Fracture initiation at sharp notches under mode I, mode II, and mild mixed mode loading

In the context of linear elasticity, a stress singularity of the type Kⁿr^δ(δ<0) may exist at sharp re-entrant corners, with an intensity Kⁿ. In general the order of the stress singularity δ and the stress intensity differ for symmetric (mode I) and antisymmetric (mode II) loading. Under general mixed-mode loadings, the magnitudes of the mode I and II intensities fully characterize the stress state in the region of the corner. A failure criterion based on critical values of these intensities may be appropriate in situations where the region around the corner dominated by the singular fields is large compared to intrinsic flaw sizes, inelastic zones, and fracture process zone sizes. We determined the mode I and II stress intensities for notched mode I tensile specimens and notched mode II flexure specimens using a combination of the Williams (1952) asymptotic method, dimensional considerations, and detailed finite element analysis. We carried out a companion experimental study to extract critical values of the mode I and II stress intensities for a series of notched polymethyl methacrylate (PMMA) tensile and flexure specimens with notch angles of 90-. The data show that excellent failure correlation is obtained, in both mode I and II loading, through the use of a single parameter, the critical stress intensity. We then analyzed and tested a series of T-shaped structures containing 90- corners. The applied tensile loading results in mixed-mode loading of the 90- corners. Failure of the specimens is brittle and can be well-correlated with a critical mode I stress intensity criterion using the results of the notched mode I tensile tests. This is attributed to large difference in the strength of the stress singularities in modes I and II: δ= -0.4555 and -0.0915 for modes I and II for a 90- notch. As a result, the mode I loading dominates the failure process for the 90- corner in the T-structure. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture assessment of Brazilian disc specimens weakened by blunt V-notches under mixed mode loading by means of local energy

The main purpose of this research is to re-analyse experimental results of fracture loads from blunt V-notched samples under mixed mode (I + II) loading considering different combinations of mode mixity ranging from pure modes I to II. The specimens are made of polymethyl-metacrylate (PMMA) and tested at room temperature. The suitability of fracture criterion based on the strain energy density (SED) when applied to these data is checked in the paper. Dealing with notched samples, characterized by different notch angles and notch root radii, the SED criterion used in combination with the concept of local mode I, valid in the proximity of the zone of crack nucleation, permits to provide a simple approximate but accurate equation for the SED in the control volume. This proposal unifies predictions for the experimental results obtained under modes I, II and mixed mode loading.     

Local strain energy density and fatigue strength of welded joints under uniaxial and multiaxial loading

In the notch stress intensity approach to the fatigue assessment of welded joints, the weld toe is modelled as a sharp V-notch and the local stress distributions in plane problems are given on the basis of the relevant mode I and mode II notch stress intensity factors (N-SIFs). These factors quantify the magnitude of asymptotic stress distribution obeying Williams’ solution. If the V-notch opening angle at the weld toe is constant and the mode II is not singular, the mode I N-SIF can be directly used to summarize the fatigue behaviour of welded joints. In all the other cases, varying the V-notch angle or including multiaxial loading conditions (where typically both Mode I and Mode III stress distributions are singular), the synthesis can be carried out on the basis of the mean value of the strain energy density over a well-defined volume surrounding the weld toe or the weld root. By using this scalar quantity, two fatigue scatterbands are obtained for structural steels and aluminium alloys, respectively. The material-dependent radius R_C of the control volume (area) is carefully identified with reference to conventional arc welding processes.Sometimes the weld toe radius is found to be very different from zero. The local strain energy approach can be extended as it stands also to these cases, providing a gradual transition from a N-SIF-based approach to a K_t-based approach.     

Analytical solutions for crack-tip sectors in perfectly plastic Mises materials under mixed in-plane and out-of-plane shear loading conditions

In this paper, analytical solutions for asymptotic crack-tip plastic sectors in perfectly plastic Mises materials are derived under mixed in-plane and out-of-plane shear loading conditions. Plastic strains in crack-tip plastic sectors are considered to be singular and non-singular. Sectors with singular plastic strains have the solution of centered fan type, and sectors with non-singular plastic strains have the solution of either centered fan or constant stress type. The requirement of stress continuity along the border between a constant stress and a centered fan sectors is then discussed. Discontinuities of the normal and out-of-plane shear stresses in the radial direction between two constant stress sectors are assumed in assembling the crack-tip fields under mixed mode II/III and I/III conditions. Crack-tip fields under mixed mode II/III and I/III conditions with small contributions of mode III are then presented to show the existence of asymptotic crack-tip fields for perfectly plastic materials under mixed in-plane and out-of-plane shear loading conditions. The trends of the angular variations of the mode III stresses under the mixed mode II/III and I/III conditions are generally in agreement with those of the available asymptotic and finite element analyses for low strain hardening materials.     

