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The Principle of Asymptotic
Proportionality
H. C. M. Chan1
Research Assistant,
Department of Civil Engineering,
Massachusetts Institute of Technology,
Cambridge, MA
The Principle of Asymptotic Proportionality, which is based on the Green's function method for equilibrium problems, is proposed. Using this principle, the induced
far-field variable due to any distribution of applied physical quantities can be approximated. This principle has been verified by considering the induced stresses due
to applied tractions and dislocations in two-dimensional linear elastic media, and
has been shown to be applicable to other physical phenomena such as electrostatics,
gravitation, and electromagnetism.
Introduction
The Green's function method has been widely used to solve
equilibrium problems (for example, see Hildebrand, 1976).
For a physical system under equilibrium with given boundary
conditions, the induced field variable (such as stress,
temperature, and electrostatic potential), due to a unit concentrated "charge" (such as traction, heat source, and electrostatic charge), is given by a Green's function (or influence
function). Using the Green's function, the resultant induced
field due to any applied distribution of charges can be found.
Capitalizing on this method of solution and postulating a
specific property of the Green's function, the induced field at
large distances from the location of the applied charges can be
approximated.
Starting with the analysis of stress fields induced in a twodimensional linear elastic medium by a certain applied traction
distribution, the Principle of Asymptotic Proportionality
(PAP) is introduced. Then it will be shown that PAP can also
be applied to other applied charges, including semi-infinite
dislocations, electrostatic charges, and heat and fluid flows.
The Principle of Asymptotic Proportionality
G(x, OP)p{x)dx
F=
\G(X,OP)\X p(x)dxY
-\ho{\XoP{X)dx)dG^P)
dx
G(h, OP)\ p(x)dxJO
\ p(-»(x)GW(x,
dx
OP)dx,
(2)
JO
where
p<~»(x)=\XoP(x)dx
(3)
and
G<»(x, OP)-
dG(x, OP)
dx
(4)
Hence,
Consider a two-dimensional linear elastic medium with
given boundary conditions (Fig. 1). The medium is at
equilibrium with an arbitrary applied traction distribution (in
the j-direction), p(x), on a segment of length h on the x-axis.
The stresses at a certain point P in the medium induced by the
applied traction, p(x), can be found by using the suitable
Green's function for the given boundary conditions:
f
where F is a certain stress component and G(x, OP) is the corresponding Green's function.
Using integration by parts, we have
F=G(h, OP)p^i\h)-
\ G^,{x,
OP)^~1Hx)dx.
(5)
(1)
Jo
i tPresently at British Petroleum Research Centre, Sunbury-on-Thames,
Middlesex TW16 7LN, U.K.
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF
MECHANICAL
MECHANICS.
ENGINEERS
for
publication
in
the
JOURNAL
OF APPLIED
Discussion on this paper should be addressed to the Technical Editor, Leon
M. Keer, The Technological Institute, Northwestern University, Evanston, IL
60208, and will be accepted until two months after final publication of the paper
itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received and accepted
by the ASME Applied Mechanics Division, April 12, 1988.
Fig. 1
A two-dimensional medium under stress equilibrium
MARCH 1990, Vol. 57/225
Journal of Applied Mechanics
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0.9
0.8
LU
n
en
o
Q
n
ITT
u
CO
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Fig. 2
0.D
Kelvin's problem for plane strain
2
4
6
8
10
Y (X = 1)
Fig. 4
Ratio of induced a yy alorig Section A
(a)
II
Section B
p(x) = 1
Equation (7) can be considered to be a set of conditions
specifying that kth {k= 1, 2, 3, . . . ) order equilibrium exists
over the applied traction distribution. When (7) is not
satisfied, zeroth-order equilibrium (£ = 0) is considered to
exist.
The Principle of Asymptotic Proportionality (PAP) is thus
stated as follows:
When the applied traction distribution is under the &thorder equilibrium, with minor exceptions, the induced stresses
F at a point P far from the region of the applied tractions are
approximately proportional to the (k+ l)th integral of the applied distribution:
Section A
(b)
P(x)-1/2x
Fig. 3 Two different applied traction distributions
By applying integration by parts repeatedly to (5), we have
F=(m
2
F=G(h, OP)P<--»(h)-G (h,
2
+ GV>(h,OP)P*-- \h)-.
