On the Concentration Criterion of Fracture
Изучена кинетика накопления микротрещин на различных стадиях циклического и статического нагружения малоуглеродистой стали, а также сталей 45 и 12Х18Н9Т в процессе растяжения. Оценены концентрационный критерий (k-параметр), характеризующий начало процесса слияния микротрещин и формирования макротрещ...
Збережено в:
| Дата: | 2017 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут металофізики ім. Г.В. Курдюмова НАН України
2017
|
| Назва видання: | Металлофизика и новейшие технологии |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/125480 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the Concentration Criterion of Fracture / L.R. Botvina, A.P. Soldatenkov // Металлофизика и новейшие технологии. — 2017. — Т. 39, № 4. — С. 477-490. — Бібліогр.: 22 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-125480 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1254802017-10-29T03:02:46Z On the Concentration Criterion of Fracture Botvina, L.R. Soldatenkov, A.P. Физика прочности и пластичности Изучена кинетика накопления микротрещин на различных стадиях циклического и статического нагружения малоуглеродистой стали, а также сталей 45 и 12Х18Н9Т в процессе растяжения. Оценены концентрационный критерий (k-параметр), характеризующий начало процесса слияния микротрещин и формирования макротрещины, а также характеристики кумулятивных кривых распределения микротрещин по длине. Показано, что k-параметр связан степенной зависимостью с остаточной долговечностью материала. Рассмотрены некоторые общие закономерности процессов взаимодействия дефектов, проявляющиеся при разрушении металлических образцов. Вивчено кінетику накопичення мікротріщин на різних стадіях циклічного та статичного навантаження маловуглецевої сталі, а також сталей 45 і 12Х18Н9Т у процесі розтягання. Оцінено концентраційний критерій (k-параметр), що характеризує початок процесу злиття мікротріщин і формування макротріщини, а також характеристики кумулятивних кривих розподілу мікротріщин по довжині. Показано, що k-параметр пов’язаний степеневою залежністю із залишковою довговічністю матеріялу. Розглянуто деякі загальні закономірності процесів взаємодії дефектів, що проявляються при руйнуванні металічних зразків. The kinetics of microcracks’ accumulation at various stages of cyclic and static loading of specimens of the low-carbon grade 20, medium-carbon grade 45, and stainless 12Kh18N10T steels is studied. Concentration criterion of fracture (k-criterion) characterizing the initiation of process of microcracks’ coalescence and macrocrack formation as well as the characteristics of cumulative microcracks’ distribution over their lengths are estimated. As shown, the k-criterion is associated with residual lifetime of a material by means of the power-law dependence. Some general regularities of defect interaction manifesting themselves at fracture of metal specimens are considered. 2017 Article On the Concentration Criterion of Fracture / L.R. Botvina, A.P. Soldatenkov // Металлофизика и новейшие технологии. — 2017. — Т. 39, № 4. — С. 477-490. — Бібліогр.: 22 назв. — англ. 1024-1809 DOI: 10.15407/mfint.39.04.0477 PACS: 62.20.mm, 62.20.mt, 75.78.Fg, 81.40.Np, 81.70.Bt, 81.70.Fy, 91.30.Px http://dspace.nbuv.gov.ua/handle/123456789/125480 en Металлофизика и новейшие технологии Інститут металофізики ім. Г.В. Курдюмова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Физика прочности и пластичности Физика прочности и пластичности |
| spellingShingle |
Физика прочности и пластичности Физика прочности и пластичности Botvina, L.R. Soldatenkov, A.P. On the Concentration Criterion of Fracture Металлофизика и новейшие технологии |
| description |
Изучена кинетика накопления микротрещин на различных стадиях циклического и статического нагружения малоуглеродистой стали, а также сталей 45 и 12Х18Н9Т в процессе растяжения. Оценены концентрационный критерий (k-параметр), характеризующий начало процесса слияния микротрещин и формирования макротрещины, а также характеристики кумулятивных кривых распределения микротрещин по длине. Показано, что k-параметр связан степенной зависимостью с остаточной долговечностью материала. Рассмотрены некоторые общие закономерности процессов взаимодействия дефектов, проявляющиеся при разрушении металлических образцов. |
