Engineering Rock Mechanics

Examples of homogeneous finite strain 73

I-

t-

1

4

4.7 ---I!-

I'

4

Contraction positive

Q Q'

I

P,P'

Negative shear- strain: P'Q'>PQ

y = tan 1)

I

Figure 5.3 Normal strain and shear strain.

of strain is the same throughout the solid. Under these circumstances:

(a) straight lines remain straight;
(b) circles are deformed into ellipses; and
(c) ellipses are deformed into other ellipses.

5.2 Examples of homogeneous finite strain

We will now consider four examples of simple homogeneous finite strain.
These are all important, both fundamentally and in terms of understanding
strain. We will discuss strain components and thence also begin to
introduce the notion of strain transformation, in terms of matrices. The four
examples are shown in Fig. 5.4.
In each of the examples in Fig. 5.4, we have given the equations relating
the new positions (eg. x') in terms of the original positions (e.g. x) of each
point. The coefficients k and y indicate the magnitudes of the normal and
shear strains, respectively. The final case in the figure, pure shear, is a result
of extensional and contractional normal strains which will be explored later,

x' = kx y' = y x' = k,x y'= kzY

Extension along x-axis Extension along x-and y-axes

x' = x + yy Y, = Y x' = kx

A Note interaction with \-ax15

Simple shear Pure shear

Figure 5.4 Four simple cases of homogeneous finite strain.