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.     

Magneto stress analysis of a strip with a semi elliptical notch under uniform magnetic field

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin strip with a semi-elliptical notch subjected to uniform magnetic field. The strip is a finite plate of a simply connected region. A linear constitutive equation is used for the stress analysis. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. Therefore, the magneto elastic stress is analyzed using Maxwell stress. In the present problem, as a result, the plane stress state does not arise, and the σ_z in the direction of the plate thickness and the shear deflection (anti-plane shear stress) arise for soft ferromagnetic material. The stress σ_z in the plate is strong compressive stress for a soft ferromagnetic material. A rational mapping function is used for the stress analysis, and the each solution is obtained as a closed form. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition is completely satisfied without any linear assumptions on the boundary. The anti-plane shear stress causes Mode III stress intensity factor when the notch is a crack. Stress concentration values are investigated for a notch problem, of which expression is given. Figures of the anti-plane shear stress distribution, Mode III stress intensity factor, and stress concentration values are shown.     

The temperature field in a crystal plate with a rectangular notch

We have used the functional sweeping method to find a solution for the nonlinear problem of the conduction of heat in the case of an infinite plate with a rectangular notch. Additionally, we have studied the temperature field.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 57, No. 4, pp. 674–680, October, 1989.     

Three-dimensional asymptotic stress field at the front of a bimaterial wedge of symmetric geometry under antiplane shear loading

Three-dimensional asymptotic stress field in the vicinity of the front of a bimaterial wedge is obtained by a new eigenfunction expansion method, subjected to three combinations of wedge-side boundary conditions – free–free, clamped–clamped and free–clamped. The bimaterial wedge has symmetric geometrical configuration with respect to the bimaterial interface. Each material is isotropic and elastic, but with different elastic properties. Additionally, numerical results pertaining to the variation of the lowest eigenvalues (or stress singularities) with respect to the half-aperture angle of the bimaterial wedge, subjected to the afore-mentioned wedge-side boundary conditions, are also presented. Dependence of the same on the material properties of the component material phases (if any) is also presented.     

Stress field near a notch root under pure bending

A finite-element analysis is carried out by using conforming quadrilateral elements in order to obtain the stress field near a rounded-tip V-notch in a beam under pure bending. Different values of the notch deptha, the radius of curvaturer at the notch root, and the angle ω between the opposite faces of the stress concentrator are considered under the assumption that the behavior of the material is either linear elastic or elastoplastic. Our results are compared with approximate solutions proposed by other authors. Dipartimento di Ingegneria Civile, Universita di Parma, Parma, Italy. Published in Fizyko-Khimichna Mekhanika Materialiv, Vol. 34. No. 5, pp. 43–48, September–October, 1998.     

Evaluation of stress concentration of U‐shaped notch based on residual magnetic field measurements

In this research, the correlation between the stress concentration and the residual magnetic field (RMF) of 30Cr steel was investigated. Tensile tests were carried out to measure the RMF signals on the surface of U‐shaped defect specimens. It was found that the tangential RMF signal at the defect area is correlated to the applied load and the stress concentration factor. A new method based on magnetic field to evaluate the stress concentration degree is proposed. This research provides a potential possibility for quantitative inspection of the stress concentration in ferromagnetic steels using the RMF measurements.     

On three-dimensional singular stress field at the front of a planar rigid inclusion (anticrack) in an orthorhombic mono-crystalline plate

A novel eigenfunction expansion technique, based in part on separation of the thickness-variable and partly on the Eshelby–Stroh type affine transformation, is developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness anticrack reinforcing an infinite orthorhombic single crystal plate, of finite thickness and subjected to far-field mode I/II loadings. Anticrack-face boundary conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the plate are exactly satisfied. The present investigation considers six through-anticrack systems reinforcing orthorhombic single crystal plates. Explicit expressions for the singular stresses in the vicinity of the front of a through-thickness anticrack reinforcing an orthorhombic plate, subjected to far-field mode I/II loadings, are presented. Finally, hitherto largely unavailable results, pertaining to the through-thickness variations of stress singularity coefficients corresponding to symmetric and skew-symmetric sinusoidal loads that also satisfy the boundary conditions on the top and bottom surfaces of an orthorhombic mono-crystalline plate under investigation, bridge a longstanding gap in the stress singularity/fracture mechanics literature.     

Fatigue-relevant stress field parameters of welded lap joints: pointed slit tip compared with keyhole notch

The notch stress intensity factor (NSIF) based analytical frame is applied to the slit tips (or weld roots) of welded joints with inclusion of the T-stress component. This T-stress can be determined from FE models evaluating the ligament stresses close to the pointed slit tip. An alternative analytical frame is presented for the corresponding keyhole notches based on analytical solutions from the literature, which are applied to the ligament stresses.
In the slit tip models, the mean local strain energy density (SED) with inclusion of the T-stress effect is determined analytically and numerically in comparison, using two different fatigue-relevant control radii,  R ₀= 0.28 mm and  R ₀= 0.15 mm, the former value well proven for thick-sheet welded joints made of structural steel. The latter smaller value is tentatively proposed for thin-sheet welded joints, in the direction suggested in the recent literature where a reduction of the microstructural support length for laser beam welds and resistance spot welds is recommended. The FEM-based and analytical stress concentration factors (SCF) for the lap joint keyhole model and also the SED values for the corresponding pointed slit tips are found to be in good agreement. The  J -integral consisting of the first and second component (the latter containing the T-stress) is compared with the corresponding SED values.     

