Measurement of Specific Fracture Energy and Surface Tension of Brittle Materials in Powder Form

Studies of the influence of specimen geometry and size–effect on the K _R –curves and the related fracture parameters were carried out by the authors (Kumar and Barai 2008b). The present paper is a supplementary contribution and reports interesting results related to the effect of the loading condition and size–effect studies on the K _R –curves associated with the cohesive stress distribution for complete fracture process, the double–K fracture parameters, the CTOD–curves and the process zone length using two different loading conditions (i.e., three–point bending test and four–point bending test). The laboratory size specimen with initial–notch length/depth ratios 0.3 and 0.5 are considered in the work. The load–crack opening displacement curves for these loading conditions are obtained using well known version of fictitious crack model.     

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.     

Probabilistic prediction of the crack resistance of nuclear pressure-vessel steels on the basis of a local approach. Part 2

On the basis of a new local probabilistic criterion of brittle fracture, a local criterion of ductile fracture proposed by the authors earlier, and the obtained approximate solution of the problem of stress-strain state near the crack tip, we develop a probabilistic model for the prediction of the crack resistance of pressure-vessel steels. The model enables one to predict the dependence of K _Ic on temperature for any given probability of brittle fracture and the influence of the thickness of the specimen on K _Ic. Bu using this model, we can also describe the temperature range of the brittle-ductile transition. The results of numerical calculations are compared with the experimental data for 15Kh2MFA pressure-vessel steel. It is shown that the proposed model fairly well describes the spread in the experimental data on the crack resistance of this type of steel. TsNII KM “Prometei,” St. Petersburg, Russia. Translated from Problemy Prochnosti, No. 2, pp. 5–22, March–April, 1999.     

Effect of rubber particle size on the impact tensile fracture behavior of MBS resin with a bimodal particle size distribution

We performed impact tensile fracture experiments on methylmethacrylate–butadiene–styrene (MBS) resin with small and large particles in a bimodal size distribution, and examined the effects of particle size on fracture behavior by fixing the total rubber content (28 wt%) and the small particle size (about 140 nm), and varying the size of large particles (about 490 nm or 670 nm). Dynamic load P′ and displacement δ′ of single-edge-cracked specimens were measured using a Piezo sensor and a high-speed extensometer, respectively. A P′−δ′ diagram was used to determine external work U _ex applied to the specimen, elastic energy E _e stored in the specimen, and fracture energy E _f for creating a new fracture surface A _s. Energy release rate was then estimated using G _f = E _f/A _s. Values of G _f were correlated with fracture loads and mean crack velocity v _m determined from load and time relationships. We then examined the effect of particle size on G _f and v _m, and results indicated that particle size plays an important role in changing the values of G _f and v _m.     

Hardness properties and microscopic investigation of crack–crystal interaction in SiO<Subscript>2</Subscript>–MgO–Al<Subscript>2</Subscript>O<Subscript>3</Subscript>–K<Subscript>2</Subscript>O–B<Subscript>2</Subscript>O<Subscript>3</Subscript>–F glass ceramic system

In view of the potential engineering applications requiring machinability and wear resistance, the present work focuses to evaluate hardness property and to understand the damage behavior of some selected glass–ceramics having different crystal morphologies with SiO₂–MgO–Al₂O₃–K₂O–B₂O₃–F composition, using static micro-indentation tests as well as dynamic scratch tests, respectively. Vickers hardness of up to 5.5 GPa has been measured in glass–ceramics containing plate like mica crystals. Scratch tests at a high load of 50 N in artificial saliva were carried out in order to simulate the crack–microstructure interaction during real-time abrasion wear and machining operation. The experimental observations indicate that the novel “spherulitic-dendritic” shaped crystals, similar to the plate like crystals, have the potential to hinder the scratching induced crack propagation. In particular, such potential of the ‘spherulitic-dendritic’ crystals become more effective due to the larger interfacial area with the glass matrix as well as the dendritic structure of each mica plate, which helps in crack deflection and crack blunting, to a larger extent. While modest damage tolerant behavior is observed in case of ‘spherulitic-dendritic’ crystal containing material, severe brittle fracture of plate like crystals were noted, when both were scratched at 50 N load.     

