Construction of quasi-brittle and quasi-ductile fracture diagrams based on necessary and sufficient criteria

Materials with a regular structure characterized by quasi-brittle and quasi-ductile fractures are considered in the case where the characteristic linear dimension of the structural element is known. Necessary and sufficient fracture criteria are constructed using the Neuber-Novozhilov approach. A modified Leonov-Panasyuk-Dugdale model for an opening mode crack is proposed where the width of the prefracture zone coincides with the width of the plasticity zone. For the critical parameters of quasi-brittle fracture (tensile stress, length of prefracture zones, stress intensity factors), relations are obtained that allow material fracture to be considered in the case where the crack length is negligible compared to the characteristic linear dimension of the structural element. A fracture diagram obtained using the critical stresses calculated from the necessary and sufficient criteria is considered in a wide range of crack lengths. The elastoplastic problem of extension of a plate with a central crack is solved using the finite-element method. The dimensions and shape of the plastic zone near the crack tip are determined for different levels of loads corresponding to quasi-brittle and quasi-ductile fracture. The obtained results are analyzed to estimate the width of the prefracture zone and the critical crack opening.     

Quasi-brittle fracture diagram of structured bodies in the presence of edge cracks

The Neuber-Novozhilov approach is used to obtain necessary and sufficient fracture criteria. Using a modified Leonov-Panasyuk-Dugdale model, simple relations for the critical fracture parameters are derived for opening mode edge cracks for the case where the diameter of the prefracture zone coincides with the diameter of the plasticity zone. These relations are suitable for studying fracture where the crack length is negligibly small. A fracture diagram using critical stresses under both criteria is proposed for a wide range of crack length. At a certain level of loading, three regions are identified, in the first of which the crack is stable, in the second, the crack extends but remains stable, and in the third, the crack is unstable. Experimental data on the fracture of specimens with edge cracks are obtained. It is established that the theoretical critical fracture curves are in good agreement with the obtained critical parameters for flat tensile specimens with two collinear edge cracks.     

An Experimental Study of Mixed Mode Crack Initiation and Growth in Functionally Graded Materials

Quasi-static mixed mode crack initiation and growth in functionally graded materials (FGMs) was studied through fracture experiments on polymer-based FGMs manufactured by selective ultraviolet irradiation poly(ethylene carbon monoxide)—a photo-sensitive copolymer that becomes more brittle and stiffer under ultraviolet irradiation. The objective of the study was to determine whether crack kinking criteria for homogeneous materials, e.g., maximum hoop stress criterion, also hold for FGMs. Single edge notched tension specimens with different spatial variations of Young's modulus, failure stress and failure strain, were tested. Near tip mode mixity was introduced either by inclining the crack to the remote loading direction, as in the case of homogeneous materials, or to the direction of material gradient, or both. A full-field digital image correlation technique was used to measure in real-time the displacement field around the crack tip while it propagated through the graded material, and to extract the fracture parameters of stress intensity factor K _I and K _II , and the T-stress. It was found that the nonsingular T-stress term in the asymptotic expansion for stresses plays a very important role in accurately measuring fracture parameters. It was also found that the maximum tangential stress criterion can be applied to the case of FGMs to predict crack kinking provided that the effect of the T-stress is accounted for and the process zone size is small compared to the intrinsic material gradient length scale. However, for accurate crack path prediction at a length scale comparable to the material gradient, detailed material property information is required. In general, the crack will propagate towards a region that exhibits less fracture toughness, but, unlike the case of homogeneous materials, along a path where K _II is not necessarily equal to zero.     

Inverse problem of fracture mechanics for a disk fitted onto a rotating shaft

A plane problem of fracture mechanics for a circular disk fitted onto a rotating shaft is considered. The disk is assumed to be fitted tightly onto the shaft, and there are N randomly located straight-line cracks of length 2l_k (k = 1, 2, ..., N) near the inner surface of the disk. The interference between the disk and the rotating shaft, providing minimization of fracture parameters (stress intensity factor) of the disk, is theoretically studied on the basis of the minimax criterion. A closed system of algebraic equations is constructed, which allows the problem of optimal design to be solved. A simplified method of minimization of disk fracture parameters is considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 201–209, July–August, 2009.     

