A critical evaluation of numerical solutions to the ‘Benchmark’ surface-flaw problem

Recent precise, elastic numerical solutions to a surface flaw in a plate under remote tension and bending have been compared to arrive at a ‘best estimate’ of the stress-intensity-factor variation along the flaw border. The geometry of the semielliptical surface flaw examined had a depth to length ratio of 0.25 (a/2c=0.25) and 25- and 75-percent relative crack depths (a/t=0.25, 0.75). The analysis methods used to determine the solutions included: Schwarz alternating technique, finite-element method and boundary-integral-equation method. The derived best-estimate curve for the stress-intensity factor is believed within 3 percent of the actual value along the crack front. The best-estimate curve compared well with scarce experimental data (±10 percent). The difference between the best-estimate curve and experiment is thought largely due to differences in geometry and Poisson's ratio.     

Stress distributions and stress intensity factors in ceramic roller bearing rings

A method is presented which allows to compute the stress intensity factors (SIFs) of a ceramic roller bearing ring with a surface flaw. First, the stresses due to mechanical loads applied to the surfaces of the undamaged ring are computed, using Michell's expression for Airy's stress function in polar coordinates ([l]). Second, the stress intensity factors of the flaw are obtained using Tada's additivity principle ([2]), namely by integration of the point force solution for the SIF of a surface crack with the aforementioned stresses as input parameter. To illustrate the procedure, the special case of a ring with a surface flaw which is loaded by six equidistantly spaced cylindrical rollers will be studied numerically.     

A domain-independent integral for computation of stress intensity factors along three-dimensional crack fronts and edges by BEM

The present work deals with an evaluation of stress intensity factors (SIFs) along straight crack fronts and edges in three-dimensional isotropic elastic solids. A new numerical approach is developed for extraction, from a solution obtained by the boundary element method (BEM), of those SIFs, which are relevant for a failure assessment of mechanical components. In particular, the generalized SIFs associated to eigensolutions characterized by unbounded stresses at a neighbourhood of the crack front or a reentrant edge and also that associated to T-stress at the crack front can be extracted. The method introduced is based on a conservation integral, called H-integral, which leads to a new domain-independent integral represented by a scalar product of the SIF times some element shape function defined along the crack front or edge. For sufficiently small element lengths these weighted averages of SIFs give reasonable pointwise estimation of the SIFs. A proof of the domain integral independency, based on the bi-orthogonality of the classical two-dimensional eigensolutions associated to a corner problem, is presented. Numerical solutions of two three-dimensional problems, a crack problem and a reentrant edge problem, are presented, the accuracy and convergence of the new approach for SIF extraction being analysed.     

Variation in stress-intensity factor and back-surface displacement for surface cracks

Knowledge of the magnitude and variation in the stress-intensity factor (SIF) around the perimeter of a surface crack is essential for an accurate analysis of a flawed component. SIFs for surface flaws of various semi-elliptical geometries were analytically determined. Three-dimensional linear-elastic finite-element analysis was performed to determine the maximum SIF for bending and tension for each of 12 crack geometries which represent deep surface flaws in finite-thickness plates. Experimental verification of one of the crack geometries was performed. Interferometry techniques were used to determine the actual variation in SIF along the curve crack front due to bending. In addition to the SIF calculations, physical characteristics are noted as observed in the analytical and experimental evaluations.     

Experimental and numerical investigation of a cracked transversely graded plate subjected to in plane bending

The variation of stress intensity factor along the thickness in a cracked transversely graded plate subjected to in plane bending is investigated in this study. A transversely graded plate having elastic modulus varying continuously along the thickness was prepared by embedding glass beads in epoxy resin. An edge crack in this plate was subjected to in plane bending and the crack tip displacement field on the surfaces of the plate was measured using digital image correlation (DIC). Using the recently reported asymptotic displacement fields for cracked transversely graded plates (Wadgaonkar, S.C., Parameswaran, V., 2009. Structure of near tip stress field and variation of stress intensity factor for a crack in a transversely graded material, Journal of Applied Mechanics 76 (1), 011014), the stress intensity factor (SIF) on the surfaces of the plate was calculated from the experimental data. The results of this part of the study indicated that the extent of variation of the SIF across the plate thickness is nearly the same as that of the elastic modulus. An expression to calculate the variation of the SIF through the plate thickness was developed assuming simple bending of the plate. The predicted variation of SIF was verified through finite element calculations. Further, the behavior of the SIF near the intersection of the crack front and the plate surfaces, the extent of dominance of the corner singular field and the exponent of the corner singularity were also investigated in detail. Finally, the effect of gradation strength and gradation type on the SIF was also studied.     

