An approximate solution for a plane strain hydraulic fracture that accounts for fracture toughness,fluid viscosity,and leak-off

The goal of this paper is to develop an approximate solution for a propagating plane strain hydraulic fracture, whose behavior is determined by a combined interplay of fluid viscosity, fracture toughness, and fluid leak-off. The approximation is constructed by assuming that the fracture behavior is primarily determined by the three-process (viscosity, toughness, and leak-off) multiscale tip asymptotics and the global fluid volume balance. First, the limiting regimes of propagation of the solution are considered, that can be reduced to an explicit form. Thereafter, applicability regions of the limiting solutions are investigated and transitions from one limiting solution to another are analyzed. To quantify the error of the constructed approximate solution, its predictions are compared to a reference numerical solution. Results indicate that the approximation is able to predict hydraulic fracture parameters for all limiting and transition regimes with an error of under one percent. Consequently, this development can be used to obtain a rapid solution for a plane strain hydraulic fracture with leak-off, which can be used for quick estimations of fracture geometry or as a reference solution to evaluate accuracy of more advanced hydraulic fracture simulators.     

Plane-strain propagation of a fluid-driven fracture during injection and shut-in: Asymptotics of large toughness

This paper considers the problem of plane-strain fluid-driven fracture propagating in an impermeable elastic medium under condition of large toughness or, equivalently, of low fracturing fluid viscosity. We construct an explicit solution for a fracture propagating in the toughness-dominated regime when the energy dissipated in the viscous fluid flow inside the fracture is negligibly small compared to the energy expended in fracturing the solid medium. The next order corrections in viscosity to this limiting solution are then derived, allowing the range of problem parameters corresponding to the toughness-dominated regime to be established. The first-order small viscosity (large toughness) solution is shown to provide an excellent approximation of the solution for the crack length in the wide range of the viscosity parameter. Furthermore, this solution, when combined with the first-order small-toughness solution of Garagash and Detournay [Journal of Applied Mechanics, 2005], provides a simple analytical approximation of the crack length solution in practically the entire range of viscosity (toughness). It is also shown that the established method of asymptotic expansion in small parameter is equally applicable to study other small effects (e.g., fluid inertia) on the otherwise toughness-dominated solution. A solution for the fracture evolution during shut-in (i.e., after fluid injection rate is suddenly stopped) is also obtained. This solution, which corresponds to a slowing fracture evolving towards the toughness-dominated steady state, draws attention to the possibility of substantial fracture growth after fluid injection is ceased especially under conditions when the fracture propagation during injection phase is dominated by viscous dissipation.     

Development of the Euro fracture toughness dataset

Ten European laboratories have generated the Euro fracture toughness dataset in order to provide an experimental data base sufficiently large to study specimen size and temperature effects on cleavage fracture toughness in the ductile-to-brittle fracture transition regime. The Euro fracture toughness dataset quantifies the fracture behaviour of the quenched and tempered pressure vessel steel DIN 22NiMoCr37. This material is frequently used in nuclear power plants. About 800 fracture toughness tests were performed using compact tension specimens with a size range from 1/2T to 4T.In the lower shelf temperature regime, no significant specimen size effects on cleavage fracture toughness scatter was observed. At higher temperatures, the lower tails of the toughness scatter bands are not significantly effected by the specimen size but with decreasing specimen size the toughness scatter increases due to the fact that the upper part of the scatter band is extended. The presence of a specimen size effect on fracture toughness scatter coincides with the appearance of single cleavage initiation sites at the fracture surface. At the lower shelf temperature both, cleavage initiation sites and size effects are not observed whereas at higher test temperatures both phenomena are present. The specimen size effect trends and the corresponding fracture surface morphology support a weakest-link type cleavage fracture mechanism in the ductile-to-brittle transition regime. A unique correlation between the amount of ductile tearing and cleavage fracture toughness was observed for the steel investigated. This result offers the possibility to determine cleavage fracture toughness from post-test fracture surface examinations.Due to the large number of tests and the wide range of testing conditions, the Euro fracture toughness dataset gives a comprehensive insight into specimen size effects and temperature effects on ductile-to-brittle transition fracture. The Euro fracture toughness dataset includes a large set of raw test data such as load versus load line displacement curves and raw tensile test data for deriving stress-strain curves. The Dataset can be downloaded from the internet via the address ftp://ftp.gkss.de/pub/eurodataset.     

