Equivalent Elastic Crack Research Articles

Disk-shaped compact tension test for fracture analysis on pharmaceutical tablets

Pharmaceutical tablets must meet a number of requirements and among them, the mechanical strength plays an important role. The diametral compression test is generally used to evaluate it but can generate unstable failures. Thanks to load-unload cycles applied to the tablets subjected to a DCT test, it was shown that the concept of equivalent linear elastic fracture mechanics usually, can be successfully applied to the Mode I fracture behavior. Within this framework, the equivalent elastic crack growth resistance, commonly called resistance curve (R-curve), of the studied material was obtained and revealed the development of a Fracture Process Zone (FPZ) which is symptomatic of a quasi-brittle behavior. Moreover, the cyclic loading applied during the fracture test revealed the existence of a second dissipative mechanism leading to residual crack opening which seems to be mainly caused by friction phenomenon in the FPZ. • Fracture behavior of pharmaceutical tablets is a key issue for their manufacturing. • An alternative to the diametral compression test is proposed here. • No instability issues using the notch opening displacement controller. • A quasi-brittle behavior identified through equivalent LEFM theory. • Presence of dissipative mechanisms like micro-friction inside the process zone.

Size-effect method to determine mode-I fracture toughness of aluminium alloys

Size-effect method is used to determine the fracture properties of three aluminium alloys. Mode-I tensile testing of thin rectangular sheets of aluminium alloys 5052−H32, 6061−T6 and 8011 is carried out. The test specimens are scaled geometrically in the ratio of 1:2:3:4 for a constant length (L) to width (D) ratio (L∕D=4). The notch depth (a) to width (D) ratio is 0.25. Size-dependency of strength is observed. The fracture toughness values and the crack growth resistance curves (R-curves) for these three alloys are determined using equivalent elastic crack model. The R-curves predict the peak loads accurately with only 0%–7% difference from the experimental values. The pre-peak load–deflection curves calculated from the R-curves closely match the experimental curves. Bažant’s size effect law could be used to fit the peak load data. Scaling laws for strains at peak loads are proposed. The transition length (D0) determined for these strain scaling laws differ from those obtained using the size effect law.

Cohesive zone modeling of thin aluminium sheets

Single edge notch (SEN) aluminium specimens of widths (D): 20, 40 and 50 mm are tested under uniaxial tensile loading for a strain rate of 0.005/min. The notch depth to width ratio (a/D = 0.25) and the gauge length to width ratio (L/D = 2.5) are constant for these tested specimens. The load versus deflection curves are analysed using cohesive crack model. The peak stress data are analysed using equivalent elastic crack model. The fracture energy Gf obtained from these approaches are compared.Uniaxial tensile testing of 50 mm width SEN specimens (L/D = 2.5) with ligament lengths of 2.5, 5, 7.5, 10, 12.5 and 15 mm is carried out. A linear regression fit between work of fracture per unit ligament area and ligament length is used to determine the essential work of fracture. An energy based approach involving fixed point iteration method to calculate the crack length at peak load is proposed.

Crack Propagation and Fracture Process Zone (FPZ) of Wood in the Longitudinal Direction Determined Using Digital Image Correlation (DIC) Technique

As a state-of-the-art method, the digital image correlation (DIC) technique is used to capture the fracture properties of wood along the longitudinal direction, such as the crack propagation, the strain field, and the fracture process zone (FPZ). Single-edge notched (SEN) specimens made of Douglas fir (Pseudotsuga menziesii) from Canada with different notch-to-depth ratios are tested by three-point-bending (3-p-b) experiment. The crack mouth opening displacements (CMOD) measured by the clip gauge and DIC technique agree well with each other, verifying the applicability of the DIC technique. Then, the quasi-brittle fracture process of wood is analyzed by combing the load-CMOD curve and the strain field in front of the preformed crack. Additionally, the equivalent elastic crack length is calculated using the linear superposition hypothesis. The comparison between the FPZ evolution and the equivalent elastic crack shows that specimens with higher notch-to-depth ratios have better cohesive effect and higher cracking resistance.

