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.     

A simplified expression for low cleavage probability calculation

This article describes an analysis made to develop a simplified stress-based criterion for brittle fracture focussed on the lowest probability of failure. For that, on the basis of fine numerical interpretation of two series of fracture-tests on 16MND5 reactor vessel steel, a number of variables were proposed:

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A stress threshold σ_th below which cleavage cannot occur. This stress is determined by testing on notch tensile specimen at low temperature.

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A minimum toughness K_min(T) required to make a crack unstable. The originality is here to consider this parameter depends on temperature.

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For K_J > K_min(T), a volume susceptible to cleavage, defined as the volume of material subjected to stress exceeding the threshold stress and noted V_th, representative of the fracture probability.

These three variables are explained in the article then used to establish a tentative criterion for expressing the risk of brittle fracture, in the presence or absence of a crack.     

Influence of statistical and constraint loss size effects on cleavage fracture toughness in the transition - A model based analysis

A database derived from tests on specimens with a large range of ligament (b) and thickness (B) dimensions was systematically analyzed to evaluate constraint loss and statistical size effects on cleavage fracture toughness. The objectives were to: (1) decouple size effects related to constraint loss, mediated by b and B, from those arising from statistical effects, primarily associated with B; and, (2) develop procedures to transfer toughness data to different conditions of constraint and B. The toughness database for a Shoreham pressure vessel steel plate, tested at a common set of conditions, was described in a companion paper. Quantification of constraint loss was based on an independently calibrated 3D finite-element critical stress-area, σ^∗-[K_Jm/K_Jc], model. The measured toughness data, K_Jm, were first adjusted using computed [K_Jm/K_Jc] constraint loss factors to the corresponding values for small scale yielding conditions, K_Jc=K_Jm/[K_Jm/K_Jc]. The K_Jc were then statistically adjusted to a K_Jr for a reference B_r = 25.4 mm. The B adjustment was based on a critically stressed volume criterion, modified to account for a minimum toughness, K_min, consistent with modest modifications of the ASTM E 1921 Standard procedure. The combined σ^∗-[K_Jm/K_Jc]-K_min adjustment procedure was applied to the Shoreham b − B database, producing a homogeneous population of K_Jr data, generally within the expected scatter. The analysis suggests that: (1) there may be a maximum B beyond which statistical size effects diminish, and (2) constraint loss in the three-point bend specimens begins at a relatively low deformation level. A corresponding analysis, based on a Weibull stress, σ_w-[K_Jm/K_Jc]-K_min, adjustment procedure, yielded similar, but somewhat less satisfactory, results. The optimized adjustment procedure was also applied to other K_Jm data for the Shoreham plate from this study, as well as a large database taken from the literature. The population of 489K_Jr data points, covering an enormous range of specimen sizes, geometries and test temperatures, was found to be consistent with the same master curve T₀ = −84 °C derived from the b − B database. Thus, calibrated micromechanical models can be used to treat size and geometry effects on K_Jm, facilitating using small specimens and data transfer to predict the fracture limits of structures.     

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.     

Evaluation of R-curve behavior of ceramic-metal functionally graded materials by stable crack growth

This paper deals with the fracture toughness and R-curve behavior of ceramic-metal functionally graded materials (FGMs). A possibility of stable crack growth in a three-point-bending specimen is examined based on the driving force and resistance for crack growth in FGMs, and the distribution of fracture toughness or R-curve behavior is evaluated on FGMs fabricated by powder metallurgy using partially stabilized zirconia (PSZ) and stainless steel (SUS 304). The materials have a functionally graded surface layer (FGM layer) with a thickness of 1 mm or 2 mm on a SUS 304 substrate. Three-point-bending tests are carried out on a rectangular specimen with a very short crack in the ceramics surface. On the three-point-bending test, a crack is initiated from a short pre-crack in unstable manner, and then it propagates in stable manner through the FGM layer with an increase in the applied load. From the relationship between applied load and crack length during the stable crack growth in the FGM layer, the fracture toughness is evaluated. The fracture toughness increases with an increase in a volume fraction of SUS 304 phase.     

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.     

