Toughening by partial or full bridging of cracks in ceramics and fiber reinforced composites

A complete solution is given for a fully or partially bridged straight crack in transversely isotropic elastic materials which may correspond to unidirectionally fiber-reinforced ceramics or other brittle composites. The stiffness of the bridging materials may have an arbitrary variation along the crack, representing partially failed fibers or ligaments. The crack may have any orientation with respect to the axis of the material symmetry. The solution is explicit in terms of the Chebychev polynomials when the bridging-forces are linearly dependent on the crack-opening-displacement. In addition, uniformly valid asymptotic solutions are developed for fully or partially bridging cracks. For the case when the crack is short relative to a length scale which depends on the material properties, the method yields a complete asymptotic solution when the bridging forces are linearly or non-linearly dependent on the crack-opening-displacement (a square-root dependence, corresponding to continuous fibers, is used for illustration). For the case of long cracks, the proposed asymptotic is effective, but the results are not presented in this work.The mechanism of crack kinking is studied for an oblique partially or fully bridged, or unbridged crack in a macroscopically transversely isotropic elastic solid. The crack is assumed to grow in the matrix material (containing unbroken strong fibers) under local driving forces which are calculated on the basis of the overall anisotropic material response. The results of various fracture criteria are studied. It is illustrated that, under far-field tensile forces normal to the crack, the criterion of the maximum opening mode stress intensity factor in the homogenized anisotropic solid (i.e., the orientation for which the strength of the singularity associated with the tensile hoop stress is maximum) produces results which suggest crack growth more or less parallel to the fibers, whereas the results based on the maximum Mode I stress intensity factor in the isotropic matrix material and/or on the local symmetry criterion (again, for the isotropic matrix) predict crack extension more or less normal to the reinforcing fibers.     

Dynamic fracture of a beam or plate under tensile loading

The dynamic fracture response of a long beam of brittle elastic material under tensile loading is studied. If the magnitude of the applied loading is increased to a critical value, a crack is assumed to propagate across the beam cross section. In a parallel analysis to [t] the crack length and applied loading at the fracture face are determined as functions of time measured from fracture initiation. The results of the analysis are shown in graphs of crack length, crack tip speed and fracturing section tensile loading vs time. As found in [1], the crack tip accelerates very quickly to a speed near the characteristic terminal speed for the material, travels at this speed through most of the beam thickness, and then decelerates rapidly in the final stage of the process. Finally, by appropriate change of the elastic modulus, the results may be applied to plane strain fracture of a plate under pure tensile loading.     

Effect of bending moment on the dynamic fracture of a beam or plate under tensile loading

The dynamic fracture response of a long beam of brittle elastic material under tensile loading is studied. If the magnitude of the applied loading is increased to a critical value, a crack is assumed to propagate across the beam cross section. As an extension of previous work, an induced bending moment generated during fracture is incorporated into the analysis and this improved formulation is presented. The crack length, crack tip speed, axial force and bending moment on the fracturing section are determined as functions of time after crack initiation. It is found that the bending moment has a significant effect on the fracture process in that it tends to retard fracture and causes a drastic change in the slope of the loading curve for large crack depths. Finally, by appropriate change of the elastic modulus, the results may be applied to plane strain fracture of a plate in pure tensile loading.     

Asymptotic stresses around a crack tip at the interface between planar or cylindrical bodies

Closed-form equations are derived for the asymptotic stresses in the neighborhood of a crack tip impinging on an interface between two isotropic materials. The symmetric problem is considered and follows from an exact elasticity solution formulated by Gupta [1]. The equations are valid for the planar problem, where the interface is straight and also for an axisymmetric problem in the presence of an annular or penny-shaped crack. The equations may serve to establish a tentative criterion that defines the subsequent direction of a crack impinging on a bimaterial interface. The ambiguity of the asymptotic stress state is highlighted and plausible application of the results is discussed.The U.S. Government right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

Fracture initiation at a crack or notch in a viscoplastic material under high rate loading

The two-dimensional problem of an edge crack in a half space or plate is considered. The body is loaded by a suddenly applied, spatially uniform normal velocity imposed on the plane boundary of the body on one side of the edge crack. Otherwise, the boundary of the body, including the crack faces, is traction free. Both cases of an initially sharp crack tip and a narrow notch with small but nonzero notch root radius are considered. The material is modeled as elastic viscoplastic, including strain hardening, rate sensitivity and thermal softening. The applied loading produces predominantly mode II loading in the crack tip region. Under these conditions it is possible to nucleate an adiabatic shear band at the crack tip as a precursor to a mode II fracture. On the other hand, because of the rate sensitivity of the material and the high rate of loading, it may be possible under certain conditions to generate tensile stresses in the crack tip region sufficiently large to nucleate brittle tensile fracture. The problem is solved numerically by means of the finite element method in order to investigate the competition between these two possible fracture initiation mechanisms. The magnitude of the impact velocity imposed on the edge of the plate and the notch tip acuity have an effect on processes near the crack tip. For given material, the inception of crack growth is determined by the competition between a stress-based brittle fracture condition, associated with rate sensitivity and strain hardening, and a strain based criterion, associated with high strain rate and thermal softening.     

