Mixed-mode Stress Field Research Articles

Connecting the essential work of fracture, stress intensity factor, Hill’s criterion in mixed mode I/II loading

Ductile sheet structures are frequently subjected to mixed mode loading, resulting that the structure is under the influence of a mixed mode stress field. Instances of interest are when stable crack growth occurs and when the crack-tip is propagating in this complex mixed-mode condition, prior to final fracture. Purposely designed apparatus was built to test thin-sheets of steel (Grade: DX51D) under mixed-mode I/II. These tests, under plane stress conditions, also investigated the effect of thickness on the specific essential work of fracture or the fracture toughness of the material under quasi-static cracking conditions. The fracture toughness is evaluated under incremental mixed-mode loading conditions. The direction of the propagating crack path and fracture type were observed and discussed as the loading mixity was varied. Whilst the specific essential work of fracture or fracture toughness was obtained using the energy approach, the theoretical analysis of the fracture type and direction of crack path were based on the crack tip stresses and fracture criterions of maximum hoop stress and maximum shear stress along with the utilisation of Hill’s theory. For mixed-mode I/II loading, the variation in the fracture toughness contributions ratios are evaluated and used predicatively using the established energy criterion approach to the crack tip stress intensity approach. The comparison between the theoretical directions of the crack path, failure mode propagation are in good agreement with those obtained from experimental testing indicating the definite link between both approaches.

Mixed-mode crack-tip stress fields for orthotropic functionally graded materials

Quasi-static mixed mode stress fields for a crack in orthotropic inhomogeneous medium are developed using asymptotic analysis coupled with Westergaard stress function approach. In the problem formulation, the elastic constants E11, E22, G12, ν12 are replaced by an effective stiffness \({E=\sqrt {E_{11} E_{22}}}\), a stiffness ratio \({\delta =\left({{E_{11}}\mathord{\left/ {\vphantom {{E_{11}} {E_{22}}}}\right. \kern-\nulldelimiterspace} {E_{22}}} \right)}\), an effective Poisson’s ratio \({\nu =\sqrt {\nu_{12}\nu _{21}} }\) and a shear parameter \({k=\left({E \mathord{\left/ {\vphantom {E {2G_{12}}}}\right. \kern-\nulldelimiterspace} {2G_{12}}}\right)-\nu }\). An assumption is made to vary the effective stiffness exponentially along one of the principal axes of orthotropy. The mode-mixity due to the crack orientation with respect to the property gradient is accommodated in the analysis through superposition of opening and shear modes. The expansion of stress fields consisting of the first four terms are derived to explicitly bring out the influence of nonhomogeneity on the structure of the mixed-mode stress field equations. Using the derived mixed-mode stress field equations, the isochromatic fringe contours are developed to understand the variation of stress field around the crack tip as a function of both orthotropic stiffness ratio and non-homogeneous coefficient.

Calculation of mixed-mode stress field at a sharp notch tip using M1?-integral

Direct computation of the mixed-mode stress field at a sharp notch tip appears to be difficult in that the mode I and mode II asymptotic stresses are in general governed by different orders of singularity. In this paper, we first present a path-independent integral termed M 1ɛ. The relation between M 1ɛ and the generalized stress intensity factors is then derived and expressed as function of the notch angle. Once the M 1ɛ-integrals are accurately computed, the generalized SIF's and, consequently, the asymptotic mixed-mode stress field can thus be properly determined. No extra complementary solutions are required in the formulation. Further, no particular singular elements are required when the integration is performed by using finite elements.

Finite element calculation of elastodynamic stress field around a notch tip via contour integrals

Direct computation of the mixed-mode dynamic asymptotic stress field around a notch tip is difficult because the mode I and mode II stresses are in general governed by different orders of singularity. In this paper, we propose a pair of elastodynamic contour integrals JkR(t). The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated with finite element solutions. Also, by defining a pair of generalized stress intensity factors (SIFs) KI,β(t) and KII,β(t), the relationship between JkR(t) and the SIF’s is derived and expressed as functions of the notch angle β. Once the JkR(t)-integrals are accurately computed, the generalized SIF’s and, consequently, the asymptotic mixed-mode stress field can then be properly determined. No particular singular elements are required in the calculation. The proposed numerical scheme can be used to investigate the dynamic amplifying effect in the near-tip stress field.

