When I was reading a book about crystal plasticity theory, I encountered
a derivation of formula. In the Author’s derivation , he used such a tensor
equation as:
AB : C =A : CBT =B : AT C
Note:
(1): All the tensor are second-order tensor
(2): dot product must come first, double dot product come second.
I try to prove the correctness of the formula, in fact it’s easy to prove it
using the component expression of the tensor.
T
Aim Bmj : Cij =Aim : Cij BTjm =Bmj : Ami
Cij
Throught the component expression, I found that such formula should
have a more universal form, we can derivate more and more tensor
equations in the same fashion.
We can find that: Aim ，Bmj ，Cij , Both of them have a index of the same.
So, both of them can make a dot product. Then make a double product
with the residual tensor.
The possible dot product between two tensors, such as:
T
Dot product Between A and B have AB ， B
AT two form
Dot product Between B and
C have BC T ， CBT two form
Dot product Between A and
C have AT C ， C T A two form
Next step is double dot product:
We take
BC T as a example, the tensor should double dot product to
*
.
BC T is Ami
In order to establish a equation as :
We let
*
Ami
 Aim ,
As we know
So we set
Then :
*
Ami
: Bmj C Tji  Aim Bmj : Cij
T
Aim =Ami
*
T
Ami
 Ami
AB : C =AT : BC T
As we know dot product between two second-order tensor can be
exchanged. Such sa:
A : B  B : A,
The component expression is:
So： AB : C
Aij Bij  Bij Aij
=AT : BC T =BC T : AT
In the same theory, we can derivate all the equations, There are total of
tewlve formulation:
AB : C
CBT : A
AT C : B
C : AB
A : CBT
B : AT C
BT AT : C T
BC T : AT
C T A : BT
C T : BT AT
AT : BC T
BT : C T A
All the component expression list below:
T
Aim Bmj : Cij  Cij BTjm : Aim  Ami
Cij : Bmj =
T
Cij : Aim Bmj  Aim : Cij BTjm  Bmj : Ami
Cij =
T
T
BTjm Ami
: C Tji =Bmj C Tji : Ami
 CijT Aim : BTjm =
T
T
C Tji : BTjm Ami
=Ami
: Bmj C Tji  BTjm : C Tji Aim