Finite Fracture Mechanics Research Articles

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.

Failure assessment of eccentric circular holes under compressive loading

AbstractThe present work aims to investigate the failure size effect on flattened disks containing an eccentric circular hole under mode I loading conditions. For this purpose, uniaxial compression tests are carried out on polymethyl methacrylate (PMMA) samples with holes. Depending on the hole radius and eccentricity, the energy release rate is either an increasing or decreasing function of the crack length, thus affecting the stability of crack propagation. Experimental results are interpreted and discussed through the coupled stress and energy criterion of Finite Fracture Mechanics. The approach lies on the assumption of a finite crack advance and it is implemented through the numerical estimation of the stress field and the Incremental Energy Release Rate functions. Finally, stability and crack speed propagation are discussed under the assumption of Linear Elastic Fracture Mechanics. Theoretical predictions reveal in agreement with experimental results thus demonstrating that the Coupled Criterion effectively captures the failure condition.

Investigating factors influencing interlaminar stresses and energy release rates of semi-elliptical cracks at free edges

Traditional approaches that rely on stress or energy-based criteria to predict delamination require the determination of stresses before crack onset or energy release rates after crack initiation, respectively. While a relatively new coupled criterion, Finite Fracture Mechanics, necessitates the computation of both stresses and energy release rates. This study discusses the numerical determination of the interlaminar stresses and energy release rates of interlaminar semi-elliptical cracks emanating at the free edge in carbon fibre-reinforced polymer materials. The method establishes a new semi-analytical framework, based on dimensional analysis, which permits obtaining a functional expression for the stresses and the energy release rates in function of material and geometrical parameters. This framework eliminates the need to re-solve the underlying boundary value problem for a different material, geometry, and load. The non-dimensional functions are obtained by performing careful interpolation from a set of finite element solutions, ensuring predictions are accurate and within acceptable numerical limits. The influence of material contrast and relative ply thickness on interlaminar stresses and energy release rates are investigated using the semi-analytical framework. Additionally, the fracture mechanics analysis is also conducted to investigate the influence of semi-axes of semi-elliptical crack on energy release rate distributions. Understanding the interlaminar stresses and energy release rates is essential for applying either stress, energy, or coupled criterion to predict the structural strength of a layered body.

Determining the Cohesive Length of Rock Materials by Roughness Analysis

In this research, the cohesive length of various rock types is measured using quantitative fractography alongside a recently developed multifractal analysis. This length is then utilized to gauge material cohesive stress through the theory of critical distances. Furthermore, the fracture process zone length of different rings sourced from identical rocks is assessed as a function of ring dimensions and experimental measurements of fracture toughness, in accordance with the energy criterion of the finite fracture mechanics theory. Subsequently, employing the stress criterion within coupled finite fracture mechanics, the failure stress corresponding to the fracture process zone is determined for various rings. Ultimately, through interpolation, the critical stress corresponding to the cohesive length, quantified via quantitative fractography, is approximated. Remarkably, the cohesive stress values derived from both methodologies exhibit perfect alignment, indicating the successful determination of cohesive length for the analyzed rock materials. The study also delves into the significant implications of these findings, including the quantification of intrinsic tensile strength in quasi-brittle materials and the understanding of tensile strength variations under diverse stress concentrations and loading conditions.

Machine learning-based determination of Mode II translaminar fracture toughness of composite laminates from simple V-notched shear tests

This paper presents a novel method for measuring the translaminar crack resistance curve of composite laminates under Mode II shear loading. A machine learning (ML)-based approach is utilized to extract the inapparent information of the crack resistance curve from the translaminar shear strength measurements obtained from simple V-notched shear tests. The entire campaign is built on the framework of the Finite Fracture Mechanics (FFM) combined with Finite Element Method (FEM). Special emphasis is made on the nonlinear mechanical behavior of composites under shear stress since the original FFM models are designed for quasi-brittle materials. With the well-trained recurrent neural network model, the Mode II R-curve of composite laminate can be obtained with un-notched and V-notched shear strength values as inputs. Experiments were conducted on carbon fiber-reinforced composites to validate the accuracy of the R-curve obtained by the proposed approach and that by the traditional compact shear test. The successful implementation of the method suggests a more convenient and low-cost way of obtaining this important damage-related parameter for composites.

