- Email: [email protected]
Nucleation of microcracks in localised slip bands: A dislocation model
MATERIALS SCIENCE & ENGIWEERIWQ
A
Materials Scienceand Engineering A234-236 (1997) 575-578
Nucleation
of microcracks in localised slip bands: A dislocation model J. Toribio,
University
of La Coruiia, Received
ETSI
4 February
Caminos,
V. Kharin *,l Campus
1997; received
de Elvitia,
in revised
form
15192 La Corufia, 26 March
Spain
1997
Abstract
Concerningdecohesivefracture initiation in coarselocalisedslip bands,the model of Stroh-type blocked array of dislocations is advancedto accountfor the possibility of separationalong a shearplane. Using a cohesiveforce approach,the coresof densely spaceddislocationsin a coarseslip band of multiple coplanar pileupsare taken into account asthe nuclei of both possibletraverse and longitudinal cleavagesin a band. The equationsof equilibrium of this dislocation population in stressfield are built up towards establishingthe fracture criteria asthe limit conditions of the stability of the band’sequilibrium. 0 1997ElsevierScience S.A. Keywords:
Fracture
micromechanism;
Microcrack;
Cleavage;
Dislocation
modelling
1. Introduction
2. Model backgrounds
Fracture has been recognised to be usually a plasticity induced phenomenon in crystalline materials having appreciable density and mobility of dislocations, like metals and alloys. Glide of dislocations, i.e. plastic shear, is a prerequisite and inevitable attribute of crystal fracture. Interaction of closely spaced dislocations in dense dislocation arrays immersed in external stressfield is the way of microcrack nucleation. Depending on certain circumstances, fracture initiates by microcracks opening either traversely to blocked shear planes (t-cracks), or by longitudinal separation of coarse shear bands along characteristic slip traces (l-cracks) [1,2]. The objective of this work is to develop the model towards a better understanding of decohesive fracture in localised slip bands. The model of the Stroh-type dislocation array is advanced to provide the equal opportunity framework for study of both t- and l-cleavages as the overcoming of cohesive forces in the crystal. This study was inspired by the works [1,3] which involved dislocation cores in modelling of fracture micromechanisms.
Analyses of edge dislocations [3,4] revealed that, apart from the leading term of their stressfields dominating the long-range repulsive (positive) force F cc Y- ’ between identical dislocations spaced by a distance Yin the same slip line, a series of negative terms does exist of the short-distance dislocations attraction force FCC -yp(2k+1) (k= 1, . ..). This arises due to dislocation cores and affects the interaction of closely spaced dislocations in a pileup up to their coalescenceto form t-cracks [3]. Concerning the l-crack, apart from the dislocation cores, no source of the tensile stress 0, # 0 exists to help separation along a plain array in a single slip line on the x-axis. Hirth [l] analysed tensile stressconcentration from the cores in a plain pileup of edge dislocations postulating the cores as dilatation centres. They produce a short-range stress gYYcc r ~ 2 hardly effective to help l-cleavage and completely unable to affect dislocation interactions in a pileup to promote cracking. However, in a bottom of a coarse shear band formed by slip lines coplanar with the x-axis, the tensile stress gY-V> 0 is elevated due to the long-range fields from the top slip planes since dislocations have the stresscomponent aYv= Dbjh just beneath itself on a distance h (here D = G/27c(l - p), G is the shear modulus, /J is Poisson’s
* Corresponding author. Tel.: + 34 81 167000, ext. 1450; fax: 8 1 167 170; e-mail: [email protected] ’ On leave from: Pidstryhach Institute for Applied Mechanics Mathematics, Lvov, Ukraine. 0921-5093/97/$17,00
Q 1997 Elsevier
PII SO921-5093(97)00254-Z
Science
S.A. All rights
+ 34 and
reserved.