| format |
Article |
| author |
Botvina, L.R. Soldatenkov, A.P. |
| author_facet |
Botvina, L.R. Soldatenkov, A.P. |
| author_sort |
Botvina, L.R. |
| title |
On the Concentration Criterion of Fracture |
| title_short |
On the Concentration Criterion of Fracture |
| title_full |
On the Concentration Criterion of Fracture |
| title_fullStr |
On the Concentration Criterion of Fracture |
| title_full_unstemmed |
On the Concentration Criterion of Fracture |
| title_sort |
on the concentration criterion of fracture |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| publishDate |
2017 |
| topic_facet |
Физика прочности и пластичности |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/125480 |
| citation_txt |
On the Concentration Criterion of Fracture / L.R. Botvina, A.P. Soldatenkov // Металлофизика и новейшие технологии. — 2017. — Т. 39, № 4. — С. 477-490. — Бібліогр.: 22 назв. — англ. |
| series |
Металлофизика и новейшие технологии |
| work_keys_str_mv |
AT botvinalr ontheconcentrationcriterionoffracture AT soldatenkovap ontheconcentrationcriterionoffracture |
| first_indexed |
2025-07-09T03:14:02Z |
| last_indexed |
2025-07-09T03:14:02Z |
| _version_ |
1837137497742639104 |
| fulltext |
477
PACS numbers: 62.20.mm, 62.20.mt, 75.78.Fg, 81.40.Np, 81.70.Bt, 81.70.Fy, 91.30.Px
On the Concentration Criterion of Fracture
L. R. Botvina and A. P. Soldatenkov
A. A. Baikov Institute of Metallurgy and Materials Science, R.A.S.,
49 Leninskii Ave.,
119334 Moscow, Russia
The kinetics of microcracks’ accumulation at various stages of cyclic and
static loading of specimens of the low-carbon grade 20, medium-carbon grade
45, and stainless 12Kh18N10T steels is studied. Concentration criterion of
fracture (k-criterion) characterizing the initiation of process of microcracks’
coalescence and macrocrack formation as well as the characteristics of cumu-
lative microcracks’ distribution over their lengths are estimated. As shown,
the k-criterion is associated with residual lifetime of a material by means of
the power-law dependence. Some general regularities of defect interaction
manifesting themselves at fracture of metal specimens are considered.
Key words: concentration criterion of fracture, damage accumulation, ten-
sion, microcracks’ density, fatigue, residual lifetime.
Вивчено кінетику накопичення мікротріщин на різних стадіях циклічно-
го та статичного навантаження маловуглецевої сталі, а також сталей 45 і
12Х18Н9Т у процесі розтягання. Оцінено концентраційний критерій (k-
параметр), що характеризує початок процесу злиття мікротріщин і фор-
мування макротріщини, а також характеристики кумулятивних кривих
розподілу мікротріщин по довжині. Показано, що k-параметр пов’язаний
степеневою залежністю із залишковою довговічністю матеріялу. Розгля-
нуто деякі загальні закономірності процесів взаємодії дефектів, що про-
являються при руйнуванні металічних зразків.
Ключові слова: концентраційний критерій руйнування, розтягання, на-
копичення пошкоджень, густина мікротріщин, утома, залишкова довго-
вічність.
Corresponding author: Lyudmila Rafailovna Botvina
E-mail: botvina@imet.ac.ru
Please cite this article as: L. R. Botvina and A. P. Soldatenkov, On the Concentration
Criterion of Fracture, Metallofiz. Noveishie Tekhnol., 39, No. 4: 477–490 (2017),
DOI: 10.15407/mfint.39.04.0477.
Ìåòàëëîôèç. íîâåéøèå òåõíîë. / Metallofiz. Noveishie Tekhnol.
2017, т. 39, № 4, сс. 477–490 / DOI: 10.15407/mfint.39.04.0477
Îòòèñêè äîñòóïíû íåïîñðåäñòâåííî îò èçäàòåëÿ
Ôîòîêîïèðîâàíèå ðàçðåøåíî òîëüêî
â ñîîòâåòñòâèè ñ ëèöåíçèåé
2017 ÈÌÔ (Èíñòèòóò ìåòàëëîôèçèêè
èì. Ã. Â. Êóðäþìîâà ÍÀÍ Óêðàèíû)
Íàïå÷àòàíî â Óêðàèíå.
478 L. R. BOTVINA and A. P. SOLDATENKOV
Изучена кинетика накопления микротрещин на различных стадиях цик-
лического и статического нагружения малоуглеродистой стали, а также
сталей 45 и 12Х18Н9Т в процессе растяжения. Оценены концентрацион-
ный критерий (k-параметр), характеризующий начало процесса слияния
микротрещин и формирования макротрещины, а также характеристики
кумулятивных кривых распределения микротрещин по длине. Показано,
что k-параметр связан степенной зависимостью с остаточной долговечно-
стью материала. Рассмотрены некоторые общие закономерности процес-
сов взаимодействия дефектов, проявляющиеся при разрушении металли-
ческих образцов.
