Local Strain Energy Density Concept

Abstract

The local strain energy density (SED) approach is elaborated for strength assessments in respect of brittle fracture and high-cycle fatigue. Pointed and rounded (blunt) V-notches subjected to tensile loading (mode 1) are primarily considered, occasionally extended to multiaxial conditions (mode 3, mixed mode 1 and 2). The application to brittle fracture is related to PMMA flat bar specimens with U-notches. The application to high-cycle fatigue comprises fillet-welded joints, weld-like shaped and V-notched base material specimens as well as round bar specimens with V-notch. The relation of the local SED concept to comparable other concepts is investigated, among them the Kitagawa and Atzori diagrams, the Neuber concept of fictitious notch rounding applied to welded joints and the J-integral approach. Alternative details of the local SED concept such as a semicircular control volume, microrounded notches and slit-parallel loading are also investigated. Coarse FE meshes at pointed or rounded notch tips are proven to be acceptable for accurate local SED evaluations. The peak stress method, which is based on a special coarse FE mesh for the assessment of the fatigue strength of welded joints, is also suitable.

List of Symbols

A

Control volume area

a

Notch depth, hole radius, crack length

$a^{*} ,\,a_{0}^{*} ,\,a_{{0{\text{k}}}}^{*}$

Crack length parameters

a _i

Initiated crack length

B _W

Parameter, dependent on 2α and n

[B]

Strain–displacement matrix

C

Coefficient in J-integral

c _W

Prestress coefficient of SED

d

Globally even FE size

d _g

Grain size

{d}

Vector of nodal point displacements

[E]

Elasticity matrix

E _fe, E ₁

Strain energy in finite element, in local volume

E, E′

Modulus of elasticity

e ₁, e ₂, e ₃

Total SED coefficient, mode 1, 2, 3

e _d1, e _d2, e _d3

Distortional SED coefficient, mode 1, 2, 3

F

Parameter, dependent on 2α

F, F _c

Force, critical force

f ₁, f ₂

Angle-dependent function related to W ₁, W ₂

f _d1, f _d2, f _d3

Angle-dependent function related to W _d1, W _d2, W _d3

f _w, f _w1, f _w2

Peak stress coefficient, mode 1, 2

G

Shear modulus

H

Parameter, dependent on 2α and R ₀/ρ

H, H _τ

Hardening coefficient, tensile and shear loading

I ₁, I ₂

Integral over f ₁, f ₂

I _d1, I _d2, I _d3

Integral over f _d1, f _d2, f _d3

I _e, I _p

Integral over elastic and plastic function f _1e, f _1p

J

J-integral

J _V, J _L

J-integral for V-notches

K _l, K ₂, K ₃

NSIF, mode 1, 2, 3

K _le, K _3e

Elastic NSIF, mode 1, 3

K _lp, K _3p

Plastic NSIF, mode 1, 3

K _lE, K _3E

Endurable NSIF, mode 1, 3

K _lρ , K _2ρ

Generalised NSIF, mode 1, 2

K _1c

Critical NSIF, mode 1

K _I, K _II, K _III

SIF, mode I, II, III

$K_{{{\text{eq}}\,}}^{{}} ,\;K_{\text{eq}}^{*}$

Equivalent SIF, without and with T-stress

K _Ic, K _c

Fracture toughness

K ₀

Threshold SIF

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

Theoretical SCF, fatigue-effective SCF

K _t1, K _t2

SCF at weld toe, weld root

K _t,B, K _t,vM

SCF of equivalent stress, Beltrami, von Mises

$\overline{K}_{\text{t,B}} ,\;\overline{K}_{\text{t,vM}}$

Fatigue-effective value of K _t,B, K _t,vM

K _W , K _t,W , K _t,Wd

SED-based SCF, total, distortional

K _f,N, K _f,W

Fatigue notch factor, based on Neuber, on SED

k

Inverse slope exponent, S–N or W–N curve

m

Elastic-plastic antiplane shear exponent

N, N _i, N _E

Number of cycles, crack initiation, endurable value

[N]

