Grain boundary influence during short fatigue crack growth using a discrete dislocation technique
We have studied the effect of a grain boundary in front of a short edge crack on its propagation under cyclic loading conditions in bcc iron. The used model is a combination of a discrete dislocation formulation and a boundary element approach where the boundary is described by dislocation dipole el...
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| Опубліковано в: : | Проблемы прочности |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2008
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| Цитувати: | Grain boundary influence during short fatigue crack growth using a discrete dislocation technique / P. Hansson, S. Melin // Проблемы прочности. — 2008. — № 1. — С. 163-166. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859903373496549376 |
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| author | Hansson, P. Melin, S. |
| author_facet | Hansson, P. Melin, S. |
| citation_txt | Grain boundary influence during short fatigue crack growth using a discrete dislocation technique / P. Hansson, S. Melin // Проблемы прочности. — 2008. — № 1. — С. 163-166. — Бібліогр.: 7 назв. — англ. |
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| description | We have studied the effect of a grain boundary in front of a short edge crack on its propagation under cyclic loading conditions in bcc iron. The used model is a combination of a discrete dislocation formulation and a boundary element approach where the boundary is described by dislocation dipole elements, while the local plasticity is modeled by discrete dislocations. The grain boundary is considered impenetrable, but dislocations positioned in the vicinity of a grain boundary give raise to high stresses in neighboring grains which, eventually, results in nucleation of dislocations and a spread of the plastic zone into the next grain.
Выполнены исследования влияния границ зерен у вершины короткой краевой трещины, распространяющейся в ОЦК-железе в условиях циклического нагружения. При этом использовалась модель, сочетающая постановку задачи с точки зрения дискретных дислокаций с гранично-элементным подходом, где граница описывается с помощью элементов диполя дислокации, а локальная пластичность моделируется с помощью дискретных дислокаций. При этом постулируется, что граница зерна является непроницаемой, однако дислокации, находящиеся в окрестности границы зерна, генерируют высокие напряжения в соседних зернах, в результате чего имеет место зарождение дислокаций и распространение пластической зоны в следующее зерно.
|
| first_indexed | 2025-12-07T15:58:51Z |
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UDC 539. 4
G r a in B o u n d a r y I n f lu e n c e d u r in g S h o r t F a t ig u e C r a c k G r o w th U s in g a
D is c r e t e D is lo c a t io n T e c h n iq u e
P . H a n sso n 1,a and S. M elin 1,b
1 Division o f Mechanics, Lund University, Lund, Sweden
a per.hansson@mek.lth.se, b solveig.melin@mek.lth.se
We have studied the effect o f a grain boundary in front o f a short edge crack on its propagation
under cyclic loading conditions in bcc iron. The used model is a combination o f a discrete
dislocation formulation and a boundary element approach where the boundary is described by
dislocation dipole elements, while the local plasticity is modeled by discrete dislocations. The grain
boundary is considered impenetrable, but dislocations positioned in the vicinity o f a grain boundary
give raise to high stresses in neighboring grains which, eventually, results in nucleation o f
dislocations and a spread o f the plastic zone into the next grain.
K e y w o rd s : fatigue, grain boundary, discrete dislocation.
In trod u ction . It is w ell know n that the behavior o f short cracks differs from that o f
long cracks due to the relative large plastic zone and strong influence o f the surrounding
microstructure. Experim ental studies have show n that short cracks grow through a single
shear m echanism [1 ] and that they can grow at load levels w ell b elow the threshold value
for long cracks at h igh rates. They can therefore not be treated b y the standard methods
used for long cracks.
For very low growth rates, in the order o f a few Burgers vectors per cycle only, it is
important to account for the discrete d islocations w ithin the material. Authors [2, 3] have
developed such a discrete d islocation m odel for a long M ode I crack to study the cyclic
crack-tip p lasticity and plastically induced crack closure. A similar m odel has also been
developed in [4] to study the influence o f grain boundaries on short m ode I cracks.
In this study, a discrete dislocation m odel, were both the geom etry and the plasticity
are described w ith discrete dislocations, is used to study a short edge crack subjected to
fatigue loading. The p lasticity is in this study restricted to tw o grains, and the change in
growth behavior due to the spread o f p lasticity betw een grains is investigated.
