An introduction to field Lagrangians for scalars, vectors, and the Einstein–Hilbert Lagrangian for gravity provides a venue to think about coupling together different field theories. The natural expression of that coupling comes from an action, and we show how the “Euler–Lagrange” field equations enforce the universal coupling of all physical theories to gravity. As an example, the combined field equations of electricity & magnetism and gravity are solved in the spherically symmetric case to give the Reissner–Nordstrøm spacetime associated with the exterior of charged, massive, spherically symmetric central bodies.

This chapter develops the Navier–Stokes equations using a Lagrangian description. In doing so, the concept of a stress tensor and its role in the overall force balance on a fluid element is discussed. In addition, the various terms in the stress tensor as well as the individual force terms in the Navier–Stokes equations are investigated. The chapter ends with a discussion on the incompressible Navier–Stokes equations.

Scales of rock heterogeneity are nano≪micro ≪ meso ≪ macro. In order to estimate effective physical parameters by using an averaging procedure over a certain representative reference volume, estimations of effective physical parameters may be done by taking the Reuss or the Voigt mean values or using the upper and lower Hashin–Shtrikman bounds. The fracture number per unit length of a rock is correlated with the rock quality or RQD index. Autorun analysis of rock fractures in scanlines yields a consistent criterion of rock quality. The stress tensor is defined in a symmetric matrix. In diagonal form, the elements of the stress matrix are principal normal stresses. The Mohr’s circle in 2D and 3D is used to define normal and tangential stress in an arbitrarily oriented plane. The strength of rocks depends on modes of deformation according to the failure criteria after Mohr-Coulomb, Griffith, and Hoek and Brown. The ISRM standard and four point bending tests of rock strength are designed to determine yield strength. Fracturing and failure modes in rocks are considered according to tensile fracture models after Griffith, Dugdale, Barenblatt and critical tip opening displacement. Focus Box 3.1: Percolation models in rocks.

The analysis of normal and shear stresses over differently oriented surface elements through a considered material point is presented. The Cauchy relation for traction vectors is introduced, which leads to the concept of a stress tensor. The analysis is presented of one-, two-, and three-dimensional states of stress, the principal stresses (maximum and minimum normal stresses), the maximum shear stress, and the deviatoric and spherical parts of the stress tensor.The equations of equilibrium are derived and the corresponding boundary conditions are formulated.