Interpolation matrix

n

Hardening exponent

n ₁, n ₂

Exponent of plate thickness effect

P _f, P _s

Probability of failure, of survival

q

Factor on π/2 for notch internal angle

R

Load ratio, nominal stress ratio

R

Radius

R ₀

Radius of integration path

R ₀, R _0d

Radius of control volume for W, W _d

R ₀₁, R ₀₃

Radius of control volume, mode 1, 3

$R_{0} ,R_{0}^{*}$

Radius of control volume, original, enlarged

R ₁, R ₂

Radii of crescent-shaped control volume

r

Polar coordinate, radial distance

r ₀

Notch root distance

r _p

Radius of plastic zone

s

Elastic-plastic eigenvalue dependent on λ ₁ and n

s

Microstructural support factor

T

Temperature or T-stress

T _σ , T _W

Scatter range index, related to σ, to W

T

Traction vector in J-integral

t, t ₀, t ₁, t ₂

Plate thickness

t

Notch depth

u ₀

Remote boundary displacement, x direction

{u}, u

Displacement vector

V

Volume

v ₀

Remote boundary displacement, y direction

W, W _c

Total SED, critical value

W ₁, W ₂, W ₃

Total SED, mode 1, 2, 3

W _d, W _d1, W _d2

Distortional SED, mode 1, 2

W _n, W _n,n, W _n,g

Nominal SED, net and gross cross-section

W _1 max

Maximum total SED, mode 1

W _1,ρ , W _1,0

Total SED, with and without microrounding

W _1d,ρ , W _1d,0

Distortional SED, with and without microrounding

$\overline{W},\;\overline{W}_{ 1} ,\;\overline{W}_{ 2}$

Average total SED, mode 1, 2

$\overline{W}_{\text{d}}$

Average distortional SED

$\overline{W}_{\text{I}} ,\;\overline{W}_{\text{II}}$

Average total SED, mode I, II

$\overline{W}_{{ 1 {\text{e}}}} ,\;\overline{W}_{{ 1 {\text{p}}}}$

Average total SED, elastic, plastic, mode 1

$\overline{W}_{{ 3 {\text{e}}}} ,\;\overline{W}_{{ 3 {\text{p}}}}$

Average total SED, elastic, plastic, mode 3

$\overline{W}_{\text{c}} ,\;\overline{W}_{\text{dc}}$

Critical average SED, total, distortional

$W_{0}$

Critical SED in unnotched specimen

$\overline{W}_{\text{E}}$

Endurable average total SED

w

Width of cross-section

w ₁, w ₂

Geometric parameters

x, y, z

Cartesian coordinates

Y, Y ₁, Y ₂

Geometry factor, mode 1, 2

α

Notch opening semi-angle

β

Notch arc related angle, coefficient in σ _eq formula

γ

V-notch internal semi-angle, coefficient in σ _eq formula

∆, ∆ _fc, ∆ _sc

Relative deviation, full circle, semicircle

δ

Gap width

δ _ij

Kronecker delta

ε, ε _ij {ε}

Strain, strain tensor, strain vector

ε _θθ, ε _rr, ε _rθ

In-plane strain components in polar coordinates

ε _zz

Out-of-plane strain component

$\theta ,\,\overline{\theta },\,\theta^{*}$

Polar angle, crescent shape angle, transition angle

θ ₀

Crack propagation angle

κ

Slope exponent

κ ₁, κ _1,0

Dimensionless shape factor

κ _fe

Dimensionless ratio, K ₁ or K ₂ related to σ _p

λ ₁, λ ₂, λ ₃

Eigenvalue at V-notch, mode 1, 2, 3

λ

Stress amplitude ratio τ _a/σ _a

ρ, ρ _c

Notch radius, notch curvature radius

ρ _f, ρ _r

Radius of fictitious notch, of reference notch

$\rho^{\ast}$

Microstructural support length

$\rho^{\ast}$

Auxiliary notch tip radius

σ, σ _ij, {σ}

Stress, stress tensor, stress vector

σ _θθ , σ _rr , σ _rθ

In-plane stress components in polar coordinates

σ _θ , σ _r , τ _rθ

In-plane stress components in polar coordinates

σ _zz , σ _z

Out-of-plane stress component

$\tilde{\sigma }_{\theta \theta } ,\;\tilde{\sigma }_{rr} ,\;\tilde{\sigma }_{r\theta }$

Angular functions of in-plane stresses

$\tilde{\sigma }_{zz}$

Angular functions of out-of-plane stress

σ _kk

Sum of principal stresses

σ _n, σ _n,g, σ _n,n

Nominal stress, in net and gross cross-section

σ _l, σ _u

Lower and upper nominal stress of load cycle

σ _{k max}

Maximum notch stress

σ _{θ max}, σ _{t max}

Maximum tangential stress

σ _a, σ _A

Stress amplitude, endurable value

σ ₀

Substitute yield limit, remote nominal stress

$\overline{\sigma }_{0} ,\;\overline{\sigma }_{{0{\text{b}}}}$

Remote stress, membrane, bending

σ _Y, σ _Y′

Yield limit, monotonic, cyclic

σ _U

Ultimate tensile strength

σ _E, σ _nE

Endurance limit stress, endurable nominal stress

σ _eq, σ _B, σ _vM

Equivalent stress, Beltrami, von Mises

σ _h

Hydrostatic stress

σ _p, σ _eq,p

Peak stress in FE mesh, equivalent peak stress

τ

Shear stress

τ _a

Shear stress amplitude

τ _xy∞

Remote shear stress

τ ₀

Substitute shear yield limit, remote nominal shear stress

τ _max

Maximum notch shear stress

τ _E

Endurance limit shear stress

ϕ

Phase shift angle, weld toe angle

φ

Coefficient of K ₁

χ _h

Multiaxiality index

ψ

Inclination angle

ω, ω _d

Dimensionless SED, total, distortional

$\overline{\omega },\;\overline{\omega }_{\text{d}}$

Dimensionless average SED, total, distortional

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

Auxiliary parameter, dependent on 2α

BE

Boundary element

BS

British Standard

CJ

Cruciform joint

FAT

Fatigue strength class

FE

Finite element

FEM

Finite element method

FNR

Fictitious notch rounding

IIW

International Institute of Welding

LEFM

Linear-elastic fracture mechanics

NSIF

Notch stress intensity factor

SCF

Stress concentration factor

SED

Strain energy density

SIF

Stress intensity factor

TAJ

Transverse attachment joint

cc, ec

Concentric, eccentric (reference notch)

fc, sc

Full circle, semicircle (average SED)

fm, cm

Fine mesh, coarse mesh

ns

Narrow section (average SED)