Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Strain Transformations
CHAPTER13
13.1 Introduction
The discussion of strain presented in Chapter 2 was useful in introducing the concept of
strain as a measure of deformation. However, it was adequate only for one-directional
loading. In many practical situations involving the design of structural or machine components,
the configurations and loadings are such that strains occur in two or three directions
simultaneously.
The complete state of strain at an arbitrary point in a body under load can be determined
by considering the deformation associated with a small volume of material surrounding
the point. For convenience, the volume, termed a strain element, is assumed to
have the shape of a block. In the undeformed state, the faces of the strain element are
oriented perpendicular to the x, y, and z reference axes, as shown in Figure 13.1a. Since
the element is very small, deformations are assumed to be uniform. This assumption
means that
(a) planes that initially are parallel to each other will remain parallel after deformation,
and
(b) lines that are straight before deformation will remain straight after deformation, as
shown in Figure 13.1b.
The final size of the deformed element is determined by the lengths of the three edges
dx′, dy′, and dz′. The distorted shape of the element is determined by the angles θ′ xy , θ′ xy , and
θ′ zx between faces.
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