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1.3 Cartesian

A second-order Cartesian is defined as a 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 . The tensor will have different components in different coordinate systems. The tensor T has components Tij with respect to {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 . 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 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) u  v of two vectors become their 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 Tii . The scalar Tii is the of T and is denoted by t rT. (It is a scalar 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)  S T  tr (TS). ij ji T T Similarly SijTkj are the components of ST where T denotes the of T (i.e., TT  T e  e ). In addition tr (ST)  tr (S TTT ) and tr (STT )  tr (S TT). ij j i 2 2 3 If S  T, then TT is denoted by T , TT by T 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