THE DIFFERENTIAL GEOMETRY OF BENDING M. J. Marcinkowski*
SOMMARIO. La deformazione associata con la flessione uniforme viene trattata usando le tecniche della geometria differenziale. B circuito di Burgers e altre quantitd tensoriali e la densitd di dislocazione, vengono determinate nei vari stati di deformazione.
SUMMARY. The deformation associated with uniform bending has been treated using the techniques o f differential geometry. The Burgers circuit as well as various related tensor quantities such as torsion and dislocation density have all been determinated with respect to these various states o f deformation.
by a cutting or tearing process so as to generate the configuration shown in Figure 2. This will be referred to as the natural state, as will become more clear shortly, and denoted by lower case Latin letters, i.e., k, £, etc. It is also apparent that Figure 2 refers to the rigid rotations of the volume elements associated with the initial state shown in Figure la. It is now possible to relate the base vectors in the (K) and (k) states as follows: eK = AkK e k
(2.1 a)
and
5
4
3
I. INTRODUCTION. The first quantitative studies of bending date back to over one hundred years ago [1] and are associated with the name of Saint-Venant. However, it was not until 1953 [2] that the first detailed analytical analysis o f the relationship between bending and disclocations was carried out. There still however remain many problems, particularly with respect to large amounts o f or non-uniform bending. The purpose of the present investigation therefore is to extend the analysis of dislocations and bending by applying the concepts of differential geometry in its most general, i.e., non-linear form. It will be shown that the relationship between bending and symmetric tilt type grain boundaries is extremely close.
_L Oo
s
J
2 ~1
o )
ao ~-
(K) STATE
5
4
3
R+p
2. DISTORTION,
METRIC
AND
STRAIN
TENSORS
ASSOCIATED WITH BENDING. Consider the perfect single crystal shown in Figure l a. After uniform bending, it takes the form shown in Figure 1b where the angle 0 was chosen as 53.1 ° for reasons which will become more clear later on. Figure la will be referred to as the initial state and denoted by upper case Latin letters i.e., K, L, etc., while Figure l b will be denoted by lower case Greek letters, i.e., K, :~, etc. This latter state will be referred to as the final state which is of course a deformed state. The elastic strains associated with the (~) state may be removed
b) 0
* Engineering Materials Group and Department of Mechanical Engineering, University of Maryland. SEPTEMBER 1978
Figure 1. Uniform bending: a) initial state; b) final state. 151
which in turn gives
R coso/2 sno/2 i)
4
5
e"--i: -
-
R+p
t~--'~ t*"--: p - ~ _ 3
i:
i-
:
R
ATe=
- -
sin 0/2
cos 0/2
.
(2.5)
R+p 0
0
Similar to Eq. 1, we can for completeness, relate the differential elements of length as follows: dx x = A~ dx k
(2.6a)
and
-0/2
(2.6b)
dx k = A~ dx K
etc. Next, utilizing the relationship A K. A X x = 6x~
(2.7)
where 6~ is the Kronecker delta defined by STATE
5~ =
1
if
X=tc
0
if
X4=K
I
(2.8)
we obtain Figure 2. Natural state associated with the uniformly bent beam of Figure lb. (2.1b)
ek = A ~ e r
/R + p Il R A rK =
(2.2)
0)
cos0/20
01
sin 0/2
0
cos 0/2
0
0
0
1
(2.9)
sin0,2i) A°l_Sio0,2 co0,20 ,cosoi2
and =/sin:/2
- R
while
[AI A~ A~/ --sin0/2
0[2
-- sin 0/2
where A~¢ is a distortion tensor which represents a right rotation and is given by
cosO/2
R + p COS
(2.10)
R 0 ~I
R +p
where 0/2 is the angle of rotation measured from the vertical x 2 axis of the (k) state. The (~) state can in turn be generated from the (k) state by means of the following distortion:
(i :) --
R
A~ =
0
\
0
(2.3)
0
where R is the radial distance measured from the origin to the neutral axis of the bent beam in Figure 1b, while R + # is some radial distance measured relative to the neutral axis. Inspection of (2.3) shows that it is simply the circumferential distortion required to close the gaps in the (k) state of Figure 2 generated by the tearing process. The distortion relating the (K) and (r) states can be expressed in terms of the two separate distortions given by (2.2) and (2.3) as - ~1k A k ~ AK--"K 152
0
1
0
0
@
(2.11)
Inspection of (2.9), (2.10) and (2.11) shows that we can write
0
p
I
Ak=
(2.4)
A K - AkA r K --
~
k"
(2.12)
The above equation simply denotes the operation in which the uniformly bent beam of Figure I b is fin'st brought to the state (k) shown in Figure 2 via the distortion (2.1 I), and is the origin of the term natural state, since the naturalization process releases all of the elastic strains. The distortion given by (2.10) then brings the natural state into the initial state (K). Thus (2.12) represents the inverse of the operations described by (2.4), as expected, since A~ is indeed the inverse of A~. The natural state shown in Figure 2 can also be designated as a tangent space [3]. And element of length associated with each of the three MECCANICA
states described in Figures 1 and 2 may be written as [3]. (ds) 2 = ag L dx K dx L
(2.13a)
(ds) 2 = gg;~dx Kdx ~
(2.13b)
(ds) 2 = b/~ dx k dx ~
(2.13c)
where agL,
4
3
etc. are metric tensors defined as
aKL = eK • eL
(2.14a)
g~x = eK" ex
(2.14b)
b ~ = ek . e~.
