Size effect on fracture toughness of freestanding copper nano-films

We conducted fracture toughness experiments on freestanding copper films with thicknesses ranging from about 800 to 100 nm deposited by electron beam evaporation to elucidate the size effect on fracture toughness in the nano- or submicron-scale. It was found that initially, the crack propagated stably under loading, and then the crack propagation rate rapidly increased, resulting in unstable fracture. The fracture toughness K_C was estimated on the basis of the R-curve concept to be 7.81 ± 1.22 MPa m^1/2 for the 800-nm-thick film, 6.63 ± 1.05 MPa m^1/2 for the 500-nm-thick film and 2.34 ± 0.54 MPa m^1/2 for the 100-nm-thick film. Thus, a clear size effect was observed. The fracture surface suggested that the crack underwent large plastic deformation in the thicker 800-nm and 500-nm films, whereas it propagated with highly localized plastic deformation in the thinner 100-nm film. This size effect in fracture toughness might be related to a transition in deformation and fracture morphology near the crack tip.     

Temperature dependence of fracture toughness J IC and ductility for BCC materials in the transition region

The temperature dependence of ductility, strength and fracture toughness for a BCC material undergoing predominantly linear elastic behavior at low temperatures and elastic-plastic behavior at higher temperatures is examined. A model, based on ductile fracture mechanisms involving void nucleation followed by cavity growth and void coalescence, is developed to relate the fracture toughness parameter J _IC with temperature. Two general equations for linear elastic and elastic plastic regimes of J _IC versus temperature T, are obtained. Applications of this model to experimental data obtained on a carbon steel show that J IC varies with T ² at low temperatures and with T at higher temperatures, thus defining a transition temperature.     

Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects

This paper presents a study on fracture of materials at microscale (∼1 μm) by the strain gradient theory (Fleck and Hutchinson, 1993; Fleck et al., 1994). For remotely imposed classical K fields, the full-field solutions are obtained analytically or numerically for elastic and elastic-plastic materials with strain gradient effects. The analytical elastic full-field solution shows that stresses ahead of a crack tip are significantly higher than their counterparts in the classical K fields. The sizes of dominance zones for mode I and mode II near-tip asymptotic fields are 0.3l and 0.5l,while strain gradient effects are observed within land 2l to the crack tip, respectively, where l is the intrinsic material length in strain gradient theory and is on the order of microns in strain gradient plasticity (Fleck et al., 1994; Nix and Gao, 1998; Stolken and Evans, 1997). The Dugdale–Barenblatt type plasticity model is obtained to provide an estimation of plastic zone size for mode II fracture in materials with strain grain effects. The finite element method is used to investigate the small-scale-yielding solution for an elastic-power law hardening solid. It is found that the size of the dominance zone for the near-tip asymptotic field is the intrinsic material lengthl. For mode II fracture under the small-scale-yielding condition, transition from the remote classical K _IIfield to the near-tip asymptotic field in strain gradient plasticity goes through the HRR field only when K _IIis relatively large such that the plastic zone size is much larger than the intrinsic material length l. For mode I fracture under small-scale-yielding condition, however, transition from the remote classical K _I field to the near-tip asymptotic field in strain gradient plasticity does not go through the HRR field, but via a plastic zone. This revised version was published online in July 2006 with corrections to the Cover Date.     

R-curve behavior in notched beam tests of rocks

The R-curve for sandstone is obtained from the load-crack mouth opening response of notched specimens subjected to three-point-bending. This approach is used to analyze the fracture behavior under monotonic and cyclic loading. The asymptotic limit of the R-curve compares well with the fracture toughness determined through an effective crack model. The analysis of the relaxation observed before the unloading-reloading cycles in the cyclic tests leads to the conclusion that the fracture toughness remains practically constant while the crack propagates slightly during the load drop.     

