Principles of virtual work and complementary virtual work for Cosserat media

PRINCIPLES OF VIRTUAL WORK AND COMPLEMENTARY VIRTUAL WORK FOR COSSERAT MEDIA C. A BERG

Department of Mechamcal Engineering, Massachusetts Instttute of Technology, CambrMge, Massachusetts (USA) (Received. 20 October, 1970)

SUMMARY

The principle of a virtual work for a Cosserat medtum is discussed. A principle of complementary vtrtual work for Cosserat media is gwen, provtdmg an extenston of the Dorn and Schild 'Converse to the Vtrtual Work Theorem for Deformable Sohds" (1956) to general Cosserat medta. As with the virtual work principle itself, the complementary virtual work principle is mdependent of the specific rheological character of the medtum. Apphcattons are brzefly discussed. INTRODUCTION

The principle of virtual work is the central theorem upon which the structure of constitutive relations, as well as the fundamental variational theorems of the various rheological branches of continuum mechanics, are based. As usually stated, the principle of virtual work says that a given symmetric stress field (tr~j = a j,) in a body (V), along with a given distribution of tractions (t 3 on its surface (S), meet the conditions of static equilibrium (a,j,~ -- 0) on the interior of the body and the Cauchy relation between the interior stress (ais), surface traction (t,) and outer normal vector (ns) to the surface (S) of the body (aijnj = t,) lf~ and only if, the virtual work equation*

Svaise,j dV = SstiuI ds

(1)

holds for every displacement field (ui) and its associated 'strain' field

(sis = k(u,,s + us,i)) defined in V and S*. Body forces and effects of inertia may be included quite simply, but will not be considered here. * Cartesian tensor notation is used here, repeated radices in products and dlfferentmtlons indicate

summation, and a comma indicates dlfferentlatmn (1 e, q,,j = Oq~[~xj),evk stands for the usual ant~symmemc alternator 261 Fibre Science and Technology (3) (1971)----C Elsevier Pubhshlng Company Ltd, England--Prmted in Great Britain

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The p r o o f of the virtual work theorem follows from the application of Green's divergence theorem to the integral on the left of (1) This requires certain smoothness properties of both the given a,j and the various d~stributlons of u, m V and S. In practice, the hnutatlons with respect to smoothness of these fields, e g., exclusion of nonintegrable singularities, does not pose strong restrlcUons upon the range of apphcatlon of the virtual work theorem The v~rtual work theorem is often used to compare two stress fields (a,J and tTUlI and two strain fields (e,j I and e,jI1), the virtual work equation is then used m the form. SV(a,3 II -- O',3I)(/;UII -- EUI) d V = $~(t,It -- t,l)(u, lI

-

-

/,/I) ds

so that only the differences in the two sets of fields considered need to meet the conditions on smoothness required by the divergence theorem; the fields individually may be strongly singular * A complementary form of the virtual work theorem was given by Dorn and Schdd (1956)4~f According to these authors, if, for a given symmetric set of functions (e,j) defined on the interior (V) of the body and a gwen set of displacements (u~) defined on the surface (S) of the body, the virtual work equation:

Svtr,je,j d V : Ss~,jnju, ds

(2)

holds for all symmetric stress fields (a,~ : try,) which meet the conditions of static equilibrium (a,j a : 0) on the interior of the body, then the e,j are the components of a compatible strain field in V and prowde internal displacements which d~ffer from the prescribed surface displacements (u,) by no more than a rigid motion. A converse statement of the Dorn and Schlld theorem--that, if e,~ is a compatible stram field derivable from a displacement field which, on S, differs from the surface d~splacements u, by no more than a rigid motion, then a virtual work equation (eqn. (2)) holds for every stress field m static e q u l h b n u m - - m a y be obtained by straightforward apphcatlon of Green's divergence theorem. With the Dorn and Schlld theorem (1956), 4 the theorem of minimum complementary elastic energy (e.g., Southwell, 1936, Langhaar, 1953; Washlzu, 1958) may be demonstrated in one step, without having to appeal to the specific structure of the elastic constitutive relations. Since neither the Dorn and Schdd theorem nor the virtual work theorem involves constltutwe relations in any way, they are of wide apphcabdlty throughout continuum mechantcs and can be used with powerful effect and slmphoty to establish useful variaUonal principles, theorems of uniqueness, techniques of estimating bounds on motton, and other practically useful results in the mechanics of materials of various rheological character. Gurtln's s * See, e g Hdl (1956) for discussion and apphcatlon of the point m plasticity. t Strictly speaking, the Dorn and Schlld theorem (1956) is a complementary virtual work theorem rather than a converse to the classical theorem of v~rtual work