Explicit modelling of meso-scale damage in laminated composites – Comparison between finite fracture mechanics and cohesive zone model

The ability of carbon fibre reinforced polymers to respect functional requirements such as permeability is strongly related to their damage state. In order to identify the fracture properties on which transverse cracking kinetics depend, we combine experimental and virtual tests by explicitly modelling all transverse cracks. Two modelling methods are compared, the cohesive zone model (CZM) and finite fracture mechanics (FFM). Both models are based on a double criterion of energy and strength. Each potential crack is assigned a couple of material properties (fracture energy and strength) derived from a probability distribution. This study focuses on how modelling choice implies assumptions on energy dissipation and the cracking behaviour represented. The relevance of both modelling methods is also discussed. A methodology is provided for identifying fracture properties and predicting damage accumulation in an isolated ply.

Prediction of thermal shock induced cracking in multi-material ceramics using a stress-energy criterion

Impact of residual stresses on the thermal shock resistance of alumina–zirconia multi-layer ceramics is investigated within the framework of finite fracture mechanics, using astress-energy criterion. The critical temperature difference (ΔTc) for crack formation is strongly dependent on the magnitude of residual stress and the material’s strength. The predicted minimal spacing between cracks, critical time for crack initiation, and initial depth of the induced cracks are compared and discussed for different designs. An increase of up to ∼40 % in ΔTc is predicted for multi-material ceramics with athin alumina surface layer with compressive stresses, compared to bulk reference alumina.

Cavitation and crack nucleation in thin hyperelastic adhesives

The present study investigates in the failure of adhesive bondings with structural silicone sealants. Point connectors of two circular metal adherends bonded with DOWSIL™ TSSA are subjected to tensile loading. We formulate and use a constitutive law that captures volumetric softening owing to the formation of cavities. Therein, cavitation is considered a process of elastic instability which is homogenized with a pseudo-elastic approach. Ultimate failure initiating from the free edges is predicted employing the framework of finite fracture mechanics. The concept requires both a strength-of-materials condition and a fracture mechanics condition to be satisfied simultaneously for crack nucleation. For the former, we use a novel multiaxial equivalent strain criterion. For the latter, we employ literature values of the fracture toughness of DOWSIL™ TSSA . The predicted onset of cavitation and ultimate failure loads are in good agreement with our experiments. The proposed model provides initial crack lengths that allow for the derivation of simple engineering models for both initial designs and proof of structural integrity while simultaneously extending the range of usability of the structural silicone compared to standardized approaches.

The effect of fibre orientation on fatigue crack propagation in CFRP: Finite fracture mechanics modelling for open-hole configuration

The demand to capture translaminar crack growth under fatigue loading scenarios led this work contribution to carry out the Finite Fracture Mechanics (FFM) method in fatigue damage growth and the application of the Paris model to generate the translaminar damage propagation prediction. The purpose of this study is to analyse the effect of fibre orientation on translaminar crack propagation rate using the FFM model, which includes cycle damage increment estimation and fractographic analysis. The results confirm the feasibility of FFM in predicting crack growth and estimating life under cyclic loading. However, C-scan analysis and the revised crack propagation direction are critical in determining the realistic crack length, considering adhesive failure along the fibre direction. Additionally, this work contribution is also related to the application of the Paris model (based on dL/dN vs ΔK) to generate the translaminar damage propagation prediction model. The most dominant damage mechanism was the splitting pattern, which changed the aspect of failure for each laminate architecture as a function of fibre orientation. The laminate with multidirectional fibre orientation exhibited higher resistance to translaminar crack propagation due to the growth of splitting and delamination in multiple directions. The fibre orientation changed the propagation path, which influenced the fracture toughness and crack propagation rate behaviour.

Influence of debonding and substrate plasticity on thin film multicracking

Multicracking of a thin brittle layer deposited on a substrate with an intermetallic layer is studied using finite fracture mechanics. Nonlinear implementation of the coupled criterion is used to predict the initiation and successive subdivisions of a periodic network of cracks considering intermetallic layer plasticity and interface debonding. Plasticity has a moderate influence on the cracking kinetics whereas debonding length has a strong influence on the saturation crack spacing. The cracking kinetics predicted numerically is more abrupt than in experiments because of the periodicity assumption, however crack spacing at saturation similar to those measured experimentally are obtained. The tensile strength and critical energy release rate of the thin brittle layer are determined by inverse identification based on the crack density variation as a function of the imposed loading. The proposed approach also enables the accurate determination of the interface critical energy release rate based on the experimentally measured debonding lengths.

Crack tip shielding and size effect related to parallel edge cracks under uniaxial tensile loading

The present work aims at investigating crack shielding and size effect related to a cracked slab under tensile loading. For this purpose, experimental tests are carried out on PMMA cracked samples. Three different geometries are taken into account, presenting one, two or three parallel edge cracks, and assuming their distance equal to their initial length. Results are interpreted through the coupled stress and energy criterion of Finite Fracture Mechanics (FFM). The approach is implemented numerically, and parametric finite element analyses are carried out to evaluate the normal stress field and the stress intensity factor for each configuration. It is found that asymmetric crack propagation has to be preferred according to the energy balance. The matching between FFM failure predictions and experimental data reveals to be satisfactory.

