Effect of initial flaw shape on crack extension in orthotropic composite materials

An analytical study is presented showing the effects of the notch tip geometry on the location and direction of crack growth from an existing notch in a unidirectional fibrous composite modelled as a homogeneous, anisotropic, elastic material. Anisotropic elasticity and the normal stress ratio theory are used to study crack growth from elliptical notches in unidirectional composites. Sharp cracks, circular holes, and ellipses are studied under far-field tension, and shear loading. Limited comparisons are made showing good correlation with experimental results.     

DETERMINATION OF CRACK SURFACE DISPLACEMENTS FOR A RADIAL CRACK EMANATING FROM A SEMI-CIRCULAR NOTCH USING WEIGHT FUNCTION METHOD

A radial crack emanating from a semi-circular notch is of significant engineering importance. Accurate determination of key fracture mechanics parameters is essential for damage tolerance design and fatigue crack growth life predictions. The purpose of this paper is to provide an efficient and accurate closed-form weight function approach to the calculation of crack surface displacements for a radial crack emanating from a semi-circular notch in a semi-infinite plate.Results are presented for two load conditions: remote applied stress and uniform stress segment applied to crack surfaces. Based on a correction of stress intensity factor ratio, highly accurate analytical equations of crack surface displacements under the two load conditions are developed by fitting the data obtained with the weight function method. It is demonstrated that the WuCarlsson closed-form weight functions are very efficient, accurate and easy-to-use for calculating crack surface displacements for arbitrary load conditions. The method will facilitate fatigue crack closure and other fracture mechanics analyses where accurate crack surface displacements are required.     

Higher order fields for damaged nonlinear antiplane shear notch, crack and inclusion problems

Damaged nonlinear antiplane shear problems with a variety of singularities are studied analytically. A deformation plasticity theory coupled with damage is employed in analysis. The effect of microscopic damage is considered in terms of continuum damage mechanics approach. An exact solution for the general damaged nonlinear singular antiplane shear problem is derived in the stress plane by means of a hodograph transformation, then corresponding higher order asymptotic solutions are obtained by reversing the stress plane solution to the physical plane. As example, traction free sharp notch and crack, rigid sharp wedge and flat inclusion, and mixed boundary sharp notch problems are investigated, respectively. Consequently, higher order fields are obtained, in which analytical expressions of the dominant and second order singularity exponents and angular distribution functions of the near tip fields are derived. Effects of the damage and hardening exponents of materials and the geometric angle of notch/wedge on the near tip quantities are discussed in detail. It is found that damage leads to a weaker dominant singularity of stress, but to little stronger singularities of the dominant and second order terms of strain compared to that for undamaged material. It is also seen that damage has important effect on the angular distribution functions of the near tip stress and strain fields. As special cases, higher order analytical solutions of the crack and rigid flat inclusion tip fields are obtained, respectively, by reducing the notch/wedge tip solutions. Effects of damage and hardening exponents on the dominant and second order terms in the solutions of the crack and inclusion tip fields are discussed.     

含有非圆形双孔的无限平板中应力的解析解研究

无限平板中含有任意形状单个孔的问题可以使用复变函数方法获得其应力解析解.对于无限平板中含有两个圆孔或两个椭圆孔的双连通域问题,也可以利用多种方法进行求解,比如双极坐标法、应力函数法、复变函数法以及施瓦茨交替法等.其中复变函数中的保角变换方法是获得应力解析解的一个重要方法.但目前尚未见到用此方法求解无限板中含有一个正方形孔和一个椭圆孔的问题.当板在无穷远处受有均布载荷和孔边作用垂直均布压力时,利用保角变换方法可以求解板中含有两个特定形状孔的问题.该方法将所讨论的区域映射成象平面里的一个圆环,其中最关键的一步是找出相应的映射函数.基于黎曼映射定理,提出了该映射函数一般形式,并利用最优化方法,找到了该问题的具体映射函数,然后通过孔边应力边界条件建立了求解两个解析函数的基本方程,获得了该问题的应力解析解.运用ANSYS有限单元法与结果进行了对比.研究了孔距、椭圆形孔大小和两孔布置方位对边界切向应力的影响,以及不同载荷下两孔中心线上应力分布规律.      

