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УДК 517
THE EXPLICIT FORMULAS FOR THE STRESS FIELD OF A PRISMATIC DISLOCATION LOOP
The objective of this paper is to obtain the explicit formulas for the stress field of a prismatic dislocation loop when Burgers vector is normal to the plane of the loop. Prismatic dislocation loop has numerous theoretical and practical applications that can benefit from knowledge of exact formulas for the components of its stress tensor. The intractability of general analytic solutions even for simple curved dislocations is well known. However, use of the comprehensive mathematical system "Mathematica" provided a necessary breakthrough resulting in the explicit formulas for the components of stress tensor of a prismatic dislocation loop. In addition, the formulas for the stress tensor of an infinitesimal prismatic loop and the asymptotic formulas at infinity are derived. Fascinating ’topographical maps’ of stress field generated by a prismatic dislocation loop are included with the paper.
1. Description of Method
Consider a circular dislocation loop C of the radius "a" located in xy - plane of the Cartesian system (x,y,z). The center of the loop coincides with the origin. Along with the Cartesian system we use the cylindrical system (г, в, z). The geometry of the problem is presented in Fig. 1. Refer interchangeably to the Cartesian coordinates of a point on the loop as (x0,yo,z0) or ((a:0)i, (^0)2, (^0)3). The stress field of a dislocation loop is given by the Peaeh-Koehler formula [2,3],
N. Langdon
k
© 2001 N. Langdon
E-mail: langdon@iit.edu
Department of Applied Mathematics, Illinois Institute of Technology, USA
Where
R% = (^o)i - Xi,i = 1,2,3, R = VRi2 + R22 + Д32,
d3R
д(хо)гд(х0)ад(х0)р
+ <hiR„ + SafjRi) +
3 RqRfjRj R5
In the above formulas eimk is the Einstein permutation operator [4], p, у are the shear modulas and the Poisson’s ratio, respectively, Sik is the Kroneeker delta, and bm is the m-th component of the Burgers vector b = (bi, b2, b3) .
The line integration in (1) is performed along the dislocation loop C with dx0 = (d(x0)v d(x0)2, 0), As in [2] the summation over the repeated indices is assumed. The Burgers vector of the prismatic dislocation loop has the form b = (0, 0, b).
2. Results
2.1. Cartesian components of stress tensor
The "Mathematica" [5] is used to develop the data base of integrals necessary to evaluate the stress field of a dislocation loop by means of Peaeh-Koehler formula. This results in the following explicit formulas for the stress tensor of a prismatic dislocation loop.
■x = ^p ^b(r2( — o6( 1 + 2y) + (r2 + z2)2(r2 + 2(r2 + z2)y)+
a4(^2z2(^3 + у) + r2( 3 + 6 y)) — a2(4r2z2(2 + y) — z4(7 + 2 y) + r4(3 + 6у))) + ( — (r2 + z2)2(r4 + 6 r2z2 + 4 z4) + 2 (r2 + z2)3(r2 + 2 z2)y + a8(^ 2 + Ay)+ a4(2r2z2(8 — 7у) + 9r4(^l + 2у) + 6z4(^3 + Ау)) + о6(г2(7 — 14г/)+
2z2(^b + 8у)) + о2(г6(5 — 10у) + 2r4z2(l — 6у)+
2г6(^7 + 8у) + rV(-5 + 14^))) eos[2в])ЕШрЫсЕ
Aar
г)2 + г2
(г2((о — г)2 + г2)3/<2((о + г)2 + z2)2{^ 1 + ^)) — ( Ь(г2(о4( 1 + 2у)+
(г2 + z2)(r2 + 2 (г2 + z2)y) + о2(^2г2(1 + 2 у) + z2{ 3 + Ау)))+
(о6(^2 + Ау) + (г2 + z2) (бг2^2 (—1 + у) + Az4(^l + у) + г4(^1 + 2 у))+ о4(г2(3 - 6у) + Az2(^2 + 3у)) + a2z2(r2(3 + Ау) + 2z2(^b + 6у))) eos[20])
EllipticK
Aar
(о — г)2 + г2
\2 | 9
а — г) + z
)3/2((а + г)2 + z2)(-1 + у))
_ 1 °ху~Ы
•r2z2(M + у) + Az4(-1 + у)+
г4(—1 + 2у)) + о4(2г2г2(8 — 7у) + 9г4(^1 + 2v) + 6z4(^3 + 4zy))+ о6(г2( 7 — 14^) + 2г2(^5 + 8у)) + о2(г6(5 — 10у) + 2rAz2(l — 6у)+
2г6(^7 + 8у) + rV(-5 + 1Ау))) ElhpticE
4 or
(о — г)2 + г2
(г2((о — г)2 + г2)3/<2((о + г)2 + z2)2(^1 + *')) — ( Ь(о6(^2 + 4zy) + (г2 + г2)
(6r2z2(_l _|_ ^ _)_ 4^4(^1 + /у) + г4(—1 + 2 г/)) + о4(г2(3 — 6^) + 4z2(^2 + Зу))
+aV(r2(3 + 4и) + 2z2(-5 + bv)))EllipticK
4 or
(о — г)2 + г2
(г2((о — г)2 + г2)3/<2((о + г)2 + z2){^ 1 + ^
(3)
о.
