T-stress Results Research Articles

On the T-stress extraction method used by current version of Abaqus

In the present note, it is shown that in the current version of Abaqus (6.10 and later), there is an error in the method used for the determination of T-stress. In particular, errors will occur in cases when there are tractions applied to the crack faces. Since T-stress is an important parameter characterizing the crack tip stress field, used widely in fracture mechanics analysis, it is important to point out this to the users of Abaqus. The reason for this error is due to the incorrect interpretation of the asymptotic crack tip stress fields. To obtain correct T-stress results, the users need to remove this error manually.

T-Stresses for Edge Cracks and Vanishing Ligaments

An edge-cracked half-plane 0 < x < A and a half-plane x > 0 with a semi-infinite crack x > a perpendicular to the edge are examined in this paper. Uniform crack-face loading is thoroughly examined, with a thorough exposition of the Koiter Wiener–Hopf approach (Koiter, 1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374); an analytical expression for the corresponding T-stress is obtained. For the additional cases of (i) nonuniform edge-crack crack-face loading σ(x/A)k (ℜ(k)>-1), (ii) concentrated loading at the edge-crack crack mouth, the Wiener–Hopf solutions and analytical T-stress expressions are provided, and tables of T-stress results for σ(x/A)k and σ(1-x/A)k are presented. A Green's function for the edge-crack T-stress is developed. The differing developments made by Koiter (1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374, Wigglesworth (1957, “Stress Distribution in a Notched Plate,” Mathematika, 4, pp. 76–96), and Stallybrass (1970, “A Crack Perpendicular to an Elastic Half-Plane,” Int. J. Eng. Sci., 8, pp. 351–362) for the case of an edge-cracked half-plane are enhanced by deducing a quantitative relationship between the three different Wiener–Hopf type factorizations. An analytical universal T-stress expression for edge-cracks is derived. Finally, the case of a vanishing uncracked ligament in a half-plane is examined, and the associated Wiener–Hopf solution and analytical T-stress expression are again provided. Several limiting cases are examined.

Numerical solution for the T-stress in branch crack problem with infinitesimal branch length

This paper investigates the T-stress in a branch crack problem with infinitesimal branch length. The branch crack is composed of a main crack and many branches. The ratio of the lengths for branch to main crack is very small. A singular integral equation method is suggested to solve the problem numerically and the stress intensity factor and T-stress can be evaluated immediately. Many computed results for T-stress under different conditions for branches are presented. It is found from the computed results that the interaction for T-stress among branches is very complicated.

Evaluation of the T-stress in branch crack problem

This paper investigates the T-stress in the branch crack problem. The problem is modeled by a continuous distribution of dislocation along branches, and the relevant singular integral equation is obtained accordingly. After discretization of the singular integral equation, the balance for the number of equations and unknowns is well designed. After the singular integral equation is solved, the equation for evaluating the T-stress is derived. The merit of present study is to provide necessary equation for evaluating T-stress, rather than to provide the integral equation. Many computed results for T-stress under different conditions for branch crack are presented. It is found from the computed results that the interaction for T-stress among branches is complicated.

Variation of stress intensity factor and elastic T-stress along the crack-front in finite thickness plates

Non-singular terms in the series expansion of the elastic crack-tip stress fields, commonly referred to as the T-stress. The T-stress is as an additional stress field characterizing parameter to stress intensity factor (K) in the analysis of cracked bodies. T-stress is used as an important constraint parameter in the fracture analysis. In this investigation, three-dimensional finite element analyses have been conducted to compute the elastic T-stress considering a single edge notched tensile (SENT) specimen with varied thickness and a/W ratio. The results indicate that the T-stress depends on the specimen thickness and significantly varies along the crackfront from surface to centre of the specimen. The T-stress results obtained in the present analysis together with corresponding KI values can be used for analysis of constraint effects in a fracture specimen.

Evaluation of the T-stress for interacting cracks

This paper investigates the T-stress for interacting cracks in general case. The cracks are located in an arbitrary position with the remote tension. If one uses the traction along crack face as unknown function, the problem for interacting cracks can be reduced to a Fredholm integral equation. The solution of integral equation provides necessary knowledge for evaluating the T-stress at a crack tip . The T-stress at the crack tip is generally composed of three parts, one from the uniform field and other two from the perturbation field. Explicit equations for the three parts are proposed. The case for two collinear cracks is taken as an example to examine the accuracy of the suggested numerical method. Many results for T-stress for interacting cracks are presented. It is shown that the interaction is significant for the closer cracks.

