**4. Comparison between test results and theoretical notions**

The model of low-cycle fatigue describing pulsing loading of a specimen with the inner macrocrack has been proposed in (Kornev, 2004, 2010). This model is appropriate to the scheme by Laird-Smith (Laird & Smith, 1962; Laird, 1967). Within the framework of the proposed model in (Kornev, 2004, 2010), information on material strain in the pre-fracture zone has been obtained in detail: processes of damage accumulation, step-wise crack tip advance, and failure of structures for pulsing loading are described. Accumulation of damages is associated with inelastic strain of materials in the pre-fracture zone. The simple relations for the critical fracture parameters and the fatigue life have been obtained. Attention should be paid to the following circumstances: in the foregoing model, the information of damage accumulation and the hypothesis concerning the crack arrest are essentially used. Emphasize that when damages are accumulated, just as linear, so nonlinear summation of damage in the pre-fracture zone material may occur in the context of the considered model. In deciding between one and another way for summation of damages, no experimental data on damage accumulation at every loading step were available. The experimental data described in the previous section make up this deficiency.

The model in (Kornev, 2004, 2010) describes occurrence of striations under fatigue fracture. Material is considered to consist of quasi-brittle fibers separated by thin layers, which possess quasi-ductile fracture type before strain, and after inelastic strain of layers, the fracture type changes to quasi-brittle. The fiber diameters coincide with diameters of grains of tested materials (the diameters of grains are ≈10-2 cm), and the fiber widths coincide with the thickness of layers separating subgrains. The width of the layers is ≈10-4 cm. Further we assign numeral 1 to the fiber material and numeral 2 to the thin layer material. Properties of the layer material allow description of occurrence of some marks (whiskers or ears) from fatigue striations. In essence, in work (Kornev, 2004, 2010) there is considered the behavior of the simplest composite medium, material of which changes its fracture type under inelastic strain.

For deriving sufficient fracture criteria (Kornev, 2004, 2010) for low-cycle fatigue, modification of the classical Leonov-Panasyk-Dugdale model (Kornev, 2004) is used, where

At the initial stage, a fatigue crack passes through material located in the region of plasticity localization near the notch (region I in Fig 11.a, regions I and II in Fig. 9). Further, when the crack continues to propagate, the zone of plasticity localization ahead of the sharp crack tip serves as a pre-fracture zone. The former zone is sufficiently small. As a result of the fact that the crack tip passes from one region to another, a pronounced fatigue striation is

The model of low-cycle fatigue describing pulsing loading of a specimen with the inner macrocrack has been proposed in (Kornev, 2004, 2010). This model is appropriate to the scheme by Laird-Smith (Laird & Smith, 1962; Laird, 1967). Within the framework of the proposed model in (Kornev, 2004, 2010), information on material strain in the pre-fracture zone has been obtained in detail: processes of damage accumulation, step-wise crack tip advance, and failure of structures for pulsing loading are described. Accumulation of damages is associated with inelastic strain of materials in the pre-fracture zone. The simple relations for the critical fracture parameters and the fatigue life have been obtained. Attention should be paid to the following circumstances: in the foregoing model, the information of damage accumulation and the hypothesis concerning the crack arrest are essentially used. Emphasize that when damages are accumulated, just as linear, so nonlinear summation of damage in the pre-fracture zone material may occur in the context of the considered model. In deciding between one and another way for summation of damages, no experimental data on damage accumulation at every loading step were available. The

The model in (Kornev, 2004, 2010) describes occurrence of striations under fatigue fracture. Material is considered to consist of quasi-brittle fibers separated by thin layers, which possess quasi-ductile fracture type before strain, and after inelastic strain of layers, the fracture type changes to quasi-brittle. The fiber diameters coincide with diameters of grains of tested materials (the diameters of grains are ≈10-2 cm), and the fiber widths coincide with the thickness of layers separating subgrains. The width of the layers is ≈10-4 cm. Further we assign numeral 1 to the fiber material and numeral 2 to the thin layer material. Properties of the layer material allow description of occurrence of some marks (whiskers or ears) from fatigue striations. In essence, in work (Kornev, 2004, 2010) there is considered the behavior of the simplest composite medium, material of which changes its fracture type under

For deriving sufficient fracture criteria (Kornev, 2004, 2010) for low-cycle fatigue, modification of the classical Leonov-Panasyk-Dugdale model (Kornev, 2004) is used, where

originated in the form of a "tooth" (marked by circle in Fig 11.d).