VIRTUAL WORK AND COMPLEMENTARYVIRTUALWORK FOR COSSERATMEDIA 263 estabhshment of variational principles for linear viscoelasticity--in particular his construction of a viscoelastic analogy to the theorem of the complementary elastic energy--is an example of powerful use of Dorn and Schdd's (1956) theorem

COSSERATMEDIA Much present day research is directed towards development of theories to describe the role of microstructure in material behavior. Classical notions of stress and strain are not adequate for this purpose because they entail a hnuting process of 'shrinking to a point' in which one obtains a system of such small size that variations of properties through the system are negligible. It is, of course, exactly such variations that must be retained to reflect the effects of microstructure.* The oldest theory of continuum mechanics capable of dealing with some of the effects of mlcrostructure was introduced by the Cosserats (1909), who proposed that the forces of interaction carded on material surface elements include distributed couples as well as tractions.* These distributed surface couples represent a new generahsed force, independent of gradients of tractions (although, m a given case, the couples may be used to represent mlcrostress gradients). Even though the Cosserat medium offers only a partial representation of all possible nucrostructural effects, it does, in certain cases, reflect these effects with sufficient accuracy to dluminate seemingly anomalous behavior of some mechamcal systems (e g., internal buckhng of laminated medm, Blot, 1965), and offers a simple approach to analysing more comphcated nucrostructured media (e g., polycrystalline plastic metals, Llppmann, 1968). 1, An important aspect of the Cosserat model of mlcrostructured materml is that the mechanical equilibrium relations are Independent of the medium's constitutive properhes. The connection between stress (a,j) and surface traction (t,) is: atjnj -----t, (3) just as in the classical (non-Cosserat) medium. The surface couple (C,) is related to the outer normal (n j) and an interior couple stress (c,j) by

c , j n j = C,

(4)

For the case of static equdlbrium m the absence of body forces, conservation of linear momentum requires: ~,j,j = 0 (5) * Numerous modern theories have been proposed to provide a system of continuum mechanics capable of reflecting microstructural effects (e g. Green and Rtvhn, 1964, Kroner, 1958, Engen, 1969).

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just as in the classical medmm. The principle of conservation of moment of momentum m the Cosserat m e d m m yields (see Jaunzlmas, 196710), e,jk~kj + c,j,j-= 0

(6)

In eqns. (5) and (6) effects of body forces, body moments and inertia have been neglected because, in the present work they are not reqmred; they could be reintroduced quite simply if necessary The complete set of eqmhbrlum relations (eqns. (3) through (6)) results from conslderaUon of conservation of m o m e n t u m and of moment of momentum. I f internal generahsed stresses other than the s~mple couple stresses of the Cosserats were considered (representing e.g effects of mlcrostress gradients in addmon to those which contribute to meehamcal couples) the conditions of e q m h b n u m , even for the simplest case, could not be given without conszderatlon of the constaut~ve properties of the materml (see e g. Green and Rivhn, 19647) Although the Cosserat continuum model is incomplete, it contains fewer amblgumes than more advanced alternatwes, and offers the possibility of a simple model for exploratory investigations of behawour believed to be assocmted with m~crostructure In view of the success with which the Cosserat model has been used in studying such phenomena as elastic and wscoelast~c internal instabilities, 1 certain aspects of plastic deformation 14'15 and of long-range interaction about dislocation cores, 15 and m pointing out directions of profitable research with more complete contmuum models 13 as well as the uses to which the Cosserat model has been put m the theory of elastic shells, It seems reasonable to suppose that the Cosserat medium wdl continue, m at least the ~mmedmte future, to provide a starting p o m t m the investigation of mlcrostructural effects in materials of various rheology. Theorems of wrtual work and complementary virtual work can be given for Cosserat medm in much the same way as for convenUonal classical models of materials in which couple stresses do not occur These theorems, because they derive solely from conditions of mechamcal equlhbnum, are independent of constitutive properties and hold for all Cosserat media irrespective of differences in rheologlcal properties In the following the principle of wrtual work for Cosserat medm is quoted, and a principle of complementary virtual work for Cosserat medm, smular to the theorem of Dorn and Schlld (1956) 4 for classical continuous media, is given. The independence of these prlnoples upon various rheologlcal aspects of the material (elasticity plasticity, VlSCoelastioty) ~s emphaslsed Vtrtual work m the Cosserat medtum The Cosserat medium supports distributed surface tractions (t,) and surface couples (C,) as well as internal stress (a,~) and couple stress (c,j) The connections between the external forces acting on the element of area having the normal vector n~ and internal stress fields were given in eqns. (3) and (4) The conditions of static