516
J. Toribio,
V. Khurin
iMateri&
Science
ratio and b is the Burgers length). Thus, formation of localised shear bands urges pull apart of the bottom crystalline planes in a band. This fits the fact that l-cracking preferably follows a coarse slip localisation, e.g. in metals under the action of hydrogen, cf. [1,2]. Then, by analogy with t-cracking [3], combining of the rs~,~stresses raised by band’s top floors and those from the neighbour dislocation cores may lead to l-separation along the coarse shear bands. Having this, the appropriate model to analyse both forms of cracking in localised shear bands seems to consider a heap of the height H formed by (N+ 1) floors of blocked plain pileups, each one having n edge dislocations. The main body of dislocations there can be treated as ordinary elastic ones interacting through their long-range stress fields. However, account for the cores of the most closely spaced dislocations in the band looks not only advantageous but necessary to elucidate both t- and l-cracking modes. In the previous studies of the t-crack nucleation in a pileup [3] the core was treated as a V-shape cavity (Fig. l(a)) represented by a distribution of infinitesimal dislocations with a density ~01) at - fi I y I 0 (i.e. p being now the core size) defining the core opening u as follows:
dY) =sY~(5) d5 and
u(O) = b
(1)
-P
The core faces attract each other as crystal planes with the interatomic force intensity gc = a,(u). To consider the l-cracking, the carrier of this sort of cleavage is incorporated considering a T-shape dislocation core as in Fig. l(b). It contains the same t-part (1) and the l-component given by a density of dislocations i(x) at - CI IX I c( so that the l-opening c’ and the condition of the Burgers vector preservation, correspondingly, are: x a u(x) = and i(t) dt ~ 1 i(t) dt = 0 (2) s -a s
and Engineering
A234-236
(1997)
575-578
Fig. 2. The discrete-continuum model of a blocked coarse shear band as a scene for development of decohesive t- and I-cracking from interaction between the cores of the most closely spaced band’s dislocations. The top presents the sketches of the dislocation densities along the slip lines in correspondent band’s parts B, and B,.
Again, the opposite faces there attract each other with the crystal cohesive stress gc = a,(v). Both types of potential cleavage in a pileup may be studied as in the mechanics of cohesive cracks. The criterion of crack nucleation will spring from the limit equilibrium condition for the dislocation band which bears all the forces: external stresses o-zX;.,a,*, and r& and internal ones from the band constituents (cf. [3]). To analyse the stability of equilibrium of the dislocation population forming the shear band it is suitable to adopt the earlier approach [3,5] with appropriate modifications accounting for the band structure [6]. Following [6] the band is considered to be formed by n straight line dislocation walls along 0 I y I H, each with (N+ 1) equidistant dislocations and the cumulative Burgers vector b, = (N + l)b, N >> 1 (Fig. 2). Thus, dislocation distributions in band’s slip lines are identical. The repulsive force between walls separated by a
distance x [6]
b
Fig. 1. Cohesive-force V-shape wedge-like
models of the core of edge dislocation: core; (b) the triple wedge T-shape core.
(a) the
1
(3)
if rehas local maximum at x * z H/2. Consequently, solved shear stress T = Ir&[ - ti (ri is the lattice friction stress) pushes the dislocations in their slip lines closer than x*, they overcome this maximum repulsion and fall down much closely to each other. The detached dense group - tilt boundary - of n, plain dislocation walls with spacing Ax <
J. Toribio,
V. Kharin
/Materials
Science
spaced (Ax > x*) plain walls situated at I, < x 5 1, I, = x* and I being the band length. At n1 <
and Engineering
A2346236
o
d< =
q(t)
s--85-Y
(1997)
575-578
-a2 ~(5) s -x1
511
tb* - t2) d( (r* +y212
P,(X)
+
E
1.23 --T a n’=N+lD b
(1.41
(6)
Similarly, the equilibrium of the l-part of the core [in, (i = 1, 2) comprising all the o,,,(x, y = 0) stresses, i.e. from its t-part, the counterpart core I+ (j # i), and so long, yields:
and in the jammed walls block B,
P,(X)= 2
g,(u) - 0x.x* D
+ 2.43 3,
(5)
