Fracture Theory Research Articles

An Over-deterministic Method Based on Atomistic Stress Fields: Higher Order Terms of the Williams power expansion

The investigation of the atomistic near crack-tip stress fields under Mixed Mode loading via molecular dynamics method implemented in a flexible classical molecular dynamics method software LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) is presented. The molecular dynamics modelling is applied for evaluation of the coefficients of higher-order terms of the Williams’ expansion -based approximation for the near crack tip fields in isotropic linear elastic materials. Fracture mechanics parameters such as stress intensity factors, higher-order coefficients of the Williams power-series for a copper plate with a central crack under Mode I and Mixed Mode (Mode I+Mode II) loadings are obtained by atomistic simulations. All computational experiments are realized in LAMMPS with the Embedded Atom Method (EAM) potential. The atomistic values of higher-order coefficients of the Williams series expansion are computed by the over-deterministic method and liken to the values obtained from classical continuum fracture mechanics solutions. It is demonstrated that the conventional continuum fracture theory successfully describes fracture quantities at extremely limited singular stress field of only several nanometers.

Multiple Reflective Cracks in Semirigid Base Asphalt Pavement under Traffic Load Using XFEM

This paper built a three‐dimensional layered structure model of semirigid base asphalt pavement with single and double transverse reflective cracks based on the Extended Finite Element Method and fatigue fracture theory. The effects of the number of cracks, crack spacing, and crack length on the stress intensity factors (KI, KII, and Keff) under moving vehicle loads were studied. The fracture life of the asphalt pavement structure was calculated based on the Pairs formula. The results demonstrate that reflective cracks in semirigid asphalt pavement are composite cracks of type I and type II under moving vehicle loads, and shear fracture is the main reason for the failure of the base. The damage to the pavement base will be accelerated with the increase in the number of cracks and the length of the cracks. As the distance between the two reflection fractures is closer, the interaction between the cracks has a superimposed enhancement effect on the crack propagation. Compared with the single nonpenetrating crack model, the fatigue life of the nonpenetrating reflective crack in the double crack pavement structure with a crack spacing of 30 cm is reduced by 46.87%. The research on the propagation mechanism of reflective cracks in this paper provides the essential theoretical and numerical basis for the design, construction, working condition evaluation, and maintenance of pavement structures.

Stress Intensity Factors of Continuum Fracture Mechanics at the Nanoscale

The paper presents the analysis of the stress field in the neighborhood of the crack tip by molecular dynamics method implemented in a classical molecular dynamics code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator). The molecular dynamics simulations are aimed at computing continuum fracture mechanics parameters. Stress intensity factors for a copper plate with a central crack under Mode I and Mixed Mode loadings are evaluated with the help of atomistic modelling. The wide class of the computational experiments in LAMMPS is realized. The atomistic values of stress intensity factors are compared with the values obtained from continuum fracture mechanics considerations. It is shown that the continuum fracture theory successfully describes fracture even at extremely confined singular stress field of only several nanometers.

Phase-field cohesive fracture theory: A unified framework for dissipative systems based on variational inequality of virtual works

This work develops a variational framework with dissipation for deriving governing equations in the phase-field cohesive fracture theory. We show that a phase-field model consistent with cohesive zone theory can be established on only two physical foundations. One of them is the law of energy conservation and the other is a variational inequality of virtual works which serves as a stability condition. The proposed phase-field model can implement a mixed-mode cohesive law of arbitrary form. The mode-dependent fracture toughness and direction-dependent crack driving force in a general cohesive fracture can be taken into account in a unified manner. Since brittle fracture can be viewed as an extreme case of cohesive fracture, the newly proposed method is also applicable to brittle fracture. Three numerical examples are presented to demonstrate the effectiveness of the presented method, including a uniaxial tension test, a mixed-mode fracture of concrete and a compressive failure of rock with flaws. In the first example, the numerical cohesive law is extracted and compared with the target analytical cohesive law. Excellent agreement is observed. In the last two examples, the crack morphology and mechanical response predicted by the model are in excellent agreement with the experimental results.