+
OP)p< >(/J)
(~\)mG^m\h,0~P)p^m-l\h)
OP)p<--m-l\x)dx
(6)
Kelvin's Problem For Plane Strain
An example application of PAP is made using Kelvin's
problem for plane strain in which a line force (with dimension
force/length) is applied in the ^-direction in an infinite linear
elastic solid (Fig. 2). The Green's function for the induced
stress ayy at point P(s, n) is given by Crouch et al., 1983:
(111 = 0 , 1 , 2 , . . . ) ,
where
pl-i~1\x)=
(9 repeated)
(The minor exceptions will be considered later.)
It should be noted that the aforementioned formulation for
PAP can be applied to other physical quantities as long as
linear superposition, as exemplified by (1), holds. Thus PAP
can be applied to many physical phenomena such as electrostatic and magnetic fields, and heat and fluid flows.
..
+ ( ~ l ) m + 1 \ G( m+1 )(x,
\)kGW{h, OP)p<-*-')(/!).
[ p(~'Xx)dx
G{x, OP) = 2(1 -V)g„-ngnn
G(/+1>(x, OP) = —G®(x,
ax
OP) (/=0, 1 , 2 , . . . )
where c = Poisson's ratio
1
\n[(s-x)2
g(s, n) =
4TT(1 - v)
pl°\x) = p(x), G®\x, OP) = G(x, W>).
Now if
p(-D(/z) =
jD (-2)( A ) =
. . . =p(-«(/!) = 0,
p(-*-"(/!)^0,(A:=l,2,3,. . . ) ,
g„ = dg/dn -(7)
then
1
47r(l-i>) (s-xf
gn„ = d2g/dn2 = -
4TT(1-^)
k i
F = ( - 1)*G<*>(A, OP)p<-- - \h)
+ ( - 1)* +1 f G<*+1>(x,
~OP)pi-k-i\x)dx.
(8)
Jo
(It has been implicitly assumed that G(x, OP) is differentiable
and p(x) is integrable with respect to x (fc+1) times.) Let
/•= IO.PI._It is postulated that, with minor exceptions,
Gik+[)(x, OP) (with 0 < x < / 0 has a lower order of magnitude
than G(k)(h, OP), when r/h is large. Assume (reasonably) that
pi~k~i)(x) is finite and bounded. Thus, for large r/h,
Fs(-\)kG<-k\h,
226/Vol. 57, MARCH 1990
OP)p<--k-l){h).
(9)
(10)
+ n2n 1/2
+ n2
(s-x)2
[(s-x)2
+ n2]2'
( H )
For a certain distribution of line forces, p{x), as shown in
Fig. 1, the induced stress, ayy at P, is given as F in equation
(1). Now suppose that zeroth-order equilibrium exists (k = 0).
It is to be shown, as an example, that (9) holds for two specific
traction distributions.
The first distribution is a constant distribution of applied
traction along the x-axis and the second a linear distribution
(Fig. 3). The induced stress, ayy, by each of the two distributions has been found in closed form (Chan, 1986) and the
stresses are compared along Section A (Fig. 3(a)). The ratio of
ayy due to the linear distribution (denoted by PD (LSDE)) to
that due to the constant distribution (PD(CSDE)) is plotted in
Fig. 4 against y. It can be seen that the ratio remains at around
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"
HI
(1)
1/3
2/3
(d) A force couple
(a)
(1)
(-1)
X (Y = X)
Fig. 5
1/3
Ratio of induced <7yy along section B
1/3
p<_1)(/0 = 0
p(-2)(/i)= [
(force equilibrium)
1
(e) Self-equilibrating forces
P (x) = -2
< 1 )I| ((-3)
- 3 )•f ( 3(3)|(-1)
)|(-1)
i.
