References CASTIGLIANO, A.: Nuova teoria intorno all'equilibrio dei sistemi elastici. Atti

accad. sci. Torino 10, 380-423 (1875). DEUTSCH, E.: Sur Ie probleme de la torsion de certains cylindres elastiques isotropes.

C. R. Acad. Sci. (Paris) 251, 2281-2283 (1960). FILON, L. N. G.: (1) On the approximate solution for the bending of a beam of rectan-

gular cross-section. Phil. Trans. (A) 201, 63-155 (1903); (2) On antiplane stress in an elastic solid. Proc. Roy. Soc. (A) 160, 137 -154 (1937).

HILBERT, D.: Uber eine Anwendung der Integralgleichungen auf ein Problem der Funktionentheorie. Verhandlungen des III. internationalen Mathematiker-Kongresses, Heidelberg 1904.

HOOKE, ROBERT: Lectures de Potentia Restitutiva, or of Spring Explaining the Power of Springing Bodies. London 1678. Facsimile reproduction in R. T. Gun-ther: Early Science in Oxford 8, 331-356 (1931).

LAME, G.: Le90ns sur la theorie mathematique de l'elasticite des corps solides Paris (1852).

LEIBENZON, L. S.: Variational methods of solution of problems in the theory of elasticity (in Russian). Moscow-Leningrad 1943.

LEKHNITSKII, S. G.: Teoriya uprugosti anizotropnogo tela. Moscow-Leningrad 1950. LOVE, A. E. H.: Mathematical theory of elasticity, 4th edition. Cambridge 1934. MCCONNELL, A. J.: Absolute differential calculus. London 1931. MILNE-THOMSON, L. M.: (1) Consistency equations for the stresses in isotropic

elastic and plastic materials. J. Lond. Math. Soc. 17, 115-128 (1942); (2) Calculo tensorial por metodos directos. Rev. Mat. Hispano-Americana, 4th series 10, 1-27 (1950); (3) Deformaciones finitas y elasticidad. Rev. Mat. Hispano-Americana, 4th series 12, 1-27 (1952); (4) The Calculus of finite differences. London: Macmillan 1960; (5) Theoretical hydrodynamics, 4th ed. London: Macmillan 1960; (6) Plane elastic systems. Berlin-Gottingen-Heidelberg: Springer 1960; (7) Bounds for the torsional rigidity of isotropic beams. Proc. Camb. Phil. Soc. 58, 216-219 (1962).

MURNAGHAN, F. D.: Finite deformations of an elastic solid. Amer. J. Math. 59, 235-260 (1937).

MUSKHELISHVILI, N. I.: Some basic problems of the mathematical theory of elasticity (translated into English from the Russian). Groningen (Holland): Noordhoff 1953.

SOKOLNIKOFF, I. S.: Mathematical theory of elasticity, 2nd edition. London-New York-Toronto: McGraw-Hill 1956.

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corpi solidi, Roma. Accad. Lincei Rend. (5) 10, 57 (1901).

Index aeolotropy 23 aerofoil profiles 152 affine transformation 220 alternating symmetry 28 analogies, hydrodynamic 115 analogy, membrane 147, 186 angle of twist 51 anisotropy, cylindrical 248 -, curvilinear 24 -, general 23 -, linear 23 antiplane 41, 42 antisymmetric tensor 4 area as contour integral 86 area theorem 85 associative law 1 associated plane problem 80 - torsion problem 173 - twist 173 applicate 13 applied forces 209 axis of alternating symmetry 33

anisotropy 248 beam 174 monotropy 31 twist 117

bending 99 by couples 89, 204

- couples 216 - without twisting 217 beta, f3 48 body-vector 12 - potential 45 boundary condition 66, 94, 135, 207 - - statical 139

cantilever, anisotropic 243 CASTIGLIANO'S theorem of minimum

strain-energy 150 variational equation 149

central perversion 35 centre of flexure 174, 245 - - symmetry 35 characteristic equation 58, 222 circular tube 257 compatibility 9, 10

complementary energy 149 - potential energy 149 complex conjugate 30 -- stresses 60 - torsion potential 112 composite profiles 151 conjugate dyad 2 - tensor 4 continued dyadic product I contraction law 1 couples M I , M i , M3 211 covering operation 27 curvilinear coordinates 96 cyclic function 65 - properties of complex stresses 65 cylindrical anisotropy 249

DE ST. VENANT, B. 79, 194 deformation, adiabatic 16 -, elastic 16

I -, isothermal 16 delta, 6.61 density of internal energy 14 determinateness of complex stress 71 DEUTSCH, E. 131 direct symmetry 28 displacement 51, 64, 93, 114, 173, 227,

250 displacements, virtual 141 distributive law 3 double scalar product 2, 5 dyad 1 dyadic multiplication 1

effective lateral load 67 - stress combinations 72 -- stress functions 72 elastic deformation 16 - homogeneity 23 elasticity, law of 17 ellipse 237 elliptic cylinder 132, 144, 188, 193, 217,

