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analyze fatigue crack applied stress characteristic length condition of plane considered as homogeneous crack front crack increment crack length crack propagation rate crack surfaces crack tip deformation crack tip opening cracked plate cyclic stresses da/dN is proportional deformation and fracture dimensional analysis displacement vector Equation 8b experimental data fatigue crack growth fatigue crack propagation Fracture Mechanics fracture processes fracture properties geometrically similar points geometrically similar solids H. W. Liu homologous points intensity factor range length parameter line force load cycles material element material homogeneity microcracks model of fatigue necking takes place normal separation mode opening displacement plane strain region plane stress plastic deformation plate thickness shear deformation shear line shear separation mode shear step shown in Figure slip bands specimen steel sheet strain gradient strains at homologous stress intensity factor stresses and strains strip necking region strip necking zone tensile stress thick plate transition layer unloading yield strength