Linear Elastic Fracture Mechanics Research Articles

Stress Field near Crack Tips in Pressurized Thick-Walled Cylinders with Single and Multiple Cracks in Linear Elastic Fracture Mechanics: Analytical and Numerical Analysis of Stress Field and Crack Behavior

<div>Arrays of radial cracks often appear at the bore of pressurized cylinders, posing potential safety risks and leading to possible structural failures. This article presents an analytical approach to evaluate the stress field arising from single or multiple uniform radial cracks in thick-walled pressurized cylinders within the context of linear elastic fracture mechanics (LEFM) under mode-I loading. This formulation is based on the fundamental equations of elasticity and approximations of stress intensity factors (SIF) reported in the literature. Hence, the SIF were revisited and their range of validity was highlighted. The study considers two types of internal pressure loading: one applied only to the cylinder’s inner surface with no pressure on the crack faces and another applied to both the inner surface and the crack faces. The influence of the number and length of cracks relative to cylinder thickness on the stress field is analyzed. A finite element model of the pressurized vessel is constructed to verify the proposed analytical solution, showing good agreement with numerical results for a limited number of cracks.</div>

A proposal for the combined analysis of bone quantity and quality of human cortical bone by quasi-brittle fracture mechanics

Quasi-brittle fracture mechanics is used to evaluate fracture of human cortical bone in aging. The approach is demonstrated using cortical bone bars extracted from one 92-year-old human male cadaver. In-situ fracture mechanics experiments in a 3D X-ray microscope are conducted. The evolution of the fracture process zone is documented. Fully developed fracture process zone lengths at peak load are found to span about three osteon diameters. Crack deflection and arrest at cement lines is a key process to build extrinsic toughness. Strength and toughness are found as size-dependent, not only for laboratory-scale experimental specimens but also for the whole femur. A scaling law for the length fracture process zone is used. Then, size-independent, tissue fracture properties are calculated. Linear elastic fracture mechanics applied to laboratory beam specimens underestimates the tissue toughness by 60%. Tissue fracture properties are used to predict the load capacity of the femur in bending within the range of documented data. The quasi-brittle fracture mechanics approach allows for the assessment of the combined effect of bone quantity and bone quality on fracture risk. However, further work is needed considering a larger range of subjects and in the model validation at the organ length scale.

Sprain energy consequences for damage localization and fracture mechanics

The 2023 smooth Lagrangian Crack-Band Model (slCBM), inspired by the 2020 invention of the gap test, prevented spurious damage localization during fracture growth by introducing the second gradient of the displacement field vector, named the "sprain," as the localization limiter. The key idea was that, in the finite element implementation, the displacement vector and its gradient should be treated as independent fields with the lowest ([Formula: see text]) continuity, constrained by a second-order Lagrange multiplier tensor. Coupled with a realistic constitutive law for triaxial softening damage, such as microplane model M7, the known limitations of the classical Crack Band Model were eliminated. Here, we show that the slCBM closely reproduces the size effect revealed by the gap test at various crack-parallel stresses. To describe it, we present an approximate corrective formula, although a strong loading-path dependence limits its applicability. Except for the rare case of zero crack-parallel stresses, the fracture predictions of the line crack models (linear elastic fracture mechanics, phase-field, extended finite element method (XFEM), cohesive crack models) can be as much as 100% in error. We argue that the localization limiter concept must be extended by including the resistance to material rotation gradients. We also show that, without this resistance, the existing strain-gradient damage theories may predict a wrong fracture pattern and have, for Mode II and III fractures, a load capacity error as much as 55%. Finally, we argue that the crack-parallel stress effect must occur in all materials, ranging from concrete to atomistically sharp cracks in crystals.