Morphology and crack toughness behaviour of nanostructured block copolymer/homopolymer blends

The crack initiation and propagation behaviour of styrene-butadiene (SB) star block copolymer/polystyrene blends (ST3/PS) forming PS-rich and polybutadiene (PB)-rich nanosized domains by self-assembling have been investigated using the essential-work-of-fracture (EWF) approach. Three morphological transitions have been observed, which are crucial to understand the crack toughness behaviour: (i) 0-30 wt.% PS homopolymer: A co-continuous domain structure of PS-rich and PB-rich domains has been observed. For PS homopolymer fraction (?_PS) < 10 wt.% PS homopolymer (i.e. only pure ST3) the rubbery PB-rich phase forms the major phase and for ?_PS > 10 wt.% the glassy PS-rich phase. (ii) At 40-60 wt.% PS homopolymer, a layer-like morphology is formed where the PS-rich layer thickness is ?50 nm, a critical dimension, which is crucial for understanding the ductile-to-semiductile transition. (iii) For 80 wt.% PS homopolymer, PS-rich phase starts to form the matrix combined with a transition from shear stress dominated (shear yielding) to normal stress dominated behaviour (PS-like crazes). The co-continuous morphology at 20 wt.% and 30 wt.% PS is capable of improving toughness of block copolymers, demonstrated by the observed maximum in the non-essential work of fracture and thus explaining a new way of toughening of polymers while retaining high transparency. The correspondence between the ductile-to-semiductile transition and the change in the shape of plastic zone from circular to elliptical as revealed from strain field analysis could be clearly reaffirmed by the observed transition from shear to normal force induced deformation in the fractured surface analysis of these blends. The conceptual correspondence of βw_p and w_e with T_J and δ_0.2 respectively reveal that resistance against crack propagation (βw_p and T_J) is morphology sensitive while the resistance against crack initiation (w_e and δ_0.2) is matrix sensitive.     

Influence of specimen size and microstructure on the fracture toughness of a martensitic stainless steel

The fracture toughness of alloy HT-9,² a martensitic stainless steel under consideration for fast reactor and fusion reactor applications, was determined from circular compact tension specimens using the multi-specimen R-curve method. Specimens with thicknesses of 11.94, 7.62 and 2.54mm and widths of 23.88 and 11.94 mm were tested to investigate the effects of specimen size on fracture toughness. The test results obtained from all specimens are in good agreement and thickness requirements for a valid J_1c test are satisfied. The experiment indicates that small specimens of HT-9 may be used for post-irradiation fracture toughness testing.Fractographic examination of the fracture surfaces reveals that fracture in HT-9 is significantly influenced by delta ferrite stringers present in the material. The fracture surface examination and crack opening displacement measurements for specimens tested at various temperatures are consistent with the temperature dependence of the J_1c results.     

Tunneling radial cracks in layered structures from contact loading

Glass/epoxy laminates glued onto a compliant substrate are indented with a hard ball. The damage is characterized by a set of transverse cracks which pop out from the subsurface of the glass layers due to flexure and propagate stably in the radial direction with load in a bell-shape front under a diminishing stress field. Compliant interlayers, even extremely thin ones, are effective in inhibiting crossover fracture. This leads to crack tunneling and crack multiplication in the hard layers, which enhances energy dissipation and reduces the spread of damage relative to the basic bilayer configuration. The experiments show that the fracture in a given layer is well approximated by a power-law relation of the form c^3/2K_C/P = δ, where P, c, and K_C are the indentation load, crack length and fracture toughness, in that order, and δ an implicit function of the layer position and material and geometric variables, derived with the aid of available tunnel crack solutions.The model specimen studied provides a useful insight into the fracture behavior of natural, biological and synthetic layered structures from concentrated loading. The analysis shows that the crack arrest capability of a thin interlayer increases in proportion to the modulus misfit ratio between the layer and interlayer, and that the spread of radial cracks in a laminate of given thickness reduces in proportion to n^1/3, where n is the number layers in the laminate.     

J integral solution for semi-elliptical surface crack in high density poly-ethylene pipe under bending

The goal of this work is to analyse the severity of semi-elliptical crack defects and to study the degree of damage in the poly-ethylene pipe in bending during the crack propagation. The semi-elliptical cracks are considered in this work located in different position in the wall of the pipe. The three finite element method based on the computation of the J integral was used to analyse the fracture behaviour of these structures. The effect of the position, shape and size of the crack on the J integral values was highlighted. The effects of strain rate and the temperature on the J integral values were also examined. The obtained results show that the strain rates have a strong influence on the J integral values especially for circumferential crack at higher bending moment. However, the energy for circumferential crack is more important compared to axial crack. The effect of the depth of the crack becomes important when the ratio (a/t) reaches a critical value of 0.6 (a/t = 0.6), especially when the ratio a/c is weak (semi-elliptical crack, a/c = 0.2) where the J integral values becomes independently of the crack depth, this conclusion is opposite to the above for the poly-ethylene pipe subjected to internal pressure. We recall finally, that the temperature effect on circumferential cracks behaviour is more important compared to the axial cracks at critical crack size (a/c = 0.2 and a/t = 0.6). It is also shown that in the wall of pipe, the internal cracks are more dangerous than the external cracks.     