VIRTUALWORK AND COMPLEMENTARYVIRTUALWORK FOR COSSERATMEDIA 265 eqmhbrium governing the distribution of stress (~r,j) and couple stress (%) wittun the body, in the absence of body forces, body couples or inertia are given in eqns. (5) and (6). The surface tractions (t,) are conjugate to the surface displacements, and if a virtual displacement u, is applied to a Cosserat medium in static equilibrium, the virtual work of the surface tractions ~s: Sst,u, ds : .[va,ju,,j dv

(7)

which results from apphcation of Green's dwergence theorem to the surface integral and use of eqn. (5). The surface couples (C,) act as forces conjugate to local rotations; one component of local rotation through wluch the local surface couple may do work is the local rotation (co,) of the displacement field given by to, = ~e,j~uk,j

(8)

In addition to the rotations co,, one may adnut additional rotations (~t~), not deterrmned by the displacement field, through which the surface couples may do work. These additional rotations may represent, for example, lattice rotations m plastic deformation of crystals* or transplanar rotations of unbalanced composite plates; for convemence the ct, will be called rmcro-rotatlons. The virtual work done by the surface couples (C,) when a virtual displacement field (u,) and a virtual micro-rotation field (ct,) occur in a Cosserat body in static equilibrium is given by LC,(~, + to,)ds By combining this expression with eqn. (4), applying the Green dwergence theorem and making use of the symmetric and rotational parts of the displacement field % = ~(u, j + u~,,) tot = ½e,jkUk, J one may write the total virtual work of the surface tractions and couples as ~s{t,u, + C,(ct, + to,) ds ---- So(tr,je,j + c,j(to,,j + ~,j) +

e,.tktYjkO~,}dv

(9)

The virtual work equation (eqn (9)) holds for every static equlhbrium stress and couple stress field and for every virtual displacement field (u,) and every virtual micro-rotation field % Suppose that for a given distribution of a,j and c,j in V, and t, and Ct on S, every virtual displacement (u,) and virtual nucro-rotatlon field (~,) satisfies the virtual work equation (eqn. (9)) In this case one may use eqns. (6), (8) and * The example suggestedby F. A McChntock.

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BERG

Green's dwergence theorem on the volume integral of eqn. (9) to transform the wrtual work equation to the form ~{(t, -- tr,,n,)u, + (C, - c,juj)(~, + to,)} ds + So{tr,,,ju, + (e,jktrk~ + C,ja)(to , + a,)} dv = 0

(10)

Since eqn. (10) holds for every virtual displacement (u,) and nucro-rotatlon (a,) field, eqn. (10) implies: trzj,j ~ e~ktTkj + C~j,j ~- t, -- tr,~n~ ~ C, - c,jn; ~- 0

which are the conditions of static eqmhbnum The wrtual work prlnople for Cosserat media can now be stated as follows: The virtual work equation (eqn. (9)) holds for every virtual displacement and micro-rotation field ff and only if the stress and couple stress distributions within the body and the surface tractmn and surface couple dlstributmn on the surface of the body meet the condltmns of static equilibrium (eqns. (3), (4), (5) and (6)). This statement of the pnnclple of wrtual work for Cosserat media is certainly not surprising, but is potentially very useful in estabhshing limit theorems in plasticity of Cosserat media. It ~s interesting to note that without the presence of the mdependent rmcro-rotatmns (~) the complete set of eqmhbrium relations (eqns. (3) through (6)) could not be obtamed from arguments of wrtual work alone, since u, and to, are not independent. In the absence of the micro-rotations the virtual work principle would be in a similar state to that of classical medm hawng no couple stress--/e, one of the internal equdibnum relations (e.g etjkakl -1- C,.l,J ~ 0 ) would have to be known m advance in order that the other could be deduced A prmctple o f complementary mrtual work f o r Cosserat media

