A new approach to determination of CTOD and axis of rotation in small scale yielding situation

A theoretical model for predicting crack tip opening displacement (CTOD) in small scale yielding situation has been developed by combining the elastic solution of Muskhelishvili with Irwin's idea of notional crack. The model is used to calculate the CTOD from the displacement at the mouth of the crack, without the use of rotational factor, r. An attempt has also been made to relate the rotational factor, r with the yield strength of the material.

The above model has been used to compute CTOD values in a low alloy steel with yield strength ranging from 400–2100 MPa and the results have been compared with Wells' as well as Xiao's CTOD values. The significance of the factor m' in the K-CTOD relationship has been investigated in the light of the results observed in the present investigation.     

承受双向等拉应力的平面应变I型裂纹线弹性断裂的I1准则与K准则一致性分析

采用线弹性有限元方法计算了承受双向等拉应力的平面应变I型裂纹的应力场,分析了裂纹尖端各应力分量间的关系,拟合了各非零应力分量关于裂纹半长度a和裂纹尖端最小网格尺寸l₁的函数,分析了应力第一不变量I₁与应力场强度因子K_I的相关性。结果表明,裂纹尖端各非零应力分量间存在稳定的比例关系;各非零应力分量值和加载应力的比值与裂纹半长度a的1/2次幂呈正比例关系、与裂纹尖端最小网格尺寸l₁的1/2次幂呈反比例关系;相同最小网格尺寸条件下,裂纹尖端的应力第一不变量与应力场强度因子的比值l₁/K_I为与加载应力和裂纹长度无关的常数,证明了承受双向等拉应力的平面应变I型裂纹线弹性断裂的I₁准则与K准则具有一致性。     

An assessment of factors influencing data obtained by the photoelastic stress freezing technique for stress fields near crack tips

Factors which influence crack tip stress field data are identified as: (1) Non-linear zone near crack tip due to crack blunting. (2) Normal stress parallel to crack surface. (3) Location of region for data retrieval.

The Kolosoff-Inglis solution is used in order to assess effects of crack tip blunting. A set of stress freezing photoelastic experiments are conducted on plates containing through cracks and results are compared with the Westergaard solution in order to assess the effect of Item 2 using appropriate Item 3 locations. A data conditioning computer program is employed to yield accurate values of the stress intensity factor from photoelastic data.     

The effect of repeating load on the crack growth initiation and crack propagation in delayed failure

The delayed failure test under repeating load was carried out with pre-cracked specimen. The incubation time and the crack propagation rate were correlated with the stress intensity factor K.

The incubation time is decreased by the superposition of repeating load, as the range of stress intensity factor ΔK or the repeating frequency f increase. The reason can be explained by the promotion of corrosion reaction due to, e.g. the destruction of oxide film on the crack tip, which facilitates the invasion of hydrogen atoms into the material.

The crack propagation rate da/dt is decreased by the superposition of repeating load, and there exist two valleys of crack propagation rate minima on the da/dt vs f and da/dt vs ΔK curves. One valley corresponds to the interaction between the cyclic movement of the region with tri-axial tensile stress and the hydrogen atoms diffused from crack tip, which disturbs the concentration of hydrogen atoms. Another seems te correspond to the generation of retained compressive stress which reduces the effective stress intensity at crack tip and supresses the invasion and diffusion of hydrogen atoms.     

Complex stress intensity factors at tips of cracks along interfaces of dissimilar media

The method of reflected caustics was used to determine the complex stress intensity factors at the tips of cracks having any shape, which lie at the interface of two dissimilar elastic media. For the evaluation of complex S.I.Fs two measurements of appropriate lengths have to be made on the caustic formed at the crack tip. These measurements allow the determination of both the absolute value and the argument of the respective stress intensity factor. The method was applied for the solution of problems which are referred to cracks at interfaces between two elastic dissimilar media. The experimental results show a good agreement between the experimental and theoretical methods of evaluating SIFs.     

Trifurcation of a crack due to plane SH-waves in an infinite elastic medium

The dynamic anti-plane problem of trifurcation of a semi-infinite crack due to incidence of two linearly varying plane SH-waves with non-parallel wave fronts in an infinite elastic medium has been considered. The semi-infinite crack is assumed to trifurcate when the plane waves intersect the crack tip. The problem has been solved using the self-similar technique, which is based on the observation that certain field variables show dynamic similarities. The results include the expressions for shear stress in the planes of the cracks and the stress intensity factors at the crack tips. Numerical calculations have been carried out to show the variations of stress intensity factors at the crack tips with the angle of skew for different values of the crack tip velocity and angle of incidence.     