Failure analysis of crack-prone joints with Finite Fracture Mechanics using an advanced modeling approach for the adhesive layer

When adhesive joints are subjected to critical loads, the components can separate at the interface between the adhesive layer and one of the adherends. Due to the strongly differing stiffness properties of the adherends and the adhesive, which cause stress peaks to arise locally at the edge of the joint, a crack grows into the structure there once the critical load is reached. The concept of Finite Fracture Mechanics allows reliable failure assessment and determination of this critical load. For this purpose, a coupled criterion consisting of an energy and a strength criterion is evaluated, both of which must be satisfied simultaneously in case of failure. In this work, Finite Fracture Mechanics is evaluated analytically and using finite element data and compared and discussed with experimental results. In the analytical approach, the focus is on extended applicability to joints with both thin and thick adhesive layers, for which an advanced modeling approach of the adhesive layer is used.

Prediction of last ply failure load of keyhole notched laminated composites using the virtual isotropic material concept

Composite laminates have become increasingly popular in industries like aerospace and shipbuilding. In practical applications, the creation of notches in composite laminates is often necessary to enable assembly and modifications. This rise in the use of notched composite laminates highlights the need for reliable methods to predict potential failures in these components. With these predictions, the safety and reliability of composite structures can be ensured by preventing catastrophic failures. The present paper investigates the use of the Virtual Isotropic Material Concept (VIMC) combined with the Finite Fracture Mechanics (FFM) criterion to predict the Last Ply Failure (LPF) load in keyhole notched composite laminates and under mode I loading conditions. Experimental tests were conducted to measure the LPF load of E-glass/Epoxy keyhole notched specimens, and the results were compared with the predicted results using the VIMC-FFM criterion. The paper also derives an expression for the Stress Intensity Factor (SIF) function for a crack that emanates from the keyhole notch tip under mode I loading conditions. The derived equation matches very well with numerical data and have a mean absolute error about 2% in the studied domain. The study shows that the VIMC-FFM criterion predicts the LPF load of keyhole notched composite laminates with acceptable accuracy and in a simple and low-cost manner.

Fatigue life assessment of notched laminated composites: Experiments and modelling by Finite Fracture Mechanics

In this paper, the coupled Finite Fracture Mechanics criterion is extended to assess the finite fatigue life of orthotropic notched laminated composites. The approach is validated through a comprehensive experimental program conducted on laminated composites under tension-tension cyclic loading conditions with two distinct lay-ups. For a given loading ratio, fatigue tests on plain and cracked specimens are first performed to provide the model inputs, the critical cyclic stress and stress intensity factor amplitudes. Fatigue tests on samples weakened by circular holes of two different radii are then used for blind predictions. Accurate predictions of the number of cycles to failure are achieved without the need for inverse calibration of material properties or deviation from standard testing procedures. Finally, a parametric study is performed to investigate the hole radius effect. It is worth mentioning that the proposed approach is general and can be applied to any notched geometry.

Discussion on three forms of coupled criterion for notch fatigue limit prediction: Comparison between point-wise and averaged stress (energy) criteria

In the framework of finite fracture mechanics, two coupled criteria were proposed by Leguillon and Carpinteri, respectively. Leguillon’s criterion uses the point-wise stress criterion, whereas Carpinteri’s criterion uses the averaged stress criterion. Both criteria employ the same incremental energy release rate criterion which can be considered as an averaged energy criterion. Mathematically, the averaged energy criterion can be transformed into an equivalent point-wise criterion. In this paper, a new coupled criterion based on the point-wise stress and energy criteria was proposed. The proposed criterion is equivalent to Carpinteri’s criterion when predicting notch fatigue limits. All the three forms of coupled criteria were checked and compared by a large number of experimental results including V-shaped and central hole specimens. When performing the coupled criteria, power functions derived from finite element analysis were used to express the energy release rate and notch stress distributions. The effect of notch size on the crack extension lengths were investigated.

