The coupled criterion for description of fatigue fracture. Material embrittlement in pre-fracture zone

Step-wise extension of a crack in quasi-brittle materials under low-cycle loading conditions is considered. Both steady and unsteady loadings in pulsating loading mode are studied. It is proposed to use quasi-brittle fracture diagrams for bodies under cyclic loading conditions. When diagrams are plotted, both necessary and sufficient fracture criteria by Neuber-Novozhilov are used. A specific implementation is made on the base of the Leonov-Panasyuk-Dugdale model for the mode I cracks when the pre-fracture zone width coincides with the plasticity zone width near the crack tip. The condition of a step-wise crack tip extension has been derived. A crack extends only in the embrittled material of the pre-fracture zone. The number of cycles between jumps of the crack tip is calculated by the Coffin equation, when damage accumulation in material in the pre-fracture zone is taken into account. Critical fracture parameters under low-cyclic loading conditions have been obtained in a closed form. Estimates of the average rate of crack tip advance for a loading cycle at step-wise crack extension and S N − curves have been obtained.


Introduction
In order to obtain estimates for the number of cycles, the diagram of quasi-brittle fracture under low-cycle fatigue is used [1,2].
The problems to be settled in this work are as follows: i) to derive relations for critical stresses above which the growth of this macrocrack takes place in specimens having a short macro-crack; ii) in the plane stresses vesus crack length plotting a subarea where damage accumulation ocсurs after single loading; iii) description of the process of step-wise propagation of short macro-cracks under fatigue when material embrittlement in the pre-fracture zone takes place ; iv) selection of parametrs for the proposed model acording to the resuls of laboratory tests. ε is the maximum material strain. Let r be the effective diameter of fracture structures. The Neuber-Novozhilov approach [3,4] makes it possible to use solutions having a singular function for structured media. Now consider an internal I mode crack. Assume that an internal plane I mode crack extends rectilinearly. In addition to the internal rectilinear crack-cut of length 0 2l , model crack-cuts of length 0 2 2 2 l l = + ∆ , each of prefracture zones ∆ being located on the continuation of a real crack ( 2l and ∆ are lengths of model crack and pre-fracture zones) are introduced into consideration. The problem of fatigue fracture has two linear scales: if a grain diameter is defined by a material structure, then the second linear size is governed by the system itself. Under low-cycle fatigue conditions, the second linear scales serve as pre-fracture zone lengths ∆ , which change in accordance with changing i) the length of a real step-wise extending crack, and ii) the intensity of loading conditions. Emphasize that under single loading conditions, the critical pre-fracture zone length * ∆ is a completely definite parameter [1,2] and When diagrams of quasi-brittle fracture under conditions of low-cycle loading are plotted in [1,2,5], sufficient fracture criteria are used when the mode I cracks are considered ∆ is the total stress intensity factor (SIF) at the tips of a model crack, I K ∞ is SIF generated by stresses σ ∞ specified at infinity, I K ∆ is SIF generated by constant stresses Y σ . The first and second summands in relation (2) are singular and smooth parts of solution, respectively. The first equality in criterion (1) controls stresses on the model crack continuation after averaging, these stresses being coincident with the yield strength Y σ , and the second equality in criterion (1) describes real blunting of a crack at its tip. The sufficient fracture criterion (1) simultaneously takes into consideration both strength and deformation fracture criteria at specific points of a pre-fracture zone. Selected specific points of the pre-fracture zone are well-adapted to description of step-wise crack tip advance and accumulation of damages in the pre-fracture zone material under fatigue conditions [1,2]. The simplest analytical representation of the field of stresses ( , 0) y x σ on the model crack continuation has the following form ( 2L is the plain specimen width) = [2]. In what follows, dimensionless variables are used ( µ is Poisson's ratio) σ ∞ and σ * ∞ Analytical representations of critical fracture parameters for quasi-brittle materials has the form for a plain specimen of the finite width [1,5] ( ) ( ) ( ) ( ) ( ) Expressions (4) make sense if 1 p Aε < .
The finite specimen width effects upon critical fracture curves on the plane "crack length versus internal load". The calculation results are given in Fig. 1 in the log-log coordinates, when 250 L = ). In this Figure

Step-wise crack propagation
When the proposed diagram under low-cycle fatigue [1,2,5] is plotted, there is no need to use SIFs. It can be plotted depending on both elastic-plastic material properties and a crack length. Consider limitations typical for one-frequency loading Here σ ∞ is amplitude. When limitations (5) are obeyed, the low-cycle fatigue may be implemented. Changes in the subarea of the diagram of quasi-brittle fracture with account for damage accumulation are described for the normal loading regime when the critical number of loading cycles N * is calculated in such a way , The number of cycles j N between 1 j − and j crack tip advances at the loading level σ ∞ is calculated as Here 0.2 1 C ≤ ≤ are Coffin's constants [6], numerical values of which depend on material properties. In fatigue failure, it is reasonable to obtain estimates of average dimensionless rate 2 / j j j V N = ∆ of crack advance per one loading cycle for plain specimens of finite width at the appropriate loading mode (7) and (8)

