Fracture analysis of base-edge-cracked reverse-tapered plates

This paper presents a fracture mechanics analysis of the base-edge-cracked reverse-tapered (RT) fracture geometry. Motivation for this study was the use of this test geometry in Phase 1 of a recently completed joint-industry-agency project entitled Large-Scale Ice Fracture Experiments. Underlying the choice of the RT fracture geometry for Phase 1 was the desirability of achieving crack propagation in a controlled and stable manner; such control would allow a number of observations to be made on one testpiece. Reverse tapering greatly improves not only crack growth stability but also crack path stability. The weight function method was used to provide accurate wide-ranging stress intensity factor (SIF), crack face displacement (COD) and crack opening area (COA) expressions for the RT subject to any loading. The required weight function was obtained through a finite element analysis of this geometry subject to a reference load case which determined the associated stress intensity factor and crack opening displacements. The Wu and Carlsson procedure was followed. A key modification to the latter procedure facilitated the attainment of the reference CMOD for all crack lengths, including the zero ligament limit; this was achieved by considering an additional reference solution. This modification is general in nature and could be pursued whenever the reference CMOD is not known analytically. An analytical solution for the crack opening area (COA) was also achieved for the special case of concentrated loading at the crack mouth. This solution can be applied to any geometry where the reference CMOD expression is known.     

Some recent fatigue and fracture mechanics research in BIAM

The paper presents a brief review of some of the major research activities on fatigue and fracture mechanics in recent years at the Beijing Institute of Aeronautical Materials. Attention is mainly given to the studies on weight function methods for analyses of two- and three-dimensional crack problems, fatigue crack growth under variable amplitude loading, small crack effects and a fracture-mechanics-based total fatigue life prediction method.Abbreviations 2(3)D two- (three-) dimensional - BIAM Institute of Aeronautical Materials, Beijing - CCT center cracked tension - COD crack opening displacement - CTOD crack tip opening displacement - FEM finite element method - LEFM linear elastic fracture mechanics - SENT single edge notched tension - SIF stress intensity factor - WFM weight function method     

A wide range weight function for a single edge cracked geometry with clamped ends

A single edge cracked geometry with clamped ends is well suited for fracture toughness and fatigue crack growth testing of composites and thin materials. Stress intensity factors may be determined by the weight function method. A weight function for the single edge cracked geometry with clamped ends is developed and verified in this paper. It is based on analytical forms for the reference stress intensity factor and crack mouth opening displacement. The analytical forms are shown to be valid, by comparison with finite element results, over a wide range of crack depths and plate aspect ratios. Use of the analytical form enables the weight function to be calculated for any plate aspect ratio without the need for preliminary finite element analysis. Stress intensity factors and crack mouth opening displacements, predicted using this weight function, correlated well with finite element results for non-uniform crack surface stress distributions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Approximate weight function technique using single-layer potential for a cracked circular disc

An efficient weight function technique using the indirect boundary integral method was presented for cracked circular discs. The crack opening displacement field was presented by a single layer whose kernel was a modified form of the fundamental solution in elastostatics. The application of a single-layer potential to the weight function method leads to a unique closed-form SIF (stress intensity factor) solution. The solution can be applied to a cracked circular discs with or without an internal hole or opening. For these crack geometries over a wide range of crack ratios, the SIF solution can be applied without any modification.

The calculation procedure of SIFs for the various cracked circular discs using only one analytical solution is very simple and straightforward. The information necessary in the analysis includes only two or three reference load cases. In most cases the SIF solution using two reference SIFs gives reasonably accurate results while the SIF solution with three reference load cases may be used to improve the solution accuracy of the crack configurations, with an internal opening or hole, compared with the solutions of the available literature.     

A numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems

A computational model based on the numerical Green's function (NGF) and the dual reciprocity boundary element method (DR-BEM) is presented for the study of elastodynamic fracture mechanics problems. The numerical Green's function, corresponding to an embedded crack within the infinite medium, is introduced into a boundary element formulation, as the fundamental solution, to calculate the unknown external boundary displacements and tractions and in post-processing determine the crack opening displacements (COD). The domain inertial integral present in the elastodynamic equation is transformed into a boundary integral one by the use of the dual reciprocity technique. The dynamic stress intensity factors (SIF), computed through crack opening displacement values, are obtained for several numerical examples, indicating a good agreement with existing solutions.     

WEIGHTED AVERAGED STRESS INTENSITY FACTORS OF CIRCULAR-FRONTED CRACKS IN CYLINDRICAL BARS

Stress intensity factors are calculated in weighted average at the surface and the deepest point of a circular-fronted surface crack in a cylindrical bar by use of the weight function method. A wide range of various crack shapes are studied, from a nearly straight-fronted edge crack to a semi-circular crack front. Use of the weight function method requires that the crack opening displacement field of a reference load has to be known. It was obtained by 3-D finite element analysis. Results are presented for the cracked cylinder subjected to a constant stress (tension) and a linear stress distribution acting perpendicular to the crack faces and they are compared with values found by other investigators.     