Stresses g-Xxand aYu caused by the band’s dislocations do help cleavage mostly at the bottom of the segregation B, in its portion with the most closely spaced dislocations where the short-range fields of their cores become important. Core effect on the band structure outside this zone is negligible, and treating the rest of dislocations as obvious elastic ones is admissible. By analogy with [3], the only two dislocations b, and b, at the head of the bottom pileup in B, worth aPmtivm ;t- t+i& am2 T&y’ P&e &it5 ~~iiii7;luri-r m-i&wiaQiwL d dtmi+&3 ‘pIGj and {(x 3~slndtzCtiTTidative stresses,correspondingly, gXX and CJ~~, opposed by lattice cohesive forces ~Ju) and a,(v). As in the study co5Ihhe’r-$eava>e!ejna G$e$9eJlln> tie cafes 05 bolh hi (i = 1,2) are supposed to be identical and symmetriLa+rwic+L rm-fict Q?J cdfc axis -y’ -vi- P&ewJix&mit 3-yei-ll iJx‘,,v’) with the or$$mpinned at the m&way of the Ipacing L be&veen &em, x = L&Z, 3:= 0. T&en, f&e cores b, and b, occupy the intervals [ - CI,, - m2]and [r,, a,] of the XI-axis, correspondingly. Since L is small ccompare8 wirh other characterjsIY)c &mensions jn %he band, this coordinate shift does not sensibly alter desc~~p2ons)4,5) 02 zhe )lea?y componennh For the t-cleavage parts of the cores b, the equation of equilibrium is derived as before [3] balancing cohesive fvicm -3&ti’,, a-&kd 3-ims Bcx and -fns~ r~~,(x = + L/2, y) from all the constituents of the dislocations configuration,. i.e. successively,. from the Icleavage component l(x) of the core itself, from its neighbaur ctis~ocatiuns of the jam 8, and the bards tail B,. Since the core size ,9 is much less than any other specific length in a band, and approximating the role of B, and B, by the stresses from dislocation
Finally, to close the system of the equations of equilibrium of the shear band, the spacing L, i.e. the equilibrium position of the dislocation b,, is required. Balancing the glide forces on b2 caused by b,, B,, B,, and resolved external stress z, we have:
L r” !” J-B J-/j .!a&~1
Eq
Y(mw
L.*-!)-1j2 d[d~-~ _.
[L2
+
b
_
Q5> and Eq. (7) ax’
5)2]2
adiw-y
bBpr x:+fz*
Eq. (3) n&J2
&&iQn% Es. {-), 5s. (1) an& d~n+S& 5s. (4) and Es. (5) form
he ched
system offhe
equations
ofdx
band
equilibrium with attention to the closely spaced disloca*km, -A lhic ~mi2~~5 +i -pm-peel t!muq~ T’zq mra’-Je atxe ta study tke ban&s equ
Two of the microfracture nucleation forms in a shear band due to stressconcentrated in its head are obvious:
super-walls placed at the gravity centres x, and x, of
(i) t-cleavage by spreading of a disbcation
the correspondent array densities p1 and pt, the integral equation for r(y) emerges:
tensile crack along negative y-semiaxis due to gXX stress; (ii) l-cleavage as a breakthrough of the cores b1
core hi as a
578
J. Toribio,
V. Kharin
/Materials
Science
and b, towards each other along x-axis due to oevystress producing a crack of the length 2c 2 (2c(, + L)-glide plane decohesion. These events may be elucidated using techniques of the fracture mechanics of cohesive cracks, cf. [3]. Beside, the third less evident form of the stability loss exists: the instability of the position of the dislocation b2 in its glide plane so that it drops down towards b, and their coalescence produces a super-dislocation with Burgers vector B, = 2b -a wedge-shape microcrack [3]. This is caused by the attraction between cores b1 and On,. Resultant interaction force between two dislocations given by the first term in Eq. (8) using Taylor’s series expansion with respect to 10, - o/L1 5 /l/L < 1 may be presented as follows: F@O,)
( )I
-0 2
(9)
As distinct from the classical formula of dislocations theory, the terms in brackets appear in Eq. (9) reflecting the role of cores. Cores attraction is controlled by their shapes, i.e. by densities I, dependent on lattice structure, interatomic potential and on the stress field. Consequently, apart from the fracture mechanics-type criteria, the next condition of the limit of the stable equilibrium in the shear band must be evaluated, too: $2)
Wb,>
aL
o
=
and Engineering
A234-236
(1997)
575-578
I2 + H/2 (again l2 << H/2) separated from the band tail which now resides at 1, I x 4 1, with 1, = 2x,. This may continue up to formation in the head of a band of several jams Bi (j = 1, _.., M) of dislocation walls followed by a tail B, of rather sparse dislocation population (Ax > x*). According to the fracture mechanics solutions [7], this will promote longitudinal cracking associated now with a sequence of collinear cohesive l-crack nuclei along the band bottom which favours their extension in contrast to the array of parallel t-crack nuclei one sideways another which cause toughening effect on each other.