4-7Г
— ^b(r2(o6(l + 2zy) — (г2 + z2)2{r2 + 2(г2 + z2)zy)^
о4(^2г2(^3 + у) + г2(3 + 6г/)) + о2(4г2г2(2 + у) — z4( 7 + 2у) + r4(3 + 6у))) + ( - (г2 + z2)2(r4 + 6rV + 4г4) + 2(г2 + z2)3(r2 + 2z2)v + о8(^2 + 4гу)+ о4(2г2г2(8 — 7у) + 9г4(^1 + 2гу) + 6г4(^3 + Ау)) + о6(г2(7 — 14гу)+
2г2(^5 + 8^)) + о2(г6(5 - Ш) + 2г4г2(1 - 6у) + 2г6(^7 + 8г^)+
r2z4{M) + 14гу))) cos[2в])EllipticE
Aar
r2((a — г)2 + z2)
3/2
(о — г)2 + г2-
((о + г)2 + z2Y(^ 1 + г/)) + ( Ь(^г2(о4(1 + 2гу) + (г2 + г2)(г2 + 2(r2 + z2)y)+
02(^2г2(1 + 2гу) + z2{ 3 + 4i/))) + (о6(^2 + 4гу) + (г2 + 22)(6rV(-l + у)+ Az4(M + у) + г4(—1 + 2гу)) + о4(г2(3 - 6у) + Az2(^2 + 3у)) + a2z2(r2(3 + 4гу)
4-/7'Г>
+2z2(^5 + 6у))) cos[2e])ElUpticK
(о — г)2 + г2
(г2((о — г)2 + г2)3/<2((о + г)2 + г2)(^1 + У
& T
^ju ^ ^2bz((a2 — r2)2(a2 + r2) + 2(a4 — 6o2r2 + r4)z2 + (a2 + r2)z4)
^ j (r((o - r)2 + z2f/2((a + r)2 + г2)2(^1 + is)) -
cos [9]EllipticE
(a — r) +
2bz((a2 - r2) + (o2 + r2)z2) cosЩЕШрЫсК (r((o - r)2 + г2)3/2((о + r)2 + z2)(^l + 1У))^
Aar
(a — r) + г2
(5)
о,
yz
EllipticE
Aar
~j“- ^2Ьг((о2 — r2)2(o2 + r2) + 2(o4 — 6o2r2 + r4)z2 + (o2 + r2)z4)
in[0]^ j(r((a — r)2 + z2)3^2((a + r)2 + z2)2(^l + is)) ■
sin[0]
(a — r)2 + г2 2 bz((a2 — r2)2 + (a2 + r2)z2)EllipticK
Aar
(a — r)2 + z2-
a
(r((o - r)2 + z2f/2((a + r)2 + z2)(-1 + ^))^,
^fx ^ ^2b((^o2 + r2)3 + 4(^2o4 + o2r2 + r4)z2 + 5(^o2 + r2)z4 + 2z6) ^ /(((° - r)2 + z2f/2((a + r)2 + г2)2(^1 + is))
_ 1 (
ZZ “j
47Г
EllipticE
(e)
(a — r) + z2
2b((a2 - r2) + (o2 + 3r2)z2 + 2z4)EllipticK ((a - r)2 + г2)3/2((о + rf + z2)(^l + is))^.
Aar
(a — r) + ^2
(7)
The cylindrical symmetry of the problem justifies the use of the cylindrical system (г, в, z) on the right-hand side of the expressions in (2)-(7), The expressions EllipticK[m] and EllipticE[m] stand for the complete elliptic integrals of the first and second kind, respectively. In accordance with ’Mathematica’ notations [5],
EllipticK[m]
(1
rnsin a)
2 da
EllipticE[m}
(1 — msin2a)2dQ:
(8)
Jo
— the complete elliptic integral of the first kind and the complete elliptic integral of the second kind, respectively.
