PSI - Issue 13

Stepanova Larisa / Procedia Structural Integrity 13 (2018) 255–260 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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2. Basic equations

A static mixed mode crack problem under plane strain conditions is considered. The equilibrium equations and compatibility condition in the polar coordinate system can, respectively, be written as     , , , , , , , 0, 2 0, 2 , , . rr r r rr r r r r r rr rr r rr r r r r r r                                 (1) The constitutive equations are described by the power law equations   1 3 / 2 ( / ) / n ij e ij B s       , (2) where ij s are the deviatoric stress tensor components; , B n are material constants;  is an integrity (continuity) parameter; ij  are the creep strain rates. The creep strain rates for plane strain conditions take the form:         2 1 1 2 2 (3 / 4) / 2 , 3 / 2 , (3 / 4) 4 . n n n n rr e rr r e r e rr r B B                                     (3) The constitutive model (2) is the phenomenological model of Kachanov and Rabotnov widely employed in creep damage theory and in damage analysis of high temperature structures (Murakami (2012), Riedel (1987), Stepanova and Yakovleva (2016)). By noting that the creep damage is brought about by the development of microscopic voids in creep process, Kachanov (1958) represented the damage state by a scalar integrity variable   0 1     where 1   and 0   signify the initial undamaged state and the final completely damaged state (or final fractured state), respectively ((Murakami (2012), Voyiadis and Kattan (2012), Voyiadis (2015), Stepanova and Igonin (2014)). Kachanov described the damage development by means of an evolution equation   / , m e A      (4) where  denotes the time derivative, while A and m are material constants. The solution to Eqs. 1 – 4 should satisfy the traction free boundary conditions on the crack surfaces     , 0, , 0. r r r               The mixed-mode loading can be characterized in terms of the mixity parameter p M (Kuna (2013)). Thus, for combine mode fracture the mixity parameter completely specifies the near-crack-tip fields for a given value of the creep exponent. One can postulate the Airy stress function and the continuity parameter as     1 1 0 0 , ( ), , 1 ( ) j j j j j j r r f r r g                    . (5) First consider the leading terms of the asymptotic expansions (5): 1 ( , ) ( ) r r f       , 1   , where  is indeterminate exponent and ( ) f  is an indeterminate function of the polar angle, respectively. The asymptotic stress field at the crack tip is derived as follows 1 ( , ) ( ) ij ij r r        , where 1   denotes the exponent representing the singularity of the stress field will be called the stress singularity exponent. The compatibility condition in Eq. 1 results in the nonlinear forth-order ordinary differential equation for the unknown function ( ) f  :             2 2 2 4 2 2 2 2 2 2 2 2 4 2 4 2 1 1 1 1 ( 1)( 3) ( 1) ' 1 4 2( 1) ' ( 1) ' 0, IV e e e e e e e e f f n g f f f n n F g n f g g f f f f g n f Fg C f g C n f F f C f f                                (6) where the notations are used:     2 2 2 2 1 2 4 , 4 ( 1) 1 , ( 1) ( 1) 2 , e f g f C n C n n                  2 1 , g f f      2 ' 4 , F gg f f      . The boundary conditions are ( ) 0, ( ) 0 f f            . The two point boundary value problem for Eq. (6) is solved numerically and the eigenspectrum is found by the proposed algorithm (Stepanova and Yakovleva (2016)). 3. Similarity variable and more general remote boundary conditions In CDM (Murakami (2012)), the damage state at an arbitrary point in the material is represented by a properly defined integrity variable. The integrity parameter reaches its critical value at fracture. According to this notion, a crack in a fracture process can be modeled with the concept of a completely damaged zone (CDZ) in the vicinity of