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The radius of the circle is (σ1 − σ2 )/2 and the centre is at the point (σ1 + σ2 )/2 on the σ -axis. The stresses σ and τ in any direction θ (Fig. 7b) correspond to a point on the Mohr’s circle. It is seen from Fig. 7a that the largest absolute value for the shear stress is (σ1 − σ2 )/2 and occurs for θ = π/4 (= 45°) and θ = 3π/4 (= 135°). The Mohr’s circle is a very useful tool in the analysis of conditions for rock failure, as will be seen in Chapter 2. 5. Principal stresses in three dimensions Now moving to three dimensions, we first have to decide how to identify a direction in space.

192) ANISOTROPY 37 Eq. 192) should be compared to the corresponding equation from thermoelasticity, Eq. 194) α ↔ 3KαT = αT 1 − 2ν solutions of problems in thermoelasticity (where a wide range of problems have been solved) may be directly applied to poroelasticity. g. by Haimson and Fairhurst (1967) in a classic paper on hydraulic fracturing. There is one important point to note. Since stress or strain do not induce significant temperature changes, the temperature field is governed by the (decoupled) diffusion equation.

First, imagine that we perform a “jacketed” test (see Fig. 16a): a porous medium is confined within an impermeable jacket, and subjected to an external hydrostatic pressure σp . The pore fluid is allowed to escape during loading, so that the pore pressure is kept constant, and hence the stress is entirely carried by the solid framework. From Eqs. 146) Since this test characterizes the stiffness of the solid part of the rock, Kfr is called the bulk modulus of the framework or the frame modulus. Since there are no shear forces associated with the fluid, we can directly identify the shear modulus of the porous system as the shear Fig.