(2.14c)
L_
It is apparent from (2.1) that axL = 8gL
(2.15a)
g,cx = A K,
(2.15b)
bt~ = A xk A~K _- ~ .
(2.15c)
We may now write a set of strain tensors for the various states in Figures 1 and 2 as follows: eKx= (g~x-- a~x)/2
(2.16a)
e ~ = ( b ~ -- a~)12.
(2.16b)
All
/
( k' ) STATE
strains vanish with the exception of e x which yields Figure 3. Uniformly diflocated state of the bent beam shown m Figure lb.
0 0
2i 2 e x=
0
.
(2.17)
0
i,
The above expression is simply the strain tensor associated with the transformation from state (K) to state (K). For small strains where R ~, p, (2.17) gives
P
(2.18)
ell c= ~-
as expected. Equation (2.17), on the other hand, applies to large amounts of bending, where the second term becomes important. Note that e n at p = O, i.e., the neutral axis, is always zero. If the overlapping matter below the neutral axis in Figure 2 is removed and inserted into the voids above the neutral axis, we obtain the disioeated (k x) state shown in Figure 3. The (k 1) state may be viewed as a uniform array of vertical edge dislocation wails, i.e., low angle grain boundaries. If these boundaries are coalesced into a single one, the high angle boundary shown in Figure 4 obtains. This will be designated as the (k 2) state. The distortion associated with the (k 2) state may be written as
A~.' = { A~' H(-- l l ) } l +{A~= H(+ I')}2
(2.19)
where H(+ x 1) are Heaviside functions given by K 0 if x l > 0 H(-- ~cl)
= I
g 1
if
while
SEPTEMBER 1978
xl<0
K
.,
I STATE
Figure 4. Non-uniformly dislocated state of the bent beam in Figure lb.
O if x l < O (2.20a)
H(+ l ) = x
1
if
g xt>O.
(2.20b)
g
The Heaviside functions essentially divide the single crystal
]53
shown in Figure la into two halves. These two halves may be treated essentially as separated as emphasized by the curly bracket notation in (2.19). The distortion A,~' may be written as
beam can be written in terms of the following line integral [5,61:
l
A~c'=
/2
1
(2.21)
0 ,
2
while A/~ is simply the inverse of (2.21). Equation (2.19) can still be2used for the (k 1) state, where it is to be understood that we are dealing with each of the small angle boundaries individually. Under these conditions (2.21) can be rewritten
b~ = --~A~ dx 1¢
(3.1)
where the integration is carried out with respect to the path defined in the initial state. It is obvious from inspection of Figure l b no closure failure is associated with the (~) state. This means that A~¢ must be chosen as 5~c as shown in (2.25). For the dislocated (k l) state shown in Figure 3, we may write
as
bk'= -- ~AkK dx K A~=
AO/2
1
0
0
which can be expanded to yield
where A0 is the change in angle across each small-angle boundary. The metric tensor associated with the (k 2) state becomes +1
0
bl/o = - - f
Aldx 1-,2
tan 20/2 + 1
0
0
1
(2.23)
-
i'
A Ida'-
f°
A dx 2.
