1.3 Tensor algebra
Cartesian tensors in terms of component transformations under change of basis were considered in Section 1.2. In Section 1.3 Cartesian tensors in its equivalent invariant form are considered. Since second-order tensors are most important in applications, attention is confined to second-order tensors in Sections 1.3.1 to 1.3.5.
1.3.1 Second-order tensors
A second-order Cartesian tensor T is a linear mapping of the vector space into itself, i.e.,
T : .
The element (vector) in to which the vector u maps is denoted Tu . The tensor is linear if,
TTT()uv+= u + v . (1.3.1)
Let ( , ) denote the set of all linear mappings from to . Then ( , ) is itself a vector space with the element S T defined by,
()()(),ST+=uu S + T u " ST, Î ( , ), , , " u Î . (1.3.2)
The inner product ST is defined by,
()STuu= S () T , " ST, Î ( , ), " u Î . (1.3.3)
The zero tensor 0 maps every vector in to the zero vector 0 and the identity tensor I maps every vector in to itself,
0Iu0==", uu uÎ . (1.3.4)
The set of all bilinear functions over forms a vector space over , denoted by ( ,
) which may be identified with ( , ). If an orthonormal basis {ei} is chosen for , then the bilinear function (ei ej ) is defined so that,
()(,)eij e uvuv ij uv , . (1.3.5)
For an arbitrary member T of ( , ), (1.3.5) yields
TTT(,uv )uvijij ( e , e ) ( e ij , e )( e i e j )(,) uv uv , . (1.3.6)
Hence from (1.3.6) the representation for T with respect to the basis {ei} is,
TT(,eeij )( e i e j ).
Comparing with T Tij ei e j , it is seen that T(ei ,e j ) is the component Tij of T relative to the basis {ei} .
December 16, 2019 1.3.1-1 Hence from (1.3.6),
T(,)uv uTiijj v . (1.3.7) i.e., TT(,uv ) u ( v ). (1.3.8)
T In general uvvu··()TT¹ (). Hence the transpose T of T is defined by,
T vuuv()TT () uv , , (1.3.9)
or equivalently,
TTT (,)vu= (,) uv .
T T With respect to an arbitrary orthnormal basis {ei }, the components of T are given by ()T ij= T ji , i.e.,
T T =ÄTjiiee j.
From the definition of TT given by (1.3.9) the following properties are obtained.
()TTTT TTT ()ST S T ST, ( , ), , . (1.3.10) TTT ()ST T S
T A second-order tensor such that TT= is a symmetric tensor. Here TTij= ji .
It is seen that I is symmetric and that I(,)uv= u· v.
A second-order tensor such that TTT =- is a skew-symmetric (or antisymmeteric) tensor. Here
TTij=- ji .
The trace of T is defined with respect to an orthonormal basis {ei } as,
trTTT==Tiiee i· i = ( ee i , i ).
The determinant of T is defined as the determinant of the matrix T of components of T with respect to an orthonormal basis.
det T ==ijkTT i123 j T k ijk TT 123 i j T k .
It can also be shown that,
1 det T 6 ijk pqrTT ip jq T kr .
December 16, 2019 1.3.1-2 When detT ¹ 0 , there exists a unique inverse tensor T-1 such that,
11 TT T T I 11 1 det(TT ) (det ) ST, ( , ), det(S ) 0 . (1.3.11) 111 ()ST T S The adjugate tensor of T denoted by adjT , is defined by (see eqn (38) Chadwick, 1976),
adj(TTT )(ab ) ( a b ) ab , .(a)
When T is invertible, i.e., when T1 exists, it can be shown from eqn (a) that
adj(TTTT1 ) (det ) . (1.3.12)
It can be easily shown that
det (adjTT ) (det )2 . (b)
Problem 1.3.1 Suppose Aijkl are as defined in Problem 1.2.4. Show that (a) if Tij are the components of a symmetric CT(2) then
ATijkl kl T ij,0 AT ijkl kl
and (b) if Tij are the components of a skew-symmetric CT(2) then
AijklTATT kl0, ijkl kl ij .
