Engineering Rock Mechanics

Infinitesimal strain 75

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Figure 5.6 The general transformation of a two-dimensional shape using

homogeneous co-ordinates.

Note that in the equation

x' = ax + cy + m

the coefficient a is related to extensional strain (as shown in Fig. 5.4), the
coefficient c is an interaction term and related to shear, and m is related to
the magnitude of the translation. Through such considerations, we can
identify the strain components associated with different parts of the matrix,
as shown below.

Shearing,

scaling, rotation

Translation

Projection,

perspective

Overall

scaling

We will see later that the ability to determine which functions are

performed by which parts of the transformation matrix is especially

helpful when interpreting the compliance matrix. This matrix relates

the strains to the stresses for materials with different degrees of

anisotropy.

5.3 Infinitesimal strain

Infinitesimal strain is homogeneous strain over a vanishingly small

element of a finite strained body. To find the components of the strain

matrix, we need to consider the variation in co-ordinates of the ends of an

imaginary line inside a body as the body is strained as illustrated in

Fig. 5.7. By this means, we can find the normal and shear components in

an analogous fashion to the finite case presented above.

In this figure, the point P with co-ordinates (x, y, z) moves when the body

is strained, to a point P* with co-ordinates (x + u, y + uy, z + uJ. The

components of movement, ux, uy and u, , may vary with location within

the body, and so are regarded as functions of x, y and z. Similarly, the point

Q (which is a small distance from P), with co-ordinates (x + ax, y + 6y,