Classical Fracture Mechanics Research Articles

Diffusive-length-scale adjustable phase field fracture model for large/small structures

In phase field fracture (PFF) method, the sharp crack is approximated by a phase field crack zone whose size is characterized by a diffusive length scale. Recently, the diffusive length scale is usually regarded as a constant material parameter determined by the fracture toughness, material strength, and young's modulus. As a result, the application of the PFF method poses challenges when dealing with structures whose sizes are much too large or small compared to the constant diffusive length scale. In details, for a large-scale structure, a significant computational burden is inevitable due to the limitation imposed by the constant diffusive crack length scale on the element size (i.e. the element size is generally at least smaller than half of the diffusive crack length scale to achieve the sufficient precision for the phase field process zone). For a small-scale structure, the crack patterns tend to be unclear and unrealistic due to the excessively large diffusive crack zone. To address these limitations, we propose a novel PFF method to make the relation between the diffusive length scale and the material parameters adjustable via modifying the energetic degradation functions. It is found that with the increase of the diffusive crack length scale, a transition from quasi-brittleness to brittleness is observed in the constitutive relationship curves, which coincides with the classical size effect on structural strength. Further, the Bažant's size effect in classical fracture mechanics can be reproduced by the present PFF method through scaling the size of a geometrically similar structure, i.e. a large-scale structure exhibits toughness-dominated fracture while a small-scale structure behavior strength-dominated fracture. The present PFF method efficiently addresses the mismatch of diffusive length scales for various material constituents by examining phase field crack growth in particle-reinforced composite plates. By adjusting the diffusive crack length scale based on structure size, it avoids high computational costs from large structures and unrealistic crack patterns from small ones. Moreover, it outperforms traditional PFF methods using a constant diffusive length scale in handling composite materials.

NOVEL APPROACHES FOR ADDRESSING EPISTEMIC UNCERTAINTY IN FRACTURE MECHANICS

This paper presents a novel optimisation approach for analysing cracked structural systems, consisting of four key steps. The first step (called the Golden Derivative) derives an equation at the discontinuity specific to cracked systems. The cracked system is transformed into an equivalent intact system for analysis in the second step. The third step converts the unique cracked system equation into a finite element equation, revealing epistemic uncertainty that leads to the formulation of Persian curves (PCs). These curves are developed using the following three methods: numerical experimentation, equivalent springs and logical reasoning, with consistent outcomes validating their effectiveness. The uncertainty is linked to the crack effect equation from fracture mechanics in the final step. It is shown by applying mathematical differentiation that this crack effect and classical fracture mechanics inherently involve epistemic uncertainty. A proposed remedy addresses this issue. The derivation of PC is based on pure mathematics and logical reasoning (bridging probabilistic and deterministic methods) making the results applicable across various fields for analysing real-world data.

Theoretical Review of Weight Functions for Rigid Line Inclusions: Implications for Stress Singularities and Crack Propagation

A comprehensive theoretical analysis of weight functions for rigid line inclusions in elastic materials is presented. Classical fracture mechanics approaches were extended to accurately predict stress intensity factors (SIFs) at the tips of these inclusions, which are crucial for understanding material failure. The analysis covered both static and dynamic loading conditions, including transient Mode-III problems. Weight functions for various deformation modes were derived, and the impact of rigid line inclusions on stress singularities and crack propagation was explored. These insights are valuable for the design and analysis of composite structures and materials subjected to dynamic loading.

Comments on standard EN-10371 and proposals for standard follow up

The small punch test technique was developed in the USA in the 1980 s, followed shortly by Japan and introduced to Europe in the 1990 s. CEN organized Workshop 21 to summarize more than ten years of research results and formulate the first technical guidelines CWA 15627. The specification work for European standard started in 2015 and the first European standard EN-10371 was published in 2022. The EN-10371 standardized the specimen size and test apparatus. It also specifies the test procedure and characteristic parameters for small punch test (SP) and small punch creep test (SPC). Nine annexes allows estimation of the values normally obtained using classical standard size uniaxial or fracture mechanics specimens. In order to make the standard perfect and usable, the standard working group proposed a five year “standard follow up” project in 2019. Responding to the proposed project, authors carefully re-examined the specification, and found some shortcomings in the standard. Proposals are put forward to improve the current methodology and extend the standard scope to include the latest development.