Mixed-mode tensile/shear fracture of the additively manufactured components under dynamic and static loads

In the practical implementation of the Additively Manufactured (AM) construction in engineering designs, Fused Deposition Modeling (FDM) is recently used to manufacture the inexpensive Acrylonitrile Butadiene Styrene (ABS) parts. The cracking behaviors of such layered components are inspected using the Semi-Circular Bending (SCB) fracture samples. In particular, the effect of anisotropy or layered nature of 3D printed specimens is only considered on the fracture manner among the affecting parameters. The dynamic and static three-point bend tests are conducted from pure mode I to pure mode II conditions. Dynamic experiments are performed using a modified Charpy instrument, and the required energies for the crack initiation are obtained. In the next step, a finite element model based on the Equivalent Material Concept-Generalized Maximum Tangential Stress (EMC-GMTS) failure theory is introduced, and its predictions are compared with the obtained experimental data. Under different loading rates, the crack path trajectories and fracture behavior is surveyed in detail concerning the isotropic counterparts in two micro and macro scales. According to the SEM images and the validity of the brittle fracture theory (with a maximum of 5% error), it is proved that the SCB samples made from ABS are fractured in a brittle manner. The fracture behavior of the FDM 3D-printed structures under dynamic loading conditions is analogous to those homogenous materials, especially where the mode I deformation is dominant. Also, the dynamic and static paths for dominantly pure mode II loading conditions are nearly identical.

Mechanistic understanding of the fracture toughening in chemically strengthened glass—experiments and phase-field fracture modeling

The use of silicate glass in engineering applications has become widespread in recent years due to its superior mechanical and optical properties. However, glass, being a brittle material, poses extremely poor resistance against fracture. Chemical strengthening through ion-exchange has emerged as a promising technique to improve its fracture resistance. In this study, the relationships between chemical strengthening parameters and the resulting fracture toughness of chemically strengthened (CS) glass are investigated using experiments, analytical and numerical simulations. Our results show that optimal fracture resistance for a CS glass requires a proper combination of chemical strengthening parameters, namely the surface compressive stress (σCS), and depth of the compression layer (DOL). The present study establishes the fact that unlike glass, the fracture toughness (KICCS) of a CS glass is not a constant material parameter, instead depends on the initial crack/notch length. The experimental data have also been supported with analytical calculations based on linear elastic fracture mechanics and numerical simulations based on the phase-field fracture theory. These findings provide a novel understanding towards the fracture toughness improvement of CS glass, which should facilitate their widespread engineering applications.

The centenary of Griffith’s theory

The year 2021 marks the 100th anniversary of Griffith's Fracture Theory. Many scholars regard Griffith as the father of the linear elastic fracture mechanics. Here, Nicola Pugno—who received the Griffith Medal and Prize in 2017—reflects on some Griffith's based approaches and quantized-related extensions of the original theory, as he has orally presented this year during his Plenary Opening Lecture at the quadrennial 25^th International Congress of Theoretical and Applied Mechanics.

A plasticity constitutive model for concrete under multiaxial compression

Structural members with confined concrete are becoming increasingly popular in civil engineering applications because of their superior strength and ductility. In these structural members, the concrete is subjected to dilation-induced (passive) lateral compressive stresses from the confining device (e.g., a steel tube). Existing research has led to theoretical models that predict closely the stress–strain behavior of concrete under uniform confinement (e.g., concrete in circular steel tubes under concentric axial compression), but theoretical models with a similar capability have not been achieved for the more common situation of concrete under non-uniform confinement (e.g., concrete in rectangular steel tubes). This paper presents a three-dimensional (3D) plasticity constitutive model that is accurate in predicting the stress–strain behavior of concrete in various scenarios of confinement. In the proposed model, a well-established open strength surface with associated open yield surfaces is combined with a hardening/softening rule compatible with both plastic volumetric compaction and dilation. In addition, a novel potential surface with a triangle-like deviatoric trace is proposed and calibrated with available experimental data of non-uniformly confined concrete. The implementation of the constitutive model in finite element analysis with an enhanced stress-return algorithm suitable for the novel potential surface is explained. While the focus of the present work is on monotonic compression-dominated loading, the model can be combined with fracture and damage theories to depict the behavior of concrete under tension-dominated and cyclic loading conditions. The performance of the proposed model is evaluated by comparing its predictions with a wide range of experimental data covering uniform active, uniform passive, and non-uniform passive confinement conditions, which demonstrates the capability and high accuracy of the proposed model.