S ta tem en t of th e P roblem . The growth o f a microstructurally short edge crack
located w ithin one grain, subjected to fatigue loading (Fig. 1), has been investigated under
plane strain and quasi-static conditions. The crack is assum ed to grow in a pure shear
m echanism due to nucleation, glide and annihilation o f discrete dislocations along slip
planes in the material. The initial crack o f length a 0 and inclined at angle a to the free
edge normal is located w ithin a sem i-infinite body. The load is applied parallel to the free
edge and is varied betw een a m axim um value, o™y max, and a m inim um value, o™y min. In
this study, tw o neighboring grains are considered, w ith both grain boundaries parallel to
the free edge. The grain boundaries are considered to be d islocation barriers, w hich the
dislocations can not pass and w ill not contribute to the overall stress field. N ucleation and
glide o f d islocations is restricted to one slip direction in each grain, inclined at angles a
and 6 to the free edge normal and w ith the slip direction in the first grain coincid ing with
the initial crack direction.
In itia l C onditions. The material in this study is pure iron and is assum ed to be linear
elastic. The material parameters at room temperature are show n in Table 1 [5] together
w ith the geom etrical data for the initial edge crack g iven in Fig. 1.
© P. H A N SSO N , S. M ELIN , 2008
ISSN 0556-171X. Проблемы прочности, 2008, № 1 163
mailto:per.hansson@mek.lth.se
mailto:solveig.melin@mek.lth.se
P. Hansson and S. Melin
T a b l e 1
Material Data for bcc-Iron and Geometrical Data for the Edge Crack
Shear modulus GPa 80 Burgers vector b, nm 0.25
Poisson’s ratio v 0.3 Lattice resistance r cr, MPa 40
Initial crack length, a Q 100006
Crack angle, a 45°
Slip plane angle in grain 2, 6 30°, 45°, 60°
Distance to grain boundary, lGB1 10000&
?GB2 5000&
Load range, 240-40MPa
o f the short edge crack.
D iscrete D isloca tion T echnique. The m odel used in this study rests so lely on a
discrete d islocation formulation, describing both the geom etry and the p lasticity in the
grains by discrete dislocations. The external boundary, defined as the free edge together
w ith the crack itself, is m odeled using d islocation dipole elem ents [6]. A dipole elem ent
consists o f tw o glide d islocations and tw o clim b dislocations, w ith equal size but opposite
direction o f the tw o dislocations o f sam e kind. The dislocations are located at the end
points o f the elem ent, w hile the stresses in the elem ent are calculated at the elem ent center
point.
The stress at an arbitrary point is calculated as the sum o f the stress contributions
from the physical d islocations describing the plasticity, the dislocations form ing the dipole
elem ents and the applied external load. The m agnitudes o f the dipole d islocations are
determ ined from an equilibrium equation, Eq. (1), describing the normal and shear stress
along the external boundary. K now ing that the normal and shear stresses m ust equal zero
along the free edge and along the parts o f the crack that is open, the m agnitudes o f the
dipole dislocations can be calculated.
G bboundary b G internal "̂ ~0 0 (1)
In Eq. (1), G is matrix containing influence functions [7], describing the stress field from
a d islocation along the external boundary, bboundary is a vector holding the m agnitudes o f
the dislocations in the dipole elem ents, b is the Burgers vector o f the material, G internal
is a vector containing the influence functions for the physical dislocations, and o is a
vector containing the contribution from the applied external load along the external
boundary.
D isloca tion N ucleation. N ucleation o f n ew dislocations is assum ed to occur i f the
resolved shear stress at a possible nucleation site exceeds the nucleation stress. D islocations
nucleate in pairs, consisting o f tw o d islocations o f equal size but opposite sign separated
by a sm all distance rnuc. The definition o f a positive and negative dislocations nucleated
at the crack tip is show n in Fig. 2a. It is assum ed that nucleation is possib le both in front
o f the crack tip and at the grain boundary betw een the tw o grains.
The nucleation stress is here defined as the low est stress at the nucleation point for
w hich the positive d islocation in the n ew ly nucleated d islocation pair travels inwards in
the material im m ediately after nucleation. This definition results in a geom etry dependence
164 ISSN 0556-171X. npo6n.eMH npounocmu, 2008, N 1
Grain Boundary Influence during Short Fatigue Crack Growth
o f the nucleation stress and, therefore, the nucleation stress is not the sam e in front o f the
crack as at the grain boundary. A more thorough discussion on the choice o f nucleation
stress is found in [6]. In order to determine w hen, and on w hich slip plane the next
nucleation o f a d islocation pair w ill take place, the resolved shear stress is calculated at all
possib le nucleation sites for each load level.
