1.3 Cartesian tensors A second-order Cartesian tensor is defined as a linear combination of dyadic products as, T Tijee i j . (1.3.1) The coefficients Tij are the components of T . A tensor exists independent of any coordinate system. The tensor will have different components in different coordinate systems. The tensor T has components Tij with respect to basis {ei} and components Tij with respect to basis {e i}, i.e., T T e e T e e . (1.3.2) pq p q ij i j From (1.3.2) and (1.2.4.6), Tpq ep eq TpqQipQjqei e j Tij e i e j . (1.3.3) Tij QipQjqTpq . (1.3.4) Similarly from (1.3.2) and (1.2.4.6) Tij e i e j Tij QipQjqep eq Tpqe p eq , (1.3.5) Tpq QipQjqTij . (1.3.6) Equations (1.3.4) and (1.3.6) are the transformation rules for changing second order tensor components under change of basis. In general Cartesian tensors of higher order can be expressed as T T e e ... e , (1.3.7) ij ...n i j n and the components transform according to Tijk ... QipQjqQkr ...Tpqr... , Tpqr ... QipQjqQkr ...Tijk ... (1.3.8) The tensor product S T of a CT(m) S and a CT(n) T is a CT(m+n) such that S T S T e e e e e e . i1i2 im j1j 2 jn i1 i2 im j1 j2 j n 1.3.1 Contraction T Consider the components i1i2 ip iq in of a CT(n). Set any two indices equal, iq ip say, and sum over ip from 1 to 3. Theses indices are contracted and the order of the tensor is reduced by two. Special cases: (a) Tensor product u v of two vectors become their dot product u v on contraction since uivj becomes uivi . April 15, 2016 1.3-1 (b) If T is a CT(2) with components Tij then this contracts to the scalar Tii . The scalar Tii is the trace of T and is denoted by trT. (It is a scalar invariant of T.) (c) IIf S and T are CT(2), then S T is a CT(4). Let SijTkl be the components of S T with respect to the basis {ei}. The indices can be contracted in many ways. For example SijTjl are the components of the CT(2) ST and this contracts to the scalar tr (ST) SijTji tr (TS). T T Similarly SijTkj are the components of ST where T denotes the transpose of T (i.e., TT T e e ). In addition tr (ST) tr (S TTT ) and tr (STT ) tr (S TT). ij j i If S T, then TT is denoted by T 2 , TT2 by T 3 etc. Consider the components vAuiijj= (where there is a repetition of only one index of the components of a tensor and a vector) and Aij= BC ik kj , (where there is a repetition of only one index of the components of two tensors). This will not be indicated by a dot when symbolic notation is used, i.e., for vu= A and ABC= , we do not write vu= A and A = BC , respectively. A double contraction of two tensors A and B will be denoted by A :B and is defined by AB: = tr()ABT = tr(BATTT ) = tr( AB ) = tr( B A ) , (1.3.1-a) = BA: See pgs. 13-14 and eqn (1.93) of Holzapfel (2001). The following properties can be deduced from eqn (1.3.1-a): A:(uvÄ= ) u(A v ), (1.3.2-b) ():()uvÄÄ= wx ()() uwvx, (1.3.3-c) See eqns (1.96) and (1.97) of Holzapfel (2001). Note: The symbol “:” denotes the contraction of a pair of repeated indices which appear in the same order, eg. AB: = AijB ij ; se= C : in component form is seij= C ijkl kl ; strain energy 1 ee::C in component form is 1 eeC . 2 2 ij ijkl kl (See pgs. 8-9 of Belytschko, Liu and Moran (2000)). April 15, 2016 1.3-2 .