Lecture 9: Measuring Strain and Mohr s Circle for Strain The last lecture explained the basic ideas about strain and introduced the strain tensor. This lecture explores a few different ways to measure strain and explains how strain, like stress, can be represented graphically using Mohr s circle. 1. Measuring Techniques a. Leveling Leveling uses a level telescope and two measuring sticks to measure the vertical distance between two points. Leveling Telescope ( level) Benchmark Figure 9.1 b. Triangulation Triangulation uses a telescope mounted on a protractor to measure the angle between two points. Triangulation Protractor θ Telescope Benchmark Figure 9.2
c. Trilateration Trilateration uses a laser mounted on a protractor to measure the distance to two objects and the angle between them. Trilateration l 2 Corner Cubes l 1 2l i = c t i Laser Figure 9.3 d. Very Long Baseline Interferometry (VLBI) and the Global Positioning System (GPS) VBLI and GPS measure the location of objects on the Earth s surface by using antennae to read signals from space. In VBLI, this signal comes from quasars. In GPS, the signal comes from a satellite. VLBI Quaser x= c t GPS (Global Positioning System) Figure 9.4
Synopsis of Measuring Techniques Angle Distance Height Orientation Leveling Yes Triangulation Yes Trilateration Yes Yes VLBI Yes Yes Yes Yes GPS Yes Yes Yes Yes For a more complete treatment of these techniques with examples, see pages 94-107 in Geodynamics by Turcotte and Schubert. 2. Indirect Measurement of Strain In addition to using the techniques listed above, strain can sometimes be measured indirectly by observing the change in shape of objects that naturally have a certain geometry. For example, a geologist may compare the deformed shape of fossils, pebbles, or veins in an outcrop to their undeformed shape to determine the presence of tension, shear, or compression. Deformed Undeformed Figure 9.5 This picture shows how deformed fossils and pebbles can be used to infer the strain in the host rock.
Veins Weak Compression Strong Weak Tension Figure 9.6 This picture shows how veins form waves under compression and boudins under tension. 3. Mohr s Circle for Strain Lecture V explained a simple and convenient way to find the stress on an arbitrary plane given the stress tensor σ ij. The technique involved writing equations for how the shear stress and normal stress on the x ˆ1plane vary when the coordinate system is rotated to x ˆi '. These equations plotted as Mohr s circle in stress space (σ,τ) and gave the tractions on plane x ' at angle θ to the most compressive principle stress. 1 Since strain is a second-order tensor like stress, the same technique can be applied. Equations for the normal strain and shear strain on a plane at angle θ to the most compressive principle strain may be derived in the same way the equations for stress were derived. Consider the following transformation of coordinates: x 2 x ' 2 x ' 1 Θ x 1 Figure 9.7
The strain tensor ε in the x ˆi coordinate system is transformed to the strain tensor ε in the x ˆi ' coordinate system by the equation T ε ' = α εα where the double underbars denote second-rank tensors, α represents the transformation matrix, and the superscript T denotes the transpose of matrix α. See Lecture V for the derivation of this equation. Since the coordinate system is rotated about the x ˆ3 axis, the transformation matrix is cosθ sinθ 0 α = sinθ cosθ 0 0 0 1 The equations for the normal strain and shear strain on the x ' plane in the new 1 coordinate system are ε ' = ε 11 + ε 22 + ε ε 11 22 11 2 2 cos 2 θ + ε sin 2 θ ε 22 ' = ε 11 + ε 22 + ε 11 ε 22 cos 2 θ ε sin 2 θ 2 2 11 ε ' = (ε ε 22 ) sin 2 θ + ε cos 2 θ 2 The derivation for these equations follows the derivation for the Mohr s circle equations of stress in Lecture V.