"5
In view of (2.22) the above equation gives
et: ~' = (cte Q?_ a t : Q~)2=
(2.24) b1 =--4+-k' 2
0
2
0'
0
(2.25)
3. BURGERS CIRCUIT ASSOCIATED WITH A BENT BEAM.
We can now define a Burgers circuit with respect to the deformed state shown by the path 1-2-3-4-5-6-1 in Figure lb. An equivalent path can also be drawn with respect to the natural and dislocated states shown in Figure 2, 3 and 4, respectively. The Burgers vector associated with the bent 154
dx 2
(3.3b)
2
0'
b 1 -- - - Ax 2 - - - Ax 2 k' 2 2-3 2 5-6
The expressions for c k, v and et, t, are similar to those above, but with tan 0/2 replaced by A0/2. It is clear that the component e22 in (2.24) measures the strain associated with a change in length of the line 1 - 4 in the (K) state shown in Figure la to the line 1 - 4 in the (k 2) state of Figure 4. It is also important to know that the plastic distortions given by A~' and At~ correspond to dragged coordinates [41, i.e., the same elements of length d x K and d x k~ in the t w o s t a t e s remain unaltered after deformation. This however is not true for the distortion given by (2.5), but could be made so by writing A~c = 8~ so that
dx '~= ~ dx K.
dx2+4 --
where 0' is the angle subtended by the arc 6-1-2 in Figure 3. The last equation can finally be written as
tan 2 0/2 0
(3.3a)
A1 d x 2 -
O)
whereas
=
(3.2)
(2.22)
(3.3c)
where Ax 2 etc. are simply the distances 2-3, etc. in Figure 1. Equatio2n3(3.3b) is simply the sum of the four closure failures shown dotted in Figure 3. It is clear that in the limit where A0 in (2.22) goes to zero, the dislocation distribution in Figure 3 becomes continuous. In the case of the discontinuous distribution of dislocations, i.e. grain boundary shown in Figure 4, we may write
b k' = - ~A~' dx x
(3.4)
which in view of (2.2 I) yields
blk, = - - 4 + t a n 0 / 2
,3
J2 dx2+4--tanO/2
i'
dx2
(3.5a)
or more simply b I = {4 tan 0/2) 2 + 4 tan 0t2} 1
(3.5b)
k=
which is terms in Figure 4 is MECCANICA
b 1 = (%'.I: ÷
(35c)
Equation (2.19) could also have been used in (3.4) but with Ak' K and by AkK and AkK respectively. This would llead to A~' replaced b I = - - 4 cos 0/2 + {4 sin 0/2} 2 + ks
(3.64)
i3
b2 = 0 +
dx 2 + 0 +
k~
K
i'
dx 2
o r more simply b 2 = {4} 2 + {-- 4} I k3
(3.9b)
+ 4 cos 0/2 + {4 sin 0/2} 1
which in terms of Figure 5 is
or in terms of Figure 4
b/e2 = {Alx:}2+ {~Xl~}]"
b I = {axl}, + {axl},.
kt
4'-Y ~
(3.94)
K
(3.9c)
(3.6b)
5'-4' "
It is important to note that the Burgers vector given by the above equations are not in terms of dragged coordinates, but rather in terms of the initial (K) state coordinates, and has been treated in detail elsewhere [7]. It is apparent that for small 0, the values given by (3.5b) and (3.64) become identical. If the uniformly bent beam of Figure l b is allowed to reach the natural state by means of a single cut, rather than by the multiple cuts shown in Figure 2, we obtain the configuration shown in Figure 5. This will be termed the (k 3) state and may be viewed as the counterpart Of the single large-angle grain boundary illustrated in Figure 4. The closure failure associated with the (k 3) state may be written as
The closure failures given by the above equations do not correspond to dislocations, but to newly created free surfaces. In the case of the (k) state, the distortion would be given in terms of some integer number N of Heaviside functions, so that b 2 would be some multiple N of that given ks by (3.9c).
4. ELUCIDATION OF THE VARIOUS TENSOR QUANTITIES ASSOCIATED WITH A BENT BEAM. It is possible, using Stokes' theorem, to convert the Line integral (3.2) into a surface integral as follows [8]:
bk'=-- ~A~' dxK=-- fatLA~ldFt'X
(4.1)
-$
bk~= ~A~r dx K.