估算裂纹应力强度因子的新方法

根据裂纹形状与裂纹尖端应力强度因子分布之间的固有关系,在线弹性断 裂力学条件下,提出了一种按已知I型裂纹应力强度因子分布规律求裂纹形状及相应应力强 度因子的无梯度迭代法. 通过有限厚度、有限宽度板穿透裂纹和表面裂纹的数值模拟实例验 证了所提出方法的有效性和实用性,并对不同应力强度因子分布规律对裂纹形状以及相应的 应力强度因子大小的影响进行了分析和讨论. 所提出的方法有助于提高实际扩展裂纹应 力强度因子的估算精度以及更合理地预测疲劳裂纹形状演化.     

裂纹垂直于双相介质界面时的应力强度因子

本文利用Ｊ积分与应力强度因子的关系,采用有限元数值方法研究了当裂纹与双相介质的界面垂直时,其裂纹的近界面端和远界面端的应力强度因子随双相介质参数和裂纹端部到界面的距离的变化规律,同时还分析了当边裂纹逐渐扩展时,应力强度因子的变化特征。     

A Zener–Stroh crack interacting with a coated inclusion with generalized Irwin plastic zone correction

The elastic–plastic fracture behavior of a Zener–Stroh crack interacting with a coated inclusion in composite materials has been investigated with crack tip plastic zone corrections. With the distributed dislocation method, the crack problem is formulated into a set of singular integral equations which are solved numerically. The plastic zone sizes at the both crack tips are determined by a generalized Irwin model where Von Mises stress yielding criterion is used. The stress intensity factor (SIF), the plastic zone size (PZS), the crack tip opening displacement (CTOD) and the effective stress intensity factor have been evaluated. In the numerical examples, the influence of the inclusion shear modulus, the coating-layer thickness and shear modulus, as well as the distance between the crack and inclusion, on the SIF, the PZS and the CTOD are discussed in detail. Numerical examples show that increasing the shear modulus or the thickness of the coating phase, the influence of the inclusion on the normalized SIF and the normalized PZS will be shielded.     

In-plane and out-of-plane constraint effects under pressurized thermal shocks

Transferability of fracture toughness data obtained on small scale specimens to a full-scale cracked structure is one of the key issues in integrity assessment of engineering structures. In order to transfer fracture toughness under different constraints, both in-plane and out-of-plane constraint effect should be considered for the specimens and structures. In this paper both in-plane and out-of-plane constraint effects of a crack in a reference reactor pressure vessel (RPV) subjected to pressurized thermal shocks (PTSs) are analyzed by two-parameter and three-parameter methods. The comparison between elastic and elastic–plastic analysis shows that the constraint effect varies with the material property. T₁₁ (the second term of William’s extension acting parallel to the crack plane) generally displays a reversed relation to the stress intensity factor (SIF) with the transient time, which indicates that the loading (SIF) plays an important role on the in-plane constraint effect. The thickness at the crack tip contributes more than the loading to the out-of-plane constraint, such that T₃₃ (the second term of William’s extension acting along the thickness) displays a similar relation to ε₃₃ (strain along the thickness direction) and a different relation to T₁₁ during the transient. The results demonstrate that both in-plane and out-of-plane constraint effect should be analyzed separately in order to describe precisely the stress distribution ahead of the crack tip.     

Why traction-free? Piezoelectric crack and Coulombic traction

The non-zero traction condition is introduced in piezoelectric crack problems with the unknown Coulombic traction acting on the crack surfaces. An analytical solution under this condition is obtained by means of the generalized Stroh formalism and by accounting for the permittivity of medium inside the crack gap. As the crack in such materials can be thought of as a low-capacitance medium carrying a potential drop, the Coulombic traction always pulls the two opposite surfaces of the crack together. It is proved that under relatively larger mechanical loading and relatively smaller electrical field, the Coulombic traction may be negligible and the previous investigations under the traction-free crack condition may be accepted in a tolerant way, otherwise the Coulombic traction may lead to some erroneous results with over 10% relative errors. It is also shown that, unlike the traction-free crack condition, the applied electric field does change the Mode I stress intensity factor (SIF) for a central crack in an infinite plane piezoelectric material, and in this way may significantly influence piezoelectric fracture. It is also concluded that the variable tendencies of the normalized SIF and the ERR against the applied electric field depend on the mechanical loading levels. This load-dependence feature may lead to a transformation of the normalized SIF and the ERR from an even functional dependence to an odd functional dependence on the applied electric field.     