A simplified approach to hydraulic fracturing of rocks based on Finite Fracture Mechanics

AbstractThis work presents a coupled approach, based on Finite Fracture Mechanics (FFM), to preliminarily investigate hydraulic fracturing of rocks. The novelty of the criterion relies on the assumption of a finite crack extension and on the simultaneous fulfillment of a stress requirement and the energy balance. Two material constants are involved, namely, the tensile strength and the fracture toughness. The FFM unknowns are represented by the critical crack advance, which results a structural parameter and not a simple material one, and the breakdown pressure. The study investigates the longitudinal failure behavior of both impermeable and permeable rocks by supposing that growing cracks are loaded by a pressure proportional to that acting on the borehole wall. The stability growth of hydraulic fractures is discussed case by case. The approach is validated against experimental data available in the literature by considering the effect of rock permeability, fluid viscosity and flow rate.

Coupled versus energetic nonlocal failure criteria: A case study on the crack onset from circular holes under biaxial loadings

The phenomenon of brittle crack onset stemming from a circular hole in an infinite plate subjected to remote biaxial loading is herein investigated. A thorough analysis on the influence of the loading biaxiality reveals the existence of a wide casuistry in the sign and trend distributions of the stress field and Stress Intensity Factor, thus rendering it an exhaustive case study for assessing different failure criteria. Subsequently, three different approaches are used to determine the biaxial safety domains, two of which rely on the coupling of stress and energy conditions for failure, namely Finite Fracture Mechanics and Cohesive Zone Model, plus the purely energy-driven Phase Field model of fracture. Noteworthy, Finite Fracture Mechanics predicts the existence of a region in the loading space where failure is exclusively governed by the energy condition. Likewise, it is mathematically proven that the system of equations governing Dugdale's Cohesive Zone Model is equivalent to the first-order minimization condition of the energy balance, the resultant predictions being fairly close to those obtained by Finite Fracture Mechanics. Lastly, the Phase Field model of fracture is numerically implemented in the context of Finite Elements while paying special attention to the choice of the energy decomposition, whereof two are implemented: No-Decomposition and No-Tension decomposition. Specifically, the latter showcases satisfactory agreement with both Finite Fracture Mechanics and Dugdale's Cohesive Zone Model, thus posing a solid contender for studying complex fracture scenarios upon combined tension-compression stress states.

Crack impinging on a curved weak interface: Penetration or deflection?

Curved weak interfaces present promising advantages to be implemented as crack arrestors in structures designed under the damage tolerant-design principles. Among other advantages, they neither add extra weight nor significantly affect the global stiffness of the structural element, in contrast with alternative crack arrestors concepts. To be employed as a crack arrestor, it is key that the interface is able to deviate the crack. If the crack penetrates across the interface, the effect of the weak interface as a crack arrestor is canceled. In view of this, this work studies how to set the interface parameters to promote crack deviation along the interface. In particular, following the dimensional analysis of the problem, the effect of three significant dimensionless parameters is studied: interface to bulk fracture toughness, interface to bulk tensile strength, and the interface curvature radius normalized with the material characteristic length. The corresponding analysis is carried out using three approaches widely applied for the prediction of cracking events: Linear Elastic Fracture Mechanics, Finite Fracture Mechanics, and a combination of Phase field and Cohesive Zone Model. The results present a clear effect of some parameters, such as the ratio of the interface to bulk fracture toughness, for which the three approaches agree. However, the results are moderately diverse in which correspond to the effect of the ratio of the interface to bulk tensile strength and quite divergent in what respect to the effect of the radius. The results are interpreted and explained as a consequence of the main assumptions behind the approaches studied.

A novel Finite Fracture Mechanics approach to assess the lifetime of notched components

A novel model is proposed to assess the fatigue lifetime of notched components under uniaxial loading conditions in the medium/high-cycle fatigue regime. It is based on the Finite Fracture Mechanics approach, previously proposed in static and fatigue limit frameworks, according to which failure occurs when a stress and an energy condition are simultaneously satisfied. Basquin’s equations are employed to catch the variation of input parameters based on SN data of a plain and a notched sample. The criterion is validated against various experimental data sets from the literature, encompassing different notch shapes, loading conditions and materials.

Ligament size effect in largely cracked tensile structures

The influence of the ligament size on the brittle failure behaviour of largely cracked elements is analysed. To this aim, the following tensile infinite geometries are taken into account: (i) a double edge cracked plate; (ii) a cylindrical bar with an external circular crack. These configurations can be considered as complementary to the Griffith crack and the Penny shaped crack, respectively. The failure size effect is investigated through a semi-analytical approach by two different methods: the coupled Finite Fracture Mechanics (FFM) criterion and the Cohesive Crack Model (CCM) implementing a Dugdale type cohesive law. Theoretical predictions are compared with each other and with experimental data obtained by tensile testing PMMA samples containing two collinear cracks. Strength estimations reveal accurate.