4. Conclusions The dislocation model is developed towards elucidation of fracture micromechanisms in slip bands. The discrete-continuum model of the traverse S&oh-type cracking in blocked plain array of dislocations is advanced to consider the shear plane separation, too. Interactions of dislocations in coarse shear band formed by plain arrays are studied with attention to the dislocation cores role. This latter is essential for microfracture nucleation in dense dislocation configurations. Further development of the proposed model, expectedly, will allow the establishment of correlations between factors affecting crystal cohesion in the core (e.g. hydrogen) and microstructural parameters such as grain size, inclusions spacing, slip localisation, etc.
(10)
The weakest of the listed criteria of the equilibrium stability loss will determine the dominant fracture mode in a slip band. It depends on a complicated interaction of a series of variables among which, beside the common parameters of dislocation models of microfracture such as the number of dislocations in a slip line n and the shear stress z, some others appear such as the band width, slip lines spacing, tensile stresses and lattice cohesion. Depending on their specific combination, different ways of cracking may be preferable in particular circumstances. In general, less localised plasticity (wider slip planes spacing) weakens the tensile stress concentration along slip planes and favours t-cracking, whereas localised shear bands promote l-cleavage. With rising z or increasing n, dislocations in a tail of a band are pushed closer and their spacing in the tail head, Ax cc (nz) ~ ‘I3 according to Eq. (4), may become less than x*. Then the second detached block B, of densely spaced walls appears in the domain H/2 5 x I
Acknowledgements The work was funded by the Spanish DGICYT (Grant UE94-001) and Xunta de Galicia (Grant XUGA 11801B95). VKh is also indebted to DGICYT for the Grant SAB95-0122 and 0122P.
References [I] J.P. Hirth, Ser. Metall. Mater. 28 (1993) 703-707. [2] Y. Takeda, C.J. McMahon, Metall. Trans. Al2 (1981) 12551266. [3] V. Kharin, Defect Assessment in Components-Fundamentals and Applications, Mechical Engineering Publishers, London, 1991, pp. 489-500. [4] J.E. Sinclair, J. Appl. Phys. 42 (1971) 5321-5329. [5] V. Kharin, J. Toribio, Anales Met. Fractura 13 (1996) 45-50. [6] V.I. Vladimirov, Sh.Kh. Khannanov, Fizika metallov i metalloved., 30 (1970) 981-988 (in Russian). [7] Y. Murakami (Ed.), Stress Intensity Factors Handbook, Pergamon Press, Oxford, 1987.
A
Materials Scienceand Engineering A234-236 (1997) 575-578
Nucleation
of microcracks in localised slip bands: A dislocation model J. Toribio,
University
of La Coruiia, Received
ETSI
4 February
Caminos,
V. Kharin *,l Campus
1997; received
de Elvitia,
in revised
form
15192 La Corufia, 26 March
Spain
1997
Abstract
Concerningdecohesivefracture initiation in coarselocalisedslip bands,the model of Stroh-type blocked array of dislocations is advancedto accountfor the possibility of separationalong a shearplane. Using a cohesiveforce approach,the coresof densely spaceddislocationsin a coarseslip band of multiple coplanar pileupsare taken into account asthe nuclei of both possibletraverse and longitudinal cleavagesin a band. The equationsof equilibrium of this dislocation population in stressfield are built up towards establishingthe fracture criteria asthe limit conditions of the stability of the band’sequilibrium. 0 1997ElsevierScience S.A. Keywords:
Fracture
micromechanism;
Microcrack;
Cleavage;
Dislocation
modelling
1. Introduction
2. Model backgrounds
Fracture has been recognised to be usually a plasticity induced phenomenon in crystalline materials having appreciable density and mobility of dislocations, like metals and alloys. Glide of dislocations, i.e. plastic shear, is a prerequisite and inevitable attribute of crystal fracture. Interaction of closely spaced dislocations in dense dislocation arrays immersed in external stressfield is the way of microcrack nucleation. Depending on certain circumstances, fracture initiates by microcracks opening either traversely to blocked shear planes (t-cracks), or by longitudinal separation of coarse shear bands along characteristic slip traces (l-cracks) [1,2]. The objective of this work is to develop the model towards a better understanding of decohesive fracture in localised slip bands. The model of the Stroh-type dislocation array is advanced to provide the equal opportunity framework for study of both t- and l-cleavages as the overcoming of cohesive forces in the crystal. This study was inspired by the works [1,3] which involved dislocation cores in modelling of fracture micromechanisms.
