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Elastic-Plastic Fatigue Crack Growth

  • Michael Vormwald
Chapter

Abstract

Conventional fatigue crack propagation approaches rely on similitude arguments and relationships between the stress intensity factor range and crack growth rate. The application limit of this approach is specified by small-scale yielding conditions. Still within these limits, an explanation and the straightforward modelling of the mean stress influence and the influence of variable amplitudes requires consideration of cyclic plasticity. Plasticity-induced crack closure greatly influences the crack growth rate. Modelling tools and algorithms are presented. Outside the small-scale yielding limits, the stress intensity factor range must be substituted by a crack driving force parameter of elastic-plastic fracture mechanics. Various proposals are presented and discussed with a focus on the ΔJ-integral. Together with an adequate consideration of crack closure, advances in simulating fatigue crack growth in this regime more realistically are presented. Multiaxial and mixed mode loading are a continuing challenge for actual research. These topics are discussed against the background of current expertise and available computational resources.

Keywords

Stress Intensity Factor Crack Growth Rate Fatigue Crack Growth Crack Closure Plastic Zone Size 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

List of Symbols

A0, A1, A2, A3

Constants in Newman’s crack opening equation

a

Crack length

a0

Initial crack length

af

Final crack length

aeff

Effective crack length

acd

Closure development crack length

Δa

Crack growth increment

B

Specimen thickness

B0, B1

Constants in DuQuesnay’s crack opening equation

C

Coefficient in Paris law

Ceff

Coefficient in ΔK eff Paris law

CJ

Coefficient in ΔJ Paris law

Cp

Wheeler’s retardation factor

D

Miner-type damage

d

Grain size, specimen thickness

eij

Components of deviatoric (plastic) strain tensor

E, E

Modulus of elasticity, modified for plane strain

f

Crack opening function, function in ΔJ expression

\( f_{\text{I, ij}} ,\,f_{\text{II, ij}} ,\,f_{\text{III, ij}}\)

Angular functions of near tip stress fields

G

Shear modulus

g

Influence function in crack opening analysis

h1, h0

Functions for geometry and hardening influence on J p

J, Je, Jp

J-Integral, elastic and plastic component

ΔJ

ΔJ-integral

ΔJeff

ΔJ-Integral with effective parameter ranges

ΔJe

Elastic component of ΔJ-integral

ΔJp

Plastic component of ΔJ-integral

ΔJI, ΔJII, ΔJIII

Mode related ΔJ-integrals

K, K

Hardening coefficient, monotonic, cyclic

Kt

Stress concentration factor

K

Stress intensity factor

KI, KII, KIII

Stress intensity factor, modes I, II, III

ΔKε

Stress intensity factor range

ΔKeff

Effective stress intensity factor range

ΔKε

Strain intensity factor range

ΔKth

Threshold stress intensity factor range

Kp

Peak stress intensity factor

Kop, Kcl

Stress intensity factor at crack opening and closure point

Kmax, Kmin

Stress intensity factor at upper and lower reversal point

KIc

Critical stress intensity factor for plane strain

Kc

Critical stress intensity factor

Ki max, Ki min

Maximum and minimum stress intensity factor of cycle i

\( K_{{{\text{i}}\max }}^{*} \)

Fictitious maximum stress intensity factor of cycle i

\( K_{{{\text{i}}\max }}^{{({\text{W}})}} ,\;K_{{{\text{i}}\min }}^{{({\text{W}})}} \)

Willenborg’s stress intensity factor of cycle i

k

Factor in expression for equivalent ΔK ε

k

Factor on grain size for microstructural crack

l0i

Bar length in strip-yield model

MJ

Coefficient in ΔJ expression

m, m

Exponents in Paris law or factor in Δδ t expression

N

Number of cycles

Nf

Number of cycles to failure

Ni

Number of cycles to failure with block i amplitude

ni

Number of cycles in load block i

n, n

Hardening exponent, monotonic and cyclic

P, P0

Load, ligament yield load

p

Pressure, empirical exponent

Δp

Pressure range

Δpeff

Effective pressure range

pmax

Pressure at upper reversal point

pmin

Pressure at lower reversal point

q

Empirical exponent

R

Stress ratio, load ratio, stress intensity factor ratio

R(W)

Willenborg’s stress intensity factor ratio

r

Radial distance

rY

Radial distance with linear-elastic stresses above yield stress

s

Path coordinate

sij

Components of deviatoric stress tensor

Ti

Components of traction vector

U

Crack opening ratio

u, ux

Displacement in x-direction

uy

Displacement in y-direction

ui

Components of displacement vector

W

strain energy density

Wx, Wxy, Wxz

Strain energy density portions related to coordinate system

W

Specimen width

v

Displacement in y-direction

Y

Geometry factor

x, y, z

Coordinates

z1, z2

Auxiliary functions in crack opening stress equation

α

Coefficient in Ramberg–Osgood relationship

α

Constraint factor in tension

β

Constraint factor in compression

γ, γxy, γxz

Shear strains

γa

Shear strain amplitude

Δδt

Crack tip opening displacement range

ε, εxx, εyy

Normal strains

εa

Normal strain amplitude

εl

Local strain

εop, εcl

Strain at crack opening and crack closure point

ε

Normal strain in shear plane

ε0

Reference strain in power-law relationship

εref

Reference strain

Δεe

Elastic strain range

Δεp

Plastic strain range

Δεeq

Equivalent strain range

η

Crack surface factor indicating effective sliding

Λ

Biaxiality ratio of far-field stresses

μ

Crack surface friction coefficient

ν

Poisson’s ratio

ρ

Notch radius

σ, σxx, σyy

Normal stresses

σx0, σy0

Far-field normal stresses

σco

Biaxiality cut-off stress

σ1, σ1,max

First principal stress, its maximum value

Δσ

Stress range

Δσeff

Effective stress range

Δσeq

Von Mises equivalent stress range

σij

Components of stress tensor

σmax

Stress at upper reversal point

σmin

Stress at lower reversal point

σ⊥max

Maximum normal stress on crack surface

σop, σcl

Stress at crack opening and crack closure point

σ0

Reference stress in power-law relationship

σres

Residual stress

σref

Reference stress

σU

Ultimate tensile strength

\( \sigma_{\text{Y}} ,\,\sigma_{\text{Y}}^{'} \)

Monotonic and cyclic yield stress

θ

Polar coordinate, polar angle

τ, τxy, τxz

Shear stresses

τfr

Friction shear stress on crack surface

τfr0

Friction shear stress due to indentation

τY

Shear yield stresses

ΔτII

Shear stress in maximum shear strain plane

φ

Plasticity correction on crack length

φcalc

Calculated critical plane angle

ω, ωc

Plastic zone size, tensile and compressive

ωmax

Maximum plastic zone size

ωcyc

Cyclic plastic zone size

ωp

Peak load plastic zone size

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Authors and Affiliations

  1. 1.Technische Universität DarmstadtDarmstadtGermany

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