In this chapter we present the finite element formulation of heat transfer problems which can be used to determine temperature distributions in solid bodies, starting with heat conduction in the 1D domain. Similar to the notion of virtual displacement in earlier chapters, a virtual temperature or an arbitrary weight function is introduced to derive an integral equivalent of the governing equation to which the finite element formulation is applied. Methods for heat conduction and convection, in 1D, 2D, and 3D domains, including time-dependent effects, will be covered. Mathematical equivalence with other scalar field problems is also discussed.

The finite element method is a powerful technique that can be used to transform any continuous body into a set of governing equations with a finite number of unknowns called degrees of freedom (DOF). In this chapter, we will introduce the fundamentals of the finite element method using a system of linear springs and a slender linear elastic body undergoing axial deformation as examples. These simple problems are chosen to describe the essential features of the finite element method which are common to analysis of more complicated structural systems such as 3D bodies.

Chapter 7 presents the soil carbon cycle. The chapter largely by-passes the still uncertain processes that occur at the molecular scale. The focus is on macroscopic properties and how they vary with space and time. Soil C storage is first examined from a box model perspective, which introduces mass balance equations and how they are useful, when coupled with data, in beginning to understanding soil C dynamics. The chapter includes an introductory perspective on the vertical trends in soil C and the transport-reaction models that are needed to fully explain these patterns. Soil organic C is largely removed from soil as CO2, and production-diffusion models are introduced to explain observable CO2 depth profiles and to calculate the fluxes to the atmosphere. Diffusion impacts the C isotope composition of soil CO2 and any CaCO3 minerals that subsequently form. These are examined through the lens of diffusion modeling, which is now common, and critical, in any examination of soil properties with depth.

The best thing that happened during this long gestation period was the outstanding work of my graduate students, which enriches this book and has transformed my understanding of soils. I mention those whose work is directly linked to this book. Gene Kelly helped expose me to the soils of the Great Plains and introduced me to biomineralization. Stephanie Ewing, Justine Owen, Kari Finstad, and Marco Pfeiffer have collectively provided a quantum leap forward in our understanding of the driest desert on Earth, in northern Chile. As a group, they have contributed to understanding the climate threshold that exists between the biotic and abiotic parts of our planet – and its profound impact on soil and landscape biogeochemistry. Erik Oerter advanced the ability of soils to provide paleoclimate information from carbonate using micro-sampling and dating techniques.

Chapter 2 is a deep dive into the threads that link the chemical and mineralogical makeup of soils to that of the surrounding cosmos. The first section examines elements across the periodic table, and how they systematically change in abundance due to a variety of cosmic processes. The second part of the chapter examines how the elements are combined into minerals, and especially the silicate minerals. Discussion of the factors that dictate mineral stability in soils is introduced, and these will be examined in even greater depth in Chapter 8. Secondary minerals are also introduced, again with a strong focus on the silicate group. Cation exchange is examined. The chapter ends with the effect of plants, and biology, on soil chemistry, which is expanded upon in Chapter 3. The activities at the conclusion of the chapter provide students with an opportunity to use spreadsheets and data analyses in order to gain experience and confidence in data analysis.

Chapter 1 is an abridged and concise overview of Jenny’s Factors of Soil Formation theory. The soil system is defined and explored from a state factor perspective. The ways in which this theory can be used to pose research questions and to design observation studies to address them are discussed. The relevance to the current Critical Zone science program is considered. The use of the state factor approach to global soil C science and to a climosequence of soils along the Mississippi River corridor is illustrated. The activities at the end of the chapter include a detailed tutorial on the solution of the state factor model for soil N in the Great Plains, first examined by Hans Jenny in a 1930 research paper.

In the finite element formulation, the body is divided into elements of various types. This chapter describes mapping functions for the description of element geometry in the undeformed configuration and shape functions for the description of displacement and thus deformed geometry in the 2D and 3D domains. We introduce the "isoparametric" formulation in which mapping functions and shape functions are identical. This is followed by discussions on integration in the mapped domains and numerical integration.

This chapter deals with the finite element formulation for thin plate and shell structures. We will review the assumptions on the kinematics of deformation from classical plate bending theories, introduce them into the finite element formulation for plates, and then extend the formulation to curved shell structures within the isoparametric formulation. For 3D solid elements that can be used for plates and shell analysis, we will first look at solid elements with three nodes through the thickness. We will then show how solid elements with two nodes through the thickness can be constructed for analysis of plate and shell structures.

Chapter 8 focuses on major advances in data acquisition and modeling of soil chemical weathering - at both the humid and the very dry end of the Earth’s climatic spectrum. The behavior of elements in aqueous solutions is examined in more detail. The now widely used mass balance method of calculating elemental gains and losses by comparison with an immobile index element is introduced and discussed. The concept of weathering fronts, and ways to understand what creates them and the rates at which they move through soils, is introduced - a relatively new concept from geoscientists who focus on soil geochemical processes and their rates.