1
i
•,(
(f) Self-equilibrating forces
(c) A uniform pressure
Fig. 6
2/3
1
(b) A unit concentrated force
0.5. This is because, as according to (9), ayy is approximately
proportional to p(_1)(/j), which is the total applied force (see
(3)). The total applied force for the linear distribution is half
of that for the constant distribution. The same ratio along Section B (Fig. 3(a)) is also plotted against x (Fig. 5). It can be
seen that the ratio approaches 0.5 as x becomes large, i.e., r/h
is large.
For self-equilibrating applied traction distributions, SaintVenant's Principle holds which states that the induced stresses
at points far from the origin are negligible (for example, see
Sternberg, 1954; Horgan and Knowles, 1983). When viewed
under the framework of PAP, we can see that the same conclusion can be reached. For self-equilibrating traction p(x),
a
<i)
i(-1)i(-1)
Two-dimensional linear elastic medium under applied tractions
p(x)=-&(x
—j,
where 5(x) is the Direc-Delta distribution. Therefore,
p^l\x)dx
In case (c),
= [xp<-"(*)! - \ xp(x)dx
L
JO
/>(*)=-2.
Jo
-\hp<-~[Hh)-0 - 0
Therefore,
(moment equilibrium)
P ( - 1 ) (l)=J o '(-2)c?x=-2.
-0.
Thus, second or higher-order equilibrium exists (k>2). From
(9) and an inspection of (10) and (11), it can be concluded that
the induced stress ayy approaches zero as r/h increases. Thus,
in this example, PAP includes Saint-Venant's Principle and
actually tells us how the induced stresses decay through (9).
Thus, a in (c) is approximately twice that in (b), which agrees
with intuition because the total applied force in (c) is twice that
in {b). Here, Saint-Venant's principle cannot be applied directly because the applied tractions in (b) and (c) are not statically
equivalent.
For case (d),
Extended Concept of "Equilibrium"
pw=-*(*-4-)+*(*--f)
The previous discussion shows that second-order
equilibrium is static equilibrium under the conventional viewpoint. Applied distributions at higher-order equilibria and the
resulting far-field stresses are examined in this section. Consider applied tractions on a straight line segment (Fig. 6) with
example cases (b) to if). Suppose we are interested in induced
oyy at point P, denoted by a, due to the applied tractions. At
first we must determine the order of equilibrium of p(x). For
p(x) given in Figs. 6(b) and 6(c), zeroth-order equilibrium
(k = 0) exists because
-1>(1>^
.-. p<-»(x)=j%(*)rfx
=
where u{x) is the unit step function.
p(-2>(x)= j"p(- l \x)dx
p(x)dx^0.
In these cases, a is approximately proportional to
„(-*-•) (I) = p<-»(1)
In case (b)
Journal of Applied Mechanics
-U(X~T)+U(X-T)'
=- (*--fM*--f)+(*-TM*-T)Thus,
MARCH 1990, Vol. 57/227
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P(X)
PM
(-1)
(1)
1/3
2/3
(1)H)
(-D
(1)
1/3
2/3
1
1
^j
H—»-x
1
,H)(x)
f (-1) x)
1
1p(" 2 )(x)
P *-2)
'•i (x)
1/3
-I
1—»-x
1/3
-1/3-Fig. 7
_)
B^X
Applied distribution at first-order equilibrium (k = 1)
-1/3
Fig. 8
Applied distribution at second-order equilibirum (k = 2)
,(-') ( 1 ) = - 1 + 1=0
0)—0-T-) + (1-T-) = -T' 40 -
n <-2>r
PM
Therefore, first-order equilibrium (k=l) occurs. This result
can also be obtained by integrating p{x) graphically as shown
in Fig. 7.
Since k = 1 in (d), a in {d) is of a lower order of magnitude
than that in (6) and (c).
For cases (e) and (/), static equilibrium exists, and by SaintVenant's principle a is small. However, by using PAP we can
see further that a in (J) is actually of a lower order than a in
(e). The integrations required to obtain the orders of
equilibrium are carried out graphically in Figs. 8 and 9.