Non-oscillatory fracture mode partition for interface crack under mixed-mode loading

The oscillatory crack-tip field in the classical interface fracture mechanics solution often leads to complications in identifying the mode mixity in bimaterial interface fracture problems. In this paper, a non-singular series solution for the continuum problem of a crack on the cohesive-spring interface under mixed-mode loading is presented and compared to the classical oscillatory solution. A Gaussian process regression model was also developed to enable quick evaluation of the phase angle for interface delamination characterization by using mixed-mode bending fracture test.

Finite element implementation of Field Crack Mechanics for brittle and ductile fracture

In this work, a Finite Element (FE) implementation of a Field Crack Mechanics (FCM) model is presented for the first time. The implementation includes general boundary conditions and application to bulk plasticity as well. The current numerical investigation adopts the standard Galerkin finite element formulation to solve the equation of linear momentum balance and finite difference framework for crack evolution. Our preliminary investigation includes the modes-I and II loading conditions, where their respective normal stress fields are compared against their analytical counterparts. The fundamental questions in classical fracture mechanics, e.g. the existence of threshold stress for the crack to move, and various stages associated with the crack propagation are investigated. Additionally, this study explores two distinct strain energy functions based on the crack surface normal and hydrostatic deviatoric energy split. Crack irreversibility is a natural phenomenon in the FCM model, and it does not require satisfying any extra irreversibility constraints. The model shows good agreement against literature and analytical framework. The current FCM implementation also recovers the basic notion of fracture propagation which satisfies both energy and stress criterion. The study also explores the crack motion under the effect of plasticity in ductile materials. The results thus obtained for both brittle and ductile fracture cases are consistent with the predictions of classical fracture mechanics. The current implementation of the FCM effectively demonstrates stable crack propagation, eliminating the requirement for surfing boundary conditions.

A novel fracture criterion for elastic‐brittle cracked body under compression and shear conditions

AbstractWhen the elastic‐brittle cracked body is subjected to compressive‐shear loading, not only does the phenomenon of wing crack initiation and propagation occur, but there is also a notable incidence of secondary crack initiation and propagation due to shear stress. Moreover, the shear fracture toughness (KIIc) predicted by the fracture criterion in the framework of classical fracture mechanics is usually smaller than the tensile fracture toughness (KIc), which is inconsistent with the actual observations. Therefore, this research firstly establishes a mechanical model of an elastic‐brittle cracked body under biaxial compression considering three kinds of crack parameters and materials' mechanical properties. Then, the stress field characteristics at the crack tip are analyzed. Next, a compression‐shear mixed fracture criterion is proposed. Finally, the effects of the lateral pressure coefficient (λ), critical plastic zone size (rc), friction coefficient (f), and deformation parameters of the crack surface on the failure modes at the crack tip are investigated.

An investigation on fracture toughness predictions from mini-sized uniaxial tensile specimens with global and local approaches

This study provides investigations on whether the fracture toughness can be accurately predicted from mini-sized uniaxial tensile specimens with existing global and local approaches. The critical toughness model, based on classical fracture mechanics, and the energy release rate (ERR) model, based on damage mechanics, were used as the two representative global approaches. While the finite element simulated compact tension (CT) specimen implemented with Gurson-Tvergaard-Needleman model was used as the local approach. Specimen size effect was judged through the viewpoints of stress–strain relationship and damage development. It was proved that the specimen size has no significant influence on stress–strain relationship, and has little influence on damage developments when the thickness reaches 1.0 mm. Fracture toughness predictions were conducted on four metals, including two steels (SA508Gr.3Cl.1 and 18MnMoNbR), one aluminum alloy (6061-T651), and one titanium alloy (TC4). For TC4 exhibiting brittle cleavage features on the fracture surface of CT specimens, neither the global nor the local approach can accurately predict its fracture toughness. For 6061-T651 exhibiting both ductile and brittle features on the fracture surface of CT specimens, there exists significant macroscopic crack propagation at the moment of conventional fracture toughness determination. The existence of significant macroscopic crack is inconsistent with the “equivalent crack” assumption in the ERR model, leading to a failure in its fracture toughness prediction. For two steels following the “equivalent crack” assumption, the ERR model and the local approach yield more accurate predictions (with the maximum error around 10 %) than their critical toughness counterpart (with the maximum error around 30 %).

