**3. Development of fatigue crack under low-cycle tension conditions of a plain specimen with edge notch**

Tests on a plain specimen loaded in tension with the narrow edge notch were conducted. During the test, direct observation of fatigue crack propagation was performed using a digitized microscope with the resolution of 22500 pixel/mm2. The testing scheme is shown in Fig. 6. The field of view is outlined by dashed line.

Fig. 6. Test scheme

412 Recent Trends in Processing and Degradation of Aluminium Alloys

tests show that this ratio for material is constant when different schemes of stationary and non-stationary loading are applied (Kornev et al., 2010). This allows one to use this ratio for comparison of results obtained on specimens with various geometrical dimensions and for

Preliminary inelastic strain of a material, from which the specimens have been made, essentially influences material resistance to cyclic fracture. As an example, Fig. 5 displays the experimental diagrams with curves of *w* versus *P* (Fig. 5 (a)) and the *δw*(*N*) diagram (Fig. 5 (b)) for D16Т duralumin with various degrees of preliminary stretching: diagram **1** for original materials, diagram **2** for materials stretched by 5%, and diagram **3** for materials stretched by 10%. All the specimens were loaded for the same *P*max value, but in Fig. 5 (a), diagrams for three tests are displaced, for convenience, from each other along the horizontal

Fig. 5. (a), (b). Low-cycle loading of aluminum alloy after preliminary plastic deformation

In Fig. 5(b), the area under the curve characterizes the limit deflection \*\* *w* , which decreases as preliminary stretching increases. However, the decrease in \*\* *w* is followed by the

*w* . This leads to increase of the limiting number of loading cycles.

Comparison of tests conducted on beams with notches of different depths has shown that if

*w N*( ) diagrams have been plotted for some notch depth *l*, the diagrams can be used to

plotted for the notch depth 1*l* , and for the notch depth 2*l* there is the only diagram with curves of *w* versus *P* for single loading. The value of maximum applied force *P P* max 1 =

cyclic loading for this notch depth being \*\* \*\* *w w*= <sup>1</sup> , and the limit deflection of the single loading being \* \* *w w*= <sup>1</sup> . In this case, the curve of damage accumulation *f*<sup>1</sup> (*N*) for *P P* max 1 <sup>=</sup> , <sup>1</sup> *l l* = can be used to obtain the curve *f*<sup>2</sup> (*N*) for *P P* max 2 = , 2 *l l* = . Here *P*2 is the force for

*w w* <sup>=</sup> , where \*\* \*\* \*\* \*

2 1 \*\* 1 2 \*\* \* 2 21 1 *w w ww w ww w* =⇒= .

which a specimen with the notch depth 2*l* has the deflection *w w*= 2 such that

δ

*l* under single loading, the limit deflection of

*w N*( ) diagrams have been

different loading regimes. For duralumin, we have \* \*\* *w w*≥ .

**(a) (b)**

**2.4 Preliminary inelastic strain** 

axis.

for D16T alloy

decrease in

δ

δ

**2.5 Variation in the notch depth** 

corresponds to some deflection *w w*= 1 for 1

obtain analogous diagrams for other *l* values. Assume that

1 2 \*\* \*\* 1 2 *w w*

The specimens were made from D16T alloy preliminary heat-treated at 500ºС to give more plastic material. Plain specimens with notches of different lengths (1–3 mm) were used. The minimum load was the same in every cycle, the maximum load was chosen such that three different loading types were provided: *i*) near yield strength, *ii*) near the limit of load capacity, *iii*) the average value of them.

Photographs in Fig. 7 illustrate stages of crack propagation near crack-like defect in two cases *i*) continuous tension with constant rate, *ii*) low-cycle tension.