-2
-1
' 0
1
2
' 3
at the crack tip (a)
C rack G rowth. It is assum ed that no d islocations exist w ithin the material prior to the
first load cycle . W hen the applied load gets sufficiently high, d islocation pairs w ill
nucleate from the crack tip. A positive d islocation glides inwards in the material
im m ediately after nucleation along its slip plane as long as the resolved shear stress at its
position exceeds t cr, whereas the negative d islocations w ill remain at the crack tip. These
dislocations shield the crack tip and the load m ust therefore be increased before more
dislocations w ill nucleate. The positive dislocations pile up at the first grain boundary,
resulting in h igh stresses on the opposite side o f the grain boundary. Eventually, these
high stresses w ill result in nucleation o f d islocations in the second grain. A lso in this grain
the positive d islocation w ill m ove inwards in the material im m ediately after nucleation
whereas the negative one remains at the grain boundary. This process o f dislocation
nucleation in the tw o grains continues until the m axim um load is reached and the load
starts to decrease. Load reversal eventually results in d islocation glide in the opposite
direction, back towards the crack. W hen a positive d islocation gets sufficiently close to its
negative counterpart, the tw o dislocations annihilate resulting in crack growth in the
corresponding direction by one b, under the assum ption that no healing o f the crack
surfaces is allowed. A more detailed description o f the crack growth m odel used can be
found in [6].
R esu lts. In order to investigate the influence o f p lasticity spread on the crack growth
behavior, the angle 6 show n Fig. 1 is varied to regulate the resistance to dislocation
nucleation in the second grain, and the results were compared to results obtained by
restricting the p lasticity to one grain only. The results o f the sim ulations are presented in
Fig. 3, w here Fig. 3a show s the number o f dislocations, N , in the tw o grains for different
applied load levels during the loading phase, and Fig. 3b - during unloading. It can be
seen that w hen allow ing the p lasticity to spread into the next grain, the number o f
nucleated d islocations in the first grain increases, as com pared to the one-grain m odel. It
can also be seen that a higher number o f dislocations are present at the m inim um load.
This happens because the negative dislocations in the second grain reduce the stress field
from the piled-up dislocations in the first grain, allow ing more d islocations to exist in this
grain. A result o f this is, som ew hat surprisingly, that the obtained crack growth rate is the
sam e in all four sim ulations. It can also be seen that at different 6, different numbers o f
dislocations are nucleated in the second grain. It w as also found that no annihilation o f
dislocations occurred in the second grain during unloading in any case studied and,
therefore, the second grain w as not included in Fig. 3b.
y
G
r > -
Fig. 2. Nucleation condition and definition of positive and negative dislocation
and at the grain boundary (b).
ISSN 0556-171X. npoôneMbi npoHHocmu, 2008, № 1 165
P. Hansson and S. Melin
Fig. 3. Number o f dislocations N as a function o f applied external load during loading (a) and
unloading (b).
The order in w hich the slip planes in the second grain were activated, according to
Fig. 2b, w as also studied. This result is obtained for Q = 45°, and it w as found that the
first d islocation for this case nucleated in plane 1, w hich is the slip plane just b elow that in
the second grain, coinciding w ith the slip plane in the first grain (Fig. 2b). The fo llow ing
order o f planes on w hich nucleation occurred w as plane 0, plane, —1, and plane 2. A s
seen, nucleation in the second grain first occurs at the slip planes c losest to the slip plane
in the first grain before occurring in more distant planes.
C on clusions. The discrete d islocation form ulation has been used to investigate the
growth behavior o f a short propagating edge crack under fatigue loading conditions.
W hen investigating the effect o f the grain boundary it w as found that more dislocations
w ere nucleated at m axim um load in the first grain, holding the crack, w hen allow ing the
plasticity to spread into the next grain, as com pared to w hen restricting the p lasticity to
the first grain only. H ow ever, the number o f dislocations in the first grain also increased
w hen allow ing the spread o f the plasticity, resulting in the sam e crack growth rate for the
sim ulated cases. It w as also found that the dislocations in the second grain first nucleated
on slip planes c lose to the slip plane in the first grain before continuing in more distant
slip planes.