A hybrid-element approach to crack problems in plane elasticity

The hybrid-element concept and the complex variable technique have been adopted for constructing a special super-element to be used jointly with conventional finite elements for the analysis of elastic stress intensity factors for plane cracks. The use of the complex variable technique permits the proper consideration of the stress intensity at the crack tip, and it also leads to very efficient programming. The use of such a super-element in the finite element solution has been shown to be highly accurate when only a very coarse element mesh is used near the crack.     

Survey of current fracture mechanics studies at AMMRC

The fracture and fatigue studies at AMMRC encompass both experimental and analytical thrusts. This presentation will cover a few representative activities.

The status of a test program on fracture behavior of unidirectionally reinforced composites using double cantilevered type specimens will be reported. Material systems include ‘S’ glass/epoxy and graphite/epoxy. Limited fracture toughness data is available. Additionally, some effort has been initiated on cross ply specimens.

Fatigue studies at AMMRC in metals reflect a continuing interest in both initiation and propagation aspects. A technique has been developed for detection of crack initiation using an automated photographic process for recording, periodically, potential initiation sites. The observations are discussed in conjunction with damage criteria. Crack propagation studies have focussed in developing a ‘law’ which accounts for the extremes of propagation rates at threshold levels of stress intensity and at the higher levels associated with unstable growth. Additionally, crack propagation experiments are described which are aimed at exploring the transient effects on crack propagation of sudden changes in stress intensity levels.

Supplementing these experimental studies are extensive analytical efforts directed toward defining the stress states in anisotropic material when used in lap and/or mechanical joint configurations. Additionally, extensive effort by one team of investigators has been devoted to analytical techniques for crack analysis. These techniques have been applied to a wide variety of geometric configurations and to a range of material types including anisotropic. This effort will be described briefly.     

Influence of interface on fatigue threshold values in elastic bimaterials

This paper describes a study of the behavior of a fatigue crack growing in one elastic material and penetrating through the interface into a second material. The study aims especially to quantify the influence of the elastic mismatch of both materials on the threshold value of a fatigue crack propagating perpendicularly to the interface. Special attention is devoted to the case of a crack touching the interface. It is shown that the corresponding threshold value is strongly influenced by the existence of an interface between the two materials. A tentative procedure suggested in the paper makes it possible to quantify the effect and determine the dependence of the threshold value of a fatigue crack growing through the interface on the Dundurs parameters and β.

The results are applied to a body with a protective layer, and the corresponding fatigue threshold value for a crack with its tip at the interface is estimated. This makes it possible to decide whether the crack will stop at the interface or continue growing into the second material.

The results generally contribute to a better understanding of the failure of bi-material bodies and of structures with protective layers.     

Growth mechanism of stress corrosion cracking in high strength steel

Stress corrosion crack growth rates are measured at sveral stress intensity levels for low-tempered 4340 steel in 0.1N H₂SO₄ solution. The characteristics of the growth rates are divided into three regions of stress intensity factors: Region I near K_1SCC; Region III near unstable fracture toughness, K_1SC; and Region II, which lies between the two. K_1SCC is the value of K at which no crack growth can be detected after 240 hr.

In order to explain these experimental results, the crack initiation analysis reported in a previous paper is extended to the growth rates. A detached crack initiates and grows at the tip of an already existing crack. When the detached crack reaches the tip of the main crack, the process repeats as a new existing crack.

A relationship between crack growth rate, v, and stress intensity factor, K, is obtained as a function of b/a and a = b + d, where b is the distance from the tip of the main crack to the detached crack, and d is the ydrogen atom saturated domain.

The experimental data are in good agreement with the theoretical values in Region II when a = 0.02 mm, b/a = 0.8, c₁/c₀ = 2.8 for 200°C tempered specimens and a = 0.015 mm, b/a = 0.7, c₁/c₀ = 3.0, ρ_b = 0.055 mm for 400°C tempered specimens, where ρ_b is a fictitious notch radius. The plateau part in Region II for 400°C tempered specimens is also successfully explained by the present theory. For Region III, the value of b/a will be almost equal to 1 because v → ∞ for b/a → 1. On the other hand, for Region I, b/a will be zero, since the value of v becomes negligibly small and no crack growth is observable.     