Crack growth in the process of unstable brittle fracture

We suggest a model of the transition of a crack from stable to unstable propagation with subsequent growth in the supercritical state. By using the energy balance equation for propagating cracks, we estimate the specific surface energy and crack growth rate under the conditions of unstable propagation. Unstable propagation starts with a rate which is lower than the maximum crack growth rate in the material. The specific surface energy and kinetics of the crack undergoing unstable fracture depend both on the maximum crack growth rate in the material and on the initial rate υ₀ of its unstable propagation. The rate υ₀ is largely determined by the critical size of the crack (critical stresses). Blagonravov Institute of Mechanical Engineering, Russian Academy of Sciences, Moscow. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 6, pp. 81–86, November–December, 1996.     

Poly(ether ether ketone) with pendent methyl groups as a toughening agent for amine cured DGEBA epoxy resin

A diglycidyl ether of bisphenol-A (DGEBA) epoxy resin was modified with poly(ether ether ketone) with pendent methyl groups (PEEKM). PEEKM was synthesised from methyl hydroquinone and 4,4′-difluorobenzophenone and characterised. Blends of epoxy resin and PEEKM were prepared by melt blending. The blends were transparent in the uncured state and gave single composition dependent T _g. The T _g-composition behaviour of the uncured blends has been studied using Gordon–Taylor, Kelley–Bueche and Fox equations. The scanning electron micrographs of extracted fracture surfaces revealed that reaction induced phase separation occurred in the blends. Cocontinuous morphology was obtained in blends containing 15 phr PEEKM. Two glass transition peaks corresponding to epoxy rich and thermoplastic rich phases were observed in the dynamic mechanical spectrum of the blends. The crosslink density of the blends calculated from dynamic mechanical analysis was less than that of unmodified epoxy resin. The tensile strength, flexural strength and modulus were comparable to that of the unmodified epoxy resin. It was found from fracture toughness measurements that PEEKM is an effective toughener for DDS cured epoxy resin. Fifteen phr PEEKM having cocontinuous morphology exhibited maximum increase in fracture toughness. The increase in fracture toughness was due to crack path deflection, crack pinning, crack bridging by dispersed PEEKM and local plastic deformation of the matrix. The exceptional increase in fracture toughness of 15 phr blend was attributed to the cocontinuous morphology of the blend. Finally it was observed that the thermal stability of epoxy resin was not affected by the addition of PEEKM.     

A time-discrete model for dynamic fracture based on crack regularization

We propose a discrete time model for dynamic fracture based on crack regularization. The advantages of our approach are threefold: first, our regularization of the crack set has been rigorously shown to converge to the correct sharp-interface energy Ambrosio and Tortorelli (Comm. Pure Appl. Math., 43(8): 999–1036 (1990); Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992); second, our condition for crack growth, based on Griffith’s criterion, matches that of quasi-static settings Bourdin (Interfaces Free Bound 9(3): 411–430, 2007) where Griffith originally stated his criterion; third, solutions to our model converge, as the time-step tends to zero, to solutions of the correct continuous time model Larsen (Math Models Methods Appl Sci 20:1021–1048, 2010). Furthermore, in implementing this model, we naturally recover several features, such as the elastic wave speed as an upper bound on crack speed, and crack branching for sufficiently rapid boundary displacements. We conclude by comparing our approach to so-called “phase-field” ones. In particular, we explain why phase-field approaches are good for approximating free boundaries, but not the free discontinuity sets that model fracture.     

The Extended Range of Reaction-layer Fatigue Susceptibility of Polycrystalline Silicon Thin Films

A linear elastic fracture mechanics analysis of a silicon dioxide-polycrystalline silicon (SiO₂–Si) bimaterial system was performed to assess the vulnerability of micron-scale silicon structures, such as microelectromechanical systems, to fatigue in ambient air. Previous research has shown that fatigue of silicon films is due to a “reaction-layer fatigue” process where silicon structural films fail due to the sequential, mechanically induced thickening and environmentally assisted cracking of the silicon dioxide reaction layer which forms on the surface upon exposure to air. This work specifically considered the stability of a crack reaching the SiO₂–Si interface. This analysis revealed a significant overestimate in the oxide thicknesses susceptible to reaction-layer fatigue reported in our previous studies. Instead, a surface oxide layer as thin as 15 nm may activate this fatigue mechanism for a polycrystalline silicon thin film whose fracture strength exceeds 5 GPa.     