Numerical simulation of loading edge cracks by edge impact using the extended finite element method

The extended finite element method is used to analyze a plate with two parallel edge cracks impacted by a cylindrical projectile. The influence of the impact speed, crack length,plate thickness and notch tip radius on the crack initiation and propagation is studied. Dynamics equations are solved by an implicit time integration scheme which is unconditionally stable. Very good agreement is achieved between numerical predictions and experimental results. The critical velocity of the crack initiation under different conditions is examined. The influence of the crack length is greater than that of the impact speed, plate thickness and notch tip radius.     

Assessment of creep crack growth due to stress relief

An analysis is conducted to predict stress relief cracking at 550 °C in notched compact tension specimens of Type 316H austenitic stainless steel. The specimens had been subjected to pre-compression to generate a tensile residual stress distribution at the notch tip. This stress distribution is represented by a uniform reference stress over the zone of tension ahead of the notch tip. Creep rupture and creep crack growth data alone are required and used to make the predictions. It is found that the shape of the crack growth curve is correctly predicted when mean data are employed. However, upper bound crack growth properties are required to accurately predict the actual extent of cracking. Sensitivity studies show that the amount of stress relief cracking predicted is relatively insensitive to the reference stress initially assumed to describe the residual stress distribution, since the reference stress relaxes to a magnitude that is almost independent of its initial value. Adoption of an initial reference stress equal to the ultimate tensile strength of the steel, when combined with mean creep rupture and upper bound crack growth properties, results in safe predictions that are not overly conservative. The analysis should only be regarded as reliable for small amounts of crack extension of less than the size of the tensile zone ahead of the crack tip.     

Crack growth in unidirectional graphite-epoxy under biaxial loading

Center-notched coupons of 16-ply unidirectional AS4/3501-6 graphite-epoxy were loaded in tension to failure in order to determine the direction of initial crack growth, the load required to cause initial crack growth, and the load required to cause failure. Fifteen-degree off-axis tensile coupons provided a far-field biaxial state of stress away from the notch. Specimens of three different length-to-width ratios were tested in order to vary the biaxial stress states. Two notch configurations were used, one perpendicular to the loading direction, the other perpendicular to the material fiber direction. In all of the cases studied experimentally, crack growth was parallel to the fiber direction. An anisotropicelasticity crack-tip stress analysis was subsequently implemented in order to compare experimental crack-growth directions to those predicted by three crack-extension-direction criteria. Of the three criteria studied, (Strain-energy density, the tensor polynomial, and the normal stress ratio), only one, the normal-stress-ratio criterion, provided crack-growth-direction predictions that agreed with the experimental results. Paper was presented at the 1985 SEM Spring Conference on Experimental Mechanics, held in Las Vegas, NV on June 9–14.     

A small-scale magnetic-yielding model for an infinite magnetostrictive plane with a crack-like flaw

The effect of an external magnetic field on the fracture toughness of magnetostrictive materials has been investigated by determining the local stress fields around the tip of a very slender elliptical flaw embedded in an infinite magnetostrictive plane subjected to magnetic loading, based on the assumption of linear magnetization. In this paper, the above-mentioned analytical approach is extended to develop a small-scale magnetic-yielding model. The magnetic saturation zone is constructed and the distributions of magnetic field and magnetization are obtained around the tip of a slender elliptical crack. Based on the complex potential theory, the stress field is obtained in the vicinity of the tip of the slender elliptical crack by implementing the continuity conditions of displacement and resultant force at the interface between the magnetic saturation and magnetoelastic zones. The stress fields near the tip of the slender elliptical crack are obtained for two kinds of soft ferromagnetic materials each with a small induction magnetostrictive modulus. The theoretical results obtained show that the stresses in the neighborhood of a crack-tip are finite even when the elliptical crack reduces to a sharp crack, and are much smaller than the yield stress or the nominal fracture stress of the material. This suggests that, generally, the magnetic field has no obvious effects on the apparent fracture toughness of soft ferromagnetic materials, which is in agreement with the existing experimental results published in the existing literature. In addition, the theoretical analysis illustrates that no crack is magnetically impermeable, and the corresponding boundary conditions are inappropriate for fracture analysis of soft ferromagnetic materials.     