(3.7)
ks The distribution A K can be written as
Af
=
+
I')}2
(3.8)
where
be' = -- (aILA~I dF LK = t
which when substituted into (3.7) yields = --
4'
4
(4.2)
.$
I 1 k' 2[aLAic - - aKA~' l d F L x .
The above equation can be evaluated for the case shown in Figure 3 to give
(4.3a) o$
which further reduces to
blk' = -- f d ( ~ 0 / 2 ) dx 2 = 40'.
(4.3b)
-$
This result is obviously the same as that obtained by the line integral method which led to (3.3b). Still a third and perhaps more fundamental method of expressing the Burgers vector associated with a given Burgers circuit is in terms of the following relation: ~V
(W3) STATE
Figure 5. Limiting case of the natural state shown in Figure 2. SEPTEMBER 1978
bk'='-- f Sin,"t,k: dF rn~ ~
(4.4)
where 155
• • kt
Sm'~'
1
= __
k'
L
K
k=
= ~ra'R'l = A m, A~t aILAK1 =
L
K
k'
(4.5)
k~
2 Am'A~'[OLAK --aKAL ]"
• k= is termed the torsion tensor, while The quantity Sin, ~, P,n k', , is the coefficient of connection. For the uniform distribution of dislocations shown in Figure 3, we may write 1
S..l= 2[AIA I 2 -2- A I A22 ] [1O l A 2 l _ a 2 A ~ ] . k,12
(4.6a)
it is possible to tear or naturalize the (k l) and (k 2) states so as to generate new states which contain both torsion as well as an anholonomic object [10]. These are shown in Figures 6a and 6b and have been termed the (k 4) and (k 5) states respectively. For these particular states, we can write [4]
FI~, e,1 = S~.~./~ -- a~,~,~•
(4.12)
Tile corresponding closure failure is given by
f
Since A0 is small, the product A I2A 21 approaches zero• Furthermore, a2Al1 = 0 since A11 is constant so that (4•6a) becomes
F[/e
d F m'~'
(4.13)
*$
= - - ( [ S ' , • , k ' - - I2",• k'] dFm'~'. S" "]-
m ~
s
(4.6b)
aI
m
When this last expression is substituted into (4.4), we obtain the same value for b I as that given by (4.3b). k~ For the symmetric tilt boundary of Figure 4, we may write, in analogy with (4.4).
In Figure 6, the tearing process has created new surfaces, part of which just balance the closure failures due to the dislocations. These are shown by the horizontal lines at the bottom of the figures. These obviously just compensate the horizontal dotted lines due to the dislocations at the top of the figures. This thus means that
f " k ~ dF m~ b k; = __ | S~n,~z,
S " l = fZ..1
(4.7)
Z
where
__1 2 6(~:)tan
k~12--1
1 6(~:)tan 0/212
(4.8)
where 5(x 1) is the Dirac delta function and arises from k~ differentiation of the Heaviside functions [9]. When (4.8) is used in conjunction with (4.7) we again obtain the same result as that given by (3.5b). The non-vanishing of the torsion tensor for the (k 1) and (k 2) states is an important result since it is synonymous with the presence of dislocations [3]. As in the case of the dislocated states, it is possible to write corresponding surface integrals for the torn of ardlolonomic states (k) and (k3). When this is done, we are able to write a quantity S2m~ktermed the object of anholonomity defined as follows [3, 4, 10, 11]: ~2~k
L r
k
= AmA~ aILAKI
(4.9)
k, 12
so that b~ given by (4.13) is zero. Similar results obtain for state (kS). The torsion tensor can also be used to determine the dislocation density ctR'ra' by means of the following relation [12]:
~'rn' = -- e~'o'P' S. -,m'
(4.15)
op
5
4
't A
h ,i'___/" .t
:A
91°--
qlo°"
6
:t
:t
41*°o
4°°*
I
Z
o)
which gives rise to the following surface closure failure:
( b k = tI2m'~ k d F m~.
(4.14)
k , 12
(k4)STATE 5
(4.10)
5'
4
5
3'
°
2
1)
.........