Analyses of edge dislocations [3,4] revealed that, apart from the leading term of their stressfields dominating the long-range repulsive (positive) force F cc Y- ’ between identical dislocations spaced by a distance Yin the same slip line, a series of negative terms does exist of the short-distance dislocations attraction force FCC -yp(2k+1) (k= 1, . ..). This arises due to dislocation cores and affects the interaction of closely spaced dislocations in a pileup up to their coalescenceto form t-cracks [3]. Concerning the l-crack, apart from the dislocation cores, no source of the tensile stress 0, # 0 exists to help separation along a plain array in a single slip line on the x-axis. Hirth [l] analysed tensile stressconcentration from the cores in a plain pileup of edge dislocations postulating the cores as dilatation centres. They produce a short-range stress gYYcc r ~ 2 hardly effective to help l-cleavage and completely unable to affect dislocation interactions in a pileup to promote cracking. However, in a bottom of a coarse shear band formed by slip lines coplanar with the x-axis, the tensile stress gY-V> 0 is elevated due to the long-range fields from the top slip planes since dislocations have the stresscomponent aYv= Dbjh just beneath itself on a distance h (here D = G/27c(l - p), G is the shear modulus, /J is Poisson’s
* Corresponding author. Tel.: + 34 81 167000, ext. 1450; fax: 8 1 167 170; e-mail: [email protected] ’ On leave from: Pidstryhach Institute for Applied Mechanics Mathematics, Lvov, Ukraine. 0921-5093/97/$17,00
Q 1997 Elsevier
PII SO921-5093(97)00254-Z
Science
S.A. All rights
+ 34 and
reserved.
516
J. Toribio,
V. Khurin
iMateri&
Science
ratio and b is the Burgers length). Thus, formation of localised shear bands urges pull apart of the bottom crystalline planes in a band. This fits the fact that l-cracking preferably follows a coarse slip localisation, e.g. in metals under the action of hydrogen, cf. [1,2]. Then, by analogy with t-cracking [3], combining of the rs~,~stresses raised by band’s top floors and those from the neighbour dislocation cores may lead to l-separation along the coarse shear bands. Having this, the appropriate model to analyse both forms of cracking in localised shear bands seems to consider a heap of the height H formed by (N+ 1) floors of blocked plain pileups, each one having n edge dislocations. The main body of dislocations there can be treated as ordinary elastic ones interacting through their long-range stress fields. However, account for the cores of the most closely spaced dislocations in the band looks not only advantageous but necessary to elucidate both t- and l-cracking modes. In the previous studies of the t-crack nucleation in a pileup [3] the core was treated as a V-shape cavity (Fig. l(a)) represented by a distribution of infinitesimal dislocations with a density ~01) at - fi I y I 0 (i.e. p being now the core size) defining the core opening u as follows:
dY) =sY~(5) d5 and
u(O) = b
(1)
-P
The core faces attract each other as crystal planes with the interatomic force intensity gc = a,(u). To consider the l-cracking, the carrier of this sort of cleavage is incorporated considering a T-shape dislocation core as in Fig. l(b). It contains the same t-part (1) and the l-component given by a density of dislocations i(x) at - CI IX I c( so that the l-opening c’ and the condition of the Burgers vector preservation, correspondingly, are: x a u(x) = and i(t) dt ~ 1 i(t) dt = 0 (2) s -a s
and Engineering
A234-236
(1997)
575-578
Fig. 2. The discrete-continuum model of a blocked coarse shear band as a scene for development of decohesive t- and I-cracking from interaction between the cores of the most closely spaced band’s dislocations. The top presents the sketches of the dislocation densities along the slip lines in correspondent band’s parts B, and B,.
