Effects of material properties on the fragmentation of brittle materials

We present a fundamental investigation of the influence of material and structural parameters on the mechanics of fragmentation of brittle materials. First, we conduct a theoretical analysis (similar to Drugan’s single wave problem, Drugan, W.J. (2001), Journal of Mechanical and Physics Solids 49, 1181–1208.) and obtain closed form solutions for a problem coupling stress wave propagation and single cohesive crack growth. Expressions for a characteristic fragment size s ₀ and a characteristic strain-rate are given. Next, we use a numerical approach to analyze a realistic fragmentation process that involves multiple crack interactions. The average fragment size s is calculated for a wide variety of strain-rates and a broad range of material parameters. Finally, we derive an empirical function that relates the normalized fragment size s/s ₀ to the normalized strain-rate and that fits all of the numerical results with a single master curve.     

Experimental Observations and Computational Modeling of Fracturing in an Anisotropic Brittle Crystal (Sapphire)

The commonly used fracture criteria-maximum K_I, zero K_II, maximum hoop-stress, and maximum energy-release rate-predict similar fracture paths in isotropic materials, but not in anisotropic materials. In the general anisotropic case, the fracture path depends on the material-symmetry properties, the nature of the applied loads, and the overall geometry of the specimen. In addition, anisotropy in the material's resistance to fracturing plays a key role in defining crack initiation and its propagation path. Experiments are performed on notched specimens made from sapphire, a microscopically homogeneous and brittle single-crystal solid. The force required for fracture initiation is measured. The experimental measurements/observations are compared with the numerical results of the FEM simulations. A stress-based fracture parameter, is shown to be a good measure of the fracture criterion, where σ and E, respectively, are the tensile stress and Young's modulus in the direction normal to the cleavage plane, with surface energy γ , and R is a characteristic length, e.g., the notch radius. This parameter takes into account the effects of the surface energy of the corresponding cleavage plane, as well as the strength of the atomic bonds in the direction normal to the cleavage plane. More than two-thirds of the notched specimens fractured at the point and along a cleavage plane where A is maximum. The measurements of the applied force made it possible to quantitatively obtain a critical value for parameter A. Finally, experiments show that for the notched sapphire specimens the weakest family of cleavage planes, , are the fracture planes, although a few specimens fractured along non-cleavage planes. This revised version was published online in July 2006 with corrections to the Cover Date.     

Investigation of failure criteria for a sharp notch

In this study, a method of evaluating the static strength of a V-shaped notch based on the singular stress field at the notch tip is studied. The singular stress fields is defined by two parameters, and , which correspond to the intensities of symmetric stress field and the skew-symmetric field, respectively. Four kinds of fracture criteria are considered; two of them are based on the tensile strength σ _B and the other two are based on the fracture toughness K _IC . The usefulness of the criteria is investigated through the experimental results carried out on plane specimens of acrylic resin having a sharp notch for various notch configurations such as the opening angle, the inclined angle and the notch depth. It is shown that the criteria using stress intensity factor and the energy release rate not sensitive to the length of the virtual crack .     

Calibration of Weibull stress parameters using fracture toughness data

The Weibull stress model for cleavage fracture of ferritic steels requires calibration of two micromechanics parameters . Notched tensile bars, often used for such calibrations at lower-shelf temperatures, do not fracture in the transition region without extensive plasticity and prior ductile tearing. However, deep-notch bend and compact tension specimens tested in the transition region can provide toughness values under essentially small-scale yielding (SSY) conditions to support Weibull stress calibrations. We show analytically, and demonstrate numerically, that a nonuniqueness arises in the calibrated values, i.e., many pairs of provide equally good correlation of critical Weibull stress values with the distribution of measured (SSY) fracture toughness values. This work proposes a new calibration scheme to find which uses toughness values measured under both low and high constraint conditions at the crack front. The new procedure reveals a strong sensitivity to m and provides the necessary micromechanical values to conduct defect assessments of flawed structural components operating at or near the calibration temperature in the transition region. Results of a parameter study illustrate the expected values of m for a typical range of material flow properties and toughness levels. A specific calibration is carried out for a mild structural steel (ASTM A36). This revised version was published online in August 2006 with corrections to the Cover Date.     