Method to calculate fatigue fracture life of control fissure in perilous rock

Rupture and safety of perilous rock are dominated by control fissure behind perilous rock block. Based on model-Ⅰand model-Ⅱstress strength factors of control fissure under acting of weight of perilous rock, water pressure in control fissure and earthquake forces, method to calculate critical linking length of control fissure is established. Take water pressure in control fissure as a variable periodic load, and abide by P-M criterion, when control fissure is filled with water, establish the method to calculate fatigue fracture life of control fissure in critical status by contributing value of stress strength factor stemming from water pressure of control fissure in Paris's fatigue equation. Further, parameters (C and m) of sandstone with quartz and feldspar in the area of the Three Gorges Reservoir of China are obtained by fatigue fracture testing.     

Erratum to: Rock Dynamic Fracture Toughness Tested with Holed-Cracked Flattened Brazilian Discs Diametrically Impacted by SHPB and its Size Effect

Dynamic fracture initiation toughness of marble was tested using two types of the holed-cracked flattened Brazilian disc (HCFBD) specimens, which were diametrically impacted at the flat end of the disc by the split Hopkinson pressure bar (SHPB) of 100 mm diameter. One type of the discs is geometrically similar with different outside diameter of 42 mm, 80 mm, 122 mm and 155 mm respectively, and with crack length being half the diameter; another type of the discs has identical 80 mm diameter and different crack length. Issues associated with determination of the stress wave loading by the SHPB system and the crack initiation time in the disc specimen were resolved using strain gage technique. The stress waves recorded on the bars and the disc failure patterns are shown and explained. The tested dynamic fracture toughness increases obviously with increasing diameter for the geometrically similar HCFBD specimens. It changes moderately for the one-size specimens of identical diameter and different crack length. The size effect of rock dynamic fracture toughness is mainly caused by the fracture process zone length l and fracture incubation time τ, the latter being an additional influencing factor for the dynamic loading as compared with the counterpart static situation. Hence a method is proposed to determine a unique value for the dynamic fracture initiation toughness, the approach takes average of the local distribution and time history for dynamic stress intensity factor in the spatial-temporal domain, which is defined by l and τ jointly. In this way the dynamic size effect is minimized.     

Two-parameter failure criterion for elastoplastic bodies with mode I cracks

The generalized Dugdale crack model is used to formulate two-parameter failure criteria for the cases of quasibrittle state and developed plastic zones at a mode I crack tip. The failure criteria relate the fracture strength characteristics and the stress mode at the crack tip through the plastic constraint factor. The critical state of bodies with cracks under uni-and biaxial loading is analyzed in the cases of plane stress and plane strain using the Tresca and von Mises yield criteria. A small-scale yield criterion, which is an analytic relation between the critical stress intensity factor and T-stresses, is established __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 47–57, July 2007.     

Prediction of mixed-mode fracture based on an elasto–plastic energy balance criterion

The maximum energy release rate criterion, i.e., G _max criterion, is commonly used for crack propagation analysis. This fracture criterion is based on the elastic macroscopic strength of materials. In the present investigation, however, the G _max criterion has been modified in order to accommodate the consideration of plastic strain energy. This modified criterion is extended to study the fatigue crack growth characteristics of mixed-mode cracks. 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 such as 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.     

Sufficient criterion of fracture in the case with a complex stress state and non-proportional deformation of the material in the pre-fracture zone

A general case of proportional loading with a complex stress state of the material in the pre-fracture zone, which is typical for polycrystalline solids with plastic deformation, is considered. A sufficient criterion of fracture is proposed for the case of a complex stress state with non-proportional deformation of the material in the pre-fracture zone. Critical parameters of fracture (pre-fracture zone length and load) for cracks propagating in quasi-brittle materials are obtained with the use of a modified Leonov-Panasyuk-Dugdale model. The pre-fracture zone width is determined by solving the problem of the plasticity theory in the vicinity of the crack tip. The proposed modification of the Leonov-Panasyuk-Dugdale model makes it possible to estimate the critical opening of the crack and the critical displacement of the crack flanks. Inequalities that describe different mechanisms of material fracture under proportional loading (predominantly shear fracture mechanism and fracture mechanism through cleavage) are derived.     

