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Appendix A Curvilinear coordinate systems Results in the main text are given in one of the three most frequently used coordinate systems: Cartesian, cylindrical or spherical. Here, we provide the material necessary for formulation of the elasticity problems in an arbitrary curvilinear orthogonal coor- dinate system. We then specify the elasticity equations for four coordinate systems: elliptic cylindrical, bipolar cylindrical, toroidal and ellipsoidal. These coordinate systems (and a number of other, more complex systems) are discussed in detail in the book of Blokh (1964). The elasticity equations in general curvilinear coordinates are given in terms of either tensorial or physical components. Therefore, explanation of the relationship between these components and the necessary background are given in the text to follow. Curvilinear coordinates ~i (i = 1,2,3) can be specified by expressing them ~i = ~i(Xl,X2,X3) in terms of Cartesian coordinates Xi (i = 1,2,3). It is as- sumed that these functions have continuous first partial derivatives and Jacobian J = la~i/axjl =I- 0 everywhere. Surfaces ~i = const (i = 1,2,3) are called coordinate surfaces and pairs of these surfaces intersects along coordinate curves. If the coordinate surfaces intersect at an angle :rr /2, the curvilinear coordinate system is called orthogonal. The usual summation convention (summation over repeated indices from 1 to 3) is assumed, unless noted otherwise. Metric tensor In Cartesian coordinates Xi the square of the distance ds between two neighboring points in space is determined by ds 2 = dxi dXi Since we have ds2 = gkm d~k d~m where functions aXi aXi gkm = a~k a~m constitute components of the metric tensor. 305 306 Handbook of elasticity solutions Base and unit vectors in orthogonal curvilinear coordinates In Cartesian coordinates Xi the base vectors can be chosen as unit vectors iI, i 2, i 3 along the coordinate axes. If dr denotes an infinitesimal vector, then dr =dxj i j =dxjij where ir and i r are the base (unit) vectors with an arbitrarily assigned indexes as covariant and contravariant respectively. Here and in the text to follow, the lower and upper indices stand for covariant and contravariant components, respectively; the difference between them vanishes in the Cartesian coordinate system. A transformation from Cartesian coordinates Xi to curvilinear coordinates ~i, based on transformation law for differentials dx j and dxj, gives the following ex- pression for dr: where are the covariant and contravariant base vectors (not unit vectors, in general), re- spectively. Thus, g j characterizes the change of the position vector r as ~ j varies; it is directed tangential to the coordinate curve, while gj is normal to the coordinate sur- face of ~i = const. In any orthogonal curvilinear coordinate system, the directions of the base vectors g j and gj coincide, so that the unit vectors defined as e j = g j / Jfjj and ei = gi / IiH (no sum over j) are identical. The following relationships hold: gj . gk = gjk. gj . gk = gjk, where gjk. gjk are covariant and contravariant com- ponents of the metric tensor. In Cartesian coordinates, gjk = gjk = 8 jk where 8 jk is Kronecker's delta. Thus, at each point a set of unit vectors (el' e2, e3) tangent to the coordinate curves of an orthogonal curvilinear coordinate system is defined. Tensorial and physical components of vectors and tensors Vectors and tensors in a given curvilinear coordinate system can be characterized by either "tensorial" or "physical" components. Tensorial components refer to a basis composed of vectors having, generally, non-unit lengths. For example, vec- tor v in terms of the tensorial components is written as: v = vj g j = v j gj. Phys- ical components refer to a basis composed of vectors of the same directions but having unit lengths. For example, vector v in terms of the physical components is v = v~j e j = V~j ej . Therefore, tensorial and physical components are interrelated as follows: V~j = vj Jfjj = Vj / Jfjj (no sum over j). Thus, the physical compo- nents U~i of any vector, say displacement vector u, in orthogonal curvilinear co- ordinates ~i are related to the tensorial components Ui (or ui ) of u as follows: U~i = ui/ ffi = ui ffi (no sum over i). Curvilinear coordinate systems 307 The physical components 8~i~j of any second rank tensor, say strain tensor 8, in orthogonal curvilinear coordinates ~i are related to the tensorial components 8ij (or 8ij) of 8 as follows: 8~i~j = 8ijl Jgiigjj = 8ij Jgiigjj (no sum over i or j). Equations of elasticity, given in the main text, are written in the physical compo- nents of