorphous material (e.g., the continuous random network model). A clear transition can be seen between steady-state cracks, where a single crack propagates in the midline of the sample, and the regime of unstable cracks, where microbranches start to appear near the main crack, in line with previous experimental results. This is seen in both a honeycomb lattice and a fully hexagonal lattice. This model reproduces the main physical features of propagating cracks in brittle materials, including the total length of microbranches as a function of driving displacement and the increasing amplitude of oscillations of the electrical resistance. In addition, preliminary indications of power-law behavior of the microbranch shapes can be seen, potentially reproducing one of the most intriguing experimental results of brittle fracture. There was found to exist a critical degree of disorder, i.e., a sharp threshold between the cleaving behavior characterizing perfect lattices and the microbranching behavior that characterizes amorphous materials.

Experimental investigation of mesoscale crack front triple line

The aim of this paper is to investigate the mesoscale behavior and structure of an adhesive near the fracture front of an asymmetric joint consisting of carbon fiber/epoxy resin composites bonded with a relatively soft, epoxy adhesive. A single cantilever beam fracture test at constant separation rate gave steady-state crack propagation, details of which were followed by digital image correlation (DIC). A deformed, triple line region was found between the adhesive, air, and the composite, somewhat resembling a “wetting ridge,” as found with a liquid meniscus in contact with a soft solid. Importantly, the partially separated bondline layer took part in (non-unidirectional) load transfer between adherends (and thus energy dissipation), contrary to common assumptions where the separated bondline is assumed no longer to play a structural role. A simple model, based on the Flamant contact mechanics approach, is proposed and compared with both a finite element solution and experimental data extracted from image correlation. The model points out the importance of two length scales: process zone extent and adhesive thickness, both being known to affect global properties of bonded structures.

Dynamic energy release rate in couple-stress elasticity

This paper is concerned with energy release rate for dynamic steady state crack problems in elastic materials with microstructures. A Mode III semi-infinite crack subject to loading applied on the crack surfaces is considered. The micropolar behaviour of the material is described by the theory of couple-stress elasticity developed by Koiter. A general expression for the dynamic J-integral including both traslational and micro-rotational inertial contributions is derived, and the conservation of this integral on a path surrounding the crack tip is demonstrated.