Singular behaviour near the tip of a sharp V-notch in a power law hardening material

This paper presents an asymptotic analysis of the near-tip stress and strain fields of a sharp V-notch in a power law hardening material. First, the asymptotic solutions of the HRR type are obtained for the plane stress problem under symmetric loading. It is found that the angular distribution function of the radial stress presents rapid variation with the polar angle if the notch angle is smaller than a critical notch angle; otherwise, there is no such phenomena. Secondly, the asymptotic solutions are developed for antisymmetric loading in the cases of plane strain and plane stress. The accurate calculation results and the detailed comparisons are given as well. All results show that the singular exponent s is changeable for various combinations of loading condition and plane problem.     

In-plane and out-of-plane stress field solutions for V-notches with end holes

Closed form expressions of stress distributions for V-notches with end holes and varying opening angles are presented. The solution for the elastic plane problem is obtained by means of the Kolosov-Muskhelishvili approach by using a reduced number of complex terms. The exponents of the potential functions are simple combinations of Williams’ eigenvalues for pointed V-notches in mode I and mode II. The degree of accuracy of the new solution, which is approximate, is found to be very satisfactory for engineering applications. When the V-notch opening angle is equal to zero, the solution matches the keyhole notch solutions already reported in the literature by Neuber (for mode I) and by Kullmer and Radaj (mode I and mode II) and based on the Airy stress function. In parallel, the out-of-plane problem is solved by means of an holomorphic function H(z) where the exponent is still linked to the leading order eigenvalue of the pointed V-notch in mode III. For this loading mode the solution is exact. When the notch opening angle is equal to zero and also the notch root radius tends to zero the solution matches Kullmer’s keyhole notch solution.     

Transformation zones, crack shielding, and crack-growth resistance of Ce-TZP/alumina composite in mode II and combined mode II and mode I loading

Crack-tip transformation zones, crack shielding and crack-growth-resistance (R-curve) behaviors of a transformation-toughened ceria-partially stabilized zirconia–alumina (Ce-TZP/alumina) composite were studied in mode II and combined mode I and mode II loading using compact-tension-shear (CTS) specimens. The mode II and mode I stress intensities for both the initial straight cracks and the subsequent kinked cracks were assessed by the method of caustics using geometrically equivalent specimens of polymethyl methacrylate (PMMA). The angle of formation of the transformation zones as well as of extension of the cracks increased systematically with increasing ratio of the mode II and the mode I stress intensities and approached a value of θ^*=−72° in pure mode II loading. This angle was close to the angle for maximum hoop tension in the stress field of a mode II crack (θ^*=−70.5°). A crack-initiation toughness envelope was constructed on a K_I–K_II diagram using the critical loads for incremental crack extension. The crack-initiation toughness in pure mode II loading was less than the corresponding toughness in mode I loading. This result was consistent with calculations that indicated no shielding from the asymmetric and elongated zones developed in mode II loading. The fracture toughness measured for the kinked cracks at long kink lengths approached the maximum fracture toughness measured for a mode I crack.     

Experimental evaluation of stress field around the sharp notches using photoelasticity

A higher order representation of the stress field around the sharp notch has been utilized for calculating notch stress intensity factors (NSIFs) as well as the coefficients of higher order terms by the technique of photoelasticity. Adding the higher order terms to the singular term makes it possible to collect the data points from a larger zone, which helps to simplify the data collection from experiments. Moreover, the effects of higher order terms in the region near the notch tip are taken into account. To utilize the advantages of whole-field photoelasticity and minimize the experimental errors, a large number of data points have been employed and the overdeterministic least squares method combined with the Newton–Raphson method have been used to solve the resulting system of nonlinear equations. The NSIFs for a laboratory specimen called V-notched Brazilian disk (V-BD) were calculated for various notch angles under pure mode I, pure mode II and mixed mode I/II loading conditions. In addition to NSIFs, the coefficient of the first non-singular term of 30° notch was calculated experimentally and the effects of this term on the stress distribution in the vicinity of notch tip were investigated. A good correlation was observed between the experimental results and the numerical results obtained from finite element analysis.     