A J-integral estimation method for C(T) specimens using the Common Format Equation

A new estimation scheme based on the Common Format Equation (CFE) is laid out for Compact Tension (C(T)) specimens. In this context, the CFE constraint factor Ω*, originally given only for the two limits plane stress, and plane strain, is discussed. A nonlinear finite element analysis of the behaviour of blunt notched C(T) specimens with varying crack length was performed. The specimen thickness B has been varied from 3.125 up to 25 mm. Furthermore the special cases plane stress and plane strain have been considered. Considering a linear elastic – ideal plastic material, a limit load analysis has been performed numerically from which Ω* has been obtained as a function of the ligament-to-thickness-ratio B/b. The -integral as a function of the load line displacement v has been determined for isotropic, nonlinear hardening material, where has been calculated using its definition as contour or surface integral, respectively. It is shown that if the obtained curves are normalized according to the Common Format Methodology, all curves fall approximately into one single curve. This allows to estimate J(v) curves for C(T) specimens using the CFE.     

Study on the behavior of elastic-plastic fracture under mixed I+II mode loading in aluminum alloy Ly12-J-integral analysis

The elastic-plastic fracture behavior of aluminum alloy Ly12 under mixed I+II mode loading was studied by finite element method and fracture test. A mixed mode elastic-plastic fracture criterion of J-integral was proposed by using the J-resistance curve, and the maximum fracture effective plastic strain _p max of different mixed ratios at crack tip were also calculated. The results show that(1) the initiation J-integral values of different mixed ratios have the equation

On the measurement and physical meaning of the cleavage fracture stress in steel

Elastic-plastic two-dimensional (2D) and three-dimensional (3D) finite element models (FEM) are used to analyze the stress distributions ahead of notches of four-point bending (4PB) and three-point bending (3PB) specimens with various sizes of a C-Mn steel. By accurately measuring the location of the cleavage initiation sites, the local cleavage fracture stress _f and the macroscopic cleavage fracture stress _F is accurately measured. The _f and _F measured by 2D FEM are higher than that by 3D FEM. _f values are lower than the _F, and the _f values could be predicted by _f=(0.8––1.0)_F. With increasing specimen sizes (W,B and a) and specimen widths (B) and changing loading methods (4PB and 3PB), the fracture load P _f changes considerably, but the _F and _f remain nearly constant. The stable lower boundary _F and _f values could be obtained by using notched specimens with sizes larger than the Griffiths–Owen specimen. The local cleavage fracture stress _f could be accurately used in the analysis of fracture micromechanism, and to characterize intrinsic toughness of steel. The macroscopic cleavage fracture stress _F is suggested to be a potential engineering parameter which can be used to assess fracture toughness of steel and to design engineering structure.     

On scattering of measured values of fracture toughness parameters

The values of fracture toughness K _1C of C-Mn steels and weld metal were calculated from values of COD measured in standard specimens tested at low temperatures. The analysis of scattering of 42 values of K _1C measured in normalized C-Mn steel showed that the highest value of K _1C could be as large as 360 percent of the lowest. The factors affecting the scattering were investigated in detail and it was found that the scattering was mainly caused by the variation of locations of cleavage initiation. The local fracture stress σ was found to be the most stable parameter and combined with the minimum cleavage distance min which is determined by the triaxiality of stress reaching a critical value, it could be used to characterize the lower boundary of fracture toughness of steels. This revised version was published online in July 2006 with corrections to the Cover Date.     

Modified four-point bending specimen for determining the interface fracture energy for thin, brittle layers

A well-known four-point bending test has been modified such that the critical energy release rate for delaminating cracks propagating at the interface of a thin, brittle layer bonded to a substrate can be measured. The energy release rate required for crack delaminating at those interfaces is obtained by attaching a stiffening layer to the layer system. Another advantage of this modification is that segmentation of the layer and plastic deformation of the substrate during bending are avoided. The interface fracture energy of a plasma sprayed ZrO₂-ceramic layer on a flame sprayed high alloyed steel substrate has been measured. This revised version was published online in August 2006 with corrections to the Cover Date.     

