Stress, Strain, and Hooke's Law

Part 1, Part 1: Elasticity and the Moduli

Learning objectives

Define stress as force per area and strain as fractional deformation, with their units
State Hooke's law: for small deformations, stress and strain are proportional, and the modulus is the slope
Compare how rocks of different stiffness respond to the same stress
Explain why seismic waves, with strains around a millionth, live deep inside the linear elastic regime

Push on a Rock

Elasticity begins with two definitions. Stress is force spread over area, $\sigma = F/A$ , measured in pascals; rocks at depth carry stresses of tens of megapascals. Strain is the deformation that stress produces, expressed as a fraction of the original size, $\varepsilon = \Delta L / L$ : dimensionless, usually tiny, often quoted in percent. Stress is what you do to the rock; strain is what the rock does back.

Hooke's Bargain

For small deformations, rocks keep a simple promise: strain is proportional to stress, $\sigma = E\,\varepsilon$ , and the energy of deformation is stored and returned rather than lost. That is Hooke's law, and the constant $E$ (here Young's modulus, the uniaxial member of the modulus family) is the slope of the stress-strain line: stress per unit strain, in pascals, because strain has no units. A modulus is nothing more mysterious than that slope, and it is a property of the material, not of the experiment. Stiff rock, steep line; soft sediment, shallow line.

Load the block and read both pictures at once: the same 30 MPa that squeezes a soft sediment by more than a percent barely dents a limestone. The deformation drawn on the block is exaggerated twentyfold to be visible at all, which is the first hint of how stiff rocks really are.

Why Seismic Lives in the Linear World

Push far enough and the promise breaks: grains crush, cracks grow, the rock yields, and the stress-strain path bends and stops retracing itself. But a passing seismic wave strains the rock by around a millionth or less, thousands of times smaller than anything on the figure's axis. At those amplitudes linear elasticity is not an approximation to apologize for; it is one of the most accurate laws in geophysics, and everything this course builds sits on it. What Hooke's law does not tell you is which modulus to use: a rock resists squeezing and shearing differently, and the next section splits stiffness into the two numbers, $K$ and $\mu$ , that carry the rest of the subject.

References

Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
Simm, R., & Bacon, M. (2014). Seismic Amplitude: An Interpreter's Handbook. Cambridge University Press.