Again, the opposite faces there attract each other with the crystal cohesive stress gc = a,(v). Both types of potential cleavage in a pileup may be studied as in the mechanics of cohesive cracks. The criterion of crack nucleation will spring from the limit equilibrium condition for the dislocation band which bears all the forces: external stresses o-zX;.,a,*, and r& and internal ones from the band constituents (cf. [3]). To analyse the stability of equilibrium of the dislocation population forming the shear band it is suitable to adopt the earlier approach [3,5] with appropriate modifications accounting for the band structure [6]. Following [6] the band is considered to be formed by n straight line dislocation walls along 0 I y I H, each with (N+ 1) equidistant dislocations and the cumulative Burgers vector b, = (N + l)b, N >> 1 (Fig. 2). Thus, dislocation distributions in band’s slip lines are identical. The repulsive force between walls separated by a
distance x [6]
b
Fig. 1. Cohesive-force V-shape wedge-like
models of the core of edge dislocation: core; (b) the triple wedge T-shape core.
(a) the
1
(3)
if rehas local maximum at x * z H/2. Consequently, solved shear stress T = Ir&[ - ti (ri is the lattice friction stress) pushes the dislocations in their slip lines closer than x*, they overcome this maximum repulsion and fall down much closely to each other. The detached dense group - tilt boundary - of n, plain dislocation walls with spacing Ax <
J. Toribio,
V. Kharin
/Materials
Science
spaced (Ax > x*) plain walls situated at I, < x 5 1, I, = x* and I being the band length. At n1 <
and Engineering
A2346236
o
d< =
q(t)
s--85-Y
(1997)
575-578
-a2 ~(5) s -x1
511
tb* - t2) d( (r* +y212
P,(X)
+
E
1.23 --T a n’=N+lD b
(1.41
(6)
Similarly, the equilibrium of the l-part of the core [in, (i = 1, 2) comprising all the o,,,(x, y = 0) stresses, i.e. from its t-part, the counterpart core I+ (j # i), and so long, yields:
and in the jammed walls block B,
P,(X)= 2
g,(u) - 0x.x* D
+ 2.43 3,
(5)
Stresses g-Xxand aYu caused by the band’s dislocations do help cleavage mostly at the bottom of the segregation B, in its portion with the most closely spaced dislocations where the short-range fields of their cores become important. Core effect on the band structure outside this zone is negligible, and treating the rest of dislocations as obvious elastic ones is admissible. By analogy with [3], the only two dislocations b, and b, at the head of the bottom pileup in B, worth aPmtivm ;t- t+i& am2 T&y’ P&e &it5 ~~iiii7;luri-r m-i&wiaQiwL d dtmi+&3 ‘pIGj and {(x 3~slndtzCtiTTidative stresses,correspondingly, gXX and CJ~~, opposed by lattice cohesive forces ~Ju) and a,(v). As in the study co5Ihhe’r-$eava>e!ejna G$e$9eJlln> tie cafes 05 bolh hi (i = 1,2) are supposed to be identical and symmetriLa+rwic+L rm-fict Q?J cdfc axis -y’ -vi- P&ewJix&mit 3-yei-ll iJx‘,,v’) with the or$$mpinned at the m&way of the Ipacing L be&veen &em, x = L&Z, 3:= 0. T&en, f&e cores b, and b, occupy the intervals [ - CI,, - m2]and [r,, a,] of the XI-axis, correspondingly. Since L is small ccompare8 wirh other characterjsIY)c &mensions jn %he band, this coordinate shift does not sensibly alter desc~~p2ons)4,5) 02 zhe )lea?y componennh For the t-cleavage parts of the cores b, the equation of equilibrium is derived as before [3] balancing cohesive fvicm -3&ti’,, a-&kd 3-ims Bcx and -fns~ r~~,(x = + L/2, y) from all the constituents of the dislocations configuration,. i.e. successively,. from the Icleavage component l(x) of the core itself, from its neighbaur ctis~ocatiuns of the jam 8, and the bards tail B,. Since the core size ,9 is much less than any other specific length in a band, and approximating the role of B, and B, by the stresses from dislocation
Finally, to close the system of the equations of equilibrium of the shear band, the spacing L, i.e. the equilibrium position of the dislocation b,, is required. Balancing the glide forces on b2 caused by b,, B,, B,, and resolved external stress z, we have:
L r” !” J-B J-/j .!a&~1
Eq
Y(mw
L.*-!)-1j2 d[d~-~ _.