The internal crack in an extended or compressed plate: Its geometric characteristics

The Muskhelishvili closed-form solution of an infinite plate containing an internal crack and submitted to either simple tension or compression was used for evaluating the exact shape of the deformed crack. The orientation of the crack-axis relative to the loading axis of the plate varied parametrically to cover all possible orientations. Using the already established fact that the deformed crack becomes an ellipse, whose principal axes change orientation from those of the unloaded crack, the dependence of the length of the deformed crack and its COD variation was plotted in terms of the angle β of inclination and the load level of the plate. It was shown that, while the purely extended crack remains symmetric to its initial axis and shrinks, the purely compressed crack presents a concordant correspondence of its deformed lips, which are closing creating a quasi-disappearance of the crack. Introduction of any amount of shear immediately introduces a discordant correspondence between respective points of the crack lips and contributes to their overlapping. Then, some amount of COD must be introduced by extending the crack in order to assure open lips of the crack. On the other hand, the length of the deformed crack is always smaller than its initial length for pure extension, whereas, for compression, it is always larger than the initial length. However, introduction of shear causes an increase of the final crack length, which may even overpass the initial crack length for some stress levels. Finally, phenomena of stick-slip and popping appear with loading when the dominant shear straining or the combination of compression and shear make the lips of the crack overlap and hinder a smooth propagation of the crack by loading. Experimental evidence with caustics in specimens under plane stress or plane strain corroborated the phenomena described by the theory.     

Cracks emanating from circular hole or square hole in rectangular plate in tension

A numerical analysis of cracks emanating from a circular hole (Fig. 1) or a square hole (Fig. 2) in rectangular plate in tension is performed by means of the displacement discontinuity method with crack-tip elements (a boundary element method) presented recently by the author. Detail solutions of the stress intensity factors (SIFs) of the two plane elastic crack problems are given, which can reveal the effect of geometric parameters of the cracked bodies on the SIFs. By comparing the SIFs of the two crack problems with those of the center crack in rectangular plate in tension (Fig. 3), in addition, an effect of the circular hole or the square hole on the SIFs of the center crack is discussed in detail. The numerical results reported here also illustrate that the boundary element method is simple, yet accurate for calculating the SIFs of complex crack problems in finite plate.     

船体钢止裂性能表征参量止裂温度和止裂韧性的对比分析

基于AH36、EH36和FH500三种船体钢的梯度温度场型双重拉伸试验结果,对止裂温度和止裂韧性分别作为止裂性能表征参量的特点进行了分析。结果表明,与止裂韧性相比,止裂温度测试稳定性好、工程适用性好,更宜作为船体钢止裂性能的工程应用表征参量。     

The Coulombic traction on the surfaces of an interface crack in dielectric/piezoelectric or metal/piezoelectric bimaterials

This paper deals with the Coulombic traction usually neglected, but inherently acting, on the surfaces of an interface crack in dielectric/piezoelectric or metal/piezoelectric bimaterials. The dielectric material phase is treated as a special kind of piezoelectric material with a little piezoelectricity, whereas the metal phase is treated as another special kind of piezoelectric material with an extremely large permittivity and an extremely small piezoelectricity. The permittivity of the medium inside the crack gap is accounted for either. The normal electric displacement component and the Coulombic traction on the crack surfaces are unknown, and are determined from a cubic equation deduced from the extended Stroh formula. Numerical results for the Coulombic traction in both kinds of bimaterials reveal that in most cases its magnitude is remarkable and cannot be entirely neglected when the applied electric field is higher. It is concluded that in most cases the Coulombic traction yields significant influence on the effective stress intensity factor at the crack tip and may influence the fracture behavior in such kinds of bimaterials. As compared to homogenous piezoelectric materials, the metal phase always decreases the Coulombic traction, whereas the dielectric phase decreases it under the negative electric field and increases it under the positive electric field. In all cases, BaTiO₃ always yields a much larger Coulombic traction than PZT-4.     