It is verified that the components of the stress tensor given by (2)-(7) satisfy the equilibrium equations of the elastic body in the absence of body force.
2.2. Cylindrical components of stress tensor
The formulas of transformation of a stress tensor under the linear transformation of the coordinate system [6,7] are used to obtain the cylindrical components of stress tensor from the Cartesian components given by formulas (2)-(7).
orr = (b//((o8(-1 + 2 is) + (r2 + z2f{2z4{^\ + is) + 2 r4is + r¥(-3 + 4 is))+
o4(rV( 11 - 8is) + 3:'(-3 + Ais) + 3r4(^l + Ais)) + o6(r2(3 - 8г^)+
z2(^ 5 + 8^)) + o2(r6(l — 8^) + z6(^7 + 8 is) + r2z4( 1 + 8 is) — r4z2( 3 + 8^)))
EllipticE \---------2-----1 — ((° + rf + z2)(a6(^l + 2 is) + a4(^2r2(M + is)+
('a — r) + z2
2z2(^2 + 3 is)) + (r2 + z2)(2z4(^1 + is) + 2 r4is + r2z2(- 3 + Ais))-
a2(z4( 5 — 6 is) + r4(l + 2 is) — r2z2( 3 + Ais)))ElliptAcK[
Aar
r)2 + г2
(2ят2((а - rf + 22)3/2((а + rf + z2)2(-l + //)),
(9)
iZ.y — 0,
(10)
CJ r
// \2 .
((a — r) + zz)
a6 — o4(r2 — 2 z2) + r2(r2 + z2)2jr 4or
o2(—r4 - 12rV + z4)) EllipticE [ — - z
^ a г ^ I z
a + r)2 + z2)(a4 + a2(^2r2 + z2) + r2(r2 + z2))
EllipticK [ — ------------]) / ((a - r)2 + z2ff2
(o — r j + 7 '
7rr((a + r) + z2) (-1 + is)),
(и)
ан = (Ы (n4(l - 2i/) + а!(-2г! + 3z2 + 2rV - 4z2i/)+
Anf
(r2 + z2)(r2 + 2:2 - 2z2is))EllipticE [------^-----
(o — r) + г2
((о + г)2 + z2)(—a2 - r2 - 2";2 + 2a2is + 2z2is)
Aar
EllipticK [
(a — rf + z2
7гг2л/(o - rf + z2((a + rf + z2)(-1 +
2)).
(12)
r’i,. = 0,
O,
4-7Г
Ц,
о2 + r2)3 + 4(^2o4 + aV + r4)z2 + 5(^a2 + r2)z4 + 2 z6)
EllipticE [
)2R + ^))
(2b((a2 - r2)2 + (o2 + 3r2)z2 + 2z4)EllipticK [ (((a - r)2 + г2)3/2((о + r)2 + z2)(^l + У
4or
r)2 + г2
(14)
Because of the cylindrical symmetry of the problem the components of the stress tensor orr, orz, о ев, <7ZZ do not depend on a polar angle в while the components ore and oez equal zero.
2.3. Asymptotic formulas
To find the asymptotic formulas for the stress field of a prismatic dislocation loop we first transform the right-hand side of formulas (2)-(7) to the spherical system (p,e,ip) and then expand each component into the infinite series about p=oo. This results in the following asymptotic formulas.
64(—1 + v)
a2bp,{b — 8u — 3(1 + 8u) eos[20]+
a.
xy
3Oeos[0]2 cos[4y>] + 24(1 — 2v) eos[2wlsin[0]2) f-
KP
1 3
3(a2bp(3 + Av + 5 cos[2y>]) sin[20]sin[(^]2) (-)
_____________________________________________P
’ 16(—1 + u) ’
7xz —
°УУ
О
yz —
1 3
3( a2 bp eos[0](6 sin[2y>] + 5sin[4y>]))(-)
______________________________________P_
32(—1 + v) ’
———\------- (a2bp,(5 — 8v + (3 + 24zz) eos[20]+
64(—1 + v) V
24(1 — 2zz)eos[0]2 eos[2<^] + 3Oeos[4(^]sin[0]2)
1 3
3(а2Ь/г8т[(9](б8т[299] + 5sin[4y>]))(-)
______________________________________P
32(—1 + u) '
(15)
The error of approximation in (15) is of at most the order of — as p approaches
P
infinity.
