Linear Elastic Fracture Mechanics Solutions Research Articles

Configurational force method enables fracture assessment in soft materials

Configurational mechanics offers a framework for quantifying the tendency of defects to alter the material configuration. When applied to fracture mechanics, configurational forces can be used to quantify the propensity of cracks to propagate. An alternative, well-established approach involves analytical solutions for crack tip displacement fields. However, these solutions typically apply to a limited range of constitutive behaviors and oftentimes to the linear small strain regime. The ease of calculating configurational forces in a numerical Finite Element implementation, along with their applicability to soft fracture at large strains, motivates the study of their performance as a standalone fracture framework. In contrast to the majority of works that remain theoretical and numerical, our study includes a robust experimental approach to configurational forces at finite strains. We report tensile experiments on a soft elastomer with pre-cuts ante fracture initiation. In a first attempt to approach the J-integral via configurational forces, we explore the performance of the linear elastic fracture mechanics solutions on Pacman-shaped domains that reproduce the crack tip vicinity. Then, we implement the entire boundary value problem with three-dimensional simulations that replicate the empirical tensile deformation of the soft elastomer samples. Subsequently, the results are benchmarked against estimations of the J-integral obtained through a bespoke finite strain analytical crack tip solution. With the successful validation of the configurational force method at finite strains, we aim to establish a pipeline for the calculation of configurational forces in a standalone manner and circumventing the need for close-form analytical solutions.

Lost circulation material design for near-wellbore transverse fractures

The use of particulate lost circulation material (LCM) in drilling fluid is a common method for mitigating lost circulation from near-wellbore fractures that develop during subterranean drilling. Fracture width estimation is the center piece of designing the particle size distribution (PSD) of LCM blends. Width estimation of near-wellbore fractures has predominantly been based on assuming axial fracture development from wellbores. However, this assumption is not valid for all wellbore orientations. Shallow vertical and highly inclined or horizontal well sections in a normal subsurface faulting regime may give rise to configurations where drill-induced fractures develop transverse to the wellbore axis. A transverse fracture yields a substantially different width compared to an axial fracture. This study aims to demonstrate the width contrast between the two fracture geometries and the impact that such contrast would have on the LCM blend design. For this purpose, the width of near-wellbore transverse fractures is estimated using a linear elastic fracture mechanics solution to an axisymmetric model of the same. Contrary to the case of hydraulic fractures, the solution must consider the effect of wellbore cavity boundary conditions on the fracture width profile. Two case studies involving horizontal and vertical boreholes are examined. Results indicate that the assumption of axial fracture geometry could lead to substantial underestimation or overestimation of the fracture width depending on the wellbore inclination and magnitudes of the in-situ stress components. Application of the solution in selecting the compositions of blends made from three LCMs is demonstrated. The discrepancy between the LCM blends PSDs obtained based on the axial and transverse models of the wellbore fracture are discussed.

Fracture and size effect analysis in concrete using 3-D G/XFEM and a CZM-LEFM correlation model: Validation with experiments

The existence of process zones at crack tips in quasi-brittle materials is responsible for the structural size-dependent fracture strength. Hence, accurate prediction tools should not only be capable of predicting the fracture propagation and its load–displacement attributes, but should render themselves capable of the size effect analysis as well. This paper applies a robust 3-D generalized/eXtended finite element method (G/XFEM) algorithm to fracture propagation in concrete and validates its capability to capture the size effect behavior based on the cohesive zone model (CZM). Equipped with the capability of mesh adaptivity, the framework utilized in discretizing the resulting system of equations allows the adoption of independent representation for the background solid mesh and the fracture surfaces without the need for conformity between the two. The fracture simulations on a series of geometrically similar concrete beams under three-point bending have shown very good agreement with available experimental results in terms of both load–displacement (peak load and post-peak) characteristics and size effect behavior. The study also models the experimental problems using Linear Elastic Fracture Mechanics (LEFM) to investigate the structure’s size threshold below which the applicability of LEFM diminishes. The effect of beam geometry and concrete material properties on the LEFM accuracy is assessed by calculating the deviations between LEFM and bilinear cohesive model solutions. The simulation results show that power functions best describe the effect of beam’s depth and tensile strength, and linear functions best fit the effect of elastic modulus and fracture toughness. Subsequently, the study presents and validates an empirical correlation model that modifies the LEFM solution to predict similar ultimate strength as the CZM in addition to capturing the concrete size effect law successfully.