Assessing fatigue crack growth thresholds for a Ti-6Al-4V (STOA) alloy using two experimental methods

Fatigue crack growth rate behavior in the threshold and near-threshold regimes for a Ti-6Al-4V solution-treated and over-aged (STOA) alloy was investigated using two test procedures: i) accelerated load reduction (ALR) utilizing faster load-shed rates than recommended in the ASTM standard E647 load-reduction (LR) test procedure, and ii) compression pre-cracking constant-amplitude (CPCA) or load-increasing (CPLI) and load-reduction (CPLR). Fatigue crack growth tests were conducted on compact, C(T), specimens (B = 6.35 mm) with two different widths (W = 51 and 76 mm) at two stress ratios (R = 0.1 and 0.7). The crack-growth-rate test data were compared to previous test data produced from the same batch of material using the ASTM LR and CPCA test procedures that were tested on C(T) specimens (W = 25, 51 and 76 mm) over a wide range in stress ratios (R = 0.1, 0.4, and 0.7).Although none of the load-reduction test procedures provided a good representation of the threshold value (ΔKth) at low R, the CPCA test procedure was the most promising. The LR, ALR, and CPLR test procedures were influenced by prior loading history, leading to higher ΔKth values and slower crack growth rates in the threshold regime due to a rise in the crack-opening loads. The LR and ALR tests at R = 0.1 and shed rates of −0.08, −0.32, and −0.95 mm−1 yielded ΔKth values that were 15 to 38 % higher than the estimated threshold stress-intensity factor range (ΔK*th)R=0.1 based on the crack-closure model (FASTRAN). The compression pre–cracking test procedures are more accurate alternatives for developing threshold and near-threshold fatigue crack growth rates. Specifically, the CPLR test procedure produced a ΔKth value that was 8–18 % higher than (ΔK*th)R=0.1. Additionally, LR and ALR tests produced different ΔKth values as a function of the specimen width for a given load ratio, which violates classical fracture mechanics principles. In addition, the spread in the ΔK-rate data in the low-rate regime was larger than the spread in the mid-rate regime as a function of R, a term referred to as “fanning”. In contrast, the CPCA test procedure produced consistent crack growth rates over the same range of ΔK values examined, independent of the specimen width and these data did not exhibit “fanning” with R. Further research is necessary to develop standard test procedures capable of providing a more definitive representation of the ΔKth value in the threshold regime. However, FASTRAN can serve as a crucial tool in aligning the actual ΔKth values with those obtained through CP testing, thereby bridging the gap and enhancing the reliability of the results.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Extension of the peak stress method to estimate the fatigue limit of welded joints by means of the cyclic R-curve method

A new simplified and effective method has been formalised to estimate the Constant Amplitude Fatigue Limit (CAFL) of stress-relieved steel welded joints subjected to uniaxial push–pull loading and failing from the weld toe. Starting from the sharp V-notch assumption of the NSIF approach and the cyclic R-curve of the material in the heat affected zone, the proposed method identifies the CAFL as threshold level of the local stress field at the V-notched weld toe in the uncracked configuration. Such threshold stress field assures the crack arrest at the V-notched weld toe, according to the cyclic R-curve analysis. The method has been validated against experimental results and proved effective for a straightforward assessment of the CAFL of welded joints, as the stable crack propagation analysis of classical fracture mechanics approaches can be avoided.

Tensile cracks can shatter classical speed limits.

Brittle materials fail by means of rapid cracks. Classical fracture mechanics describes the motion of tensile cracks that dissipate released elastic energy within a point-like zone at their tips. Within this framework, a "classical" tensile crack cannot exceed the Rayleigh wave speed, [Formula: see text]. Using brittle neo-hookean materials, we experimentally demonstrate the existence of "supershear" tensile cracks that exceed shear wave speeds, [Formula: see text]. Supershear cracks smoothly accelerate beyond [Formula: see text], to speeds that could approach dilatation wave speeds. Supershear dynamics are governed by different principles than those guiding "classical" cracks; this fracture mode is excited at critical (material dependent) applied strains. This nonclassical mode of tensile fracture represents a fundamental shift in our understanding of the fracture process.

