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```1.2.2b,
three stress components will act on each of the three perpendicular
planes. Now, if we cut a plane ii across these three planes, we will
form a small tetrahedron around P (Figure 1.2.2~). The stress t,
t
‘ i
2
ELASTICSOLID I 9
on plane ii can be determined by a force balance on the tetrahedron.
The force on ii is the vector sum of the forces on the other planes
-fn = fx + fy + fz (1.2.2)
Because force equals stress times area, the balance becomes
- fn = ant, = axt, +arty + a,tz (1.2.3)
where a, is the area of the triangle MOP as indicated in Figure
1.2.2~. From geometry we know that the area a, can be calculated
by taking the projection of a,, on the f plane. The projection is
given by the dot or scalar product of the two unit normal vectors to
each plane
(1.2.4) A * a, = ann x
and similarly for ay and a,. Thus, the force balance becomes
ant,, = (a,\$ f)t, + (anfi . f)t, + (a,\$ * %)tz (1.2.5)
In the limit as the area shrinks down to zero around P, the
stresses become constant, and we can divide out a,, to give
tn = ii [ft, + ity + %tz] (1.2.6)
Performing the dot operation gives
where n, is the magnitude of the projection of f onto 2. Figure
1.2.2b indicates the three components of each stress. These com-
ponents with their directions can be substituted into the balance
above to give
tn = 6 . [jET,x + f f T , y +f%T,,
+ 9fTy.x + f iTyy + i i T y , (1.2.8) + f f T , + 2iTzy + %T,,]
which, when we take the dot products, reduces to
tn = f h T , , + nyTyx + n,T,,) + f(n,Txy + nyTyy + n,TZy)
(1.2.9)
It is rather cumbersome to write out all these components
each time, so a shorthand was invented by Gibbs in the 1880s (see
Gibbs, 1960). He defined a new quantity called a tensor to represent
all the terms in the brackets in eq. 1.2.8. Following Gibbs and
modem continuum mechanics notation, we generally use boldface
capital letters to denote tensors, while boldface lowercase letters
10 / RHEOLOGY
are vectors. Thus, the stress tensor becomes T, and when we use it
eq. 1.2.8 certainly looks less forbidding
t,, = i i * T (1.2.10)
Here the dot means the vector product of a tensor with a vector to
generate another vector.
Perhaps the simplest way to think of a tensor dot product with
a vector is as a machine (see Figure 1.2.3) that linearly transforms a
vector to another vector. Push the unit vector ii into one side of the
stress tensor machine and out comes the stress vector t,, acting on
the surface with the normal vector h. T is a mathematical operator
that acts on vectors. It is the quantity that completely characterizes
the state of stress at a point. We can not draw it on the blackboard
like a vector, but we can see what it can do by letting it act on any
plane through eq. 1.2.10.
1.2.1. Notation
By comparing eq. 1.2.10 with eq. 1.2.6, we see that the stress tensor
can be viewed as the sum of three “double vectors”
(1.2.1 1) T = i t , + 3ty + ft,
directions, the first being that of the plane on which the stress vector
is acting and the other the direction of the vector itself. Thus another
way to visualize the stress tensor is as the dyad product, the special
combination of the forces (or stress) vectors with the surface that
they act on.
In eq. 1.2.8 we see the tensor represented as the sum of nine
scalar components, each associated with two directions. This is the
usual way to write out a tensor because the dyads are now expressed
in terms of the unit vectors
Figure 1.2.3.
The tensor as a machine for
transforming vectors.
ELASTICSOLID / 11
Often matrix notation is used to display the scalar compo-
nents.