2.4. Infinitesimal loop
The stress field of an infinitesimal prismatic loop is obtained in the identical manner. First express the right-hand side of the expressions in (2)-(7) in terms of spherical coordinates (p,6,(p) and then expand the results into infinite series about a = 0. Here "a" is the radius of the dislocation loop.The following are the asymptotic formulas for the stress field of a prismatic infinitesimal loop.
Fig. 1. The geometry of the problem
crxx
1.5 -1 -0.5 0 0.5 1 1.5
7Г
Fig. 2. Contour plot of oxx in vertical plane 0 = — and its cross-section at z = 0 (in the plane of dislocation loop)
Fig. 3. Contour plot of (Txx in
7Г
vertical plane 0 = — and its
cross-section at z = 0 (in the plane of dislocation loop)
Fig. 5. Contour plot of <jxy in
7Г
vertical plane 0 = — and its
cross-section at z = 0.1 (above the plane of dislocation loop)
Fig. 4. Contour plot of oxx in vertical plane 0 = 0 and its cross-section at z = 0.1 (above the plane of dislocation loop)
Fig. 6. Contour plot of gxz in vertical plane 0 = 0 and its cross-sections at z=0.2 and z=-0.2
СГ\
Fig. 7. Contour plot of сгуу in vertical plane 0 = ^
coincides with contour plot of cfxz for 0 = 0 (Fig. 4)
coincides with contour plot of cfxz for 0 = 0 (Fig. 6)
cross-section at z = 0.1 (above the plane of dislocation
loop)
°xx = + ^3(M5 -8v- 3(1 + 8v) cos[26>] +
30cos[#]2 cos[4</?] + 24(1 — 2v) cos[2</?]sin[#]2)a2),
3(6/i(3 + Au + 5 cos[2</?]) sin[2$]sin[ф\2)a2 axy = 16((-l + z/)p3) ’
3(bp cos[#](6 sin[2<^] + 5 sin [4^])) a2
^ = 32((—1 + v)p3) ’ (16)
аУУ = 64(_11+г/-)рз(^(5 - + (3 + 2Au) cos[20]+
24(1 — 2z/)cos[#]2 cos[2<^] + 30 cos[4<^]sin[^]2)a2),
3(6/isin[^](6sin[2v9] + 5 sin[4<^]))a2 °yz = 32((—1 + v)p3) '
The error of approximation above is of at most the order of o4 as the radius of an infinitesimal loop approaches zero. As expected from St.Venant’s principle formulas (15) and (16) coincide for sufficiently large p and sufficiently small radius "a" of an infinitesimal loop,
2.5. Graphs
A number of contour plots and their cross sections is given to show the complexity of the stress distribution in the vicinity of a prismatic dislocation loop. The contour plots in "Mathematica" are essentially "topographic maps" of a function. The contours join the points that have the same value of stress. The more rapidly the stress changes the higher is the density of the contours in this region. Figs, 2-9 show the Cartesian components of a stress tensor in different vertical planes в = в о and their respective cross sections. All plots show that stress changes very rapidly in the vicinity of the dislocation loop. When the cross section of the contour plot is taken in the plane of the loop (z = 0) the absolute value of stress rapidly approaches infinity when the point P approaches the loop,The reason is that the Peach - Koehler formula is not applicable in the very close vicinity of the core of the dislocation.
The cross sections of the contour plots with z ф 0 (in planes above or below the plane of the dislocation loop) show that all stresses attain their absolute extrema directly above or below the dislocation loop.
Because of the cylindrical symmetry of the problem the contour plot of oyy for в = 7г/2 (Fig, 7) coincides with that of oxx for в = 0 (Fig, 4) and the plot oyz for в = 7Г/2 (Fig, 8) coincides with the plot of oxz for в = 0 (Fig, 6),
References
1. N. Langdon, Explicit expressions for stress field of a circular dislocation loop // Theor. Appl. Fract. Mech. 2000. V.33. P.219-231.
2. J.P. Hirth, J. Lothe, Theory of Dislocations. New York: John Wiley , 1982.
3. R.W. Lardner, Mathematical Theory of Dislocations and Fracture. Toronto: University of Toronto Press, 1974.
4. R. de Wit, The Continuum Theory of the Stationary Dislocations // Solid State Phvs. 1960. V.10 , P.249-292.
5. S. Walfram, The Mathematica Book. Walfram Media, Champaign, II / Cambridge: Cambridge University Press, 1996.
6. J. Weertman, J.R. Weertman, Elementary Dislocation Theory. New York: The Macmilan Company, London: Collier-Macmillan Limited, 1964.
7. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity. New York: Dover Publications, 1944.