An alternative formulation of the Ritchie-Knott-Rice local fracture criterion

In the paper an alternative formulation of the RKR local fracture criterion is proposed. It is based on the features of the stress distribution in front of a blunted crack in an elastic-plastic material. The stress distribution is computed using the finite strain option in the finite element method. It is postulated that the opening stress in front of the crack should be greater than the critical one, σ_c, over the distance l ? l_c, where l_c is considered as a material parameter. The hypothesis is applied to estimate the influence of the in-plane constraint on fracture toughness. New formulas to compute the critical value of the J-integral are derived both for the small scale yielding and large plastic deformations in front of the crack. The results obtained are compared with the Sumpter and Forbes experimental results and with the O’Dowd analytical formula concerning the J_c = J_c(J_IC,Q) relation.     

Method of constraint loss correction of CTOD fracture toughness for fracture assessment of steel components

This paper presents a procedure for transferring the CTOD fracture toughness obtained from laboratory specimens to an equivalent CTOD for structural components, taking constraint loss into account. The Weibull stress criterion is applied to correct the CTOD for constraint loss, which leads to an equivalent CTOD ratio, β, defined as β = δ/δ_WP, where δ and δ_WP are CTODs of the standard fracture toughness specimen and the structural component, respectively, at the same level of the Weibull stress. The CTOD ratio β is intended to apply to the fracture assessment of ferritic steel components to stress levels beyond small-scale yielding. Nomographs are given to determine the β-value as a function of the crack type and size in the component, the yield-to-tensile ratio of the material and the Weibull shape parameter m. Examples of the fracture assessment using β are shown within the context of a failure assessment diagram (FAD). An excessive conservatism observed in the conventional procedure is reduced reasonably by applying the equivalent CTOD ratio, β.     

Crack opening displacements of Vickers indentation cracks

Vickers indentation cracks are an appropriate tool to determine the crack-tip toughness K_I0 and, possibly, the bridging relation of ceramics with an R-curve behaviour from the total crack opening displacements. Two contributions to the total crack opening displacement field are addressed. First, the residual stresses occurring in the uncracked body are considered and then, the contact stresses generated by preventing crack face penetration are computed. The COD solution resulting from the superposition of residual and contact displacements is given and an analytical expression is provided. Near-tip displacements are represented by the first terms of series expansions. As an example of application, an evaluation of the actual stress intensity factor is presented for a window glass 1 h after Vickers indentation.     

Fracture toughness of unidirectional fiber-metal laminates: crack orientation effect

Fiber-metal laminates (FMLs) are structural composites developed for aeronautical applications. The application of FMLs to structures demands a deep knowledge of a wide set of properties, including fracture toughness. The objective of this work was to evaluate the effect of crack orientation on the fracture toughness (critical J-integral and CTOD δ₅) of unidirectional FMLs. Small C(T) and SE(B) specimens with notches parallel and perpendicular to the fibers direction were tested. A study of the relation and equivalence between J_C and δ_5C, which heavily depend on the yield strength and on the stress state, was performed motivated by apparently contradictory experimental results. These results can be explained by the direction-dependent yielding properties of unidirectional FMLs. The best overall equivalence between J_C and δ_5C was obtained considering plane stress state and using the effective yield strength, both for unidirectional FMLs notched parallel and perpendicular to the fibers direction.     

The fracture toughness of Ni/Al2O3 laminates by digital image correlation II: Bridging-stresses and R-curve models

A composite of metal and brittle ceramic layers have increased fracture toughness as compared to ceramic monoliths. The property controlling the toughness enhancement is the, ‘bridging-stress’, exerted by the ductile phase astride the crack in the ceramic. This bridging-stress is a function of the crack-opening displacement (COD) which is a function of the size of the crack and the position along its profile. Depending on the accuracy of estimation of the bridging-stress, the modeled R-curve and experimental one match. In this study, a weight function based approach to generate the R-curve is reported and compared with the experimental results for Al₂O₃/Ni multilayer laminates.     

Experimental and numerical investigation of thickness effect on ductile fracture toughness of steel alloy sheets

Effect of thickness on ductile fracture toughness of plates made of steel alloy GOST 08Ch22N6T is investigated experimentally. Multiple specimen tests for determining fracture toughness have been conducted using compact tension (CT) specimens with thicknesses of 1.25, 1.64 and 4.06 mm according to standard test method ASTM E813. The results show the significant effect of thickness on fracture toughness. It is observed that in low thickness, J_c increases with the thickness increase until it reaches a maximum; however, further increase in the thickness causes the J_c-value to decrease. Two-dimensional finite element analysis is also performed to reproduce the experimental results. The comparison shows a very good agreement.     