The virtual work principle shows that, for a gwen displacement field (u,) with its associated strain (e,j) and rotation (to,) fields, and for a given micro-rotatmn field (ct,), all fields being defined both on the interior (V) and the boundary surface (S) of a body of Cosserat material, the virtual work equation: j',(a,su, + c,j(~, + co,))nj ds = ~v{tr,je,~ + c,l(toia + ~,,j) + e,~ktrjk~l}dv

(11)

holds for every stress and couple stress dlstrlbuUon whach satisfies the conditions of static equilibrium, (eqns. (5) and (6)), m V and S. We now demonstrate that a pnnciple of complementary virtual work analogous to that of Dorn and Schdd (1956), 4 holds for Cosserat media. Suppose, for a given set of surface displacements (u,) and surface rotations (fit) defined on S, and a gwen set of symmetric functions

VIRTUAL WORK AND COMPLEMENTARY VIRTUAL WORK FOR COSSERAT MEDIA 267

(e,j = s j,), a given set of six (in general) asymmetric functions ¢,j and a set of three functions ct, defined in V, the virtual work equation: ~,(a,jnju, + c,nlf~,) ds = ~v(tr,je,j + c,jcp,j + eukgjkOt,) d V

(12)

holds for every stress and couple stress field which satisfies the conditions of equlhbrlum (eqns. (5) and (6)) in g and S. Then (A) the e,j are the components of a compatible strain field, the dispacements of which differ from the given surface displacements (u,) by no more than a ngid motion; (B) the t , are simply the sum of boundary values of the rotaUons of the internal displacement field (co,) and the micro-rotaUons (cq), i.e. : and (C): (Pu =

cot,J + ~t,l

To verify (A), (B) and (C) one may first consider the stress functions of Dorn and Schfld4: Y m j r a = - - Y t n j m = - Prom1 ~ - P j,,,,,~ (13) whtch generate arbitrary symmetnc stress fields in static eqmllbnum (O'tj (1) = O'lt(1); atl,l(l)= O) vla

at~(1) = P.tj . . . .

(14)

By using the atj (1) of eqn. (14) and setting c,j = 0, one has classical symmetric stress fields which satisfy eqns. (5) and (6); introducing these into eqn. (12) reduces that virtual work equation to the form:

~vs,jPnurn,nm d V : SsPnum,nmnjUt d S

(15)

As Dorn and Schild have shown, ff (15) is to hold under arbitrary choice of the P.,jm from the class of functmns which satisfy (13) the sq must obey Caesaro's compatibility relations (S,j,kt + ek~,,j- e , k , O - e0,,k = 0), and may therefore be written as e,j = 1/2(a,,j + aj,,) (16) Dorn and Schdd also showed that the boundary values of the fi, may differ from the prescribed surface displacements u, by no more than a ngjd moUon. The results of Dorn and Schild may be venfied by direct calculation using the dwergence theorem and (13) in 15, and confirm A above. It is convenient to eliminate the rigid motion by which the internal displacement field (~,) nught differ from the surface displacements (u,) on S, and set s,j = 1/2(fi,,j + fij,,), throughout V. Now, to verify B and C, one may consider general

268

c. A BERG

antlsymmetnc stress fields (a,f 2) = _0-j t2)) winch meet eqns. (5) and (6). The condition ofequlhbrlum for the stress components (eqn (5)) will be met by setting: (17)

cr,j (2) = e,,,~Q,s

where Q is a scalar function of position in v, the equlhbrium condition for couple stresses (eqn (6)) is now met by setting: c,j = 2Q6,~

(18)

+ ejktflk,t

where the f,k are a set of nine scalar functxons m V. When the stress and couple stress fields of eqns. (16) and (17) are used m the virtual work equation (eqn (21)) one obtains, after using the divergence theorem: Sv{ZQcpkk -- e.tktqg,j,tf~t. " q- 2 Q , , ~ , ) d V = Ss{e,js(Q,su , - f , . , , f ~ , .

