Analysis of a mixed mode fracture specimen: the asymmetric double cantilever beam

An asymmetric double cantilever beam (ADCB) is a simple but effective specimen for the measurement of polymer/polymer and polymer/non-polymer bimaterial interface fracture toughness. In order to characterize fully the bimaterial interface strength, and to control the crack trajectory, the critical energy release rate, G _c, and the phase angle, , of the applied stress field as functions of loading and geometry of the specimen should be obtained. For most practical cases, has to be evaluated numerically. In this work, a boundary element analysis is carried out to obtain G and for the ADCB specimen at different material and geometry combinations. An expression for the energy release rate, G, based on Kanninen's beam on elastic foundation model is compared with the numerical results. Limitations on the use of the ADCB specimen are also discussed.     

Measurement of the heat of fusion of tungsten by a microsecond-resolution transient technique

A microsecond-resolution pulse-heating technique was used for the measurement of the heat of fusion of tungsten. The method is based on rapid (100 to 125s) resistive self-heating of a specimen by a high-current pulse from a capacitor discharge system and measuring current through the specimen and voltage across the specimen as functions of time. Melting of a specimen is manifested by changes in the slope of the electrical resistance versus time function. The time integral of the power absorbed by a specimen during melting yields the heat of fusion. Measurements gave a value of 48.7 kJ · mol^–1 for the heat of fusion of tungsten with an estimated maximum uncertainty of ±6%. The electrical resistivity of solid and liquid tungsten at its melting temperature was also measured.Paper presented at the Third Workshop on Subsecond Thermophysics, September 17–18, 1992, Graz, Austria.     

Experimental and numerical studies in model composites Part II: Numerical results

A numerical approach using the boundary element method for strength and toughness of a composite with long aligned fibers is reported. The three-dimensional problem is reduced to a two-dimensional one by substituting the rows of fibers with layers of appropriate width and elastic constants. The configuration examined in this work is a compact tension specimen similar to that used in the experimental studies (Part I, [1]). The experimental results on strength and apparent fracture toughness are compared with the numerical results. For the particular geometry and fiber spacing, the numerical simulations are in good agreement with the experimental findings, i.e. the composite's strength _A , scales with the fiber spacing , in the form of _A . Using the numerical formalism a number of different geometries was examined. The simulations suggested that if the external specimen characteristics remain the same and the fiber spacing in the direction of crack advance is changed, then the strength of the composite specimen can be expressed _A . If the fiber spacing varies in both directions simultaneously, for a certain range of , it can be considered that the composite's strength _A , is proportional to _A .     

On the determination of the cohesive zone parameters for the modeling of micro-ductile crack growth in thick specimens

The paper deals with the determination of the cohesive zone parameters (separation energy, , and cohesive strength, T _max) for the 3D finite element modeling of the micro-ductile crack growth in thick, smooth-sided compact tension specimens made of a low-strength steel. Since the cohesive zone parameters depend, in general, on the local constraint conditions around the crack tip, their values will vary along the crack front and with crack extension. The experimental determination of the separation energy via automated fracture surface analysis is not accurate enough. The basic idea is, therefore, to estimate the cohesive zone parameters, and T _max, by fitting the simulated distribution of the local crack extension values along the crack front to the experimental data of a multi-specimen J _IC-test. Furthermore, the influence of the cohesive zone parameters on the crack growth behavior is investigated. The point of crack growth initiation is determined only by the magnitude of . Both and T _max affect the crack growth rate (or the crack growth resistance), but the influence of the cohesive strength is much stronger than that of the separation energy. It turns out that T _max as well as vary along the crack front. In the center of the specimen, where plane strain conditions prevail, the separation energy is lower and the cohesive strength is higher than at the side-surface.     

Fracture behavior of ultra-fine grained steel bar under dynamic loading

Ultra-fine grained steel bars were recently developed by thermo-mechanical controlled rolling with rapid cooling for increasing the strength of low carbon and low alloy steels. The developed steels are characterized by fine ferrite grains of less than 1 m and high strength as a result of grain refinement. However, their correlations between tensile properties and impact behavior are not well understood. In this paper, impact absorbed energy (E _p) and dynamic fracture toughness (J _Id) were used to evaluate the dynamic fracture behavior of the ultra-fine grained steels, and the fracture mechanisms were also investigated. For the ultra-fine grained steels, tensile stress-strain curve was shown to be correlated with the impact curve of load vs. time, and to be related to the dynamic fracture toughness. The steel with large ferrite grains, small ferrite grain colony and martensite was found to have a good combination of strength and toughness.     