Predicting fracture and fatigue crack growth properties using tensile properties

The safe-life assessment of components requires information such as the plane stress (K_c), plane strain (K_Ic), part-through fracture toughness (K_Ie), and the fatigue crack growth rate properties. A proposed parametric/theoretical approach, based on an extended Griffith theory is used to derive fracture toughness properties and generate fatigue crack growth rate data for a range of alloys. The simplicity of the concept is based on the use of basic, and in most cases available, uniaxial stress-strain material properties data to derive material fracture toughness values. However since the methodology is in part based on an empirical relationship a wide ranging validation with actual data is required. This paper uses steel, aluminum and titanium based alloys from a pedigree database to quantify material properties sensitivity to the predictions for K_Ic and K_c and the subsequent estimation of ΔK_th threshold and the Paris constants, C and n values. A sensitivity analysis using experimental scatter bounds show the range of da/dN predictions can be achieved. It is found K_Ic/ΔK_th ratios designated as α has a range of 5-25 irrespective of tensile ductility, ε_f, and is insensitive to it. The value of ΔK_th for all the alloys considered was found to be proportional to the final elongation, ε_f, and an empirical relationship describing ΔK_th as a function of ε_f was established. Furthermore it is suggested that, with the knowledge of appropriate tensile properties and the estimated range of K_Ic/ΔK_th ratios for the different alloys applying this method could be an appropriate tool that can be used to conservatively predict fracture and fatigue in similar alloy categories. Thus helping to reduce costs and optimize the number of experimental tests needed for alloy characterizations.     

Fracture of quasi-isotropic composite sheets with sharp notches

Continuous fibre composites are materials that exhibit rather linear elastic deformation behaviour: suggesting brittleness and notch sensitivity. However, notched composites may sustain significant mechanical load. The notch resistance of composites is investigated on quasi-isotropic composite sheets with sharp crack like notches. This allows the use of analytic solutions of the stress field around a crack in a similar way as is used for linear elastic fracture mechanics (LEFM) in homogeneous isotropic solids. Similar to the small scale yielding boundary condition in fracture mechanics, applied on homogeneous isotropic solids, a small-scale non-linear damage condition should be fulfilled for valid LEFM application on quasi-isotropic composites. Indeed, it appeared to be possible to define critical stress intensity factors (K_1c) for the quasi-isotropic composite. Moreover, K_1c values can quantitatively be related to laminate parameters and to the related damage and deformation processes occurring in a small near crack tip zone with intense non-linearity and strain gradients in the thickness direction. Before the final explosive fracture occurred, stable crack growth was observed. This could be described with R-curves, as done for homogeneous metal sheet specimens. Indeed, also in this case, the R-curves were identical, independent of the length of the initial crack-like notch. The R-curves can be estimated adopting a crack-bridging model. Crack growth occurs at the notch tip in the 0° plies. The other plies bridge the fractured 0° plies. The fracture mechanisms, determining the K_1c-values and the shape of the R-curve, are quite different for composites and metals. Yet, the method of fracture mechanics, well established for metals, can obviously also be applied to quasi-isotropic composites.     

The relation between the fracture toughness of plates and the thickness of the shear lips

Single edge-cracked plate tension fracture toughness specimens were machined from Comsteel 3140 and an Atlas steel of approximately the same grade. Specimens were fatigue-precracked, quenched. and then tempered at various temperatures. Tensile specimens were machined from the same stock and heat treated with the fracture toughness specimens. Yield stress (0.2% offset), σ_y, was determined from the various tensile specimens, and fracture toughness. K_c, from the fracture toughness specimens. The average thickness of the shear lips, t₈, was measured on the fracture toughness specimens. Over the range of relative fracture toughness (0.12 ? K_c/σ_y ? 0.55) it was found that t₈ could be related to K_c and σ_y through t₈ = 0.18 (K_c/σ_y)^1·9. A rationale for this result is that t₈ is approximately equal to the radius of the plane stress plastic zone at and near the free surface of the plate specimens, r_y, which is estimated from analysis as $\text{r}{}_{\mathrm{y}}\text{= (}\text{1}\text{2}\text{π)}$ (K_c/σ_y)² = 0.16 (K_c/σ_y)². The application of the result to failure analysis is discussed.     

Influence of microcracking on crack-tip toughness of alumina

Al₂O₃-samples with different grain sizes were produced and the crack-tip toughness K_I0, also called intrinsic fracture toughness, was determined using two different measurement techniques. It appeared that K_I0 depends strongly on specimen grain size with the mechanistic link provided by the increase of microcrack density with grain size. Results obtained using measurements of crack opening displacements (COD) lie considerably lower than values based on a method using bending tests of pre-notched bend bars. It is suggested that the latter method relies on the use of a notch with a microcrack, which in fact is different than the behavior of a long crack alone.