The influence of elastic anisotropy on the propagation of fracture

The model of a propagating crack introduced by Craggs for an isotropic solid is extended to the case of general elastic anisotropy. The general theory for propagation under tensile or shear stresses is derived. As for most two dimensional problems in anisotropic elasticity, the solution involves the roots of a sixth degree polynomial so that it is necessary to proceed numerically at some stages. A computer program has been written to do this. This is used to show that the shape of the square cracks which are produced in the interior of silicon-iron by the internal pressure of electrolytic hydrogen may be due to elastic anisotropy. On this basis, predictions can be made as to the shape of cracks in other metals, in particular molybdenum, vanadium and tantalum. These metals are representative of the four combinations of fracture plane and deviation from isotropy. Explicit formulae are given for the special orientations where the sixth degree polynomial can be explicitly factorized.     

An assessment of fracture toughness in the ductile to brittle transition regime using the Euro fracture toughness dataset

The effect of specimen size on the fracture toughness of a ferritic steel in the transition regime has been investigated in a joint European Project. The project involved the testing of 25, 50, 100 and 200 mm wide compact specimens over the temperature range −154°C to 20°C with the aim of evaluating techniques for assessing the fracture toughness data.This paper evaluates the data at, or close to, the onset of stable tearing instead of at cleavage. The approach, which is applicable to structural assessment procedures, results in a temperature shift of less than 12°C between the specimen widths. The approach also enables simplified recommendations to be made for fracture toughness testing in the transition regime and the onset of upper shelf behaviour to be quantified.     

Fatigue fracture toughness of metals

The paper considers the peculiarities of fatigue crack propagation and final fracture of metals under cyclic loading. It is shown that the value of the fatigue fracture toughness of steels in an embrittled state is appreciably lower than that of the fracture toughness under static loading. A model of the transition from stable to unstable fatigue crack propagation is justified.     

Anisotropy in the fracture properties of apple flesh as investigated by crack-opening tests

The texture of apple flesh is important in assessing the eating qualities of the fruit. Texture is in turn related to the structure of the parenchyma. Crack-opening tests (wedge penetration tests and notch tensile tests) have shown the tissue to have marked anisotropy in its fracture properties. These differences can be detected by sensitive mechanical instruments and also in the mouth. The flesh of the apple is split much more easily along the fruit's radius than, for example, in a direction parallel to the fruit's tangent. This was shown with the fracture tests as well as discriminated by a taste panel. In tangential orientation the wedge, or teeth, have to penetrate to a greater distance exerting a greater force to initiate a free-running crack, and fracture toughness is about 50% greater than in radial orientation. The mechanical behaviour of apple parenchyma is directly related to its structural composition. The radially elongated intercellular spaces ease the passage of radially travelling cracks, i.e. along the direction of the spaces, and act as crack stoppers and crack deflectors to tangentially travelling cracks, i.e. at right angles to the spaces. This increases the energy requirement for crack propagation for tangential cracks hence increasing the fracture toughness in that orientation.     

Dynamic fracture of tungsten base heavy alloys

A recently developed short beam experimental technique has been applied to the characterization of the mode I dynamic fracture toughness (KId) of a commercial tungsten base heavy alloy (w/o-90W-7Ni-3Fe). The specimens were taken from a cylindrical swaged alloy bar and tested at a typical loading rate of the order of 10⁶ MPa\sqrtm/s. Three different crack orientations (one longitudinal and two radial) were investigated. The KIdvalues obtained for the three crack orientations are compared with the corresponding values obtained under quasi-static loading conditions (KIc). Our results show that the dynamic fracture of heavy alloys is both anisotropic and rate sensitive. For specimens containing radial cracks (LR, RR), the dynamic fracture toughness is higherthat its static counterpart. By contrast, for longitudinal cracks (RL), the dynamic fracture toughness is lowerthan the static one. Also, for radial cracks, both the (average) static and the dynamic fracture toughness are higher than in the longitudinal orientation. These new results about the anisotropy of the dynamic fracture toughness of the heavy alloys are reported and correlated with metallographic and fractographic examinations.     

Decrease in fracture toughness of chert by heat treatment

The fracture toughness of Ocala chert, as measured with short rod specimens and with the microhardness indenter, decreases to 60% of its original value as a result of heat treatment to 500° C while the elastic modulus increases 22%. The change in fracture toughness is associated with a transition from crack propagation around particles in the porous boundaries of densely packed zones in the chert to propagation through the zones. The transition is related to an increase in particle/particle bonding within the porous boundaries. Consolidation of a silica gel in the boundary regions, which resulted in a loss of water of 1.12% by weight, is apparently responsible for the increased bonding.     