1. D. S. Suresh, Fatigue o f Materials, Second Edition, University Press, Cambridge (1998).
2. F. O. Riemelmoser, R. Pippan, and O. Kolednik, “Cyclic crack growth in elastic plastic solids:
a description in terms o f dislocation theory,” Comput. Mech., 20, 139-144 (1997).
3. F. O. Riemelmoser and R. Pippan, “Mechanical reasons for plasticity-induced crack closure
under plane strain conditions,” Fatigue Fract. Eng. Mater. Struct., 21, 1425-1433 (1998).
4. C. Bjerken and S. Melin, “A study of the influence of grain boundaries on short crack growth
during varying load using a dislocation technique,” Eng. Fract. Mech., 71 (15), 2215-2227
(2004).
5. D. R. Askeland, The Science and Engineering o f Materials, Third Edition, Stanley Thornes
(Publishers) Ltd (1998).
6. P. Hansson and S. Melin, “Dislocation-based modeling of the growth o f a microstructurally
short crack by single shear due to fatigue loading,” Int. J. Fatigue, 27, 347-356 (2005).
7. D. A. Hills, P. A. Kelly, D. N. Dai, and A. M. Korsunsky, Solution o f Crack Problems: The
Distributed Dislocation Technique, Kluwer Academic Publisher (1996).
Received 28. 06. 2007
166 ISSN 0556-171X. npo6neMbi npoHuocmu, 2008, № 1
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| id | nasplib_isofts_kiev_ua-123456789-48416 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T15:58:51Z |
| publishDate | 2008 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Hansson, P. Melin, S. 2013-08-19T13:02:24Z 2013-08-19T13:02:24Z 2008 Grain boundary influence during short fatigue crack growth using a discrete dislocation technique / P. Hansson, S. Melin // Проблемы прочности. — 2008. — № 1. — С. 163-166. — Бібліогр.: 7 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/48416 539.4 We have studied the effect of a grain boundary in front of a short edge crack on its propagation under cyclic loading conditions in bcc iron. The used model is a combination of a discrete dislocation formulation and a boundary element approach where the boundary is described by dislocation dipole elements, while the local plasticity is modeled by discrete dislocations. The grain boundary is considered impenetrable, but dislocations positioned in the vicinity of a grain boundary give raise to high stresses in neighboring grains which, eventually, results in nucleation of dislocations and a spread of the plastic zone into the next grain. Выполнены исследования влияния границ зерен у вершины короткой краевой трещины, распространяющейся в ОЦК-железе в условиях циклического нагружения. При этом использовалась модель, сочетающая постановку задачи с точки зрения дискретных дислокаций с гранично-элементным подходом, где граница описывается с помощью элементов диполя дислокации, а локальная пластичность моделируется с помощью дискретных дислокаций. При этом постулируется, что граница зерна является непроницаемой, однако дислокации, находящиеся в окрестности границы зерна, генерируют высокие напряжения в соседних зернах, в результате чего имеет место зарождение дислокаций и распространение пластической зоны в следующее зерно. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Grain boundary influence during short fatigue crack growth using a discrete dislocation technique Исследование влияния границ зерен на развитие коротких усталостных трещин с помощью метода дискретных дислокаций Article published earlier |
| spellingShingle | Grain boundary influence during short fatigue crack growth using a discrete dislocation technique Hansson, P. Melin, S. Научно-технический раздел |
| title | Grain boundary influence during short fatigue crack growth using a discrete dislocation technique |
| title_alt | Исследование влияния границ зерен на развитие коротких усталостных трещин с помощью метода дискретных дислокаций |
| title_full | Grain boundary influence during short fatigue crack growth using a discrete dislocation technique |
| title_fullStr | Grain boundary influence during short fatigue crack growth using a discrete dislocation technique |
| title_full_unstemmed | Grain boundary influence during short fatigue crack growth using a discrete dislocation technique |
| title_short | Grain boundary influence during short fatigue crack growth using a discrete dislocation technique |
| title_sort | grain boundary influence during short fatigue crack growth using a discrete dislocation technique |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/48416 |
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