A 3-D ellipsoidal flaw model for brittle fracture in compression

Engineering materials are rarely free of flaws. Mode I cracking from pre-existing flaws is the major cause of the brittle fracture in compression of materials such as concrete and rock. A 3-D ellipsoidal flaw model is used to show the significant influence of flaw geometry on crack initiation in uniform uniaxial, biaxial and triaxial compression. The model shows that the governing criterion for crack initiation may change from energy to stress with increasing crack size, and that for voids of similar size a spherical void is the most critical shape for crack initiation. The model thus provides a basis for a better understanding of both the phenomenon and the mechanism of brittle fracture in compression.     

Modeling dynamic brittle behavior of materials with circular flaws or pores

Compressive failure of brittle materials is driven primarily by crack growth from pre-existing flaws in the material. These flaws, such as grain boundaries, pores, preexisting cracks, inclusions and missing grains, are randomly spaced and have a range of possible shapes and sizes. The current work proposes a micromechanics-based model for compressive dynamic failure of brittle materials with circular pore flaws, which incorporates both the number density and the size distribution of flaws. Results show that the distribution of flaw sizes is very important, particularly at moderate strain rate, since analyses based solely on the mean flaw size overpredict strength. Therefore, in order to increase dynamic strength at low to moderate strain rates, it is most effective to control the presence of large flaws. At very high strain rates, however, crack growth is activated even in small flaws and therefore controlling the total number density rather than the size of the flaws is effective for increasing dynamic strength. Finally, the model shows that neglecting very small flaws in the pore population may not have significant effects on the results in many cases, suggesting that the model is a useful tool for identifying a minimum resolution required for experimental characterization of microstructure.     

Equations and a fast algorithm for determining the probability of failure initiated by flaws

Powerful equations and an efficient algorithm are proposed for determining the probability of failure of loaded components with complex shape, containing multiple types of flaws. The equations are based on the concept ‘conditional individual probability of initiating failure’ characterising a single flaw given that it is in the stressed component. The proposed models relate in a simple fashion the conditional individual probability of failure characterising a single flaw (estimated by a Monte Carlo simulation) to the probability of failure characterising a population of flaws. The derived equations constitutes the core of a new statistical theory of failure initiated by flaws in the material, with important applications in optimising designs by decreasing their vulnerability to failure initiated by flaws during overloading or fatigue cycling.Methods have also been developed for specifying the maximum acceptable level of the flaw number density and the maximum size of the stressed volume which guarantee that the probability of failure initiated by flaws remains below a maximum acceptable level. An important parameter referred to as ‘detrimental factor’ is also introduced. Components with identical geometry and material, with the same detrimental factors are characterised by the same probability of failure. It is argued that eliminating flaws from the material should concentrate on types of flaws characterised by large detrimental factors.The equations proposed avoid conservative predictions resulting from equating the probability of failure initiated by a flaw in a stressed region with the probability of existence of the flaw in that region.     

Scattered-light determination of mode I stress-intensity factors by a fringe gradient independent technique

Integration of the scattered-light stress-optic law and established bending and singular crack-tip relationships yielded new experimental equations for calibration and for the determination of mode I stress-intensity factors which are independent of a stress-fringe gradient. Scattered-light stressoptic coefficients determined from four-point bending tests and an integrated scattered-light bending equation show good agreement with values based on stress-fringe gradients computed with polynomials. Excellent agreement was also shown between mode I stress-intensity factors predicted by the integrated stress-optic equation and analytical solutions available in the literature. Favorable comparisons were also made with predictions based on a polynomial-finite-difference method of determining a stress-fringe gradient. Analyses were limited to flaw geometries and locations where there was minimal rotation of the refraction tensor.     

Stress-intensity distributions for corner cracks emanating from open holes in plates of finite width

A series of frozen stress photoelastic tests is carried out in the present investigation. Linear Elastic Fracture Mechanics is used to analyze the problem of a part-through corner crack at an open hole in a plate of finite width loaded in tension. The photoelastic modeling capability of three-dimensional subcritical crack growth problems is assessed. This is achieved by comparing monotonically grown flaw shapes in epoxy models with crack profiles relative to fatigue crack propagation tests in a different material. Photoelastic stress-intensity factor distributions are checked against numerical results obtained for quarter elliptical geometries. The dependence of stress intensity factors on flaw shape is also discussed.     

