Griffith Theory Research Articles

An energy-equilibrium model for complex stress effect on fatigue crack initiation

Based on Tanaka and Mura’s fatigue model and Griffith theory for fracture, an energy-equilibrium model was proposed to explain the complex stress effect on fatigue behavior. When the summation of the elastic strain energy release and the stored strain energy of accumulated dislocations reach the surface energy of a crack, the fatigue crack will initiate in materials. According to this model, for multiaxial stress condition, the orientation of the crack initiation and the initiation life can be deduced from the energy equilibrium equation. For the uniaxial fatigue loading with mean stress, the relation between the maximum stress or the minimum stress and the stress amplitude is in agreement with an ellipse equation on the constant life diagram. If the ratio of the mean stress to stress amplitude is less than a critical value −0.17, and the stress amplitude keeps constant, the fatigue crack initiation life will decrease with the increase of the compress mean stress. In this model, the mean stress does not cause damage accumulation with the fatigue cycles in crack initiation. For this reason, the loading sequence of different load levels would induce the cumulative damage to deviate from the Palmgren-Miner cumulative damage rule. The procedure of estimating the damage under random loading is also discussed.

Prediction of Cutting Forces in Mill-Grinding SiCp/Al Composites

In this study, an analytical cutting force model has been presented for mill-grinding SiCp/Al composites. In this model, except the chip formation force and frictional force components, debonding and particle fracture force components are considered in modeling the total force. The energy theory of particle fracture and Griffith theory of fracture were applied in developing the debonding and particle fracture force components during mill-grinding SiCp/Al composites. Results of verification experiments for the developed cutting force model reveal that the mean forecast error of this developed cutting force model is less than 8% excluding few experimental values which deviate far from the corresponding theoretical model values, and has acceptable agreement with the experimental values under different machining conditions.

60 Years of Hall-Petch: Past to Present Nano-Scale Connections

Pioneering research results reported in the early 1950’s by E. O. Hall and N. J. Petch on iron and steel materials have led to an expanded description of the grain size dependence of the complete stress­strain behavior of a wider range of materials and including assessments of other mechanical properties such as the ductile to brittle transition behavior and the hardness of materials, particularly, of nanocrystalline materials. The dislocation pile-up model that was presented originally for the inverse square root of grain diameter dependence of material strength has endured. Most recently, the pile-up model description has been more definitely associated with the Griffith theory of achieving a critical stress concentration at the tip of a crack; and, the Hall-Petch analysis has been connected to the macro-scale description of the fracture mechanics stress intensity parameter. These topics and other “60 years of Hall-Petch” type researches are tracked over time in the present report while giving special emphasis to current order-of-magnitude strength improvements that are reported for metals with nanopolycrystalline grain diameters.

Evaluation of fracture toughness of fine-grained isotropic graphites for HTGR

In order to investigate the effects of the experimental methodology and the notch angle on the fracture toughness of the fine-grained isotropic nuclear graphites IG-110 and IG-430, the three-point-bending test, which has been recently proposed as the methodology to evaluate the fracture toughness of graphite for high temperature gas-cooled reactors (HTGRs), was performed using two types of the specimens with different notch angles. The results obtained in this study could be summarized as follows: (1) the values of the fracture toughness of IG-110 and IG-430 measured in this study were 0.890MPam1/2 and 1.031MPam1/2, respectively. It was also found that the value of the fracture toughness of IG-110 was nearly equal to or smaller than the values obtained by the other method reported previously. (2) The values of the fracture toughness of the fine-grained isotropic graphites were not affected between the notch angles introduced by the incisive razor blade. (3) The ratio of the tensile strengths of IG-110 and IG-430 was estimated from Griffith Theory using the experimental data obtained in this study. The estimated strength ratio was in good agreement with the strength ratio obtained from the supplier's data.

Fracture failure of asphalt binder in mixed mode (Modes I and II) by using phase-field model

