PSI - Issue 13

Vera Turkova / Procedia Structural Integrity 13 (2018) 982–987 Vera Turkova/ Structural Integrity Procedia 00 (2018) 000 – 000

984

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under mixed-mode loading is given. With the similarity variable and the self-similar representation of the solution for a power-law creeping material and the Kachanov-Rabotnov power-law damage evolution equation, the near-crack-tip stresses, creep strain rates and continuity (integrity) distributions for plane stress conditions are obtained. The self similar solutions are based on the hypothesis of the existence of a completely damaged zone near the crack tip. It is shown that the asymptotical analysis of the near crack-tip fields gives rise to nonlinear eigenvalue problems. A technique enabling all the eigenvalues to be found numerically is proposed, and numerical solutions to nonlinear eigenvalue problems arising from the mixed-mode crack problems in a power-law medium under plane stress conditions are obtained. Using the approach developed, eigenvalues different from the eigenvalues corresponding to the Hutchinson-Rice-Rosengren (HRR) problem have been found. Having obtained the eigenspectra and eigensolutions, the geometry of the completely damaged zone in the vicinity of the crack tip all values of the mixity parameter is found. In the present paper a damage model, based on continuum damage mechanics, is presented. The material law is implemented computationally as a user defined subroutine (UMAT) in a commercially available FEM package Simulia Abaqus. The active damage accumulation zones in the vicinity of the crack tip are analyzed. In recent work we use the defined by Dubé et al. [23] damaged material model, based on continuum damage mechanics. The intact undamaged material was considered to be linear, elastic, isotropic. The stiffness degradation arising from the damage accumulation was simulated by adding a damage tensor into the constitutive equation initially proposed by Sun and Khaleel [7]: ( ) ( ) { } { } e d ij ijkl ijk th kl l kl T T K K        , (3) where ij  and kl  are the stress and strain tensor components, ( ) e ijkl K T is the temperature-dependent fourth order stiffness tensor representing the undamaged isotropic material, ( ) d ijkl K T is the temperature-dependent fourth order stiffness tensor representing the added influence of damage, th kl  are the thermal strain tensor components. The components of the stiffness tensors are given by: ( ) ( )( ) e ijkl ij kl ik jl il kj K T T            , (4) 2. Material model based on the damage tensor where ( ) T  and ( ) T  are Lame’s constants, ij D are the damage parameters, 1 ( ) C T and 2 ( ) C T are constants of the material in [7]. We are interested in analysis of active damage accumulation zone. The components of the damage tensor ij D are functions of the stress state, 0 1 ij D   . The original damage model by Sun and Khaleel [7] took into account the effect of the shear stresses on the damage evolution, only the diagonal terms of the damage tensor, so the damage parameters due to tensile principal stresses, were accounted for in the paper by Dubé et al. [23]. Their values follow a linear evolution law: 0 ( ) ( ) ( ) 1 i th i th ii th i c c th i c T D T T                         , (6) where 1, 2,3 i  , i  is the i th principal tensile stress, th  is the temperature-dependent threshold stress under which no damage occurs, c  is the temperature-dependent critical stress above which the material is fully damaged [23]. The damage tensor will thus be naturally defined as follows. From equations (3) – (5) and neglecting the non diagonal terms of the damage tensor, the constitutive equations, given by Doquet et al. [18], Dubé et al. [23] and Doquet et al. [24] of the material are:  1 il jk K C T D D C T D D         , 2 ( )( ) ( )( ) d ijkl ij kl kl ij jk il (5)