An effective method of stress intensity factor calculation for cracks emanating from a triangular or square hole under biaxial loads

This paper is concerned with stress intensity factors for cracks emanating from a triangular or square hole under biaxial loads by means of a new boundary element method. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfied and the crack‐tip displacement discontinuity elements proposed by the author. In the boundary element implementation, the left or the right crack‐tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called a Hybrid Displacement Discontinuity Method (HDDM). Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors for plane elastic crack problems. In addition, the present numerical results can reveal the effect of the biaxial loads on stress intensity factors.     

The effect of an inclusion or a hole on an approaching crack in the three-point bend specimen

A numerical finite element analysis of the influence of a disturbance: an inclusion or a hole on the stress intensity factor at the tip of an edge crack in a three-point bend specimen is presented. The effect of four parameters is considered: the crack length, the disturbance size and its distance from the crack tip, and the original neutral axis. It is found that both disturbances have an inverse effect on the stress intensity factor: the hole increases it, whereas the inclusion causes it to decrease. The magnitude of the effect depends on the above-mentioned parameters. The numerical results are found to be in good agreement with preliminary experimental findings.     

Size effect on buckling strength of eccentrically compressed column with fixed or propagating transverse crack

The strength and size effect of a slender eccentrically compressed column with a transverse pre-existing traction-free edge crack or notch is analyzed. Rice and Levy’s spring model is applied to simulate the effect of a crack or notch. An approximate, though accurate, formula is proposed for the buckling strength of the column of variable size. Depending on the eccentricity, the crack at maximum load can be fully opened, partially opened or closed. The size effects in these three situations are shown to be different. The exponent of the power-law for the large-size asymptotic behavior can be −1/2 or −1/4, depending on the relative eccentricity of the compression load. Whether the maximum load occurs at initiation of fracture growth, or only after a certain stable crack extension, is found to depend not only on the column geometry but also on its size. This means that the definition of positive or negative structural geometry (as a geometry for which the energy release rate at constant load increases or decreases with the crack length) cannot be extended to stability problems or geometrically nonlinear behavior. Comparison is made with a previous simplified solution by Okamura and coworkers. The analytical results show good agreement with the available experimental data.     

Stress intensity factors for cracks emanating from a triangular or square hole in an infinite plate by boundary elements

This paper concerns stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure calculated by means of a boundary element method, which consists of constant displacement discontinuity element presented by Crouch and Starfied and crack tip displacement discontinuity elements proposed by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors of plane elasticity crack problems. Specifically, the numerical results of stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure are given.     

Effect of underloads or overloads in fatigue crack growth by crack-tip blunting

Crack-tip blunting under tensile loads and re-sharpening of the crack-tip during unloading is one of the basic mechanisms for fatigue crack growth in ductile metals. Here, based on an elastic-perfectly plastic material model, crack growth computations are continued up to 700 full cycles by using re-meshing at several stages of the plastic deformation. A compressive underload in one of the cycles tends to increase the rate of cyclic crack growth, and this effect is studied in detail for a single underload, based on the blunting re-sharpening mechanism. Subsequently, the increased rate of crack growth due to periodically occurring underloads is analysed. A single overload has the opposite effect of giving a significant delay in the subsequent fatigue crack growth. An analysis is carried out to compare the effect of a small overload to that of a larger overload.     

A remeshing algorithm for three-dimensional crack growth and intersection with surfaces or cracks in non-coplanar planes

A recent procedure developed for crack growth in which three-dimensional stress intensity factors are calculated by boundary integral equations is used to compute fatigue crack growth as cracks grow in accordance with Paris’ law and also intersect each other or other surfaces. This paper concentrates on describing the remeshing algorithms needed as the cracks grow and when they intersect each other or other surfaces. The algorithms produced treat crack growth; prior to intersections, after intersection with a free surface, after intersection of a crack edge with a crack surface away from its edge, and after intersection of coplanar cracks.The method is applied here to the growth of an initially circular crack at the centre of a block under uniform tensile traction on the faces parallel to the crack. The crack grows to intersect either a free surface of the block or the centre of a square crack in an orthogonal plane.     

Criterion for brittle fracture of notched or cracked specimens based on combined micro- and macro crack mechanics—II

Brittle fracture criterion based on combined micro- and macro crack mechanics was proposed in the previous paper. In the present paper more improved criterion for the brittle fracture based on this approach has been obtained as a function of ferrite grain diameter, over much wider range of crack tip radius including the case of small crack tip radius. The criterion is in good agreement with the experimental results on brittle fracture of notched or cracked specimens of low carbon steel.     

