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Please follow the pdf’s index, and why the index shown doesn’t match this 2-d function….42
CHAPTER 4. VECTORS AND TENSORS
where A, B are the projections of Ā onto the coordinate axes. The right
panel of figure 4.1 shows the same vector Ā and an orthogonal Cartestian
coordinate system, but now this system has been rotated relative to the
system in the left panel. In this new system, designated with a prime,
Ā= A’ + B’j’
Thus
Aſ + Bj = A’i’ + B’j’.
(4.2)
Let the primed axes be rotated by an angle & from the unprimed axes. Each
primed base vector can be expressed in terms of unprimed base vectors, as
in figure 4.2,
히 = cos di + sin 07,
j’= – sin 67 + cos 6j.
Using these in (4.2) produces
A = A’ cos 0 – B’ sin ,
B = A’ sin 6 + B’ cos 0.
Matrix notation is sometimes handy:
A
cose – sin
sin e
] ()
cos A
This can be inverted to produce
Α’
B’
() = [
cos sin
– sin cos
30] ()
While useful here, matrix notation becomes cumbersome for higher dimen-
sional cases, and will generally be avoided in favor of tensor notation.
4.2
More standard vector transformation
The notation of the previous section, where each component of A has its
own symbol, is cumbersome. Instead we choose tensor notation, also called
indicial notation, where the components of Ā are A1, A2, glibly refered to as
Aj. Correspondingly, the base vectors will now be called ēl, ēz instead of 7,j.
4.2. MORE STANDARD VECTOR TRANSFORMATION
43
ſ’
او
Figure 4.2: Base vector conversion.
AX2
AR
012)
022
X’i
021
011
11
Figure 4.3: Base vector conversion.
In order to generalize the transformation, it is convenient to refer to bij’
the angle between ē; and ēj, shown in figure 4.3. In two dimensions, all
these angles shown can be related to the angle 6 of the previous section, e.g.,
011′ = 6221 = 0, 021 = 1 – 0, etc. The cosine of these angles are the direction
cosines,
lij’
= cos Oija.
As above, the vector Ã may be expressed with components in either the
unprimed or primed system,
Ā= Ajēi + Azē2 = Alē + Alsē.
thus
Ajē; = A’e’,
remembering to use Einstein’s summation convention. Each primed base
vector can be expressed in terms of unprimed base vectors, as in figure 4.2,
7 = cos 011 ēj + cos 6211 ēm,
ē = cos 012, ēi + cos 622, 72,
44
CHAPTER 4. VECTORS AND TENSORS
012
o’11
011
012
021
022
Figure 4.4: Differential triangle.
Finally
A; = A’ cos bij!.
Again, the components A’
; are converted into A; with only the angles between
coordinate axes. This transformation rule may be expressed as
A; = lijı A’;;
which also applies to three dimensions.
4.3
Transformation of stress
As discussed in a previous chapter, the stress on an oblique face of a differ-
ential tetrahedron is found with a balance of forces on the tetrahedron. We
will consider the two-dimensional case first, which the tetrahedron is merely
a triangle. Choose the oblique face of the tetrahedron to be oriented with
one of the primed axes, e.g., choose the normal to be parallel to ē’, making
the normal and tangential stress components on this oblique face the same
as o’11 and 0-12, as shown in figure 4.4.
Stress is converted to the primed coordinate system by balancing the
forces in a convenient direction. Choose the x’ı direction, for example, and
remember that stress must be multiplied by area to get force:
011A1 – 011Aj cos 011 – 012A1 cos 621 021 A2 cos 0 12 – 022 A2 cos 6221 = 0,
4.4. EXERCISES
45
where A1, A2, Aſ are the three areas in figure 4.4. These areas are geometri-
cally related:
A1 = Aj cos 011′, A2 = Aſ cos 012′.
The result is
011 = 011 cos 011′ cos 011′ + 012 Cos 011cos 621
+021 Cos 012, cos 012+022 Cos 012 cos 622. (4.3)
This transformation is the same as
d’ab = lailB;’O ij.
This expression again is valid for three dimensions.
SI 5 wolus or less
Chapter 4 Vectors and Tensors
 Conaider a transformation that consists of a fototion of 180 degrees about the x-ars as shown in figure
(a) Find the direction Cosines.
Rotation of 180 degrees:
&
L
=
O
1 оо
Г1
1 o
O COSB Sino = O COBI80°Sinis
O Sin180° cos180°
Y’ –
Y
o-sino Cose
d 것
=
1 o
0
10=L
о
=LT
(b) Presume that a differential element in a state of plane stress, where Oxx = Oxy = Ox2=0, has value of
Oxy=10 0yz=2 8zz = 5
– 5
in some units. Find the stress components in the primed systein by performing a tensor transforma
1
Sirice Oxx = Sxy=0xz =O , we have
0
To ool
6=0
LO
og
1
Trio
o o=LTOL= 0 -1 0 0 2
0-111025 oo-1
2
5
2
La
2 5

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