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 Copyright © 1990 by ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/27/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/27/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use " 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 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/27/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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) Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/27/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/27/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use / 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 Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/27/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 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 Order No. H00540 ISBN 0-7918-0406-2 372 pp. $90 List / $45 ASME Members To order, write ASME Order Department, 22 Law Drive, Box 2300, Fairfield, NJ 07007-2300 or call 1-800-THE-ASME (843-2763) or FAX 1-201-882-1717. Journal of Applied Mechanics MARCH 1990, Vol. 57/231 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/27/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

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