各向异性介质中SH波引起的裂纹扩展

本文利用Green函数法,求解各向异性介质中半无限长裂纹在SH波作用下,以任意速度扩展的问题。首先,利用Laplace变换和Cagniard-de Hoop反演法求解各向异性介质中反平面问题的Green函数,并利用它建立了求解裂纹扩展问题的积分方程。因为方程为Abel型的,所以可得到在SH波作用下,半无限长裂纹扩展问题的解析解。还可求得裂纹端点附近的应力和裂纹表面上位移的表达式。并对裂纹端点附近的奇异性进行讨论。最后讨论了裂纹尖端附近任一点的能量关系。并应用Griffith的能量准则,对裂纹扩展规律进行了讨论。     

An Application of the Kane and Mindlin Theory to Crack Problems in Plates of Arbitrary Thickness

Classical plane solutions of the theory of elasticity, which are sometimes more than 100 years old, are still used today and provide a framework for the analysis of many practical problems. But, strictly speaking, these analytical solutions are only applicable to plates with vanishing thickness or infinite thickness, where the stress state could be classified as plane stress or plane strain, respectively. However, the through-the-thickness stresses that exist in a plate of given thickness have a significant impact in a number of practical applications; and these stresses are often inevitably ignored due to the lack of analytical tools. This paper presents new analytical results for crack tip opening displacement (CTOD) for the through-the-thickness crack in infinite plates with various thicknesses. These results are based on the solution for an edge dislocation in infinite plate of arbitrary thickness and an application of the distributed dislocation technique. The analytical predictions of the CTOD and the constraint factor are compared with the three-dimensional elasto-plastic finite element (FE) results. It is shown that both analytical and numerical results are in good agreement when the numerical calculations are not affected by the size of the FE mesh and by the boundaries of the FE model.     

Stable growth of a crack interacting with a circular inclusion

The problem of failure of a plate containing a circular inclusion and a crack is studied. The crack is oriented along a diameter of the inclusion and the plate is subjected to a remote uniaxial stress perpendicular to the crack axis. The process of slow stable crack growth from initiation to termination is studied by the strain energy density theory. The crack growth is simulated by predicting finite increments of crack extension when material elements near the crack tip absorb a critical amount of strain energy density level, . Unstable crack growth occurs when the strain energy density factor S reaches a critical value where r_c is the critical size of the final crack increment prior to instability. The stress at crack initiation and the critical stress and crack length at failure are determined. The influence of the mechanical properties of the plate and the inclusion, the relative position of the inclusion and the crack and the crack length on the characteristic quantities of stable crack growth is analyzed. The dependence of the stable crack growth process on the loading rate is also investigated. Results are displayed in graphical form.     