Damage zone around crack tip and fracture toughness of rubber-modified epoxy resin under mixed-mode conditions

Damage zones that form around crack tips before the onset of fracture provide significant data for evaluating the fracture behavior of polymeric materials. The size of the damage zone correlates closely with the fracture toughness of the resin. In this study, we investigate the relationship between the fracture toughness and damage zone size around crack tips of a rubber-modified epoxy resin under mixed-mode conditions. The fracture toughness, G_C, based on the energy release rate, is measured using an end-notched circle type (ENC) specimen. The deformation of rubber particles in the damage zones is also observed using an optical microscope. The results show that the fracture toughness, G_C, of the rubber-modified epoxy resin is closely related to the area of the damage zone. In the specimen with a loading angle of 30°, the rubber particles were deformed ellipsoidally due to the difference between the first and second principal stresses.     

Modeling the brittle–ductile transition in ferritic steels: dislocation simulations

We present a model for the brittle–ductile transition in ferritic steels based on two dimensional discrete dislocation simulations of crack-tip plasticity. The sum of elastic fields of the crack and the emitted dislocations defines an elasto–plastic crack field. Effects of crack-tip blunting of the macrocrack are included in the simulations. The plastic zone characteristics are found to be in agreement with continuum models, with the added advantage that the hardening behavior comes out naturally in our model. The present model is composed of a macrocrack with microcracks ahead of it in its crack-plane. These microcracks represent potential fracture sites at internal inhomogeneities, such as brittle precipitates. Dislocations that are emitted from the crack-tip account for plasticity. When the tensile stress along the crack plane attains a critical value σ _F over a distance fracture is assumed to take place. The brittle–ductile transition curve is obtained by determining the fracture toughness at various temperatures. Factors that contribute to the sharp upturn in fracture toughness with increasing temperature are found to be: the increase in dislocations mobility, and the decrease in tensile stress ahead of the macrocrack tip due to increase in blunting, and the slight increase in fracture stress of microcracks due to increase in plasticity at the microcrack. The model not only predicts the sharp increase in fracture toughness near the brittle–ductile transition temperature but also predicts the limiting temperature above which valid fracture toughness values cannot be estimated; which should correspond to the ductile regime. The obtained results are in reasonable agreement when compared with the existing experimental data.     

Prediction of fracture toughness for heat-resistant steels considering specimen dimensions. Report 3. Brittle fracture

A model has been developed for predicting fracture toughness characteristics on the lower shelf of the fracture toughness temperature dependence considering the influence of specimen dimensions. The model is based on the physicomechanical model of brittle fracture governed by microcleavage critical stress which in its turn depends on plastic prestraining. This model is also based on the assumption that the change in the body dimensions influences fracture toughness as much as the change in the stress state caused by it, which is evaluated by the introduced parameter χ, and affects plastic deformation preceding brittle fracture by cleavage. According to the given model, in some cases an increase in specimen thickness can lead only to a shift of the brittle-to-ductile transition temperature, and in other cases to both a shift of the brittle-to-ductile transition temperature and a decrease in the critical SIF value at which this transition occurs. The paper also presents further elaboration of the probabilistic approach to the prediction of size effect for the cases of brittle fracture based on Weibull's three-parametric distribution. A distinctive feature of this approach is the determination of one of the parameters of the cumulative distribution function, namely, the parameter of location K_{c min} , directly in the experiment and not by conventional statistical methods. This reduces appreciably the requirements for the size of the sample and simplifies the body of mathematics. In this case, the parameter of location K_{c min} corresponds to the critical SIF value during the first unstable jump of a fatigue crack K _fc ⁽¹⁾ which, according to a large number of experimental data, is minimal and the most invariant among all other critical SIF values obtained experimentally. Translated from Problemy Prochnosti, No. 2, pp. 21–31, March–April, 1997.     

Experimental T33-stress formulation of test specimen thickness effect on fracture toughness in the transition temperature region

This paper describes a study of the test specimen thickness effect on fracture toughness of a material, in the transition temperature region, for CT specimens. In addition we studied the specimen thickness effect on the T₃₃-stress (the out-of-plane non-singular term in the series of elastic crack-tip stress fields), expecting that T₃₃-stress affected the crack-tip triaxiality and thus constraint in the out-of-plane direction. Finally, an experimental expression for the thickness effect on the fracture toughness using T₃₃-stress is proposed for 0.55% carbon steel S55C. In addition to the fact that T₃₃ (which was negative) seemed to show an upper bound for large B/W, these results indicate the possibility of improving the existing methods for correlating fracture toughness obtained by test specimen with the toughness of actual cracks found in the structure, using T₃₃-stress.     

