弾性定数の対称性について

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() by T. oyama

() ij C ij = () () C, C, C () ij ji ij ijlk ij ij () C C C C C C * C C C C C * * C C C C = * * * C C C * * * * C C * * * * * C () * P (,, ) P (,, ) lij = () P (,, ) P(,, ) (,, ) P (, 00, ) ( 0,, 0) ( 00,, ) (, 00, ) ( 0,, 0) ( 00,, ) (, 00, ) ( cos( θ ), cos( θ ), cos( θ )) θ ij j i ( 0,, 0) ( 00,, )

( cos( θ ), cos( θ ), cos( θ )) ( cos( θ ), cos( θ ), cos( θ )) = cos( θ ) + cos( θ ) + cos( θ ) cos( θ ) + cos( θ ) + cos( θ ) cos( θ ) + cos( θ ) + cos( θ ) = (5) () (5)l ij (6) = (6) (6)θ l ij l l ik ik l l = 0( i j) ik l l jk ki ki l l = 0( i j) ki kj = = (7) ll i i = ll + ll + ll = l l = l l + l l + l l + l l + l l + l l = 0 i j l ij (8) (9) = l l (8) ij ik jl = l l k l = l l + l l + l l + l l + l l + l l + l l + l l + l l ij l ik l jl = (9),,

- =, =, = Cij l ij = = cos( 0) cos( 90) cos( 90) cos( 90) cos( 0) cos( 90) = cos( 90) cos( 90) cos( 80) 0 0 0 0 0 0 (0) (8)(9) = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l () () ij ij () ()

= + C + C + C + C + C + C + C + C + C + ( C ) + C + C + ( C ) + ( C ) + ( C ) + C + C + C + C + C + C + C + C + C C = 0, C = 0 C = 0, C = 0, C = 0, C = 0, C = 0, C = 0 = + C + C + C + C + C + C + C + C + C + ( C ) + C + C + ( C ) + ( C ) + ( C ) + C = ( C + C + C + C + C + C + C + C + C ) C = 0, C = 0, C = 0, C = 0, C = 0 C = 0, C = 0, C = 0, C = 0, C = 0 C = 0, C = 0, C = 0, C = 0, C = 0, C = 0, C = 0, C = 0 () () C C C 0 0 C * C C 0 0 C * * C 0 0 C = * * * C C 0 * * * * C 0 * * * * * C () - = = =,, Cij l ij

= = cos( 80) cos( 90) cos( 90) 0 0 cos( 90) cos( 0) cos( 90) 0 0 = cos( 90) cos( 90) cos( 0) 0 0 (5) (8)(9) = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l (6) (7) ij ij (6) (7) = + C + C + C + C + C + C + C + C + ( C ) + ( C ) + ( C ) + C + C + ( C ) + C + C + C + C + C + C + C + C + C + C C = 0, C = 0 C = 0, C = 0, C = 0, C = 0, C = 0, C = 0 5

= + C + C + C + C + C + C + C + C + ( C ) + ( C ) + ( C ) + C + C + ( C ) + C + C = ( C + C + C + C + C + C + C + C + C ) C = 0, C = 0, C = 0, C = 0, C = 0 C = 0, C = 0, C = 0, C = 0, C = 0 C = 0, C = 0, C = 0, C = 0, C = 0, C = 0, C = 0, C = 0 (8) () C C C C 0 0 * C C C 0 0 * * C C 0 0 = * * * C 0 0 * * * * C C * * * * * C (9) - =, =, = Cij l ij = cos( 0) cos( 90) cos( 90) = cos( 90) cos( 80) cos( 90) = cos( 90) cos( 90) cos( 0) 0 0 0 0 0 0 (0) (8)(9) 6

= l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l () () ij ij () () = + C + C + C + C + C + C + C + C + ( C ) + C + ( C ) + C + ( C ) + C + ( C ) + C + C + C + C + C + C + C + C + C C = 0, C = 0 C = 0, C = 0, C = 0, C = 0, C = 0, C = 0 = + C + C + C + C + C + C + C + C + ( C ) + C + ( C ) + C + ( C ) + C + ( C ) + C = ( C + C + C + C + C + C + C + C + C ) C = 0, C = 0, C = 0, C = 0, C = 0 C = 0, C = 0, C = 0, C = 0, C = 0 C = 0, C = 0, C = 0, C = 0, C = 0, C = 0, C = 0, C = 0 () 7