单边裂纹通电瞬间裂尖处应力场的复变函数解

本文应用复变函数中的Schwarz-Christoffel变换方法，在具单边裂纹的导电薄板通电瞬间温度场复变函数解的基础上，推导出用复变函数表示的应力场的表达式，并且给出算例，通过理论计算得知；当对具有单边裂纹的导电薄板通入适当密度的电流时，裂尖处温度急剧升高并熔化便裂尖变钝。同时，在裂尖周围形成了有利于遏制裂纹扩展的压应力场，有效地防止了裂纹沿其主方向和其它方向延伸。从理论上证明了电磁热效应在裂尖处产生高温形成焊口的同时，压应力场的形成是遏制裂纹扩展的主要因素之一。理论计算结果与实验结果比较吻合，为这一止裂方法的应用打下了理论基础。     

Mixed-mode crack growth in plates under three-point bending

A combined theoretical and experimental study of the problem of crack growth in a plate subjected to unsymmetrical three-point bending was undertaken. The opening-modeK _I and sliding-modeK _II stress-intensity factors describing the local stress field around the crack tip were determined by a finite-element computer program. The crack growth was analyzed by the maximum circumferential stress and the minimum strain-energy density criteria. The critical loads for crack growth and the crack trajectories were determined both by theory and experiment. The experimental results corroborated the theoretical predictions.     

A discrete-fibers model for bridged cracks and reinforced elliptical voids

A model of crack bridging and reinforced elliptical voids is proposed, in which the fibers joining the surfaces of the void or crack are modelled as discrete, linear elastic bars. We show that a theory recently developed by us to analyze structural interfaces permits analytical attack and solution of multiple important previously unsolved problems of stress concentration and fracture. In particular, an analytical solution is provided for a reinforced elliptical void, which, by superposition, allows treatment of arbitrary fiber distributions, which can be even randomly distributed and oriented. In the special case of small or null ratio between a void's axes, new stress intensity factor expressions are obtained, which account for fibers’ inclination and geometry.     

EXACT SOLUTION FOR ORTHOTROPIC MATERIALS WEAKENED BY DOUBLY PERIODIC CRACKS OF UNEQUAL SIZE UNDER ANTIPLANE SHEAR

Orthotropic materials weakened by a doubly periodic array of cracks under far-field antiplane shear are investigated, where the fundamental cell contains four cracks of unequal size. By applying the mapping technique, the elliptical function theory and the theory of analytical function boundary value problems, a closed form solution of the whole-field stress is obtained. The exact formulae for the stress intensity factor at the crack tip and the effective antiplane shear modulus of the cracked orthotropic material are derived. A comparison with the finite element method shows the efficiency and accuracy of the present method. Several illustrative examples are provided, and an interesting phenomenon is observed, that is, the stress intensity factor and the dimensionless effective modulus are independent of the material property for a doubly periodic cracked isotropic material, but depend strongly on the material property for the doubly periodic cracked orthotropic material. Such a phenomenon for antiplane problems is similar to that for in-plane problems. The present solution can provide benchmark results for other numerical and approximate methods.     

Prediction and observation of crack tip microstructure in shape memory CuAlNi single crystals

The formation of martensite at a notch tip in a CuAlNi shape memory alloy loaded in tension is studied. The geometry of the initial martensite plate to form at the notch is predicted theoretically, using the stress field at a crack tip in an anisotropic linearly elastic body together with a listing of all possible austenite-martensite interfaces from the Crystallographic Theory of Martensite (CTM). The stress field and CTM analyses are combined through a selection criterion based on computing the work available from the stress field to transform to each austenite-martensite interface. The resulting predictions are compared to experimentally observed microstructures in notched specimens of single crystal CuAlNi loaded in tension for eight notch orientations. Results show that the available work criterion accurately predicts the orientation, number and order of the austenite-martensite interfaces that initially form near a crack.     

Analytical representation of the non-square-root singular stress field at a finite angle sharp notch

The stress field near the tip of a finite angle sharp notch is singular. However, unlike a crack, the order of the singularity at the notch tip is less than one-half. Under tensile loading, such a singularity is characterized by a generalized stress intensity factor which is analogous to the mode I stress intensity factor used in fracture mechanics, but which has order less than one-half. By using a cohesive zone model for a notional crack emanating from the notch tip, we relate the critical value of the generalized stress intensity factor to the fracture toughness. The results show that this relation depends not only on the notch angle, but also on the maximum stress of the cohesive zone model. As expected the dependence on that maximum stress vanishes as the notch angle approaches zero. The results of this analysis compare very well with a numerical (finite element) analysis in the literature. For mixed-mode loading the limits of applicability of using a mode I failure criterion are explored.     

