连续介质力学练习2

(c) The expression ti = Sij nj is a valid indicial notation expression, for which i is a free index (giving 31 = 3 equations) and j is a dummy index pair (giving 1 + 31 = 4 terms per equation). The corresponding expanded equations are t1 = S11 n1 + S12 n2 + S13 n3 , t2 = S21 n1 + S22 n2 + S23 n3 , t3 = S31 n1 + S32 n2 + S33 n3 . (d) The expression ti = Sij ni is not a valid indicial notation expression. The index i looks like a free index on the left-hand side and looks like a dummy index pair on the right-hand side and the index j appears in only the term on the right-hand side. (e) The expression Sij = 2µEij + λEkk δij is a valid indicial notation expression, for which i and j are free indices (giving 32 = 9 equations) and k is a dummy index pair (giving 1+1+31 = 5 terms per equation). The corresponding expanded equations are S11 = 2µE11 + λ(E11 + E22 + E33 )δ11 , S12 = 2µE12 + λ(E11 + E22 + E33 )δ12 , S13 = 2µE13 + λ(E11 + E22 + E33 )δ13 , . . . S33 = 2µE33 + λ(E11 + E22 + E33 )δ33 . (f) The expression xi xi = ρ2 is a valid indicial notation expression, with no free indices (giving 30 = 1 equation) and for which i is a dummy index pair (giving 31 + 1 = 4 terms). The corresponding expanded equation is x1 x1 + x2 x2 + x3 x3 = ρ2 , or, equivalently,
ijk ai bj ek ) ·
(
pqr cp dq er )
ijk pqr ai bj cp dq (ek ijk pqr ai bj cp dq δkr ijk pqk ai bj cp dq
Fra Baidu bibliotek
· er )
= (δip δjq − δiq δjp )ai bj cp dq = ai bj ci dj − ai bj cj di = (ai ci )(bj dj ) − (ai di )(bj cj ) = (a · c)(b · d) − (a · d)(b · c) .
ijk pjk
= δip δjj − δij δjp = 3δip − δip = 2δip .
(c) If follows from contraction of part (b) that
ijk ijk
= 2δii = 6 .
Problem 2.12 Using indicial notation and scalar components with respect to a right-handed orthonormal basis {ei }, and taking care not to use the same character for two different dummy index pairs, (a × b) · (c × d) = ( = = =
ME2004 Elasticity Homework #2 Solution Set Problem 2.7 (a) For the expression Ams = bm (cr − dr ) [which can be rewritten as Ams = bm cr − bm dr ], the index m is a free index since it appears once in each term. However, the indices s and r are inconsistent—they are neither free indices nor do they form dummy index pairs. Thus, there is no unambiguous way to expand this expression. This expression is not a valid indicial notation expression. (b) The expression Ams = bm (cs − ds ) is a valid indicial notation expression, for which m and s are free indices (giving 32 = 9 equations) and there are no dummy index pairs (so there are 3 terms per equation). The corresponding expanded equations are A11 = b1 (c1 − d1 ) , A21 = b2 (c1 − d1 ) , A31 = b3 (c1 − d1 ) , A12 = b1 (c2 − d2 ) , A22 = b2 (c2 − d2 ) , A32 = b3 (c2 − d2 ) , A13 = b1 (c3 − d3 ) , A23 = b2 (c3 − d3 ) , A33 = b3 (c3 − d3 ) .
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Problem 2.11 (a) Using contraction, the substitution property of the Kronecker delta, and the identity δkk = 3, it follows from (2.97) that δip δiq δik ijk pqk = det δjp δjq δjk δkp δkq δkk = 3(δip δjq − δiq δjp ) + δkq (δik δjp − δip δjk ) + δkp (δiq δjk − δik δjq ) = 3(δip δjq − δiq δjp ) + δiq δjp − δip δjq + δiq δjp − δip δjq = δip δjq − δiq δjp . (b) If follows from contraction of part (a) that
ijk pqr
= ei , ej , ek ep , eq , er ei · ep ei · eq = det ej · ep ej · eq ek · ep ek · eq δip δiq δir = det δjp δjq δjr δkp δkq δkr
ei · er ej · er ek · er .
2 2 2 x2 1 + x2 + x3 = ρ .
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(g) The expression Brs = hr (ds − hs Krr ) [which can be rewritten as Brs = hr ds − hr hs Krr ] is not a valid indicial notation expression, because the index r is not consistent. (In the second term on the right-hand side, is r a free index of a dummy index pair? It cannot be both.) (h) The expression Bij cj = 0 is a valid indicial notation expression, for which i is a free index (giving 31 = 3 equations) and j is a dummy index pair (giving 31 + 1 = 4 terms per equation). Note that this is an example of the one exception to the general rule that a free index must appear once and only once in every term of a valid indicial notation expression—the numerical value “0” on the right-hand side is assumed to carry over to each equation. The corresponding expanded equations are B11 c1 + B12 c2 + B13 c3 = 0 , B21 c1 + B22 c2 + B23 c3 = 0 , B31 c1 + B32 c2 + B33 c3 = 0 . Problem 2.9 Noting that det[A] det[B ] = det([A][B ]T ), it follows from (2.95) that a1 a2 a3 d1 d2 d3 a, b, c d, e, f = det b1 b2 b3 det e1 e2 e3 c1 c2 c3 f1 f2 f3 a1 a2 a3 d1 e1 f1 = det b1 b2 b3 d2 e2 f2 c1 c2 c3 d3 e3 f3 ai di ai ei ai fi = det bi di bi ei bi fi ci di ci ei ci fi a· d a· e a· f = det b · d b · e b · f . c· d c· e c· f Problem 2.10 Using (2.92) and (2.96), it follows that
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