Fracture surface analysis of ceramics

Flaw size, c, fracture mirror boundaries, r, fracture stress, , and critical fracture energy were measured for glasses, glass ceramics, and single and polycrystalline ceramics. The relationship r ^1/2 = constant was verified for all these materials. The mirror constants, A, in these materials were shown to be directly proportional to the average critical stress intensity factor for crack propagation, K _IC. Based on the A — K _IC relationship, the outer mirror to flaw size ratio is shown to scatter about a value of 131. Thus, the mirror constants were used to predict critical flaw sizes in these materials. The observed flaw sizes in most cases correlated well with those calculated. The cases in which poorer correlation was obtained are those in which flaw sizes were smaller than the grain size, flaws were pores or surrounded by porous regions, or where severe microcracking existed. It is shown that the elastic modulus is proportional to the mirror constant and probably to the critical fracture energy, but that the latter is highly dependent on local microstructure. The smaller inner to outer mirror ratios for polycrystalline ceramics over glasses is attributed to the difference in available paths for crack propagation.     

Crack-tip damaged zones in rubber-toughened amorphous polymers: a micromechanical model

The various stages of crack propagation in rubber-toughened amorphous polymers (onset and arrest, stable and unstable growth) are governed by the rate of energy dissipation in the cracktip damaged zone; hence the relationship between the applied stress intensity factorK ₁ and the damaged zone size is of utmost importance. The size of the crack-tip damaged zone has been related toK ₁ via a parameter which is characteristic of the material in given conditions: this factor is proportional to the threshold stress for damage initiation in a triaxial stress field, and has been denoted by ^*. Theoretical values of ^* have been calculated by means of a micromechanical model involving the derivation of the stresses near the particles and the application of damage initiation criteria. The morphology, average size and volume fraction of the rubbery particles have been taken into account together with the nature of the matrix. The calculated values of ^* have been successfully compared with the experimental ones, for a wide set of high-impact polystyrenes (HIPS) and rubber-toughened poly(methyl methacrylate) (RTPMMA).Nomenclature PS; HIPS polystyrene; high-impact polystyrene - PMMA; RTPMMA poly(methyl methacrylate); rubber-toughened PMMA - MI; CS/H; CS/R particle morphologies (multiple inclusion; hard core - rubber shell; rubber core - rigid shell) - K _r;K _g bulk moduli of rubber and glassy materials - G _r;G _g shear moduli of the same materials - v _p particle volume fraction - L mean centre-to-centre distance between neighbouring particles - B; H; W standard names for the dimensions of the compact tension specimen - R _y size of the crack-tip plastic zone in a homogeneous material - h half thickness of the crack-tip damaged zone - r; polar coordinates around the crack tip (Fig. 1) - r;r _p distance from particle centre; particle radius - p normalized distance from the particle (Equation 5) - K ₁;K _1c;K _1p stress intensity factor; critical values ofK ₁ at the onset of and during crack growth - G _1c plane strain energy release rate - _y yield stress in uniaxial tension - _th macroscopic threshold stress for the onset of local damage initiation in a composite material - ^* characteristic parameter (Equation 3) - ⁰; ₁ ⁰ ; ₂ ⁰ ; ₃ ⁰ applied stress tensor and its three principal stresses - ⁰ uniaxial applied stress - ; ₁; ₂; ₃ local stress tensor and its three principal stresses - A tensor which elements are the ratios of those of over those of ⁰ (Equation 4) - v Poisson's coefficient of the matrix - g triaxiality factor of the crack-tip stress field - _e; _p Mises equivalent stress; dilatational stress (negative pressure) - I ₁;I ₂ invariants of the stress tensor - U ₁;U ₂ material parameters for argon and Hannoosh's craze initiation criterion (Equation 12)     

Analysis of brittle fracture under the combined stresses of tension and compression

Brittle fractures occurring under biaxial stress states were analysed based on the weakest link model using the mixed mode fracture criterion. Expressions for the mixed mode fracture criterion were chosen for application to the negativeK _l region, corresponding to the compressive stress for the crack. Calculations for biaxial strength with randomly oriented constant-length cracks from the mixed mode fracture criterion were made in the region ofK _I>0 because an unstable fracture seems to occur in this region. The results indicated that the tensile stress component in the combined tension and compression stress state remains constant when the compressive component is smaller than the critical value, which is given by [1 –(K _c/K _c)²]_t derived from the mixed mode fracture criterion, (K /K _c) + (K /K _c)² = 1. Considering the statistical effects, however, calculation of the biaxial strength is modified to result in: (1) lowering the biaxial tensile strength, and in (2) a smooth transition from the constant tensile strength region to the decreasing strength region under the combined tension and compression stress. This suggests that the highK _IIc/K_Ic ratio results in the increase in the compressive strength relative to the tensile strength.     