Ключевые слова: концентрационный критерий разрушения, накопление
повреждений, растяжение, плотность микротрещин, усталость, остаточ-
ная долговечность.
(Received December 13, 2016; in final version, March 31, 2017)
1. INTRODUCTION
The process of material fracture at any type of loading begins with the
localization of plastic deformation near the structure inhomogeneity
and stress concentrators. This process leads to the formation of plastic
deformation zone, the defect accumulation in the zone and the subse-
quent initiation of macrocracks, whose growth regularities are deter-
mined by the fracture mechanics parameters. The development of this
science field in recent decades has led to an understanding of these regu-
larities under different loading conditions. However, regularities of
damage accumulation in the deformation localization zone remain in-
sufficiently studied, despite the importance and necessity of such stud-
ies related to the safe operation of structures and machine elements.
Significant contribution to the study of the regularities of mi-
crocracks accumulation process was made by researchers of Zhurkov’s
school. They obtained [1, 2] direct experimental evaluations of the ki-
netics of crack accumulation in polymers. The opalescence phenome-
non at specimen loading was discovered, due to the light scattering be-
cause of the submicroscopic discontinuities appearance in physical sys-
tem. These discontinuities are close to the light wavelength in size.
Concentration, size and shape of these discontinuities in amorphous
and crystalline polymers as well as the cracks’ accumulation rate,
which increases with the applied stress exponentially, were evaluated.
Before fracture, the submicrocracks’ concentration reached a critical
value (1015–1016
cm
3) that does not depend on the type of polymer and
the stress state, the applied stress and the specimen durability.
In addition, it was found that the average size of cracks at the max-
imum concentration becomes comparable to the distance between them
and the stress fields generated by these cracks overlap. It contributed
to cracks’ coalescence when reaching the maximum discontinuities’
ON THE CONCENTRATION CRITERION OF FRACTURE 479
concentration. These studies led to the establishment of an important
concentration parameter characterizing the damage accumulation pro-
cess and determined as:
3
av
1/( ),k L n (1)
where Lav is the average microcrack length, n is density.
To assess the criterion, which characterizes this process, the authors
used the computational results on the cracks’ interaction characteris-
tics obtained by V. V. Panasyuk and B. L. Lozovyi [3, 4]. As a result,
the interaction of two plane collinear cracks in an elastic medium be-
gins at the moment when the ratio (a/L) of the distance between the
cracks to their length becomes less than 3. Experimental evaluation of
criterion a/L (or R/L) in the polymer specimens coincided with the
predicted values. Thus, when a/L 3 cracks are isolated and do not af-
fect each other and when a/L 3, the cracks interact and coalesce. T. L.
Chelidze [5] obtained a similar criterion based on the percolation theo-
ry. Considering that in the three-dimensional case for the number N of
defects’ concentration places with a radius R of their influence sphere,
the connectivity of individual spheres and an infinite cluster for-
mation for these spheres occur in case when:
3
C C S
(4/3) 2.7 or / 1.4,NR R r (2)
where RC is the critical percolation radius, and
3
S
3 /4r N is the av-
erage distance between the origins of concentration N.
The authors [2, 6] have shown experimentally that, before the frac-
ture, the average normalized distance between the cracks is constant
and equal to:
rS/l const 4.5,
where l is the average distance between the cracks. From Eq. (2), it fol-
lows that rS/l RC/(1.4l).
It means the experimental Eq. (2) is similar to Eq. (1), and
RC/(1.4l) 4.5. Therefore, the critical percolation radius (or the inter-
action region of defects) is equal to 1.4rS.
In this paper, according to the data of the damage accumulation in
structural grade 20, grade 45, and 12Kh18N9T steels at various stages
of static and cyclic loading, the results of the damage characteristics
evaluation and k-criterion are considered.
2. MATERIALS AND RESEARCH METHODS
The damage development analysis was carried out on specimens of
480 L. R. BOTVINA and A. P. SOLDATENKOV
structural low-carbon grade 20, medium-carbon grade 45 and stainless
12Kh18N9T steels (in wt.%, 0.07 C, 18 Cr, 9.5 Ni, 0.35 Ni) [7–9],
which are widely used in various technique fields and possessing dif-
ferent strength and plastic properties. Mechanical properties are
shown in Table 1.