Rectangular tensile sheet with single edge crack or edge half-circular-hole crack

By using the displacement discontinuity method with crack-tip elements (a boundary element method) proposed recently by the author, this note presents the stress intensity factors (SIFs) of a rectangular tensile plate with single edge crack. Further this note studies the SIFs of crack emanating from an edge half-circular hole. By comparing the calculated SIFs of the single edge half-circular-hole crack with those of the single edge crack, a shielding effect of the half-circular hole on the SIFs of the single edge crack is discussed. It is found that the boundary element method is simple, yet accurate for calculating the SIFs of complex crack problems in finite plate.     

Comparison of predictions by mode II or mode III criteria on crack front twisting in three or four point bending experiments

Whatever the external loading, a crack front in a solid tries to reach mode I loading conditions after propagation. In mode I + II, the crack kinks to annihilate mode II, kinking angle being well predicted by the principle of local symmetry (PLS) or by the maximum tangential stress criterion (MTS). In presence of mode III, the problem becomes three-dimensional and the proposed propagation criterion are not yet well proved and established. In particular in three point bending experiments (3PB) with an initially inclined crack, the crack twists around the direction of propagation to finally reach a situation of pure mode I. The aim of the paper is to compare the propagation paths predicted by two different criteria for 3PB fatigue experiments performed on PMMA. The first criterion developed by Schollmann et al. (Int J Fract 117(2):129–141, 2002), is a three-dimensional extension of the MTS criterion and predicts the local angles that annihilates mode II and III at each point of the front. The second one developed by Lazarus et al. (J Mech Phys Solids 49(7):1421–1443, 2001b), predicts an abrupt and then progressive twisting of the front to annihilate mode III. Due to presence of sign changing mode II and almost uniform mode III in the experiments, both criteria give good results. However, since mode III is predominant over mode II in the case under consideration, the global criterion gives better results. Nevertheless, the local type criterion seems to be of greater universality for practical engineering applications.     

What happens after a fibre breaks — pull-out or resin cracking?

We consider tensile fracture of a specimen consisting of a single rigid fibre embedded in a cylindrical block of a linearly-elastic resin. When the fibre breaks, two possible modes of failure can occur. A circular crack may propagate outwards into the resin, leading to fracture of the specimen. Alternatively, a cylindrical crack can propagate along the fibre-matrix interface, starting from the break in the fibre, leading to fibre pull-out. The question is: which mode of failure will occur in practice? Finite-element analysis is used here to calculate the pull-out force and the force causing growth of a circular crack outwards into the resin, for samples containing fibres of different radius. A general criterion is obtained to predict the mode of failure. Even for samples with perfect adhesion between resin and fibre, pull-out of the fibre is expected when the fibre radius is less than about one-fifth of the sample radius. For fibres of larger radius, either pull-out or resin cracking can take place, depending on the relative levels of interfacial fracture energyG _a and resin fracture energyG _c.     

Multilayered piezomagnetic/piezoelectric composite with periodic interfacial Yoffe-type cracks under magnetic or electric field

The problem of periodic mode-III Yoffe-type cracks propagating subsonically along the interfaces in a multilayered piezomagnetic/piezoelectric composite under in-plane magnetic or electric field is studied. By means of periodic conditions, the analysis of the multilayered problem is simplified to a bilayer model with an interfacial Yoffe-type crack, which can be reduced to the Cauchy singular integration equation of the first kind, by utilizing the Fourier transform. The normalized dynamic stress intensity factor (NDSIF) can be obtained numerically. Results show that the NDSIF generally depends on the layer thickness ratio, crack moving speed, electric or magnetic loading as well as material properties. In regard to the curve monotony of the NDSIF versus the passive layer thickness, there generally exist three different cases distinguished by a parameter, θ, which depends on the crack moving speed as well as material mismatch parameters. Similar behavior has been reported for the periodic static cracks where the different monotonies are judged by the material mismatch parameter G (Wan et al. in Eng. Fract. Mech. 84:132–145, 2012). The present results reduce to those of periodic static cracks when considering the vanishing crack speed. As far as the curve monotony and the judging parameter are concerned, no matter what the crack moving speed is, the Yoffe-type crack problem is identical to the static crack problem when the piezomagnetic and piezoelectric layers share the same shear wave velocity. In addition, detailed analyses are also provided for NDSIF versus crack moving speed for different layer thickness ratio and material mismatch parameters. This study can be considered as an extension of the previous analysis of periodic static cracks problem and is expected together to provide some guidelines for the optimal design of a multilayered PM/PE composite.