Application of the weight function method to solving multiparametric three-dimensional fracture mechanics problems

A system of equations of the weight function method is obtained for the calculation of the stress intensity factors (SIF) along the crack front under arbitrary normal loading, as well as the crack opening displacement field (CODF) for cracks of arbitrary shape. The equations, in particular, make it possible to estimate the error of the known empirical formulas for the SIF, as well as to calculate the variation of the SIF values with the crack shape. It is shown that the SIF values for the problems with the mixed boundary conditions can be calculated making use of the weight functions obtained for the problems with the boundary conditions in stresses.     

Determination of dynamic fracture parameters using a semi-circular bend technique in split Hopkinson pressure bar testing

Fracture initiation toughness, fracture energy, fracture propagation toughness, and fracture velocity are key dynamic fracture parameters. We propose a method to simultaneously measure these parameters for mode-I fractures in split Hopkinson pressure bar (SHPB) testing with a notched semi-circular bend (SCB) specimen. The initiation toughness is obtained from the peak load given dynamic force equilibrium. A laser gap gauge (LGG) is developed to monitor the crack surface opening displacement (CSOD) of the specimen, from which the fracture velocity and the fracture energy can be calculated. The feasibility of this methodology for coarse-grained solids is demonstrated with the SHPB-SCB experiments on Laurentian granite.     

Stress intensity factor for center-cracked plate with crack surface interference

This paper presents a simple and physically acceptable analysis of stress intensity factor (SIF) for the center-cracked infinite and finite-width plates. The analysis includes the effect of crack surface interference (i.e., the upper and lower crack surfaces are not allowed to overlap) that influences both the SIF at the tension-side crack tip and the crack opening displacement (COD) profile. For an infinite plate, exact solutions are obtained by superimposing the classical (overlapping) solutions. For a finite-width plate, where the SIF solutions cannot be found in closed form, the solutions are carried out numerically. The overlapping SIF solutions from the weight function method are used. An example is given for the case of a finite-width plate under bending. It was found that the overlapping solutions underestimate the stress intensity factor at the tension-side crack tip up to 15%. The analysis results are also compared with the finite element solutions for verification purpose.     

聚丙烯纤维水泥稳定碎石断裂性能试验研究

通过30个尺寸为100mm×100mm×515mm的聚丙烯纤维水泥稳定碎石和普通水泥稳定碎石三点弯曲试件断裂试验,探讨了聚丙烯纤维对水泥稳定碎石断裂韧度(KIC)、断裂能(GF)、临界裂缝嘴张开位移(CMODC)、临界裂缝尖端张开位移(CTODC)、极限裂缝嘴张开位移(CMODmax)和极限裂缝尖端张开位移(CTODmax)的影响。试验结果表明:聚丙烯纤维的掺入可增大水泥稳定碎石的断裂韧度、断裂能、临界裂缝嘴张开位移、极限裂缝嘴张开位移、临界裂缝尖端张开位移和极限裂缝尖端张开位移;随着聚丙烯纤维体积掺量的增加,断裂韧度、临界裂缝嘴张开位移和临界裂缝尖端位移的变化无明显规律,但断裂能、极限裂缝嘴张开位移和极限裂缝尖端位移基本上呈线性增加的。     

Evaluation of elastic fracture mechanics parameters for bend specimens

Existing solutions for stress intensity factor and mouth opening displacement of three-point bend specimens are shown to overestimate these quantities for shallow cracks by up to ±4.5 percent, because they do not account for the disturbance of the bending stress distribution by the concentrated force at the loading point. This error is far larger than the accuracy claimed by these solutions (0.2 to 0.5 percent).New expressions are therefore developed for stress intensity, crack mouth opening displacement and crack mouth open angle of single edge notched bend specimens loaded in three-point bending. As a reference, and to show the accuracy of the solutions, also the pure bending situation is treated. Rigorously derived asymptotic solutions are used for the shallow and deep crack limits, in order to prescribe both the proper limit values and gradients to the interpolation functions, of which the intermediate values are derived from refined finite element analyses.The crack mouth opening angle solutions are primarily intended to facilitate crack mouth opening measurement at other locations then the specimen surface, i.e. at an offset from the specimen surface as is the case when removable knife edges are applied. No solutions of the crack mouth opening angle of three-point bend specimens were available until now. For use with unloading compliance crack length measurement, also an inverse crack mouth opening relation is developed. This also includes crack mouth opening measurement at an offset from the specimen surface, which is lacking in presently available expressions of this kind.     

A generalized model for dynamic mixed-mode fracture via state-based peridynamics

A generalized model is developed to investigate dynamic crack propagation in isotropic solids under mixed-mode I/II conditions using state-based peridynamics. The critical stretch and the critical strain energy release rate (ERR) are related within the state-based peridynamic framework to construct a computational model capable of capturing fracture energy of the kinked cracks. A novel formulation is presented to predict crack growth trajectory and pattern by combining the traditional expression of ERR and the peridynamic states of the crack opening and sliding displacements. The proposed model is used to predict dynamic fracture behavior in polymethyl methacrylate (PMMA) and soda-lime glass using various test specimens, including cracked semi-circular bending (SCB), cracked rectangular plate, and single edge-notched tensile (SENT) specimens, and under different dynamic loading conditions. The developed model is examined against the numerical and experimental data available in the literature, and a very good agreement is observed.     