Sub-critical crack extension and crack resistance in polycrystalline alumina

Sub-critical crack extension can readily be observed in controlled fracture tests in fourpoint bending. A natural crack of any desired lengthc which exceeds the notch depthc ₀ by the amount c =c –c ₀ can be introduced into bend specimens by stable crack propagation. The stress intensity factor to achieve c increases considerably with increasing c. In pre-cracked specimens the stress intensity factorK _I0 to start the crack and the critical valueK _IC strongly depend on the natural crack length c whereasK _I0 andK _IC are independent ofc ₀ in solely notched specimens. From a quasi-continuous evaluation of the load-deflection curve recorded during controlled fracture, the differential work of fracture can be obtained as a function of the achieved crack length. It may be regarded as the crack extension resistanceR of the material because the balance between the energy release rateg ₁ andR is maintained throughout the experiment. By that, a formal analogy to theR-curve concept of fracture mechanics is given. The steady increase ofR is explained by multiple crack formation and by the interference of the fracture surfaces due to the angular development of the crack front.     

Phase Transitions in the Gross–Neveu Model on a Sphere for High-T c Materials

Phase transitions were studied in the 3D Gross–Neveu model on a sphere at T 0. The zeta function of the Dirac operator of the model was represented with a double sum, and its derivative was calculated using recurrent formulas for generalized Epstein–Hurwitz functions. The effective potential Vwas determined as a function of the field, temperature T, and inverse radius of curvature r. Phase transitions first-order in temperature at R= 0 (T _c = 1/2ln2) and second-order in curvature R= 2/r ² at T= 0 (r _c = 1/1.5) were identified. For the general case, T 0 and R 0, the critical phase dependence T _c(r _c) was obtained, and the V(, T, r) surface was constructed. External stimuli such as temperature and curvature were shown to restore the symmetry of the system in the 3DGross–Neveu model.     

Correlation between resin material variables and transverse cracking in composites

In this paper, the correlation between the resin material variables and the transverse cracking in composites is established. A theoretical model based on the fracture mechanics principle is built to describe thein situ failure process of transverse cracking. The central concept of the model is that the fracture is controlled by the plastic zone developed at the crack tip. Based on an approximate crack tip stress distribution, a quantitative representation is found to relate the laminate transverse cracking fracture toughness,G _c(comp), to certain resin properties: fracture toughness,G _c(resin), yield stress, _y, Young's modulus,E, and residual stress build-up, _R.G _c(comp) values of several fibre-glass/epoxy laminate systems were measured using the double torsion technique. The experimental results are found to be interpreted reasonably well by the theory. As a result, a clear picture of transverse cracking emerges. It seems that _y ²/E plays a more dominant role thanG _c(resin) in controllingG _c(comp). The residual stress _R can weaken the laminate significantly when its level is high. It is also shown that the failure model discussed here can be readily applied to laminate delamination failure as well as adhesive bond fracture.     

The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints

Analytical solutions for beam specimens used in fracture-mechanics testing of composites and adhesively-bonded joints typically use a beam on an elastic foundation model which assumes that a non-infinite, linear-elastic stiffness exists for the beam on the elastic foundation in the region ahead of the crack tip. Such an approach therefore assumes an elastic-stiffness model but without the need to assume a critical, limiting value of the stress, _max, for the crack tip region. Hence, they yield a single fracture parameter, namely the fracture energy, G _c. However, the corresponding value of _max that results can, of course, be calculated from knowledge of the value of G _c. On the other hand, fracture models and criteria have been developed which are based on the approach that two parameters exist to describe the fracture process: namely G _c and _max. Here _max is assumed to be a critical, limiting maximum value of the stress in the damage zone ahead of the crack and is often assumed to have some physical significance. A general representation of the two-parameter failure criteria approach is that of the cohesive zone model (CZM). In the present paper, the two-parameter CZM approach has been coupled mainly with finite-element analysis (FEA) methods. The main aims of the present work are to explore whether the value of _max has a unique value for a given problem and whether any physical significance can be ascribed to this parameter. In some instances, both FEA and analytical methods are used to provide a useful crosscheck of the two different approaches and the two different analysis methods.     