Tensile tests of standard smooth specimens (220 40 6 mm3
with the
working portion of the specimen 80 20 6 mm3) were performed on In-
stron 3382 tensile-testing machine at room temperature and the de-
formation rate was 2 mm/min. The obtained ‘load–displacement’ dia-
grams were used for estimation of the specific work of plastic defor-
mation (w) as the area under the deformation curves (without the elas-
tic part) divided into the volume of the working portion of the speci-
men.
The observations of lateral polished specimen surface by means of
the optical microscope ‘Olympus GX-51’ equipped with a digital video
camera were conducted at various tension stages. Using image analysis
software, the microcracks’ length (LC), their number ( NC), and the
relative area (S
*) occupied by them, i.e., the total area of microcracks
divided into the image area, were measured in the neck region of the
specimens. Crack length exceeding 0.5 m was measured because it
was difficult to distinguish the cracks of shorter length from the large
pores or pore chains. Based on the measurement results, damage accu-
mulation curves in the coordinates ‘the total number of cracks ( NC)
with length greater than the current length (LC)–the current crack
length’ were plotted, and the k-parameters were determined according
to the modified equation (1): av
1/( ),k L n using the square root of
the surface defect density instead of the cubic one, since the damage
analysis was performed on the specimen surface.
3. RESULTS OF THE STUDY
3.1. Damage Analysis of Low-Carbon Steel at Various Stages of Fatigue
Figure 1, a shows the microcracks’ patterns observed on the polished
surface of the specimens of low-carbon steel [10]. Using measurements
on the patterns, the cumulative microcracks’ number distributions on
TABLE 1. Mechanical properties of studied steels.
Steel grade , % 0.2, МPа ult, МPа
Grade 20 36.7 3.1 283 4.7 435.3 4.2
Grade 45 24.5 2.8 362.9 16.5 641.7 22.0
12Kh18N9T 71.3 3.1 234.2 4.1 593 6.1
ON THE CONCENTRATION CRITERION OF FRACTURE 481
their lengths (Fig. 1, b) [11, 12] corresponding to different relative
lifetime were plotted.
Apparently, the damage development leads to an increase in the
number of microcracks on the early fatigue stages (upper plateau of
the cumulative distribution curves increases) and their length on the
final stage because of the adjacent defects’ coalescence. At the relative
durability N/Nf 0.85 (i.e. the ratio of the number of loading cycles N
to the number of cycles Nf corresponding to the specimen fracture), the
microcracks start to interact, thereby forming a macrocrack. It has an
effect on the curves of cumulative damage (Fig. 1, b) described by the
exponential function
NC Cexp( cLC) (3)
at the early stages of fatigue fracture, and by the power-law one
Fig. 1. Fatigue microcracks’ patterns in a specimen of low-carbon steel (at
333 MPa, N/Nf 0.17, 0.43, 0.85, and 0.97) [10] (a), the cumulative mi-
crocracks’ number distributions over their length plotted on these patterns
[11–14] (b), and the dependences of k-parameter on the residual lifetime (bot-
tom axes) and c-exponent in Eq. (3) (top axes) plotted according to [10, 15] (c).
482 L. R. BOTVINA and A. P. SOLDATENKOV
C
C C
( )
b
N B L (4)
at the stage preceding the fracture; the exponents c and bC in these re-
lations decrease with a lifetime increasing; C and B are constants. On-
set of microcracks’ coalescence leads to a sharp almost three-fold de-
crease in k-parameter obeying the power-law dependence (black curve
in Fig. 1, c):
f
( / ) ,
dk A N N (5)
where A is constant. The rate of this reduction increases with the steel
degradation caused by the development of delamination on the bound-
aries of the structural elements [15]. As follows from the grey curve in
Fig. 1, c, the reduction of k-parameter is accompanied by a decrease in
c-exponent in (3).
3.2. Damage Analysis of Low-Carbon Grade 20 Steel at Various
Tension Stages
Figure 2 shows the microcracks’ patterns observed on the polished sur-
face of the specimen of low-carbon grade 20 steel at various tension
stages (a–c) [8] characterized by ratio of the current load P to the yield
load P0.2, and the k-parameter dependences on the damaged surface
portion (d) and c-exponent (e) in relation (3) plotted on these patterns.
Figure 2, d shows that the sharp decrease in k-parameter takes place
at the initial stage of discontinuities accumulation to 10% damage.
After reaching the value, the parameter varies little and is almost in-
dependent on the grain size and the specimen thickness. However, the
influence of specimen thickness affects the k-parameter dependence on
c-exponent in (3) (Fig. 2, e). In this case, the rate of increase in the k-
parameter for the 16 mm-thick specimens is higher than for 4 mm-
thick specimens. This one shows the parameter sensitivity to the mate-
rial local stress state, or rather, to the rise of the plastic strain con-
straint due to increase in the specimen thickness.