In Fig. 8, after obtainingp(_2)(x), it is clear that p <_3) (x)^0.
Thus, k-2. In Fig. 9, &=3. Thus, one is tempted to say that
the applied tractions in Fig. 9 is "more at equilibrium" than
those in Fig. 8.
(1) (-3)
,
(3)
(- 1)
.
11
1/3
1
2/3
1
pH)(x)
-1
-2 +
Theorem of Equivalent Expansions
Fracture opening and slip displacements can be modeled by
semi-infinite dislocations (Chan, 1986). A semi-infinite
dislocation occurs when there is a displacement discontinuity
across the two surfaces of a slit which begins inside an infinite
medium and extends to the boundary at infinity. Figure 10
shows a normal and a shear semi-infinite dislocation at the
origin (Dd and Db, respectively). Only the opening mode is
considered in the following discussion.
Let d„(s) be the negative of the opening displacement along ,
the fracture axis (Fig. 11). Within the infinitesimal element of
length ds at s, the opening increases by an infinitesimal
amount dd„(s), i.e., the applied infinitesimal dislocation at s is
dd„ (s). The applied dislocation distribution is then given by
dd„(s)
P(s)ds
For an embedded fracture the closure condition demands that
228/Vol. 57, MARCH 1990
\—»~x
Fig. 9
Applied distribution at third-order equilibrium (k = 3)
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1000
-Db/2
75
free surface
-D .12
d
Section A
-• s
Dd/2
Fig. 10
Db/2
925
Fracture
S*—Ti 1
400
200
i
Normal and shear semi-infinite dislocations
1»
475
i
y.n
rollers
X
P (s,n)
E = 5 x 1 0 4 , v=0.25
Fig. 12
-dn(s)
B>
0.05
Collapse of underground fracture
I
Fig. 11
Fracture opening modeled by normal dislocations
-0.10
-
-0.15
-
-0.05
'" dd„(s)
=0
\o ds
I
I
i
"~~7~~
0.00
~~-~-
"W/>
~~~'
-
-0.20
i.e., p<-'>(/!) = 0.
Thus, at least first-order equilibrium exists. When first-order
equilibrium does exist, according to PAP the induced
stresses/displacements at distances far from the crack are approximately proportional to p{~2)(h) and
p(-2)(/!)= ( pi~>\s)ds= \ d„(s)ds.
Jo
-0.25
-0.30
I
i
i
i
Fig. 13 Induced displacements along Section A due to closure of
embedded fracture
Jo
The volume of expansion E of the fracture is defined as
E=-\
Jo
d,l(.s)ds=-p^(h).
Thus, the Theorem of Equivalent Expansions, as a corollary
of PAP, is formulated as follows:
Each induced stress or displacement component at large
distances from a crack due to its opening, with minor exceptions, is approximately proportional to the expansion and is
independent of the opening shape.
A numerical experiment was carried out to verify the
Theorem of Equivalent Expansions in Chan (1986). The induced displacements and stresses due to the collapse of an
underground fracture were modeled using a computer program called FROCK (acronym for Fractured ROCK). FROCK
is based on a hybridized Displacement Discontinuity Element
and Fictitious Stress Element scheme (Chan et al., 1988), in
which exact, closed-form influence functions for the elements
were used.
Fig. 12 shows a square medium with an embedded fracture
under plain strain. If the fracture closes uniformly by 1 unit,
the expansion is - 1 x 2 0 0 = - 2 0 0 square units. The induced
displacements and stresses (as calculated using FROCK) along
Section A (Fig. 12) are, respectively, shown in Figs. 13 and 14
in solid curves. When the expansion takes another shape in
which the central half of the fracture closes uniformly by 2
units, the induced displacements and stresses are shown in
Figs. 13 and 14 in dotted curves. One can see that the corresponding solid and dotted curves are very close to each
other. This example shows that the induced displacements and
stresses at locations far from the fracture mainly depend on
the expansion of the fracture and not on the opening shape.