Residual thermal stresses in the pull-out specimen: A finite element calculation

The residual thermal stress field in the pull-out specimen is calculated in the case of a high properties thermoset system (carbon-bismaleimide). The calculation is performed within the framework of the linear theory of elasticity by means of a finite element method. The specimen is modelled as a three-phase composite (holder-fibre-matrix). The meniscus which forms at the fibre entry is taken into account in order to provide a realistic stress concentration. The latter is far higher than the matrix strength. Evidence that fibre debonding propagates from the fibre end during cooling is then produced.Nomenclature T thermal load - L _e embedded length - r _f fibre radius - _c curvature radius of the meniscus (fibre entry) - r _c radial dimension of the finite element mesh - E _m,E _h matrix and holder moduli - E _A,E _T fibre axial and transverse moduli - _m, _h matrix and holder thermal expansion coefficients - _A, _T fibre axial and transverse thermal expansion coefficients - _rr, , _zz, _rz non-zero components of the residual stress field - _rr ⁱ , ^im , _zz ^im , _rz ⁱ stresses at the interface in the matrix (r=r _f + ) - _rr ⁱ , ^if , _zz ^if , _rz ⁱ stresses at the interface in the fibre (r=r _f–) - _p1 maximum principal stress - _zz ^f mean axial stress over the fibre section - _rupt ^m matrix strength - u _r ,u _z non-zero components of the displacement field     

Why K? High order singularities and small scale yielding

Singular terms in the crack tip elastic stress field of order r ^-3/2, r ^-5/2, ... are often neglected, thus rationalizing the use of the K field, r ^-1/2, as the dominant term for fracture mechanics. We find the common explanation for neglecting the more singular terms in the series solution for the crack tip stress field unsatisfying. Further, the more singular terms are non-zero and are needed to understand the energetics of fracture, i.e., J and {ie97-1}. Given that the singular terms are generally present, the rationale for the validity of the small scale yielding assumption (the basis of linear elastic fracture) is more subtle than any argument which depends on the elimination of terms with stress r ^-3/2, r ^-5/2, ... Our explanation for the validity of small scale yielding is as follows. First, with or without small scale yielding, the stress field outside of the nonlinear zone does contain more singular terms. In the limit as the nonlinear zone at the crack tip shrinks to zero size (SSY) we show that the r ^-1/2 term in the Williams expansion dominates both the more singular and the non-singular terms in an annular region somewhat removed from this zone. Further, in this limit the magnitude of the r ^-1/2 term is almost entirely determined by tractions on the outer boundary. Our theory and examples are for representative problems in mode III anti-plane shear fracture. We expect, however, that the general results also apply to mode I and mode II fracture.     

Transport properties of antiferromagnetic superconductors

Transport properties of antiferromagnetic superconductors with T _N<T _chave been investigated. Detailed numerical results are given for SmRh₄B₄ by using the following model. The paramagnetic phase (T _NT _c)is described by using Abrikosov-Gorkov theory of magnetic ions in an ordinary superconductor. In the AF phase (TT _N)the effects of the molecular field H _Q (T) and the elastic scattering of conduction electrons from spin fluctuations are included. Expressions for H _Q (T) and the scattering rate from spin fluctuations have been derived. The aim has been to see if properties are enhanced or depressed by the AF ordering occurring below T _N.It is found that whereas the electronic thermal conductivity K _s,nuclear spin relaxation rate R _s,and the longitudinal ultrasonic attenuation _s,are depressed by the AF ordering, the inverse of the magnetic penetration depth, [(T)]^–1 is enhanced below T _N.The effect of all types of impurities is included in the study. Theoretical results for K _sand [(T)]^–1 agree with the experimental values for SmRh₄B₄ (experimental data for other properties are not available in literature). In the above, T _Nis the Néel temperature and T _cis superconducting transition temperature.     

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     

Formation and properties of amorphous (Fe1−x Nb x ) l B100−l

Amorphous (Fe_1–x Nb _{x l} B_100–l alloys with 0 x 0.15 and 74 T _g, crystallization temperatureT _x, and microhardnessH _v, but to decrease the magnetization and Curie temperatureT _c. The effects of niobium onT _x,H _v, andT _c in iron-based amorphous alloys are similar to those of chromium, manganese, molybdenum, tungsten and vanadium.     