In the pavement industry, asphalt binder fracture failure is one serious problem that may be caused by excessive loading. In the state-of-the-art research of asphalt cracking, pure single mode cracking, especially Mode I has been studied by many researchers. However, there is a lack of theoretical and experimental research on the mixed-mode cracking, which is more reasonable and realistic. In this paper, a new approach, namely, the phase-field method (PFM) is presented for modelling the mixed mode (Modes I and II) cracking failure in the asphalt binder. The traditional way is to use the Griffith's theory for analysing the initiation and propagation of cracks. Our PFM model describes the whole cracking system using a phase-field variable which assumes a negative one in the void region and a positive one in the solid region. Considering the fact that asphalt is brittle when the temperature is below the glass transition temperature, the fracture toughness is considered as the material property and modelled as the surface energy stored in the diffuse interface between the intact solid and the crack void. The non-conserved Allen–Cahn equation is adopted to evolve the phase-field variable to account for the growth of cracks. The energy-based formulation of the PFM handles the competition between the growth of the surface energy and the release of elastic energy in a natural way: the crack propagation is a result of the energy minimisation in the direction of the steepest descent. Both the linear elasticity and phase-field equation are solved in a unified finite element framework, which is implemented in the commercial software COMSOL. The mixed mode cracking experiment is then performed for validation. It is discovered that PFM can give a good result in predicting the onset of crack propagation.

Three-dimensional brittle fracture: configurational-force-driven crack propagation

This paper presents a computational framework for quasi-static brittle fracture in three-dimensional solids. The paper sets out the theoretical basis for determining the initiation and direction of propagating cracks based on the concept of configurational mechanics, consistent with Griffith's theory. Resolution of the propagating crack by the FEM is achieved by restricting cracks to element faces and adapting the mesh to align it with the predicted crack direction. A local mesh improvement procedure is developed to maximise mesh quality in order to improve both accuracy and solution robustness and to remove the influence of the initial mesh on the direction of propagating cracks. An arc-length control technique is derived to enable the dissipative load path to be traced. A hierarchical hp-refinement strategy is implemented in order to improve both the approximation of displacements and crack geometry. The performance of this modelling approach is demonstrated on two numerical examples that qualitatively illustrate its ability to predict complex crack paths. All problems are three-dimensional, including a torsion problem that results in the accurate prediction of a doubly-curved crack. Copyright © 2013 John Wiley & Sons, Ltd.

A study of a micro-indentation technique for estimating the fracture toughness of Al6061-T6

An indentation test is a non-destructive and easy to implement test studied in this paper to estimate the fracture toughness of ductile materials, specifically the Al6061-T6 aluminium alloy. The Berkovich indenter is adopted to generate a crack-like indent; this established technique is based on the principal concept of continuum damage mechanics (CDM) and the model of indentation energy to fracture. It is found that the relation between the effective elastic modulus and the indentation plastic depth hp during the indentation process can be exponentially fitted. Thus the critical plastic indentation depth hp* is evaluated based on the elastic modulus degradation technique and the effective elastic modulus behaviour. Consequently, the critical indentation energy per unit area γ* is determined by using the power law behaviour between the applied load and the plastic indentation depth hp. Considering the stress intensity approach and the Griffith theory, this energy per unit area value is used to calculate the fracture toughness KIC of the ductile material. The fracture toughness of Al6061-T6 is evaluated by indentation tests. Traditional compact tension (CT) tests are performed to attest the accuracy of the indentation energy approach and a numerical analysis is adopted to investigate the stress state in the indentation test. The fracture toughness obtained by the indentation test stands in good agreement with the experimental CT tests.

High toughness in the intercritically reheated coarse-grained (ICRCG) heat-affected zone (HAZ) of low carbon microalloyed steel

Motivated by the small lattice mismatch between ferrite and vanadium nitride (VN), we describe here the welding thermal cycle simulation that provides high toughness in the ICRCG HAZ of low carbon V–N steel. This unique behavior is attributed to the formation of ultra-fine grained ferrite along prior austenite grain boundaries generated by the first pass welding thermal cycle with high misorientation boundaries, where V(C, N) precipitates provide potential nucleation sites for ferrite, leading to extraordinary refinement of martensite/austenite (M/A) constituent. Nitrogen stimulates the precipitation behavior of V(C, N). The nucleation of high density of V(C, N) precipitates consumes carbon-content in the austenite, leading to decrease in the carbon-content in the M/A constituent, with consequent decrease in hardness. The increase in toughness is explained in terms of Griffith's crack propagation theory.

The Zonal Disintegration Law within Coal in Front of Working Face in Deep Coal Mine

The mining coal in the front of long wall face consists of elastic zone, plastic zone and fracture zone. Based on the research of zonaldisintegration of surrounding rock mass in deep tunnel, the zonal disintegration phenomenon of deep coal mine was analyzed by using the theoretical analysis method. On the basis of the Griffith theory, the internal and external boundary formulas of fracture zone and the mechanical criteria of zonal disintegration were obtained. The research shows that the essence of zonal disintegration is the tensile failure and brittle tension crack failure under the abutment pressure, the main influence factors of the width variation of fracture zone are original rock stress and un-axial tensile strength of coal.