Crack tip solution for Mode III cracks in spring interfaces

Considering an infinite linear elastic isotropic solid in antiplane shear, the Mode III crack-tip solution for a semi-infinite crack located in a straight spring interface is systematically studied for the first time. A new analytic expression for this crack-tip solution is given in the form of a double asymptotic series of the main and the so-called associated shadow terms. It is shown that the series of the shadow terms associated with a main term is infinite, and all shadow terms include logarithmic terms. Thus, although the interface tractions are bounded, the linear elastic solution at this crack-tip has a logarithmic stress singularity which is comprehensively analysed. Noteworthy, the character of this stress singularity is very different from the well-known square root singularity at the crack tip in the classical fracture mechanics. A key advantage of the present approach is its simplicity, as only elementary mathematical tools are employed, and also its easy implementation in a computer algebra software. The latter fact is very relevant because the expressions of higher-order shadow terms become increasingly complicated, so their generation by a computer code becomes crucial. The present results allow the implementation of new enriched or singular crack-tip finite elements for such cracks, and the automatic generation of analytic solutions for benchmark problems for testing the finite-element codes using these special elements. Such codes can be applied to efficient numerical modelling of interface cracks, e.g., in adhesively bonded joints with a thin adhesive layer.

Finite element study of V-shaped crack-tip fields in a three-dimensional nonlinear strain-limiting elastic body

We study the crack-tip fields in a three-dimensional (3D) plate with a V-shaped crack employing Rajagopal’s theory of elasticity with the strain-limiting effect. The classical linear elastic fracture mechanics (LEFM) theory bears a well-documented inconsistency, i.e., for a static crack problem under the traction-free boundary condition, the LEFM theory predicts singular crack-tip strains. Such results are inconsistent with the constitutive relation derived using the assumption of uniform bound for the strain values. Instead, Rajagopal’s recent theory of elasticity provides implicit constitutive relations between the linearized strain and Cauchy stress in which one can impose an a-priori upper bound for strains. We formulate the strain-limiting theory comprehensively in this article. The linearized strain is expressed as a nonlinear function of the Cauchy stress. The equation for the equilibrium of the linear momentum under a special type of nonlinear constitutive relation reduces to a second-order quasilinear elliptic boundary value problem (BVP). Using a Picard-type linearization and a continuous Galerkin-type finite element technique, we solve several BVPs with tensile and shear displacement loading. The numerical results for the strains, the stresses, the stress intensity factor (SIF), and strain energy density (SED) are obtained for different Poisson values, strain-limiting parametric values, and the V-shaped angles. The growth of the near-tip strains is far less than that of stresses, and the results for both the SIF and SED are also consistent with the results from LEFM. Our work demonstrates that the framework of Rajagopal’s theory of elasticity can provide a basis for developing physically meaningful and mathematically well-posed BVPs to study evolution of cracks, damage, nucleation, or failure in elastic bodies.

Endowing Griffith’s fracture theory with the ability to describe fatigue cracks

In this paper, a possible extension of Griffith’s fracture theory to describe fatigue-induced crack propagation is proposed. To this end, an energy-based model is presented, taking Griffith’s model as the point of departure and employing the concept of state-dependent fracture toughness. Therein, fatigue degradation is achieved through a suitable history variable endowed with a functional form that is able to consider crucial aspects of fatigue, including the crack-tip singularity, the identification of fatigue-inducing loading conditions, and mean load effects. Simple paradigmatic examples indicate that the model provides a unified description of different fatigue responses and unveils peculiar regimes in the crack propagation process, always preserving the link to classical fracture mechanics. In this context, analytical results establish a clear relation between Griffith’s fracture theory and Paris’ law. The proposed modeling framework paves the way for future developments in modern fracture mechanics, e.g., to derive a novel generation of variational fatigue phase-field fracture models suitably rooted in a Griffith-based theory.

Electrochemically induced fracture in LLZO: How the interplay between flaw density and electrostatic potential affects operability

Fracture and short circuit in the Li7La3Zr2O12 (LLZO) solid electrolyte are two key issues that prevent its adoption in battery cells. In this paper, we utilize phase-field simulations that couple electrochemistry and fracture to evaluate the maximum electric potential that LLZO electrolytes can support as a function of crack density. In the case of a single crack, we find that the applied potential at the onset of crack propagation exhibits inverse square root scaling with respect to crack length, analogous to classical fracture mechanics. We further find that the short-circuit potential scales linearly with crack length. In the realistic case where the solid electrolyte contains multiple cracks, we reveal that failure fits the Weibull model. The failure distributions shift to favor failure at lower overpotentials as areal crack density increases. Furthermore, when flawless interfacial buffers are placed between the applied potential and the bulk of the electrolyte, failure is mitigated. When constant currents are applied, current focuses in near-surface flaws, leading to crack propagation and short circuit. We find that buffered samples sustain larger currents without reaching unstable overpotentials and without failing. Our findings suggest several mitigation strategies for improving the ability of LLZO to support larger currents and improve operability.