In the case of continuous tension, the following stages may be observed. First, intense plastic deformation ahead of the notch tip occurs and two zones of strain localization are formed with a delta-shaped area between these zones. The area we term as a pre-fracture zone since further it will define the crack extension direction. At this stage, several focuses of fatigue crack initiation are formed, which are located at notch angles as it is seen in the photograph (a.1). The pre-fracture zone is formed not by a prospective crack, but by the notch itself and its shape is unclear. Further (a.2), one of microcracks develops as a crack propagating within the zone of plastic strain localization irrespective of the pre-fracture zone specified by the notch. This zone becomes more structured and its tip is separated from the crack tip. Here the pre-fracture zone tip starts to shift towards the developing crack. At the next stage (a.3), crack branching takes place, the branches being formed just as near the crack, so at its faces. This evidences the significant extent of material embrittlement in the vicinity of crack extension. The angle at the pre-fracture zone tip starts to decrease. Then (a.4) the branch nearest to the pre-fracture zone tip has some advantages and defines the final direction of crack extension. When the crack tip joins the pre-fracture zone tip (a.5), the critical state is achieved after which the crack starts to extend very fast. The final failure of a specimen is preceded by a short stage (a.6), at which the angle at the pre-fracture zone crack becomes similar to the crack opening angle and one of pre-fracture zone edges defines a path of the subsequent crack extension unambiguously. The crack is very short before the critical state: its length is less than the notch width.

Under repeated low-cycle loading conditions, the pre-fracture zone created by a notch plays no noticeable role especially in the cases when cycle loading starts at insignificant plastic

Interrelation Between Failure and

length.

notch tip.

its appearance.

given. At the right, (*P-*

ε

heterogeneity being leveled out at its further propagation.

Here a crack at the notch 3 mm in length is shown.

Damage Accumulation in the Pre-Fracture Zone Under Low-Cycle Loading 415

deformations of a specimen. In this case, zones of localization of maximum plastic strains at notch angles are slightly structured and the crack can start to develop from any point of the front bound of the notch (b.1). When one of microcracks starts to extend, a delta-shaped prefracture zone is formed ahead of the microcrack (b.2). When the crack is short, the length of the pre-fracture zone is defined by that of the edge notch. The angle between whiskers in the pre-fracture zone tip is close to the right angle, and visible sizes of this zone are of the order of surface roughness occurring due to plastic flow of material. Then, at some distance from the front edge of the notch, temporal crack arrest takes place in all the considered cases. Keeping constant length, the crack begins to open at the expense of blunting and increasing the length of its front edge, and then it changes both the direction and rate of propagation. The path, which the crack takes, deviates from the mean direction of its propagation and the former coincides with the upper or lower edge of the pre-fracture zone. After this, the crack returns to the mean path of propagation. The "tooth" formed at a crack face is clearly seen (b.3). Then the rate of crack growth is steadily increased, crack opening being continued. The pre-fracture zone length increases (b.4). The next to last stage is characterized by continuous growth of visible sizes of the pre-fracture zone and by decrease of the angle between whiskers (b.5). At last, the critical state is achieved (b.6): the crack is blunted, significant crack opening takes place that is comparable with the original notch width, then the crack produces a path branch sharply deviated from the mean path of propagation and then final failure of material occurs after which loading must be ceased. As opposed to continuous loading, the crack length to the instant of the critical state is close to the notch

Given in Fig. 8 are plots characterizing crack propagation as a function of the number of cycles. At the left, some geometric characteristics of cracks versus the number of cycles are

overall elongation of a specimen. Both minimum and maximum loads of loading cycles are shown on these plots, as well as the maximum strain on the first loading cycle. The value of maximum force in a cycle was chosen such that to provide areas of plastic strain ahead the

Plots a.1 and a.2, b.1. and b.2, and c.1. and c.2 correspond to specimens with the notch of 1 mm, 2 mm and 3 mm in length, respectively. Curves 1, 2, 3 and 4 on the left plots correspond to the overall length of crack along its face, the distance of the crack tip from the front notch edge, the difference between these values, and the notch width increase of which shows opening of the crack mouth, respectively. All these values demonstrate nonlinear growth of cracks for which hyperbolic functions are applicable for description of both this growth and damage accumulation at the notch tip in the case of three-point bending of a beam. The general structure of material undoubtedly influences crack propagation as it can be seen on plots in Fig. 8 (a.1, b.1, and c.1), deviations of a crack due to structural

Fig. 9. illustrates separate parts of a crack corresponding to different stages of cyclic loading.

