The Strain Compatibility Equations in Polar Coordinates RAWB, Last Update 27/12/07

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1 The Stain Compatibility Equations in Pola Coodinates RAWB Last Update 7//7 In D thee is just one compatibility equation. In D polas it is (Equ.) whee denotes the enineein shea (twice the tensoial shea) and indices afte a comma denote patial diffeentiation with espect to the coespondin coodinate. This equation is simple enouh to deive u tial and eo statin fom the explicit expessions in polas fo the stains in tems of the pola displacements. The same cannot be said fo 3D (spheical) polas. Howeve they can be deived fom the Riemann tenso. Cuiously this method does not equie the explicit expessions fo the stains in tems of the pola displacements to be known. In 3D thee ae six independent compatibility equations. It tuns out that one of them is identical to Equ. (whee is now the pola anle to the z-axis and the azimuthal anle is ). As an example one of the othe equations is (Equ.) cos cos Nasty isn t it! The othe fou equations may be deived if desied u the same method (details below). Howeve it is a tedious execise and pesonally I cannot be botheed. On the othe hand the manne in which the poblem is popely fomulated does have intinsic inteest.

2 Deivation of the Compatibility Equations in Full Geneality This is a puely eometical poblem. An abitay coodinate system is imained as embedded within a medium. Upon defomin the medium the coodinate system also defoms to what may be consideed as a new coodinate system. Specifyin a displacement field clealy defines the new coodinate system and the associated stain field. On the othe hand a stain field cannot be specified abitaily and still be compatible with a displacement field. Mathematically if we wok in tems of stains we must ensue that the stains satisfy cetain inteability conditions which uaantee the existence of a compatible displacement field. The poblem may also be posed in tems of the intinsic cuvatue of the defomed coodinate system. Due to the physical manne in which we have defined ou defomed coodinate system it is clea that thee exists a continuous coodinate tansfomation to a Catesian system. The intinsic cuvatue is zeo because the intinsic cuvatue of the undelyin space is zeo. Convesely a stain field which was not compatible with displacements fom an initial Euclidean space would possess intinsic cuvatue. But intinsic cuvatue is measued by the Riemann tenso. Hence the compatibility equations ae simply the equiement that the Riemann tenso vanish. In tems of the metic tenso of the defomed coodinate system ( ) the Riemann tenso can be witten R (3) whee (4) and whee the epeated index implies summation (ove = 3). The compatibility equations ae then simply R fo all (= 3) (5) Equ.(5) is the most eneal expession of the compatibility equations applicable fo an abitay coodinate system and fo abitaily lae stains. Befoe tunin to paticula special cases fistly conside how many independent equations Equ.5 epesents. Since each of the indices takes 3 values it miht appea that Equ.5 is actually 3 x 3 x 3 x 3 = 8 equations! Of couse this is not so (it would be massively ove-constained). The expessions (3 4) imply cetain symmeties amonst the 8 components of the Riemann tenso. These ae most simply expessed in tems of the fully covaiant tenso ( R R ) as follows Assumin that we ae not dealin with a eion of intense avitational field othewise space(time) eally would be cuved. Incidentally this would povide an appoach to the de of the hull of a spacecaft intended fo close appoach to an event hoizon. If the usual compatibility equations ae witten X = then we would eplace this with X = Y whee Y epesents the avitationally induced cuvatue.

