Stress Field Near a Dislocation Slip Plane in an Orthotropic MediumNazeer Ahmed Citation: Journal of Applied Physics 37, 9 (1966); doi: 10.1063/1.1707898 View online: http://dx.doi.org/10.1063/1.1707898 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/37/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Peierls stress of a screw dislocation in a piezoelectric medium Appl. Phys. Lett. 85, 2211 (2004); 10.1063/1.1790030 Elastic fields of a dislocation loop near a stressfree surface J. Appl. Phys. 49, 4953 (1978); 10.1063/1.325503 Unconventional behavior of edge dislocations in a plane shockstress field J. Appl. Phys. 47, 22 (1976); 10.1063/1.322352 Elastic Stress Field of an Angular Dislocation J. Appl. Phys. 34, 2337 (1963); 10.1063/1.1702742 Linear Dislocations in Nonuniform Stress Fields J. Appl. Phys. 33, 3312 (1962); 10.1063/1.1931161

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JOURNAL OF APPLIED PHYSICS VOL U M E 3 7. N U:M B E R 1 JANUARY 1966

Stress Field Near a Dislocation Slip Plane in an Orthotropic Medium.

NAZEER AmmD

Department of Theoretical and Applied Muhanics, Cornell University, Ithaca, New York (Received 21 May 1965)

The elastic stress field in the vicinity of a dislocation slip plane in an infinite orthotropic medium subjected to uniform shear at infinity is determined using a formulation in integral equations. It is assumed that a slip plane is different from a traction-free surface crack in that it has uniform shear on its surface and remains closed under the applied stress field. However, the use of these mixed boundary conditions under the re­strictions of plane stress in linear elasticity is found to give the same stress distribution as that for a shear crack. Although a shear crack under an applied uniform shear opens up into an ellipse, the opening per unit length is shown to be so small that for most engineering applications it can be considered closed and the equivalence of a shear crack and a slip plane is established. It is pointed out that this correspondence is not valid between a shear crack in a thick elastic plate and a slip plane in an infinite elastic medium. Results for an isotropic body are given as a limiting case of an orthotropic body.

I. INTRODUCTION

T HE elastic stress field in the vicinity of a traction­free surface crack under uniform applied shear

has been studied extensively,l-3 A dislocation slip plane is different from a shear crack in that it can support a shear stress on its surface and, unlike the shear crack, does not open up under the applied stress. The use of these mixed boundary conditions on the surface of the dislocation slip plane complicates the analysis of the stress field in its vicinity. It is the purpose of this paper, therefore, to investigate the effect of these more appro­priate boundary conditions on the stress field of a slip plane.

E., Eyz, E.", are zero, and this is used to eliminate CTz, Til"

Tz", from the expressions for E"" Ell, Exy which are now written

II. MATHEMATICAL FORMULATION

Figure 1 shows a dislocation slip plane in the x-z

plane of a right-handed Cartesian coordinate system x,y,z, at y=O, -l<x<+1. It is assumed that the slip plane is straight and infinitely long in the z direc­tion. A uniform shear Tzy is applied at infinity in the x-y plane. On the surface of the slip plane the y displace­ment v is specified as identically zero and a shear stress T xll= Tf assumed uniform, acts on it.

For such a state one can assume a plane-strain situation in which the z displacement w vanishes and the x and y displacements are independent of z. In an orthotropic elastic body, the strains are related to the stresses by

Ex=Ex-1U x--' IIYXEy-1Uy-lIzxEz-1uz,

Ey= -IIX yEX- 1CF x+ Ey-lCFy-lIzyEz-lCFz,

E.= -lIx.Ez-1CFx-lIyzE1I-ICFy+E.-IU.,

Ey.= GII.-1T yz , Ezx= G.z-1Tzx, Ezy= G",y-1Tzy, (1)

where, because of the symmetry of the stress-strain matrix, EzIl II", = Eyllzy , Eyll.y=Ezllyz, Ezllzz=E",lIz",' For plane strain in the x-y plane the components of strain

1 A. T. Starr, Proc. Cambridge Phil. Soc. 24, 489 (1928). 2 J. D. Eshelby, Proc. Roy. Soc. (London) A241, 376 (1957). 3 D. D. Ang and M. L. Williams, J. Appl. Mech. 28, 371 (1961).

9

where

Ez= SUCFz+S12CFIII

Ey = S 12CF '" + 822CF y,

Ezy=866TXY'

8u =Ex-1- vzxllxzE",-l,

8 12 = -lIl1xEy-1_lIzzllyzEy-1,

8 22= E1I-1_IIZyllyzEy-t,

866= GXy-1.