Plotting the cumulative microcracks’ number distributions on their
length in specimens of different thicknesses from grade 20 steel (Fig.
3) confirmed the previously noticed regularity at the analysis of cyclic
damage (Fig. 1, b) associated with the change of the distribution func-
tion during damage development. At the initial tension stages, the cu-
mulative microcracks’ distributions are described by a simple expo-
nential function, which changes by power-law function at the pre-
fracture stage because of microcracks’ coalescence and increase in the
number of long microcracks. The solid curves 1 and 2 in Fig. 3 corre-
spond to the Eqs. (3) and (4), respectively. The graphs show that the
increase in specimen thickness results in a reduction of maximum mi-
ON THE CONCENTRATION CRITERION OF FRACTURE 483
crocrack lengths on both early and late stages of damage development.
In addition, the analysis of accumulated damage curves led to the
conclusion [8] that the nature of approximating dependence is associ-
ated with a maximum length of microcracks. Therefore, at the stage of
separate noninteracting-defects’ accumulation with a maximum
length not exceeding the grain size (lmax d), the cumulative curves are
described by an exponential relation. When the maximum length of
microcracks becomes larger than grain size of about 3 times (lmax 3d),
the accumulated damage curves are most likely to be described by a
power-law function. In the region of intermediate microcrack lengths
varying in the range of (1–3)d, the cumulative curves are approximat-
ed with equal probability by both exponential and power-law depend-
ences.
In Refs. [7–9, 12], it was shown that the change in the cumulative
microcracks’ distributions is reflected in the similar change in the cu-
mulative amplitude distributions of acoustic emission (AE) signals,
which described by an exponential function at the beginning of loading
Fig. 2. Microcracks in a specimen with thickness of 4 mm of low-carbon grade
20 steel (grain size d 97 m) observed on various loading stages at P/P0.2
equal to: 1.19 (a), 1.38 (b), and 1.52 (c) [8], and the k-parameter dependences
on the damaged surface portion (d) and c-exponent (e) in equation (3) plotted
for specimens with thickness of 4 mm (1—d 27 m, 2—d 97 m) and
16 mm (3—d 97 m).
484 L. R. BOTVINA and A. P. SOLDATENKOV
lgNAE D exp( cAEAAE), and then approached the power-law function
lgNAE B bAElgAAE, where NАЕ is the total number of acoustic emis-
sion signals with an amplitude AAE exceeding a given one.
3.3. Comparative Analysis of the Damage Accumulation
in Three Steels with Different Microstructures under Tension
As seen from Fig. 4, a–c, the microcracks’ patterns in the low-carbon
grade 20 and medium-carbon grade 45 steels differ from the mi-
crocracks’ patterns observed in stainless steel. In specimens of ferrite-
pearlitic steels, the microcracks are oriented at an angle of 45 to the
loading axis, while in specimens of the austenite steel they are parallel
to the loading axis. However, the k-parameter dependence on the dam-
aged surface portion observed in Fig. 2, d is the same and corresponds
to the power-law dependence
*
( ( ) )
mk S exhibiting a plateau at
achieving 10% damage.
The power-law dependences also connect k-parameter and a specific
work of plastic deformation estimated on area under the deformation
curve divided into the volume of the working portion of the specimen
(Fig. 4, e).
It is worth noting that the most parts of the diagrams in Figs. 2 and
4 are below the level defined by the k-criterion and equal to 3.
4. DISCUSSION OF THE RESULTS
Some of the above-described dependences of microcracks’ accumula-
tion in metal specimens are revealed at other scale levels that allows
assuming the existence of general regularities characterizing the de-
Fig. 3. The cumulative microcracks’ distributions in the specimens of low-
carbon grade 20 steel with thickness of 6 mm (a) and 15 mm (b) at early (1) and
final (2) stages of tension [8].
ON THE CONCENTRATION CRITERION OF FRACTURE 485
fects’ accumulation that will be considered below.