3 R R ; R R ; R R (6) Thus the Riemann tenso is antisymmetic in both the fist and second pai of indices and also symmetic with espect to exchane of the two pais. In 3D space this means that the Riemann tenso has only 6 independent components. This of couse is the coect numbe of compatibility equations in 3D. In D thee is only one independent component of the Riemann tenso aain the coect numbe. Hold Up Whee Ae The Stains? Subtact the unit matix fom the metic tenso and thee you have the stain tenso in its full lae stain loy. Howeve the eneal fomulation of Riemannian eomety pemits any makes to be used as coodinates. The consequence of this is that the components of the stain tenso which esult fom this pesciption will not necessaily confom to the enineein definition. This is because diffeentiatin with espect to an anle (say ) is not the same as diffeentiatin with espect to a coodinate with the dimensions of lenth in the diection of. But the enineein definition equies that stains be dimensionless with displacements popely diffeentiated with espect to quantities with the dimensions of lenth. Thee is a futhe complication. Riemannian eomety contains two vesions of the metic tenso: the covaiant and the contavaiant foms. Which ae we to use to define the stains? The pesciption used hee is fistly to define covaiant tensoial stains as follows (7) We now estict attention to small stains. In this case the contavaiant metic can be witten to fist ode in stain O( ) (8) Note that the undefomed metic is not necessaily the unit matix but depends upon the coodinate system chosen. Multiplication of (7) by (8) shows that the defomed contavaiant metic is the invese of the defomed covaiant metic to fist ode in the stain as equied. We now estict attention to spheical pola coodinates in tems of which the undefomed metic tenso can be witten and (9) Relativists miht wonde why I have not included the familia cyclic symmety condition: R R R. This is because in 3D this equation may be deived fom Equs.(6). Only in fou o moe dimensions does this become an independent symmety. In 4D spacetime this toethe with Equs.(6) imply that the Riemann tenso has independent components.

4 The dimensionless (enineein) stains in spheical polas ae ; ; () ; Compatibility Equations in Catesian Coodinates Do Equs. (5) educe to the familia Catesian equations? Yes they do. Fo example we find R And R These and thei symmetical equivalents educe to the usual Catesian compatibility equations when the Riemann tenso is equated to zeo. Note that in Catesian coodinates and small stain theoy the tems quadatic in in Equ.(3) can be inoed. Compatibility Equations in D (Cylindical) Polas Do Equs. (5) educe to Equ.()? Yes they do. But ce it tuns out that the 3D (spheical) pola expession fo R is identical we leave the deivation to Explicit Evaluation of Compatibility Equations in 3D Spheical Polas The non-linea tems quadatic in in Equ.(3) now play an essential pat even in small stain theoy. The eason is that the Chistoffel symbols ( ) can be lae due to the coodinate system even thouh the stain is small. Howeve when evaluatin tems in the Riemann tenso thee odes of tems must be distinuished:- [] Zeo th ode tems which depend only upon the undefomed coodinates and not upon the stains. Since the undefomed coodinate system also has zeo intinsic cuvatue the sum of all such tems contibutin to a iven component of the Riemann tenso must be zeo. We theefoe just inoe them. [] Fist ode tems: these ae the tems linea in the stain. These ae the tems we wish to find. [] Tems quadatic in stain : These will be nelected assumin stains ae small compaed with unity. The technique to explicitly evaluate the Riemann tenso in the small stain appoximation can be illustated by a couple of examples. In Equ.3 the fist two tems ae linea in and hence ae equied to fist ode in stain. An example is (a)

5 whee we have used Equs.(78) to expess the defomed metic in tems of the undefomed metic and the tensoial stain. Note the use of the symbol in (). This is a eminde that () omits the laest (zeo th ode) contibution to the th Chistoffel symbol namely ode. It also means that the second ode tems have been inoed. Thus denotes the fist ode pat of. Appealin now to the explicit expessions fo the undefomed metic Equs.(9) Equ.() simplifies as follows (b) whee we have conveted to conventional enineein stains u Equs.(). When evaluatin the second pai of tems in the Riemann tenso i.e. those quadatic in e.. one of the must be evaluated to zeo th ode and the othe to fist ode. So we also need the Chistoffel symbols to zeo th ode. An example is th ode A selection of othe esults (thouh not exhaustive) is iven below () st ode pats:- (3) ; ; cos th ode appoximations:- (4) ; ; cos ; Expessions (-4) ae all that ae equied to evaluated the Riemann components R

6 and R fom Equ.(3). Equatin these to zeo yields Equs.() and () espectively. The emainin fou compatibility equations would follow fom equatin the othe fou independent Riemann components to zeo i.e. R R R and R =. This is left as an execise fo the eade.

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