Generalised Neuber Concept of Fictitious Notch Rounding

Abstract

Sharp or pointed notches reduce the fatigue strength or life of structural components drastically, but not to the extent of the elastic notch stress increase. Microstructural support is observed for crack initiation at the notch root. The support effect may be described by averaging the maximum notch stresses in a small material volume (length $\rho^{\ast}$) at the notch root (radius ${\rho}$), which can be expressed by the maximum stress of a corresponding notch of a slightly enlarged, fictitious radius, ρ _f  = ρ + $s\rho^{\ast}$ (Neuber 1937, 1968). The support factor s is derived for elementary notches and V-notches in the three loading modes: in-plane tensile and shear loading as well as out-of-plane shear loading (modes 1, 2 and 3). Out-of-bisector crack initiation and propagation is basic for mode 2 loading. The dependency of s on the notch opening angle 2α is recognised, besides its correlation with multiaxiality conditions and failure criteria. The Neuber concept of fictitious notch rounding is thus generalised. Application-relevant issues such as reference notches, design S–N curves, non-singular stress components, seam-welded cruciform joints and spot-welded lap joints are also dealt with.

List of Symbols

a

Notch depth, semilength of slit or crack

a _i

Initiated crack length

a, b

Semiaxes of elliptical hole

b

Net section semiwidth

C

Auxiliary parameter of S–N curve

C ₁, C ₂

Constants in Neuber’s stress field equations

d

Weld spot diameter

F

Shear force

F _a

Load amplitude

K ₁, K ₂, K ₃

Notch stress intensity factors, mode 1, 2, 3

K _1,ρ , K _2,ρ

Generalised notch stress intensity factors, mode 1, 2

$K_{1,\rho }^{\ast}$

T-stress-corrected K _1,ρ

K _f, K _{f red}

Theoretical fatigue notch factor, reduced K _f

$K_{\text{f}}^{\ast}$, $K_{\text{f}}^{\ast\ast}$

Fatigue notch factor, reduced and modified

K _f,l, K _f,u

Fatigue notch factor, lower and upper slit face

K _O

Stress intensity factor, transverse singular mode

K _t

Theoretical (equivalent) stress concentration factor

$\overline{K}_{\text{t}}$

Stress concentration factor of locally averaged notch stresses

K _t(ρ _f)