Crack-tip cleavage/dislocation emission competition behaviors/mechanisms in magnesium: ALEFM prediction and atomic simulation

Structural properties and reliability of materials can be improved by increasing fracture toughness. At the atomic scale, the fracture is a material separation process, and the fracture toughness of materials is associated with the atomic-scale crack-tip behaviors/mechanisms. The crack-tip behaviors are relevant to the energy state of atoms in the system. Atomic thermal oscillation increases with increasing temperature, which may affect/alter the crack tip behaviors. This work is the first to investigate the temperature-dependent crack-tip cleavage/dislocation emitting competition in magnesium (Mg) using anisotropic linear elastic fracture mechanics theory, Density Functional Theory (DFT), and atomic simulation. Crack-tip behaviors are examined using a specially designed ‘K-field’ loads model. DFT calculations show that a single crystal system with lower entropy and higher Gibbs free energy implies stronger interatomic bonding that favors a higher KIc. Changes in the stress distribution initiate a brittle-ductile transition in crack-tip behavior. The ductile crack tip can be blunted by continuous crack-tip dislocations nucleation/slip, and the evolution of the ductile crack-tip geometry from sharp to semicircular structure significantly decreases the stress concentration at the crack tip. A new criterion of the crack-tip force vector is established, which reasonably explains the geometrical evolution of ductile crack tip where the angle θ between the crack plane and the slip plane is 0∘<θ<90∘ and θ=90∘. This work expands the atomic-scale brittle/ductile crack-tip behaviors/mechanisms of Mg, which provides a reference for crack-tip behavior analysis in engineering research.

Fracture and size effect in mechanical metamaterials

We resort to variational methods to evaluate the asymptotic behavior of fine metamaterials as a function of cell size. To zeroth order, the metamaterial behaves as a micropolar continuum with both displacement and rotation degrees of freedom, but exhibits linear-elastic fracture mechanics scaling and therefore no size effect. To higher order, the overall energetics of the metastructure can be characterized explicitly in terms of the solution of the zeroth-order continuum problem by the method of Γ-expansion. We present explicit expressions of the second-order correction for octet frames. As an application, we evaluate the compliance of double-cantilever octet specimens to second order and use the result to elucidate the dependence of the apparent toughness of the specimen on cell size. The analysis predicts the discreteness of the metamaterial lattice to effectively shield the crack-tip, a mechanism that we term lattice shielding. The theory specifically predicts anti-shielding, i. e., coarser is weaker, in agreement with recent experimental observations.

3D characterization of kinematic fields and poroelastic swelling near the tip of a propagating crack in a hydrogel

AbstractIn fracture mechanics, polyacrylamide hydrogel has been widely used as a model material in experiments due to its optical transparency, the brittle nature of its failure, and low Rayleigh wave velocity. Indeed, linear elastic fracture mechanics has been used successfully to model the fracture of polyacrylamide hydrogels. However, in soft materials such as hydrogels, the crack opening can be extremely large, leading to substantial geometric and material nonlinearity at the crack tip. Furthermore, poroelasticity may also modify the local mechanical state within the polymer network due to solvent migration. Direct characterization of the kinematic fields and the poroelastic response at the crack tip is lacking. Here we use a hybrid digital image correlation—particle tracking technique to retrieve high-resolution 3D particle trajectories near the tip of a slowly propagating crack, and measure the near-tip 3D kinematic fields in-situ. With this method, we charactherize the displacement fields, rotation fields, stretch fields, strain fields, and swelling fields. These measurements confirm the complex multi-axial stretching near the crack tip and the substantial geometric nonlinearity, particularly in the wake of the crack, where material rotation exceeds $$30^{\circ }$$ 30 ∘ . Comparison between the measured fields and the corresponding prediction from linear elastic fracture mechanics highlights an increasing disagreement in the direct vicinity of the crack tip, particularly for displacement component $$u_x$$ u x and the through-thickness strain component $$\varepsilon _{zz}$$ ε zz . Significant swelling occurs due to solvent migration, with a strong correlation to the local stretch.

Evaluation of impact toughness of glass-reinforced composite materials immersed in water

The mechanical and physical properties of glass fiber composite materials can be modified when immersed in water. This change in properties can result from internal instability of the material, interaction with the environment, mechanical stresses, or a combination of these effects. In this study fiber glass reinforced polyester resin plates have been manufactured by the contact molding method with 20% of reinforcement percentage. Prismatic specimens have been cut then notched to evaluate the impact toughness by applying the Charpy impact test. The Williams method based on the principle of linear elastic fracture mechanics was used to deduce the impact toughness of composite. These specimens have been immersed in water for 30 and 90 days to study absorption effect and diffusion on the impact behavior of this material. From the results, it is found that there is a direct correlation between water penetration in the glass-polyester composite and its impact toughness.