The Theorem of Equivalent Expansions for each of the induced stress components in an infinite medium has been
Journal of Applied Mechanics
Fig. 14 Induced stresses along Section A due to closure of embedded
fracture
proved in Chan (1986). There is an exception for induced axy
at points P near the vertical axis (i.e., rf»s). However, in this
case, the induced axy is small compared with the two other induced stress components and, hence, the exception is considered to be minor.
Similarly, by considering shear dislocation, the Theorem of
Equivalent Distortions has also been established (Chan, 1986).
These two theorems have important applications in fracture
mechanics including monitoring of underground fracture expansion using only surface measurements, indirect measurement of the expansion of a crack in a plate, and derivation of
the equivalent moduli for fractured media.
Electrostatic Fields
The electric field strength (or electric intensity), E, in an infinite free medium with permittivity, e, at a point P due to a
point charge q is
MARCH 1990, Vol. 57/229
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/
by the application of q. For free space $ ref = 0, and the equation for the electric field in the previous section can be obtained by (14). When <J?ref # 0 , both *„ and * ref have to be considered in (14).
In the previous section it has been shown that PAP is applied to the partial derivative (E^ and Ey) of * 9 , here it can
also be shown that PAP applies to * ? itself since its Green's
function is
P(s,n)
A!L
pM
mi
p dx
=dq
h
Gq(x,OP) =
Fig. 15 Electric field due to applied charge distribution p(x)
E=
1
Q
Aire
d2
where d is the distance between q and P. For an applied charge
distribution p(x) on the x-axis (Fig. 15), the Green's function
of the ^-component of the electric field Ex is then
—
cos0
1
s—x
Gx(x, OP) = -j—p- = —
[ ( i S _ x ) 2 + „2 ] 3 /2yn = v) rar irom tne applied cnarges. in this case,
1
1
Aire (s — x)2
The postulate that (recall r= \OP I)
Gx(x,OP)--
Gf +lKx, O P ) « G « ( / * , OP), 0<x<h,
r/h is large,
(12)
is clearly satisfied and PAP applies. If the point P is on the
vertical axis, postulate (12) may not be satisfied. However, in
this case Ex is small compared with Ey, and it can be shown
that PAP applies to Ey whose Green's function is
—sin0
GJx,OP) =
1
1
Am [(s-x)2 + n2}V2'
Similar arguments can be used to show that PAP also applies to E^, with minor exceptions.
Aired
Other Potential Fields
After PAP has been shown to be applicable to electrostatics, generalization to other fields covered by potential
theory is immediate. At first we examine how PAP can be applied in electrostatic potential theory:
The governing equation of the electrostatic potential $(s, ri)
is the Laplace's equation
V2* =
a2$
ds2
~JriT
- = 0.
(13)
The electric field (E x , E,) is given by
3$
(14)
~~dn~
An applied concentrated charge q at the point (x, 0) of an infinite medium with permittivity, e, induces the potential
Ev =
Us - . E , =
1
<?
$a(s, ri) =
—.
4
*
« J(s-x)2
+ n2
The electrostatic potential $ is then
* = * ? + *ref
where $ ref is the reference potential, which is the potential existing before q is applied. It is assumed that # ref is not affected
230/Vol. 57, MARCH 1990
1
1
4ire ~J(s-x)2 + n2
which satisfies postulate (12) with some exceptions.
Note that PAP may not apply to <J? because its Green's function, if any, depends on $ r e f .
At this point it is interesting to notice that the asymptotic
behavior of the far-field potential has been widely examined
(for example, see Owen, 1963, II. J). Usually isolated concentrated charges are considered and Taylor's series expansion is
used on Gq (x, OP) to arrive at an infinite series. PAP differs
from this usual approach because repeated integration by
parts, instead of the Taylor's series, is used to arrive at the infinite series in equation (6).
Since now that PAP has been shown to be applicable to
electrostatic fields, PAP can also be applied to other potential
fields such as gravitation, magnetism, temperature, and fluid
velocity potential because they can all be represented by potential functions satisfying Laplace's equation. However, in the
case of gravitation negative masses (if any) are required to
create first and higher-order equilibria.