Fracture mechanics of plates and shells applied to fail-safe analysis of fuselage Part I: Theory

In this paper, a general theory on the asymptotic field near the crack tip for plates and shells with and without shear deformation effect is established. It is found that four stress intensity factors, two for symmetrical and antisymmetrical stretching and two for symmetrical and antisymmetrical bending, are required to describe arbitrary asymptotic fields near the crack tip for plates without shear deformation. An additional stress intensity factor is required for the transverse shearing force induced by antisymmetrical bending when the shear deformation is included in the analysis. It is also proven by means of the complex variable technique that for problems of plates with shear deformation, there exist similarities in the asymptotic expressions of moments and membrane forces and also in the asymptotic expressions of in-plane displacements and rotations of the mid-surface. The energy release rate associated with crack growth in the direction of the crack line can be expressed in terms of stress intensity factors by means of Irwin's method of work and energy associated with a virtual crack extension. A combined stress intensity factor can be defined through the total energy release rate. The theory of the fracture of plates is generalized and applied to the study of problems in the fracture of shells. An example of an infinitely long cylindrical shell with a circumferential crack subjected to remote axial tension is given to demonstrate the application of the theory and to test the accuracy of the numerical analysis used for the problem.     

Equivalence of the notch stress intensity factor,tip opening displacement and energy release rate for a sharp V-notch

For an infinite elastic plane with a sharp V-notch under the action of symmetrically loading at infinity, the length of crack initiation ahead of the V-notch’s tip is estimated according to a modified Griffith approach. Applying a new conservation integral to the perfectly plastic strip (Dugdale model) ahead of the V-notch’s tip, the relationship between notch stress intensity factor (NSIF) and notch tip opening displacement (NTOD) is presented. Also, the relationship between NSIF and perfectly plastic strip size (PPSS) is found. Since there are three fracture parameters (NSIF, NTOD, and PPSS) with changeable notch opening angle in two basic relationships, it is necessary to select one critical parameter with changeable notch opening angle or two independent critical parameters, respectively. With the help of a characteristic length, it is found by this new conservation integral that the NSIF, NTOD and energy release rate are equivalent in the case of small-scale yielding. Especially, the characteristic length possesses clear physical meaning and, for example, depends on both the critical NSIF and SIF or both the NTOD and CTOD, respectively, in which SIF and CTOD are from the tip of a crack degenerated from the sharp V-notch. The dependence of NSIF on NTOD and PPSS is presented according to the equivalence, and the critical NSIF depending on the critical NTOD with a notch opening angle is also predicted.     

Elliptical crack normal to functionally graded interface of bonded solids

This paper presents an analysis of an elliptical crack that is perpendicular to a functionally graded interfacial zone between two fully bonded solids. The functionally graded interfacial zone is treated as a non-homogeneous solid layer with its elastic modulus varying in the thickness direction. A generalized Kelvin solution based boundary element method is employed for the calculation of the stress intensity factors associated with the three-dimensional crack problem. The elliptical crack surface is subject to either uniform normal traction or uniform shear traction. The stress intensity factors are examined by taking into account the effects of the non-homogeneity parameter and thickness of the functionally graded interfacial zone, as well as the crack distance to the zone. The SIF values are further incorporated into the S-criterion for prediction of crack growth. The paper presents the most possible direction and location of the elliptical crack growth under an inclined tensile (or compressive) load. The paper further presents results of the critical external loads that would cause the elliptical crack to grow at the most possible location and along the most possible direction. The paper also examines the effects of external load direction and material and geometrical parameters on the critical loads.