Small Scale Yielding Research Articles

The Improved Irwin’s Model for Crack Problems in Power-Law Hardening Materials

In this paper, an improved Irwin’s model for power-law hardening materials is established through the static equivalence between the linear elastic stress solution and the stress redistribution in the plasticity-affected zone (PAZ) under small-scale yielding (SSY) conditions. HRR solution is adopted to describe the stress field within the crack-tip plastic zone (CTPZ). Analytical expressions of the effective stress intensity factor (SIF) and the size of PAZ are derived to illustrate the effect of CTPZ on the fracture behavior for power-law hardening materials. Influences of external load and hardening exponent on fracture behavior are explored. The proposed model can assess the toughening effect of CTPZ accurately using the SIF ratio, which is always smaller than 1. This model can be applied to various materials obeying power-law hardening constitutive equations. It will reduce to the extended Irwin’s model for elastic-perfectly plastic materials in the limiting case.

Modeling hydrogen dragging by mobile dislocations in finite element simulations

Finite element simulation modeling permits to predict hydrogen concentration for various initial boundary-values problems, but the results depend on the underlying transport mechanisms accounted for. Trapping process is a key factor in the apparent hydrogen diffusion, and the case of mobile traps as dislocations needs modification of the hydrogen transport equation usually considered in the literature. An extension of this model is proposed where hydrogen dragging by mobile traps is modeled by reaction-diffusion equations, involving trapping and detrapping kinetic, and is applied for evolving trap density with plastic strain. The consequences of trapped hydrogen mobility on diffusive hydrogen repartition in a reference Small Scale Yielding configuration are focused on, especially in term of acceleration of hydrogen transport. The potentiality of the model is illustrated by the modeling of the trapped hydrogen breakaway from fast moving dislocations. • An extension of the hydrogen transport equation is proposed. • This extension includes trap mobility and hydrogen dragging. • The model is implemented in Finite Element Abaqus software. • A Parametric study is conducted on a Small-Scale Yielding configuration. • An extension to include the loss the Cottrell atmosphere is proposed.

Evaluation of small‐scale yielding boundary using digital image correlation results

AbstractIn this work, the boundaries of small‐scale yielding (SSY) and large‐scale yielding (LSY) have been experimentally evaluated from the analysis of crack tip opening displacement (CTOD) measured by digital image correlation (DIC). According to a previous numerical work, the ratio between the elastic CTOD range and the total CTOD range (Δδe/Δδt) defines the boundaries of SSY conditions and LSY. Three materials have been studied, commercially pure titanium and 2024‐T3 and 7050‐T6 aluminum alloys, tested at different stress ratio values. SSY conditions are shown to dominate when Δδe/Δδt ≥ 79% and ≥78% for titanium and the two aluminum alloys, respectively. In addition, LSY can be established when Δδe/Δδt ≤ 66.3% and ≤67.2% for titanium and for 2024‐T3 and 7050‐T6 aluminum alloys, respectively. Transition or LSY conditions are more probable in fatigue tests conducted at low R ratio and for crack lengths above 40% with respect to the width of the specimen.

Fatigue crack growth of a railway wheel steel and fatigue life prediction under spectrum loading conditions

In this paper, fatigue life and crack growth retardation of railway wheel specimens due to spectrum loading have been investigated. For this purpose, numerical and experimental methods are used to investigate the behavior of crack growth in the wheel specimens. The crack propagation is examined by using the stress intensity factor (SIF) values achieved from boundary element (BE) analyses. The non‐linear Kujawski‐Ellyin (KE) model has been adopted for the crack propagation under small‐scale yielding (SSY) conditions. The periodic overloads have remained constant and the effect of overload ratio (OLR) and occurrence ratios (OCR) on the fatigue life has been investigated.

