Basic Ideas about Strain

Stresses cause solids to deform; that is, the stresses produce changes in the distances separating neighboring small elements of the solid. In the discussion that follows we describe the ways in which this deformation can occur. Implicit in our discussion is the assumption that the deformations are small.

A small element of the solid in the shape of a rectangular parallelepiped. Prior to deformation it has sides δx, δy, and δz. The element may be deformed by changing the dimensions of its sides while maintaining its shape in the form of a rectangular parallelepiped. After deformation, the sides of the element are δx−εxxδx, δy−εyyδy, and δz−εzzδz. The quantities εxx, εyy, and εzz are normal components of strain; εxx is the change in length 2.7 Basic Ideas about Strain 155. A deformation that changes the dimensions of a rectangular parallelepiped but not its shape of the side parallel to the x axis divided by the original length of the side, and εyy and εzz are similar fractional changes in the lengths of the sides originally parallel to the y and z axes, respectively. The normal components of strain εxx, εyy, and εzz are assumed, by convention, to be positive if the deformation shortens the length of a side. This is consistent with the convention that treats compressive stresses as positive. If the deformation of the element in Figure 2–20 is so small that squares and higher order products of the strain components can be neglected in computing the change in volume of the element, the fractional change in volume (volume change divided by original volume) is εxx + εyy + εzz. This quantity is known as the dilatation ∆; it is positive if the volume of the element is decreased by compression.

Uplift and subsidence of large areas are also accompanied by horizontal or lateral strain because of the curvature of the Earth’s surface. Show that the lateral strain ε accompanying an uplift ∆y is given by ε = ∆y R , (2.74) where R is the radius of the Earth. Problem 2.22 The porosity φ of a rock is defined as its void volume per unit total volume. If all the pore spaces could be closed, for example, by subjecting the rock to a sufficiently large pressure, what would be the dilatation? For loose sand φ is about 40%, and for oil sands it is usually in the range of 10 to 20%. Table 2–2 gives the porosities of several rocks. The strain components of a small element of solid can be related to the displacement of the element. In order to implify the derivation of this relationship, we consider the two-dimensional. Prior to deformation, the rectangular element occupies the position pqrs.