Correlation between Weibull moduli for tensile and bending strength of brittle ceramics: a numerical simulation analysis based on a three-parameter Weibull distribution

A numerical simulation was designed and performed to produce uniaxial tensile strength and three-point bending strength data. It was assumed that the specimen could be divided into many small units of volume for which the tensile strength followed a three-parameter Weibull distribution, characterized by the parameters m, ₀ and _u. Statistical analysis of the strength data showed that the variation of the bending strength could be characterized by the explicit form of the three-parameter Weibull function as deduced from only the tensile test data. A strong correlation was found to exist between the Weibull modulus, m _E,B, estimated for the bending strength and that for tensile strength, m _E,T. The difference between m _E,B and m _E,T is dependent on m but independent of the ratio of ₀ to _u.     

Crack propagation studies in brittle materials

An analysis was made of cantilever beam specimens used for crack propagation studies, Included in this analysis were the effects of a plastic zone at the crack tip, beam rotation, and the viscoelastic response of the material. This analysis showed that application of a constant bending moment to the specimen rather than a constant load provides a test in which the strain energy release rate,G, is independent of crack length. Other advantages of this test configuration are that corrections for shear or beam rotation effects are not necessary. Results of this test on both glass and ceramics are reported.List of symbols a crack length - A cross-sectional area of beam - b total thickness of specimen - d deflection of loading arm - E elastic modulus of material - E ₁ dynamic modulus - E ₂ transient response modulus - G shear modulus of material - G strain energy release rate - G _vE strain energy release rate of viscoelastic material - h half height of specimen - I moment of inertia of cantilever beam =bh ^/12 - k modulus of elastic foundation - K stress intensity factor - L distance from point of load application to fulcrum of loading arm - L distance from point at which arm deflection is measured to fulcrum - M applied bending moment - P force applied to beam - r length of plastic zone - t thickness of specimen at groove - T force applied to loading arm - u displacement of beam - V crack velocity - w half height of groove - W stored elastic energy - characteristic length of beam on elastic foundation - reciprocal of the characteristic length of beam - rotation of beam - X viscoelastic creep compliance function - time - inherent opening distance as defined by Wnuk [10] - _y yield strength of material - Poisson's ratio     

Strength improvement of cemented carbides by hot isostatic pressing (HIP)

HIP treatment after sintering increases the strength of the investigated cemented carbide alloy by a factor of two whereas hardness, fracture toughness, and work of fracture remain unchanged. HIP does not affect the microstructural parameters of the carbide skeleton and the binder phase, but the residual pores are eliminated entirely. Failure of both the as-sintered and post-densified material occurs by a pure Griffith mechanism. The strength-flaw size relationship is established experimentally and is shown to obey exactly Griffith's basic strength equation. The strength is controlled by the largest microstructural defects, i.e. pores in the as-sintered material, and coarse WC grains and inclusions in the HIP-treated specimens.Nomenclature a size of the fracture initiating flaw - a _th theoretical flaw size - b sample thickness - c length of the pre-crack - C contiguity of the carbide phase - D _WC mean carbide grain size - E Young's modulus - F fracture surface - G _IC critical energy release rate - h sample height - H _V Vicker's hardness - K _IC critical stress intensity factor - l span length - l _Co mean Co layer thickness - m Weibull parameter - P load - r _p1 radius of the plastic zone - R crack resistance - S probability of failure - U fracture energy - X relative crack length - Y K-calibration - _F specific work of fracture - _I specific energy for fracture initiation - spread of grain size distribution - compliance of the pre-cracked specimen - ₀ compliance of the uncracked specimen - v Poisson's ratio - fracture stress - ₀ maximum stress - _B bend strength - _Y Yield strength - _eff maximum local stress     

The failure of fibre composites and adhesively bonded fibre composites under high rates of test

