Rotation of Axes

55 The rule for changing second order tensor components under rotation of axes goes as follow:

= ajTjqaqpvp From Eq. 1.33

But we also have Ui = Tipvp (again from Eq. 1.39-a) in the barred system, equating these two expressions we obtain

hence

Tip

= a aq T— ai apT jq

in Matrix Form

[T ] =

[A]T [T ][A]

(1.73)

T-

ai apTip

in Matrix Form

[T ] =

[A][T][A]T

(1.74)

By extension, higher order tensors can be similarly transformed from one coordinate system to another. 56 If we consider the 2D case, From Eq. 1.38

cos a — sin a

0

xx

Txy

Txy

Tyy

0

xy xy 0

yy 0

sin 2aTx

2 (— sin 2aTxx + sin 2aTyy + 2 cos 2aTxy sin2 aTxx + cos a (cos aTyy — 2 sin aTxy 0

alternatively, using sin 2a = 2 sin a cos a and cos 2a = cos2 a — sin2 a, this last equation can be rewritten as

i T xx

cos2 d

sin2 0

2 sin 0 cos 0

i Txx 1

^ T yy

=

sin2 d

cos2 0

—2 sin 0 cos 0

^ Tyy

Txy

— sin 0 cos 0

cos 0 sin 0

cos2 0 — sin2 0

0 0

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