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Coordination Transformations for Strain & Stress Rates
To keep the presentation as simple as possible, we will look at
purely two-dimensionalstress-strain rates. Given an original
coordinate system (x,y ) and a rotated system
( yx, ) as shown below:
x
x'y
y'
Recall that the strain rates in the x-y coordinate system
are:
u 1 u v v = = =xx xy yyx 2
y+
x y
Or, in index notation:
=ij1
2
x
ui
j
+u
xi
j
Also, we note that the unit vectors for the rotated axes
are:
K
iK = cosi
K+ sinj
KK K
j= sini+ cosj
Thus, the location of a point in ( yx, ) is:
cos sin x= y y sin cos
Similarly, the velocity components are related by:
cos sin u u=
v v sin cos For differential changes, we also have
cos sin dx dx= dy dy sin cos
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Coordination Transformations for Strain & Stress Rates
16.100 2002 3
If yx are the principal strain directions, then yyyxxxxy &,0
and= are
)
22
22
(cossin
cossin
sincos
xxyyyx
yyxxyy
yyxxxx
=
+=
+=
if 0xy =
The next step is to relate the stresses in ( ) ( )yxtoyx ,, .
Consider a differential surfacewith y normal:
The resultant stress is given as the vector
and the force on the surface is .ds
Decomposing the stress vector into the coordinate axes
gives:
( )
( ) ( )
yx yy
xx yx yy xy
d x i j d x
d y d x i d x d y j
= +
= + +
Note that:
cos
sin
cos sin
sin cos
dx dx
dy dx
i i j
j i j
=
=
=
= +
dx
dy
yx
yy
xx
xy
yy
yx
dx
x
x'y
y'
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Coordination Transformations for Strain & Stress Rates
16.100 2002 4
Thus, the second line becomes:
( sin cos )(cos sin )
( cos sin )(sin cos ) ( )
xx yx
yy xy yx yy
i j dx
i j dx i j dx
+
+ + = +
So collecting all the ji & terms (and enforcing yxxy = )
gives:
2 2
2 2
( )sin cos (cos sin )
sin cos 2 sin cos
yx yy xx yx
yy xx yy yx
= +
= +
For the principal strain axes,
( )0
22
=
++=
++=
xy
yyxxyyyy
yyxxxxxx
Plugging this into yyxy and gives
( )
( ) ( )
2 2
2 sin cos
2 sin cos
xy
xx yyyy
yx yy xx
yy xx yy xx yy
+
=
= + + +
Thus, we arrive at the known result:
( )
2
2
yx xy
yy yy xx yy
=
= + +
A similar derivation would give:
( ) 2xx xx xx yy = + +