Fracture parameters of ferroelectric ceramics connected with the roughness of the edges of the crack and its deflection

A numerical experiment and a model of the region of contact of the edges of a crack are used as the basis for evaluating different parameters of fracture for two type of ferroelectric ceramics—PbTiO₃ and BaTiO₃—obtained by hot pressing. Consideration is given to features associated with inhibition of anomalous grain growth, deviation of the crack from a straight line due to structural nonuniformity, and the roughness of the fracture surface. The effect of roughness is independent of the parameter that characterizes inhibition of grain growth and causes the results for both ceramics to be close to one another. Translated from Problemy Prochnosti, No. 1, pp. 113–120, January–February, 1997.     

Damage modelling of adhesively bonded joints

A cohesive zone model (CZM) has been used in conjunction with both elastic and elasto– plastic continuum behaviour to predict the response of a mixed mode flexure and three different lap shear joints, all manufactured with the same adhesive. It was found that, for a specific dissipated CZM energy (Γ₀) there was a range of CZM tripping tractions (σ_u) that gave a fairly constant failure load. A value of σ_u below this range gave rise to global damage throughout the bonded region before any crack propagation initiated. A value above this range gave rise to a discontinuous process zone, which resulted in failure loads that were strongly dependent on σ_u. A discontinuous process zone gives rise to mesh dependent results. The CZM parameters used in the predictions were determined from the experimental fracture mechanics specimen test data. When damage initiated, a deviation from the linear load–displacement curve was observed. The value for σ _uwas determined by identifying the magnitude that gave rise to the experimentally observed deviation. The CZM energy (Γ ₀) was then obtained by correlating the simulated load-crack length response with corresponding experimental data. The R-curve behaviour seen with increasing crack length was successfully simulated when adhesive plasticity was included in the constitutive model of the adhesive layer. This was also seen to enhance the prediction of the lap shear specimens. Excellent correlation was found between the experimental and predicted joint strengths.     

Prediction of fracture toughness for heat-resistant steels considering specimen dimensions. Report 2. Ductile fracture

A model of ductile failure of a body with a crack has been developed which enables predicting fracture toughness on the upper shelf of the fracture toughness temperature dependence taking into account the influence of the stress state. The model is based on the physical-mechanical model of ductile failure which is controlled by the critical value εf reached by plastic strain at the crack tip ε _i ^ρ . In this case it is assumed that both the ε _i ^ρ value, which precedes the crack growth onset by the mechanism of pore coalescence, and the critical strain εf are functions of specific stress state parameters, namely: the critical strain is a function of the stress state triaxiality σ _m /σ _n (σ _m is the hydrostatic stress, σ _i is the stress intensity), and ε _i ^ρ is a function of the parameter χ introduced, which is an explicit function of all three principal local stresses in the process zone at the crack tip and which defines the degree to which the stress state approaches the plane strain conditions for a body of specified thickness. The model developed has two modifications one of which enables predicting fracture toughness of large-size bodies from the results of testing only small cylindrical specimens without cracks (smooth and with a circular recess) and the other from the results of testing small cylindrical specimens and small specimens with a crack. Translated from Problemy Prochnosti, No. 2, pp. 5–19, March–April, 1997.     

Improving the fracture toughness of MgO–Al2O3–SiO2 glass/molybdenum composites by the microdispersion of flaky molybdenum particles

The flake-forming behaviour of powders of molybdenum, niobium, nickel, BS 316 S 12, Ni–17Cr–6Al–0.6Y, iron, titanium and Ti–6Al–4V, using a wet ball mill, was investigated. MgO–Al₂O₃–SiO₂ (MAS) glass composites reinforced with these flaked particles were fabricated, and improvements in flexural strength evaluated. The MAS glass composites reinforced with flaky metallic particles such as molybdenum, niobium, iron, nickel and Ni–17Cr–6Al–0.6Y, showed an improvement. The effect of molybdenum particle size on the flake-forming behaviour of molybdenum, flexural strength and fracture toughness of MAS glass/molybdenum composites, were investigated. The flake-forming behaviour shows a high degree of dependence on molybdenum particle size and, upto a size of 32 μm, becomes conspicuous with increasing particle size. At 32 μm, the aspect ratio reaches a value of 17 and, above 32 μm, flake forming saturates. Fracture toughness is closely related to flake-forming behaviour and the more marked the flake forming, the greater is the increase in fracture toughness. A composite of MAS glass with flaky molybdenum particles has a greater improvement effect on fracture toughness than composites with SiC whiskers, SiC platelets or ZrO₂ particles. This is closely linked to plastic deformation of the flaky metallic particles at the crack tip at the time of fracture. This revised version was published online in November 2006 with corrections to the Cover Date.     