225, 230, 247 - hole 238, 241 ends 77 energy, complementary 149 -, internal 13

264 Index

energy of deformation 14 epsilon, e 47 EULER-BERNOULLI law of bending 101,

206 EULERIAN deformation tensor 9 ezz 22 e2z 50

FILON, L. N. G. 41, 42, 102, 219 first scalar invariant 5 flexural rigidity 101 flexure 89 -, BERNOULLI'S lemniscate 182 -, cardioid 181 -, centre of 174, 245 -, circle 175 -, circular annulus 177 -, ellipse 187, 193, 199, 247 -, equilateral triangle 188 - function, DE ST. VENANT'sI94 - , limayon 179 --- problem 170 -, rectangle 196, 200, 230 fourth rank idemfactor 21 fundamental stress combinations 44

gamma, 'Yv 61 GAUSS'S theorem 15, 189 generalized plane deformation 234 geometrical parameters 172

half-plane 235 half-sections 183 HAMILTON, Sir W. R. 9, 12 HOOKE, ROBERT 21 HOOKE's law 20 - moduli 23 - tensor 21 hydrodynamic analogies 115

idemfactor 3 -, fourth rank 21 indefinite integration 57 induced circuit 64 - mappings 237 infinitesimal deformation 10 integration by parts 86, 189 internal energy 13 invariant, first scalar 5 inverse moduli 23 isotropic beams 79 - point 8 isotropy 35 - transverse 31

KELVIN'S hypothesis 13

LAGRANGE'S variational equation 141 Lagrangian deformation tensor 9, 11, 13,

16 LAME'S constants 36 - problem 258 LEIBENZON, L. S. 146, 152 line force 235 linear vector operator 3 lines of shearing stress 113, 176 - - principal stress 7, 105 load point 87, 170

m 47 mapping on a semicircle 131 maximum shearing stress 116 - torsional rigidity 122, 124 MCCONNELL, A. J. 9 membrane analogy 147, 186 MILNE-THOMSON, L. M. 3, 10, 17, 30,

41,43,60,65,69,73,80,85,96, 105, 107, 115, 116, 154, 189, 219, 234

minimum strain-energy 150 - torsional rigidity 122 modified inverse moduli 53 moduli, HOOKE'S 23 - inverse 23 moment Mg 87 - twisting 118, 120 monotropy 31 motion, equation of 11

I mu,!tv 59

nabla 9, 12 neutral plane 100 norm of a tensor 10

orthotropy 29, 228

PASCAL'S limayons 179 plane deformation, generalized 234 - of bending 100 - of the couple 100 POISSON'S ratio 37, 229 potential 45 - energy 125, 141 - --, complementary 149 principal axis 35 - plane 7 - stress 6 principle of minimum complementary

energy 150 virtual stresses 148

- - work 141

Index 265

psi, 'P 44, 47 pure antiplane problem 80, 219 - torsion 217,219 push 89, 102

rank I, 2 rectangular beam 136, 145, 196, 200,

230 reduced load 171 root 77

scalar law 1 - product 2 semicircle, mapping on 131 semicolon 1 shear function 74, 79 - stress functions 184 simple extension 89, 97 skew tensor 4 SOKOLNIKOFF, I. S. 148, 181 solution of the antiplane problem 95 state of a system 13 statical boundary condition 139, 144 STEVENSON, A. C. 131, 174, 197 STOKES'S theorem 85 strain coefficients 46 strain-energy function 18, 24, 53 stress components 6, 12, 62 - functions 44 - in terms of stress functions 55 -, principal 6

quadric 7 tensor 5 - in cartesians 12 vector 6

stresses at infinity 69, 70 subharmonic functions 117 suspended cylinder 98 symmetric tensor 4 symmetry, alternating 28 -, axis of 28, 29 -, direct 28 -, elastic 26 -, plane of 28

tau, T 51 tensor, antisymmetric 4 -, conjugate 4 -, definition of 3 -, deformation 9

4erivative 17 - H(4) 19 - H(4) 20 -, HOOKE'S 21

tensor, RIEMANN-CHRISTOFFEL 9 -, skew 4 -. stress 5 -, symmetric 4 TIMOSHENKO'S stress fu~ction 190, 224 torsion 89 -, BERNOULLI'S lemniscate 128 -, BOOTH'S lemniscate 135 -, cardioid 127 -, circle 127 -, circular shaft with insert 163 -, compound bar 160 -, confocal elliptic ring 138 -, ellipse 132,144,217,229 -, epitrochoid 134 -, equilateral triangle 133 -, homothetic elliptic ring 133 -, isosceles triangle 159 -,lune 134 -, parabola 146 -. parabolic crescent 154 --, rectangle 136, 145 -, segmental cross-section 155 -, semicircle 130 - shear stress function 113 -, triangle 157 - without bending 217 torsional rigidity 121, 216, 218

-, maximum of 122 - -, maximum property 124 - -, minimum property of 122 transformation of integrals 208 transverse isotropy 31 twisting and bending 216

by axial couple 213 - couples 89,213,216,217 - moment 118, 120, 226 two-tensor, 2-tensor 3

uniqueness theorem 139 unstressed state 19

variational equation, CASTIGLIANO'S 149 - -, LAGRANGE'S 141 virtual displacements 141

stresses 148, 150 work 142, 197 work of stress 15

warping ll5, 259

YOUNG'S modulus 37

zz method 132