Elastic-Plastic Fatigue Crack Growth

  • Michael Vormwald


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.


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.


List of Symbols

A0, A1, A2, A3

Constants in Newman’s crack opening equation


Crack length


Initial crack length


Final crack length


Effective crack length


Closure development crack length


Crack growth increment


Specimen thickness

B0, B1

Constants in DuQuesnay’s crack opening equation


Coefficient in Paris law


Coefficient in ΔK eff Paris law


Coefficient in ΔJ Paris law


Wheeler’s retardation factor


Miner-type damage


Grain size, specimen thickness


Components of deviatoric (plastic) strain tensor

E, E

Modulus of elasticity, modified for plane strain


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


Shear modulus


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-Integral with effective parameter ranges


Elastic component of ΔJ-integral


Plastic component of ΔJ-integral


Mode related ΔJ-integrals

K, K

Hardening coefficient, monotonic, cyclic


Stress concentration factor


Stress intensity factor


Stress intensity factor, modes I, II, III


Stress intensity factor range


Effective stress intensity factor range


Strain intensity factor range


Threshold stress intensity factor range


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


Critical stress intensity factor for plane strain


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


Factor in expression for equivalent ΔK ε


Factor on grain size for microstructural crack


Bar length in strip-yield model


Coefficient in ΔJ expression

m, m

Exponents in Paris law or factor in Δδ t expression


Number of cycles


Number of cycles to failure


Number of cycles to failure with block i amplitude


Number of cycles in load block i

n, n

Hardening exponent, monotonic and cyclic

P, P0

Load, ligament yield load


Pressure, empirical exponent


Pressure range


Effective pressure range


Pressure at upper reversal point


Pressure at lower reversal point


Empirical exponent


Stress ratio, load ratio, stress intensity factor ratio


Willenborg’s stress intensity factor ratio


Radial distance


Radial distance with linear-elastic stresses above yield stress


Path coordinate


Components of deviatoric stress tensor


Components of traction vector


Crack opening ratio

u, ux

Displacement in x-direction


Displacement in y-direction


Components of displacement vector


strain energy density

Wx, Wxy, Wxz

Strain energy density portions related to coordinate system


Specimen width


Displacement in y-direction


Geometry factor

x, y, z


z1, z2

Auxiliary functions in crack opening stress equation


Coefficient in Ramberg–Osgood relationship


Constraint factor in tension


Constraint factor in compression

γ, γxy, γxz

Shear strains


Shear strain amplitude


Crack tip opening displacement range

ε, εxx, εyy

Normal strains


Normal strain amplitude


Local strain

εop, εcl

Strain at crack opening and crack closure point


Normal strain in shear plane


Reference strain in power-law relationship


Reference strain


Elastic strain range


Plastic strain range


Equivalent strain range


Crack surface factor indicating effective sliding


Biaxiality ratio of far-field stresses


Crack surface friction coefficient


Poisson’s ratio


Notch radius

σ, σxx, σyy

Normal stresses

σx0, σy0

Far-field normal stresses


Biaxiality cut-off stress

σ1, σ1,max

First principal stress, its maximum value


Stress range


Effective stress range


Von Mises equivalent stress range


Components of stress tensor


Stress at upper reversal point


Stress at lower reversal point


Maximum normal stress on crack surface

σop, σcl

Stress at crack opening and crack closure point


Reference stress in power-law relationship


Residual stress


Reference stress


Ultimate tensile strength

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

Monotonic and cyclic yield stress


Polar coordinate, polar angle

τ, τxy, τxz

Shear stresses


Friction shear stress on crack surface


Friction shear stress due to indentation


Shear yield stresses


Shear stress in maximum shear strain plane


Plasticity correction on crack length


Calculated critical plane angle

ω, ωc

Plastic zone size, tensile and compressive


Maximum plastic zone size


Cyclic plastic zone size


Peak load plastic zone size


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Authors and Affiliations

  1. 1.Technische Universität DarmstadtDarmstadtGermany

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