Scattered-light analysis of a surface-flawed plate subjected to cylindrical bending

The scattered-light photoelastic technique was utilized to determine Mode I stress-intensity factors associated with a semi-elliptical surface flaw in a plate subjected to cylindrical bending. Stress-intensity factors were experimentally determined for the point of maximum flaw penetration and the point of intersection of the flaw border with the free surface of the plate. Experimental results compare favorably with those obtained in a three-dimensional finite-element analysis.     

冲击载荷下两裂纹间的连通规律

脆性材料内部含有大量裂纹,当某一裂纹扩展时,其他裂纹会对扩展裂纹产生影响。为了研究冲击载荷下,脆性材料内两裂纹的相互影响、连通规律及裂纹尖端应力强度因子的变化规律,利用有机玻璃板制作了含非平行双裂纹的实验试件,利用落板冲击设备进行了中低速冲击实验,结合有限元分析软件ABAQUS计算出裂纹尖端应力强度因子,利用有限差分软件AUTODYN进行了动态数值模拟研究,并将其模拟结果与实验结果进行对比分析。实验及模拟结果表明:裂纹破坏形态与AUTODYN数值模拟破坏形态基本一致;试件的断裂形态随着两裂纹间距不同而不同;裂纹间的相互影响程度随着裂纹间间距增大而减小;裂纹尖端应力强度因子K_I随着裂纹间距的增大而减小,而K_II随着裂纹间距增大而增大。     

Stress-strain state of an anisotropic plate with curved cracks and thin rigid inclusions

We solve the bending problem for an anisotropic plate with flaws like smooth curved nonoverlapping through cracks and rigid inclusions. The problem is solved by the method of Lekhnitskii complex potentials specified as Cauchy type integrals over the flaw contours with an unknown integrand density function. We use the Sokhotskii—Plemelj formulas to reduce the boundary-value problem to a system of singular integral equations with the additional conditions that the displacements in the plate are single-valued when going around the cut contours and the equilibrium conditions for stress-free rigid inclusions. After the singular integrals are approximated by the Gauss-Chebyshev quadrature formulas, the problem is reduced to solving a system of linear algebraic equations. We study the local stress distribution near flaw tips. We analyze the mutual influence of flaws on the stress distribution character near their vertices and compare the well-known solutions for isotropic plates with the solutions obtained by passing to the limit in the anisotropy parameters (“weakly anisotropic material”) and by using the method proposed here.     

Experimental study on cracking behaviour of moulded gypsum containing two non-parallel overlapping flaws under uniaxial compression

Failure of rock mass that is subjected to compres-sive loads occurs from initiation, propagation, and linkage of new cracks from preexisting fissures. Our research inves-tigates the cracking behaviour and coalescence process in a brittle material with two non-parallel overlapping flaws using a high-speed camera. The coalescence tensile crack and tensile wing cracks were the first cracks to occur from the pre-existing flaws. The initiation stresses of the primary cracks at the two tips of each flaw were simultaneous and decreased with reduced flaw inclination angle. The following types of coalescence cracks were identified between the flaws: pri-mary tensile coalescence crack, tensile crack linkage, shear crack linkage, mixed tensile-shear crack, and indirect crack coalescence. Coalescence through tensile linkage occurred mostly at pre-peak stress. In contrast, coalescence through shear or mixed tensile-shear cracks occurred at higher stress. Overall, this study indicates that the geometry of preexisting flaws affect crack initiation and coalescence behaviour.     

Numerical computation and analytical estimation of stress-intensity factors for strips with multiple edge cracks

In this paper, stress-intensity factors for a two-dimensional problem are determined. Strips with multiple symmetrical edge cracks in tension are investigated. A simple analytical estimation is compared to numerical results. The influence of penetration of the crack faces and mixed-mode loading on the numerical results is investigated. A simple method to estimate stress-intensity factors for strips with multiple edge cracks is proposed.     