A bi-linear cohesive law-based model for Mode II fracture analysis: Application to ENF test for unidirectional fibrous composites

The complicated toughening mechanism of Mode II cracking in end notch flexure (ENF) test for fibrous composites prevents the existence of clear crack-tip. Hence the concept of equivalent linear crack length was proposed to avoid detecting the crack-tip in ENF test, which assumes that the increase of compliance due to the development of fracture process zone (FPZ) or crack propagation is attributed to the equivalent elastic crack length giving the same compliance as the one of the actual crack with its FPZ; thus the equivalent crack length can be estimated according to the associated compliance obtained from experiment. The objective of the presented work was to develop an analytical model for the analysis of ENF test specimen without need for crack-tip identification and complicated data reduction. Bilinear cohesive law was employed to model the cohesive force across the potential crack interfaces. Analytical solutions for the governing equation were derived. Furthermore, methods to determine the cohesive parameters, the length of fracture process zone (FPZ), and the energy release rate were developed. The critical loads corresponding to the damage and crack onset can also be estimated by the model. Experimental validation was implemented and a good agreement was achieved between the test results and model predictions. It can be concluded that bilinear cohesive law may achieve a good approximation for the tangential traction-separation constitutive relationship in Mode II fracture; the shear strength of the material can be taken as the critical stress of cracking onset; the energy release rate depends on the initial crack length to some extent, and this effect could be reduced as the length of initial crack increases.

Influence of the Extent of Fracture on the Resistance against Chloride Penetration into Cementitious Composite

The paper presents data obtained in own experimental campaign focused on the investigation of the effect of fracture propagation on the change in values of electrical resistivity and ultrasound pulse passing time in notched beam specimens. Test geometry of three-point bending was utilized to propagate the crack through the specimen ligament in several loading−unloading cycles, between which the electrical resistivity and ultrasonic measurements over the fractured region in the centre of the beam were performed. Equivalent elastic crack length was estimated at those loading stages. Relationships among the changes in the investigated, non-destructively gained parameters, the effective crack length and the distance from crack tip are discussed with respect to two variants of treatment of the specimens’ surface during testing: the pre-dried and the wet one.

Influence of crack propagation on electrical resistivity and ultrasonic characteristics of normal concrete assessed by sequential TPB fracture test

The paper is focused on the investigation of the effect of tensile failure propagation on the change in values of electrical resistivity and ultrasound pulse passing time within the central section of beam bend specimens made of normal concrete. These two physical characteristics may be employed as a quality indicator of the concrete cover within the procedures of the assessment of the structural durability. The three-point bending test configuration was utilised to drive the crack propagation process through several loading–unloading cycles, between which, the electrical resistivity and ultrasonic measurements over the fractured region were performed. Equivalent elastic crack model was used for estimation of the fracture progression (described via the effective crack length increment) at those loading stages. The relationships between changes of the non-destructively determined parameters, the effective crack length and the distance from crack tip are discussed with respect to two variants of treatment of the test specimens’ surface: the pre-dried surface and the wet surface. A difference between the ultrasonic and electrical resistivity measurements ability with respect to crack width is found in the case of wet samples.

Determining mode I cohesive law of Pinus pinaster by coupling double cantilever beam test with digital image correlation

The direct identification of the cohesive law in pure mode I of Pinus pinaster is addressed. The approach couples the double cantilever beam (DCB) test with digital image correlation (DIC). Wooden beam specimens loaded in the radial-longitudinal (RL) fracture propagation system are used. The strain energy release rate in mode I (GI ) is uniquely determined from the load-displacement ( P ?? ) curve by means of the compliance-based beam method (CBBM). This method relies on the concept of equivalent elastic crack length ( eq a ) and therefore does not require the monitoring of crack propagation during test. The crack tip opening displacement in mode I ? ? I w is determined from the displacement field at the initial crack tip. The cohesive law in mode I I I (? ? w ) is then identified by numerical differentiation of the I I G ? w relationship. Moreover, the proposed procedure is validated by finite element analyses including cohesive zone modelling. It is concluded that the proposed data reduction scheme is adequate for assessing the cohesive law in pure mode I of P. pinaster.

Electrical resistivity and ultrasonic measurements during sequential fracture test of cementitious composite

Cracks in cover of reinforced and pre-stressed concrete structures significantly influence the ingress of deleterious species causing decrease in durability of these structures. The paper is focused on the effect of fracture process on two selected physical parameters of concrete – the electrical resistivity and the ultrasonic pulse passing time – which might be employed as the quality indicator of concrete cover within (nondestructive) procedure(s) of assessment of the structural durability. The concrete electrical resistivity and ultrasonic passing time were investigated here with respect to two variants of treatment of the test specimens’ surface (the pre-dried surface and the wet surface). Test configuration of three-point bending of notched beam was utilized to control the crack propagation; the fracture process passed through several loading–unloading sequences between which the electrical resistivity and ultrasonic passing time readings over the fractured region were performed. Equivalent elastic crack model was used for estimation of the fracture advance (described via the effective crack length) at the loading stages corresponding to the resistivity and ultrasonic measurements. Relationships between changes of both the concrete resistivity and ultrasonic pulse passing time and the effective crack length is determined and discussed.