Analytical evaluation of J-integral for elliptical and parabolic notches under mode I and mode II loading

In the present work the J-integral (indicated here as J_Vρ because two parallel flanks are not present) was calculated by using, along the free border, the exact analytical stress distribution for the ellipse and the asymptotic one for parabolic notches. The material was assumed as homogeneous isotropic and linear elastic. First, for an ellipse under remote tensile loading, the expression of J_Vρ has been analytically calculated on the basis of Inglis’ equations. The equations have been used to prove that, in terms of J-integral, the crack is the limit case of an equivalent elliptic notch. Furthermore, by distinguishing the symmetric and skew-symmetric terms, the well-known Stress Intensity Factors (SIF) of mode I and II for a crack in a wide plate under tension are obtained by adding a limiting condition. Second, by means of Creager–Paris’ equations, J_Vρ has been analytically calculated for a parabolic notch of assigned tip notch radius ρ. The asymptotic value of J_Vρ and the relationship between the peak stress and the relative SIF are the same as the ellipse. Finally, as an engineering application, we provide an accurate formula for the evaluation of the Notch Stress Intensity Factors of a crack, mainly subjected to tensile stress, from the peak stress of the equivalent ellipse under the same loading.     

Calculation of mixed-mode stress field at a sharp notch tip using <Emphasis Type="Italic">M</Emphasis><Subscript>1ɛ</Subscript>-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 M₁. The relation between M₁ and the generalized stress intensity factors is then derived and expressed as function of the notch angle. Once the M₁-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. This work has been partially supported by National Science Council Grant No. NSC90-2211-E-008-040 to National Central University.     

An approximate, analytical approach to the `HRR'-solution for sharp V-notches

The well-known so-called `HRR-solution' (Hutchinson, 1968 and Rice and Rosengren, 1968) considers the elasto-plastic stress field in a power-law strain hardening material near a sharp crack. It provides a closed form explicit expression for the stress singularity as a function of the power-law exponent `n' of the material, but the stress angular variation functions are not found in closed form. More recently, similar formulations have appeared in the literature for sharp V-notches under mode I and II loading conditions. In such cases not only is the angular variation of the stress fields obtained numerically, but so is the singularity exponent of the stress field. In the present paper, approximate but accurate closed form solutions are first reported for sharp V-notches with an included angle greater than /6 radians. Such solutions, limited here to Mode I loading conditions, allow a very satisfactory estimate of the angular stress components in the neighbourhood of the notch tip, in the entire range of notch angles and for the most significant values of n (i.e. from 1 to 15). When the notch opening angle tends towards zero, and the notch approaches the crack case, the solution becomes much more complex and a precise evaluation of the parameters involved requires a best-fitting procedure which, however, can be carried out in an automatic way. This solution is also reported in the paper and its degree of accuracy is discussed in detail.     

Failure criteria for mixed mode delamination in glass fibre epoxy composites

Critical strain energy release rate of glass/epoxy laminates using the virtual crack closure technique for mode I, mode II, mixed-mode I + II and mode III were determined. Mode I, mode II, mode III and mixed-mode I + II fracture toughness were obtained using the double cantilever beam test, the end notch flexure test, the edge crack torsion test and the mixed-mode bending test respectively. Results were analysed through the most widely used criteria to predict delamination propagation under mixed-mode loading: the Power Law and the Benzeggagh and Kenane criteria. Mixed-mode fracture toughness results seem to represent the data with reasonable accuracy.     

Interlaminar fracture characterization for plain weave fabric composites

For the analysis of laminated composite plates under transverse loading and drilling of composites, all the elastic, strength and fracture properties of the composite plates are essential. Interlaminar critical strain energy release rate properties in mode I, mode II, mixed mode I/II and mode III have been evaluated for two types of plain weave fabric E-glass/epoxy laminates. The double cantilever beam test and the end notch flexure test have been used for mode I and mode II loading. The mixed mode bending test and split cantilever beam test have been used for mixed mode I/II and mode III loading. It is observed that the plain weave fabric composite with lesser strand width has higher interlaminar fracture properties compared to the plain weave fabric composite with more strand width. Further, crack length versus crack growth resistance plots have been presented for mode III loading. In general, it is observed that total fracture resistance is significantly higher than the critical strain energy release rate.     

Investigation of blunting line and evaluation of fracture toughness under mixed mode I/III loading in commercially pure titanium

Abstract

The blunting line and fracture toughness in commercially pure titanium under mode I and mixed mode I/III loading was studied. A modified compact tension geometry was used for determining the blunting line as well as mixed mode I/III fracture toughness. The results showed that the constraint factor m in the blunting line equation under mode I loading was 1.84. Also, there was no effect of notch root radius on the slope of the blunting line. The blunting line slope under mixed mode I/III loading was found to be lower than that under mode I loading and agreed with empirical correlations. The fracture toughness under mode I loading was found to be higher for specimens with larger notch root radius. However, notch root independent fracture toughness could be obtained from blunt notch specimen tests using stretch zone width measurements. The fracture toughness was found to decrease with increasing mode III loading.     