**4. Comparison between test results and theoretical notions** 

experimental data described in the previous section make up this deficiency.

Fig. 11. Cracks and pre-fracture zones

inelastic strain.

the pre-fracture zone is a rectangle ahead of the crack tip. The modification of the classical Leonov-Panasyk-Dugdale model allowed one to describe not only the pre-fracture zone length Δ1σ <sup>+</sup> at every loading cycle, but a magnitude of inelastic strain under tension σ ε <sup>+</sup> for material of the pre-fracture zone fiber nearest to the macrocrack center

$$
\varepsilon\_{\sigma+} = \frac{1 - \frac{\sigma\_{m1}}{\sigma\_a} \sqrt{\frac{r\_1}{2l}} \frac{k\_1}{\sqrt{h\_1}}}{\frac{5}{\pi(\eta+1)} \frac{G\_1}{\sigma\_{m1}}},
$$

$$
\frac{\Lambda\_{1\sigma+}}{r\_1} = \frac{k\_1^2}{2n\_1} \frac{\left(\frac{5\sqrt{2}}{4(\eta+1)} \frac{G\_1}{\sigma\_{m1}} \varepsilon\_{\sigma+}\right)^2}{\left(1 - \frac{5}{\pi(\eta+1)} \frac{G\_1}{\sigma\_{m1}} \varepsilon\_{\sigma+}\right)^2} \tag{1}
$$

Here σ *<sup>a</sup>* is the amplitude of pulse loading; σ *<sup>m</sup>*1 is the limit of elasticity; *n*1 and 1 *k* are integers ( 1 11 *n kk* ≥ , is the number of damage-free material fibers); *n r*1 1 is the averaging interval for the first material; 1*r* is the specific linear dimension of the first material structure; 0 2*l* and 0 1 22 2 *l l* = +Δ σ <sup>+</sup> are lengths of initial and fictitious cracks, respectively; *G*1 is the shear modulus of fibers; η = 3 4 − μ and η = (3 1 − + μ μ ) ( ) are coefficients for plane strain and plane stress state, respectively, where μ is the Poisson ratio; for relations (1), the restriction 1 5 − *G*ε πσ η σ<sup>+</sup> ( *<sup>m</sup>*<sup>1</sup> ( + > 1 0 )) holds.

Under cyclic pulse loading conditions, when the scheme of three-point bending is used, hysteresis loops take the form given in Fig. 12.

These hysteresis loops with translation differ from the standard statement in the model in (Kornev, 2004, 2010), in which the scheme of rigid loading under unloading is accepted. In Fig. 12, ( ) 1 *i* ε is the limit elongation of original materials for 0 *i* = ; that after the first inelastic strain is 1 *i* = ; that after the second inelastic strain is 2 *i* = , and etc. First three loops are depicted with lines widen from one loop to another, and the onset of the fourth loop is depicted with dots.

Fig. 12. Scheme of material damage

Interrelation Between Failure and

relations

2004, 2010).

**5. Conclusion** 

may withstand.

χ= 1 or

refined hypothesis 1

the pre-fracture zone Δ<sup>1</sup>

of beams after unloading.

χ

χ>

layer interface at the distance 1 1

χ

σ

σ

structure with the crack and in analyzing situations after overloading.

second and third stages of the Paris curve.