Predicting damage in aggregates due to the volume increase of the alkali-silica reaction products

Volume increase between the reactants and the products of alkali-silica reaction could reach up to 100%. Taking place inside the aggregates, ASR imposes internal pressure on the surrounding material. In the current paper, the possibility of crack growth due to such internal loading is studied. This study is done by employing a semi-analytical mechanical model comprising an elastic solution to a well-known Eshelby problem and a linear elastic fracture mechanics solution to a ring-shaped crack encircling a spheroidal inclusion. The proposed method implies the presence of pre-existing micro-fissures within the aggregate.The study reveals the dependence of the crack growing potential on the spheroid's shape: the larger the ASR pocket - the longer crack opens. The two most critical shapes, causing the highest stress intensity factor and developing the longest crack, are a sphere and a spheroid with a 1/4 aspect ratio respectively. The size analysis of the problem suggests a critical spheroid's radius below which no crack growth is expected. For a chosen material properties and expansion value, such radius lies in the range between 0.1μm and 1μm. Independently of the expansion value and the shape of the pocket of the ASR product, the developed crack length has a power-law dependence on the size of a spheroid.All the theoretical predictions are confirmed by a numerical model based on the combination of the finite element method and the cohesive zone model.

A NURBS-discontinuous and enriched isogeometric boundary element formulation for two-dimensional fatigue crack growth

A new extended isogeometric boundary element method (XIGABEM) formulation is proposed for simulating multiple fatigue crack propagation in two-dimensional domains. The classical use of NURBS in isogeometric formulations is further extended by repeated knot insertion to introduce a C−1 continuity within the approximation space as an elegant approach to representing geometrical discontinuities at crack intersections. This strategy is also used to restrict the enrichment term to the portion of the NURBS defining the tip, where it is necessary. At this near-tip zone, the linear elastic fracture mechanics solutions are embedded into the displacement approximation to represent the theoretical square root behaviour. The enrichment procedure introduces just two degrees of freedom per crack tip, and a tying constraint is used to yield a square linear system. In this direct approach, the stress intensity factors (SIFs) are found as terms in the solution vector without requiring post-processing techniques.Several examples are presented to illustrate the application of the XIGABEM. The accuracy of the results compares favourably against those from the literature, and also against solutions obtained from unenriched and enriched indirect methods that employ the J-integral for SIF extraction. Furthermore, the proposed direct approach is capable of significantly reducing the execution time.

A poro-damage phase field model for hydrofracturing of glacier crevasses

Hydraulic fracture (or hydrofracture) can promote the propagation of meltwater-filled surface crevasses in glaciers and, in some cases, lead to full-depth penetration that can enhance basal sliding and iceberg calving. Here, we propose a novel poro-damage phase field model for hydrofracturing of glacier crevasses, wherein the crevasse is represented by a nonlocal damage zone and the effect of hydrostatic pressure due to surface meltwater is incorporated based on Biot’s poroelasticity theory. We find that the elastic strain energy decomposition scheme of Lo et al. (2019) with an appropriate fracture energy threshold can adequately represent the asymmetric tensile–compressive fracture behavior of glacier ice subjected to self-gravity loading. We assessed the performance of the model against analytical linear elastic fracture mechanics solutions by comparing their predictions of maximum crevasse penetration depth. The model simulates both surface crevasse propagation in the interior region of the glacier, as well as cliff failure in the terminus region. The excellent performance of the proposed model for air/water-filled surface crevasses in idealized land- and marine-terminating grounded glaciers illustrates its applicability to studying the dynamic response of glaciers to atmospheric warming.

On the effect of the loading apparatus stiffness on the equilibrium and stability of soft adhesive contacts under shear loads

The interaction between contact area and frictional forces in adhesive soft contacts is receiving much attention in the scientific community due to its implications in many areas of engineering such as surface haptics and bioinspired adhesives. In this work, we consider a soft adhesive sphere that is pressed against a rigid substrate and is sheared by a tangential force where the loads are transferred to the sphere through a normal and a tangential spring, representing the loading apparatus stiffness. We derive a general linear elastic fracture mechanics solution, taking into account also the interaction between modes, by adopting a simple but effective mixed-mode model that has been recently validated against experimental results in similar problems. We discuss how the spring stiffness affects the stability of the equilibrium contact solution, i.e. the transition to separation or to sliding.