It seems likely that origination of a "tooth" is not a consequence of a heterogeneous structure of the material, but this is associated with sizes of defects from which a fatigue cracks start to propagate. In this case, a residual durability of the structure can be defined by

Let us discuss interaction between plasticity zones formed in the vicinity of the notch tip and propagating cracks under single and cyclic loading conditions. Fig. 11 shows: a) schemes of zones of plasticity localization in the vicinity of the notch tip (region I) with

) plots of specimens are given, where Р is applied force, ε is the

Fig. 7. Stages of crack propagation for D16T alloy; a.1. occurrence of incipient cracks under single loading conditions; b.1. the same under low-cycle loading conditions; a.2. growth of one of incipient cracks under single loading conditions; b.2. the same under low-cycle loading conditions; a.3. crack branching and moving of pre-fracture zone of a notch close together with the crack tip under single loading conditions; b.3. occurrence of "tooth" under low-cycle loading conditions; a.4. development of crack branch nearest to the pre-fracture zone of a notch under single loading conditions; b.4. growth of a crack beyond pre-fracture zone of a notch, formation of the proper pre-fracture zone for a crack under low-cycle loading conditions; a.5. merging pre-fracture zones of the crack and notch under single loading conditions; b.5. final stage of crack development under low-cycle loading conditions characterized by continuous growth of visible sizes of the pre-fracture zone and by decrease of the angle between whiskers; a.6. crack in critical state followed by fast final fracture under single loading conditions; b.6. the same under low-cycle loading conditions

Fig. 7. Stages of crack propagation for D16T alloy; a.1. occurrence of incipient cracks under single loading conditions; b.1. the same under low-cycle loading conditions; a.2. growth of one of incipient cracks under single loading conditions; b.2. the same under low-cycle loading conditions; a.3. crack branching and moving of pre-fracture zone of a notch close together with the crack tip under single loading conditions; b.3. occurrence of "tooth" under low-cycle loading conditions; a.4. development of crack branch nearest to the pre-fracture zone of a notch under single loading conditions; b.4. growth of a crack beyond pre-fracture zone of a notch, formation of the proper pre-fracture zone for a crack under low-cycle loading conditions; a.5. merging pre-fracture zones of the crack and notch under single loading conditions; b.5. final stage of crack development under low-cycle loading conditions characterized by continuous growth of visible sizes of the pre-fracture zone and by decrease of the angle between whiskers; a.6. crack in critical state followed by fast final fracture under

single loading conditions; b.6. the same under low-cycle loading conditions

deformations of a specimen. In this case, zones of localization of maximum plastic strains at notch angles are slightly structured and the crack can start to develop from any point of the front bound of the notch (b.1). When one of microcracks starts to extend, a delta-shaped prefracture zone is formed ahead of the microcrack (b.2). When the crack is short, the length of the pre-fracture zone is defined by that of the edge notch. The angle between whiskers in the pre-fracture zone tip is close to the right angle, and visible sizes of this zone are of the order of surface roughness occurring due to plastic flow of material. Then, at some distance from the front edge of the notch, temporal crack arrest takes place in all the considered cases. Keeping constant length, the crack begins to open at the expense of blunting and increasing the length of its front edge, and then it changes both the direction and rate of propagation. The path, which the crack takes, deviates from the mean direction of its propagation and the former coincides with the upper or lower edge of the pre-fracture zone. After this, the crack returns to the mean path of propagation. The "tooth" formed at a crack face is clearly seen (b.3). Then the rate of crack growth is steadily increased, crack opening being continued. The pre-fracture zone length increases (b.4). The next to last stage is characterized by continuous growth of visible sizes of the pre-fracture zone and by decrease of the angle between whiskers (b.5). At last, the critical state is achieved (b.6): the crack is blunted, significant crack opening takes place that is comparable with the original notch width, then the crack produces a path branch sharply deviated from the mean path of propagation and then final failure of material occurs after which loading must be ceased. As opposed to continuous loading, the crack length to the instant of the critical state is close to the notch length.