A transferability model for brittle fracture including constraint and ductile tearing effects: a probabilistic approach

This study describes a computational framework to quantify the influence of constraint loss and ductile tearing on the cleavage fracture process, as reflected by the pronounced effects on macroscopic toughness (J _c , _c). Our approach adopts the Weibull stress _w as a suitable near-tip parameter to describe the coupling of remote loading with a micromechanics model incorporating the statistics of microcracks (weakest link philosophy). Unstable crack propagation (cleavage) occurs at a critical value of _w which may be attained prior to, or following, some amount of stable, ductile crack extension. A central feature of our framework focuses on the realistic numerical modeling of ductile crack growth using the computational cell methodology to define the evolution of near-tip stress fields during crack extension. Under increased remote loading (J), development of the Weibull stress reflects the potentially strong variations of near-tip stress fields due to the interacting effects of constraint loss and ductile crack extension. Computational results are discussed for well-contained plasticity, where the near-tip fields for a stationary and a growing crack are generated with a modified boundary layer (MBL) formulation (in the form of different levels of applied T-stress). These analyses demonstrate clearly the dependence of _w on crack-tip stress triaxiality and crack growth. The paper concludes with an application of the micromechanics model to predict the measured geometry and ductile tearing effects on the cleavage fracture toughness J _c of an HSLA steel. Here, we employ the concept of the Dodds-Anderson scaling model, but replace their original local criterion based on the equivalence of near-tip stressed volumes by attainment of a critical value of the Weibull stress. For this application, the proposed approach successfully predicts the combined effects of loss of constraint and crack growth on measured J _c -values.     

Influence of threshold parameters on cleavage fracture predictions using the Weibull stress model

This study explores applications of three-parameter Weibull stress models to predict cleavage fracture behavior in ferritic structural steels tested in the transition region. The work emphasizes the role of the threshold parameters (_th and _{w – min}) in cleavage fracture predictions of a surface crack specimen loaded predominantly in tension for an A515-70 pressure vessel steel. A recently proposed procedure based upon a toughness scaling methodology using a modified Weibull stress (^* _w) extends the calibration scheme for the Weibull modulus, m, to include the threshold parameters. The methodology is applied to calibrate the Weibull stress parameter for the tested material and then to predict the toughness distribution for the surface crack specimen. While the functional relationship between ^* _w and m suggests a strong effect of the threshold stress, _th, on the calibrated m-parameter, the results show a remarkably weak dependence of fracture predictions on _th as does the dependence of fracture predictions on _w–min for this specimen.     

Theory of the Self-Organized Critical State in Nonequilibrium 4He

A near-critical sample of superfluid ⁴He subjected to a combination of a heat current Q and a gravitational field g has recently been confirmed experimentally to exhibit a self-organized critical (SOC) state, in which the local temperature profile T(z) parallels the local superfluid transition temperature, T (z), so that T(z)–T (z) is independent of the vertical coordinate z. We show that there is a particular heat current Q ₀(g)g such that the SOC state lies in the normal phase only when Q<Q ₀(g). The SOC state is shown to be dynamically stable and, surprisingly, to possess a propagating mode in which perturbations travel upstream at a particular speed c _SOC(Q). For Q>Q ₀(g) the SOC state is pushed below the superfluid transition. We present a model based on the Model F equations that shows that the resulting superfluid state is intrinsically dynamical and supports an average temperature gradient by way of an approximately uniform density of recurring phase slips. The role of a certain local dynamical instability temperature, T _c(Q, z), in the nucleation of the phase slips is emphasized.     

Constraint effects on the ductile-to-brittle transition temperature of ferritic steels: a Weibull stress model

This study examines crack front length and constraint loss effects on cleavage fracture toughness in ferritic steels at temperatures in the ductile-to-brittle transition region. A local approach for fracture at the micro-scale of the material based on the Weibull stress is coupled with very detailed three-dimensional models of deep-notch bend specimens. A new non-dimensional function g(M) derived from the Weibull stress density describes the overall constraint level in a specimen. This function remains identical for all geometrically similar specimens regardless of their absolute sizes, and thus provides a computationally simple approach to construct (three-dimensional) fracture driving force curves _w vs. J, for each absolute size of interest. Proposed modifications of the conventional, two-parameter Weibull stress expression for cumulative failure probability introduce a new threshold parameter _w–min. This parameter has a simple calibration procedure requiring no additional experimental data. The use of a toughness scaling model including _w–min>0 increases the deformation level at which the CVN size specimen loses constraint compared to a 1TSE(B) specimen, which improves the agreement of computational predictions and experimental estimations. Finally the effects of specimen size and constraint loss on the cleavage fracture reference temperature T ₀ as determined using the new standard ASTM E1921 are investigated using Monte Carlo simulation together with the new toughness scaling model.     

