Elastic-plastic Crack-tip Fields Research Articles

A numerical investigation of loss of crack tip constraint in a dynamically loaded ductile specimen

The objectives of this paper are to study the applicability of a two-parameter ( J– Q) characterisation of the elastic–plastic crack tip fields in a dynamically loaded ductile specimen and to ascertain the effect of loading rate J ̇ on the triaxiality parameter Q. To this end, 2D (plane strain) large deformation finite element analyses of a single edge notched specimen (SEN(T)) subjected to a tensile stress pulse are conducted. The material is assumed to obey the J 2 flow theory of plasticity with small elastic strains and both rate independent and rate dependent behaviour are considered. The results demonstrate that even a deeply cracked SEN(T) specimen which maintains high triaxiality under quasi-static loading can exhibit progressive loss of triaxiality as the loading rate increases. Since this behaviour is observed for both rate independent and rate dependent case, it is attributed to inertial effects. The above phenomenon can satisfactorily explain retarded micro-void growth rates near a notch tip, and strong elevation in the dynamic fracture toughness at high loading rates, which have been reported in recent computational and experimental studies.

The 3D Surface Crack-Front Constraints in Welded Joins

초록 The validity, of a single parameter such as stress intensity, factor K or J-integral in traditional fracture mechanics depends strongly on the geometry, and loading condition. Therefore the second parameter like T-stress measuring the stress constraint is additionally needed to characterize the general crack-tip fields. While many, research works have been done to verify, the J-T description of elastic-plastic crack-tip stress fields in plane strain specimens, limited works (especially. for bimaterials) have been performed to describe the structural surface crack-front stress fields with the two parameters. On this background, via detailed three dimensional finite element analyses for surface-cracked plates and straight pipes of homogeneous materials and bimaterials under various loadings, we investigate the extended validity or limitation of the two parameter approach. We here first develop a full 3D mesh generating program for semi-elliptical surface cracks, and calculate elastic T-stress from the obtained finite element stress field. Comparing the J-T predictions to the elastic-plastic stresses from 3D finite element analyses. we then confirm the extended validity of fracture mechanics methodology based on the J-T two parameters in characterizing the surface crack-front fields of welded plates and pipes under various loadings.

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.

The development of a Java engineering application for higher-order asymptotic analysis of crack-tip fields

In this paper an experience in development of an engineering application using Java is presented. Higher-order asymptotic analysis of elastic–plastic crack-tip fields includes a computationally intensive part related to obtaining the three-term asymptotic expansion. It also requires the input of a large amount of data for fitting the asymptotic field to numerically determined stresses in the near crack-tip region. The developed Java applet implements a graphical user interface using Abstract Windowing Toolkit components. The applet can be downloaded through the Internet and can be run on different computer platforms inside a Java enabled Web browser. Using simple examples, we explain how to overcome some drawbacks of Java API classes, which can arise during the development of engineering applications.

The application of constraint based fracture mechanics to engineering structures

Elastic-plastic crack tip fields can be characterised by two parameters, J and T/ Q which describes crack tip constraint. This forms the foundation of a constraint based fracture mechanics in which toughness is expressed as a function of constraint in the form of a J–( T/ Q) locus. The enhanced toughness associated with shallow cracks and defects can be used in a systematic manner by constraint matching, and implemented through a simple modification to a failure assessment diagram. This methodology enables the conservatism associated with the use of deeply cracked fracture toughness measurements to be relaxed. In this work these methods have been applied to a tubular welded joint.

Crack-tip fields in elastic-plastic material under plane stress mode I loading

New results on the crack-tip fields in an elastic power-law hardening material under plane stress mode I loading are presented. Using a generalized asymptotic expansion of the stress function, higher-order terms are found which have newly-discovered characteristics. A series solution is obtained for the elastic-plastic crack-tip fields. The expansion of stress fields contains both the \( r^{t_i } \sigma _{pq}^{(i)} (\theta ;t_i ) \) and \( \operatorname{Re} [r^{t_k } \sigma _{rs}^{(k)} (\theta ;t_k )] \) terms where ti is real and tk is complex; the terms σ(i)pq(θti) and σ(k)rsθtk) are real and complex functions of θ respectively. Comparing the results with that for the plane strain mode I loading shows that: (1) the effect of higher-order solutions on the crack-tip fields is much smaller; and (2) the path-independent integral J also controls the second-order or third-order term in the asymptotic solutions of the crack-tip fields for most of the engineering materials (1 3. These theoretical results imply that the crack-tip fields can be well characterized by the J-integral, and can be used as a criterion for fracture initiation under plane stress mode I loading. This is in agreement with existing full-field solutions and experimental data that J at crack growth initiation is essentially independent of in-plane specimen geometry. The comparison confirms the theoretical asymptotic solutions developed in this study.

Prediction of Cleavage Probability Using Higher Order Terms of the Crack Tip Field

The dependence of the Weibull stress is investigated on the parameters of the elastic-plastic crack tip field. The general form of the Weibull stress is given for a three-parameter approximation of the elasto-plastic crack tip field. This is the basis to define a constraint correction of the critical value of the J-integral for cleavage fracture.