Magnetic Field of a Current-Carrying Wire
Consider a long straight wire in an infinite space carrying a
current I pointing perpendicularly out of the x-y plane. The
current I can be regarded as a concentrated charge applied at
the position of the wire. If the current is applied according to
the distribution p(x) on the x - a x i s , the Green's function for
the x and y - components of the induced magnetic field at P
are, respectively,
Gv=-
V-
2ir {s-x)2 + n2
ix.
s—x
y
~ 2w (s-x)2 + n2
where ft. is the permeability of the infinite space.
It can be shown that PAP applies to the magnetic field induced by the current-carrying wire, with minor exceptions.
One application (which may not be economical at present) is
that the induced magnetic field of current-carrying wires can
be greatly reduced by arranging the wires so that a high-order
equilibrium exists. For examples of higher-order equilibria,
see Figs. 7 to 9. It is quite commonly known that a higherorder equilibrium distribution can be obtained by subtracting
the shifted distribution from a given distribution: If p(x) is at
Ath-order equilibrium, p(x)-p(x-c)
is at (£+l)-order
equilibrium, where c is a nonzero constant.
Conclusion and Further Comments
• A general physical principle (PAP) has been established
which uses the Green's function method to approximate the
induced far field due to any applied charge distribution. For
most physical problems (especially for problems with finite
boundaries), the corresponding Green's functions have not
been found. However, even for those problems, PAP is useful
because (9) gives at least an approximate proportional
relationship.
This paper has dealt specifically with an applied charge
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distribution on a straight line segment. For two and threedimensional geometries of applied charge distribution, similar
formulations can be established.
Acknowledgments
Most of the present research results were obtained while the
author was a research assistant in the Department of Civil
Engineering, the Massachusetts Institute of Technology,
under the supervision of Drs. V. Li and H. H. Einstein.
The research at the Massachusetts Institute of Technology
was sponsored by the U.S. Army Research Office under grant
No. DAAG24-83-K-0016 and was later continued at the
California State University at Fullerton.
The author also appreciates the discussions with his colleagues at California State University, Fullerton, including
Dr. G. Cohn on the topic of electrostatics.
References
Chan, H. C. M., 1986, "Automatic two-dimensional multi-fracture propagation modelling of brittle solids with particular application to rock," Sc.D.
dissertation, Department of Civil Engineering, Massachusetts Institute of
Technology, Cambridge, Mass.
Chan, H. C. M., Li, V., and Einstein, H. H., 1988, "A hybridized displacement discontinuity and indirect boundary element method to model fracture
propagation," submitted to International Journal of Fracture, accepted for
publication.
Crouch, S. L., and Starfield, A. M., 1983, Boundary Element Methods in
Solids Mechanics, G. Allen and Unwin, pp. 46-47.
Hildebrand, F : B . , 1976, Advanced Calculus for Applications, Prentice-Hall,
Englewood Cliffs, N.J., pp. 652-658.
Horgan, C. O., and Knowles, J. K., 1983, "Recent developments concerning
Saint-Verant's principle," Advances in Applied Mechanics, Vol. 23, pp.
179-269.
Owen, G. E., 1963, Introduction to electromagnetic theory, Allyn and Bacon.
Sternberg, E., 1954, "On Saint-Verant's principle," Quarterly Appl. Math.,
Vol. 11, pp. 393-402.
.Readers of
lecnaiiics
AMD-Vol. 103
Computational Techniques for Contact, Impact,
Penetration and Perforation of Solids
Editors: LE. Schwer, N.J. Salamon & W.K. Liu
This volume covers low velocity impact, contact and impact, penetration and perforation, and
earth penetration. It addresses the impact-related response of structures, computational
techniques and strategies for treating the low to intermediate velocity impact of solids when the
deformations are significant, analysis techniques for the penetration and perforation of materials
impacted at velocities in the intermediate to hypervelocity range, and computational approaches
to solving modeling problems.
1989
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ISBN 0-7918-0406-2
372 pp.
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