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.

Chapter 4 is an introduction to the architecture and makeup of soils as observed during study and sampling. This chapter is abridged in such a way as to provide an understandable foundation for the following chapters. In the classroom, this chapter can be presented and illustrated with any number of soil profile photos, and associated data, all of which are readily accessible on the internet. Additionally, even in urban settings, a class can be taught outdoors near the classroom, and a soil core can be extracted, laid sequentially on a tarp, and examined using the concepts and terms introduced in the chapter.

In this chapter we first describe how to construct the element stiffness matrix and load vector in 2D domains. The mapping and shape functions derived in the previous chapter are introduced to express strain components in terms of nodal DOF. Extension of the finite element formulation to 3D domains is demonstrated using the eight-node hexahedron as an example. For dynamic problems, the element mass matrix can be formed by treating the inertia effect as a body force applied to the element. The global mass matrix is then assembled to construct the equation of motion for analyses of free vibration and forced vibration. In the last section, we briefly discuss important aspects of finite element modeling and analysis that often arise in 2D and 3D problems where the number of DOF can be large. We discuss issues, such as sparse matrices and mesh generation, which early students of the finite element method may find helpful for future reference.

Chapter 9 is devoted to one of the most important new developments in soil research in the last 50 years: the linkage of geomorphological models with soils on hillslopes. Soils on hillslopes are dynamic entities that are constantly in motion and that exhibit aspects of complex systems, with feedbacks that stabilize them and help ensure their presence on all but the more extreme conditions on Earth. The processes (and models) to describe hillslope mass balance (erosion and production) are introduced. The importance of hillslope erosion and soil production to biogeochemistry is considered. Finally, soils on depositional landforms, which receive the material from the hillslopes, are also considered.

In this chapter we consider the finite element formulation for bending of slender bodies under a tensile or compressive axial force. In order to capture the effect of axial force we look at the force and moment equilibrium in the deformed configuration, but still assuming small translational displacement and rotation of the cross-section. In the finite element formulation, it is shown that the effect of axial force on bending manifests as an effective bending stiffness. It will be shown that the finite element formulation of a slender body under compressive axial force results in a matrix equation for eigenvalue analysis from which we can determine the static buckling load and the buckling mode. Subsequently, we consider the finite element formulation for vibration analysis of slender bodies to investigate the effect of axial force on the natural frequencies and modes. Finally, we introduce the finite element formulation of slender bodies subjected to a compressive follower force in which the direction of the applied force is always parallel to the body axis in the deformed configuration.

Chapter 3 is devoted to the biology of soil biogeochemistry. This is a rapidly evolving field. The chapter begins with our present understanding of the tree of life, and how little of it we have been able to detect. The second section examines the role of biology, its enzymatic impacts on the nature of chemistry over geological time, and the impacts it has had via oxidation-reduction pathways. The section considers how minerals and compounds that are now common on Earth are present largely (or only) because of biological processes. The next section examines what is presently known about the geography of soil microbiology, and what that means for the spatial diversity of metabolic capabilities. The chapter considers the challenges and emerging opportunities of explicitly embedding microbial parameters into biogeochemical models. Finally, the role and impact of vascular plants on nutrient cycling, distributions, and weathering are introduced.