Microstructure and microwave properties of {CaT{i}O}<Subscript>3</Subscript>–LaGaO<Subscript>3</Subscript> {solid solutions}

The phase assemblage, microstructure and microwave (MW) properties of xCaTiO₃-(1-x)LaGaO₃ (CT-LG) ceramics prepared by solid state synthesis from raw oxides and carbonates have been investigated. LG was predominantly single phase perovskite with space group, Pnma, but a small quantity of pyrochlore structured La₂Zr₂O₇ second phase was present due to the contamination from the ZrO₂ milling media. At MW frequencies, the temperature coefficient of resonant frequency (τ_{f}) of LG was --80~ppm/\({\circ}\)C, its permittivity, \(\varepsilon_{r} {=} \)27 and MW quality factor, \(Q^{*}f_{o} {=} \)97,000 (@5~GHz). As CT concentration increased, \(\tau_{f}\) and \(\varepsilon_{r}\) increased to The phase assemblage, microstructure and microwave (MW) properties of xCaTiO₃-(1-x)LaGaO₃ (CT-LG) ceramics prepared by solid state synthesis from raw oxides and carbonates have been investigated. LG was predominantly single phase perovskite with space group, Pnma, but a small quantity of pyrochlore structured La₂Zr₂O₇ second phase was present due to the contamination from the ZrO₂ milling media. At MW frequencies, the temperature coefficient of resonant frequency (τ_{f}) of LG was --80~ppm/ C, its permittivity, 27 and MW quality factor, 97,000 (@5~GHz). As CT concentration increased, and increased to $+$850~ppm/ C and 160, respectively but decreased to 20,000. Zero was achieved at 0.65 with 47, and 40,000, properties comparable with commercial compositions. However, for 0.5, a different second phase was observed which was rich in Ca and Ga. Electron diffraction patterns could be indexed according to a body centred cubic lattice, 12.5 ?. It is suggested that the presence of second phases in the CT-LG compounds may, in part, be responsible for the deterioration in .     

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     

Conversion of transgranular to intergranular fracture in NiCr steels

The paper is focused on quantification of causes and characteristics that govern the intergranular fracture initiation and propagation of this fracture micromechanism in competition with cleavage one. A NiCr steel of commercial quality and the same steel with an increased content of impurity elements, Sn and Sb, have been used for this investigation. Step cooling annealing was applied in order to induce intergranular embrittlement and brittle fracture initiation in both steels. Standard bend and the pre-cracked Charpy type specimen geometries were both tested in three-point bending to determine the fracture toughness characteristics. Charpy V notch specimens tested statically in three-point bending supported by FEM calculation have been used for local fracture stress and other local parameters determination. Relation of cleavage fracture stress and critical stress for intergranular failure has been followed showing capability of this parameter for quantification of the transgranular/intergranular fracture conversion. In order to characterise the quantitative roughness differences in fracture surfaces fractal analysis was applied. A boundary level of fractal dimension has been determined to be 1.12 for the investigated steel; the fracture surface roughness with a higher value reflects high level of intergranular embrittlement and thus fracture toughness degradation.     

Dynamic crack propagation and unloading behavior of brittle polymers

The method of caustics in combination with a Cranz–Schardin high-speed camera was utilized to study dynamic crack propagation and unloading behavior of epoxy, PMMA and Homalite-100 specimens. Dynamic stress intensity factor K _ID and crack velocity were evaluated in the course of crack propagation. Caustic patterns at the loading points were also recorded to estimate load P applied to the specimen. Unloading rate , the time derivative of P, was determined as a function of time t, and its time correlation with K _ID or was examined. The findings showed that the change in was qualitatively in accord with the change in K _ID or . However, there existed slight differences among the values of t giving the maximum , and K _ID, so that their order was , and K _ID. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of loading rate on the local cleavage fracture stress σf in notched specimens

Four point bending (4PB) notched specimens with different notch sizes are tested at various loading rates at a temperature of −110 °C for a C-Mn steel. An elastic-plastic finite element method (FEM) is used to determine the stress and strain distributions ahead of notches. By accurately measuring the distances of the cleavage initiation sites from the notch roots, the local cleavage fracture stress σ_f is measured. The results show that the local cleavage fracture stress σ_f does not essentially change with loading rate V and notch size. The reason for this is that the cleavage micromechanism does not change in the different specimens at various loading rates. The cleavage micromechanism involves competition of two critical events of crack propagation and crack nucleation in the high stress and strain volume ahead of notch root. The large scatter of σ_f and notch toughness are mainly caused by the different critical events in different specimens.     