[L2
+
b
_
Q5> and Eq. (7) ax’
5)2]2
adiw-y
bBpr x:+fz*
Eq. (3) n&J2
&&iQn% Es. {-), 5s. (1) an& d~n+S& 5s. (4) and Es. (5) form
he ched
system offhe
equations
ofdx
band
equilibrium with attention to the closely spaced disloca*km, -A lhic ~mi2~~5 +i -pm-peel t!muq~ T’zq mra’-Je atxe ta study tke ban&s equ
Two of the microfracture nucleation forms in a shear band due to stressconcentrated in its head are obvious:
super-walls placed at the gravity centres x, and x, of
(i) t-cleavage by spreading of a disbcation
the correspondent array densities p1 and pt, the integral equation for r(y) emerges:
tensile crack along negative y-semiaxis due to gXX stress; (ii) l-cleavage as a breakthrough of the cores b1
core hi as a
578
J. Toribio,
V. Kharin
/Materials
Science
and b, towards each other along x-axis due to oevystress producing a crack of the length 2c 2 (2c(, + L)-glide plane decohesion. These events may be elucidated using techniques of the fracture mechanics of cohesive cracks, cf. [3]. Beside, the third less evident form of the stability loss exists: the instability of the position of the dislocation b2 in its glide plane so that it drops down towards b, and their coalescence produces a super-dislocation with Burgers vector B, = 2b -a wedge-shape microcrack [3]. This is caused by the attraction between cores b1 and On,. Resultant interaction force between two dislocations given by the first term in Eq. (8) using Taylor’s series expansion with respect to 10, - o/L1 5 /l/L < 1 may be presented as follows: F@O,)
( )I
-0 2
(9)
As distinct from the classical formula of dislocations theory, the terms in brackets appear in Eq. (9) reflecting the role of cores. Cores attraction is controlled by their shapes, i.e. by densities I, dependent on lattice structure, interatomic potential and on the stress field. Consequently, apart from the fracture mechanics-type criteria, the next condition of the limit of the stable equilibrium in the shear band must be evaluated, too: $2)
Wb,>
aL
o
=
and Engineering
A234-236
(1997)
575-578
I2 + H/2 (again l2 << H/2) separated from the band tail which now resides at 1, I x 4 1, with 1, = 2x,. This may continue up to formation in the head of a band of several jams Bi (j = 1, _.., M) of dislocation walls followed by a tail B, of rather sparse dislocation population (Ax > x*). According to the fracture mechanics solutions [7], this will promote longitudinal cracking associated now with a sequence of collinear cohesive l-crack nuclei along the band bottom which favours their extension in contrast to the array of parallel t-crack nuclei one sideways another which cause toughening effect on each other.
4. Conclusions The dislocation model is developed towards elucidation of fracture micromechanisms in slip bands. The discrete-continuum model of the traverse S&oh-type cracking in blocked plain array of dislocations is advanced to consider the shear plane separation, too. Interactions of dislocations in coarse shear band formed by plain arrays are studied with attention to the dislocation cores role. This latter is essential for microfracture nucleation in dense dislocation configurations. Further development of the proposed model, expectedly, will allow the establishment of correlations between factors affecting crystal cohesion in the core (e.g. hydrogen) and microstructural parameters such as grain size, inclusions spacing, slip localisation, etc.
(10)
The weakest of the listed criteria of the equilibrium stability loss will determine the dominant fracture mode in a slip band. It depends on a complicated interaction of a series of variables among which, beside the common parameters of dislocation models of microfracture such as the number of dislocations in a slip line n and the shear stress z, some others appear such as the band width, slip lines spacing, tensile stresses and lattice cohesion. Depending on their specific combination, different ways of cracking may be preferable in particular circumstances. In general, less localised plasticity (wider slip planes spacing) weakens the tensile stress concentration along slip planes and favours t-cracking, whereas localised shear bands promote l-cleavage. With rising z or increasing n, dislocations in a tail of a band are pushed closer and their spacing in the tail head, Ax cc (nz) ~ ‘I3 according to Eq. (4), may become less than x*. Then the second detached block B, of densely spaced walls appears in the domain H/2 5 x I
Acknowledgements The work was funded by the Spanish DGICYT (Grant UE94-001) and Xunta de Galicia (Grant XUGA 11801B95). VKh is also indebted to DGICYT for the Grant SAB95-0122 and 0122P.
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