Discussion on the SIF for points on border of elliptical flat crack inside infinite solid under uniform tension

Using the results of crack surface displacement field in Green-Sneddon’s solution[1] and coordinate transformation, this paper has derived an expression K₁(x₁,z₁,a) for SIF at any point and at any orientation on the border of elliptical flat crack inside infinite solid under uniform tension. As a complement of Irwin’s work[3], it is shown that for any pointed point on the elliptical border the SIF defined on normal plane takes the maximum value. And it should be pointed out that in some works some idea concerning Irvin’s contents is open to question. An expression K₁ in terms of polar angle which is more intuitional than centrifugal angle is proposed for SIF at any point on the elliptical border.     

Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).     

Fracture of Edge Cracked Layered Plates Subjected to In-plane Bending

Layered structures are used in protection systems such as personal and heavy armor, windshields and also in functionally graded thermal barriers. The focus of this study is to understand the behavior of cracks in such systems, especially when the crack orientation is such that there are property changes along the crack front. Layered plates were prepared by bonding together sheets of epoxy and Polymethylmethacrylate (PMMA) using an epoxy adhesive. Among the two, epoxy has higher elastic modulus and lower fracture toughness compared to PMMA. Two different layer configurations; a single sheet of epoxy bonded to a single PMMA sheet (two layer) and a single sheet of epoxy sandwiched between two PMMA sheets (three layer) were considered. Single edge notched specimens were loaded in three point bending and the thickness averaged stress intensity factor (SIF) was estimated through photoelasticity. Subsequently, the behavior of crack-extension in these plates was also investigated. In both configurations, crack growth initiated in the epoxy layer first and extended stably before the start of crack extension in the PMMA layer. Once the crack extension started in the PMMA layer, the plate looses its structural integrity. It was observed that the onset of crack extension in the epoxy layer can be predicted using the thickness averaged SIF. A method using analysis of patched cracks is presented for estimating the load at which the plate completely looses its load carrying capacity. The estimates from this method match well with experimental results.     

On the stress state of a piezoceramic body with a flat crack under symmetric loads

A static-equilibrium problem is solved for an electroelastic transversely isotropic medium with a flat crack of arbitrary shape located in the plane of isotropy. The medium is subjected to symmetric mechanical and electric loads. A relationship is established between the stress intensity factor (SIF) and electric-displacement intensity factor (EDIF) for an infinite piezoceramic body and the SIF for a purely elastic material with a crack of the same shape. This allows us to find the SIF and EDIF for an electroelastic material directly from the corresponding elastic problem, not solving electroelastic problems. As an example, the SIF and EDIF are determined for an elliptical crack in a piezoceramic body assuming linear behavior of the stresses and the normal electric displacement on the crack surface __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 67–77, November 2005.     

Thermostressed state of a piezoelectric bodywith a plane crack under symmetric thermal load

The paper addresses a thermoelectroelastic problem for a piezoelectric body with an arbitrarily shaped plane crack in a plane perpendicular to the polarization axis under a symmetric thermal load. A relationship between the intensity factors for stress (SIF) and electric displacement (EDIF) in an infinite piezoceramic body with a crack under a thermal load and the SIF for a purely elastic body with a crack of the same shape under a mechanical load is established. This makes it possible to find the SIF and EDIF for an electroelastic material from the elastic solution without the need to solve specific problems of thermoelasticity. The SIF and EDIF for a piezoceramic body with an elliptic crack and linear distribution of temperature over the crack surface are found as an example __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 96–108, March 2008.     