Cycles of sub-critical tensile and shear alternating fracturing in diminishing dimensions,under tensile loading

A microscopic study reveals that when the curvature of striae that mark the fracture surface of PMMA glass with a chevron pattern increase beyond the critical angle, μc = 3° ± 2°, a breakdown into alternating tensile dark zones, and bright, ragged shear zones occurs. This breakdown was repeated in primary, secondary and tertiary cycles in diminishing scales. The secondary and tertiary breakdowns occurred exclusively in the shear zones. Similar breakdowns were found in chevron patterns on the fractured surface of a silicate glass ceramics. Due however to their different properties, certain differences were identified between the two materials in their breakdown characteristics, e. g. in the glass ceramic μc = 20° ± 2°. A similar primary breakdown was also identified on tensile fractures cutting rocks in geological outcrops. In the glass ceramic the interface angle , which the striae form with the fracture boundary, decreased from 32° ± 2°in the early stage of the striae growth at relatively low velocity, to 13° ± 2° during their advanced growth, at greater velocity, demonstrating that is a good tool for monitoring the change in fracture velocity in a given material. It was found that four interconnected factors determine the geometries and breakdown styles of the chevron pattern: (1) the curvatures of the fracture front and that of the striae which intersect each other orthogonally, (2) the influence of the fracture boundaries, (3) the material properties, such as stiffness, and (4) the fracture velocity in the material.     

Constraint Loss under Dynamic Loading in Rate Independent Plastic Solids

The objectives of this paper are to examine the loss of crack tip constraint in dynamically loaded fracture specimens and to assess whether it can lead to enhancement in the fracture toughness at high loading rates which has been observed in several experimental studies. To this end, 2-D plane strain finite element analyses of single edge notched (tension) specimen and three point bend specimen subjected to time varying loads are performed. The material is assumed to obey the small strain J ₂ flow theory of plasticity with rate independent behaviour. The results demonstrate that a valid J–Q field exists under dynamic loading irrespective of the crack length and specimen geometry. Further, the constraint parameter Q becomes strongly negative at high loading rates, particularly in deeply cracked specimens. The variation of dynamic fracture toughness K _dc with stress intensity rate K for cleavage cracking is predicted using a simple critical stress criterion. It is found that inertia-driven constraint loss can substantially enhance K _dc for .     

On locations initiating cleavage fracture in precracked specimens of low alloy steel and weld metal

On the basis of a model of an `active zone' for initiating cleavage fracture proposed by authors, the distributions of cleavage initiation location in precracked specimens are explained, and the factors affecting cleavage initiation locations are analyzed. The change of the length of the active zone with applied load determines the distributions of cleavage initiation locations. With increasing temperatures, the distance X _f from precrack tip to locations initiating cleavage fracture and its scatter increase, and the lower boundary of X _f increases slowly, and the scatter is mainly caused by the rapid increase of the upper boundary. With decreasing the strength of the weakest constituent in steels and increasing their number, the minimum distance X _\min and the average distance for initiating cleavage fracture will decrease and the maximum distance X _\max will increase, and the corresponding toughness values will be decreased.     

Construction of binary minimal product parity-check matrices

A parity-check matrix H of a given code is called minimal if it has minimum number of nonzero entries among all parity-check matrices representing . Let and be two binary linear block codes with minimal parity-check matrices H ₁ and H ₂, respectively. It is shown that, using H ₁ and H ₂, one can efficiently generate a minimal parity-check matrix for the product code .     

Effect of fibre concentration, strain rate and weldline on mechanical properties of injection-moulded short glass fibre reinforced thermoplastic polyurethane

The effect of fibre concentration, strain rate and weldline on tensile strength, tensile modulus and fracture toughness of injection-moulded thermoplastic polyurethane (TPU) reinforced with different concentration levels of short glass fibres was investigated. It was found that tensile strength, σ_c, of single-gated mouldings increased with increasing volume fraction of fibres, ϕ_f, according to a second order polynomial function of the form and increased linearly with natural logarithm of strain rate (). Tensile modulus and fracture toughness (at initiation) of single-gated mouldings increased linearly with increasing ϕ_f (rule-of-mixtures) and . A linear dependence was obtained between fibre efficiency parameter for composite modulus, η_E, and . The presence of weldline in double-gated mouldings reduced tensile strength, tensile modulus and fracture toughness of TPU composites but had no significant effect upon properties of the TPU matrix. All the aforementioned properties increased with increasing fibre concentration and showed a linear dependence with respect to . Weldline integrity factor for all three properties decreased with increasing fibre concentration showing no strain-rate effect of any significance. Results indicated that tensile strength was more affected by the presence of weldline than tensile modulus or fracture toughness. It was noted that composite properties in the presence of weldline were still much greater than those for the unweld matrix. Weldline integrity values close to unity indicated that measured properties for the matrix were not significantly affected by the weldline.