Given in Fig. 8 are plots characterizing crack propagation as a function of the number of cycles. At the left, some geometric characteristics of cracks versus the number of cycles are given. At the right, (*P*ε) plots of specimens are given, where Р is applied force, ε is the overall elongation of a specimen. Both minimum and maximum loads of loading cycles are shown on these plots, as well as the maximum strain on the first loading cycle. The value of maximum force in a cycle was chosen such that to provide areas of plastic strain ahead the notch tip.

Plots a.1 and a.2, b.1. and b.2, and c.1. and c.2 correspond to specimens with the notch of 1 mm, 2 mm and 3 mm in length, respectively. Curves 1, 2, 3 and 4 on the left plots correspond to the overall length of crack along its face, the distance of the crack tip from the front notch edge, the difference between these values, and the notch width increase of which shows opening of the crack mouth, respectively. All these values demonstrate nonlinear growth of cracks for which hyperbolic functions are applicable for description of both this growth and damage accumulation at the notch tip in the case of three-point bending of a beam. The general structure of material undoubtedly influences crack propagation as it can be seen on plots in Fig. 8 (a.1, b.1, and c.1), deviations of a crack due to structural heterogeneity being leveled out at its further propagation.

Fig. 9. illustrates separate parts of a crack corresponding to different stages of cyclic loading. Here a crack at the notch 3 mm in length is shown.

It seems likely that origination of a "tooth" is not a consequence of a heterogeneous structure of the material, but this is associated with sizes of defects from which a fatigue cracks start to propagate. In this case, a residual durability of the structure can be defined by its appearance.

Let us discuss interaction between plasticity zones formed in the vicinity of the notch tip and propagating cracks under single and cyclic loading conditions. Fig. 11 shows: a) schemes of zones of plasticity localization in the vicinity of the notch tip (region I) with

Interrelation Between Failure and

nonlinearity (region IV)

Damage Accumulation in the Pre-Fracture Zone Under Low-Cycle Loading 417

distance to a "tooth" is close to one third of the overall length of each crack. The segment on the horizontal axis in Fig. 8 is assigned to origination of a "tooth". This segment is nearly the same for all the cracks with respect to the total number of cycles. These segments

Fig. 9. Propagation of crack in pre-fracture zone for D16T alloy. Given at the top are the numbers of loading cycles, dashed line at the bottom outlines the visible area of maximum plastic strains. This area is divided into four regions: a crack initiates and propagates from the beginning in the closed state, and then it takes the mean propagation path (region I); the stage of stable crack propagation before a "tooth" arises (region II); stable crack propagation after "tooth" proceeds with increasing rate and continuous growth of the pre-fracture zone (region III); the final stage of crack propagations that is characterized on plots by clear

Fig. 10. Cracks and pre-fracture zones for various notch lengths

correspond to about 600, 1100 and 4800 cycles, respectively, on a.1, b.1, c.1.

features near angles (regions II and III), b) schemes of crack initiation under single loading conditions, c) scheme of crack initiation under cyclic loading conditions, d) scheme of fatigue crack propagation with origination of a "tooth" at the boundary of the zone of plasticity localization for a notch. Only in the case when load capacity is exhausted, initiation of cracks on the front notch edge at several points takes place: in original material under single loading conditions (Fig. 11.b), in material accumulated damages under fatigue loading (Fig. 11.c).

