The influence of the T-stress on the directional stability of cracks

The sign of the T-stress has been widely used for deciding whether directional stability prevails for straight cracks subjected to mode I loading under small scale yielding. However, there is little evidence for the reliability of such a criterion. On the contrary, it is shown that a local criterion is not applicable and that directional stability generally depends on body and load geometry as well as on material parameters, whereas the sign of the T-stress is totally irrelevant in most cases.     

The effective T-stress estimation and crack paths emanating from U-notches

The concept of the T-stress as a local constraint factor has been extended to U-notch tip stress distribution as the effective T-stress. The effective T-stress has been estimated as the average value of the T-stress in the region corresponding to the effective (characteristic) distance ahead of the notch tip. The T-stress is evaluated by finite element method using the experimental load for crack initiation and computing the difference between principal stresses along ligament. A large range of critical effective T-stress values is investigated for different specimen configurations and notch aspect ratios. Crack stabilisation and crack bifurcation for fracture emanating from notches according to the critical effective T-stress is discussed. A model involving the influence of the critical effective T-stress on void growth for ductile failure in the vicinity of the notch tip has been proposed.     

T-effect for the interaction problem of an interface macrocrack with a near-tip microvoid

The paper deals with the analysis of the T-effect in the problem of an interface macrocrack interacting with a microvoid in the process zone of the main crack tip. With the aid of the `pseudo-traction-edge-dislocation' method, the interaction problem with the T-effect under the remote field loading conditions can be reduced to singular Fredholm integral equations with Hilbert kernel. After solving the singular integral equations numerically, the T-effect on the main crack tip parameters is analyzed and the numerical results are given in figures.     

Crack path selection in adhesively bonded joints: the roles of external loads and speciment geometry

This paper investigates the roles of external loads and specimen geometry on crack path selection in adhesively bonded joints. First, the effect of mixed mode fracture on crack path selection is studied. Using epoxy as an adhesive and aluminum as the adherends, double cantilever beam (DCB) specimens with various T-stress levels are prepared and tested under mixed mode fracture loading. Post-failure analyses on the failure surfaces using X-ray photoelectron spectroscopy (XPS) suggest that the failure tends to be more interfacial as the mode II fracture component in the loading increases. This fracture mode dependence of the locus of failure demonstrates that the locus of failure is closely related to the direction of crack propagation in adhesive bonds. Through analyzing the crack trajectories in failed specimens, the effect of mixed mode fracture on the directional stability of cracks is also investigated. The results indicate that the direction of the crack propagation is mostly stabilized when more than 3% of mode II fracture component is present at the crack tip regardless of the T-stress levels in the specimens for the material system studied. Second, using a high-speed camera to monitor the fracture sequence in both quasi-static and low-speed impact tests, the effect of debond rate on the locus of failure and directional stability of cracks is investigated. Post-failure analyses including XPS, Auger electron spectroscopic depth profile, and scanning electron microscopy indicate that as the crack propagation rate increases, the failure tends to be more cohesive and the cracks tend to be directionally unstable. Last, as indicated by the finite element analyses results, the T-stresses, and therefore the directional stability of cracks in adhesive bonds, are closely related to the thickness of the adhesive layer and also the thickness of adherend. This specimen geometry dependence of crack path selection is studied analytically and is verified experimentally.     