Direct Evaluation of Cohesive Law in Mode I of Pinus pinaster by Digital Image Correlation

The direct identification of the cohesive law in pure mode I of Pinus pinaster is addressed in this work. The approach couples the double cantilever beam (DCB) test with digital image correlation (DIC). Wooden beam specimens loaded in the radial-longitudinal (RL) fracture propagation system are used. The strain energy release rate in mode I (G I) is uniquely determined from the load–displacement curve by means of the compliance-based beam method (CBBM). This method relies on the concept of equivalent elastic crack length (a eq) and therefore does not require the monitoring of crack propagation during test. DIC measurements are processed with two different purposes. Firstly, the physical evidence of a eq is discussed with regard to actual estimation of the crack length based on post-processing full-field displacement measurements. Secondly, the crack tip opening displacement in mode I (w I) is determined from the displacements near the initial crack tip. The cohesive law in mode I (σ I − w I) is then identified by numerical differentiation of the G I − w I relationship. The methodology and accuracy on this reconstruction are addressed. Moreover, the proposed procedure is validated by finite element analyses including cohesive zone modelling. It is concluded that the proposed data reduction scheme is adequate for assessing the cohesive law in pure mode I of P. pinaster.

An application for the fracture characterisation of quasi-brittle materials taking into account fracture process zone influence

The paper introduces a Java application programmed for the advanced determination of the fracture characteristics of silicate-based materials failing in a quasi-brittle manner. The tool reconstructs the progress of a quasi-brittle fracture from the measured load–displacement curve and the knowledge of basic mechanical properties of the material. The main contribution of the proposed approach is that it takes the characteristics of the Fracture Process Zone (FPZ, particularly its extent, i.e. its size and shape) evolving at the tip of the propagating crack during the failure process into account and incorporates them into the fracture-mechanical parameter evaluation procedure(s). This approach is expected to substantially diminish the influence of the test specimen’s size/shape and the test geometry on the values of the parameters of nonlinear fracture models determined from the records of fracture tests on laboratory specimens. The application implements a developed technique for estimation of the size and shape of the FPZ. The technique is based on an amalgamation of several modelling concepts dealing with the failure of structural materials, i.e. multi-parameter linear elastic fracture mechanics, classical nonlinear fracture models for concrete (equivalent elastic crack and cohesive crack models), and the plasticity approach. The knowledge of the FPZ’s extent is employed for the relation of a part of the entire work of fracture to its characteristics within the presented approach. The verification and validation of the developed technique is performed via numerical simulations using the authors’ own computational code based on physical discretization of continuum and selected sets of experimental evidence published in the literature. Reasonable agreement is observed between the outputs of the presented semi-analytical technique and both the simulation results and the experimental data.

Identification of a cohesive zone model from digital images at the micron-scale

We present a new methodology for the identification of a zone cohesive model that describes material failure. The material under consideration fails by crazing. The study is conducted at the micron scale in order to capture and analyze the fracture mechanism. The crack tip displacement fields are measured optically by Digital Image Correlation. The local stress intensity factors (mode I and II) and the location of the equivalent elastic crack tip are calculated during the loading. The variation of the location of the equivalent crack tip is used to track the initiation and growth of the process zone, up to the onset of crack propagation. These experimental measurements are used to define the appropriate parameters in a cohesive zone model. The methodology addresses the onset of crazing, the traction–separation profile and the maximum opening corresponding to the local nucleation of a crack. The cohesive parameters that are derived from the experimental data are consistent with results available in the literature. In addition, the model enables the characterization of the normal and tangential mode of the cohesive model.

Biaxial tension fatigue response of concrete

Abstract Concrete structures such as rigid airport pavements are subjected to repeated high-amplitude loads resulting from passing aircraft. The resulting stress-state in concrete is a biaxial combination of compression and tension. It is of interest to understand the response of plain concrete to such loading conditions, which will enable development of realistic material models for implementation in mechanistic pavement design procedures. The objective of this work is to characterize the quasi-static and low-cycle fatigue response of concrete subjected to biaxial stresses in the biaxial tension region, where the principal tensile stress is larger than or equal in magnitude when compared with the principal compressive stress. An experimental investigation of material behavior in the biaxial tension region is conducted. The experimental setup consists of the following test configurations: (a) notched concrete beams tested in three-point bend configuration, and (b) hollow concrete cylinders subjected to torsion. Failure of concrete in the biaxial tension region is shown to be a local phenomenon under quasi-static and fatigue loading, wherein the specimen fails owing to a single crack. The crack propagation is studied using the equivalent elastic crack concept. It is observed that the crack growth rate in constant amplitude fatigue loading exhibits a two-phase process: a deceleration phase followed by an acceleration stage. The crack growth in the acceleration stage is shown to follow Paris law. The model parameters obtained from uniaxial fatigue tests are shown to be sufficient for predicting the considered biaxial fatigue response.