Problem 1.3.2 If Tij are the components of an arbitrary CT(2), show that ijkT jk are the
components of a CT(1). Deduce that Tij is symmetric if and only if ijkT jk 0.
Problem 1.3.3 If Wij are the components of an antisymmetric CT(2) W then the vector w 1 with components wWiijkkj 2 is called the axial vector of W . Show that ipqwW i qp and that, for an arbitrary vector aw, aW a. Deduce that uv is the axial vector of vuuv.
Problem 1.3.4 If
00T12 T T 00 12 000
is the matrix representing the components of an antisymmetric CT(2) T with respect to basis {ei },
show that, for any change of basis eeiiijj Q e such that ee33 , the matrix representing the components of T is unchanged.
December 16, 2019 1.3.1-3 1.3.1-A1 Useful identities
The following properties can be deduced from the previous equations:
()()()uvwuwvw+Ä=Ä+Ä , (1.3.1.A1.1) uvwÄ+()()() =Ä+Ä uvuw , (1.3.1.A1.2)
T ()()uvÄ=Ä vu, (1.3.1.A1.3)
()()()()uvwx vwux , (1.3.1.A1.4) AA()()uvÄ= u Ä v, (1.3.1.A1.5) T ()uvÄ=ÄAA u () v, (1.3.1.A1.6)
tr(uv ) uv , (1.3.1.A1.7) where uvwx,, , are arbitrary vectors and A is an arbitrary second-order tensor. See pgs. 21-22, eqns (42a,b), (43), (45), (46a,b) and (47a) of Chadwick (1976).
December 16, 2019 1.3.1-A1-1 1.3.1-A2 Inverse of the 3x3 matrix obtained from the definition of the adjugate tensor
The components of the adjugate tensor adjT are (pg. 47, Chadwick (1976)),
1 (adjT )ij= 2 ipq jrsTT pr qs .
T 11 \ (adjT )ij==22 ipq jrsTT rp sq irs jpq TT pr qs .
adj(TT ) From eqn (1.3.12), T1 , (detT ) 11 \ ()TT1T (adj) 1 TT. (1.3.1.A2.1) ij(detTT ) ij (det ) 2 irs jpq pr qs
From matrix algebra (pg. 210, Noble (1969)),
adjA A1 , (1.3.1.A2.2) det A where adjA is the adjoint of the matrix, defined as the transpose of the matrix of cofactors.
Consider the 3x3 matrix,
A11AA 12 13 A A AA. 21 22 23 A31AA 32 33
()()()AA22 33 AA 32 23 AA 21 33 AA 31 23 AA 21 32 AA 31 22 The matrix of cofactors B ()()()AA AA AA AA AA AA . 12 33 32 13 11 33 31 13 11 32 31 12 ()()()AA12 23 AA 22 13 AA 11 23 AA 21 13 AA 11 22 AA 21 12
()()()AA22 33 AA 32 23 AA 12 33 AA 32 13 AA 12 23 AA 22 13 adjABT (AA AA ) ( AA AA ) ( AA AA ) . 21 33 31 23 11 33 31 13 11 23 21 13 ()()()AA21 32 AA 31 22 AA 11 32 AA 31 12 AA 11 22 AA 21 12
1 Check whether, [adjA]ij 2 irs jpqA prA qs .
1 1 [adjA]112 1rs 1 pqA prAAAAA qs 22 33 32 23 , [adjA]122 1rs 2 pqA prAAAAA qs ( 12 33 32 13 ), … 1 [adjA]332 3rs 3 pqA prAAAAA qs 11 22 21 12 .
Hence, from eqns (1.3.1.A2.1) and (1.3.1.A2.2) it can be concluded that the inverse of a 3x3 matrix obtained from the definition of the adjugate tensor for second-order tensors and the inverse of a 3x3 matrix obtained from matrix algebra are the same.
Ref: Noble, B., 1969, Applied Linear Algebra, Prentice-Hall, Inc. New Jersey. (BN)
December 16, 2019 1.3.1-A2-1