If a stress function {is chosen such that

the equations of equilibrium

CFii,j=O, i, j=x, y, z,

(2)

(3)

(5)

are satisfied. Combining (1)-(5) with the compatibility condition

a2Ez a 2Ey a 2E",y

-+-=­aj ax2 axay

gives the governing field equation

(6)

8 22Xzzxx+ (2812+866)Xzxyy+8u xYYYIl= 0. (7)

This can be factored as

(8)

where

(9)

The total stress field can be separated into two parts, one due to the uniform shear field at infinity, the other

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10 NAZEER AHMED

! Lim. 'txy (x,y)= 'Tf Iyl- 0

Lim. Vt(x,y) =v-ex,y)- 0

IvJ-o , plane strain condition

we x/y) ·const.

FrG. 1. Boundary conditions for a dislocation slip plane.

due to the slip plane so that the stress function can be written as

(10)

where x""=-'i',,,uXY and X gives the stress field due to the slip plane which satisfies (7) and the boundary conditions

Ixl<1, (11)

lim V+(x,y) =r(x,y) =0, Ixl <1. 1111...0

(12)

As 1x2+y211~ lXI, Tij(X,y)~O, (13)

where the + and - superscripts indicate limits ob­tained by approaching the x axis from above and below, respectively. In the region outside the slip plane, in order that the fourth derivative of X exist, X and its first three derivatives should be continuous. In par­ticular along y= 0,

o"x(x,o+) onx(x,(r-) ---+ , n=O, 1,2,3, Ixl ~ 1. (14)

oy" oy"

m. SOLUTION

Let X be written as a Fourier transform

x= i co

R(y,t) cosxtdt+ i co

R(Y,t) sinxtdt. (15)

When (15) is substituted in (8) and the constants are rearranged, a solution that satisfies the governing field

equation (7) can be written as

2 fCO x=:E Gi(t) costxe-K;tlllldt i-I 0

+t r G/(t) sintxe-Kitlllidt. (16) i=lJO

Substitution of (16) in the continuity requirements (14) gives

iCO[Gt'(t)+G2'(t)]sintxdt=0, Ixl~1, (17)

lCOt[KlGl(t)+Ki]2(t)]costxdt=o, Ixl~1, (18)

l co

t2[KI2G1' (t)+K22G2' (t)] sintxdt=O, I xl ~ 1, (19)

i"" t3[Kl3G1 (t) +K23G2 (t)] costxdt=O, I xl ~ 1. (20)

Similarly, the boundary condition (11) gives

lim (t - r Kit2Gi(t) sinlxe-K,tlllidt 1111...0 ;=1 Jo

=Tf-'fz1l• Ix/ <1. (21)

To express (12) in terms of X one observes from (2) that

where a/x represents a rigid body rotation about the z axis and is hence discarded. From (12) and (22) one obtains, therefore,

lim {t r tKrIG.(t) costxe-Kil11lldt 1111...0 ;=1 J 0

21"" + :E tKrlG/ (t) sintxe-K,111Ildt ... 1 0

+ t [ao KitG/ (t) sintxe-Kitfllldt]} = 0, I x 1< 1. .-1. 0

(23)

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STRESS NEAR SLIP PLANE IN ORTHOTROPIC MEDIUM 11

A possible solution of (18) is

l'" tG1(t) costxdt=O

and

l'" tG2(t) costxdt=O, Ixl ~ 1. (24)

Then (20) is satisfied by differentiation. Similarly, a possible solution of (19) is

Jo'" t2GI' (t) sintxdt=O

and

l'" t2Gl (t) sintxdt= 0, ! x I ~ 1. (25)

Then (17) is satisfied by integration. The constants of integration represent a rigid body motion at most and are discarded.

To solve for the unknown G.(t) and G/(t) one defines a function uW such that

(26)

where UiW=O and u/W=O for Ixl ~ 1, i=1 or 2. Then, taking the inverse Fourier integral transforms of (24)-(25),

and

211 G;.'(t)=- u/w cos~td~, i=1 or 2. 7rt2 0

(27)

When Eqs. (27) are substituted into (21) and (23) and the limits are taken, one obtains

101

uIWL1(~)d~+ 101

u2WL2(~)d~

+ 101

u/ (~)L3Wd~+fol U2' WL4(~)d~

and

i1

uIWL5Wd~+ i 1 u2WL6Wd~

+ i1

U/(~)L7(~)d~+ 101

u2'(~)L8Wd~=O, (29)

respectively, where

Kl~ LaW= ~_~'

a~ L7(~)=-­

,&-~'

K2~ L4(~)=-­

~-e'

fj~ LsW=--,

,&_~2

a=Kcl-KlS12/S22, fj=K2-1_KJ312/S22.