Determination of horizontal stress orientation in the areas of the Tomsk region

Introduction. The World Stress Map project proves that horizontal stress orientation determination is a global task essential for the majority of geomechanical calculations. However, there is scant data on stress orientations in the territory of Russia at the moment. The task is therefore highly relevant. Research objective is to determine the orientations of maximum and minimum horizontal stresses by separate areas of the Tomsk region and build a map of horizontal stresses. Method of research. The basis for determining the orientations of horizontal stresses is the theory of drilling-induced fracture and borehole breakouts occurrence. The maximum stress orientation coincides with the drilling-induced fracture orientation, whereas the minimum stress orientation coincides with the borehole breakouts orientation or is perpendicular to the maximum stresses. Results. Research results are compiled in a summary table containing mean orientations of horizontal stresses by areas and a map of horizontal stress orientations. Conclusions. A summary map of maximum horizontal stress strike azimuths has been constructed. The stresses are of similar orientation in every well under consideration, except for a well in the North-Shingin area. The average value of maximum horizontal stress orientation has made up 337° northwest and 157° southeast.

Factors governing segmentation crack characteristics in air plasma sprayed ceramics

Air Plasma Spray (APS) is an industrially popular coating technique implemented in vast domains of applications. Among the inherent defects in an APS coating, segmentation cracks have been recognized to enhance the strain tolerance and durability of thermal barrier coatings. However, segmented coatings also exhibit higher thermal conductivity and are susceptible external debris infiltration. Hence, to optimize the coating properties, it is necessary to have processing control over the incorporation of such segmentation cracks.In this study, segmentation characteristics (thickness to cracking and crack spacing) of multiple Yttria Stabilized Zirconia (YSZ) and Gadolinium Zirconate (GZO) coatings were analyzed. The correlations between the processing parameters and the mechanical properties of the coatings were identified, and the cumulative effect of such variations on the segmentation characteristics were observed. In addition, the experimental observations were reconciled with fracture theories in thin films to provide a comprehensive phenomenological interpretation of segmentation in APS coatings.

An investigation into the role of non-metallic inclusions in cleavage fracture of medium carbon pearlitic steels for high-speed railway wheel

Non-metallic inclusions are often considered to be detrimental to the toughness of steels. Here, we found that, in a cleavage-dominated fracture mode, the toughness, particularly the room temperature fracture toughness of pearlitic wheel steel can be improved by refining oxide inclusions or by increasing sulfur content. The roles of these inclusions in crack initiation and propagation were carefully investigated. Intriguingly, no spherical oxides or elongated sulfides were found to be the direct sources of cracks in the cleavage fracture within both Charpy V-notch and compact tension specimens. Cleavage cracks were found to be originated only from the steel matrix. Based on the theory of cleavage fracture and direct microscopic observations, the mechanisms were proposed for understanding the inclusion-toughness relations: (i) the oxide inclusions could indirectly induce crack initiation by enhancing local stress concentration, therefore promoting cleavage fracture and being detrimental to the toughness; (ii) in addition to enveloping the oxides to reduce the local stress concentration, the sulfide inclusions could also break the continuous propagation of cleavage cracks in the longitudinal direction by sulfide cracking or debonding, which released the stress concentration at the crack tip and enhanced the plastic work of fracture, thereby being beneficial to the toughness improvement.