裂尖具线性分布约束应力的运动裂纹模型及其解析解

以恒定速度运动的Griffith裂纹解析解为著名的Yoffe解。静止裂纹的条状屈服模型即Dugdale模型,将其推广到运动裂纹模型时发现,当裂纹运动速度跨越Rayliegh波速时,裂纹张开位移COD趋于(∞,且表现为间断。通过在裂尖引入一个约束应力区及两个速度效应函数,假设约束应力为线性分布,采用复变函数方法,求得动态应力强度因子SIF与裂纹张开位移COD的解析解。新的结果,在Rayleigh波速下裂纹张开位移连续且为有限值。给出裂纹张开位移的一些数值结果,获得了一些有意义的结论。     

Determination of crack-tip stress intensity factors in complex geometries by the composition of constituent weight function solutions

Linear elastic fracture mechanics (LEFM) is the science frequently used to understand the stable and progressive fatigue crack growth that often occurs in engineering components under varying applied stress. The stress intensity factor (SIF) is its basis and describes the stress state at the crack tip. This can be used with the appropriate material properties to calculate the rate at which the crack will propagate in a linear elastic manner. Unfortunately, the SIF is difficult to compute or measure, particularly if the crack is situated in a complex three‐dimensional geometry or subjected to a non‐simple stress state. This is because the SIF is not only a function of the crack and component geometry but is also dependent on the applied stress field. In the last 20 years, the SIF weight function has gained prominence as a method for calculating and presenting SIFs independent of applied stress. This paper demonstrates that the real promise of the SIF weight Function lies in its use to rapidly generate SIF solutions for cracks in complex geometries by simple composition of geometric influences from reference constituent solutions.     

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIF (stress intensity factor) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIF using only one SIF formula without any modification of the crack geometries symmetric in a two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential crack or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straghtforward with only one SIF formula. The necessary information in the analysis is two reference SIFs. The analysis results, using several examples, verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.     

Fracture Analysis of Double-Side Adhesively Bonded Composite Repairs to Cracked Aluminium Plate Using Line Spring Model

A line spring model is developed for analyzing the fracture problem of cracked metallic plate repaired with the double-sided adhesively bonded composite patch. The restraining action of the bonded patch is modeled as continuous distributed linear springs bridging the crack faces provided that the cracked plate is subjected to extensional load. The effective spring constant is determined from 1-D bonded joint theory. The hyper-singular integral equation (HSIE), which can be solved using the second kind Chebyshev polynomial expansion method, is applied to determine the crack opening displacements (COD) and the crack tip stress intensity factors (SIF) of the repaired cracked plate. The numerical result of SIF for the crack-tip correlates very well with the finite element (FE) computations based on the virtual crack closure technique (VCCT). The present analysis approaches and mathematical techniques are critical to the successful design, analysis and implementation of crack patching.     

On the calculation of crack opening displacement from the stress intensity factor

For the application of the weight function method the crack opening displacements for a reference case have to be known. An approximate method to derive the crack opening field from the stress intensity factor was proposed by Petroski and Achenbach [Engng Fracture Mech. 10, 257 (1978)]. The limited accuracy of their method becomes evident in cases where the stresses differ strongly from the homogeneous loading case (σ = const.). By expanding the crack opening displacement field in a power series it is demonstrated here how the approximative solutions can be improved by simple additional conditions.     

On the influence of reference load case on the crack face weight functions

The influence of reference load case on the crack face weight functions for different finite cracked bodies was investigated by examining the Green's functions which are related to the crack face point force loading. The weight functions were analytically determined based on a general crack opening displacement equation. It was found that the degree of reference-load-case-dependence is not only a function of the reference stress itself, but also the crack length, overall cracked configuration and the number of terms contained in the crack opening displacement equation. Some useful guidelines for the choice of reference load cases in generating the weight functions have been introduced, and the way to improve the weight function accuracy at extended crack lengths recommended.     

Loading‐rate dependence of mode I crack growth in concrete

Mode I crack propagation process of concrete under relatively low loading rates which cover four orders of magnitude (0.2 μm/s to 2.0 mm/s) is investigated with three‐point bending (TPB) beams. All measured material properties exhibit rate sensitivity and follow a log‐linear relationship with the loading rate. A rate‐sensitive softening curve is established. The complete load‐crack mouth opening displacement (P‐CMOD) curve, crack propagation length, and fracture process zone (FPZ) length are simulated based on crack growth criterion with the fitted material parameters under those loading rates. Results show that the simulated P‐CMOD curves agree well with those of experimental measurements. It is clear that the peak load increases with the loading rate and so is the critical crack mouth opening displacement. Moreover, under the same load level, the length of the FPZ and the cohesive stress at the initial crack tip also increase with the increasing loading rate.