A layered composite containing a crack in its lower layer loaded by a rigid stamp

The elastostatic plane problem of a layered composite containing an internal or edge crack perpendicular to its boundaries in its lower layer is considered. The layered composite consists of two elastic layers having different elastic constants and heights and rests on two simple supports. Solution of the problem is obtained by superposition of solutions for the following two problems: The layered composite subjected to a concentrated load through a rigid rectangular stamp without a crack and the layered composite having a crack whose surface is subjected to the opposite of the stress distribution obtained from the solution of the first problem. Using theory of Elasticity and Fourier transform technique, the problem is formulated in terms of two singular integral equations. Solving these integral equations numerically by making use of Gauss–Chebyshev integration, numerical results related to the normal stress σ_x(0,y), the stress-intensity factors, and the crack opening displacements are presented and shown graphically for various dimensionless quantities.     

Elastodynamic inversion for shape reconstruction and type classification of flaws

Two linearized inverse scattering methods are investigated to reconstruct the shape of flaws in the elastic solid. One is based on the Born approximation and the other is based on the Kirchhoff approximation. The Born inversion is sensitive to the volumetric flaw but not to a crack-like flaw. On the other hand, the Kirchhoff inversion reacts to both boundaries of volumetric and crack-like flaws. The combined use of Born and Kirchhoff inversions leads to the classification method of flaw type. The performance of the proposed classification procedure is demonstrated by the numerical simulations and then by the experimental measurements for the two-dimensional models of flaws. An example for the three-dimensional shape reconstruction is also shown by using the numerically calculated backscattering amplitudes.     

Dynamic fracture toughness determined from load-point displacement

The paper presents a method to determine dynamic fracture toughness using a notched three-point bend specimen. With dynamic loading of a specimen there is a complex relation between the stress-intensity factor and the force applied to the specimen. This is due to effects of inertia, which have to be accounted for to evaluate a correct value of the stress-intensity factor. However, the stress-intensity factor is proportional to the load-point displacement if the fundamental mode of vibration is predominant in the specimen. The proportionality constant depends only on the geometry and stiffness of the specimen. In the present method we have measured the applied force and load-point displacement by a modified Hopkinson pressure bar, where two-point strain measurement has been used to evaluate force and displacement for times greater than the transit time for elastic waves in the Hopkinson bar. We have compared the method with the stress-intensity factor derived from strain measurement near the notch tip and good agreement was obtained. The method is well suited for high-temperature testing and results from fracture toughness tests of brittle materials at ambient and elevated temperatures are presented.     

An interacting micro-crack damage model for failure of brittle materials under compression

A model is developed for brittle failure under compressive loading with an explicit accounting of micro-crack interactions. The model incorporates a pre-existing flaw distribution in the material. The macroscopic inelastic deformation is assumed to be due to the nucleation and growth of tensile “wing” micro-cracks associated with frictional sliding on these flaws. Interactions among the cracks are modeled by means of a crack-matrix-effective-medium approach in which each crack experiences a stress field different from that acting on isolated cracks. This yields an effective stress intensity factor at the crack tips which is utilized in the formulation of the crack growth dynamics. Load-induced damage in the material is defined in terms of a scalar crack density parameter, the evolution of which is a function of the existing flaw distribution and the crack growth dynamics. This methodology is applied for the case of uniaxial compression under constant strain rate loading. The model provides a natural prediction of a peak stress (defined as the compressive strength of the material) and also of a transition strain rate, beyond which the compressive strength increases dramatically with the imposed strain rate. The influences of the crack growth dynamics, the initial flaw distribution, and the imposed strain rate on the constitutive response and the damage evolution are studied. It is shown that different characteristics of the flaw distribution are dominant at different imposed strain rates: at low rates the spread of the distribution is critical, while at high strain rates the total flaw density is critical.     