Mixed mode (I and II) fracture behavior of nanoglass and metallic glass

Nanoglasses (NGs) have shown large tensile ductility and hardness than metallic glasses (MGs) used to synthesized them. The mixed mode (I and II) fracture experiments and complementary finite element (FE) simulations reported significant influence of applied mode mixity on the crack tip plasticity and fracture toughness of MGs. However, no such study has been under taken for NGs which are synthesized from MGs only. Therefore, two-dimensional, plane strain, FE analysis on the stationary crack in nanoglass (NG) and metallic glass (MG) subjected to mixed mode (I and II) loading conditions are performed under small scale yielding (SSY) conditions by employing the constitutive model for MGs. It is found that the crack tip plasticity is markedly affected by mode mixity in both NGs and MGs, and the shear bands ahead of crack tip appears to be more diffused in the former than the latter. Further, the plastic zone size ahead of the crack tip in NG is larger than that for MG for identical loading conditions. Also, FE simulations predict enhancement in fracture toughness of both the alloys with increase in the mode I contribution. In addition, the results suggest that the NGs may not exhibit larger fracture toughness than MGs with identical composition, although they exhibit larger tensile ductility.

Fatigue crack propagation studies based on the plastic component of the CTOD evaluated from Digital Image Correlation data

This work presents a new approach based on the Crack Tip Opening Displacement (CTOD) for studying the fatigue crack growth. The plastic component of the CTOD is employed for predicting the crack propagation on austenitic stainless steel. This is in contrast to Linear Elastic Fracture Mechanics analysis based on the Stress Intensity Factor (SIF) (Ewalds and Wanhill 1984). The current method is also valid beyond Small Scale Yielding (Antunes et al. 2016)(Antunes et al. 2017a)(Borges et al. 2020) as it incorporates naturally the plasticity effects taking place at the crack tip (Lopez-Crespo et al. 2018). The approach is based on post-processing the full-field displacement information generated by Digital Image Correlation technique (Chernyatin et al. 2018). Such technique is used to monitor the crack advance throughout the test (Cruces et al. 2020). The post-processing procedure employed for extracting the plastic component of the CTOD is described in detail. The results on steel CT specimens appear very promising.

Numerical simulations of mixed-mode (I and II) notch tip fields in basal-textured magnesium alloys

Crystal plasticity based finite element (CPFE) simulations employing Taylor homogenization for a basal-textured Mg alloy are performed under plane strain, small scale yielding (SSY) conditions to study mixed-mode (I and II) notch tip fields. Two notch orientations are considered. In one case, the notch line and normal to the flat surfaces of the notch are taken parallel to the transverse direction (TD) and rolling direction (RD), whereas in the other, they are aligned with normal direction (ND) and TD. The results are compared against those obtained from an isotropic (von Mises) plasticity model with identical stress–strain behavior under RD tension. The maximum plastic zone size at a given level of effective stress intensity factor |K|, enhances with mode II component (i.e., with decrease in mode mixity parameter Me), and is larger for ND–TD orientation. Further, {101̄2} or tensile twinning (TT) occurs near the stretched part of the notch surface, and is more profuse as Me reduces especially for the ND–TD case. It is found that for a given Me, the notch orientation can profoundly affect the distribution of equivalent plastic strain and hydrostatic stress around the notch tip, which underscores the importance of plastic anisotropy of Mg alloys. Finally, by employing appropriate fracture criteria and the computed near-tip fields, the variations of fracture toughness and operative fracture mechanism with Me are predicted for the TD–RD case, which corroborate well with a recent experimental study.

Validity of small-scale yielding regime in notched-cracked geometries

This paper proposes a procedure based on crack tip opening displacement (CTOD) to define the boundary of the small-scale yielding (SSY) regime. Numerical modelling was applied to notched samples with different notch radii, under both plane stress and plane strain conditions, with and without crack flank contact. Results indicate that an increase in notch radius promotes the validity of SSY. The increase in crack length promotes a progressive decrease of SSY's validaty, particularly in notched specimens. Most of the situations studied fall outside the SSY regime, which reinforces the importance of verifying the applicability of LEFM to fatigue crack growth rate calculations.