Averaged rates of macro-crack tip advance
Intsead of the discontinous function j V in relation (9)  . In Fig. 3, light and dark dots on curves correspond to the left-and right-side inequalities in relation (10). Middle sections of curves are lengths of cracks for which the Paris law "works" well. To the right of asterisks depicted on curves 1 -5, failure of the specimen takes place in accordance with relation (4), therefore, the conditional sections of curves 1 -5 are continued as dotted lines. Curve 1 almost completely falls into the region of short macro-cracks.  Fig.4 correspond to the left-and right-side inequalities in relation (10). To the right of asterisks depicted on these curves, failure of the specimen takes place in accordance with relation (4), therefore, the conditional sections of the curves are continued as dotted lines. Each of these curves descrides initiation, evolution, and completion of the process of macro-crak tip advance. Middle sections of all the curves correspond to Paris' curves with some degree of certainty.
The exellent representation of all stages of the process is shown in monograph [6, p.27, Fig. 1.18]. In this mongraph and handbook [7], all three stages of the process and transition modes are discussed in detail. Howeever, the main attention of these authors [6,7] is concentrated on the representation of these diagrams in the form of dependencies on SIF that does not agree with the results obtained for small-scale yielding in quasibrittle materials (9). Given in Fig. 5  ). Light and dark dots on the curves in Fig, 5 correspond to the left-and right-side inequalities from relation (10). To the rigth of asteriks depicted on the curves, failure of the specimen takes place in accordance with relation (4), therefore, the conditional sections of these curves are continued as dotted lines.
The margin of plastic strain after preliminary plastic deformation of material 1 p ε may be a small part of the plastic strain margin of basic material 0 p ε , i.e., . Therefore, taking into account relation (11), the ratio of rates of crak tip advance 0 1 / V V for basic material and material after significant plastic deformation can reach from one to two orders of magnitude. then there are no sections on these curves, which can be treated as Paris' curves.

The interrelation between the Coffin constant and characteristic rate of crack tip advance
The proposed approximate relation (9) permits relate the Coffin constant with the charcteristic rate of crak tip advance drawing on their association at the second stage of crack tip advance when the Paris law "works" well.
Let the characteristic rate Relation (12) associates the constant C and characteristic rate 0 V % based on the Paris law, i.e.
( ) It is proposed to use in relation (9) the experimentaly determined C constant from relation (12) for description of all three stages of crack tip advances including closely adjacent value of parameters σ ∞ and L . It is to be recalled that derived relation (9) permits one to study behavior of cracks, lengths of which can be assignd just as to short, so to long cracks. Three stages of the process of crack tip advances are roughly estaimated by relations (10). Implementation of some or other stages of the process should be agreed with constrains imposed by the presence of two certain threshold values of crack lengths with the given loading intensity σ ∞ .
Relation (9) can be used for variable loading condititions. With a gradual decrease of loading intensity, the system can fall several times into the region, which corresponds to the region of short crack tip advance (10) (as compared with the behavior of curves 3 and 4 with that of curves 5 and 6 in Fig. 4).   growth rate V are taken into account, then it will be possible to estimate transition from relatively secure operating conditions of a structure with a crack to dangerous loading mode.

Conclusion
Step-wise crack propagation in quasi-brittle materials under cyclic loading conditions has been considered. For a model of deformable body, the model of elastic ideally-plastic material having the ultimate tensile strain is taken. When basic material undergoes nonlinear deformation, the material is embrittled.
For analysis of the process mentioned above, it is proposed to use modified diagrams of quasi-brittle fracture of deformable bodies. Reference diagrams have been plotted within the context of quasi-linear fracture mecanics when the small-scale material yelding is realized in the vicinity of a crack tip and the singularity of the stress field at the tip of a crak is partally retained.
Two theshold values of critical parameters are proposed for description of the process of step-wise crack tip advance in responce to fatigue loading modes. Both theshold values are given in the explicit form including that for short macro-cracks. The second theshold value is a variable and it is directly associated with material damage in a pre-fracture zone. Under nonlinear deformation of material in the pre-fracture zone, the plasticity margin is spent due to material embrittlement during each loading cycle.
Constants in the described model are proposed to be selected for individual material from the data obtained during three laboratory experiments based on i) approximation of a real diagram stress versus strain, ii) searching critical SIF, and iii) Coffin's constants or the characteristic rate of crack tip advance for the second process stage.
The analysis of all three process stages of crack tip afvances and their dependence on geomentric parameters of cracks and specimens, characterisics of material, and intensity of fatigue loading under pulstaing loading appliance has been performed in detail. For the averaged process, crude rwo-sided estimates for transition from the first to the second stage and from the second to the third stage have been pointed out, fairly simple analitical expressions for describing the whole process, including the case when some stages are absent have been derived. The plotted 0 N S l − − diagrams charcterize lifetime of specimens having initial defects. All the constructions have peformed in terms of cracks. There is the fracture mechanics approach, which specifically treats growing cracks by the methods of fracture mechanics.