Identification of Linear Elastic Fracture Mechanics Parameters Through the Atomistic Simulation Based Analysis

The contribution presents atomistic modeling of the proximate crack tip and notch stress fields through molecular dynamics method. The study is aspired to comparing atomistic stress distributions for monocrystalline fcc copper and aluminum with continuous stress fields obtained on the basis of classical continuum fracture mechanics. Atomistic simulations were implemented for pure tensile loadings and mixed (I+II mixed loadings) mode loadings in Large-scale Atomistic/ Molecular Massively Parallel Simulator (LAMMPS). In atomistic computations specimens with central cracks and single edge notches made of crystalline fcc copper and aluminum under mixed mode loadings are considered. The elastic tensor components of the selected crystalline materials are obtained via molecular dynamics simulations and the circumferential distributions of stress fields in the straight away proximity of the crack and notch tips in aluminum and copper are constructed. The main regularities arising from the comparative analysis of the atomistic and continuum approaches are revealed. The major conclusion that can be reached after atomistic modeling is that the stress apportionments given by classical linear fracture mechanics qualitatively and quantitatively describe the molecular dynamics stress distributions.

Mixed-mode fracture of compacted tailing soils. II: Crack properties from full-field displacement fields

This paper deals with the acquisition of the crack evolution and propagation behaviors of the compacted tailing soils from full-field displacement fields by incorporating two-dimensional optical technique-digital image correlation (2D DIC). Meanwhile, the fracture properties, such as mode I+II fracture toughness were also measured and compared to the results obtained in Part I. Cracking properties, such as strain energy rate, cracking initiation angle, and size of critical distance were investigated from the perspective of full-field displacement field and compared to the classic fracture mechanics. It is more intuitive to recognize the mixed mode fracture properties of the compacted tailing soils under asymmetric loading conditions by using optical techniques in full-field displacement fields. This work largely enriched the database of the fracturing properties of compacted tailings soils.

Mixed-mode fracture of compacted tailing soils. I: Fracture toughness

Mine tailings are man-made soils and leftovers from the mineral processing that share the similar geotechnical properties with natural soils. In areas with abundant mineral reserves and developed mining industries, tailings could be utilized as an alternative of natural soils for subgrade materials of pavement and adobes for building and construction constructions. The tailing soils are also the based materials for tailing dams. The compacted tailing soils might thus be subject to combined opening or shearing deformations that were unexplored in the current academic society. In this paper, a test configuration called asymmetric semi-circular bending (ASCB) was adopted to investigate the mixed I/II mode fracture properties of the compacted tailing soils. First, the load–displacement relationships of the SCB specimens for different notch depths and span ratios were presented. The mode I and II fracture toughness components were then calculated by referring to the theoretical solutions derived from classical fracture mechanics. The fracture envelope from mode I and II fracture toughness components were presented. Meanwhile, the mixity parameter was introduced to evaluate the mode mixity of the ASCB specimens. Finally, the crack initiation angles of the SCB specimens were measured through experimental tests and compared to the theoretical results obtained from generalized maximum tangential stress criterion (GMTS). Results show that the mode I fracture toughness component of the compacted tailing soils increased with the span ratio, while the mode II fracture toughness component increased with the decrease of span ratio. The mode mixity and crack initiation angles of the SCB specimens under mixed mode loading conditions increased with the decrease of span ratio.

Issue Information

International Journal of Applied Glass ScienceVolume 14, Issue 1 p. 1-2 ISSUE INFORMATIONFree Access Issue Information First published: 02 December 2022 https://doi.org/10.1111/ijag.16586AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Graphical Abstract Cover Photograph: (a) Snapshot of molecular model of a 0.1Na2O-0.9SiO2 sodium silicate glass structure with a loaded slit crack simulated with the ReaxFF forcefield. Bonds along a half plane (depicted with a dashed line) were severed to create the initial slit crack. The displacements of atoms in an outer annulus were prescribed by an external boundary condition to generate far-field loading conditions. Atom colors: Si (yellow), O (red), Na (blue). (b) The same sodium silicate molecule model demonstrating the radial displacement field generated by the far field loading conditions. As the distance from the crack tip increases so does the total displacement, consistent the classical linear elastic fracture mechanics continuum solution for a semi-infinite slit crack under mode I loading. Volume14, Issue1Special Issue: Glasses for tomorrowJanuary 2023Pages 1-2 RelatedInformation