Fig. 8. Characteristics of fatigue crack propagation for D16T alloy

Given in Fig. 10 are forms of three cracks for notches of various lengths (notch lengths in mm are shown inside contours of final pre-fracture zones). Here it is seen that a "tooth" is inherent to all the cracks, the distances from the crack onsets to "teeth" are in direct proportion to lengths of notches. This "tooth" can be identified with pronounced fatigue striations originated on crack faces. In Fig. 10, all three "teeth" are marked with circles. The

features near angles (regions II and III), b) schemes of crack initiation under single loading conditions, c) scheme of crack initiation under cyclic loading conditions, d) scheme of fatigue crack propagation with origination of a "tooth" at the boundary of the zone of plasticity localization for a notch. Only in the case when load capacity is exhausted, initiation of cracks on the front notch edge at several points takes place: in original material under single loading conditions (Fig. 11.b), in material accumulated damages under fatigue

Fig. 8. Characteristics of fatigue crack propagation for D16T alloy

Given in Fig. 10 are forms of three cracks for notches of various lengths (notch lengths in mm are shown inside contours of final pre-fracture zones). Here it is seen that a "tooth" is inherent to all the cracks, the distances from the crack onsets to "teeth" are in direct proportion to lengths of notches. This "tooth" can be identified with pronounced fatigue striations originated on crack faces. In Fig. 10, all three "teeth" are marked with circles. The

loading (Fig. 11.c).

distance to a "tooth" is close to one third of the overall length of each crack. The segment on the horizontal axis in Fig. 8 is assigned to origination of a "tooth". This segment is nearly the same for all the cracks with respect to the total number of cycles. These segments correspond to about 600, 1100 and 4800 cycles, respectively, on a.1, b.1, c.1.

Fig. 9. Propagation of crack in pre-fracture zone for D16T alloy. Given at the top are the numbers of loading cycles, dashed line at the bottom outlines the visible area of maximum plastic strains. This area is divided into four regions: a crack initiates and propagates from the beginning in the closed state, and then it takes the mean propagation path (region I); the stage of stable crack propagation before a "tooth" arises (region II); stable crack propagation after "tooth" proceeds with increasing rate and continuous growth of the pre-fracture zone (region III); the final stage of crack propagations that is characterized on plots by clear nonlinearity (region IV)

Fig. 10. Cracks and pre-fracture zones for various notch lengths

Interrelation Between Failure and

length Δ1

Here σ

Fig. 12, ( ) 1 *i* ε

depicted with dots.

σ

Damage Accumulation in the Pre-Fracture Zone Under Low-Cycle Loading 419

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

<sup>+</sup> at every loading cycle, but a magnitude of inelastic strain under tension

( )

+

( )

5 2 4 1

η

<sup>Δ</sup> <sup>+</sup> ⎝ ⎠ <sup>=</sup>

πη

σ

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

> μ and η

Under cyclic pulse loading conditions, when the scheme of three-point bending is used,

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

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

is the limit elongation of original materials for 0 *i* = ; that after the first inelastic

( )

1

⎛ ⎞ ⎜ ⎟ <sup>−</sup> <sup>+</sup> ⎝ ⎠

 σ

⎛ ⎞ ⎜ ⎟

5 1

*m a*

σ

πη

1 1 <sup>1</sup>

<sup>2</sup> <sup>5</sup> <sup>1</sup>

*r n G*

σ

1 1 1

 σ

*r k l n G*

2

1 1 1

,

2

2

<sup>+</sup> are lengths of initial and fictitious cracks, respectively;

 = (3 1 − + μ

μ

*<sup>m</sup>*1 is the limit of elasticity; *n*1 and 1 *k* are

 μ

*m*

*G*

1

*m*

 σ

1

σ

+

ε

σ

+

ε

*m*

material of the pre-fracture zone fiber nearest to the macrocrack center

σ

+

=

2 1 1 1

*k*

η= 3 4 −

<sup>+</sup> ( *<sup>m</sup>*<sup>1</sup> ( + > 1 0 )) holds.

σ

*<sup>a</sup>* is the amplitude of pulse loading;

plane strain and plane stress state, respectively, where

 πσ η

ε

hysteresis loops take the form given in Fig. 12.

σ

structure; 0 2*l* and 0 1 22 2 *l l* = +Δ

(1), the restriction 1 5 − *G*

*G*1 is the shear modulus of fibers;

Fig. 12. Scheme of material damage

+

σ

ε

1

−

σ ε<sup>+</sup> for

(1)

) ( ) are coefficients for

is the Poisson ratio; for relations

Fig. 11. Cracks and pre-fracture zones

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