Chapter 1 Stress and strain 1.1 Stress (T&M pp.128 164) 1.1.1 Introduction In geology, we seldom observe the forces responsible for the deformations that we are interested in. In fact, it turns out that you can’t actually measure stress directly (stress measurements are made by observing the deformation of reference materials whose response to stress is known). Nevertheless, one of the main objectives of earth science is to try and understand the mechanisms whereby observed deformations (faults, folds, mountain ranges and so on) are produced. The key concept in this regard is stress, which is related to the familiar concept of a force in a reasonably straightforward way: Stress = Force / Area By this deﬁnition, it is clear that a given force acting on a larger area results in a smaller stress than the same force acting on a smaller area. Since Force is a vector quantity (i.e. its direction and magnitude are deﬁned), we denote the above relation as → − F → − σ= A (1.1) → − σ is termed a stress vector, or a traction, and is deﬁned with reference to a par ticular plane. Stress has units of force (Newtons) divided by area (square meters). A N is a Pascal (Pa). In the Earth, most stress of interest are of the order of 106 109 m2 Pa, and stresses are commonly reported in megapascals (MPa). In engineering and material science, the convention is that positive stresses are tensional, and compressional stresses are negative. In the Earth, true tensional stresses are exceedingly rare, however, so we use the opposite sign convention. 1.1.2 The stress tensor An arbitrary stress acting on a plane can be resolved into three components: one normal to the plane (normal stress) and two mutually orthogonal components tan gential to the surface of the plane (shear stresses). Since we are interested in contin uous volumes (the interior of the earth), this is further generalized: Consider a small volume whose faces are oriented with the coordinate axes. We keep track of the faces by identifying each face x, y, z with the coordinate axis normal to that face. The reason we deal with this volume is that the surface stresses on each pair of faces is independent of the surface stresses on the other two pairs of faces. As it turns out, if we know the surface stresses on any three orthogonal faces deﬁning an inﬁnitesimal volume, we have completely characterised the stress and we can calculate the surface stress on any plane crossing this region.