Deformation and fracture of cork in tension

Various properties related to the deformation and fracture of cork in tension were experimentally determined, including the Young's modulus, the stress and strain at fracture and the fracture toughnessK _Ic. The transverse isotropy of cork implies that there are three independent systems of mode I crack propagation andK _Ic was measured for each. The mechanisms of deformation and fracture were identified by SEM microscope observation ofin situ deformation and of the fracture surfaces and crack paths. Two fundamental mechanisms of fracture occur: crack propagation along the lateral cell walls in non-radial tension, withK _Ic = 94±16 kPam^1/2 and crack propagation by breaking the cell walls in radial tension withK _Ic=125±14 kPam^1/2. In radial tension, local fractures that do not propagate due to crack stopping were observed which lead to serrations in the tensile curves for that direction. The strain to fracture in this direction is considerably larger than in the perpendicular (non-radial) directions.     

Cohesive zone modelling of interface fracture near flaws in adhesive joints

A cohesive zone model is suggested for modelling of interface fracture near flaws in adhesive joints. A shear-loaded adhesive joint bonded with a planar circular bond region is modelled using both the cohesive zone model and a fracture mechanical model. Results from the models show good agreement of crack propagation on the location and shape of the crack front and on the initial joint strength. Subsequently, the cohesive zone model is used to model interface fracture through a planar adhesive layer containing a periodic array of elliptical flaws. The effects of flaw shape are investigated, as well as the significance of fracture process parameters. The results from simulations of fracture in a bond containing circular flaws show that localization of crack propagation in the vicinity of a flaw has significant effect on the joint strength and crack front shape. The localization effects are highly dependent on the fracture process zone width relative to the flaw dimensions. It is also seen that with increasing fracture process zone width, the strength variation with the flaw shape decreases, however, the strength is effected over a wider range of propagation.     

The elastic stress fields of elliptic and tapered cracks with predefined shapes

The shape of a tapered crack is more alike cracks in brittle materials than an elliptical crack. The deformation and stress fields for a tapered crack are therefore estimated for hydrostatic pressure and tensional stress by applying the method of complex potentials. The stress fields for the tapered and elliptical cracks are quite similar, which suggests that the elliptical crack can be used as a model for the stress fields for cracks in general. However, the tapered crack has a larger tensional stress at the crack tip, which show that fracture propagation occur at lower applied stresses than for the elliptical crack. A tapered shape of fluid filled fractures can account for their discontinuous propagation. The discontinuous fracture propagation is observed in a large scale by volcanic eruptions where the fracture propagation generates seismic activity.     

Propagation of a penny-shaped hydraulic fracture parallel to the free-surface of an elastic half-space

In this paper we analyze the problem of a penny-shaped hydraulic fracture propagating parallel to the free-surface of an elastic half-space. The fracture is driven by an incompressible Newtonian fluid injected at a constant rate at the center of the fracture. The flow of viscous fluid in the fracture is governed by the lubrication equation, while the crack opening and the fluid pressure are related by singular integral equations. We construct two asymptotic solutions based on the assumption that either the solid has no toughness or that the fluid has no viscosity. These asymptotic solutions must be understood as corresponding to limiting cases when the energy expended in the creation of new fracture surfaces is either small or large compared to the energy dissipated in viscous flow. It is shown that the asymptotic solutions, when properly scaled, depend only on the dimensionless parameter cal R cal, the ratio of the fracture radius over the distance from the fracture to the free-surface. The scaled solutions can thus be tabulated once and for all and the dependence of the solution on time can be retrieved for specific parameters, through simple scaling and by solving an implicit equation.     

The lower bound toughness procedure applied to the Euro fracture toughness dataset

The validity of a statistical method for estimating an engineering lower bound fracture toughness in the ductile-to-brittle transition region is investigated using the Euro fracture toughness dataset generated in the European SM&T Project “Fracture Toughness of Steel in the Ductile-to-Brittle Transition Regime”. The lower bound method is based on the empirical evidence that, in the low probability regime, the cumulative failure probability function tends to be a straight line rather than a curve as is the case for Weibull distributions. The investigation demonstrates that the engineering lower bound toughness values as predicted by the method are related to a cumulative cleavage failure probability lower than 2.5%. Such bound predictions can be achieved on the basis of a small number of cleavage toughness values measured at the temperature of interest. The results confirm the validity of the method.     

A radially cracked,cylindrical fracture toughness specimen

A new fracture toughness specimen in the shape of a hollow circular cylinder has been characterized, allowing more economical and representative use of as-formed cylinders in toughness tests. Several configurations of a cylinder with radial cracks parallel to the axis of revolution are described with respect to crack mouth compliance and stress intensity factor. Specimen geometries useful for fracture toughness tests are analyzed in detail, based on results derived from finite-element codes. Results of experiments using 707S-T6 aluminum and uranium alloys of known toughness showed good agreement with the analytical results.     