Effect of composition on fracture behavior of polypropylene–wollastonite–polyolefin elastomer system

The fracture behavior of polypropylene (PP)–wollastonite–polyolefin elastomer (POE) in the mixed mode region was studied using the essential work of fracture (EWF) method. The relationship between the microstructure and the fracture parameters was analyzed. The effect of wollastonite content on the essential work of fracture and the work of plastic deformation was discussed. The energy dissipation during a double-edge-notched tension (DENT) test was calculated with the EWF method. It was found in the mixed mode region that σ_n increases with shortening of the ligament length region as plastic constraint effect rises and variation of the specific total work of fracture with ligament length was still reasonably linear within the mixed mode region. With increasing wollastonite content, w _e (specific essential work of fracture) increases, while the βw _p (specific non-essential work of fracture) decreases. The measurements of energy dissipation show that improvement in the fracture toughness of PP–wollastonite–POE is mainly due to the increase in crack propagation resistance during the necking and tearing processes after yielding, while the plastic deformation capability of the material depends mainly on the properties of fracture behavior before yielding. It is also found that the impact strength of the material decreases with increasing wollastonite content. However, the composition with high impact strength has lower specific essential energy of fracture and lower long-term fracture resistance, indicating that EWF is a better indicator of long-term fracture properties than the impact strength. DSC results show that the presence of wollastonite hinders crystallization of the PP.     

On the “local approach” to the brittle fracture of structural materials

We consider the principles and sources of an important scientific direction in fracture mechanics-the so-called “local approach” to the problem of fracture of materials. This approach provides a possibility to describe their crack resistance based on conventional mechanical properties and opens up a direct way to determining the limiting state of structural elements. As the first example of the application of this approach, we present the principal propositions of the two-parameter K _μ-model of fracture, which relates the “global” K _Ic and “local” ρ_c and σ_c parameters via the yield condition on the contour of a small plastic zone near the tip of a crack. For the last decades, this direction was developed substantially and extended to the regions of quasi-brittle and ductile fracture. We also present the basic statements of this approach that were corroborated for the last decades. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 2, pp. 39–44, March–April, 2006.     

Determination of the effective mode-I toughness of a sinusoidal interface between two elastic solids

A finite element model of crack propagation along a sinusoidal interface with amplitude A and wavelength λ between identical elastic materials is presented. Interface decohesion is modeled with the Xu and Needleman (J Mech Phys Solid 42(9):1397, 1994) cohesive traction–separation law. Ancillary calculations using linear elastic fracture mechanics theory were used to explain some aspects of stable and unstable crack growth that could not be directly attained from the cohesive model. For small aspect ratios of the sinusoidal interface (A/λ ≤ 0.25), we have used the analytical Cotterell–Rice (Intl J Fract 16:155–169, 1980) approximation leading to a closed-form expression of the effective toughness, K _Ic , given by where is the work of separation, E is Young’s modulus, and ν is Poisson’s ratio. For A/λ > 0.25, both the cohesive zone model and numerical J-integral estimates of crack tip stress intensity factors suggest the following linear relationship: Parametric studies show that the length of the cohesive zone does not significantly influence K _Ic , although it strongly influences the behavior of the crack between the initiation of stable crack growth and the onset of unstable fracture. An erratum to this article can be found at     

Effect of material plasticity on fatigue crack propagation under complex stress state

The maximum energy release rate criterion, i.e., G _max criterion, is commonly used for crack propagation analysis. However, this fracture criterion is based on the elastic macroscopic strength of materials. In the present investigation, a modification has been made to G _max criterion to implement the consideration of the plastic strain energy. This criterion is extended to study the fatigue crack growth characteristics of mixed mode cracks in steel pipes. To predict crack propagation due to fatigue loads, a new elasto-plastic energy model is presented. This new model includes the effects of material properties like strain hardening exponent n, yield strength σ_y and fracture toughness and stress intensity factor ranges. The results obtained are compared with those obtained using the commonly employed crack growth law and the experimental data.     