焊趾表面裂纹的形态发展曲线与疲劳寿命预测

以作者建立的焊地椭圆表面裂纹应力强度因子数据库以及复杂应力场中焊践半随圆表面裂纹前缘应力强度因子分布计算的基本模式法为基础上,给出了复杂应力场中焊践表面表纹在疲劳扩展过程中形态变化规律及寿命的工程分析方法。     

Evaluation of stress intensity factors for bi-material interface cracks using displacement jump methods

The aim of the present work is to investigate the numerical modeling of interfacial cracks that may appear at the interface between two isotropic elastic materials. The extended finite element method is employed to analyze brittle and bi-material interfacial fatigue crack growth by computing the mixed mode stress intensity factors (SIF). Three different approaches are introduced to compute the SIFs. In the first one, mixed mode SIF is deduced from the computation of the contour integral as per the classical J-integral method, whereas a displacement method is used to evaluate the SIF by using either one or two displacement jumps located along the crack path in the second and third approaches. The displacement jump method is rather classical for mono-materials, but has to our knowledge not been used up to now for a bi-material. Hence, use of displacement jump for characterizing bi-material cracks constitutes the main contribution of the present study. Several benchmark tests including parametric studies are performed to show the effectiveness of these computational methodologies for SIF considering static and fatigue problems of bi-material structures. It is found that results based on the displacement jump methods are in a very good agreement with those of exact solutions, such as for the J-integral method, but with a larger domain of applicability and a better numerical efficiency (less time consuming and less spurious boundary effect).     

Multiple axial cracks in a coated hollow cylinder due to thermal shock

The transient thermal stress problem of an inner-surface-coated hollow cylinder with multiple pre-existing surface cracks contained in the coating is considered. The transient temperature, induced thermal stress, and the crack tip stress intensity factor (SIF) are calculated for the cylinder via finite element method (FEM), which is exposed to convective cooling from the inner surface. As an example, the material pair of a chromium coating and an underlying steel substrate 30CrNi2MoVA is particularly evaluated. Numerical results are obtained for the stress intensity factors as a function of normalized quantities such as time, crack length, convection severity, material constants and crack spacing.     

A mode III crack crossing the magnetoelectroelastic bimaterial interface under concentrated magnetoelectromechanical loads

A mode III crack cutting perpendicularly across the interface between two dissimilar semi-infinite magnetoelectroelastic solid is studied under the combined loads of a line force, a line electric charge and a line magnetic charge at an arbitrary location. The impermeable conditions are implied on the crack faces. The technique developed in literature for the elastic bimaterial with a crack cutting interface is exploited to treat the magnetoelectroelastic bimaterial. The Riemann-Hilbert problem can be formulated and solved based on complex variable method. Analytical solutions can be obtained for the entire plane. The intensity factors around crack tips can be defined for the elastic, electric and magnetic fields. It shows that, no matter where the load position is, the electric displacement intensity factors (EDIFs), as well as the magnetic induction intensity factors (MIIFs), are identical in magnitude but opposite in sign for both crack tips, on condition that a line force is solely applied. Alternatively, if only a line electric charge is considered, then the stress intensity factors (SIFs) and the MIIFs exhibit the behavior. Likewise, if only a line magnetic charge is applied, it turns to the SIFs and the EDIFs instead. In addition, the dependence of the intensity factors is graphically shown with respect to the location of a line force. It is found that the SIF for a crack tip tends to be infinite if the applied force is approaching the tip itself, but the EDIF, with the complete opposite trend, tends to be vanishing. Finally, focusing on the more practical case of piezoelectric/piezomagnetic bimaterial, variation of the SIF along with the moduli as well as the piezo constitutive coefficients is explored. These analyses may provide some guidance for material selection by minimizing the SIF. It is also believed that the results obtained in this paper can serve as the Green’s function for the dissimilar magnetoelectroelastic semi-infinite bimaterial with a crack cutting the interface under general magnetoelectromechanical loads.     

Toughening of ferroelectrics by the out-of-plane poling

Subjected to the prior out-of-plane poling, the ferroelectrics can be toughened considerably. The present paper describes the variation of the stress intensity factor (SIF) by 90° switching in ferroelectrics. The analysis is carried out for the combined mechanical and electrical loading, with simple relations obtained for the case of the purely electrical loading. The out-of-plane poling is found to raise the SIF for the crack initiation, but appreciably reduces the SIF for the crack growth in a steady state. More stable fracture resistance curves can be achieved by the out-of-plane poling. This prediction is supported quantitatively by the testing data of SENB specimens of PZT-5 samples, when the toughening effects of polings in three orthogonal directions are compared. The project supported by Sino-Germany Center for Science and Technology (G2050/8)