The non-linear solution of the mixed-mode caustics in birefringent materials

The purpose of this note is to study the transmitted and rear face reflected caustics in birefringent plates for a mixed-mode stress field at the crack tip.

Calculation of mixed-mode stress intensity factors for a crack normal to a bimaterial interface using contour integrals

A pair of contour integrals J kε are proposed in this paper. The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated without using any particular singular finite elements. Also, the relationship between J kε and the generalized stress intensity factors (SIFs) is analytically derived and expressed as functions of the bimaterial mechanical constants. Once the J kε -integrals are accurately computed, the generalized SIFs and, consequently, the asymptotic mixed-mode stress field can then be properly determined. Numerical results in this study show that the contribution from mode II stress component appears to be more dominant when the uncracked material is relatively stiffer, and vice versa.

Digital Photoelasticity: Advanced Techniques and Applications

Digital Photoelasticity: Advanced Techniques and Applications

Evaluation of the stress field around a notch tip using contour integrals

The mode I and mode II asymptotic stresses around a notch tip are in general governed by different orders of singularity. Direct computation of the mixed-mode near-tip stress field therefore appears to be difficult. In this paper, we propose a pair of contour integrals J kR . The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated with finite element solutions. As an aside, by defining a pair of generalized stress intensity factors (SIFs) ( K I) β and ( K II) β , the relationship between J kR and the SIFs is derived and expressed as functions of the notch angle β. Once the J kR -integrals are accurately computed, the generalized SIFs and, consequently, the asymptotic mixed-mode stress field can then be properly determined. The feasibility of our formulation is demonstrated in two numerical examples, where various instances with different notch angles are considered. No particular singular elements are used in this study.

Dynamic shear bands: an investigation using high speed optical and infrared diagnostics

This paper presents an experimental investigation of the initiation and propagation characteristics of dynamic shear bands in C300 maraging steel. Pre-fatigued single edge notched specimens were impacted on the edge under the notch to produce shear dominated mixed mode stress fields. The optical technique of coherent gradient sensing (CGS) was employed to study the evolution of the mixed mode stress intensity factors. Simultaneously, a newly developed two-dimensional high speed infrared (IR) camera was employed to observe the temperature field evolution during the initiation and propagation of the shear bands. Possible criteria for failure mode selection are discussed. The IR images, for the first time, revealed the transition of crack tip plastic zone into a shear band and also captured the structure of the tip of a propagating shear band. These thermographs support the notion of a diffuse shear band tip and reveal “hot spots” distributed along the length of a well-developed shear band.

Elastic and elastic–plastic fields on circular rings containing a V-notch under inclined loads

Elastic and elastic–plastic finite element analyses are described for circular rings containing a V-notch on the internal radius. The circular rings are subjected to a pair of compressive loads at different angles to the plane of the notch axis. In the elastic analysis, the mixed-mode stress field near the notch root is calculated for a sharp tip V-notch with or without a short precrack. The analyses include calculations of the stress intensity factors of mode I and mode II, and of the energy release rate for crack extension. In the elastic–plastic analysis, the stress distribution near the notch root is presented. The relationship between the notch root radius and the plastic zone is described. The influence of the angle of loading on the distribution of elastic–plastic stresses is discussed. In order to justify the results of the elastic–plastic numerical method, a test of constant loading with a stainless steel specimen is carried out by using a Moiré interferometry method. Furthermore, 10 tests of crack initiation by cyclic and inclined loading are performed to examine the adaptation of the finite element analysis to the experiments. The results of the calculation are found to correspond to the experimental data.

Mixed mode near-tip fields for cracks in materials with strain-gradient effects

Large strain gradients exist near the tip of a crack due to stress singularity. The strain-gradient effect becomes significant when the size of the fracture process zone around a crack tip is comparable to the intrinsic material length, l, typically on the order of microns. Fleck and Hutchinson's [(1993) A phenomenological theory for strain-gradient effects in plasticity. J. Mech. Phys. Solid 41, 1825–1857], strain-gradient plasticity theory is applied to investigate the asymptotic field near a mixed mode crack tip in elastic as well as elastic-plastic materials with strain-gradient effects. It is established that the dominant strain field is irrotational. For an elastic material, stresses and couple stresses have the square-root singularity near the crack tip, and are governed by three variables (two for mode I and II stress fields, and the third, resulting from higher order stresses, for the couple stress field). Stresses ahead of a crack tip in elastic materials with strain-gradient effects could be more than 50% higher than their counterparts in materials without strain-gradient effects. For an elastic-power law hardening strain-gradient material, an analytical solution is obtained. The mixed mode stress field in strain-gradient plasticity is the linear superposition of their counterparts in mode I and II. The angular distribution of stresses and couple stresses for several near-tip mode mixities clearly indicate that the new near-tip field in strain-gradient plasticity differs significantly from the HRR field. Stresses ahead of a crack tip in elastic-plastic solids with strain-gradient effects could be more than 2.5 times their counterparts in the HRR field. The asymptotic analysis compares favorably with available finite element results. The relevance of this solution to materials, in particular the size of the dominant zone, is discussed.