On the application of the damage work density as a new initiation criterion for ductile fracture

In this paper the ‘damage work’ proposed by Chaouadi et al. is used to formulate an energy crack initiation criterion to describe ductile crack initiation. The traditional assessment of structural integrity by the J-integral, a property of elastic-plastic fracture mechanics is compared. Two free-cutting and one structural steel are investigated. The measured values for the critical damage work density at initiation W_di are compared with values for copper and RPV steel. As the fracture mechanical approach is limited to sharp cracks in the material (high-constraint stress state) the present damage mechanics approach is regarded as important as a more general concept closer to reality. While old void growth models of damage mechanics cannot formulate a simple criterion for crack initiation the applied damage work reaches a constant value at initiation W_di which is independent of the stress state during the deformation process. We recommend W_di as a material property of toughness for testing and engineering purposes.     

An integrated approach based on acoustic emission and mechanical information to evaluate the delamination fracture toughness at mode I in composite laminate

This paper addresses a new method based on the combination of mechanical behavior and acoustic emission (AE) information of composite materials during mode I delamination. The method is based on a special purpose function, called sentry function, which is defined as the logarithm of the ratio between mechanical energy and acoustic energy (f = Ln(E_s/E_a)). The sentry function is used to study the delamination process and to evaluate the delamination fracture toughness in mode I. The relationship between cumulative fracture toughness energy release rate (G_I) and the integral of the sentry function during crack propagation showed a transition point with two sensitive regions below and above it. This behavior can be followed to obtain the critical strain energy release rate value (G_Ic). Results obtained by means of the sentry function are compared with results obtained by a methodology proposed by other authors.     

Fracture assessment of damaged square hollow section (SHS) K-joint using BS7910:2005

A pre-cracked square hollow section K-joint was tested under static loads up to failure. It is found that the load-displacement curves are in good agreement with the finite element results. Ductile tearing was observed to initiate from the crack front parallel to the chord side wall where fracture toughness is smaller. Using plastic collapse load obtained via twice elastic compliance technique and fracture toughness obtained from crack tip opening displacement, the two fracture parameters K_r and L_r are plotted on the standard failure assessment diagram. It shows a conservative assessment for the cracked K-joint subjected to brace end axial loads.     

The jump-like crack growth model, the estimation of fracture energy and JR curve

In this paper the jump-like crack growth model for monotonic loading is applied to re-examine both the onset of crack growth and process of stable crack growth. In the former case the fracture energy associated with a new surface creation is estimated and the in-plane constraint influence on this quantity is examined using the J-A₂ approach. In the later case the formula to compute the J-resistance curve is re-examined and compared with the one known from the standards. In the analysis the plane strain model of a structural element made of elastic-plastic material is assumed.     

Early age fracture properties of microstructurally-designed mortars

This paper compares the fracture properties as well as crack initiation and propagation of real and equivalent mortars. The development of the elastic modulus, tensile strength, and fracture energy at different hydration stages were determined by inverse analysis of load-displacement curves obtained by the compact tension test (CTT). Further, the impact of the moisture content on the aforementioned material properties was also tested on oven-dried equivalent mortars. Digital image correlation (DIC) was used to follow the crack initiation and propagation.The elastic modulus, tensile strength, and fracture energy support the validity of the equivalent mortars approach. The load-displacement curves obtained by the CTT were also compared to those simulated by finite element method showing excellent correlations. DIC revealed the formation of similar crack patterns at comparable load levels between the two mortars. At early age, the moisture content has a considerable influence on the tensile strength and the fracture energy.     