Criterion for residual strength and brittle-ductile transition of brittle rock under triaxial stress conditions

The critical condition with respect to the brittle-ductile transition (BDT) is of great significance in understanding the distinction between brittle and ductile behavior for rocks. In this paper, we develop the two criteria for the residual strength and BDT point of brittle rock based on linear elastic fracture mechanics. The effect of the microcrack friction and slip friction of brittle rock on the residual strength and BDT point is analyzed. Two models are derived to predict the critical confining stress and critical compressive stress at the BDT point for brittle rock, and a new brittleness index is proposed. The reliability of models is validated by using a great number of test data from previous publications. The results show that the stress of the BDT point increases with increasing microscopic friction on the microcrack, and decreases with increasing macroscopic sliding friction. The critical confining stress and critical compressive stress at the BDT point are predicted, which is qualitatively consistent with laboratory tests. The results also show that the two criterion have the capability to predict the residual strength and BDT point, which has provided useful insights into the mechanics of brittle-ductile transition in brittle rocks.

The effect of stress barriers on unconventional-singularity-driven frictional rupture

Whether or not energy dissipation is localized in the vicinity of the rupture tip, and whether any distal energy dissipation far from the crack tip has a significant influence on rupture dynamics are key questions in the description of frictional ruptures, in particular regarding the application of Linear Elastic Fracture Mechanics (LEFM) to earthquakes. These questions are investigated experimentally using a 40-cm-long experimental frictional interface. Three independent pistons apply a normal load with a fourth piston applying a shear load, enabling the application of a heterogeneous stress state and stress barriers. After loading the frictional interface to a near-critical state, subsequent unloading of one normal-load piston leads to dynamic ruptures which propagate into the heterogeneous stress fields. The ruptures in these experiments are found to be driven by unconventional singularities, characterised by an ever-increasing breakdown work with slip, and as a result do not conform to the assumptions of LEFM. As these experimental stress barriers inhibit slip, they therefore also reduce the breakdown work occurring outside of the cohesive zone. It is shown that this distal weakening, far from the crack tip, must be considered for the accurate prediction of rupture arrest length. These experiments are performed in the context of a proposed stimulation technique for Enhanced Geothermal Systems (EGSs). It has previously been suggested, through theoretical arguments, that stress barriers could be induced through the manipulation of pore pressure such that there is reduced seismic hazard during the shear stimulation of EGSs. This stimulation technique, known as preconditioning, is demonstrated here to reduce the mechanical energy flux to the crack tip, G, while also increasing the fracture energy, Gc. Preconditioning is shown to be capable of arresting seismic rupture and reducing co-seismic slip, slip velocity, and seismic moment at preconditioning stresses which are reasonably achievable in the field. Due to the fully-coupled nature of seismic rupture and fault slip, preconditioning also reduces distal weakening and its contribution to the propagation of induced seismic ruptures. In a similar vein, heterogeneous pore pressure fields associated with some seismic swarms can be used to explain changes in stress drop within the swarm without recourse to material or total-stress heterogeneity.

The length of fracture process zone deciphers variations of rock tensile strength

Tensile strength is one of the most critical design factors in many rock engineering projects. However, despite many available testing techniques, an accurate estimation of the true tensile strength of quasi-brittle rock-like materials is yet a controversial problem since it can vary by the shape and size of a test specimen, the adopted test method, and applied loading conditions. Different studies have tried to address this issue by providing (mainly empirical) laws for determining variations of rock tensile strength as a function of a particular test parameter such as specimen size. In this study, however, a new general approach is presented that can decipher the tensile strength variations of rock under various testing conditions. Using coupled Finite Fracture Mechanics (FFM), it is first proved that the length of the Fracture Process Zone (FPZ) can be determined with accuracy and ease using the energy criterion of coupled FFM. Then, the length of FPZ is used in the stress criterion of coupled FFM to determine rock tensile strength. The failure stress of a material is then proved to be mainly a function of the FPZ length following a power law originated from the Linear Elastic Fracture Mechanics (LEFM). The results assist in deciphering variations of rock tensile strength related to the sample size and test method.

