Significance of the elastic peak stress evaluated by FE analyses at the point of singularity of sharp V‐notched components

Abstract

A B S T R A C T The paper presents an expression useful to estimate the notch stress intensity factor (NSIF) from finite element analyses carried out by using a mesh pattern with a constant element size. The evaluation of the NSIF from a numerical analysis of the local stress field usually requires very refined meshes and then large computational effort. The usefulness of the presented expression is that (i) only the elastic peak stress numerically evaluated at the V-notch tip is needed and no longer the whole stress-distance set of data; (ii) the adopted meshes are rather coarse if compared to those necessary for the evaluation of the whole local stress field. The proposed expression needs the evaluation of a virtual V-notch tip radius, i.e. the radius which would produce the same elastic peak stress than that calculated by FEM at the sharp V-notch tip by means of a given mesh pattern. Once such a radius has been theoretically determined for a given geometry, the expression can be applied in a wide range of notch depths and opening angles.Keywords elasticity; finite element analysis; notch stress intensity factor; stress intensity factor; V-notches. N O M E N C L A T U R Ea = crack length, V-notch depth or component's reference dimension d = adopted finite element size ρ = notch tip radius ρ * = virtual notch tip radius r, θ = polar coordinates K I = mode I Stress Intensity Factor K V I = mode I Notch Stress Intensity Factor for a sharp V-notch σ r , σ θ = stress components in a polar frame of reference σ g = nominal stress on the gross section σ max = linear elastic maximum principal stress at the tip of a rounded notch (exact stress) σ peak = linear elastic peak stress calculated by FEM at the sharp V-notch tip by means of a given mesh pattern 2α = opening angle of the V-shaped notch I N T R O D U C T I O NNotch stress intensity factors (NSIFs) play an important role in static strength assessments of components made of brittle or quasi-brittle materials and weakened by sharp V-shaped notches. 1-8 This holds true also for components made of structural materials undergoing high cycle fatigue Correspondence: P. Lazzarin. loading 9,10 as well as for welded joints both under tension or bending loading 11-16 and multiaxial loading. 17 In plane problems, the Mode I NSIF for sharp V-notches can be expressed by means of the Gross and Mendelson's definition 18 :where (r, θ ) is a polar coordinate system centred at the notch tip as shown by Fig. 1, σ θ is the stress component c 2006 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 95-106 95

Some relationships between the peak stresses and the local strain energy density for cracks subjected to mixed-mode (I+II) loading

“…It has been based on a link between the exact value of mode I NSIF K1, see Eq. (1), and the linear elastic opening peak stress σ peak calculated at the V-notch tip according to the following expression [9]: 1 * 1 1…”

“…(4) and (5) are valid under the following conditions:  use of 4-node linear quadrilateral elements, as implemented in ANSYS ® numerical code (PLANE 42 of Ansys element library or alternatively PLANE 182 with K-option 1 set to 3);  the pattern of finite elements around the V-notch tip must be that shown in Fig. 2b (see also [9,10]); in particular, four elements share the node located at the crack tip;  concerning Eq. (4), V-notches characterised by an opening angle 2 ranging from 0° to 135°;  the ratio a/d must be greater than 3 in order to obtain % 3 38 .…”

“…In particular Nisitani and Teranishi [7,8] were able to show that the ratio K I /σ peak depends only on the element size for a given element type, so that the σpeak value can be used to rapidly estimate KI, provided that the adopted mesh pattern has been previously calibrated on geometries for which the exact value of KI is known. This approach has been theoretically justified and extended also to sharp V-shaped notches subject to mode I loading [9] giving rise to the so-called Peak Stress Method (PSM), which can be regarded as an approximate FE-based method to estimate the NSIFs. Later on, the PSM has been extended to cracks subjected to mode I as well as mode II stresses [10].…”

Fatigue damage criteria classification, modelling developments and trends for welded joints in marine structures

“…Peak stress σ p does not include a material characteristic contribution and is a geometry and loading & response defined parameter like K N . Comparing the scatter band parameter of fatigue resistance data involving weld toe induced failures (6 < t p < 100 [mm]) using respectively K N and σ p shows similar values (Meneghetti and Lazzarin 2006), but slightly larger for σ p . The NSIF issue concerning weld toe and weld root data combinations because of different scaling parameter and units have been solved, providing the possibility to develop 1 material and environment dedicated σ p -N resistance curve.…”

“…t p > 3 and a n > 3 for a weld toe or weld root to describe sufficiently accurate a linear far field stress distribution). Accordingly, the (K N /σ p ) ratio is a constant as well (Meneghetti and Lazzarin 2006) and σ p can be used as local intact geometry and area or volume equivalent point fatigue damage criterion itself, quantifying the intensity of the local linear elastic stress field of the weld notch. In comparison to the K N = k·σ·t p 1−λ or K N = k·σ·a n 1−λ relation (Section 2.4), the physical scaling parameters t p and a n have been replaced by a numerical one e s ; σ p substitutes the nominal stress.…”