1. It is known that the basic equation in seismology is the Guten-
berg–Richter law [16], which relates the number of seismic events with
energy:
, lg lg ,
bN AE N b E C bM C
where N is the number of seismic events (earthquakes) with energy
equal to or greater than E, M is magnitude (M lgE), C is constant,
and b is slope of the accumulated earthquakes number dependence on
magnitude. In seismology, the curve plotted in the coordinates ‘accu-
mulated events’ number–magnitude’ and obeying this law in its middle
part is referred to as ‘cumulative distribution curve’ or ‘recurrence
curve’. Usually, the Gutenberg–Richter law plotted for the most re-
gions in the world is linear in the semi-log axes. However, the recur-
rence curve plotting for the individual regions or different time peri-
ods prior to the earthquake revealed a knee on the curve with a de-
Fig. 4. Microcracks’ patterns on the polished surfaces of specimens of the low-
carbon grade 20 (a), medium-carbon grade 45 (b) and stainless 12Kh18N9T (c)
steels observed under tension conditions at close relative strain values ( / f)
and the k-parameter dependences on the damaged surface portion (S
*) (d) and
specific work of plastic deformation w (e). Markers 1, 2, and 3 correspond to
the low-carbon grade 20, medium-carbon grade 45 and stainless 12Kh18N9T
steels, respectively.
486 L. R. BOTVINA and A. P. SOLDATENKOV
crease in slope before a seismic event with large magnitude (Fig. 5). It
is similar to inflection point in the cumulative microcracks’ distribu-
tion curve (Fig. 1, b) before the fatigue macrocrack formation [11–14].
Since the energy released during an earthquake is proportional to
the length of fault arisen, the cumulative faults’ number distributions
over length are described by the relation similar to the Gutenberg–
Richter law (and equation (4) of the microcracks’ accumulation) with
an exponent values varying in different regions of the world.
Recurrence curves plotted on the acoustic emission data registered
during deformation of metal specimens also show a decrease in the
slope of the curves before the fracture [7–9, 12].
2. The authors of [18] used a concentration criterion of fracture for
assessing the degree of clusterization, i.e., grouping of seismic events
or acoustic emission signals in laboratory testing of rocks. A cluster is
assumed to be formed by two or more acoustic signals, if the distance
between signal hypocentres and time between the signals are less than
the critical values of Rcr (Rcr 3R B) and Тcr (
i 0( )
cr
10
K K
T ), where
R is the crack size calculated by the formula: lgR DlgE C, K0 is the
minimum energy class of event used in the experiment, 0.5. Factor
3 corresponds to the concentration criterion, according to which the
critical density of cracks spaced by three times of the crack length ac-
cumulates before the macrofracture. At the fracture of the rock speci-
mens, the value of Rcr varied from 1.8 to 3 mm for all acoustic signals.
The authors found that, at the beginning of loading, the signal hypo-
Fig. 5. Recurrence curve characterizing seismic activity in the New Madrid
zone: S, I, L are small, intermediate and large events, respectively; bS is expo-
nent in the Gutenberg–Richter law [17].
ON THE CONCENTRATION CRITERION OF FRACTURE 487
centres arose randomly in the specimen volume, and after reaching the
maximum load, the hypocentre localization occurred in the area of fu-
ture macrorupture. Furthermore, it was shown that, when approach-
ing to macrorupture, the acoustic activity of clusters increases and ac-
tivity of single signals decreases. In this work, the direct measure-
ments of the crack lengths were not carried out, and the authors did
not compare the concentration criterion values on the data of both
acoustic emission and direct measurements. According to preliminary
authors’ estimates, the concentration criterion for these tests varies
from 3 to 5.
Concentration criterion of crack accumulation was tested for seis-
mic regime description of the seismoactive Kamchatka zone and other
regions; based on the implemented assessments, the maps of this crite-
rion changes were plotted [18]. The authors noted another important
and informative parameter indicating the earthquake approaching,
namely, the area of the accumulated faults, which is estimated as the
difference between the accumulated area of seismogenic ruptures over
the last year and the average annual value of the area. The epicentres
of strong earthquakes practically coincide with the locations of the pa-
rameter maxima.
The k-criterion dependences on the damaged surface portion of the
metal specimens shown in Figs. 2, d and 4, d confirm this relationship.
The general tendency to decreasing concentration criterion with in-
crease in deformation observed for ductile (metals) and brittle (rocks)
materials appears to reflect the general physical regularity associated
with defects’ coalescence at pre-fracture stage. The differences in the
properties and microstructures of the studied materials have an effect
on a change in the values of the k-parameter corresponding to a given
deformation, i.e., in the position of the estimated value of the k-
parameter on its dependence on deformation.
3. Apparently, the observed change in the microcracks’ distribution
functions on length before a critical event connected with the localized
crack appearance is the condition of transition from one hierarchical
level of fracture process to the next one because of the previous de-
fects’ coalescence. Similar change in the defects’ distribution function
is observed at multiple fractures under dynamic and cyclic loading
conditions.
