**3. Influence of fractures and discontinuities on the strength of brittle materials**

Hydraulic fracturing can be considered as a technique to overcome the strength of a rock mass in situ, initiation and propagation of a crack through a system of pre-existing fractures, essentially planar discontinuities (e.g., bedding planes), and intact rock.

In examining the fracture propagation process, the pioneering work of Griffith (1921, 1924) is a logical point of departure. Griffith had identified planar discontinuities, or flaws, in fabri‐ cated materials as the reason why the observed technical strength of brittle materials was about three orders of magnitude lower than the theoretical inter-atomic cohesive (tensile) strength. 4 Using an analytical solution by Inglis (1913) for the elastic stresses generated around an elliptical crack in a plate, Griffith observed that the maximum tensile stress at the tip of the crack σ<sup>t</sup> = σ<sup>0</sup> (1+ 2a/b), where a and b are the major and minor semi-axes of the ellipse, and as the ellipse degenerated to a sharp crack or flaw (i.e., as the ratio a/b became very high)5 , the stress σ<sup>t</sup> could rise to a value high enough to reach the inter-atomic cohesive strength sufficient to cause the original crack to start to extend.

<sup>3</sup> Future of Geothermal Energy (2005)Synopsis and Executive Summaryp.1-5 (5).

<sup>4</sup> A fractured rock mass is typically about two orders of magnitude lower in strength than the strength of a laboratory specimen taken from the rock mass [Cundall (2008); Cundall et al, (2008)].

But would the crack continue to extend and lead to macroscopic failure? To address this question, Griffith invoked the *Theorem of Minimum Potential Energy,* which may be stated as "The stable equilibrium state of a system is that for which the potential energy of the system is a minimum." For the particular application of this theorem to brittle rupture, Griffith added the statement*,* "The equilibrium position, if equilibrium is possible, must be one in which rupture of the solid has occurred, if the system can pass from the unbroken to the broken condition by a process involving a continuous decrease of potential energy."6

Griffith's classical work has provided the foundation for the field of "Fracture Mechanics" [Knott (1973); Anderson (2005)] responsible for major continuing advances in the development of high-performance fabricated materials.

Since we will make reference later to this specific definition by Griffith, it is useful to re-state it here.