The dynamic effects which are commonly encountered during high-rate DCB tests with fibre composite and adhesively bonded fibre composite arms have been studied in detail. This paper, Part II of the series, follows Part I, which described the experimental aspects of the high-rate testing. Part III will report the results from mode II and mixed-mode I/II tests on the fibre-composite materials.Nomenclature a crack length - a ₀ initial crack length - a crack speed - ä crack acceleration - c longitudinal wave speed - h thickness of single arm of test specimen - p crack length perturbation (i.e. the measured value of the crack length minus the value predic ted by steady-state theory) - p crack velocity perturbation - crack acceleration perturbation - t time - t ₀ time taken for crack to initiate during the mode I test - u ₀ load-line vertical displacement of single arm of test specimen (/2 in Part I) - u(x) vertical displacement of specimen at distance x from the load-line - u(x) vertical displacement rate of specimen at distance x from the load-line - x distance along the test specimen from the load-line - A constant relating the steady state crack length to root time - B width of specimen - C compliance of the specimen (u ₀/P) - E ₁₁ axial modulus of the fibre-composite beam - G mode I energy release rate - G _Ic mode I critical energy release rate or fracture toughness - G ₁ half the value of G _Ic during steady-state propagation (i.e. calculated for half the beam as shown in Fig. 1) - G ₂ half the value of G _Ic at crack initiation - P end load applied to specimen - U _ext external work done - U _s strain energy - U _k kinetic energy - V velocity of a single arm of test specimen (i.e. half the measured test velocity) - dynamic term, governed by the ratio of the energy to initiate versus that to propagate a crack - _I mode I crack shear deflection and root rotation correction term - crack length correction term, evaluated by the negative intercept on the a versus t ^1/2 plot - dynamic term controlling the form of the computed perturbations - Poisson's ratio for the fibre-composite beams density of the fibre-composite beams - time, normalized by the initiation time, t ₀ and thus equivalent to (t/t ₀) - values of at which crack arrest occurs. n = 1,2,3... - ratio of distance along beam to crack length (x/a)     

Unit fracture energy of certain metals and alloys in thermal shock

An energy-based approach is used to examine the problem of cleavage fracture. The approach is based on a comparison of the store of energy in the specimen and the work done in its fracture. A series of approximate calculations is performed with the use of the program UP-OK to determine the critical unit fracture energy _* necessary to complete the fracture work and fragment the specimen over a period within the submicrosecond range. The specimens were made of titanium (VT1-0), brass (L62, L63), bronze (BrB2, BrB2M), and molybdenum (MCh-1) and had a thickness within the range 0.01–1 mm. They were subjected to a brief pulse of x radiation from a nuclear explosion. The estimates that were obtained showed that the critical unit fracture energy _* is not a material constant but instead depends on the loading conditions. It increases with an increase in the time of action of the tensile stresses in the cleaved cross section.Translated from Problemy Prochnosti, No. 6, pp. 27–32, November–December, 1996.     

J andG canalysis of the tearing of a highly ductile polymer

The use of a conventionalJ analysis to describe the ductile tearing of thin low density polyethylene sheet is described. This is a measure of the total energy required to cause fracture. The use of the current energy release rate to determine the local dissipation rate is then given and it is shown that an initiation (plane strain) and reasonably constant propagation (plane stress) values are obtained. Input energy of system - A Area of specimen - B Thickness of specimen - W Width of specimen - a Crack length - J J-integral - G _c Energy release rate - P Load - U Energy - C Compliance - M Constraint factor - _y Yield stress - u Displacement - Dimensionless factor     

Size dependence of concrete fracture energy determined by RILEM work-of-fracture method

The paper analyzes the size dependence of the fracture energy of concrete obtained according to the existing RILEM recommendation proposed by Hillerborg and based on the work-of-fracture method of Nakayama, Tattersal and Tappin, in which the energy dissipated at the fracture front is evaluated from the measured load-displacement curve. The analysis is based on the size effect law proposed by Baant, which has been shown to be applicable to the size ranges up to about 1:20 and apply in the same form for all specimen geometries. The analysis utilizes the previously developed method for calculating the R-curve from the size effect, and the load-deflection curve from the R-curve. The R-curve is dependent on the geometry of the specimen. The results show that the fracture energy according to the existing RILEM recommendation is not size-independent, as desired, but depends strongly on the specimen size. This dependence is even stronger than that of the R-curve. When the specimen size is extrapolated to infinity, the fracture energy according to the RILEM recommendation coincides with the fracture energy obtained by the size effect method. It is also found that, in fracture specimens of usual sizes, the pre-peak contribution of the work of the load to the fracture energy is relatively small. Finally, as a by-product, the analysis also verifies the fact that, in three-point bend fracture specimens, the fracture energy according to the RILEM definition is dependent on the notch depth.     