Sudden twisting of a penny-shaped flaw in a certain non-homogeneous elastic medium

The small-time axisymmetric elastic field due to a sudden or impulsive twist applied to a penny-shaped flaw in a radially non-homogeneous isotropic linear clastic medium is investigated. A Green's function approach is adopted leading to an integral equation which is solved analytically by the Wiener-Hopf technique. Using this approach a theorem which relates the solutions of the problem under stress and displacement boundary conditions is proved and applied. General and explicit results for the displacements and normal displacement gradients of the plane of the flaw and also the stress-intensity factors are given.

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Zusammenfassung Das kurzzeithlich axialsymmetrische elastische Feld verursacht durch eine ploetzliche oder impulsive Drehung, die auf einen groschenfoermigen Riss in einem radial unhomogenen isotropischen linearen elastischen Medium augewendet wird, wird untersucht. Ein Green Funktionsansatz wird angewendet der zu einer integralen gelichung fuehrt, die analytisch durch die Wiener-Hopf-Technik geloest wird. Mit dicsem Ansatz wird ein Theorem bewiesen und angewendet das die Loesungen des Problems unter Stress und Verdraengungsrand bedingungen verbindet. Generelle und explizite Resultate Fuer die Verdraengungen und normale Vergraengungsgradienten der Ebene des Risses und auch die Stress intensitaetsfaktoren werded gegeben.

     

Computational micromechanics of dynamic compressive loading of a brittle polycrystalline material using a distribution of grain boundary properties

A two-dimensional finite element model is used to investigate compressive loading of a brittle ceramic. Intergranular cracking in the microstructure is captured explicitly by using a distribution of cohesive interfaces. The addition of confining stress increases the maximum strength and if high enough, can allow the effective material response to reach large strains before failure. Increasing the friction at the grain boundaries also increases the maximum strength until saturation of the strength is approached. Above a transitional strain rate, increasing the rate-of-deformation also increases the strength and as the strain rate increases, fragment sizes of the damaged specimen decrease. The effects of flaws within the specimen were investigated using a random distribution at various initial flaw densities. The model is able to capture an effective modulus change and degradation of strength as the initial flaw density increases. Effects of confinement, friction, and spatial distribution of flaws seem to depend on the crack coalescence and dilatation of the specimen, while strain-rate effects are result of inertial resistance to motion.     

Some propositions on caustics and an application to the biaxial-fracture problem

Some fundamental problems on the optical method of caustics, which is used to determine the value of the stress-intensity factor, are studied. The paper shows that the radiusr _o of the initial curve considerably affects the results and describes a method to eliminate this effect. This method is also applied to the biaxial-stress fracture problem. It is shown that the biaxial stress affects the fracture toughness of thin specimens of Plexiglas sheets and that the fatigue-crack-growth rate is increased due to the compressive stress along the crack line.Paper was presented at 1982 SESA/JSME Spring Meeting held in Oahu and Maui, HI on May 23–29, 1982.     

Combined fracture and delamination: A simple idealized example

Under static pre-stress, small voids located between constituent layers in a laminated composite may extend both as interface flaws and as cracks into the layers themselves. Thus, the material may fail due to the combined effects of delamination (separation) and fracture of the layers. As a first step in understanding this complex process, while minimizing mathematical difficulties, a simple idealized example of such a process is examined. A cruciform brittle crack-flaw combination is assumed to initiate at the interface of perfectly-bonded half-spaces under a uniform anti-plane shear field and to extend with constant speeds. When the crack and wave speeds satisfy a simple relation, straight-forward solutions to the resulting wave propagation problem can be obtained and several crackflaw interaction effects noted. In particular, the stress intensities and energy flux rates at the crack and flaw edges are studied.     

Dynamic photoelastic and dynamic finite-element analyses of dynamic-tear-test specimens

The transient response of the dynamic-tear-test specimen of a brittle material, Homalite-100, was investigated by dynamic photoelasticity and dynamic finite-element method. The dynamic stress-intensity factors obtained from dynamic photoelasticity and dynamic finite-element analyses were in reasonable agreement with each other. The dynamic finite-element analysis also showed that the dynamic-fracture-initiation toughness could be determined from the dynamicstrain response of a strain gage located near the crack tip in conjunction with a simple static analysis. Dynamic-fracture-toughness vs. crack-velocity relation was also obtained.