Notched fatigue testing of Inconel 718 prepared by selective laser melting

AbstractSelective laser melting (SLM) was used to prepare notched high‐cycle fatigue test specimens made from nickel‐based superalloy Inconel 718. Samples were designed to have 1 of 3 different notch geometries, including V notches with Kt of 2.2 or 3.1, a U notch with Kt of 2.0, and were printed in either vertical or horizontal orientations. Samples were tested with as‐printed dimensions and surfaces after heat treatment, but a separate set of SLM samples were printed as plates and machined to final dimensions comporting to the V‐notch specimen with Kt = 3.1. High‐cycle fatigue testing showed that machined SLM specimens behaved similar to wrought Inconel 718 plate specimens, but testing with as‐produced surfaces led to a decrease in fatigue life. The explanation for this difference is based on approximations of linear elastic fracture mechanics solutions for short cracks emanating from notch roots, with intrinsic surface features of SLM materials serving as the cracks. Analysis of the actual notch geometries after SLM fabrication indicates that stress intensity in the presence of these features plays a prominent role in determining number of cycles before fatigue crack initiation and propagation occurs.

A comparative and parametric study of dynamic cohesive and linear elastic fracture mechanics models

In cohesive fracture mechanics (CFM), fundamental nondimensional parameters are the ratios of spacetime domain geometries and loadings to corresponding intrinsic scales implied by the cohesive fracture traction–separation relations (TSRs). One of these parameters is the nondimensional load-to-strength parameter which is the ratio of the applied loads, expressed in stress form, to an intrinsic strength scale implied by a TSR. Herein the radii of stress singularity from asymptotic Linear Elastic Fracture Mechanics (LEFM) solutions are derived to normalize cohesive process zone (CPZ) sizes from CFM. By approximating these nondimensional CPZ sizes, a simple small-scale yielding (SSY) indicator is derived for dynamic fracture which in turn is shown to be proportional to the square of the load-to-strength parameter. Thus, the load-to-strength parameter serves two purposes. First, increasing this ratio is shown to correspond to more ductile response for families of cohesive fracture self-similar solutions. Second being related to SSY condition, it is used to evaluate the validity of an LEFM model. Numerical results compare characteristic differences between these groups of CFM solutions, investigate the accuracy of the proposed SSY indicator, demonstrate LEFM solutions underestimate crack length and speed even when the SSY condition is satisfied, and study the evolution of the CPZ size.

In-situ SEM investigation of sub-microscale deformation fields around a crack-tip in silicon

A combination of in-situ scanning electron microscopy (SEM) and geometric phase analysis (GPA) was used to study the deformation fields around a crack-tip in single-crystal silicon under uniaxial tensile load. The sub-microscale silicon pillars grating was fabricated using holographic lithography followed by inductively coupled plasma etching. A series of SEM images of dynamic crack with the sub-microscale grating were obtained during tensile testing. The strain fields around the crack-tip were mapped by GPA. The strain fields were compared with the linear elastic fracture mechanics solutions. It was determined that the deformation is performed around the crack-tip area. The normal strain εxx and shear strain εxy are nearly zero, and the strain fields are dominated by the normal strain εyy component. With the increase of displacement load, the crack propagated mainly along the [010] crystal direction and the strains around the crack-tip increased gradually. It is noted that the theoretical prediction is lower than the experimental results from 0 to 2μm ahead of the crack-tip. However, the agreement between experimental results and theoretical prediction is very good far from the crack-tip (>2μm).

A study on the influence of cohesive zone interface element strength parameters on mixed mode behaviour

This paper presents a detailed study of the influence of maximum interfacial stress on interface element analyses for composites delamination. The development of the non-linear cohesive zone ahead of a crack tip is analysed with respect to length, stress distribution and mode ratio. The energy absorbed by interface elements is compared with the crack tip strain energy release rate from fracture mechanics analyses. These studies are performed initially on standard fracture toughness specimens, where mode-ratio is fixed by the applied displacement constraints. Results show close agreement with linear elastic fracture mechanics solutions. A simple ply drop specimen is then modelled, where the mode ratio is not constrained by the boundary conditions, and results are compared with the Virtual Crack Closure Technique. In this case maximum interfacial stress has a far greater influence on the numerical results, due to its significant influence on cohesive zone length, mode ratio and energy absorbed.