Atomic Models of Strong Solids Interfaces Viewed as Composite Structures

This paper looks back through the 1960s to the invention of carbon fibres and the theories of Strong Solids. In particular it focuses on the fracture mechanics paradox of strong composites containing weak interfaces. From Griffith theory, it is clear that three parameters must be considered in producing a high strength composite:- minimising defects; maximising the elastic modulus; and raising the fracture energy along the crack path. The interface then introduces two further factors:- elastic modulus mismatch causing crack stopping; and debonding along a brittle interface due to low interface fracture energy. Consequently, an understanding of the fracture energy of a composite interface is needed. Using an interface model based on atomic interaction forces, it is shown that a single layer of contaminant atoms between the matrix and the reinforcement can reduce the interface fracture energy by an order of magnitude, giving a large delamination effect. The paper also looks to a future in which cars will be made largely from composite materials. Radical improvements in automobile design are necessary because the number of cars worldwide is predicted to double. This paper predicts gains in fuel economy by suggesting a new theory of automobile fuel consumption using an adaptation of Coulomb’s friction law. It is demonstrated both by experiment and by theoretical argument that the energy dissipated in standard vehicle tests depends only on weight. Consequently, moving from metal to fibre construction can give a factor 2 improved fuel economy performance, roughly the same as moving from a petrol combustion drive to hydrogen fuel cell propulsion. Using both options together can give a factor 4 improvement, as demonstrated by testing a composite car using the ECE15 protocol.

The Reliability of Micro-Spring under Dynamic Circumstances

This article mainly focuses on the reliability of micro-spring, which as a typical component of MEMS device, applies electrodeposited nickel material and is fabricated by LIGA process, being used under the circumstances like transporting and restoring which may induce shock and crash. Griffith theory of brittle fracture and Abaqus FEA simulation software are applied for analyzing the probable failure modes of the micro-spring as overloaded. And the principle of elastic deformation, energy-time and displacement-time variation curves are given by three groups of simulation experiments. It proves to be one novel method to evaluate the reliability and accelerate exposing the failure mode of these MEMS components under dynamic circumstances in short period. After comparison, this method is proved reliable for analyzing and providing evidence for enhancing the reliability of the micro-spring, also improving the performance in micron size.

Star-Shaped Crack Pattern of Broken Windows

Broken thin brittle plates like windows and windshields are ubiquitous in our environment. When impacted locally, they typically present a pattern of cracks extending radially outward from the impact point. We study the variation of the pattern of cracks by performing controlled transverse impacts on brittle plates over a broad range of impact speed, plate thickness, and material properties, and we establish from experiments a global scaling law for the number of radial cracks incorporating all these parameters. A model based on Griffith's theory of fracture combining bending elastic energy and fracture energy accounts for our observations. These findings indicate how the postmortem shape of broken samples are related to material properties and impact parameters, a procedure relevant to forensic science, archaeology, or astrophysics.

Numerical simulation of particle breakage in dry impact pulverizer

A novel method to simultaneously simulate particle motion and its breakage in a dry impact pulverizer was developed. The motion of particles in the pulverizer was calculated using a discrete phase model (DPM)‐computational fluid dynamics (CFD) coupling model. When the particle impacts against a vessel wall, impact stress acting on the particle is calculated from Hertz's theory as a function of the impact velocity. At the same time, the particle strength as a function of the particle size is calculated from Griffith's theory. If the impact stress is larger than the particle strength, the particle is broken and replaced with smaller fragments. The size distribution of the fragments is obtained from a breakage function proposed. The motion of the fragments is calculated again by using the DPM‐CFD coupling model. By repeating the above calculations over the whole particles, the grinding phenomenon can be simulated. The calculated results showed good agreement with the experimental one, and validity of the proposed method was confirmed. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3601–3611, 2013

Experimental Data Highlighting the Role of Surface Fracture Energy in Quasi-Static Confined Comminution

Since the pioneering work of Griffith, Linear Elastic Fracture Mechanics has been widely experimentally validated and successfully developed in solid mechanics modeling. However, recent theoretical models applying the energy balance found in Griffith theory specifically for quasi-static confined comminution have until now not been systematically confronted to experiments. In this study, we analyze data of compression tests on crushable sand, where grain breakage has been triggered by flooding the initially dry material at constant stresses. We consider a partition of the dissipation between surface fracture energy and the rearrangement of fragments and grains surrounding crushed particles. Our results show that the role of the surface fracture energy is stressdependent and that its influence becomes less significant at high stresses.

