13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- The specific free energy of the system can be calculated as F n Q lnZ 0 or . s / Q dsd p s p n Q ln exp 2 1 F p 3 2 i k ik ik ik 2 ik (10) The equation (10) request a determination materials constants at micro level and complex calculation. At macroscopic level the equation (10) can be approximated by corresponding function [4]. To consider the influence of diffusion processes on defect nucleation and evolution and to study the localization effects of the damage accumulation, we have introduced in the expression of the total free energy F the term describing spatially-nonuniform distribution of microcrack density tensor ik p 2 ik ik 2 ik e ll e ik e 2 ll r p 2 1 F p 3 1 K 2 1 F= . (11) In order to evaluate tensor pF , we have to consider the equation (11) as a functional determined for a representative material volume. For one dimension problem we can write l ik l ik ik p p p x p / x F F F F . (12) The system (8)–(9) in the case of uniaxial cyclic loading ( zz , e e zz , p p zz ) takes the form x p D p x F l l p p p p , (13) p p p l x p D p x F p l , (14) where D is the coefficient of self-diffusion which is known to obey the Arrenius law, D D exp E / T sd 0 ( sd E is the activation energy of self-diffusion) and largely depends on the defect concentration. 4. Defect evolution under cyclic loading To describe the defect evolution in bulk and near specimen surface let us to reduce the equations (13,14). Under high cyclic and gigacyclic fatigue we can propose a weak interaction of defect accumulation and microplastisity processes ( 0 l pp ) and write the equation (14) in the form
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