A PFA methodology to investigate UD composites in fatigue comprising a KTF-based model for matrix damage and stochastic fiber failure prediction

A physics-based, multiscale, progressive fatigue damage analysis methodology for continuous fiber-reinforced polymer composites was further developed and applied to the problem of fatigue damage prediction in open hole tension (OHT) coupons comprised of multidirectional laminates undergoing tension–tension loading. Based off lamina level calibration data, cumulative damage analysis of matrix and fiber constituents under cyclic loading condition was implemented with the objectives to: (i) predict the onset and the progression of subcritical matrix failures as well as the ultimate failure due to fiber fracture, and (ii) observe the effects of damage mode interactions on the progressive failure process. A kinetic theory of fracture (KTF) based durability prediction methodology was utilized to keep track of the matrix damage transpiring in the intra- and inter-laminar layers of the laminate. Tensile strengths were randomly assigned to the fibers using two-parameter Weibull statistics, while the maximum stress theory in tandem with the linear damage accumulation rule was applied to capture the tensile rupture of the reinforcements. Progressive fatigue damage prediction of OHT coupons consisting of the unidirectional composite IM7/977-3 and layup schedules of [0/45/90/−45]2S and [60/0/−60]3S were carried out and benchmarked against available experimental data. It was observed that the KTF-based fatigue model captured the damage state at individual layers, overall loss of mechanical performance, and stochastic nature of the ultimate failure well. The effects of interactions among the damage modes, namely matrix cracking, delamination, and fiber rupture were also investigated and it was recognized that such interactions are a critical feature of the progressive failure process in composite structures.

Experimental and numerical simulations on compound stress intensity factor of semi-elliptical cracks on the exchanger outer walls with inclined angles

Based on the theory of elastic fracture and using direct modeling method, a numerical model was developed for solving the composite stress intensity factor of semi-elliptical cracks on exchanger outer walls with inclined angles. The model exchanger had an outer diameter of φ50 mm, inner diameter of φ44 mm and length of 300 mm. The values of crack plane inclination β (0°, 10°, 20°, 30°, 40° and 45°), crack depth a (0.5 mm, 0.75 mm, 1 mm and 1.25 mm), and crack shape ratio a/b (0.4, 0.5, 0.55, 0.6, 0.7, 0.8, 0.9 and 1.0) were obtained and 3D surface crack model was established. The results were significance for the evaluation of the remaining life of exchanger components.

Cumulative Shear Damage Mechanism to Short Fiber Type C/SiC

A cumulative damage mechanism for short fiber type C/SiC during shear loading–unloading testing was examined and quantified using Iosipescu specimens parallel in the in-plane and through-thickness plane, and by using modified fracture and damage mechanics theory referring to measured damage characteristics (crack length, number and angle). A nonlinear stress–strain relation was found for both specimens. Decrease in the apparent modulus was confirmed with increased peak stress, although permanent strain increased. Inelastic strain of the decomposed tensile direction derived from shear stress was greater than that of the compressive one. Cracks propagated perpendicularly to the tensile direction, intruding on the boundary of the transverse fibers and connecting to other cracks. The theoretical damage mechanics model succeeded to predict the stress–strain relation, proposing that the shear mechanical properties are predictable by measuring the damage characteristics.

Irreversible phase field models for crack growth in industrial applications: thermal stress, viscoelasticity, hydrogen embrittlement

Three new industrial applications of irreversible phase field models for crack growth are presented in this study. The phase field model for irreversible crack growth in an elastic material is derived as a unidirectional gradient flow of the Francfort–Marigo energy with the Ambrosio–Tortorelli regularization, which is known to be consistent with classic Griffith fracture theory. The obtained compact parabolic-elliptic system of PDEs including two regularization parameters for space and time enables us to simulate various kinds of crack behaviors with standard finite element software, without any geometric restriction on the crack path. We extend the irreversible phase field model to thermal cracking in solder and to cracking in a viscoelastic material, keeping the compact forms of the PDEs and the energy consistency. On the other hand, for hydrogen-assisted cracking in metal, we propose a compact phase field model focusing on a kinematic jamming effect of the hydrogen by a weak coupling approach. Several numerical experiments for these three models show not only their reasonableness and usefulness but also flexible extendability of the phase field approach in fracture mechanics.