(30)

(31)

(32)

Since the coefficients of G1(t) and G2(t) are identically zero in both (28) and (29), to avoid further complica­tions one chooses G1(t)=G2(t)=0. Coordinates x and ~ are now transformed such that X2='1/, ~2=X, and u'W=u(>..). From (27)-(31), u.(>..) are found to satisfy

where

f1U;,(X)d}..

. --=m., i=1 or 2, o X-'1/

ml= i'n"fj( f"y-1',)/ (K~-K 20i) ,

m2= 17ra(f"y- T,)/ (K~-K~).

(33)

(34)

Equation (33) is a Cauchy's singular integral equation of the first kind. A solution for u; is5

From physical considerations it is known that the solution is bounded everywhere between -1 <x< + 1 and in particular at x=O. This requirement is satisfied by choosing C= -m;[27r( -1)t]-\ i= 1 or 2. Integration of (35) then gives

Uia) = -m.~/ (e-l)-i, i= 1 or 2. (36)

From (27) and (36) one obtains

G/(t) = (mi/7rf2)fx(t), i= lor 2. (37)

The stress function is then given from (10) and (16) as

X=7r-1!oco t-2J 1(t) sintx

X [mle-Kttllll+m2e-KSIIYIJdt- T"'I/X)" (38)

The complete stress field in the orthotropic body con­taining the dislocation slip plane is known from this equation. A region of particular interest is along the x

4 H. W. Liu, J. Appl. Phys. 36, 1468 (1965). 6 S. G. Mikhlin,Integral Equations, Intemational Series of Mono­

graphs on Pure and Appl. Math. (1957).

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12 NAZEER AHMED

axis. Evaluating the integral in (38) along this axis,

u",(x,O) = - (Kl+K2)('f",y- r,)x(1-x2)-t,

u",(x,O) =0,

uix,O)=O

r",y(x,O) = (f"y-r,)

Ixl <1, (39)

Ixl~1, (40)

for all x, (41)

X {1+ (x2-1)-I[x+ (x2-1)t]-l},

Ixl ~ 1, (42)

Uz(x,O)=E.E.,vuu.,(x,O) for all x. (43)

The displacement field along the x axis is

u(x,O) = ('f",y-T,)E.,-I(Kl+K2) (1_X2)t, /x/ < 1, (44)

u(x,O)=O, Ixl ~1, (45)

v (x,O) = [EII-I (KCIK2-1+S12/S22) +!G"II-1](f.,y-r,)x, Ixl <1, (46)

Ixl ~ 1. (47)

For an isotropic medium K 1=K2= 1. It can be verified that as Ki-7 1, the stress function (38) becomes

X (isotropic) = ('f"'II-r,)

xl'" r 2J1(t) (1-tly/) sintxe-tlllldt-'f"'lIxy (48)

and the stress and displacement fields are easily ob­tained from (39)-(47). It is observed that the stress fields in an orthotropic medium and an isotropic medium have the same qualitative behavior; they have the characteristic square-root singularity at the tip of the slip plane.

IV. CORRESPONDENCE BETWEEN SHEAR CRACK AND DISLOCATION SLIP PLANE

The correspondence between linear dislocation arrays and shear cracks in isotropic media has been pointed out by Liu.4 For plane-stress state it can be verified that the stress field obtained here for a slip plane in an orthotropic medium reduces to that for a shear crack in a thin plate under uniform applied shear as given by Ang and Williams.s Consequently, the results obtained by Liu4 for an isotropic meditiln can be extended to an or tho tropic medium. For exatnple, if one considers a double-ended linear edge dislocation array gliding on the slip plane, the dislocation density on it can be determined from

f(x) = [E.,(X,o+)-E.,(X,o-)]b, (49)

where E",(X,O+) and E.,(X,o-) illdicate the strains on the surface of the slip plane aboye and below the x axis, respectively, and b is the But-gers vector. Substituting

y,y' y"

x"

e ----~"'--'--~---x.x'

FIG. 2. Coordinate transformations of a point on the surface of the slip plane.

from (44),

f(x) = - [(fXy -TI)E.,-1(Kl+K2)x](1-x2)-tb (SO)

and this is identical to the strain in the x direction on the surface of a shear crack in an infinite orthotropic medium under a uniform applied stress as given by Ang and Williams.s It should be pointed out, however, that the stress field in a thick plate in the vicinity of a shear crack depends on the thickness of the plate and there­fore there is no correspondence between a shear crack in a thick plate and a dislocation slip plane in an infinite elastic medium.