An analysis of the interaction between surface and basal crevasses in ice shelves

Abstract. The prescription of a simple and robust parameterization for calving is one of the most significant open problems in ice sheet modelling. One common approach to the modelling of crevasse propagation in calving in ice shelves is to view crevasse growth as an example of linear elastic fracture mechanics. Prior work has, however, focused on highly idealized crack geometries, with a single fracture incised into a parallel-sided slab of ice. In this paper, we study how fractures growing from opposite sides of such an ice slab interact with each other, focusing on different simple crack arrangements: we consider either perfectly aligned cracks or periodic arrays of laterally offset cracks. We visualize the dynamics of crack growth using simple tools from dynamical systems theory and find that aligned cracks tend to impede each other's growth due to the torques generated by normal stresses on the crack faces, while periodically offset cracks facilitate simultaneous growth of bottom and top cracks. For periodic cracks, the presence of multiple cracks on one side of the ice slab, however, also generates torques that slow crack growth, with widely spaced cracks favouring calving at lower extensional stresses than closely spaced cracks.

Microcrack behavior in bone: Stress field analysis at osteon cement line tips.

Bone microstructure governs microcrack propagation complexity. Current research, relying on linear elastic fracture mechanics, inadequately considers authentic multi-level structures, like cement lines and osteons, impacting stress intensity at cracks. This study, by constructing models encompassing single or multiple osteons, delves into the influence of factors like crack length, osteon radius, and modulus ratio on the stress intensity factor at the crack tip. Employing a fracture mechanics phase-field approach to simulate crack propagation paths, it particularly explores the role of cement lines as weak interfaces in crack extension. The aim is to comprehensively and systematically elucidate the critical factors of bone microstructure in the context of crack propagation.

A Relationship between Fracture Toughness Kc and Energy Release Rate Gc According to Fracture Morphology Analysis

This study investigated the relationship between fracture toughness (Kc) and energy release rate (Gc) through fracture morphology analysis, emphasizing the critical role of fractal dimensions in accurately characterizing fracture surfaces. Traditional linear elastic fracture mechanics (LEFM) models relate Gc to Kc by combining energy principles with the nominal area of the fracture surface. However, real materials often exhibit plasticity, and their fracture surfaces are not regular planes. To address these issues, this research applied fractal theory and introduced the concept of ubiquitiform surface area to refine the calculation of fracture surfaces, leading to more accurate estimates of Gc and Kc. The method was validated through standard compact tensile specimen tests on a nickel-based superalloy at 550 °C. Additionally, the analysis of fractal dimension differences and dispersion in various fracture regions provides a novel perspective for evaluating the fracture toughness of materials.

The analysis of scaled cracked components

A recent arrival in the open literature is the scaling theory finite similitude bringing into existence a countably infinite number of new similitude rules. The application of the theory to fracture mechanics has revealed that the first-order similitude rule is applicable in the sense that information gleaned from two fractured scaled components can be combined to capture the response of a third. This paper examines an unexplored aspect of the finite similitude theory, in that, it is shown how it can be re-configured for the purposes of scaling analysis. This is achieved by retaining the first derivative (with respect to scale) in the first-order rule and on involving an additional transport equation. The approach assumes the first-order rule to be exact, which is confirmed here, but in addition, has the advantage of automatically accounting for any defect-size related size effects that are present. It is a top-down scheme with analysis performed under the constraint of an imposed rule. To instigate the methodology, fracture mechanics for the first time is defined on a new scaling space, where all physical quantities are deemed dependent on a single dimensional scaling parameter β. This approach is demonstrated to facilitate the imposition of similitude rules and has led to the discovery of new relationships that connect fatigue responses at two sizes. Although analysis here is limited to linear elastic fracture mechanics (LEFM), it is demonstrated through proof, analytical, and experimentally validated numerical models, that the approach provides an exact scaling theory for fatigue.