Effect of crack closure on the experimentally measured characteristics of cyclic crack-growth resistance of steels

On the basis of the proposed approach and an original procedure for determination of the current value of crack tip opening displacement, we obtain new characteristics of cyclic crack-growth resistance for which the effect of crack closure disappears. We experimentally discovered a phenomenon of natural pulsations of crack closure with constantly decreasing amplitude and duration of pulses to the minimum stress intensity factor K _min in a cycle for any positive asymmetry of cycles. We construct a model of damped pulsations of crack closure depending on the maximum stress intensity factor K _max in a cycle and asymmetry of cycles. This model is used to show that the kinetic diagrams of fatigue fracture are invariant for any positive asymmetry of cycles within the limits of the presence of the effect of crack closure. The observed effect of asymmetry quantitatively takes into account the range of pulsating stress intensity factors of crack closure ΔK _cl ^op in the range of stress intensity factors ΔK in a cycle. Practical experience shows that the procedure for quantitative investigation of the kinetics of growth of fatigue cracks extensively used in linear fracture mechanics and based on analysis of the coefficients of crack opening displacement U and γ and functional dependences of the stress intensity factor of crack opening displacement K_op on k _max is inconsistent. We propose a new experimental approach to the qualitative analysis of the kinetics of growth of fatigue cracks based on evaluation of the effect of crack closure and expressed in terms of the effective range of stress intensity factors ΔK_eff in the case of its simple determination from the kinetic diagrams of fatigue fracture together with K_op. We hope that the accumulated experimental data and the proposed model of damping pulsations of crack closure will lay a foundation for a new understanding of the natural resistance of materials to fracture under cyclic loading. Scientific and Engineering Center of Materials-Science Support of the Production and Certification of Equipment of Nuclear Power Plants at the Institute for Problems of Strength of the National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 161 – 171, March – April, 1998.     

Statistical description of the effect of Zr addition on the behavior of microcracks in Cu–6Ni–2Mn–2Sn–2Al Alloy

As possible substitutes for high-strength Cu–Be alloys, Cu–6Ni–2Mn–2Sn–2Al alloys have been developed. To clarify the physical background of the effect of trace Zr on the fatigue strength of such alloys, the initiation and propagation behavior of a major crack that led to the fracture of the tested specimens was monitored. When the stress amplitude was less than σ _a = 350 MPa, the fatigue life of the alloys with Zr was about 2–2.5 times larger than that of the alloy without Zr. When σ _a > 350 MPa, the effect of Zr addition on the fatigue life dramatically decreases as the stress amplitude increases. The increased fatigue life due to Zr addition resulted from an enhancement of the crack initiation life and microcrack growth life. The enhanced crack initiation life was mainly attributed to the strengthening of grain boundaries due to the precipitation of SnZr compounds. A statistical analysis of the behavior of multiple cracks was made to quantitatively evaluate the scatter in fatigue behavior. The statistical analysis supported the conclusions obtained from the behavior of a major crack.     

The mechanism of the failure of the dispersion-strengthened Cu–Al<Subscript>2</Subscript>O<Subscript>3</Subscript> nanosystem

The method of “in situ tensile testing in SEM” is suitable for investigations of fracture mechanisms because it enables to observe and document deformation processes directly, thank to which the initiation and development of plastic deformation and fracture can be reliably described. The deformation and fracture mechanisms of Cu–Al₂O₃ nanomaterials with 5 vol.% of Al₂O₃ phase has been analyzed using technique of the “in situ tensile testing in SEM.” It has been shown that the deformation process causes break-up of large Al₂O₃ particles and decohesion of smaller ones. The final fracture path is influenced also by boundaries of nanograins, through which the principal crack propagates towards the sample exterior surface. Based on the experimental observations, a model of damage and/or fracture mechanisms has been proposed.