:¢
/¢ $
"'~1 -d . . . . . .
l
For the (k 3) state, we find with the help of (3.8)
k 3 12
"4"
P
(4.l 1)
which when substituted into an equation of the form given by (4.10), yields the same result as (3.9b). The states considered thus far have contained either torsion or an anholonomic object, but not both. However, 156
6
q.***
I
b)
4,o..4o.°,4...
I'
( k 5 ) STATE
Figure 6. Natural state of the dislocated state shown in a) Figure 3, b) Figure 4. MECCANICA
where e ~' o' # is the permutation tensor defined by eg'°*P' = e~'°'P' /%/rgg
(4.16)
and where e ~'°'p' are permutation symbols and g is the determinant of the metric tensor, which in the present case is equal to unity. Thus, in the particular case of the dislocated state (k 1) in Figure 3, we obtain c~31 = 2S::.1 = k' k ,'L
a f °t
(4.17)
l ~-~-1 •
The above is simply the change in angle with X 1 . The indices k* 21 and m 1 in o~~' m' refer to the dislocation line direction and Burgers vector component respectively. On the other hand, the dislocation density associated with the symmetric tilt boundary of Figure 4 is cx31k'= {/tC~) tan 0/2} 1 +
{~(:~)tan 0/2}2
k,
/A.~2 1'
- -
(4.18b)
We could also write, similar to (4.15) o~Sm~
=
e R~o3p I a o , ;, m ~
(4.19)
which with the aid of (4.11) yields
c( 2 k~12 --
_(~(~) 1 + 5(~.~)2
A detailed differential geometric analysis has been carried out for a uniformly bent beam with and without dislocations. The Burgers circuit, and various tensor quantities such as torsion, dislocation density, distortion, etc. have all been treated in detail with respect to these various dislocated and undislocated states.
ACKNOWLEDGMENTS.
.
I A l ~ 12
SUMMARY AND CONCLUSIONS.
(4.18a)
or more explicitly in terms of Figure 4 ~3~ = 1 s"--2-4 1 +
The above equations simply measure the density of newly created surfaces.
(4.20a)
or in terms of Figure 5
The present research effort was carried out at the Institut ftir Theoretische und Angewandte Physik der Universit~t Stuttgart in The Federal Republic of Germany under a Senior U.S. Scientist Award presented to the author by the Alexander yon Humboldt Stiftung in conjunction with a one-year sabbatical leave. Financial support for the present study was also provided in part by the National Science Foundation under Grant No. DMR-7202944. The author is indebted to Professor Ehhehart Kr6ner for his kind assistance during the course of this investigation.
Received: 15 March 197%
REFERENCES. [ 1] FUNGY. C., Foundations of Solid Mechanics, Prentice Hall, New Jersey, 1965. [2] NYE J.F., Some Geometrical Relations in Dislocated Crystals, Acta MetaUurgica,Vol. 1, 1953, pp. 153-162. [3] KONDOK.; Memoirs of the Unifying Study of the Basic Problems in Engineerfng Sciences by Means of Geometry, Vol. 1, Gakuyutsu Bunken Fukyu-kai, Tokyo, 1955. [4] SCHOUTEN J.A., Ricci-Calculus, Springer-Verlag, Berlin, 1954. [5] KR6NER E., Allgeraeine Kontinuumstheorie der Versetzungen und Eigenspannungen, Archive for Rational Mechanics and Analysis,Vol. 4, 1959, pp. 273-334. [6] HIRTH J.P., LOTHE J., Theory of Dislocations, McGraw-Hill Book Company, New York, 1968. SEPTEMBER 1978
[7] MARCINKOWSKIM.J., The InterrelationshipBetween Internal Boundaries of Large and Small Misfit, Physica Status Solidi, Vol. (a) 41, 1977, pp. 213-224. [8] SCHOUTENJ.A., Tensor Analysis for Physicists, Oxford at the Clarendon Press, London, 1951. [9] DEWrr R., Theory o f Disclinations: IV. Straight Disclinations, Journal of Research of the National Bureau of Standards-A. Physics and Chemistry, Vol. 77A, 1973, pp. 607-658. [10] ZORAWSKIM., Thdorie Math~matique des Dislocations, Dunod, Paris, 1967. [II] ERINGEN C.A., Continuum Physics, Vol. I, Academic Press, New York, 1971. [12] KR(~NER E., Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer-Verhg,Berlin, 1958. 157