Mode I fatigue crack growth under biaxial loading

This paper is centred on the role of the T-stress during mode I fatigue crack growth. The effect of a T-stress is studied through its effect on plastic blunting at crack tip. As a matter of fact, fatigue crack growth is characterized by the presence of striations on the fracture surface, which implies that the crack grows by a mechanism of plastic blunting and re-sharpening (Laird C. The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In: Fatigue crack propagation, STP 415. Philadelphia: ASTM; 1967. p. 131–68 [8]). In the present study, plastic blunting at crack tip is a global variable ρ, which is calculated using the finite element method. ρ is defined as the average value of the permanent displacement of the crack faces over the whole K-dominance area. The presence of a T-stress modifies significantly the evolution of plastic deformation within the crack tip plastic zone as a consequence of plastic blunting at crack tip. A yield stress intensity factor K_Y is defined for the cracked structure, as the stress intensity factor for which plastic blunting at crack tip exceeds a given value. The variation of the yield stress intensity factor was studied as a function of the T-stress. It is found that the T-stress modifies significantly the yield point of the cracked structure and that the yield surface in a (T, K_I) plane is independent of the crack length. Finally, a yield criterion is proposed for the cracked structure. This criterion is an extent of the Von-Mises yield criterion to the problem of the cracked structure. The proposed criterion matches almost perfectly the results obtained from the FEM. The evolution of the yield surface of the cracked structure in a (T, K_I) plane was also studied for a few loading schemes. These results should develop a plasticity model for the cracked structure taking into account the effect of the T-stress.     

An improved technique for the solution of edge crack problem for finite plate

This paper provides a detailed examination for the edge crack problem of finite plate. The Williams expansion for the crack problem is used first. Secondly, the complex potentials for the central crack problem are used in the present study, which is called the improved technique hereafter. In both techniques, the eigenfunction expansion variational method (EEVM) is used for evaluating the undetermined coefficients in the expansion form. The ratio of height versus width of plate (h/w) is varying from 1.5, 1.0, 0.75, 0.5, 0.4, 0.3 to 0.25. The ratio of edge crack length versus width of plate (a/w) takes two sets: (1) a/w = 0.1, 0.2, … to 0.9, (2) a/w = 0.01, 0.02, … to 0.09. The detailed computation proves that for moderate cases of the a/w ratio, for example, 0.2 < a/w < 08, the deviations for SIFs and T-stress from two techniques are minor. However, for the case of short edge crack length, for example, a/w < 0.05, the deviations for SIFs and T-stress from two techniques are significant. It is found that the Williams expansion may not be suitable for the short edge crack, for example, a/w < 0.05.     

Stress intensity factors and T-stresses for offset double edge-cracked plates under mixed-mode loadings

A particular weight function method is used in this study to determine the stress intensity factors (SIFs) and T-stresses for offset double edge-cracked plates (ODECPs). By using reference loading conditions prescribed on the crack flanks for finite element analyses, the coefficients of weight functions are derived and compiled in the form of tables. With the weight functions, the SIFs and T-stresses for several loading cases are calculated. The results compare well with those obtained using the displacement method. Applying the derived weight functions, the SIFs and T-stresses for ODECPs under several loading conditions are determined. The results can be used as references for related applications.     

Numerical assessment of T-stress computation using a p-version finite element method

Two path independent integrals for T-stress computations, one based on the Betti-Rayleigh reciprocal theorem and the other based on Eshelby's energy momentum tensor are studied. Analytical as well as numerical equivalence between the two integrals is found. To quantify and assess the accuracy of computed values, error analysis for the proposed numerical computation of the T-stress is presented. Specifically, it is found that the error of the computed T-stress is proportional to the ratio of the stress intensity factor divided by the square root of the characteristic dimension of the integration domain where the path independent integral is evaluated. Using a highly accurate hierarchical p-version finite element method, the convergence and accuracy of computed values are easily monitored, and it is shown for numerical examples that the error of the computed T-stress complies with the described error analysis. We conclude that path independent integrals, in conjunction with hierarchical p-version finite element methods, provide a powerful and robust tool to obtain highly accurate numerical results for the T-stress.     

Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions

Finite element analyses have been conducted to calculate elastic T-stress solutions for cracked test specimens. The T-stress solutions are presented for single edge cracked plates, double edge crack plates and centre cracked plates. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were used to derive weight functions for T-stress for the corresponding specimens. The weight functions for T-stress are then verified against several linear and non-linear stress distributions. The derived weight functions are suitable for the T-stress calculation for cracked specimens under any given stress field.     