Fracture toughness and absorbed energy measurements in impact tests on brittle materials

Previous work on impact testing has shown that the energy/unit area (w) normally measured in notched impact tests is dependent on specimen geometry. A fracture mechanical analysis has now been developed to account for the observed dependence ofw on notch size. A correction factor () has been derived to accommodate notch effects and this allows for the calculation of the strain energy release-rateG directly from the measured fracture energies.Tests on PMMA have shown that corrected results are independent of specimen geometry and theG _c for PMMA has been evaluated as 1.04 × 10³ J m^–2. The experimental results show that there is an additional energy term which must be accounted for and this has been interpreted here as being due to kinetic energy losses in the specimens. A conservation of momentum analysis has allowed a realistic correction term to be calculated to include kinetic energy effects and the normalized experimental results show complete consistency between all the geometries used in the test series.It is concluded that the analysis resolves many of the difficulties associated with notched impact testing and provides for the calculation of realistic fracture toughness parameters.     

Combined weakest link and random defect model for describing strength variability in fibres

A mathematical model which describes the strength variability along the length of a fibre was developed. The model is a combination of the modified weakest link and random defect models. This combined model describes very well the strength variability data of aramid fibres.Nomenclature L Specimen length - F(s) Cumulative frequency distribution of link strengths - 1 — F(s) Survival function of a link - F _L(s) Cumulative frequency distribution of strengths of a specimen of length L - 1 — F _L(s) Survival function of a specimen of length L - s Strength variable - s ₀ Fibre defect-free strength for a random defect or combined model - s ₁, s ₂... Fibre strength at the point of a defect - s ₁, s ₂ ... Strength a fibre must have at the location of the defect to have a strength of s at the location of the defect - Length of a hypothetical link in a weakest link model - ₂, ₂ ... Defect frequencies (mean number per unit length) - v ₁, v ₂ ... Defect severities, 0 v 1 - (s) Defect frequency distribution function defined in terms of the strength at the defect - (v) Defect frequency distribution function defined in terms of the defect severity - , Defect frequency distribution parameters (Equation 14) - a, b Weibull distribution parameters (Equation 4) - P(m) Probability that m defects will occur in a given specimen length - m Number of defects occurring - ¯s Mean strength - CV Coefficient of variation of strength     

Carbon fibre-reinforced cement

This review describes fabrication processes for aligned fibre and random fibre carbonreinforced cement and links important process parameters with composite theory. The way in which the material fits into the general framework of crack constraint and matrix cracking theories is discussed. A broad survey is made of the mechanical properties, durability and dimensional stability of a variety of carbon-reinforced cement composites, and economic constraints on potential applications are considered.List of symbols b breadth of three-point bend specimen - d depth of three-point bend specimen - E _c composite Young's modulus - E _f fibre Young's modulus - E _m matrix Young's modulus - l fibre length - l _c fibre critical transfer length - l _s specimen span in three-point bend test - m Weibull modulus - r fibre radius - P applied load - V _f fibre volume fraction - V _m matrix volume fraction - x length of fibre needed to transfer load _mu V _m - x _d crack spacing in a composite with short, aligned fibres - _fu fibre ultimate strain - _mu matrix ultimate strain - _fu fibre ultimate strength - _mu matrix ultimate strength - _cu composite ultimate strength - _MOR modulus of rupture - _T tensile strength - interlaminar shear strength - _i interfacial shear strength - _m matrix work of fracture - _F work of fracture     

Elasto-plastic analysis of a mode II fracture specimen

An elasto-plastic analysis of a compact mode II fracture specimen is performed. Both an approximate approach with rigid plastic material and a more exact elasto-plastic finite element calculation are carried out. From this analysis, an -factor is determined relating the J-integral to the internal energy measured along the specimen crack faces. It is shown through the finite element computation that it is justifiable to define an -factor. With this result, it is now possible to perform tests on aluminium specimens so as to determine J _IIc.     