Slip and stress in block-in-matrix shear zones: 2. numerical modelling of a serpentine-filled dilational jog

We use 2-D numerical models to explore slip and stress dynamics in a fault-bound dilational jog, informed by a field example from the Dun Mountain Ophiolite in New Zealand. The jog is hosted in a metre-scale phacoid of massive serpentinite embedded in a “block-in-matrix” creeping serpentinite shear zone. The models show how periodic exceedance of the tensile strength of the contact between the sealed jog and host rock leads to episodic opening and deposition of a new crack-seal band, with a thickness limited by the release of stress around the tips of the stepover-bounding faults. Jog stress release in the model is lower than that predicted from linear elastic fracture mechanics because additional crack opening can occur slowly due to post-failure creep on the bounding faults. For 10 km overburden and constant high fluid pressure, we predict event stress release of ca. 16 MPa and total crack opening of ∼22 μm on either side of the jog, consistent with crack-seal band widths in the field example. Because our models show that total crack width reflects both initial cracking and subsequent creep, we suggest caution when using crack-seal band widths to directly infer stress release in similar shear zone settings.

Micro/Nanomechanical Characterization of ScAlMgO4 Single Crystal by Instrumented Indentation and Scratch Methods.

ScAlMgO4 (SCAM), which can be used as an epitaxial substrate material of GaN in power devices, faces the challenge of achieving a high-quality surface by ultra-precision polishing due to its brittle and easily cleaved characteristics, which are closely associated with its mechanical properties. The micromechanical properties of SCAM single crystals were evaluated by nanoindentation and microscratch tests using different indenters. The elastic modulus EIT and the indentation hardness HIT of SCAM obtained by nanoindentation were 226 GPa and 12.1 GPa, respectively. Leaf-shaped chips and the associated step-like planes of SCAM can be found in the severely damaged regime during scratching by Berkovich and Vickers indenters with sharp edges due to the intersection of intense radial and lateral cracks. The fracture toughness (Kc = 1.12 MPa·m1/2) of SCAM can be obtained by using a scratch-based methodology for a spherical indenter based on linear elastic fracture mechanics (LEFM) under an appropriate range of applied loads. An optimal expression for calculating the fracture toughness of easily cleaved materials, including SCAM, via the Vickers indenter-induced cracking method using a Berkovich indenter was recommended.

On the anisotropic fracture behaviour of AA2050-T84 thick plate

Anisotropy in the fracture behaviour of AA2050-T84 was investigated using 2D digital image correlation (DIC)-assisted 3-point bending fracture toughness test on specimens with crack propagation direction oriented along the normal direction (ND) and rolling direction (RD). The tensile properties along the RD indicated slightly higher yield strength (482 ± 8 MPa) than ND sample (469 ± 9 MPa), while the fracture toughness in terms of energy release rate was higher when the notch was oriented along the ND (35.3 ± 1 kJ/m2) compared to the RD (18.5 ± 0.9 kJ/m2). Fractography of the fractured surfaces and detailed electron backscatter diffraction (EBSD) analysis of the microstructure along crack paths unveiled higher crack tortuosity with extensive secondary cracking in the TD-ND specimen than in the TD-RD. DIC measurements clearly showed a larger deformation zone for the TD-ND specimen also as which corroborated the plastic zone size determined using linear elastic fracture mechanics (LEFM). Additionally, the crystallographic model of tilt and twist angle differences between the neighbouring grains was employed to explain the increase in crack path tortuosity and higher dissipation of plastic work contributing to superior fracture toughness for the TD-ND sample.

Numerical Investigation of Fatigue Behavior in Ti-6Al-4V Orthopedic Hip Implants Subjected to Different Environments.