Fracture mechanism in short fibre reinforced thermoplastic resin composites

The properties of two types of short carbon fibre (CF) reinforced thermoplastic resin composites (CF-PPS and CF-PES-C), such as strength (_y). Young's modulus (E) and fracture toughness (K _1c), have been determined for various volume fractions (V _f) of CF. The results show that the Young's modulus increases linearly with increasingV _f with a Krenchel efficiency factor of 0.05, whereas _y andK _1c increase at first and then peak at a volume fraction of about 0.25. The experimental results are explained using the characteristics of fibre-matrix adhesion deduced from the load-displacement curves and fractography. By using a crack pinning model, the effective crack tensions (T) have been calculated for both composites and they are 57 kJ m^–1 for CF-PPS and 4.2 kJ m^–1 for CF-PES-C. The results indicate that the main contribution to the crack extension originates from localized plastic deformation of the matrix adjacent to the fibre-matrix interface.     

Stress-strain behaviour of nitrogen bearing austenitic stainless steels in the temperature range 298–473 K

The stress-strain behaviour of three nitrogen-bearing low-nickel austenitic stainless steels has been investigated via a series of tensile tests in the temperature range 298–473 K at an initial strain rate of 1.6×10^–5s^–1. Experimental stress-strain data were analysed employing Rosenbrock's minimization technique in terms of constitutive equations proposed by Hollomon, Ludwik, Voce and Ludwigson. Ludwigson's equation has been found to describe the flow behaviour accurately, followed by Voce's equation. The resultant strain-hardening parameters were analysed in terms of variations in temperature. A linear relationship between ultimate tensile stress and the Ludwigson parameters has been established. The influence of nitrogen on the Ludwigson modelling parameters has also been explained.Nomenclature True stress - _t True strain - _f True fracture strain - Strain rate - T Temperature - K _H, n _H Hollomon parameters - K _L, n _L Ludwik parameters - K _1L, k _2L, n _1L, n _2L Ludwigson parameters - _s, K _V, n _V Voce parameters - _{u relation} Uniform strain computed from a particular relation - _L Transient strain - ₀ Flow stress at zero plastic strain (Ludwik) - _L Transient stress - _y Yield stress - _u Ultimate tensile stress     

Cutting cheese with wire

Wire cutting involves fracture, plastic deformation and surface friction effects so that, in principle, it provides a method for determining parameters for all these properties. This paper describes an analysis in terms of the fracture toughness, G _c, the yield stress, _y, and the coefficient of friction, . By measuring the cutting force as a function of wire diameter, G _c and (1 + )_y can be found. These values are compared with direct measurements in notched bending for G _c, in simple compression for and _y, and in sliding tests for . A comparison of values obtained for a range of cheeses shows encouraging agreement.     

Estimation procedure for the Weibull parameters used in the local approach

The local approach was recently proposed by Beremin and Mudry for evaluating the statistical behaviour of toughness results of materials. This approach introduces a stress parameter _w , termed the Weibull stress, as a measure of the fracture resistance of materials instead of the conventional toughness parameters such as K _c, _c and J _cl (critical stress intensity factor, CTOD and J-integral, respectively). The Weibull stress _w obeys the Weibull distribution with the two parameters m and _u (the shape and the scale parameter, respectively). The first parameter m is normally estimated to be 22 irrespective of the kind of material. In this paper a procedure for the determination of the Weibull parameters m and _u is developed. This procedure consists of the determination of the plastic zone ahead of the crack tip, from which cleavage fracture originates, and of the maximum likelihood estimation of the parameters m and _u based on the stress distribution in the plastic zone. Calculations using this procedure confirm that the distribution of the Weibull stress _w is a material property independent of specimen thickness, and in particular that the shape parameter m depends on the material, e.g. m12 for a German reactor pressure vessel steel (20 Mn Mo Ni 5 5). Using these parameters for the distribution of the Weibull stress the size effect in fracture toughness values is predicted and an improved agreement between theory and experiments is obtained compared to the Weakest Link model.