Damage Accumulation in the Pre-Fracture Zone Under Low-Cycle Loading 421

is the parameter, describing inelastic (residual) deflection of beams. In experiments, the residual deflection of beams characterizes both advance of the real crack tip and development of the pre-fracture zone after each loading cycle. The experimental results obtained agree with the diagram of fatigue failure with various kinds of relief elements formed as a result of fatigue fracture seen in Fig. 3.23 from (Shaniavski, 2003), just as the theory described in (Kornev, 2004, 2010), so the experimental data being correlated with the

The model in (Kornev, 2004, 2010) gives no preference to linear or nonlinear summation of damages in metals (Romaniv et al., 1990; Coffin & Schenectady, 1954) under cyclic loading. The performed tests show that damage accumulation, in general, is always nonlinear, however, if the loading is performed for *P*max near to the limit of the specimen plasticity, there exists, as a rule, the large interval on the axis of the number of cycles *N*, where the *δw* value is close to its minimum for given *P*maxvalue. Within this interval, the damage accumulation may be considered to be linear as it is under non-stationary loading with *P*max values close to the limit of plasticity. In both cases, the model proposed in (Kornev, 2004, 2010) and the obtained experimental results agree well. In this case, the hypothesis concerning a crack arrest holds (Kornev, 2004, 2010): a crack arrests at the interface between a fiber and a layer and then it is blunted. This layer is the first one located beyond a pre-fracture zone

∆1σ+, and the zone itself is located in material not subjected to some plastic strain.

σ

When a loading regime with increased *P*max is considered (this regime models catastrophic overload of a structure with crack), the model like that in (Kornev, 2004, 2010) is proposed, and the experimental results obtained agree essentially worse. Such disagreement of theoretical notions (Kornev, 2004, 2010) with the experimental results is explained by the

damage accumulation process in material significantly increases. In this loading regime, the

some plastic strain and embrittlement (Romaniv et al., 1990; Laird & Smith, 1962; Kornev,

Recording the loading diagrams for beams with the edge crack allows description of both damage accumulation at the macroscale and failure of constructions at every loading cycle. The damage accumulation in the pre-fracture zone is associated with the residual deflection

Preliminary plastic deformation of aluminum alloy leads to decrease in its durability. However, here the intensity of damage accumulation also reduces under low-cycle threepoint bending of beams. This results in increase in the number of loads, which structure

In the chapter, loading regimes with gradual overloading and with increased loading have been considered. These regimes model a common situation and catastrophic overloading under cyclic loading. The results obtained may be useful for prediction of fatigue life of a

Direct observation in the vicinity of the notch tip during the process of low-cycle tension of a plain specimen allows one to trace the behavior of a fatigue crack at various stages of its

> 1 . After the first catastrophic series of loadings, the intensity of

> 1 concerning crack arrest holds: a crack arrests at the fiber-

<sup>+</sup> , and the last is in material that has already been subjected to

<sup>∗</sup> Δ >Δ <sup>+</sup> <sup>+</sup> and it is blunted, this interlayer is located beyond

Consider a typical situation. Let failure occur at the fourth loading cycle. During the strain process, the material elongations at the real crack tip coincide with the limit material elongation (3) 1 ε after the third inelastic strain of the fiber nearest to the real crack tip (Fig. 12), and the crack tip being advanced.

Recall that the crack length in the model in (Kornev, 2004, 2010) changes by the pre-fracture zone length Δ1σ <sup>+</sup> after the step-wise real crack tip advance, and under the repeated loading, materials in the pre-fracture zone become brittle (Romaniv et al., 1990; Laird & Smith, 1962; Kornev, 2004, 2010). Each such advance is associated with the certain number of cycles when linear and non-linear damages are summed. The performed tests show that the initial state of material, which falls into the pre-fracture zone, influences the process of the step-wise crack tip advance. The magnitude of inelastic strain under stretching σ ε <sup>+</sup> and the prefracture zone length Δ1σ <sup>+</sup> depend on: *i*) load amplitude σ *<sup>a</sup>* ; *ii*) initial crack length *2l*; and *iii*) preliminary inelastic strain of material. If the basic parameter of inelastic strain σ ε <sup>+</sup> slightly depends on *2l*, then for the pre-fracture length Δ<sup>1</sup>σ <sup>+</sup> , the analogous dependence is pronounced. This corresponds to the passage of the second fracture process to the third one on the Paris curve (Romaniv et al., 1990; Shaniavski, 2003).