The Mode III elastic stress distribution near the root of (a) an intrusion-type notch and (b) a key-hole notch

An important element of a research programme aimed at developing a fracture mechanics methodology for blunt notches, is the representation of the elastic stress distribution in the immediate vicinity of a notch root. Earlier work for a general two-dimensional blunt notch model has shown that, for Mode III deformation, the stress at a distance x ≪ ρ (notch root radius of curvature) ahead of the notch root only depends on x, ρ and σ_p (peak stress), irrespective of the notch shape and the loading characteristics. This uniqueness has been confirmed in earlier work for an elliptically cylindrical notch, a special case being a circular cylindrical notch. This paper provides further underpinning for the uniqueness conclusion by analysis of the models of (a) an intrusion-type notch and (b) a key-hole notch.     

Three-dimensional asymptotic stress fields at the front of a trimaterial junction

A recently developed eigenfunction expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of the junction corner front of an infinite pie-shaped trimaterial wedge, of finite thickness, formed as a result of bimaterial (matrix plus reaction product or contaminant) deposit over a substrate or reinforcement. The wedge is subjected to extension/bending (mode I), inplane shear/twisting (mode II) and antiplane shear (mode III) far field loading. Each material is isotropic and elastic, but with different material properties. The material 2 (substrate) is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying composition of the bimaterial deposit. Numerical results pertaining to the variation of the mode I, II, III eigenvalues (or stress singularities) with various moduli ratios as well as the wedge aperture angle of the material 1 (reaction product/contaminant), are also presented. Hitherto unavailable results, pertaining to the through-thickness variations of stress intensity factors for symmetric exponentially decaying distributed load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the trimaterial plates under investigation, bridge a longstanding gap in the stress singularity/interfacial fracture mechanics literature.     

Asymptotic characterisation of nearly-sharp notch root stress fields

A solution procedure is developed for characterising the stress state at the root of a notionally sharp notch, but possessing a small root radius, using two nested asymptotic solutions: an outer asymptote representing a sharp semi-infinite V-notch and an inner solution representing a semi-infinite rounded notch. The two asymptotes are matched to each other remote from the notch root, and to an example finite notch using a generalised stress intensity factor. It follows that the characteristic, singular, sharp-notch field diverges from the rounded-notch solution very near the root. On the other hand, the notch in a finite body diverges from the sharp semi-infinite notch in the far field. Providing that the notch root radius is sufficiently small, it follows that there is an intermediate field where the singular field does characterise the behaviour of the finite radiused notch, and this is quantified.Note that this topic has also been investigated in the very recent literature by Gomez and Elices (2004) for brittle components, using the same analytical frame presented here     

Fatigue resistance of weld ends – Analysis of the notch stress using real geometry

Modern welded structures often contain weld start and end points which are the failure critical location. In particular, for special investigations into thin sheet structures, no approach for the determination of the fatigue life has been established thus far. In this research, the primary aim was to obtain the real geometry of weld ends with high precision using a three dimensional scanner to find a general approach using the notch stress concept. Going one step further, analyses have been performed regarding to unify the notch stress concept. The existing results of Olivier – who examined long welds with no start and end points – were re‐evaluated to unify the results of long regular welds with the local weld end under one scatter band.     

A conservative integral for determining stress intensity factors of a bimaterial notch

This work presents regular and singular asymptotic solutions for non-planar quasi-circular cracks. Regular asymptotic solutions, which do not give rise to hyper-singular stresses near the crack tip, are developed in the general three-dimensional setting. Singular asymptotic solutions are considered for axisymmetric problems only, because those problems do not involve hyper-singularity. The validity of asymptotic solutions for planar and non-planar cracks is investigated by comparing them with detailed numerical and analytical solutions. Also asymptotic solutions are applied to analysis of quasi-static crack growth in three dimensions.     

Asymptotic deformation and stress fields at the tip of a sharp notch in an elastic-plastic material

The singular elastic-plastic stress, strain and the displacement fields at the tip of a sharp notch for both plane stress and plane strain conditions are investigated analytically. The material is assumed to be governed by the deformation theory of plasticity with linear strain hardening characteristic. Since the elastic strain is retained in the analysis, the final strain and displacement fields can be separated into the elastic and the plastic parts. In the case with zero notch angle, the results reduce to the classical crack problem. The relationship of the amplitude of the near crack tip elastic-plastic field to the elastic far field is obtained. Both mode I and mode II cases are investigated. The mixed mode case is also discussed.     

Antisymmetric stress distribution in an elastic body with a sharp or a rounded V-shaped notch

Using the method of singular integral equations, we have obtained a solution of the plane problem of the theory of elasticity for a plane with a semiinfinite rounded V-shaped notch under antisymmetric loading. On this basis, we have determined relations between the stress intensity factor at the vertex of a sharp V-shaped notch, the maximal stresses on the boundary contour or stress gradient at the vertex of the corresponding rounded V-shaped notch, and its rounding-off radius. It is shown that such dependences are ambiguous: for the same curvature at the notch vertex, they are significantly different for various shapes of its neighborhood. For finite bodies with V-shaped notches, the obtained solutions are asymptotic dependences for small rounding-off radii of their vertices. Such relations can be used in passing to the limit for determination of the stress intensity factors at the vertices of sharp notches based on the solutions for the corresponding rounded stress concentrators.     

Crack tip stress intensity factors for a crack emanating from a sharp notch

Accurate calibrations are provided for the crack tip stress intensity factor for a crack of finite length emanating from the symmetric tip of a sharp notch, of arbitrary angle, in terms of the generalised stress intensity quantifying remote loading of the notch. The solution is applied to example problems and shown to be accurate for cases where the crack is much shorter then the notch depth.