The fracture mechanics of finite crack extension

This paper describes a modification to the traditional Griffith energy balance as used in linear elastic fracture mechanics (LEFM). The modification involves using a finite amount of crack extension (Δa) instead of an infinitesimal extension (da) when calculating the energy release rate. We propose to call this method finite fracture mechanics (FFM). This leads to a change in the Griffith equation for brittle fracture, introducing a new term Δa/2: we denote this length as L and assume that it is a material constant. This modification is extremely useful because it allows LEFM to be used to make predictions in two situations in which it is normally invalid: short cracks and notches. It is shown that accurate predictions can be made of both brittle fracture and fatigue behaviour for short cracks and notches in a range of different materials. The value of L can be expressed as a function of two other material constants: the fracture toughness K_c (or threshold ΔK_th in the case of fatigue) and an inherent strength parameter σ₀. For the particular cases of fatigue-limit prediction in metals and brittle fracture in ceramics, it is shown that σ₀ coincides directly with the ultimate tensile strength (or, in fatigue, the fatigue limit), as measured on plain, unnotched specimens. For brittle fracture in polymers and metals, in which larger amounts of plasticity precede fracture, the approach can still be used but σ₀ takes on a different value, higher than the plain-specimen strength, which can be found from experimental data. Predictions can be made very easily for any problem in which the stress intensity factor, K is known as a function of crack length. Furthermore, it is shown that the predictions of this method, FFM, are similar to those of a method known as the line method (LM) in which failure is predicted based on the average stress along a line drawn ahead of the crack or notch.     

Limitations of elastic definitions in Al-Si-Mg cast alloys with enhanced plasticity: linear elastic fracture mechanics versus elastic-plastic fracture mechanics

Linear elastic fracture mechanics describes the fracture behavior of materials and components that respond elastically under loading. This approach is valuable and accurate for the continuum analysis of crack growth in brittle and high strength materials; however it introduces increasing inaccuracies for low-strength/high-ductility alloys (particularly low-carbon steels and light metal alloys). In the case of ductile alloys, different degrees of plastic deformation precede and accompany crack initiation and propagation, and a non-linear ductile fracture mechanics approach better characterizes the fatigue and fracture behavior under elastic-plastic conditions.To delineate plasticity effects in upper Region II and Region III of crack growth an analysis comparing linear elastic stress intensity factor ranges (ΔK_el) with crack tip plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) is presented. To compute plasticity corrected stress intensity factor ranges (ΔK_pl), a new relationship for plastic zone size determination was developed taking into account effects of plane-strain and plane-stress conditions (“combo plastic zone”). In addition, for the upper part of the fatigue crack growth curve, elastic-plastic (cyclic J based) stress intensity factor ranges (ΔK_J) were computed from load-displacement records and compared to plasticity corrected stress intensity factor ranges (ΔK_pl). A new cyclic J analysis was designed to compute elastic-plastic stress intensity factor ranges (ΔK_J) by determining cumulative plastic damage from load-displacement records captured in load-control (K-control) fatigue crack growth tests. The cyclic J analysis provides the true fatigue crack growth behavior of the material. A methodology to evaluate the lower and upper bound fracture toughness of the material (J_IC and J_max) directly from fatigue crack growth test data (ΔK_FT(JIC) and ΔK_FT(Jmax)) was developed and validated using static fracture toughness test results. The value of ΔK_FT(JIC) (and implicitly J_IC) is determined by comparing the plasticity corrected elastic fatigue crack growth curve with the elastic-plastic fatigue crack growth curve. A most relevant finding is that plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) are in remarkably good agreement with cyclic J analysis results (ΔK_J), and provide accurate plasticity corrections up to a ΔK corresponding to J_IC (i.e. ΔK_FT(JIC)). Towards the end of the fatigue crack growth test (above ΔK_FT(JIC)) when plasticity is accompanied by significant tearing, the cyclic J analysis provides a more accurate way to capture the true behavior of the material and determine ΔK_FT(Jmax). A procedure to decouple and partition plasticity and tearing effects on crack growth rates is given.Three cast Al-Si-Mg alloys with different levels of ductility, provided by different Si contents and heat treatments (T61 and T4) are evaluated, and the effects of crack tip plasticity on fatigue crack growth are assessed. Fatigue crack growth tests were conducted at a constant stress ratio, R = 0.1, using compact tension specimens.     