Considering the fragmentation process of the material as caused by
extensive influence of independent centres, A. N. Kolmogorov [19]
showed that, regardless of fragmentation mechanism, this process in
the limiting case leads to a log-normal (i.e., exponential) particle size
distribution. For such a case, the process must satisfy two conditions:
the different parts of the body are fragmented independently on each
other, and the fragment sizes are proportional to the process time.
At the initial stage of fracture, the damage accumulation process
488 L. R. BOTVINA and A. P. SOLDATENKOV
characterized by the appearance of single microcracks with length sub-
stantially smaller than a grain size probably corresponds to such condi-
tions.
The established fact of the change in microcracks’ distribution func-
tions, having deep fundamentals and being a common feature of many
processes prior to the critical event appearance, is supported by known
Ising model describing the phase transition in magnetic systems. The
model considers a magnetic material consisting of particles with posi-
tive and negative spins. The force of interaction between the particles
is limited by the nearest neighbours, and the magnetic polarization of
the system is determined by the amount of spins. The scheme in Fig. 6
[20] illustrates the behaviour of such system as a function of tempera-
ture in the absence of an applied magnetic field.
At T 0 K, all the particles have a positive (M M0) or negative
(M M0) spins. For T TC, the system is completely random and non-
magnetic. The temperature increases over time. If, at T 0 K, all the
particles have a positive spins, then, at the low finite temperature
(point 1 in Fig. 6, a), the clusters of particles with a negative spin form
randomly, without spatial correlation, and the particle clusters’ dis-
tribution on area corresponds to an exponential distribution (Fig. 6, b).
Near the critical temperature (point 2), the clusters of particles with a
negative spin become correlated, and a power-law (fractal) cluster dis-
tribution on area appears. In this transition region below the critical
temperature (T TC), the magnetic polarization obeys a power-law re-
lation M (TC T) , and the particle clusters’ distribution is illustrat-
ed by the scheme in Fig. 6, c. At T ТС, magnetic polarization occurs
Fig. 6. Temperature dependence of the magnetic moment (M) for the Ising
model (a), exponential cluster distribution on area with a negative magnetic
moment at point 1 (b), and power-law distribution in point 2 (c) near the criti-
cal point of the second-order phase transition at T TC [20].
ON THE CONCENTRATION CRITERION OF FRACTURE 489
(M 0 at T TC).
4. Coalescence of cracks at their interaction characterized by the
critical value of k-parameter can probably be considered from the point
of coagulation theory described on the basis of the Boltzmann and
Smoluchowski equations. It follows from [21, 22], the authors of which
pay attention to the possibility of using the Smoluchowski equation to
describe the physical kinetics of different coagulation processes, in-
cluding, in particular, the formation of microstructure in polycrystal-
line materials and processes of cracks’ growth due to their intercross-
ing (coagulation) at the macrocrack formation and fracture. According
to V. A. Galkin [21], the coagulation ‘is directly related to the problems
of both the fracture dynamics of structural parts and the prediction of
defect development in order to prevent possible accidents’.
5. CONCLUSIONS
1. The concentration criterion of fracture characterizing the predomi-
nant mechanism of damage development at different stages of static
and cyclic loading (accumulation at k 3 or coalescence at k 3) was
estimated. Under cyclic loading of specimens of low-carbon steel, the
power-law dependence of k-parameter on relative durability was
found.
2. Under static and cyclic loading of low-carbon steel, the exponential
function approximating the cumulative microcracks’ distribution
curves at the initial loading stage changes by the power-law function at
the pre-fracture stage.
3. The power-law k-parameter dependences on both the damaged sur-
face portion and the specific work of plastic deformation were found
for the three structural steels under tension.
4. Some general regularities of defect accumulation on different scale
levels were considered.
The Russian Science Foundation (project No. 15-19-00237) sup-
ported this study.
REFERENCES
1. S. N. Zhurkov, V. S. Kuksenko, and A. I. Slutsker, Fizika Tverdogo Tela, 11,
No. 1: 296 (1969) (in Russian).
2. S. N. Zhurkov, V. S. Kuksenko, V. A. Petrov, V. N. Savel’ev, and
U. S. Sultanov, Fizicheskie Protsessy v Ochagakh Zemletryaseniy [Physical
Processes of Earthquake Source] (Eds. M. A. Sadovskiy and V. I. Myachkina)
(Moscow: Nauka: 1980), p. 78 (in Russian).
3. V. V. Panasyuk and B. L. Lozovyy, DAN UkrSSR, No. 11: 1444 (1962) (in
Ukrainian).