Crack observation methods, their application and simulation of curved fatigue crack growth

The observation of cracks with curved crack path has to be done optically. It is shown that a modified commercial flat bed scanner and a combination of a high-resolution scanner camera system with 10,000 × 10,000 pixels with a telecentric lens are appropriate for high-resolution (up to 8 μm/pixel) optical recording of multiple crack ends on sample areas up to 210 mm × 290 mm. The high-resolution photographs are suitable for determination of crack lengths. It is also possible to observe crack paths or geometrical crack tip parameters and strain fields by image correlation. The method is used to determine static crack resistance and cyclic crack growth curves on center crack tension and biaxial cruciform samples.Furthermore, the paper presents an improved finite element technique for the simulation of curved fatigue crack growth in a multiple arbitrarily pre-cracked isotropic sheet under biaxial plane stress loading applying a predictor-corrector procedure in combination with the modified virtual crack closure integral (MVCCI) method including the consideration of the plastic limit loads. For this, the program PCCS-2D was extended to analyse the crack growth and the plastic limit load for each crack propagation step in a fully automatic simulation. The proposed solution algorithm provides a powerful tool for flaw assessment with the failure assessment diagram procedure in combination with a numerical crack path simulation. Finally, the simulation is verified experimentally.     

Influence of T-stress, cohesive strength and yield strength on the competition between decohesion and plastic flow in a crack edge vicinity

The discussion is based on the cell model of materials. Each cell contains one dominant kernel of micro-separation, for instance a particle. A cell is either in a cohesive or a decohesive state, the latter implying instability at load control. The process region consists of cells which have reached the decohesive state. The cells are characterized by their linear size and their cohesion-decohesion relation. The process region develops either in an elastic or in a plastic environment. In the latter case, it may be more or less deeply embedded in the plastic region. In some cases there will be no process region, only plastic flow. The ratio between the cohesive strength and the yield strength is an important parameter for describing this competition between decohesion and plastic flow, but the T-stress also plays a part. The fracture toughness depends on the area under the cohesion-decohesion curve and on the embedment of the process region.     

A note on <Emphasis Type="Italic">T</Emphasis>-stress determination using dislocation arrays

In general, numerical determination of T-stresses requires careful handling, because of their location in the vicinity of singular points. Discretization methods, such as common finite element methods, may not lead to accurate results. However, the well known technique of using dislocation arrays for determination of stress intensity factors may also be used for determination of T-stresses. Except for a few simple cases, this technique leads to a Fredholm equation, that can be solved very accurately. The general method is described and the technique is demonstrated by examples.     

A non-singular boundary integral formula for determining the T-stress for cracks of arbitrary geometry

A simple, yet accurate 2-D boundary integral equation (BIE) for determining the T-stress for cracks of arbitrarily geometry is introduced in this paper. The formulation is based upon the asymptotic expansion for the stress field in the vicinity of a crack tip. It can be conveniently implemented in the post-processing stage of a boundary element fracture analysis. As demonstrated in this work, the proposed BIE is non-singular, and thus it can directly be collocated at the crack tip under consideration. The technique requires a similar computational effort as that used in calculating the stress components at an interior point of a domain. Consequently, this new approach is very computationally effective and accurate for evaluating the elastic T-stress. Five test examples, involving straight, kink and curved cracks, are studied to validate the proposed technique and to assess its accuracy.     

Quantifying crack tip displacement fields with DIC

Crack paths under both fatigue and fracture conditions are governed by the crack tip displacement field and the material deformation characteristics, including those influenced by metallurgical anisotropy. Experimental techniques such as thermoelasticity and photoelasticity have been successfully used to characterise the elastic stress fields around cracks but they do not take into account either plasticity or anisotropy. Considerable work has been carried out to characterise crack tip stress fields from displacement measurements. The current method of choice for obtaining displacement field data is digital image correlation (DIC) which has undergone significant advances in the recent years. The ease of use and capabilities of the technique for full field displacements has led to improved methods for characterising crack tip displacement fields based on data obtained from DIC. This paper gives an overview of some of the applications of DIC for crack tip characterisation such as K, T-stress and crack tip opening angle (CTOA) measurements as well as data obtained from 3D measurements of a propagating crack.     