Analytical forms of higher-order asymptotic elastic-plastic crack-tip fields in a linear hardening material under antiplane shear

An analytical study of the higher-order asymptotic solutions of the stress and strain fields near the traction-free crack tip under antiplane shear in a linear hardening material is investigated. The results show that every term of the asymptotic fields is controlled by both elasticity and plasticity and all the higher-order asymptotic fields are governed by linear nonhomogeneous equations. The first four term solutions are presented analytically and the first four terms are described by two independent parameters J and K2. The amplitude of the second order term solution is only dependent on the material properties, but independent of loading and geometry. This paper focuses on the case with traction-free crack surface boundary conditions. The effects of different crack surface boundary conditions, such as clamped and mixed surfaces, on the crack-tip fields are also presented. Comparison of multi-term solution with leading term solution, and finite element solution in an infinite strip with semi-infinite crack under constant displacements along the edges is provided.

Constraint-based failure assessment diagrams

Constraint-dependent toughness has been addressed by characterizing elastic-plastic crack tip fields and the associated toughness in terms of two parameters:J, and a parameter that indexes the level of constraint (T/Q). In the past, failure assessment diagrams have been developed on the basis of a single-parameter characterization of toughness. The present work modifies these diagrams to incorporate constraint effects and indicates the loadings where advantage can be taken of constraint enhanced toughness.

Stresses in a thin centrally cracked circular disk

Full field solutions, based on small deformation three-dimensional finite element analysis of a center-cracked circular thin disk under mode I, and mixed mode I and II loadings have been performed. Under mode I loading, the solution for the stresses under small scale yielding has been compared with the HRR plane stress solution. The existence of a high triaxial state of stress at the near crack tip in the mid-plane of the disk has been possible due to the thickness effect. This is in contrast to the loss of triaxiality of stresses in the plane strain condition in the presence of a significant amount of compressive T-stress along the crack flanks [C. Betegon and J. W. Hancock, Two parameter characterization of elastic-plastic crack-tip fields. J. appl. Mech. 58, 104–110 (1991)]. The extent of the three-dimensional zone surrounding the crack front has been mapped and compared with the available experimental results of Chiang and Hareesh [Three dimensional crack tip deformation: an experimental study and comparison to HRR-field. Int. J. Fracture 36, 243–253 (1988)]. Under mixed mode I and II loading, the annular region surrounding the plastic zone in which the K-field dominates has been explored. This has been performed at a specific load level at which the fatigue crack extension experiment has been performed in the thin disk [T. K. Paul, Three-dimensional stress field near the crack front in a thin circular disk of elastic-plastic material (submitted for publication)]. The plastic zone shapes and sizes are quite different from the plane stress shape as obtained in the mode I analysis of a thin cracked sheet by Hom and McMeeking [Large crack tip opening in thin elastic-plastic sheet. Int. J. Fracture 103–122 (1990)] and plane stress mixed mode I and II by Shih [Elastic-plastic analysis of combined mode fracture problems, Ph.D. Dissertation, Harvard University (1973)].

Constraint-based failure assessment diagrams

The loss of crack tip constraint leads to enhanced resistance to ductile tearing and cleavage. However, current defect assessment schemes use lower bound toughness obtained from highly constrained testpieces. Although this approach is safe, it is conservative, and may lead to the imposition of prohibitive repair and inspection policies when applied to defects which develop unconstrained flow fields. Recent theoretical developments indicate that geometry dependent fracture toughness effects can now be rationalised and united through a two-parameter characterisation of elastic-plastic crack tip fields. In this context a simple modification to existing failure assessment diagrams (FAD) is proposed, using a constraint matched toughness to modify one of the axes of the FAD. The procedure recovers the original failure assessment line and unifies the constraint dependent fracture toughness data.

Two-Parameter Characterization of Elastic-Plastic Crack-Tip Fields

Plane-strain elastic-plastic crack-tip fields have been modeled with modified boundary layer formulations based on the first two terms K and T, of the elastic field. These formulations match the appropriate full field solutions. Compressive T stresses reduce the stresses by an amount which is independent of radial distance, corresponding to the introduction of a second term in addition to the dominant plastic singularity. Geometries which maintain J-dominance are characterized by zero or positive T stresses, while geometries with negative T stresses can be described by a two-parameter characterization using J and T into full plasticity.

The characterization of a crack-tip field

The stress intensity factor, K has been successfully used as a measure of fracture toughness, and stress intensity factor range, Δ K, correlates well with fatigue crack growth rate. This paper gives a deductive logic to demonstrate rigorously that, in the case of small-scale yielding, K and Δ K are capable of characterizing the elastic-plastic crack-tip field both under a monotonic loading and cyclic fatique loading. This capability to characterize the entire crack-tip field is the underlying reason for the successful application of K and Δ K to linear fracture mechanics.

A Comparison of Crack-Tip Field Parameters for Large and Small Fatigue Cracks

A comparison of the elastic-plastic crack-tip fields of large and small fatigue cracks has been made using the ΔJ-integral approach. Using experimental strain and displacement range measurements obtained by means of the stereoimaging technique, the ΔJ-integral has been computed for an aluminum alloy containing either large or small fatigue cracks, by performing line-contour integration along a variety of rectangular paths around the crack tip. These calculations reveal that for both large and small cracks the ΔJ-integral is path-dependent, and that the value of ΔJ increases with decreasing distance from the crack tip. Using an average local ΔJ or one estimated from the CTOD, the enhanced crack growth rates associated with small fatigue cracks can be explained on the basis of the large average ΔJ-integral within the cyclic plastic zone. The effect of crack closure on the computed local ΔJ (or ΔK) and crack growth is discussed.