Fracture mechanics analysis of row shear failure in dowelled timber connections

A quasi-nonlinear fracture mechanics model is presented for the analysis of row shear failure in timber connections with multiple fasteners in a row. A prerequisite for use of the model is a known distribution on the fasteners of the total applied load. It is shown that the ideal plastic and linear elastic fracture mechanics solutions appear as special cases of the quasi-nonlinear model. The model offers strength predictions that include the effect of among others the number of fasteners in a row, fastener spacing, row spacing, end-distance, edge-distance, fastener diameter, and material properties, such as shear strength, fracture energy, and modulus of elasticity. Simple explicit expressions are obtained from the analysis for any known fastener load distribution. For bolted connections, where bolt-hole gaps cause high loads on certain bolts at random, the model may be used in simulation procedures or in probabilistic models.

Free vibration of an edge-cracked beam with a Dugdale–Barenblatt cohesive zone

This paper presents a theoretical investigation into the effect of crack-plane plasticity on the vibration of a transversely edge-cracked beam. The crack-plane is presumed to be loaded statically in either bending or shear such that Dugdale–Barenblatt type cohesive zones are activated in the crack-plane ahead of the crack tip. Then the beam undergoes small amplitude vibrations about this state. Euler–Bernoulli beam theory is used for the vibration analysis in which an infinitesimally thin beam element surrounding the crack-plane is replaced with a bending and a shear line-spring. The line-springs by nature account for the discontinuities in slope and deflection, respectively, across the crack-plane. Previous studies of the elastic problem (no cohesive zones) calculate the compliances of the line-springs from tabulated linear elastic fracture mechanics solutions and the compliances depend only on the flexural rigidity of the beam and on the crack length to beam depth ratio. Here the compliance of the line-spring for the mode loaded statically in the nonlinear range is calculated using the Boundary Element Method (BEM) in an iterative nonlinear analysis and the compliance depends nonlinearly on the load across the crack plane. This nonlinear load–deformation relation is linearized about the applied static load level, and the resulting compliance used in the vibration analysis. Among other results for mode I cohesive zones, there is a strong reduction of the fundamental frequency as both load and crack length increase and as yield strength decreases. The results show that there is some potential for designing a non-destructive material characterization technique which would use the changes in frequency to infer the properties of material behavior laws in the cohesive zone.

Direct Measurements of Fracture Toughness and Crack Growth in Polysilicon MEMS

ABSTRACTDirect measurements of Mode-I critical stress intensity factor and crack tip displacements were conducted in the vicinity of atomically sharp edge cracks in polycrystalline silicon MEMS using our in situ Atomic Force Microscopy (AFM)/Digital Image Correlation (DIC) method. The average Mode-I critical stress intensity factor for various fabrication runs was 1.00 ± 0.1 MPa√m. The experimental crack tip displacement fields were in very good agreement with linear elastic fracture mechanics solutions. By means of an AFM, direct experimental evidence of incremental crack growth in polycrystalline silicon was obtained for the first time via spatially resolved crack growth measurements. The incremental crack growth in brittle polysilicon is attributed to its locally anisotropic polycrystalline structure which also results in different local and macroscopic (apparent) stress intensity factors.

Analysis of crack propagation in welded stiffened panels

Models are developed to predict the growth of long fatigue cracks in stiffened panels. An analytical model is described that estimates the stress intensity factor based on superposition of linear-elastic fracture mechanics solutions, taking into account the typical residual stress patterns in stiffened panels. The results were confirmed by finite element analyses. The computed stress intensity factors, alongwith the Paris law, are used to predict the crack propagation in box girders with welded stiffeners with reasonable accuracy.

Numerical evaluation of the rotating crack model

The paper discusses typical fracture mechanics problems which have been analyzed in order to evaluate the performance of the rotating crack model. The outcome indicates that the model is a promising tool for nonlinear analysis of concrete structures. For model-I cracking, the model is capable of simulating the whole range of size covered by limit analysis and linear elastic fracture mechanics, but when a slant mesh is used, a mesh bias is observed. For mixed-mode cracking, the model is capable of predicting the ultimate load for small concrete structures; the results of simulations for large concrete structures show a good coincidence with linear elastic fracture mechanics solutions.