Fatigue fracture and fatigue life assessment of railway wheel using non‐linear model for fatigue crack growth

In this paper, the fatigue behavior of railway wheel specimens subjected to cyclic loading have been investigated via experimental tests and numerical analysis. The material behaviour of the wheel specimen has been modelled by means of the Ramberg‐Osgood equation. The non‐linear Kujawski‐Ellyin (KE) model has been adopted for the fatigue crack propagation under small‐scale yielding (SSY) conditions. The formulation of the rigid-insert crack closure (RICC) model has been applied to a specimen of railway wheel that has the same conditions as a real instance. The results indicate that the fatigue life of specimens decreases with any increase in load level.

Predicting failure of cracked aluminum plates with one-sided composite patch

Composite patch repairs are an alternative to weld repair methods to arrest crack growth in metal plates. While these repairs have been effective, it has been difficult to use basic linear elastic fracture mechanics (LEFM) theories and design methods based on LEFM to predict the efficacy of patch repairs because the repaired plate exhibits bending and significant crack tip plasticity. In this investigation, we have studied the effect that one-sided low modulus composite patch repairs have on the development of crack tip plasticity on the free surface and through the thickness using large strain area and volume, total axial stress balance, and the free surface J-integral. Digital Image Correlation (DIC) and three-dimensional finite element analysis with first order elements were used to understand the evolution of crack tip plasticity and strain energy release rate to develop a simplified prediction method identifying a characteristic crack opening displacement prior to crack tip behavior of a one-sided composite patch repaired Center-Cracked Tension (CCT) specimen transitioning from Small Scale Yielding (SSY) to Large Scale Yielding (LSY), and eventually the re-initiation of crack growth. This method was then validated by comparing the predicted characteristic transition load to observed ultimate patched CCT specimen load capacity. Results correlated well with predictions when the composite patch was perfectly bonded to the aluminum and predicted characteristic transition loads more than 90 $$\%$$ greater than LEFM based failure predictions and 25 $$\%$$ lower than observed ultimate failure loads, demonstrating the potential to use this method for patch repair optimization.

Effect of transient trapping on hydrogen transport near a blunting crack tip

The paper revisits the way transient trapping is introduced in the literature based on the Sofronis and McMeeking model (Sofronis and McMeeking, 1989) [1] of hydrogen transport. It is shown that the direct use of the improved formulation made by Krom et al. (1999) [2] for transient trapping may lead to non-physical results of hydrogen concentration in case of an insulated system. The use of McNabb and Foster trapping kinetic equation is more relevant, and its ability to model both trap creation and kinetic trapping is investigated on a Small Scale Yielding configuration for the sake of comparison with a reference case from the literature. A parametric study is conducted, exhibiting differences with literature, and emphasizes on the significant effect of trapping kinetics on the hydrogen distribution.

Numerical study of stationary cracks in bulk metallic glass composites under Mode I, small scale yielding conditions

In this study, finite element analyses of Mode I loading of stationary cracks in in-situ bulk metallic glass composites (BMGCs) are performed under plane strain, small scale yielding (SSY) conditions. In the first part of this work, a thermodynamically consistent finite strain constitutive theory for BMGCs is employed to represent the entire domain. In this homogenized model, the soft crystalline dendrites are considered to obey J2 flow theory of plasticity with isotropic power law hardening, while the bulk metallic glass (BMG) matrix is taken to follow a free-volume based constitutive model. The effects of volume fraction Vf and hardening of the crystalline dendrites on the macroscopic stress and plastic strain fields are systematically studied. The predicted trend of the fracture toughness versus Vf corroborates well with experiments. In the second part, multi-scale SSY analyses are performed, with discrete soft dendrites in the BMG matrix close to notch tip explicitly simulated, while the surrounding region is represented by the homogenized BMGC model. It is found that low strength and/or low hardening dendrites with high Vf dissipate more plastic work in the neighborhood of the notch tip, thereby shielding the relatively brittle BMG matrix.

Fatigue crack growth prediction for small-scale yielding (SSY) and non-SSY conditions

A model to predict fatigue crack growth (FCG) for small-scale yielding (SSY) and non-SSY conditions where J-integral applies is established. The ΔJ-characterized elastic-plastic fracture mechanics (EPFM) analysis is performed based on a rearranged FCG process to achieve the crack-tip geometry and stress state. The root stress of blunted crack and the FCG rate are calculated from the cycle-wise local and global energy balance. To consider effects of geometry, load level and strain hardening, a correction factor is further developed in the governing equation of global energy balance. The model is validated using the test data of several types of metals.