Eigenfunction expansion variational method for stress intensity factor and T-stress evaluation of a circular cracked plate

In this paper, the eigenfunction expansion variational method (Abbreviated as EEVM) is developed to solve the T-stress problem of the circular cracked plate. In the traction boundary value problem, EEVM is equivalent to the theorem of least potential energy in elasticity. Therefore, EEVM possesses a clear physical meaning. EEVM does not need any boundary collocation scheme. For the circular cracked plate, the following boundary value problems are solved: (a) with a uniform normal loading on the boundary, (b) with a partial loading on the boundary, (c) under mixed boundary condition. For the circular cracked plate with applied concentrated forces, after using the superposition principle and EEVM, the boundary value problem is solved. In the numerical examples, many computed results for stress intensity factor (SIF) and T-stress are presented. Some of computed results for T-stress are first presented in this paper.

T-stress evaluation for slightly curved crack using perturbation method

A general formulation for evaluating the T-stress at crack tips in a curved crack is introduced. In the formulation, a singular integral equation with the distribution of dislocation along the curve is suggested. For a slightly curved crack, a small parameter is generally assumed for the crack configuration. By using the assumption for the small parameter, the perturbation method is suggested and it reduces the singular integral equation into many successive singular integral equations. If the cracked plate has a remote loading and the curve configuration is a quadratic function, the mentioned successive singular integral equations can be solved in a closed form. Therefore, the solution for the T-stress in a closed form is obtained. The obtained results for T-stress are shown by figures. It is found that if the involved parameter is not too small, the influence of the curve configuration is significant. Comparison for T-stresses obtained from a quadratic-shaped curved crack and an arc crack is presented.

Weight Functions for T-Stress for Edge Cracks in Thick-Walled Cylinders

This paper presents T-stress solutions for an internal edge crack in thick-walled cylinders. Elastic fracture mechanics analysis using the boundary element method (BEM) is performed to determine the T-stress solutions for a wide range of radius ratios and relative crack lengths. The loading cases considered in the BEM analysis for the cracked cylinder are crack-face pressures with polynomial stress distributions acting on the crack face. T-stress results for the uniform and linearly varying crack-face pressure cases are subsequently used as the reference solutions to derive weight functions for T-stress. Boundary element results of T-stress for other stress distributions, namely, other nonlinear crack face loading, internal pressure, and steady-state thermal loading, are used to validate the derived T-stress weight functions. Excellent agreement between the results from the weight function predictions and those directly computed is shown to be obtained. The weight functions derived are suitable for obtaining T-stress solutions for thick-walled cylinders with an internal edge crack under any complex stress fields.

T-stress solution for DCDC specimens

A finite element study is performed in order to determine the T-stress term for DCDC specimens with holes set off from the specimen centre line. In addition, mixed-mode stress intensity factors K I and K II are given and the biaxiality ratio β is computed from T and K I. The T-stress results are fitted and presented in a simple relation.

Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates

Three-dimensional finite element analyses have been conducted to calculate the elastic T-stress for semi-elliptical surface cracks in finite thickness plates. Far-field tension and bending loads were considered. The analysis procedures and results were verified using both exact solutions and approximate solutions. The T-stress solutions are presented along the crack front for cracks with a/ t values of 0.2, 0.4, 0.6 or 0.8 and a/ c values of 0.2, 0.4, 0.6 or 1.0. Based on the present finite element calculations for T-stress, empirical equations for the T-stress at three locations: the deepest, the surface and the middle points of the crack front under tension or bending are presented. The numerical results are approximated by empirical formulae fitted with an accuracy of 1% or better. They are valid for 0.2⩽ a/ c⩽1 and 0⩽ a/ t⩽0.8. These T-stress results together with the corresponding K or J values for surface cracks are suitable for the analysis of constraint effects for surface cracked components.

T-stresses in rectangular plates and circular disks

The aim of the present note is to compile T-stress results for a number of rectangular components and circular disks under different loadings. In addition, Green’s function representations are proposed, which enable the determination of T for arbitrarily given stress states (residual stresses, thermal stresses) in the uncracked body and a method is applied, which allows the computation of T-stresses for any traction distribution at the plate ends. The results are presented in the form of tables, diagrams and polynomials.