Sudden Fracture from U-Notches in Fine-Grained Isostatic Graphite Under Mixed Mode I/II Loading

The U-notched maximum tangential stress (UMTS) criterion, proposed originally and utilized previously by the author and his co-researcher for predicting mixed mode I/II fracture in plexi-glass (PMMA) and also pure mode II fracture in PMMA and soda-lime glass, was employed to estimate the experimental results reported in literature dealing with brittle fracture of many U-notched fine-grained isostatic graphite plates under combined tensile/shear loading conditions. By using the fracture curves of the UMTS criterion, which can predict the onset of brittle fracture in terms of the notch stress intensity factors (NSIFs) in the entire domain from pure mode I to pure mode II, the mixed mode fracture toughness (i.e. the load-bearing capacity) of U-notched graphite plates was successfully estimated.     

A criterion for brittle fracture in U-notched components under mixed mode loading

A failure criterion is proposed for brittle fracture in U-notched components under mixed-mode static loading. The criterion, called UMTS, is developed based on the maximum tangential stress criterion and also a criterion proposed in the past for mode I failure of rounded V-shaped notches [Gomez FJ, Elices M. A fracture criterion for blunted V-notched samples. Int J Fracture 2004;127:239-64]. Using the UMTS criterion, a set of fracture curves are derived in terms of the notch stress intensity factors. These curves can be used to predict the mixed mode fracture toughness and the crack initiation angle at the notch tip. An expression is also obtained from this criterion for predicting fracture toughness of U-notched components in pure mode II loading. It is shown that there is a good agreement between the results of UMTS criterion and the experimental data obtained by other authors from three-point bend specimens.     

Experimental studies on mixed‐mode dynamic fracture behaviour of aluminium alloy plates with narrow U‐notch

Mixed‐mode dynamic fracture behaviour of cast aluminium alloy ZL205A thin plates with narrow U‐notch was studied by split Hopkinson tensile bar apparatus. Specimens with different loading angles were designed to realize different fracture modes. The same loading condition was maintained during the tests. Recovery specimens show that crack propagates along the notch direction. Force–elongation relations show that with the loading angle increasing, the fracture force increases while the final elongation decreases. Deformation and fracture process was observed by a high‐speed camera. Displacement distribution around the crack was calculated through digital image correlation technique. Based on the photos and displacement results, initiation time of the crack was derived. Besides, two stress components (normal stress and shear stress) applied on the fracture surface were investigated. Results show that crack initiation stresses at different loading angles satisfy the ellipse equation. Pure mode I and II fracture stresses are 425.3 and 236.7 MPa, respectively. Furthermore, specific fracture energy of different specimens was calculated. The energy data vary with loading angle and located on an approximate upward parabolic curve. From the curve, the minimum specific fracture energy of the thin plate specimen is 42.0 kJ/m² under loading angle of 76.3°.     

Experimental evaluation of stress field around crack tip by caustic method

The caustic method is an optical technique which is useful to determine stress intensity factor values. In this paper, the caustics method was applied to specimens which have an oblique crack, various thicknesses and an open notch to investigate the stress field around the crack tip. The results are summarized as follows:

1. The caustic method is a useful technique to determine the stress intensity factor values of the specimens which have an oblique crack or various thickness and an open notch.

2. The conventional theory of measurement concerning this method is effective when the initial curve r₀ is larger than the minimum initial curve r₀^min which was obtained in this study. It is observed that the values of r₀^min decrease as the ratio of K_II to K_I increases under mixed-mode loading, the one increases with an increase of thickness and notch opening angle.

3. The 3D stress field exists in the vicinity of crack tip; however, the stress state is nearly plane strain deformation in the case of mode I loading. In the case of mixed-mode loading, the stress state approximates to plane stress deformation as the ratio of K_II to K_I increases.

4. A method based on the distribution of the three-dimensional (3D) stress field is proposed to expediently yield the values of K_I using the caustic method in the case of r₀<r₀^min.

A generalised notch stress intensity factor for U-notched components loaded under mixed mode

A novel notch stress intensity factor (NSIF) for U-notched specimens loaded under mixed mode is examined in this article. The concept is based on the averaged strain energy density criterion, or alternatively on the cohesive zone model, as well as the equivalent local mode approach. To a certain extent, it is a generalisation of Glinka’s NSIF for mode I, where σ_tip is replaced by σ_max.The applicability of a fracture criterion based on this new NSIF is checked against 171 fracture tests with PMMA (at −60 °C) performed on U-notched specimens, with different notch root radii and loaded under mixed mode. The asymptotic behaviour of the new NSIF as the notch becomes a crack (when the notch root radius tends to zero) or when the notch disappears (when the notch root radius tends to infinity) is also discussed.