Dynamic fracture mechanics—A scientific tool for the prevention of catastrophic failure

The major area of research in dynamic fracture has been the extension of the concept of static fracture toughness to predict crack arrest for a propagating crack. In this work crack propagation due to a ductile (microvoid) mechanism and cleavage (brittle) mechanism, as well as transition from one mode to another, has been analysed theoretically. Dynamic fracture toughness as a function of crack velocity has been determined. Temperature distribution near a propagating crack tip has been predicted for plane stress condition. The effect of reflected stress wave in a single edge notch specimen under transient crack growth conditions has also been analysed.     

Master curve analysis of the “Euro” fracture toughness dataset

Brittle fracture in the ductile to brittle transition regime is connected with specimen size effects and - more importantly - tremendous scatter of fracture toughness, which the technical community is currently becoming increasingly aware of. The size effects have the consequence that fracture toughness data obtained from small laboratory specimens do not directly describe the fracture behavior of real flawed structures. Intensive research has been conducted in the last decade in order to overcome these problems. Different approaches have been developed and proposed, one of the most promising being the master curve method, developed at VTT Manufacturing Technology.For validation purposes, a large nuclear grade pressure vessel forging 22NiMoCr37 (A508 Cl.2) has been extensively characterized with fracture toughness testing. The tests have been performed on standard geometry CT-specimens having thickness 12.5, 25, 50 and 100 mm. The a/W ratio is close to 0.6 for all specimens. One set of specimens had 20% side-grooves. The obtained data consists of a total of 757 results fulfilling the ESIS-P2 test method validity requirements with respect to pre-fatigue crack shape and the ASTM E-1921 pre-fatigue load. The master curve statistical analysis method is meticulously applied on the data, in order to verify the validity of the method. Based on the analysis it can be concluded that the validity of all the assumptions in the master curve method is confirmed for this material.     

Study of mode I crack dynamic propagation behaviour and rock dynamic fracture toughness by using SCT specimens

To study crack dynamic propagation behaviour and rock dynamic fracture toughness, a single cleavage triangle (SCT) specimen was proposed in this paper. By using these specimens and a drop‐weight test system, impact experiments were conducted, and the crack propagation velocity and the fracture time were measured by using crack propagation gauges. To examine the effectiveness of the SCT specimen and to predict the test results, finite difference numerical models were established by using AUTODYN code, and the simulation results showed that the crack propagation path agrees with the test results, and crack arrest phenomena could happen. Meanwhile, by using these numerical models, the crack dynamic propagation mechanism was investigated. Finite element code ABAQUS was applied in the calculation of crack dynamic stress intensity factors (SIFs) based on specimen dimension and the loading curves measured, and the curves of crack dynamic SIFs versus time were obtained. The fracture toughness (including initiation toughness and propagation toughness) was determined according to the fracture time and crack speeds measured by crack propagation gauges. The results show that the SCT specimen is applicable to the study of crack dynamic propagation behaviour and fracture toughness, and in the process of crack propagation, the propagation toughness decreases with crack propagation velocity, and the crack arrest phenomena could happen. The critical SIF of an arrest crack (or arrest toughness) was higher than the crack propagation toughness but was lower than the initiation toughness.     

Determination of dynamic rock Mode-I fracture parameters using cracked chevron notched semi-circular bend specimen

Rock dynamic fractures are common in many geophysical processes and engineering applications. Characterization of rock dynamic fracture properties such as the initiation fracture toughness, the fracture energy, and the fracture velocity, is thus of great importance in rock mechanics. A novel method is proposed in this work to measure dynamic Mode-I rock fracture parameters using a cracked chevron notched semi-circular bend (CCNSCB) specimen loaded by a split Hopkinson pressure bar (SHPB) apparatus. A strain gauge is mounted on the sample surface near the chevron notch to detect the fracture onset, and a laser gap gauge (LGG) is used to monitor the crack surface opening distance (CSOD) during the dynamic test. With dynamic force balance achieved in the tests, the stable–unstable transition of the crack propagation crack is observed and the initiation fracture toughness is calculated from the dynamic peak load. The average dynamic fracture energy as well as the fracture propagation toughness are calculated based on the first law of thermodynamics. The measured dynamic fracture properties of Laurentian granite using CCNSCB method are consistent with those reported in the literature using other methods.