+ fpsq~p,) + 2 Q ~ ) n j ds

(19)

Smce Q and the f,j may be chosen arbitrarily (19) implies that e~kl~O,~,lvanish at each point of V, thus the ~o,~are the gradients of a vector field; ~o,j = ~o,.j Now one may use the gradients of the rotatmn field (o9,) of the internal displacements (u,) to split the gradients q~,~as follows: ~,,1 = ~,a + &,,~

(20)

Since both &,a and og,.j are gradients of vector fields, then so are the a,,j The split up of eqn. (20), together with the fact that the e,j are derived from an internal displacement field, hawng boundary values winch coincide with the prescribed u, on S, may be used to apply the wrtual work prinople (eqn. (11)) to reduce eqn. (12) to" .f~c,j(f~, - (09, + &,) -

(or, - &,))n~ ds -

[.~c,j(~, -

&,),j d V = 0

(21)

Arbitrary asymmetric stress fields (0-,j ~ (3)) with associated couple stress fields (c,j) which obey the condltmns of eqtullbrium (eqns (5) and (6)) may be constructed via.

0",3(3)= ejktA,j,t

(22a)

c,~ ~ A j, - Akl, f , j + ejktB,k.l

(22b)

and where A,j and B,k are sufficiently smooth scalar functions defined on Vand S By choosing couple stress states for winch the B,k vanish everywhere and the A,j, along with all t h o r grachents, vanish outside of a suitably small neighborhood v(6) of an arbitrary point (6) in V (in particular, the B,k, A,~ and all their gradients vamsh on S) one has static equdibrmm stress and couple stress fields for which eqn. (21) becomes: S,,(,~)(Aj, - Akkf,j)(Ot, -- &,),~ d V = 0 (23)

VIRTUAL WORK AND COMPLEMENTARY VIRTUAL WORK FOR COSSERAT MEDIA

269

With A,j = A3,~, A being a scalar function in V(6), and since the point (3) is arbitrary, eqn. (23) implies: (~, - & , ) , , - 0 (24) at each point in V Thus, eqn (23) may be written as" 5v~a)Aj,(~, - &,),, dV = 0

(25)

and since the Aj, may be assigned arbitrarily m v(3), eqn. (25) lmphes that: (~, - &,) = 2, (uniform in V)

(26)

Since =, and &, can differ only by a uniform rotation 2, in V, eqn (20) reduces to

which proves part C of the theorem. Now, using eqn. (26) m eqn. (21) I,c,~nj(~, -- (co, + cq))ds = 0

(27)

and since c,~nj may be assigned arbitrarily on S (by, for example the use ofeqn. (22)), one has" f~, = co, + ~, (28) which is part B of the theorem. This completes the demonstration of the complementary virtual work principle for Cosserat media; the results may now be briefly summarlsed. Given the six symmetnc ('strata') functions (e,j) nine functions (q),j) and three (micro-rotation) functions (~,) defined on the interior (V) and surface (S) of the body, and given three ('displacement') functions (u,) and three other ('rotation') functions (t2,) defined on the surface (S) of the body, then the virtual work equation" Ss(a,jnju, + c,jnjD,) ds = Sv(a,je,j + c,j~o,j + etjkffjk{Xt) d V

(12)

holds for every stress (e,j) and couple stress (c,j) field which satisfies the conditions of statm equdlbrlum. a,j a = %kakj + e l j , j ~ - 0 m V and S, i f and only if0): The functions e,j are the components of a compatible strain field within V, derivable from an internal displacement field (~,) hawng boundary values which differ from the given surface displacements (u,) on S by no more than a rigid motion (Dorn and Schdd, 1956).4 (ll): In addition, the qhj consist of the sum of the gradients of the rotation (co, = ½e,jkUk,j) of the internal displacement field and the gradients of the micro-rotations (~,): q0,j =- co'a + 0q,j, i n ( V )

(Ul): Finally, the boundary rotations t2, differ from the sum of the boundary values of rotations (co,) of the internal displacement field and the boundary values of the

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internal micro-rotations (~,) by no more than a uniform rotanon When the internal displacement field derived from the e,j is constructed so that its boundary values c o m o d e with the given surface &splacements then the f~, colnode with the sum of co, and ~, f~ = o9, + ~,, on S