Aspects of equivalence between contact mechanics and fracture mechanics: theoretical connections and a life-prediction methodology for fretting-fatigue

We identify aspects of quantitative equivalence between contact mechanics and fracture mechanics via asymptotic matching. An analogy is invoked between the geometry of the near-tip regions of cracked specimens and that of the sharp-edged contact region between two contacting surfaces. We then demonstrate that the asymptotic elastic stress and strain fields around the rim of the contact region, as derived from classical contact mechanics analyses, are identical to those extracted from linear-elastic fracture mechanics solutions for analogous geometries. Conditions of validity of this model for contact mechanics are established by recourse to the singular fields and small-scale-yielding concepts of fracture mechanics. With this method, the geometry of the contact-pad/substrate system naturally introduces a fictitious crack length, thereby providing a physical basis to analyze contact-fatigue fracture initiation and growth. Possible extensions of this crack analogue model are then suggested, for situations where static or cyclic mechanical loads are superimposed parallel to the direction of interfacial friction, using the two-parameter fracture characterization that involves the stress intensity factor K and the non-singular T-stress, or alternatively, the J integral and the triaxiality parameter, Q. The predictions of the crack analogue model are then compared with fretting-fatigue experiments and are shown to be in agreement with a variety of independent experimental observations. A new life-prediction methodology for fretting fatigue is also proposed on the basis of the present approach.

Comparison of shear-lag theory and continuum fracture mechanics for modeling fiber and matrix stresses in an elastic cracked composite lamina

This study analyzes fiber tensile and matrix shear stresses near the crack tip in a transversely cracked, unidirectional, fiber-reinforced lamina under a remote tensile stress applied in the fiber direction. The two-dimensional lamina consists of parallel, equally-spaced elastic fibers with elastic matrix in-between, and contains a row of up to a few hundred contiguous fiber breaks aligned transverse to the fiber direction forming a central transverse crack. Using the break-influence superposition (BIS) technique, a recently developed method for analyzing a shear-lag model first introduced by Hedgepeth, we calculate the tensile and shear stress concentrations in the fibers and matrix, respectively. These are compared to tensile and shear stresses calculated using Linear Elastic Fracture Mechanics (LEFM) and the complete elasticity solution both for the continuum limit of a homogeneous, orthotropic elastic material with a transverse central crack loaded in Mode I. For the shear-lag model a critical scaling parameter for examining the stress behavior away from the crack tip along the fiber direction is √E∗/G∗, where E∗ and G∗ are composite in-plane stiffness constants along the fiber direction and in shear, respectively. In addition to these parameters, the LEFM and complete elasticity solutions also involve the effective transverse stiffness and Poisson's ratio. For a sizable crack (consisting of 100 or more fiber breaks), the fiber tensile stresses ahead of the crack tip along the crack plane calculated from the BIS approach achieve excellent agreement with the LEFM solution down to the scale of one fiber diameter and even better agreement with the complete solution both in the near crack tip field and far field, regardless of the composite stiffness constants. The profiles of the fiber tensile and matrix shear stresses along the fiber direction show generally good agreement, with the agreement improving as the composite stiffness transverse to the fiber direction grows.

Subcritical crack growth at bimaterial interfaces: Part I. flexural peel technique

A flexural peel (FP) technique was developed to study the crack growth behavior along a bimaterial interface. The technique was based on a sandwich specimen where one arm of the specimen was peeled away from the interface under a combined tensile and shear mode. An approximate linear elastic fracture mechanics solution was derived analytically from theJ integral formulation. The solution was compared to finite element calculations based on the crack closure method and experimental measurements. From the finite element analysis, ratios of the mode-I and mode-II components of the crack tip field were determined for a wide range of modulus and thickness ratios.

An experimentally verified criterion for propagation across unbounded frictional interfaces in brittle, linear elastic materials

Understanding how a propagating fracture interacts with existing fractures, lithologic boundaries and other material interfaces is essential to the interpretation of fracture network geometries. In this paper, a first order analysis of the stresses near a mode I fracture impinging upon a frictional interface oriented normal to the growing fracture results in a simple criterion that predicts whether a growing fracture will terminate at or cross the interface. The analysis uses the linear elastic fracture mechanics solution for the stresses near a fracture tip to determine the compressive stress required to prevent slip along the interface at the moment when the stress on the opposite side of the interface is sufficient to initiate a fracture. A series of experimental investigations designed to assess the conditions required for crossing are presented and shown to be consistent with the criterion. Using data from previously published experiments, the criterion is shown to accurately predict the occurrence of compressional crossing in nine different brittle materials including three types of synthetic materials and six types of natural rock.