System for Measuring the Spectral Distribution of Normal Emissivity of Metals with Direct Current Heating

A system for measuring time variations of the normal spectral emissivity at wavelengths ranging from 0.55 to 5.3 m was developed and applied to metal specimens in vacuum and oxidizing environments in the temperature range from 780 to 1200° C. The specimen was heated to high temperatures by passing a direct current in a vacuum chamber, and the surface oxidation was controlled by a low-pressure oxidizing gas. The specimen temperature was measured by a single-band (0.9-m) radiation thermometer viewing at a cavity formed in the specimen from the rear side. The front surface of the specimen was observed by a multiband (112-wavelength) radiation thermometer to measure the normal spectral emissivity. The effective normal spectral emissivity of the specimen cavity was evaluated to be 0.94±0.05 at a wavelength of 0.9 m in comparison with a metal tube having a small blackbody hole on the rear. The measurement uncertainty of the normal spectral emissivitiy by the system was estimated to be 5 to 10% of the emissivity value in most of the interesting ranges of emissivities, temperatures, and wavelengths.     

Statistical analysis of the brittle fracture of sintered tungsten

A Weibull analysis was applied to the fracture data of sintered tungsten round bar specimens. Fracture data were obtained by performing flexural and tensile tests on these components. Two quantities were obtained which characterized the material variability and strength for each test method. The correlation between these quantities for the two test methods was found to be close.Nomenclature C least square line intercept - P _f failure probability - P _f ⁱ experimental failure probability - V specimen volume - W applied load - X horizontal coordinate of least square line - Y vertical coordinate of least square line - d specimen diameter - L distance between supporting knife edges - m Weibull modulus - v unit volume - stress - ₀ normalizing stress - _fV failure stress of specimen - _fv unit volume failure stress of specimen - _fv ^B unit volume failure stress in bending - ⁱ experimental stress - gamma function     

Fracture characterization of short glass fibre reinforced thermoplastic polyester by theJ-integral

Fracture initiation of short glass fibre reinforced thermoplastic polyester was characterized by theJ-integral measurement based on the energy release rate interpretation ofJ. The criticalJ value (J _c) is shown to be a fracture characterizing parameter for the onset of the crack initiation in the injection moulded short glass fibre reinforced composite material. TheJ _c value of the composite is estimated by be 6.0kJ m^–2. This value is in good agreement with the linear elastic strain energy release rate (G _c), since the composite exhibited a fairly linear stress-strain relationship. The estimated ratios ofJ _c to the total energy absorbed per unit uncracked area are in good agreement with the analytically obtained values after the remote energy dissipation due to fibre and matrix interaction away from the crack tip has been subtracted from the total energy.Nomenclature J J-integral - J _c The criticalJ - G Elastic strain energy release rate - G _c The criticalG - K _l Opening mode stress intensity factor - The criticalK _l - P Applied load - x Load-point displacement - B Specimen thickness - E Young's modulus - v Poisson's ratio - F Force - Y Central deflection - a/W Ratio of the crack length to the specimen width - _y Yield stress - U _t Total strain energy in loading a specimen - U _d Remotely dissipated strain energy after unloading - U _t–d U _t–U _d - _t Ratio ofJ _c toU _t per unit uncracked ligament - _t–d Ratio ofJ _c toU _t–d per unit uncracked ligament.     

The impact toughness of discontinuous boron-reinforced epoxy composites

Previous theories for the impact strength of discontinuously-reinforced composites predict that the toughness is a maximum when critical transfer length fibres are used. Experiments utilizing mini-Charpy specimens of unidirectional boron-fibre-reinforced epoxy composites have been conducted which corroborate this prediction. However, calculations of the fracture energy, based on a uniform interfacial shear stress during fibre pull-out, proved inadequate for the reinforced epoxy composites. Revisions to existing theories are presented to take into account the non-uniformity of the interfacial shear stress distribution along the fibre length and catastrophic failure of the interfacial bond.Nomenclature A _f fibre cross-sectional area - E _f fibre Young's modulus - G _m matrix shear modulus - l fibre length - L fibre pull-out length - l _c fibre critical length - r fibre radius - R half fibre centre-to-centre spacing - V _f fibre volume fraction - W mean work of fracture per unit area of specimen cross-section - x distance from fibre end - y dummy variable of integration - surface energy - strain in composite - tensile stress on fibre - _f fibre fracture strength - interfacial shear stress