Elastic-Plastic Fatigue Crack Growth

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.

List of Symbols

A ₀, A ₁, A ₂, A ₃

Constants in Newman’s crack opening equation

a

Crack length

a ₀

Initial crack length

a _f

Final crack length

a _eff

Effective crack length

a _cd

Closure development crack length

Δa

Crack growth increment

B

Specimen thickness

B ₀, B ₁

Constants in DuQuesnay’s crack opening equation

C

Coefficient in Paris law

C _eff

Coefficient in ΔK _eff Paris law

C _J

Coefficient in ΔJ Paris law

C _p

Wheeler’s retardation factor

D

Miner-type damage

d

Grain size, specimen thickness

e _ij

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

h ₁, h ₀

Functions for geometry and hardening influence on J _p

J, J _e, J _p

J-Integral, elastic and plastic component

ΔJ

ΔJ-integral

ΔJ _eff

ΔJ-Integral with effective parameter ranges

ΔJ _e

Elastic component of ΔJ-integral

ΔJ _p

Plastic component of ΔJ-integral

ΔJ _I, ΔJ _II, ΔJ _III

Mode related ΔJ-integrals

K, K′

Hardening coefficient, monotonic, cyclic

K _t

Stress concentration factor

K

Stress intensity factor

K _I, K _II, K _III

Stress intensity factor, modes I, II, III

ΔK ε

Stress intensity factor range

ΔK _eff

Effective stress intensity factor range

ΔK _ε

Strain intensity factor range

ΔK _th

Threshold stress intensity factor range

K _p

Peak stress intensity factor

K _op, K _cl

Stress intensity factor at crack opening and closure point

K _max, K _min

Stress intensity factor at upper and lower reversal point

K _Ic

Critical stress intensity factor for plane strain

K _c

Critical stress intensity factor

K _{i max}, K _{i 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

l _0i

Bar length in strip-yield model

M _J

Coefficient in ΔJ expression

m, m′

Exponents in Paris law or factor in Δδ _t expression

N

Number of cycles

N _f

Number of cycles to failure

N _i

Number of cycles to failure with block i amplitude

n _i

Number of cycles in load block i

n, n′

Hardening exponent, monotonic and cyclic

P, P ₀

Load, ligament yield load

p

Pressure, empirical exponent

Δp

Pressure range

Δp _eff

Effective pressure range

p _max

Pressure at upper reversal point

p _min

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

r _Y

Radial distance with linear-elastic stresses above yield stress

s

Path coordinate

s _ij

Components of deviatoric stress tensor

T _i

Components of traction vector

U

Crack opening ratio

u, u _x

Displacement in x-direction

u _y

Displacement in y-direction

u _i

Components of displacement vector

W

strain energy density

W _x , W _xy , W _xz

Strain energy density portions related to coordinate system

W

Specimen width

v

Displacement in y-direction

Y

Geometry factor

x, y, z

Coordinates

z ₁, z ₂

Auxiliary functions in crack opening stress equation

α

Coefficient in Ramberg–Osgood relationship

α

Constraint factor in tension

β

Constraint factor in compression

γ, γ _xy , γ _xz

Shear strains

γ_a

Shear strain amplitude

Δδ _t

Crack tip opening displacement range

ε, ε _xx , ε _yy

Normal strains

ε _a

Normal strain amplitude

ε _l

Local strain

ε _op, ε _cl

Strain at crack opening and crack closure point

ε _⊥

Normal strain in shear plane

ε ₀

Reference strain in power-law relationship

ε _ref

Reference strain

Δε _e

Elastic strain range

Δε _p

Plastic strain range

Δε _eq

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, σ _xx , σ _yy

Normal stresses

σ _x0, σ _y0

Far-field normal stresses

σ _co

Biaxiality cut-off stress

σ ₁, σ _1,max

First principal stress, its maximum value

Δσ

Stress range

Δσ _eff

Effective stress range

Δσ _eq

Von Mises equivalent stress range

σ _ij

Components of stress tensor

σ _max

Stress at upper reversal point

σ _min

Stress at lower reversal point

σ _⊥max

Maximum normal stress on crack surface

σ _op, σ _cl

Stress at crack opening and crack closure point

σ ₀

Reference stress in power-law relationship

σ _res

Residual stress

σ _ref

Reference stress

σ _U

Ultimate tensile strength

$\sigma_{\text{Y}} ,\,\sigma_{\text{Y}}^{'}$

Monotonic and cyclic yield stress

θ

Polar coordinate, polar angle

τ, τ _xy , τ _xz

Shear stresses

τ _fr

Friction shear stress on crack surface

τ _fr0

Friction shear stress due to indentation

τ _Y

Shear yield stresses

Δτ _II

Shear stress in maximum shear strain plane

φ

Plasticity correction on crack length

φ _calc

Calculated critical plane angle

ω, ω _c

Plastic zone size, tensile and compressive

ω _max

Maximum plastic zone size

ω _cyc

Cyclic plastic zone size

ω _p

Peak load plastic zone size