Evaluation of Fracture Parameters by Double-G, Double-K Models and Crack Extension Resistance for High Strength and Ultra High Strength Concrete Beams

This paper presents the advanced analytical methodologies such as Double- G and Double - K models for fracture analysis of concrete specimens made up of high strength concrete (HSC, HSC1) and ultra high strength concrete. Brief details about characterization and experimentation of HSC, HSC1 and UHSC have been provided. Double-G model is based on energy concept and couples the Griffith's brittle fracture theory with the bridging softening property of concrete. The double-K fracture model is based on stress intensity factor approach. Various fracture parameters such as cohesive fracture toughness (4), unstable fracture toughness (K-Ic(c)), unstable fracture toughness (K-Ic(un)) and initiation fracture toughness (K-Ic(ini)) have been evaluated based on linear elastic fracture mechanics and nonlinear fracture mechanics principles. Double-G and double-K method uses the secant compliance at the peak point of measured P-CMOD curves for determining the effective crack length. Bi-linear tension softening model has been employed to account for cohesive stresses ahead of the crack tip. From the studies, it is observed that the fracture parameters obtained by using double - G and double - K models are in good agreement with each other. Crack extension resistance has been estimated by using the fracture parameters obtained through double - K model. It is observed that the values of the crack extension resistance at the critical unstable point are almost equal to the values of the unstable fracture toughness K-Ic(un) of the materials. The computed fracture parameters will be useful for crack growth study, remaining life and residual strength evaluation of concrete structural components.

Analytical model for force prediction when machining metal matrix composite

This paper presents an analytical force model to predict the forces generated during machining of metal matrix composites and investigates the effect of particle size on machining forces. Several aspects of the cutting mechanics such as shear force, ploughing force, and particle fracture force are considered to estimate the generated cutting forces. Chip formation force is obtained using the Johnson–Cook constitutive model. The ploughing force is formulated using the slip-line field theory, while the fracture force is calculated using Griffith Theory. The predicted results are compared to experimentally measured data under different conditions. The results show acceptable agreement between the theoretically predicted and experimentally measured cutting forces.

Editorial for special issue on Fracture mechanics: from Griffith theory, numerical modelling to applications

Editorial for special issue on Fracture mechanics: from Griffith theory, numerical modelling to applications

Force-induced breakdown of flexible polymerized membrane

We consider the fracture of a free-standing two-dimensional (2D) elastic-brittle network to be used as protective coating subject to constant tensile stress applied on its rim. Using a molecular-dynamics simulation with a Langevin thermostat, we investigate the scission and recombination of bonds, and the formation of cracks in the 2D graphenelike hexagonal sheet for different pulling force f and temperature T. We find that bond rupture occurs almost always at the sheet periphery, and the first mean breakage time <τ> of bonds decays with membrane size as <τ> ∝N(-β), where β≈0.50±0.03 and N denotes the number of atoms in the membrane. The probability distribution of bond scission times t is given by a Poisson function W(t)∝t(1/3)exp(-t/<τ>). The mean failure time <τ(r)> necessary to rip off the sheet declines with growing size N as a power law <τ(r)>∝N(-φ(f)). We also find <τ(r)>∝exp(ΔU(0)/k(B)T), where the nucleation barrier for crack formation ΔU(0)∝f(-2), in agreement with Griffith's theory. <τ(r)> displays an Arrhenian dependence of <τ(r)> on temperature T. Our results indicate a rapid increase in crack spreading velocity with growing external tension f.

Reserch of Dislocation Theory in Micro Cutting

This paper analysis Griffith 's Theory and the metal theoretical fracture strength, thus introducing a line defect that existence in actual crystal — dislocation; And in-depth analysis the metal fracture dislocation mechanism in the micro cutting, and the formation and expand of the Fracture cracks , provides the theory basis for micro cutting

Damage localization and rupture with gradient damage models

We propose a method of construction of non homogeneous solutions to the problem of traction ofa bar made of an elastic-damaging material whose softening behavior is regularized by a gradient damage model.We show that, for sufficiently long bars, localization arises on sets whose length is proportional to the materialinternal length and with a profile which is also characteristic of the material. The rupture of the bar occurs at thecenter of the localization zone when the damage reaches there the critical value corresponding to the loss ofrigidity of the material. The dissipated energy during all the damage process up to rupture is a quantity c G whichcan be expressed in terms of the material parameters. Accordingly, c G can be considered as the usual surfaceenergy density appearing in the Griffith theory of brittle fracture. All these theoretical considerations areillustrated by numerical examples.