() C C C 0 C 0 * C C 0 C 0 * * C 0 C 0 = * * * C 0 C * * * * C 0 * * * * * C () ()(9)() C C C 0 0 0 * C C 0 0 0 * * C 0 0 0 = * * * C 0 0 * * * * C 0 * * * * * C (5) =, =, = =, =, = =, =, = - =, =, = l ij = = cos( 90) cos( 0) cos( 90) cos( 80) cos( 90) cos( 90) = cos( 90) cos( 90) cos( 0) 0 0 0 0 0 0 (6) (8)(9) 8

= l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l = l l = l l = k l (7) (8) ij ij (7) (8) = + C + C + C + C + C + C + C + C C, C = + C + C + C + C + C + C + C + C C, C 9

= + C + C + C + C + C + C + C + C C = ( C ) = = = C = C = C == = C C C, C, C (9) 0

C C C 0 0 0 * C C 0 0 0 * * C 0 0 0 = * * * C 0 0 * * * * C 0 * * * * * C (0) () () C C C 0 0 0 * C C 0 0 0 * * C 0 0 0 = * * * C 0 0 * * * * C 0 * * * * * C 66 () - =, =, = -,, (9),, C, C, C () C C C 0 0 0 * C C 0 0 0 * * C 0 0 0 = * * * C 0 0 * * * * C 0 * * * * * C () - =, =, = -,, (9),, C, C, C ()

C C C 0 0 0 * C C 0 0 0 * * C 0 0 0 = * * * C 0 0 * * * * C 0 * * * * * C (5) C C C 0 0 0 * C C 0 0 0 * * C 0 0 0 = * * * C 0 0 * * * * C 0 * * * * * C (6) ()()(5) C C C 0 0 0 * C C 0 0 0 * * C 0 0 0 = * * * C 0 0 * * * * C 0 * * * * * C 66 (7) C C C 0 0 0 C C C 0 0 0 * C C 0 0 0 * C C 0 0 0 * * C 0 0 0 * * C 0 0 0 = = * * * C 0 0 * * * C66 0 0 * * * * C55 0 * * * * C 0 * * * * * C 55 * * * * * C (8)

C C C 0 0 0 C C C 0 0 0 * C C 0 0 0 * C C 0 0 0 * * C 0 0 0 * * C 0 0 0 = = * * * C66 0 0 * * * C 0 0 * * * * C55 0 * * * * C66 0 * * * * * C 66 * * * * * C (9) y l ij = = cos( 60) cos( 0) cos( 90) cos( 50) cos( 60) cos( 90) = cos( 90) cos( 90) cos( 0) / / 0 / / 0 0 0 (0) (8)(9) = l l = l l + l l + l l + l l = + + + = l l = l l + l l + l l + l l = + = l l = l l = k l = ll = ll + ll = + = l l = l l + l l = + = l l = l l + l l + l l + l l = + + k l ()

= l l = l l + l l + l l + l l = + + + = l l = l l + l l + l l + l l = + = l l = l l = k l = ll = ll + ll = + = l l = l l + l l = + = l l = l l + l l + l l + l l = + + () ij ij () () + C + C L = M O + + + P L + M O C C + P + C N Q N b g b g b g + C + C C + C + C + C = + + b g b g bc C Cg C C + C + C + C = + + + + = bc + C g + C + bc + C g + bc + C g C, C, C C Q

= C C + C = + b g b g b = C + C + C C + C + C + C + C b g b g b g = C C C + C + C + C C C C, C C, C g + C = + + C C C, C, C, C C () C C C 0 0 0 * C C 0 0 0 * * C 0 0 0 = * * * C 0 0 * * * * C 0 C C * * * * * () (5) 5

C C C 0 0 0 * C C 0 0 0 * * C 0 0 0 = * * * C 0 0 * * * * C 0 C C * * * * * (5) 6

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