Determination and representation of the stress coefficient terms by path-independent integrals in anisotropic cracked solids

Because the elastic T-stress and other coefficients of the higher-order terms play an important role in fracture mechanics such as the stability of crack kinking, crack path, and two-parameter characterization of elastic-plastic crack tip fields, determination of all the coefficients in the crack tip field expansion in an anisotropic linear elastic solid is presented in this paper. Utilizing conservation laws of elasticity and Betti's reciprocal theorem, together with selected auxiliary fields, T-stress and third-order stress coefficients near the crack tip are evaluated first from path-independent line integrals. To determine the T-stress terms using the J-integral and Betti's reciprocal work theorem, auxiliary fields under a concentrated force and moment acting at the crack tip are used respectively. Through the use of Stroh formalism in anisotropic elasticity, analytical expressions for all the coefficients including the stress intensity factors are derived in a compact form that has surprisingly simple structure in terms of one of the Barnett-Lothe tensors, L. The solution forms for degenerated materials, monoclinic, orthotropic, and isotropic materials are also presented.     

Numerical simulation of fatigue crack growth

A numerical simulation of fatigue crack growth which uses currently available crack tip stress and strain fields is described. The essential features of the numerical model are the concepts of damage accumulation cycle by cycle and repeated re-initiation at the tip of the growing crack. The failure criteria employed are a combination of a failure condition and a critical distance over which this condition must be achieved. This critical distance, the material size parameter, has a magnitude which depends on the failure mechanism.

The use of the model to illustrate the effects of stress ratio and environmental effects is described and the ability of the model to predict the onset of bursts of crack growth due to static failure mechanisms is demonstrated. The phenomenon of self-arresting cracks is also displayed.

Material characteristics are included in the model and comparisons with experimental data are presented for a C-Mn steel used in the fabrication of offshore structures.     

Fatigue crack closure study based on whole-field displacements

This study is concerned with crack tip strain field fluctuations at loads below the point of crack closure in fatigue cycling. Moiré interferometry was used to investigate crack tip fields in compact tension specimens, cracked under constant stress intensity range and fixed R-ratio conditions. An elastic-plastic finite element model of simulated closure was developed to provide a theoretical cross-reference for the moiré studies. The ‘stretched zone’, which is believed to be the most significant source of closure effects, was simulated by inserting a constant thickness strip of elements into the crack before unloading from the maximum load point. Analysis of the crack tip fields in the experimental and theoretical cases was made in terms of crack face opening profiles, compliance changes and elastic stress intensity parameters. The latter were inferred through stress and displacement measurements made along circular and radial paths relative to the crack tip. Closure on the stretched zone was found to generate non-proportional loading in the crack tip field, so that the resulting stress changes were not well characterized by the asymptotic elastic equations. It is concluded firstly, that significant strain fluctuations occur below the point of closure load and that these should not be ignored in crack propagation studies. Secondly, the effective stress intensity range in fatigue cycling is not simply related to the open-crack stress intensity range and the need therefore remains for R-ratio and geometry effects to be treated as variables in crack propagation data collection programmes.     

Practical use of the ‘equivalent’ measured stress intensity factor to control fatigue crack propagation rates in aircraft full-scale fatigue tests—First assessment of the method in testing of a pressurized aircraft fuselage

In the case of circumferential cracks in a cylindrical fuselage, the comparison of some analysis and test results shows that the theoretical stress intensity factor is a suitable correlation parameter of fatigue crack propagation rates, both in aircraft fuselages and in plane panels. Values of the ‘equivalent’ stress intensity factor, computed by applying the Barrois-Bhandari method to slot-opening measurements performed under decreasing loading levels, agree well with the values computed from two dimensional Theory of Elasticity, using the method of finite elements.

In the case of longitudinal cracks, the experimental values of the ‘equivalent’ stress intensity factor, i.e. the stress intensity factor of the infinite plane sheet containing a centre crack with the same elastic strain and stress distributions near the boundary of the plastically strained region around the crack tip, yield a good correlation of fatigue crack propagation rates of the cracked fuselage and of cracked plane structures. The values of the ‘equivalent’ stress intensity factor are lower than those of the theoretical stress intensity factor, the interest of which disappears, but are also far higher than the bidimensionally computed values, which are no longer to be considered.

Some meant of safety provided to limit crack openings will make it possible, in the near future, to investigate test conditions reaching the ultimate residual static strength of cracked structures, while avoiding, however, catastrophic failures.     