Mixed-mode interfacial fracture toughness for thermal barrier coating

A new interfacial fracture test method was developed for measuring the mixed-mode interfacial fracture toughness of thermal barrier coated material over a wide range of loading phase angles. The principle of this developed method is based on peeling the coating from the substrate due to compressive loading to the coating edge, as forming a shear loading to the interface, and slinging loading such as beam bending, as normal loading to the interface. The complete closed form of the energy release rate and associated complex stress intensity factor for our testing method is shown. An yttria stabilized zirconia (YSZ) coating, which was sprayed thermally on Ni-based superalloy, was tested using the testing device developed here.The results showed that the energy release rate for the coating-interfacial crack increased with loading phase angle, which is defined by tan⁻¹ for a ratio of stress intensity factor K₂ to K₁. It was noticed that the interfacial energy release rate increasing with mode II loading could be mainly associated with the contact shielding effect due to crack surface roughness rubbing together.     

Interlaminar and intralaminar fracture modes in 0/90 cross-ply glass/epoxy laminate

Delamination of a cross-ply 0/90 glass fibre-reinforced composite laminate with an epoxy-phenol matrix was studied using a double cantilever beam test. Fracture toughness was determined by measurement of bend angle of the cantilever beams. Results obtained with this method were in agreement with those from conventional compliance and area methods. Two different fracture modes were observed: interlaminar and intralaminar. In the interlaminar fracture mode, crack jumps in the space between two neighbouring 0° and 90° plies were observed. With the interlaminar fracture mode, during crack initiation G_Ic decreased with crack length. Intralaminar fracture mode consisted of the gradual growth of a crack through a 0° ply. Fibres bridging the opposite sides of the crack were observed in this case, and fracture toughness G_Ic did not change with crack length. G_Ic (420 J m⁻²) at intralaminar fracture mode was approximately twice that at interlaminar fracture mode (220 J m⁻²). The difference in fracture toughness was explained by the dissipation of energy by fibres bridging the opposite sides of the crack at intralaminar fracture mode.     

The fracture mechanics of surface layer removal

A general solution is given for the removal of a surface layer either by a sharp tool or by a peeling force. The creation of the separated layer is treated as a fracture process and analysed using an energy approach. For the cutting case frictional work at the tool-layer interface is included. Elastic cutting is first analysed and then that involving plastic bending which results in the layer curling. The transition from elastic to elastic-plastic bending occurs when \fracs_y²h2EG_c{\frac{\sigma_{y}^{2}h}{2EG_{c}}} . For small values of this parameter and high cutting angles there is a further transition to plastic shearing in the layer which leads to lower cutting forces. It is shown that both in this, and in some bending cases, there can be touching of the tool at the fracture which changes the nature of the process. The peeling case is shown to be a special, zero friction, version of cutting in which both elastic and elastic-plastic bending can occur. However plastic shearing is shown not to occur as it is energetically less favourable. Some comments on crack veering in high thickness cuts from mode II effects are made. Such deviations in direction can lead to none steady propagation.     

Development of a reference strain approach for assessment of fracture response of reeled pipelines

As a result of recent increase in exploitation of hydrocarbon resources in harsher environments and also installation techniques which utilize the materials plastic deformation capacity, accurate assessment of fracture response of pipelines subject to large plastic strains (e.g., typical of reeled pipes) has attracted particular interest nowadays. In this paper, an approach, based on the evaluation of the J-integral, is developed for assessing the integrity of such pipelines, manifested in a model of a pipeline with a circumferential part-through crack subjected to plastic bending. The proposed approach is an extension of the reference strain method developed earlier by other researchers, and takes advantage of the displacement controlled loading nature in such pipes (thus being suitable for Strain Based Design methodologies), and the resulting high strain levels, which often cause fracture response of the material in the plastic regime. The developed formulation relates the fracture response of the pipe (in terms of the non-dimensionalized J-integral) as a linear function of the axial strain in the pipe at its uncracked state. A series of 300 3D nonlinear finite element models using the ABAQUS software were analyzed in preparation of the equation that could assess the fracture response of such pipes with great accuracy. The resulting equation, calibrated by the finite element results, can predict the fracture response of pipes with a maximum error of 2% for a practical uncracked material strain range of 1.5% ? ε_unc ? 4%.