displacements, strain and stress tensors, and tensor of the elastic constants. Equations of elasticity in general orthogonal curvilinear coordinates Tensorial components 8ij of the tensor of (small) strains in terms of the tensorial components Ui of the displacement vector u are given by: where aUj k Uij = a~i - UkIij where Ii' are the so-called Christoffel's symbols (they are not components of any tensor) that are zeros for the Cartesian coordinate system; their expressions for sev- eral curvilinear orthogonal coordinate systems are given in the text to follow. Dilatation (relative volume change) in terms of the tensorial components of strains: () = 8ii or, in terms of the physical components of dispJacements, 1 [ a a () = -al- (U~I Jg22g33) + -al- (U~2Jgllg33) .jgllg22g33 <;1 <;2 + a:3 (U~3JgIIg22)J where components gij of the metric tensor are given in the text to follow for several coordinate systems. Tensorial components of the tensor of (small) rotations: 1 UJij = 2(Uij - Uji) or, in terms of the physical components of displacements (no sum over i or j) where ij = 12,23,31. 308 Handbook of elasticity solutions Six compatibility conditions in terms of the tensorial components of strains: a2smq a2snp a2snq a2smp (k s k s) a~na~p + a~ma~q - a~ma~p - a~na~q - 2Sks rqmrpn - rmprqn + 2rn~ Emqk +2rqkm Enpk -2rn~ Empk -2r~p Enqk= 0 where mnpq = 1212, 1313,2323,1213,2123,3132 and _ 1 (aCPk aCkn asnp ) Enpk="2 a~n + a~p - a~k Equations of motion (equilibrium) of any continuous medium in terms of the ten- soria1 stress components: aukm __ + rk u jm + rmukj + Fm a~k Jk Jk a2um = P ----at2 (= 0 in equilibrium), m = 1, 2, 3 where F m are the tensorial components of body force vector F per unit volume, um are the tensorial components of displacement vector u and P is the mass density. Hooke's law for the isotropic material in terms of the physical components: Strains in terms of stresses 1 + v 3v C~i~j = ~U~i~j - [jUOij where u == (U~l ~l + U~2~2 + U~3~3) /3 is the mean normal stress and oij is Kronecker's delta. Inversely, stresses in terms of strains U~i~j = MOij + 2Gc~i~j where),., G(== p,) are Lame constants (see chapter 1). Equations of motion of isotropic elasticity in terms of the physical components of displacements (Lame equations): gjjgkk ae [ a a ] (),. + 20) g;;- a~i - 20 a~j (W~i~j../ikk) - a~k (W~k~i..;gjj) + F~i,jgjjgkk aU~i = y'gjjgkkP at2 (no sum over i or j or k) (= 0 in equilibrium) where ijk = 123,231,321. Below, components gij of the metric tensor and Christoffel's symbols If~ in four orthogonal curvilinear coordinate systems are given. The following symme'try rela- tions hold: gij = gji, If~ = rj;. Curvilinear coordinate systems 309 Elliptic cylindrical coordinates (a, f3, z == X3) b sinha Figure A.i. Traces of the coordinate surfaces in (XI, x2)-plane of the Cartesian coordinate system, shown in Fig. A.I, are confocal ellipses and confocal hyperbolas. The relationship between Cartesian (XI, X2) and elliptic cylindrical (a, f3) coordi- nates is: XI =bcoshacosf3 X2 = b sinha sinf3 (0";; a, 0";; f3 ,,;; 2n, -00";; z,,;; 00) The equations of coordinate curves a = const, f3 = const in Cartesian coordinates (XI, X2) are: X2 x2 ---::---,-1~ + 2 = I b2 cosh2 a b2 sinh2 a x2 x2 --=---,-1-;:;-- _ 2 = 1 b2 cos2 f3 b2 sin2 f3 Components of the metric tensor are: b2 gil = g22 = 2 (cosh2a - cos2f3) g33 = I g23 = g31 = g12 = 0 where subscripts 1,2,3 denote coordinates a, f3, z, respectively. 310 Handbook of elasticity solutions The (non-zero) Christoffel's symbols are: I 2 I sinh(2a) r -r --r, ------- 11 - 21 - 22 - cosh(2a) - cos(2fJ) r,2 _ rl _ _ r2 _ sin(2fJ) 22 - 12 - 11 - cosh(2a) - cos(2fJ) Bipolar cylindrical coordinates (a, fJ, z == X3) b P=sin~ Figure A.2. Traces of the coordinate surfaces in (XI, x2)-plane of the Cartesian coordinate are circles shown in Fig. A.2. The relationship between Cartesian (XI, X2) and bipolar (a, fJ) coordinates is: bsinha XI=----- cosh a + cos fJ b sinfJ X2=----- cosh a + cos fJ ( -00 ::;; a ::;; 00, -7r ::;; fJ ::;; 7r, -00 ::;; z ::;; 00) The equations of coordinate curves a = const, fJ = const of the bipolar coordinate system in Cartesian coordinates (XI, X2) are: b2 (XI - bcotanha)2 +xi = -'-2- smh ex b2 + (X2 + b cotan fJ)2 = -'-2- xt sm fJ Curvilinear coordinate systems 311 Components of the metric tensor are: b2 gIl = g22 = ----------=- (cosh a + cos tl)2 g23 = g31 = g12 = 0 where subscripts 1,2,3 denote coordinates a, tl, Z, respectively. The (non-zero) Christoffel's symbols are: rl -r2 --rl _ -sinha II - 21 - 22 - cosha + cos tl r 2 _ rl _ _ r2 _ sintl 22- 12- 11- cosha+costl Toroidal coordinates (a, tl, e) j3=const a=const Figure A.3.
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