A simplified model for hydraulic fracturing and its role in seismicity

In this study we propose a simple mathematical model to evaluate the seismic energy (magnitude) generated by the injection of a pressurized liquid in a geothermal area. The aim is to distinguish between induced and triggered seismicity. We compute the induced seismicity by a simple equation that allows to evaluate the energy contribution of the human activity. By using the classical linear elastic fracture theory, we obtain a formula that correlates the strain energy to the injection pressure, to the material parameters of the rock under consideration and to the characteristic seismic length. We compare the results of the model with a series of recorded data available in the literature, showing that our model is capable of defining the maximum magnitude of seismic events caused by human activity. Consequently, the model puts in evidence the “triggered” events with larger magnitude that are produced by the elastic energy accumulated in the rocks at depth by other natural means (e.g. tectonic).

Investigation and comparison of three noval strain rate-dependent fracture criteria

In this study, three new strain rate-dependent fracture criteria were proposed and studied upon DP900 steel. The damage threshold was rewritten as a function related to strain rate on the basis of Lou’s fracture theory. Three different uncoupled strain rate base forms were also given: (i) two polynomial forms and (ii) one power form. The model parameters were identified and validated through uniaxial tensile and double bridge shear tests. The prediction accuracies of each model within four different strain rates were compared by contrasting and analysing the experimental and numerical results. In conclusion, the proposed fracture criteria of power form induce the results with the best agreements with experimental ones. These results prove that the proposed fracture criteria can predict ductile failure under different strain rate loadings.

A Damage Constitutive Model for a Rock under Compression after Freeze-Thaw Cycles Based on the Micromechanics

The freeze-thaw cycles will cause continuous damage to the rock, which is much related to the microcrack length, rock permeability, and frost heaving pressure. However, the failure mechanism of the rock under compression after freeze-thaw cycles is not very clear; therefore, it is studied with the damage theory here. First of all, according to the hydraulic pressure theory, the relationship between the frost heaving pressure and the microcrack propagation length in one single microcrack is established based on the elastoplastic mechanics and fracture theory. Second, by assuming the total strain of the rock under compression is comprised of the initial damage strain, elastic strain, additional damage strain, and plastic damage strain, a constitutive model for a rock based on the deformation and propagation of the microcrack under compression after freeze-thaw cycles is established. Finally, the proposed model is verified with the test result. In all, the proposed model can perfectly reflect the deterioration of the rock mechanical behavior under compression after the freeze-thaw cycles.

Nonsteady fracture of transient networks: The case of vitrimer

We have discovered a peculiar form of fracture that occurs in polymer network formed by covalent adaptable bonds. Due to the dynamic feature of the bonds, fracture of this network is rate dependent, and the crack propagates in a highly nonsteady manner. These phenomena cannot be explained by the existing fracture theories, most of which are based on steady-state assumption. To explain these peculiar characteristics, we first revisit the fundamental difference between the transient network and the covalent network in which we highlighted the transient feature of the cracks. We extend the current fracture criterion for crack initiation to a time-evolution scheme that allows one to track the nonsteady propagation of a crack. Through a combined experimental modeling effort, we show that fracture in transient networks is governed by two parameters: the Weissenberg number [Formula: see text] that defines the history path of crack-driving force and an extension parameter Z that tells how far a crack can grow. We further use our understanding to explain the peculiar experimental observation. To further leverage on this understanding, we show that one can "program" a specimen's crack extension dynamics by tuning the loading history.

Energetics of cracks and defects in soft materials: The role of surface stress

In classical fracture theory, the resistance to fracture is governed by the fracture toughness Gc, which is defined as the energy required to create new crack surfaces. To propagate a crack, the energy released to the crack tip (i.e. the energy release rate J) must overcome the fracture toughness Gc. Previous studies have demonstrated that surface stress can significantly modify the crack opening displacement and the stress field in soft solids and hence alter the energy release rate J, assuming small deformation. In this paper, we present a theoretical work on how surface stress affects fracture of soft solids under large deformation. We provide an expression of a modified J-integral for the energy release rate that accounts for surface stress. The expression is verified using large-deformation finite element analysis (FEA) with surface stress elements. Our results show that when the elastocapillary length, defined as the ratio of surface stress σ to the Young’s modulus of the solid E, is comparable to the relevant material size, surface stress noticeably blunts the crack opening profile and reduces the energy release rate.