Fig. 13. Schematic drawing of material damage

As explained the step-wise advance of the real crack tip by the length 1 1 σ χ σ <sup>∗</sup> Δ =Δ <sup>+</sup> <sup>+</sup> , where χ is such a coefficient that for quasi-ductile layers χ = 1 , and for quasi-brittle layers χ > 1 . The most important question is as follows: what is a distance from the crack onset where the crack will arrest? In the model in (Kornev, 2004, 2010), the case of original material loading is considered when the layers possess the quasi-ductile fracture type. Because of this, the ratio χ = 1 holds. After preliminary inelastic strain of a material from which specimens have been made, embrittlement of layer materials occurs, and the ratio χ > 1 can hold. The step-wise crack tip advance in composite materials with quasi-ductile and embrittled layers elucidates the schematic drawing in Fig. 13. At the scheme top, the right-hand tip of a blunted crack is shown before its start. In the middle part of the scheme, the crack tip advances by the segment Δ<sup>1</sup>σ <sup>+</sup> , then the crack is blunted at the mesoscale, see the ratio 1 χ = . At the scheme bottom, the crack tip advances by the segment χΔ1σ <sup>+</sup> , and then the crack is blunted at the mesoscale, see the ratio 1 χ> .

In essence, just as in the model in (Kornev, 2004, 2010), so in foregoing experimental results, the distinct influence of the single parameter on the process of damage accumulation under cyclic pulse loading is traced. In the model in (Kornev, 2004, 2010), it is the parameter that characterizes inelastic strain of material in the pre-fracture zone σε<sup>+</sup> , and in experiments, it

is the parameter, describing inelastic (residual) deflection of beams. In experiments, the residual deflection of beams characterizes both advance of the real crack tip and development of the pre-fracture zone after each loading cycle. The experimental results obtained agree with the diagram of fatigue failure with various kinds of relief elements formed as a result of fatigue fracture seen in Fig. 3.23 from (Shaniavski, 2003), just as the theory described in (Kornev, 2004, 2010), so the experimental data being correlated with the second and third stages of the Paris curve.

The model in (Kornev, 2004, 2010) gives no preference to linear or nonlinear summation of damages in metals (Romaniv et al., 1990; Coffin & Schenectady, 1954) under cyclic loading. The performed tests show that damage accumulation, in general, is always nonlinear, however, if the loading is performed for *P*max near to the limit of the specimen plasticity, there exists, as a rule, the large interval on the axis of the number of cycles *N*, where the *δw* value is close to its minimum for given *P*maxvalue. Within this interval, the damage accumulation may be considered to be linear as it is under non-stationary loading with *P*max values close to the limit of plasticity. In both cases, the model proposed in (Kornev, 2004, 2010) and the obtained experimental results agree well. In this case, the hypothesis concerning a crack arrest holds (Kornev, 2004, 2010): a crack arrests at the interface between a fiber and a layer and then it is blunted. This layer is the first one located beyond a pre-fracture zone ∆1σ+, and the zone itself is located in material not subjected to some plastic strain.

When a loading regime with increased *P*max is considered (this regime models catastrophic overload of a structure with crack), the model like that in (Kornev, 2004, 2010) is proposed, and the experimental results obtained agree essentially worse. Such disagreement of theoretical notions (Kornev, 2004, 2010) with the experimental results is explained by the relations χ = 1 or χ > 1 . After the first catastrophic series of loadings, the intensity of damage accumulation process in material significantly increases. In this loading regime, the refined hypothesis 1 χ > χ > 1 concerning crack arrest holds: a crack arrests at the fiberlayer interface at the distance 1 1 σ σ <sup>∗</sup> Δ >Δ <sup>+</sup> <sup>+</sup> and it is blunted, this interlayer is located beyond the pre-fracture zone Δ<sup>1</sup>σ <sup>+</sup> , and the last is in material that has already been subjected to some plastic strain and embrittlement (Romaniv et al., 1990; Laird & Smith, 1962; Kornev, 2004, 2010).