Evaluation of stress field parameters in fracture mechanics by photoelasticity—Revisited

Multi-parameter stress field equations have been reported in the past two decades to model the elastic stress field in the neighbourhood of a crack lying in finite bodies. Notably among them are the generalised Westergaard equations proposed by Sanford, Williams' eigen function expansion and the recent one of Atluri and Kobayashi. The equivalence among these equations is brought out. Using the equations of Atluri and Kobayashi an over-deterministic least squares technique is proposed to evaluate the mixed-mode stress field parameters by the technique of photoelasticity. It is shown that the use of Atluri and Kobayashi's equations leads to a very simple software where one can idependently increase the number of terms in Mode I or Mode II series depending on the specific problem requirement until the experimental fringes are correctly modelled. The software is validated by solving three example problems. The example problems emphasise that the use of multi-parameter stress field equations is not just an academic curiosity but a practical necessity to apply concepts of Fracture Mechanics to solve real life problems. Further, it is shown that the multi-parameter solution allows the collection of data from a larger zone which helps to simplify the data collection from experiments.

Damage mechanics approach for predicting high-temperature crack growth under mixed-mode loading conditions

Recent studies of the factors that control high-temperature crack growth under mixed-mode loading conditions are reviewed. In particular, measurements of crack growth direction and crack growth rate in specially designed specimens under mode I, mode II and mixed-mode loading conditions are discussed. Attempts have been made with the aid of finite element results to determine how both the crack path and the crack growth rate are influenced by the nature of loading. Discrepancies as to the ability of the von Mises equivalent stress in correlating the experimental data have been observed. A damage mechanics approach is proposed that brings mixed-mode crack growth data into better agreement when plotted against an effective value of C ∗ . This approach is also shown to correlate mixed-mode stress field calculations with reported observations of crack growth direction. The results demonstrate the importance of addressing damage mechanics ahead of the crack tip for accurate predictions of high-temperature crack growth under mixed-mode loading conditions.

Analysis of dynamic mixed mode stress fields in bimaterials by the method of caustics

The equations of caustics for dynamically extending curvilinear interface cracks are derived where optical isotropy and anisotropy of the material have been considered, respectively. The equations have been obtained by applying the method of generalized complex potentials. Moreover, an algorithm for the determination of stress intensity factors from experimentally obtained caustics is presented for the case of dynamic crack extension.

Analysis of dynamic mixed-mode isochromatics

The mixed-mode, elastodynamic state of stress in the neighborhood of a constant-velocity crack tip is used to generate numerically unsymmetric isochromatics. Unsymmetry associated with the third-order terms of a mixed-mode stress field, with and without the Mode II singular stress term, is also investigated. In extractingK I from an unsymmetric isochromatic pattern, errors in the Mode I fracture parameters due to the assumed presence ofK II in aK I stress field were found to be significant when data are taken more than 4 mm from the crack tip.

Mixed mode stress field effect in adhesive fracture

Numerical or analytical analyses were performed on seven different test specimens including blister test, 90-degree peel test, torsion test, and various cone tests. These specimens are in general subjected to complex stress fields having various amounts of Mode I, Mode II, and Mode III loads. The specimens were then constructed using polymethylmethacrylate (PMMA) for the adherends and a transparent polyurethane elastomer (Solithane® 113)1 for the adhesive. This combination permitted direct observation of the bondline as load was applied. Although initial debonds as well as bond end termination singularities were present in all specimens, in some cases the debond did not initiate at the singularity points as would normally have been expected. An explanation for this behavior is presented, as well as a comparison of loading mode effect on those specimens for which the debond did propagate from a bond terminus singular point. In addition, the effect of adhesive compressibility, adhesive thickness, and bondline geometric curvature are discussed for many of the test specimens.