The equivalent elastic crack procedure as applied to internally pressurised crack problems

Analysis of a specific idealised model demonstrates the limitations of the equivalent elastic crack procedure when it is applied to internally pressurised crack problems.

Kr– Lr failure assessment diagrams as applied to a flaw in the vicinity of a stress concentration: the large Lr regime

This paper is concerned with K r– L r failure assessment diagrams when they are applied to a flaw in the vicinity of a stress concentration. Earlier work by the author has highlighted the strong effect of geometrical parameters on the K r– L r failure assessment curve for such a situation. The earlier work had a particular focus on the small L r regime, which was studied via use of the equivalent elastic crack procedure. The present paper proceeds to the other extreme and focuses on the large L r regime for which, as with the small L r regime, there is a strong effect of geometrical parameters on the failure assessment curve.

Toughness and microstructural degradation at high temperature in SiC fiber-reinforced ceramics

The fracture behavior of three different SiC fiber-reinforced ceramics at ambient and elevated temperature in air is studied. The fracture properties were obtained by flexure tests on notched beams and the increase in fracture resistance with crack length was determined using an equivalent elastic crack approach. The microstructural changes during elevated temperature exposure, which could be divided into interfacial and fiber degradation, were analyzed in each material and related to the fracture micromechanisms and to the degradation in toughness and fracture resistance. Finally, the different strategies to improve the high temperature toughness of fiber-reinforced ceramics are briefly discussed.

Fracture Mechanics Parameters of Concrete An Overview

Abstract The purpose of this paper is to set a scenario for discussing and comparing different models of concrete fracture. Using as a reference the cohesive crack model, as conceived by Hillerborg, it was found that simpler models—the equivalent elastic crack, the R-Δa curve approach, or models based on two parameters such as Bazant's and Jeng-Shah's—fit inside this scheme and are hierarchically related. For each approach, the theoretical background is reviewed, the experimental aspects are discussed, and the relevance to size effect predictions is considered. This review supplies criteria for comparing the different models and provides interpretation for the measured parameters with a unified language.

Fracture mechanics parameters of concrete

The purpose of this paper is to set a scenario for discussing and comparing different models of concrete fracture. Using as a reference the cohesive crack model, as conceived by Hillerborg, it was found that simpler models—the equivalent elastic crack, the R - Δa curve approach, or models based on two parameters such as Bazant's and Jeng-Shah's—fit inside this scheme and are hierarchically related. For each approach, the theoretical background is reviewed, the experimental aspects are discussed, and the relevance to size effect predictions is considered. This review supplies criteria for comparing the different models and provides interpretation for the measured parameters with a unified language.

The equivalent elastic crack: 2. X-Y equivalences and asymptotic analysis

The use of crack growth resistance curves (R-Δa) to predict the behaviour of cracked specimens is a well-established practice for ceramics and materials. In Part 1 of this work, the authors showed that the use of R-Δa curves may imply a certain elastic equivalence between an actual specimen governed by a cohesive crack model and a virtual equivalent linear elastic specimen. Part 1 included the analysis of certain classes of equivalences (P-Y equivalences) where the loads acting on the actual and on the equivalent specimen were forced to be equal. This paper analyzes more general equivalences in which two arbitrary variables X and Y are forced to be identical in the actual and in the equivalent specimens. In particular, the J-CTOD equivalence and the size-effect-based equivalence put forward by Bazant are analyzed. The first part of the paper deals with the bases and general applicability of these equivalences. The second part presents the results of asymptotic analyses intended to assess the applicability of the equivalences to specimens of relatively large size.

Asymptotic analysis of a cohesive crack: 1. Theoretical background

A method to analyze the evolution of a cohesive crack, particularly appropriate for asymptotic analysis, is presented. Detailed descriptions of the zeroth order and first order asymptotic approaches are given and from these results the far field equivalent elastic crack theorem is derived. An analytically soluble example, the Griffith crack, and a simple model, the Dugdale model, are used to exemplify the results.