490 L. R. BOTVINA and A. P. SOLDATENKOV
4. V. V. Panasyuk, Predel’noe Ravnovesie Khrupkikh Tel s Treshchinami [Limit
Equilibrium of Brittle Solids with Cracks] (Kiev: Naukova Dumka: 1968) (in
Russian).
5. T. L. Chelidze, Pure Appl. Geophys., 124, Nos. 4–5: 731 (1986).
6. V. P. Tamuzh and V. S. Kuksenko, Mikromekhanika Destruktsii Polimernykh
Materialov [Micromechanics of Destruction of Polymeric Materials] (Riga:
Zinātne: 1978) (in Russian).
7. L. R. Botvina, A. P. Soldatenkov, and M. R. Tyutin, Dokl. Earth Sci., 446,
No. 1: 1127 (2012).
8. L. R. Botvina, N. A. Zharkova, M. R. Tyutin, A. P. Soldatenkov,
Yu. A. Demina, and V. P. Levin, Zavodskaya Laboratoriya. Diagnostika
Materialov, 79, No. 5: 46 (2013) (in Russian).
9. L. R. Botvina, M. R. Tyutin, T. B. Petersen, V. P. Levin, A. P. Soldatenkov,
and Yu. A. Demina, Russian Metallurgy (Metally), 2017, No. 1: 10 (2017).
10. C. M. Suh, R. Yuuki, and H. Kitagawa, Fatigue Fract. Eng. M, 8, No. 2: 193
(1985).
11. L. R. Botvina, I. M. Rotvayn, V. I. Keylis-Borok, and I. B. Oparina, Doklady
RAN, 345, No. 6: 809 (1995) (in Russian).
12. L. R. Botvina, Razrushenie: Kinetika, Mekhanizmy, Obshchie Zakonomernosti
[Fracture: Kinetics, Mechanisms, General Regularities] (Moscow: Nauka:
2008) (in Russian).
13. I. Rotwain, V. Keilis-Borok, and L. Botvina, Phys. Earth Planet. In., 101,
Nos. 1–2: 61 (1997).
14. L. R. Botvina, I. B. Oparina, and O. V. Novikova, Met. Sci. Heat Treat., 39,
Nos. 3–4: 151 (1997).
15. L. R. Botvina, I. M. Petrova, I. V. Gadolina, V. P. Levin, Yu. A. Demina,
A. P. Soldatenkov, and M. P. T’utin, Inorg. Mater., 46, No. 14: 1570 (2010).
16. B. Gutenberg and C. F. Richter, Seismicity of the Earth and Associated
Phenomena (Princeton, New Jersey: Princeton University Press: 1949).
17. I. G. Main, S. Peacock, and P. G. Meredith, Pure Appl. Geophys., 133, No. 2: 283
(1990).
18. G. A. Sobolev and A. V. Ponomarev, Fizika Zemletryaseniy i Predvestniki
[Physics of Earthquakes and Precursors] (Moscow: Nauka: 2003) (in Russian).
19. A. N. Kolmogorov, DAN USSR, 31, No. 2: 99 (1941) (in Russian).
20. J. B. Rundle, W. Klein, D. L. Turcotte, and B. D. Malamud, Pure
Appl. Geophys., 157, No. 11: 2165 (2000).
21. V. A. Galkin, Analiz Matematicheskikh Modeley: Sistemy Zakonov
Sokhraneniya, Uravneniya Boltsmana i Smolukhovskogo [Analysis
of Mathematical Models: Set of Conservation Laws, the Boltzmann and
Smoluchowski Equations) (Moscow: BINOM, Laboratoriya Znaniy: 2011)
(in Russian).
22. S. V. Shevchenko, Nanosistemi, Nanomateriali, Nanotehnologii, 13, No. 2: 371
(2015) (in Russian).
https://doi.org/10.1007/BF00879607
https://doi.org/10.1134/S1028334X12090188
https://doi.org/10.1134/S1028334X12090188
https://doi.org/10.1134/S0036029517010037
https://doi.org/10.1111/j.1460-2695.1985.tb01203.x
https://doi.org/10.1111/j.1460-2695.1985.tb01203.x
https://doi.org/10.1016/S0031-9201(96)03224-4
https://doi.org/10.1016/S0031-9201(96)03224-4
https://doi.org/10.1007/BF02469070
https://doi.org/10.1007/BF02469070
https://doi.org/10.1134/S0020168510140190
https://doi.org/10.1007/BF00877164
https://doi.org/10.1007/BF00877164
https://doi.org/10.1007/PL00001079
https://doi.org/10.1007/PL00001079
|