Biot (1956) derived theoretical formulas for predicting the frequency-dependent seismic velocities of saturated rocks in terms of the dry-rock properties. His formulation incorporates some, but not all, of the mechanisms of viscous and inertial interaction between the pore fluid and the mineral matrix of the rock.

It was established experimentally by Darcy () that the fluid flow rate in a fluid-saturated porous medium is linearly related to the pressure gradient by the following equation

If we wish to predict theoretically the effective elastic moduli of a mixture of grains and pores, we generally need to specify: (1) the volume fractions of the various phases, (2) the elastic moduli of the various phases, and (3) the geometric details of how the phases are arranged relative to each other. If we specify only the volume fractions and the constituent moduli, the best we can do is predict the upper and lower bounds (shown schematically in Figure 4.1.1).

In an isotropic, linear elastic material, the stress and strain are related by Hooke’s law as follows (e.g., Timoshenko and Goodier, 1934)

The Earth is continuously being deformed due to forces related to earthquakes, surface loads such as ice-sheets, the gravitational attraction to other planets, and many other phenomena. The relation between forces and the resulting style of deformation defines the rheology of a material. While the Earth behaves nearly like elastic rubber at short timescales from seconds to years, large parts of it can be treated as a liquid when studied over geologic timescales exceeding millennia. In this chapter we provide a detailed introduction to the various deformation styles of the Earth and their relevance to a wide range of processes, including the propagation of seismic waves, the deformation of the lithosphere near mountain ranges and subduction zones, and convective flow in the mantle.

Rocks contain a tiny proportion of magnetic minerals that make them weakly magnetic. Some are magnetized in the direction of the magnetic field in which they formed. By analyzing this direction, the position of a virtual geomagnetic pole (VGP) at the time the rock formed can be located. Connecting the VGP for rocks with different ages from the same continent gives a curve of apparent polar wander (APW), which results from the motion of the continent relative to the rotation axis. Comparing APW paths for different continents reveals a history of relative motions and allows the reconstruction of past supercontinents. The paleomagnetic field has reversed polarity numerous times in the geological past, leaving a record of geomagnetic polarity in the magnetizations of rocks. Polarity reversals cause oceanic magnetic anomalies, which are important for understanding plate tectonics and also form the basis of a geomagnetic polarity timescale for the past 230 Myr.

The geomagnetic field is generated by complex motions of electrically conducting liquid iron in the Earth’s outer core. It has been studied for centuries at observatories and for decades from orbiting satellites. Outside the Earth, the magnetic field deflects the solar wind, a stream of charged particles from the Sun, thereby shielding the planet from harmful radiation. Some radiation penetrates deep into the outer atmosphere, where it ionizes air molecules. These cause the aurora and also form spherical shells of ions encircling the Earth that produce their own magnetic fields. We have learned about the magnetic fields of the other planets from earth-bound observations and from space missions. The geomagnetic field provides an important tool for exploring the mineral wealth of our planet. We learn how to interpret the shapes of the magnetic anomalies of simple geometric bodies, including block models of oceanic magnetic anomalies.

The shape of the Earth is determined primarily by two forces. Gravitational attraction, directed toward the center of the planet, results in an almost spherical shape. The Earth’s rotation produces a centrifugal force away from the rotation axis that flattens the sphere to a rotational ellipsoid. Differences in internal mass distribution produce bumps and hollows in the ellipsoid, forming a smooth but uneven surface called the geoid. Gravity at any place acts in the vertical direction, which is everywhere perpendicular to the local geoid. We explain how the gravitational attractions of the Moon and Sun deform the Earth’s free surface, creating tides in the oceans and in the solid planet. The Earth’s ellipsoidal shape allows the gravitational attractions of other planets to modulate both its rotation and its orbit cyclically with periodicities of 21,000 to 405,000 years, which are correlated to long-term climatic changes.

The lithosphere – the thin outer shell of the Earth – is stronger than the underlying mantle. Geodynamic activity driven by heat in the planet’s interior has caused it to subdivide into a number of thin plates, several hundred to several thousand kilometers in horizontal extent. They are in constant motion at speeds of a few centimeters per year. The relative motion at plate margins where they adjoin results in tectonic activity, characterized by earthquakes and volcanism. We distinguish three types of margin – spreading centers, subduction zones, and transform faults – and describe the seismic, gravity, and magnetic data that characterize them. We explain how reconstructions of plate positions in the geological past are obtained and what they tell us about the planet’s geodynamic history.