Decisions made by trustees and third parties (e.g. employers, actuaries, protectors) in relation to trusts are important. The decision-maker will want to get it right and others may look to the courts to review the decision. Generally, such decision-making powers are not ‘absolute’ but are subject to some limited review by the courts (whether or not being exercised by a fiduciary). Although not a trust case, Braganza [2015] UKSC 17 in the UK Supreme Court marks what looks likely to be a definitive shift in favour of applying public law Wednesbury reasonableness (or rationality) standards in both limbs (due consideration and perversity) to review many trust-related decisions. This is despite indications to the contrary in some previous cases, including Pitt v. Holt [2013] UKSC 26. The Braganza tests are in addition to review tests based on scope, proper purposes, honesty and (for fiduciary powers) unauthorised conflicts of interest. This chapter looks at the implications of Braganza and Wednesbury, mainly for trustees, but also of relevance for other decision-makers under a trust, including employers and protectors.

Chapter 9 reviews fold terminology, describes fold profiles at outcrop scales, uses differential geometry to quantify folds in three dimensions at crustal scales, and builds canonical models for folding due to bending and buckling. For two-dimensional profiles of folds in sedimentary, metamorphic, and igneous rocks, we introduce the concept of curvature to characterize fold shapes. Then we describe the crustal-scale fold at Sheep Mountain, WY, and use a modern technology called Airborne Laser Swath Mapping (ALSM) to map the fold geometry in three dimensions. We employ differential geometry to characterize a three-dimensional bedding surface in this fold in terms of the principal curvatures at every point. The fundamental shapes at a given point on such a surface can be planar, elliptic, parabolic, or hyperbolic. These distinct shapes are used to define a classification scheme for folded geological surfaces based on curvature.

Chapter 7 focuses on fractures in rock by describing outcrops of joints, veins, and dikes, introducing a canonical model for opening fractures, and considering fracture initiation and propagation using linear elastic fracture mechanics. The outcrop descriptions serve to highlight the characteristic geometric features of these structures, and provides the background necessary to build a conceptual model for opening fractures in rock. The canonical fracture model is based on a pure opening fracture in an elastic rock mass. With attention focused on the fracture tips, we explore the stress concentration there that leads to fracture propagation, and identify the three modes of fracture tip deformation. We explain how fractures initiate at flaws in rock and propagate when the stress intensity in the near-tip region reaches a critical value called the fracture toughness. Although many opening fractures are approximately planar, we describe how minor amounts of shearing can alter the propagation path and lead to kinked or echelon fractures. These interesting geometries provide evidence for interpreting the state of stress at the time the fractures formed.

Chapter 10 describes rock fabrics at outcrop and thin section scales in sedimentary, metamorphic, and igneous rock, introduces the kinematics of ductile deformation, builds a canonical model for a viscous shear zone, and relates fabric development to plastic deformation at fault steps. We revisit the spherical particles called ooliths that were introduced in Section 2.1 and use their alignment in the South Mountain–Blue Ridge Uplift to quantify the deformation that produced a planar fabric called cleavage. Next, we describe outcrops from the Sierra Nevada where shear zones initiated along aplite dikes that localized the strain, and where a fabric developed between the overlapping tips of echelon fault segments. To quantify the kinematics relevant to fabric development, we introduce the concepts of progressive deformation, and the rate of deformation and spin in a shear zone. Then, we develop the canonical model for a viscous shear zone from the equations of motion using the constitutive law for a linear viscous material. Finally, we investigate fabric development at a fault step in granitic rock, and show how a model that includes elastic deformation of the surrounding rock, frictional slip on the fault segments, and plastic deformation of rock within the step matches geometric observations from the outcrop.

Chapter 4 introduces elastic–brittle deformation of rock using field observations of geologic structures, laboratory tests of mechanical behaviors, and theoretical constructs that relate stress to strain. We use solid Earth tides to demonstrate that linear elastic behavior is characteristic of rock deformation in much of Earth’s lithosphere. Then, we use field and laboratory observations to show how elastic deformation of rock culminates in brittle fracture, which sets a limit on rock strength. Brittle deformation occurs as opening fractures and as shearing fractures. Laboratory tests measure the elastic stiffness and strength of rock, and examples show how strength depends on rock type, confining pressure, and pore fluid pressure. To quantify elastic deformation we introduce the stress tensor and small strain tensor. We link the components of these tensors using Hooke’s law for linear elasticity. Then, we show how Newton’s second law, embodied in Cauchy’s equations of motion, is used to model elastic deformation and the development of fractures and faults.

Chapter 5 introduces elastic–ductile deformation of rock using field observations of geologic structures, laboratory tests of mechanical behaviors, and theoretical concepts relevant to strength and ductility. We show that ductile deformation takes over from brittle deformation as confining pressure and temperature increase, and as the rate of deformation decreases, so ductile behavior is characteristic of rock deformation in the deeper levels of Earth’s lithosphere. To quantify ductile deformation, we introduce the idealized elastic–plastic solid, a mathematical construct in which strain is linearly proportional to stress and is recoverable up to the yield point. Thereafter, the stress remains constant while permanent strain accumulates. We review the mechanisms of plastic deformation that involve the motion of dislocations in the crystal lattice. The dependence of plastic deformation on time motivates consideration of flow laws that relate strain rate to the state of stress. We conclude by showing how Newton’s laws, embodied in Cauchy’s equations of motion, model ductile deformation of salt flowing toward a rising diapir.

Chapter 1 sets the stage for a quantitative introduction to structural geology. We begin by identifying forces that cause deformation in Earth’s lithosphere and asthenosphere. Then, we describe three different styles of deformation, and five broad classes of geologic structures that result from this deformation. To lay out the methodology for studying geologic structures, we introduce what we mean by a complete mechanics and by canonical models of structural geology. Then we examine the roles of physics and mathematics in studying the origins of geologic structures. Finally, we describe applications of structural geology to problems facing our society and the careers that utilize structural geology to solve those problems.

Chapter 11 begins by defining intrusions, then describes the different characteristic forms of intrusions, and ends by deriving the solution for the rise of a spherical body of viscous liquid in a more dense viscous liquid. This classic solution from fluid dynamics is the canonical model for the intrusion of salt in sedimentary basins. In general, an intrusion is a body of rock that, in a former more mobile state, was injected into and deformed the surrounding host rock. Intrusions take the form of dikes, sills, laccoliths, stocks, plutons, and diapirs. The intruded material could be magma, rising due to buoyancy from deep in Earth’s asthenosphere (Section 1.1.2), or magma injected laterally from a shallow pressurized chamber in Earth’s lithosphere (Section 6.8.2). The intruded material could be salt, moving upward in a diapir due to buoyancy (Section 5.8), or a mobilized slurry of sand injected into the surrounding sedimentary rock (Section 11.1). The intruded material also could be molten rock, formed due to frictional heating on a fault during an earthquake (Section 8.6.3). The diversity of intrusions makes them an interesting and challenging topic for structural geologists.