Here we have left out the unit dyads that belong with each scalar
component, so the = sign does not really signify “equals” but rather
should be interpreted as “scalar components are.” Usually the unit
dyads are understood. Matrix notation is convenient because the
“dot” operations correspond to standard matrix multiplication. In
matrix notation eq. 1.2.10 becomes a row matrix times a 3 x 3
matrix.
nxTxy + nyTyy + nzTzy T x x Txy Tx, n,Txx i- nyTyx + nzTzx
[ T z x T,, :: 1 = [ n J x z + nyTyz + n,Tz, tn = i.T = (n , n y n z ) . Tyx Tyy
Remember again that we have left out the unit dyads (ff, etc). In
matrix notation the vector scalar product of eq. 1.2.4 becomes the
multiplication of a row with a column matrix.
(1.2.15) A A
Usually in rheology we use numbered coordinate directions.
Under this notation scheme the unit vectors f, f , 2 become 21,22,
i 3 , and the components of the stress tensor are written
TI1 TI2 TI3
Tij = T21 T22 (1.2.16)
[T31 T32
This numbering of components leads to a convenient index notation.
As indicated the nine scalar components of the stress tensor can be
represented by Tij, where i and j can take the values from 1 to 3
and the unit vectors 21, 22, f 3 become 2;. Thus, we can write the
stress tensor with its unit dyads as
(1.2.17)
If we evaluate the summation, we will obtain all nine terms in
eq. 1.2.8. Using index notation, the “dot” operations can be written
as simple summations, and eq. 1.2.4 or 1.2.15 becomes
12 / RHEOLOGY
and eq. 1.2.9 or eq. 1.2.14 becomes
3 3
When index notation is used, usually the summation signs
are dropped and the unit vectors and unit dyads are understood.
Here is how it works:
3
t = Ci;f; = f;
i= I
(1.2.20)
(1.2.21)
(1.2.22)
(1.2.23)
If an index is not repeated, multiplication of each component by
a unit vector is implied (e.g., t; or niTij) . If two indices are not
repeated, we will have two unit vectors or a unit dyad (e.g., T; j ) .
If an index is repeated, summation before multiplication by a unit
vector, if any, is implied. Since the indices all go from 1 to 3, the
choice of which index letters is arbitrary, as indicated in eq. 1.2.23.
To summarize, three types of notation are used in vector and
tensor manipulations. The simplest to write is the Gibbs form (e.g.,
n T ) , which is convenient for writing equations and seeing the
physics of things quickly. The index notation in its expanded form
(e.g., xi i; C j njTj;) , or as abbreviated (e.g., njT,;) , indicates
all the components explicitly, but it is harder to write down and to
read all the indices. Matrix notation (e.g., eq. 1.2.14) is convenient
for actually carrying out “dot” operations but is even more tedious
to write out and tends to obscure the physics.
All this notation associated with tensors has been known to
been no reported fatalities. In fact, when students realize that it
is mostly notation, they usually attack tensor analysis with new
confidence.
Perhaps the following examples will serve as a helpful aspirin
tablet. Several more examples appear at the end of the chapter.
ELASTICSOLID / 13
Figure 1.2.4.
Stress on a plane cutting
through a cylindrical rod.
f
t
Example 1.2.1 Stress on a Shear Plane in a Rod
It is helpful to consider the simple example of the force acting on
a cylindrical rod of cross-sectional area a as illustrated in Figure
1.2.4.
(a) What is the state of stress at point P?
(b) What are the normal and shear stresses acting on a plane that
cuts across the rod? The normal il to the cutting plane lies in the
2 1 2 2 plane and makes an angle 0 with 21; fi = cos O i l + sin O i 2 .
The tangent 0 is the intersection of the 2 1 2 2 plane and the cutting
plane; \$ = sin O i l - cos 8%2.
Solution
(a) Using Cartesian coordinates the stress tensor everywhere in
the rod is just (from eq. 1.2.1 1)
T = i l t l + 2 2 t 2 + i 3 t 3 (1 -2.24)
Since tl = ?I( f l u ) and t 2 = t 3 = 0, then
T = (6) ilt, (1.2.25)
or
.1=[ f l u 0 0 0 01 0
0 0 0
Note that there are mostly zero components in this matrix. This is
typical in rheological measurements. The rheologist needs simple
stress```