Dynamic fracture mechanics analysis of failure mode transitions along weakened interfaces in elastic solids

Dynamic fracture mechanics theory was employed to analyze the crack deflection behavior of dynamic mode-I cracks propagating towards inclined weak planes/interfaces in otherwise homogenous elastic solids. When the incident mode-I crack reached the weak interface, it kinked out of its original plane and continued to propagate along the weak interface. The dynamic stress intensity factors and the non-singular T-stresses of the incident cracks were fitted, and then dynamic fracture mechanics concepts were used to obtain the stress intensity factors of the kinked cracks as functions of kinking angles and crack tip speeds. The T-stress of the incident crack has a small positive value but the crack path was quite stable. In order to validate fracture mechanics predictions, the theoretical photoelasticity fringe patterns of the kinked cracks were compared with the recorded experimental fringes. Moreover, the mode mixity of the kinked crack was found to depend on the kinking angle and the crack tip speed. A weak interface will lead to a high mode-II component and a fast crack tip speed of the kinked mixed-mode crack.     

On crack path evolution rules

The models of crack growth in mixed mode conditions are reviewed for the plane and three-dimensional (3D) states of stress. Both critical load value and crack path or surface growth are predicted by different criteria in terms of elastic singular stress states and T-stress component. Monotonic and cyclic loading induced crack growth is considered. The energy and critical plane criteria based on local or non-local measure of stress and strain are most useful in developing predictive crack growth simulation. The finite critical distance from the crack tip should be specified to provide averaged or local stress and strain states. The application of MK-criterion of crack growth expressed in terms of volumetric and deviatoric strain energies is presented for several specific cases of monotonic and cyclic loading. The concepts of smooth and rough crack surface are discussed with application to 3D crack surface growth.     

Material constraint parameters for the assessment of shallow defects in structural components. Part II: Constraint-based assessment of shallow defects

This Part II paper describes a series of constraint-based R6 assessments of shallow-cracked specimens under equibiaxial loading using material constraint parameters obtained from lookup tables presented in the companion Part I paper. Parameters are derived for an A533B-1 steel forging from a knowledge of the yield and flow behaviour of the material and a calibration of the Beremin model parameter m. Resulting assessments, which use both T- and Q-stress to quantify constraint, are found to be conservative with respect to the available experimental data. The results are also used to predict the influence of T-stress on the Master Curve reference temperature. Available data are broadly in agreement with the predicted trend curve. Finally, the results demonstrate that out-of-plane biaxial loading increases constraint to the extent that the inherent conservatism of the elastic T-stress at high L_r is eroded. Out-of-plane constraint effects are only properly accounted for by the hydrostatic Q-stress and for this reason, the use of T to assess biaxial loading situations should be undertaken with a degree of caution.     

Crack-tip constraint in mode II deformation

Studies of cracked specimens loaded in mode I have shown that the stresses near the crack tip depend significantly on the level of constraint. The stresses can be determined near the crack tip using the HRR solution, but only for high constraint specimens. For other levels of constraint, O'Dowd and Shih's Q parameter may be used to adjust the stresses derived from the HRR solution. Only limited research has been carried out to study the effect of constraint in mode II. In this paper a mode II boundary layer formulation is used to study the effect of far field elastic stresses on the size and shape of the plastic zone around the crack tip and on the stresses inside the plastic zone. It is shown that in mode II, both positive and negative values of remote T-stress influence the tangential stress along the direction of maximum tangential stress. In the spirit of O'Dowd and Shih, a dimensionless parameter Q _II is introduced to quantify the constraint for mode II specimens failing by brittle fracture. The relation between Q _II and T/₀ is determined for different values of the strain hardening coefficient n. To investigate the range of validity of the Q–T diagram for real specimens, the constraint parameter Q _II is calculated directly from finite element analysis for three mode II specimens and compared with the evaluation using the Q–T diagram.