A numerical study of the effects of overload on fatigue crack growth in plastically compressible hardening and hardening-softening-hardening solids

Variation of cyclic loading effect on fatigue crack growth is investigated under plane strain and small scale yielding (SSY) conditions. The material is characterized by a finite strain elastic viscoplastic constitutive model with hardening and hardening-softening-hardening hardness functions. Displacements corresponding to the isotropic linear elastic mode I crack field are prescribed on a remote boundary. The influence of cyclic stress intensity factor range (), load ratio (R), number of cycles (N), plastic compressibility ( overload and material softening on near tip deformation was studied in detail. For comparison purpose, a few results pertaining to plastically incompressible solids are also considered. The crack tip opening displacement (CTOD), plastic crack growth, deformed crack tip shape, plastic zone shape and size near the crack tip appear to depend significantly on R, N, overload and material softening. The overload produces strong plastic deformation ahead of the crack tip and decreases the fatigue crack growth substantially. Highlights Variation of cyclic loading effect on fatigue crack growth is investigated for materials characterized by a finite strain elastic viscoplastic constitutive model with hardening and hardening-softening-hardening hardness functions. Both plastically compressible and incompressible solids are considered. The influence of cyclic stress intensity factor range (), load ratio (R), number of cycles (N), plastic compressibility ( overload and material softening on near tip deformation was studied in detail. The crack tip opening displacement, plastic crack growth, plastic zone shape and size near the crack tip appear to depend significantly on R, N, overload and material softening. The combination of softening or softening-hardening material response and plastic compressibility leads to major deviations in crack tip blunting and fatigue crack growth from those that prevail for a hardening material.

Analysis of crack-tip constraint due to V- and U- side grooves in C(T) specimens

The effect of side grooves in standard compact tension (C(T)) specimens on the constraint due to plasticity under small scale yielding is presented in this paper. The Elastic-plastic crack-tip stress state is analysed through three dimensional finite element method. Ramberg-Osgood model is used to characterize the smooth transition of elastic-plastic behavior at the onset of crack-tip under Small Scale Yielding. Side grooves of type V and U are analysed and compared with plain specimen. Generally, it is observed that the plane stress effect on the surface of the plain specimen is restricted by the introduction of the side grooves. And this effect is studied with the in-plane crack-tip constraint parameter Q and out-of-plane constraint parameters Tz and h through the specimen thickness. It is found that localized plastic deformation at the side grooves increases the constraint level of the cracked specimen at the side grooves whereas it shows less at the middle of the specimen. It is also concluded that compared with U- groove specimen, V- groove specimen shows better constraint level.

Non‐linear fatigue propagation of multiple cracks in an aluminium metal matrix composite (AlMMC) with silicon‐carbide fibre reinforcement

The fatigue crack growth (FCG) in a mesoscale metal matrix composite (MMC) sample has been simulated by the analyses of a 3D finite elements (FEs) model of a sample volume cell of the composite. Several FCG numerical simulations have been performed on MMC specimens made of aluminium (Al) alloy and silicon carbide (SiC). First, in the microscale, the homogenization of the single MMC unit cell has been performed in order to find the crack initiation sites. Subsequently, in the mesoscale, a tensile remote load with appropriate boundary conditions has been applied on the MMC unit volume. Simulation of FCG has been carried out using the Paris law and two non-linear laws for small-scale yielding (SSY), namely, the modified Kujawski-Ellyin (KE) and UniGrow laws. Finally, the FCG rates provided by numerical simulations have been compared with each other.