In this paper, hip implants made of Ti-6Al-4V titanium alloy are analyzed numerically using Extended Finite Element Method XFEM. The combined effect of corrosion and fatigue was considered here since this is a common cause of failure of hip implants. Experimental testing of Ti-6Al-4V alloy was performed to determine its mechanical properties under different working environments, including normal, salty, and humid conditions. The integrity and life of the hip implant were assessed using the Linear Elastic Fracture Mechanics (LEFM) approach. For this purpose, the conditional fracture toughness Kq using CT specimens from all three groups (normal, humid, salty conditions) were determined. This provided insight into how different aggressive environments affect the behavior of Ti-6Al-4V alloy; i.e., how much its resistance to crack growth would degrade depending on conditions corresponding to the real exploitation of hip implants. Next, analytical and XFEM analyses of fatigue behavior in terms of the number of cycles were performed for all three groups, and the obtained results showed good agreement, confirming the validity of the integrity assessment approach shown in this work, which also represented a novel approach since fatigue and corrosion effects were investigated simultaneously.

Size effect analysis of mode I fracture performance of hot mix asphalt

The diameters of asphalt concrete cylindrical specimens typically measure either 100 mm or 150 mm, as these are the standard sizes for almost all conducted tests. However, the impact of specimen size on the fracture performance of asphalt mixtures has not been extensively explored. This study aimed to address this gap by performing laboratory tests on semi-circular bending (SCB) specimens. These tests were designed to apply the Size Effect Law (SEL) in evaluating the fracture performance of two asphalt mixture types (AC10 and AC16) at −10 ℃. The experiment selected four different diameters (76 mm, 100 mm, 125 mm, and 150 mm) and two notch-radius ratios (α0 = 0.2 and 0.4) to investigate the relationship between the specimen dimensions and both the nominal stress and fracture toughness (KIC). Results indicated that the nominal stress and KIC adhered to the SEL. It was observed that both aggregate gradation and notch-radius ratio (α0) significantly influenced the SEL’s effect on nominal stress. An increase in α0 from 0.2 to 0.4 shifted the fracture failure towards a more linear elastic fracture behavior. Specifically, the coarser gradation in AC16, compared to AC10, predisposed the fractures towards linear elastic behavior. Moreover, an increase in α0 notably augmented the brittleness number. For specimens with identical dimensions and α0, AC16 exhibited higher D/D0 values, indicating a stronger alignment with Linear Elastic Fracture Mechanics (LEFM) principles for the tested specimen sizes compared to AC10. As the size of the specimen increased, there was a gradual rise in KIC, with the enlargement from 76 mm to 150 mm resulting in KIC increases of 25.2 % and 12 % for α0 values of 0.2 and 0.4, respectively, in the AC10 mixture. Conversely, the AC16 mixture saw respective enhancements of 15.8 % and 7.6 % for the same α0 values. The influence of α0 on the apparent fracture toughness became less pronounced with its increase; KIC gradually decreased for AC10, yet increased for AC16 with rising α0 levels. For the same asphalt mixture type, KIC was found to escalate with both D/D’0 and α0. At an α0 of 0.4, the disparities between KIc and KIcm, – expressed as (KIcm- KIc)/ KIcm, – stood at 13.9 % and 9.2 % for AC10 and AC16, respectively, with a diameter of 150 mm. Based on these findings, it is recommended to utilize an SCB specimen with a diameter of 150 mm and an α0 of 0.4 for fracture performance testing, as these parameters appear to provide the reliable assessment of the materials’ fracture behavior.

Fatigue life prediction of friction-stir-welded aluminum bridge decks – Experimental tests and numerical framework

This paper presents the results of experimental tests and proposes a numerical framework for the prediction of the fatigue behaviour of friction stir welded (FSW) butt-lap joints of aluminium bridge deck extrusions. Fatigue test specimens made of welded real roadway bridge deck extrusions were subjected to cyclic bending under different loading conditions. The fatigue failure modes were thoroughly analyzed. A numerical framework incorporating a 2D finite element (FE) model based on the theory of critical distances (TCD) and a simplified linear elastic fracture mechanics (LEFM) model was developed to predict the fatigue failure initiation location and the fatigue life of the specimens under the test conditions. The location of the failure initiation played a key role in the choice of the numerical approach adopted in the prediction of fatigue life. For fatigue failures initiating from the base metal, the framework utilizes the TCD formulations while for failures in the weld joint, the LEFM model was applied. The proposed numerical framework showed a good agreement with the experimental data.