Modification of the fracture criterion for V-shaped notches (plane problem). Relationship between toughness and strength and structural parameters

A fracture criterion of the type of the Neuber-Novozhilov criterion is proposed to describe the fracture in the vicinity of the tip of a V-shaped notch under tensile and shear loading. In the proposed criterion, the limits of averaging of the stresses along the notch axis depend on the presence, location, and size of the initial defects in the material. The crystal lattice parameter of the initial material is chosen for the characteristic linear size. For a V-shaped notch subjected to tension and shear, simple equations are obtained that relate the stress intensity factors for the modified singularity coefficients, the singularity coefficients themselves, and the theoretical tensile and shear strengths of a single crystal of the material taking into account the damage to the material in the vicinity of the notch tip. The equations obtained allow a passage to the limit from a notch to a crack. It is shown that the classical critical stress intensity factor used in the strength analysis of cracked solids is not a material constant.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 106–115, January–February, 2005.     

Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by consideration of various contact zone models

Summary An interface crack with an artificial contact zone at the right-hand side crack tip between two piezoelectric semi-infinite half-planes is considered under remote mixed-mode loading. Assuming the stresses, strains and displacements are independent of the coordinate x ₂, the expression for the displacement jumps and stresses along the interface are found via a sectionally holomorphic vector function. For piezoceramics of the symmetry class 6 mm and for electrically permeable crack faces, the problem is reduced to a combined Dirichlet-Riemann boundary value problem which can be solved analytically. Further, analytical expressions for the stresses, electrical displacements, derivatives of elastic displacement jumps, stress and electrical intensity factors are found at the interface. Real contact zone lengths and the well-known oscillating solution are derived from the obtained solution as well. Analytical relationships between the fracture-mechanical parameters of various models are found, and recommendations are suggested concerning the application of numerical methods to the problem of an interface crack in the discontinuity area of a piezoelectric bimaterial. Received 16 March 1999; accepted for publication 31 May 1999     

On the effects of characteristic lengths in bending and torsion on Mode III crack in couple stress elasticity

The problem of a stationary semi-infinite crack in an elastic solid with microstructures subject to remote classical K_III field is investigated in the present work. The material behavior is described by the indeterminate theory of couple stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion and thus it is able to account for the underlying microstructure of the material as well as for the strong size effects arising at small scales. The stress and displacement fields turn out to be strongly influenced by the ratio between the characteristic lengths. Moreover, the symmetric stress field turns out to be finite at the crack tip, whereas the skew-symmetric stress field displays a strong singularity. Ahead of the crack tip within a zone smaller than the characteristic length in torsion, the total shear stress and reduced tractions occur with the opposite sign with respect to the classical LEFM solution, due to the relative rotation of the microstructural particles currently at the crack tip. The asymptotic fields dominate within this zone, which however has limited physical relevance and becomes vanishing small for a characteristic length in torsion of zero. In this limiting case the full-field solution recovers the classical K_III field with square-root stress singularity. Outside the zone where the total shear stress is negative, the full-field solution exhibits a bounded maximum for the total shear stress ahead of the crack tip, whose magnitude can be adopted as a measure of the critical stress level for crack advancing. The corresponding fracture criterion defines a critical stress intensity factor, which increases with the characteristic length in torsion. Moreover, the occurrence of a sharp crack profile denotes that the crack becomes stiffer with respect to the classical elastic response, thus revealing that the presence of microstructures may shield the crack tip from fracture.     

Crack initiation in a stiffened plate

The fracture mechanics problem of crack initiation in a stiffened plate is considered. The crack nucleus is modeled by a prefracture zone with bonds between the crack faces, which is treated as a region of weakened interparticle bonds of the material. The boundary-value problem of the equilibrium of a stiffened plate with a crack nucleus reduces to a nonlinear singular integrodifferential equation with a Cauchy type kernel. The strains in the crack initiation zone are found by solving this equation. The case of the stress state of the plate with a periodic system of prefracture zones is considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 111–120, July–August, 2007.     