Stress concentration factor after fictitious notch rounding

$K_{{\sigma^{\prime}}} ,K_{\tau }$

Stress concentration factor relating to $\sigma^{\prime}_{\max } ,\tau_{ \max }$

K _I, K _II, K _III

Stress intensity factor, mode I, II, III

K _Ic

Fracture toughness

K _eq

Equivalent stress intensity factor

K _w

Stress concentration factor of weld notch

k

Inverse slope exponent, S–N curve

k _I, k _II, k _III

Normalised stress intensity factors of modes I, II and III

l

Semilength of plate

m

Parameter in complex stress function

N, N _i

Endurable number of cycles, the same up to crack initiation

n

Elastic notch support index

q, r

Stress field parameters linked to V-notch opening angle

P _s

Survival probability

R

Ratio of lower to upper stress

R

Parameter in complex stress function

r

Polar coordinate, radial distance

${r}^{\ast}$

Radius of circular core region

r ₀

Distance between notch tip and origin of polar coordinate system

S

Endurable nominal stress

S

Sih’s strain energy density factor

$s,\;\overline{s}$

Microstructural support factor, plateau value

s _vM, s _B

Support factor, von Mises and Beltrami equivalent stress

T

T-Stress

t

Notch depth, semiaxis of ellipse, plate thickness

t _w

Weld throat thickness

u, u ₀

Horizontal displacement, remote value

v, v ₀

Vertical displacement, remote value

w

Plate semiwidth, joint face width, specimen width

x ₀

Distance between notch tip and origin of polar coordinate system

x, y, z

Cartesian coordinates

α

Notch opening semi-angle

Δ

Relative deviation

ζ

Complex coordinate

$\theta ,\overline{\theta }$

Polar angle, value for $\sigma^{\prime}_{\max }$

κ

Multiaxiality jump factor

κ ₀

Free surface factor

$\kappa_{{\sigma^{\prime}}} ,\,\kappa_{\tau }$

Geometry factor, relating to σ’ and τ

κ _vM, κ _B

Jump factor, von Mises and Beltrami equivalent stress

κ ₁, κ ₂, κ ₃

Short crack reduction factor on K _f, mode I, II, III

λ ₁, λ ₂

Eigenvalue of stress distribution at V-notch, mode 1, 2

μ

Exponent in Filippi’s stress equations

ν

Poisson’s ratio

$\rho$

Real notch radius

$\rho^{\prime}$

Radius of curvature at point of maximum stress $\sigma^{\prime}_{\max }$

${\rho} _{}^{\ast}$

Microstructural support length

$\rho_{\text{f}} ,\rho_{\text{r}}$

Fictitious and reference notch radius

$\rho_{\text{f}}^{\ast}$

Degree of cross-sectional weakening, $\rho_{\text{f}} /t$

$\sigma_{}^{\ast}$

Lower to upper surface stress ratio, $\sigma_{\text{l}} /\sigma_{\text{u}}$

$\overline{\sigma }$

Locally averaged notch stress

$\sigma_{\text{a}}$

Stress amplitude

$\sigma _{{\rm{b}}}^{\ast}$

Structural bending stress at level of hole radius

$\overline{\sigma }_{{\rm{c}}}$

Locally averaged notch stress at fracture

$\sigma_{\text{E}} ,\sigma_{\text{kE}}$

Endurance limit, notch stress endurance limit

$\overline{\sigma }_{{\rm{eq}}}$

Locally averaged equivalent stress

$\sigma_{{{\text{eq}}\,{ \max }}}$

Maximum equivalent stress

$\sigma_{{{\text{i}},{\text{l}}}} ,\sigma_{{{\text{i}},{\text{u}}}}$

Lower and upper plate inner side stresses

σ _k

Notch stress

σ _l, σ _u

Stresses on lower and upper plate surface

σ _n, σ _ng

Nominal stress, value in gross cross-section

σ _m

Static mean stress or membrane stress

σ _max, $\sigma '_{\max }$

Maximum notch stress, symmetric and antimetric component

σ _r , σ _θ

Stresses in the polar coordinate system

σ _s

Structural stress

σ _th

Theoretical (equivalent) notch stress

σ _vM, σ _B

Equivalent stress, von Mises and Beltrami

σ _U

Ultimate tensile strength

σ _Y

Yield limit

$\sigma_{0}^{\ast}$

Structural stress parallel to slit front

σ ₀

Structural basic stress parallel to slit

σ _0s

Structural stress at slit flank

σ ₁, σ ₂, σ ₃

Principal stresses

σ _{θ max}

Maximum tangential stress

$\overline{\tau },\;\overline{\tau }^{\ast}$

Locally averaged notch shear stress, in-plane and out-of-plane

τ ₀

Reference shear stress

τ _max, $\tau_{\max }^{\ast}$

Maximum notch shear stress, in-plane and out-of-plane

τ _n, $\tau_{\rm{n}}^{\ast}$

Nominal shear stress, in-plane and out-of-plane

$\tau_{{ {\text{ng}}}} ,\tau_{\text{ng}}^{\ast}$

Nominal shear stress, gross cross-section, in-plane, out-of-plane

τ _rθ

Shear stress in polar coordinate system

$\tau_{{ {\text{th}}}} ,\tau_{\text{th}}^{\ast}$

Theoretical notch shear stress, in-plane and out-of-plane

$\tau_{ yz}$

Shear stress along notch bisector

$\tau_{{z\,{ \max }}}^{\ast}$

Maximum out-of-plane shear stress produced by $\tau_{0}^{\ast}$

$\tau_{0}^{\ast}$

Non-singular out-of-plane structural shear stress

$\phi,\psi$

Complex stress function

$\tilde{\omega }_{1}$

Auxiliary parameter in Filippi’s stress equations

B

Beltrami (criterion)

BE

Boundary element

FAT

Fatigue strength class in IIW design recommendations

FE

Finite element

IIW

International Institute of Welding

NSIF

Notch stress intensity factor

MSED

Minimum strain energy density (criterion)

MTS

Maximum tangential stress (criterion)

SCF

Stress concentration factor

SIF

Stress intensity factor

ns

Normal stress (criterion)

ps

Plane stress

pn

Plane strain

vM

von Mises (criterion)