One observes that the equivalence of a shear crack and a slip plane in a plane-stress state has been estab­lished although the boundary conditions for the two are seemingly different. To investigate this further, one considers the displacements of a point on the surface of the slip plane. They are related by

u2(x,0)

so that the position of this point after deformation is given by

x'=x+u, y'=v,

where u and v can be written as

(52)

u (x,O) = A sincI>, v (x,O) = B coscI>. (53)

A and B are respectively the semimajor and semiminor axes of the ellipse (51) and cI> is a polar coordinate. Re­ferring to Fig. 2 there is, in general, an additional rotation of the glide plane described by 0 so that the final position of a point on its surface is given by

x"=x' cosO+y' sinO, y"=y' cosO-x' sinO. (54)

For an isotropic elastic body A=B=2(f.,y-T,)/E and elimination of x', y', 0 and cI> between (51)-(54) gives

(55)

so that the opening of a slip plane of unit length in an

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STRESS NEAR SLIP PLANE IN ORTHOTROPIC MEDIUM 13

isotropic medium is approx A 4, Typical values, for example, for single-crystal zinc are E= 15X 106 psi, if "'11- T J= 30 psi. Here the total opening of a dislocation slip plane 1 in. long is approximately 1.6X 10 .... 11 in. This is so small that the linear theory of elasticity does not distinguish between the boundary conditions for a traction-free surface crack and the more appropriate boundary conditions used here for a dislocation slip plane. They have, therefore, the same stress concentra-

tion factor, and the nature of stress singularity at their tips is the same.

In conclusion, the stress field in the vicinity of a slip plane in an orthotropic medium is qualitatively the same as that in an isotropic medium and is identical to that for a shear crack under the restrictions of plane stress. This equivalence does not hold between a shear crack in a thick plate and a slip plane in an infinite medium.

JOURNAL OF APPLIED PHYSICS VOLUME 37, NUMBER 1 JANUARY 1966

Further Considerations on a Theory of Superlinearity in CdS . and Related Materials*

GUSTAVO A. DUSSELt AND RICHARD H. BUllE

Department of Materials Science, Stanford University, Stanford, California (Received 22 April 1965)

A theory of superlinearlty by Cardon and Bube is extended by considering the effect of a high density of shallow traps, either discrete Of with a quasiexponential distribution. New conditions fOf the breakpoints of superlinearity are introduced. These new conditions allow the explanation of several features of super­linearity in sintered layers, including the "anomalous" observation of lifetime decrease above the superlinear region, as described in investigations on CdSe by Stupp. A possible relationship between such trap dis­tributions and an apparent decrease in sensitizing center hole ionization energy in highly impure single crystals of CdS is suggested. A summary of all the basic superlinearity conditions is given, with principal emphasis on the physics of the involved mechanisms.

INTRODUCTION

By superlinearity is meant a variation in photo­conductivity with a power of excitation intensity

greater than unity. Such an effect requires an increase in free-electron lifetime with increasing excitation intensity. A typical variation of photocurrent with excitation intensity through a superlinear range is given in Fig. 1, together with an indication of the common terminology.

A simple model for superlinearityI-3 has been generalized by Cardon and Bube! to include the effects of a quasicontinuous trap distribution on the form of the dependence of photocurrent on excitation intensity in the superlinear range. The model is shown in Fig. 2 with such additions as are also to be included in the present treatment. A notation consistent with Cardon and Bube is used, defined as follows:

F=density of free electron-hole pairs created per second,

* Work sponsored partially by the U. S. Army Research Office (Durham).

t Fellow of the University of Buenos Aires. Permanent address: Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, University of Buenos Aires, Argentina.

1 R. H. Bube, J. Phys. Chem. Solids 1, 234 (1957). 2 H. A. Kiasens, J. Phys. Chem. Solids 7, 175 (1958). 3 J. Voigt, Phys. Status Solidi. 2, 1689 (1962); 5, 123 (1964). 4 F. Cardon and R. H. Bube, J. Appl. Phys. 35,!3344 (1964).

N.= total density of states (i= 1, 2, 3, 4),

ni= density of states occupied by electrons (i= 1, 2, 3, 4),

{3 •• = capture probability for electrons at empty centers i,

(3hi= capture probability for holes at occupied centers i,

n=density of free electrons,

p= density of free holes,

E.= electron ionization energy to the conduction band from centers i for i=3, 4; or hole ionization energy to the valence band from centers i for i= 1, 2,

P.i=probability for thermal excitation of electrons from centers i to the conduction band for i= 3, 4; Pei=NJJ.i exp( -Ei/kT),

N c= effective density of states in conduction band,

Phi=probability for thermal excitation of boles from centers i to the valence band for i= 1, 2; Phi = N J1hi exp( - E;jkT),

N,,=effective density of states in valence band.

Neglecting hole recombination step {3h3 and omitting traps N4 completely, Cardon and Bube showed that the temperature dependence of the free electron density at

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