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Macosko, Christopher W - Rheology - Principles, Measurements and Applications-John Wiley & Sons (1994)

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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, 
These double vectors are called dyads. The dyad carries two 
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 
cause a severe headache upon first reading; however, there have 
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