PSI - Issue 13

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

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

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Kachanov [2] and Rabotnov [3] in their pioneer works proposed the continuity and damage parameters for the mathematical description of damage accumulation processes. Since then the set of damage evolution equations has been considerably developed by Murakami [1], Voyiadjis and Kattan [4], Altenbach and Sadowski [5], Riedel [6] and many works describe mutual influence of stress-strain state and damage evolution near-stress concentrators: Sun and Khaleel [7], Stepanova et al. [8 – 15], Wen et al. [16], Belnoue et al. [17], Doquet et al. [18], Cousigné et al. [19], Chen et al. [20], Gao et al. [21]. The directions of continuum damage mechanics development can be classified. First of all, different damage evolution equations have been proposed which take into account different aspects of damage accumulation by Murakami [1]. Then several models for anisotropic damaged materials have been suggested by Fengxia Ouyang [22], Dubé et al. [23], Doquet et al. [24], Lee et al. [25], Salavatian and Smith [26], Hazar et al. [27], Jun et al. [28]. Finally, one can note that many asymptotic solutions to the coupled elasticity-damage, plasticity damage and creep damage problems have been obtained in [1,7,23,25,26]. To elucidate the effect of damage in structures working in real conditions it is necessary to apply finite element method for cracked bodies under complex loading as in Turkova and Stepanova [29], Kim et al. [30], Lua et al. [31]. Damage accumulation process in solids can be described by scalar or tensor damage parameter. In the simplest case damage is presented as scalar value 1 0    in [2]. In initial state, when the structure is undamaged 1   , and eventually function  decreases. It is possible to interpret function  as continuity. In [3] function 0   is established, zero meaning that the material is undamaged and one meaning that the material is fully damaged. It is essential to interpret function  as damage and then the equation 1     is true. The damage parameter  represents the relative cross-sectional area of the specimen occupied with cracks. Damage rate  depends on stress and  . This assumption let us consider  as one of the structural parameters of the material model. The simplest hypothesis assumes that  is a power function of the relation / (1 )    . This relation is explained as medium stress on cross-sectional area, free from cracks. Constitutive equations of the material are based on steady-state creep theory power law of Baily-Norton: 1 (3 / 2) ( / ) / n ij e ij B s       , (1) where  is continuity parameter, evolving according to the power law of damage accumulation: / ( / ) m e d dt A      . (2) Nowadays the damage and failure of the material are of particular interest to designers of engineering structures. While composites applications in the aircraft and spacecraft industry are rapidly increasing, there is a lack of accurate failure prediction and progressive damage analysis. The efficient design of a composite structure depends on developing accurate analytical and numerical material models. Critical to this advancement is a thorough understanding of damage mechanisms and their interactions [26]. Especially for early stages of the design, a quicker way of estimating complicated material behavior is needed. Engineers must be able to predict the strength of the future structure element and design on the whole. However, the numerical modeling of these materials poses several challenges. There is a need for a material model coupled with a damage evolution law. For example, Wen et al. [16] presents a rate-dependent crystallographic creep constitutive theory coupled with a two-state-variable creep damage model, which takes simultaneously the rafting damage and void damage into consideration. A circular notched structure, containing a micro-void was taken as a specimen. Distribution of total creep damage for different crystallographic orientations was obtained. In the paper was stated that both the crystallographic orientation and the distance from the notch to the void have a great influence of the crack initiation. Stepanova and Igonin [9,10] used perturbation technique for solving a nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium. Then they also performed an asymptotic analysis of growing fatigue near-crack-tip fields in a damaged material [9,10]. These researches are fine prerequisite works for this paper because they provide an approach for acquiring stress intensity factor which is important for crack growth prediction. Creep crack problems in damaged materials under mixed-mode loading in the creep-damage coupled formulation are considered in [14]. The class of self-similar solutions to the plane creep crack problems in a damaged medium