NOTES ON APPLICATIONS The two wrtual work principles given above can be used immediately m the theory of elastic deformation of a Cosserat body to establish vananonal prmoples of potential energy and complementary energy (see e g. Relssner and Wan, 1967, who gave a restricted form of the principle of complementary wrtual work for elastic Cosserat media in their p r o o f of the theorem of complementary elastic energy for Cosserat elasticity). Since no restrictions concerning rheologlcal properties of the materml apply to the wrtual work pnnc~ples they can be used to estabhsh variational pnnclples for Cosserat media showmg nonhnear elastic constitutive behawor at mfinatleslmal strata levels, as easdy as for stnctly hnear matenals. In addmon, basic hmlt theorems for Cosserat theories of plastlctty (e.g. Llppmann, 196814) can be estabhshed directly by use of the virtual work principles gwen above. One may reasonably expect that as Cosserat theories of wscoplasnctty, creep, wscoelasnclty, deformation of granular medm and fiber reinforced lanunates are developed, the virtual work prlnoples wtll prove useful.

ACKNOWLEDGEMENT The author gratefully acknowledges the financial support of the National Science Foundation (NSF GK1875-XI) and the MIT Center for Materials Science and Engineering REFERENCES 1 M A BLOT,Further developments of the theory of internal buckhng of multdayers, Geol Soc, of Amertca Bulletin, 76 (1965) 833-40. 2 D.B. Bo6y, An 'opttmal' solution of Samt-Venant's flexure problems for a circular cylinder. J. AppL Mech., 34 (1967) 175-86 3. BULLOUGH,Orowan Anmversary Volume (to be pubhshed MIT Press). 4 W. S. DORNand A SemLv, A converse of the wrtual work theorem for deformable sohds, Quarterly AppL Math, 14 (1956) 209-13. 5 E & F COSSE~,T,Theone des Corps Deformables, Pans, 1909 6 CEreALA. ERINGEN, Theory of mlcropolar elasticity, 622-736 in Vol II, Fracture--An Advanced Treatise, E&ted by Harold Ltebowltz, Academic Press, New York, 1969. 7. A. E G~EN and R S RIVLIN,MulUpolar continuum mechanics, Arch Rat. Mech. Anal, 17 (1964) 113-47 8 M E GugrrN, Variational prmmples m the linear theory of vlscoelasnclty, Arch Rat. Mech. Anal., 17 (1964) 313

VIRTUAL WORK AND COMPLEMENTARY VIRTUAL WORK FOR COSSERAT MEDIA 271 9. R. HILL, On the problem of uniqueness m the theory of a rigid plast,c sohd I, J. Mech. Phys. Solids, 4 (1956) 247-55. 10. WALTERJAUNZ_ZMAS,Continuum Mechanics, MacMillan Company, New York, 1967. 11. O. D. KELLOC_,O,Foundations of Potential Theory, Dover Pubhcatlons, New York, 1953 12. W. T. KO1TER, Couple-stresses m the theory of elastlcRy, Proc., Konenkl, Nedul Akademw Van Wetanchappen--Amsterdam, 67 (1964) No. 1. 13. E. KRONER, Kontmuumstheonc der Versetzungen und Elgenspannungen, Sprmger-Vcrlag, 1958 14 H L LAUGHAARand M. J STn'PES, Frankhn Inst. 258, pp. 371-82, 1954 15. HORST Ln~PMANN, 12th International Confcrenc~ Apphed Mech., Stanford Uruvers~ty, 1968 16 F. A M c ~ o c K , Contribution of interface couples to the energy of a dislocation, Letter to the EdRor of ,4cta Metallurgica, 8 (1960) p 127. 17 F A. McCLn~rrocK, Interface couples in crystals, w~th Patrlcla A Andre, Kurt R Schwerdt, and Robert E. Stoeckly, Letter to the Editors of Nature, 8 (1958) p, 127 18 REISSNERand WAN, A note on Gunther's analysis of couple stress, Proc I U T A M Symp on the Generalized Cosserat Continuum, Freudenstadt-Stuttgard 19. R V. SOUTHWELL,Proc. Roy Soc. A154 pp. 4-21 (1936) 20 K A WASmZU, A note of the condmons of compatibility, Journ. of Math. and Phys, 4, p 306-12.