A micromechanical model for predicting the fracture toughness of functionally graded foams

In this paper, finite element method based micromechanical analysis is used to understand the fracture behavior of functionally graded foams. The finite element analysis uses a micro-mechanical model in conjunction with a macro-mechanical model in order to relate the stress intensity factor to the stresses in the struts of the foam. The stress intensity factor at the crack tip of the macro-mechanical model can be evaluated using either the J-contour integral or the stresses in the singularity-dominated zone. The fracture toughness is evaluated for various crack positions and length within the functionally graded foam. Then the relationship between the fracture toughness of the graded foam and the local density at the crack tip is studied. Convergence tests for both macro-mechanical and micro-mechanical model analysis were conducted in order to maintain adequate accuracy with reasonable computational time. Fracture toughness of homogenous foams and functionally graded foams for various cases are presented as a function of relative density. This study indicates that the fracture toughness of functionally graded foams mainly depends on the relative density at the crack-tip.     

Yielding on inclined planes at the tip of a crack loaded in uniform tension

Plane-strain yielding from a crack in an infinite elastic body is represented here by a distribution of edge dislocations on two planes inclined at angles ±ga to the crack plane, and the equilibrium condition is solved numerically. Approximate analytical expressions are obtained for the plastic-zone length, the crack opening displacement, and the J-integral, as functions of the applied stress and α. A comparison with a co-planar model of the plastic zone gives very similar results for α ≈ 65°. It is shown that fracture criteria based either on a critical crack opening displacement (COD) or on a critical value of J are always different, and the use of the former may lead to critical defect-sizes which are twice as large as those given by the latter. Furthermore, COD appears not to be a well-defined material property. The critical J criterion gives a fracture stress which is α-dependent : this may be responsible for deviations towards results of linear elastic fracture mechanics when post-yield fracture mechanics is used to analyse extensive yielding. The changes in the stress field of the crack due to the existence of the plastic zone are also discussed.     

Brittle fracture: Variation of fracture toughness with constraint and crack curving under mode I conditions

The effect of constraint on brittle fracture of solids under predominantly elastic deformation and mode I loading conditions is studied. Using different cracked specimen geometry, the variation of constraint is achieved in this work. Fracture tests of polymethyl methacrylate were performed using single edge notch, compact tension and double cantilever beam specimens to cover a bread range of constraint. The test data demonstrate that the apparent fracture toughness of the material varies with the specimen geometry or the constraint level. Theory is developed using the critical stress (strain) as the fracture criterion to show that this variation can be interpreted using the critical stress intensity factorK _Cand a second parameterT orA ₃,whereT andA ₃are the amplitudes of the second and the third term in the Williams series solution, respectively. The implication of this constraint effect to the ASTM fracture toughness value, crack tip opening displacement fracture criterion and energy release rateG _Cis discussed. Using the same critical stress (strain) as the fracture criterion, the theory further predicts crack curving or instability under mode I loading conditions. Experimental data are presented and compared with the theory.     

Edge fracture in cone-plate and parallel plate flows

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N _2c = KΓ/a, where N _2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N _2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N ₂) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N ₂ is negative, and a secondary flow tending to suppress crack formation if N ₂ is positive. Received: 4 January 1999 Accepted: 19 May 1999     

Study of the tensile fracture of plates with a rigid linear inclusion

The limit equilibrium of elastoplastic body is studied under the conditions of a plane problem. The body contains a linear inclusion, which is rigid but of finite rupture strength. The plastic or prefracture zones develop near the ends of the inclusion and are modeled by slip cracks along the matrix—inclusion interface. A new interpretation of the boundary conditions is proposed to solve a model problem for such a composition, and its analytical solution is derived. Two possible mechanisms of local fracture are considered: (a) fracture of the inclusion and (b) separation of the inclusion. The critical length of the inclusion is determined. This length together with the elastic and strength parameters of the composition determines the mechanism of local fracture. The limit loads are found for each mechanism of fracture. State Academy of Water Industry, Rovno, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 123–129, July, 2000.