An analytical method for mixed-mode propagation of pressurized fractures in remotely compressed rocks

An analytical method for mixed-mode (mode I and mode II) propagation of pressurized fractures in remotely compressed rocks is presented in this paper. Stress intensity factors for such fractured rocks subjected to two-dimensional stress system are formulated approximately. A sequential crack tip propagation algorithm is developed in conjunction with the maximum tensile stress criterion for crack extension. For updating stress intensity factors during crack tip propagation, a dynamic fictitious fracture plane is used. Based on the displacement correlation technique, which is usually used in boundary element/finite element analyses, for computing stress intensity factors in terms of nodal displacements, further simplification in the estimation of crack opening and sliding displacements is suggested. The proposed method is verified comparing results (stress intensity factors, propagation paths and crack opening and sliding displacements) with that obtained from a boundary element based program and available in literatures. Results are found in good agreements for all the verification examples, while the proposed method requires a trivial computing time.     

A model of brittle fracture propagation part I—continuum aspects

The micromechanism of crack propagation in steel is described and analyzed in continuum terms and related to the macroscopic fracture behavior. It is proposed that propagation of cleavage microcracks through favorably oriented grains ahead of the main crack tip is the principal weakening mode in brittle fracture. This easy cleavage process proceeds in the Griffith manner and follows a continuous, multiply connected, nearly planar path with a very irregular front which spreads both forward and laterally and leaves behind disconnected links which span the prospective fracture surface. A discrete crack zone which extends over many grains thus exists at the tip of a running brittle crack. Final separation of the links is preceeded by plastic straining within the crack zone and occurs gradually with the increasing crack opening displacement. It is suggested that in low stress fracture, straining of the links is the only deformation mode. However, it is recognized that under certain conditions plastic enclaves may adjoin the crack zone. This deformation mode is associated with high stress fracture, energy transition and eventually with crack arrest.

Energy dissipation resulting from the two deformation mechanisms is related to crack velocity, applied load and temperature and the crack velocity in a given material is expressed as a function of the external conditions. Fracture initiation and crack arrest are then discussed in terms of the conditions which are necessary to maintain the propagation process. Finally, the dimensions of a small scale crack tip zone for a steady state, plane strain crack are evaluated as functions of material properties and the elastic stress intensity factor.

The microstructural aspects of brittle fracture will be discussed in a separate Part 2 [1].     

Energy density approach for calculation of three-dimensional inelastic strain and stress at the crack tip in compact tension specimens of polycarbonate

An energy-based method is utilized for calculating elastic-plastic strains and stresses near fatigue crack tip in specimens of Merlon polycarbonate. The stress redistribution caused by the plastic yielding around the crack tip is taken into account so that theoretical crack tip strain is improved. The estimated values of crack tip strain based on an energy density approach are compared with experimental results obtained from an embedded grid moire technique and embedded strain gages. Large-scale yielding seems to dominate near the crack tip. In fact, the measured strain is in agreement with the elastic solution, which means, in reality, only small-scale yielding takes place near the crack tip. The strain in the mid-plane (plane strain) is found to be higher than in the surface plane (plane stress). The experimental and theoretical results are in good agreement.     

A finite element method for calculating stress intensity factors and its application to composites

A concept which allows for the development of efficient finite element techniques in the analysis of plane elastic structures containing cracks is discussed. It consists of combining a specially defined finite element in the region surrounding each crack tip with conventional CST elements describing the remaining portion of the geometry considered. For the special element a pair of displacement functions is chosen which adequately represents the singular character of the elastic solution at the crack tip. The application of this concept is illustrated through a specific numerical method developed by W. K. Wilson for the calculation of mode I stress intensity factors.

Wilson's method was coded and used to analyze an infinitely long strip under tension with a line crack perpendicular to its axis of symmetry. Circular inclusions of different material properties were assumed to be present near the tips of the crack and their effect on the mode I stress intensity factor was investigated.

It was found that more flexible inclusions increase the intensity factor while more rigid inclusions decrease it. These results are quite similar to those obtained by analytical methods in an analogous problem involving an infinite sheet, but in the case of a strip, the influence of inclusions on the intensity factor was found to be more pronounced.     

Numerical analysis of the inclusion-crack interactions using an integral equation

 A general-purpose integral formulation is proposed for the analysis of the interaction between inclusions and cracks embedded in an elastic isotropic homogeneous infinite medium subjected to a remote loading. This formulation is tailored for the inclusions of arbitrary shapes with the presence of cracks. The discretization is limited to the inclusions (with continuous quadratic triangular and quadrilateral elements) and the cracks (using discontinuous quadratic elements). For the calculation of the stress intensity factors at the crack tips, special crack tip elements are used to model the variation of the displacements near the crack tips. Maximum circumferential stress criterion is adopted to determine the crack propagating direction. Numerical results of benchmark examples are compared with other available methods. Received: 8 January 2002 / Accepted: 24 September 2002