Evaluation of ΔJ-integral for a shallow crack in steel for pressure vessels under large scale yielding (LSY) condition

Fracture mechanics studies on fatigue crack growth under elastic-plastic condition is usually characterized by J integral range, ΔJ determined by an approximation from area under load-displacement hysteresis loop developed by Dowling and Begley. Despite being supported by some experimental data which reported that fatigue crack growth rates under elastic-plastic conditions have good correlations with ΔJ, the applicability of ΔJ on crack growth in a material subjected to cyclic bending remains inconclusive. In this study, finite element analysis of a four point-bending rectangular plate model with an edge crack length a/W between 0.1 to 0.5 was carried out under displacement amplitude δa between 0.1 to 1.5 mm to represent small to large scale yielding conditions. The elastic component was found to be the driven force of fatigue crack growth parameter in small scale yielding (SSY) condition while in large scale yielding (LSY) condition, the plastic component must be considered. It was also found that the simple method of ΔJ-integral calculation using load-displacement approach is overestimated for shallow crack under LSY condition.

Crack closure in friction stir weldment using non‐linear model for fatigue crack propagation

AbstractThe fatigue crack propagation in a friction stir‐welded sample has been simulated herein by means of two 3‐dimensional finite element method (FEM)‐based analyses. Numerical simulations of the fatigue crack propagation have been carried out by assuming a residual stress field as a starting condition. Two initial cracks, observed in the real specimen, have been assessed experimentally by performing fatigue tests on the welded sample. Hence, the same cracks have been placed in the corresponding FE model, and then a remote load with boundary conditions has been applied on the welded specimen. The material behaviour of the welded joint has been modelled by means of the Ramberg‐Osgood equation, while the non‐linear Kujawski‐Ellyin (KE) model has been adopted for the fatigue crack propagation under small‐scale yielding (SSY) conditions. Owing to the compressive nature of the residual stress field that acts on a part of the cracked regions, the crack closure phenomenon has also been considered. Then, the original version of the KE law has been modified to fully include the closure effect in the analysis. Later, the crack closure effect has also been assessed in the simulation of fatigue propagation of three cracks. Finally, an investigation of the fracture process zone (FPZ) extension as well as the cyclic plastic zone (CPZ) and monotonic plastic zone (MPZ) extensions have been assessed.

Characterisation of fatigue crack tip field in the presence of significant plasticity

Characterisation of a fatigue crack tip in the presence of significant plasticity has been challenging due to the lack of suitable tools and lack of knowledge of material constitutive information under cyclic loading. In this paper, Digital Image Correlation (DIC) and integrated finite element (FE) analyses have been used to characterise the crack-tip field beyond the small-scale yielding (SSY) regime in a stainless steel 316L of a compact-tension (CT) specimen under mode I loading conditions. The non-linear characteristics of the near-tip deformation field were verified by the poor fit to the William’s regression and the overestimation of the stress intensity factor K. The extent of the crack tip plasticity was estimated using a detailed constitutive material model and compared with the estimated by Irwin. The displacement fields local to a stationary fatigue crack were mapped using DIC, and inputted into the FE model as boundary conditions so that an integrated FE analysis was carried out. Fatigue pre-cracking was simulated in the FE analysis prior to the full-field analysis of the fatigue crack tip, including stress/strain distributions ahead of the crack tip and the crack opening displacement (COD) under selected loading conditions. Although a distinct “knee” was captured as an indication of crack opening from the compliance curves in both the DIC measurements and the FE analyses, consistent with the existing knowledge on the phenomenon of crack closure, it does not appear to correlate with the crack driving force measured by the J-integral.

Nonlinear fatigue crack propagation in a baffle module of Wendelstein 7‐X under cyclic bending loads

AbstractSimulation of the fatigue crack propagation in a Wendelstein 7‐X baffle module is performed in this study using both a finite element method‐based software and the UniGrow nonlinear model for small‐scale yielding (SSY) conditions. Some experimental fatigue tests of several cracked baffle modules have been performed through a servo‐hydraulic machine. One of these experimental tests has been considered to simulate fatigue crack propagation in the baffle module. Before starting the experimental test, a first crack partly contained in the welding seam and partly in the steel pipe is found. Subsequently, owing to the applied load, the crack propagated both into the welding seam and into the steel pipe until the plastic zone in the near field attains SSY conditions. Finally, owing to the increase in the extension of the plastic zone, SSY conditions are not more valid, and the breakage of the steel pipe is produced by plastic collapse.