Linear Elasticity#
For a linear, isotropic solid, under an applied stress, \(\mathbf{\sigma}\) we can write the strain \(\mathbf{\varepsilon}\) generated in the body using Hooke’s Law as :
where \(i\) and \(j\) represents the face of the body and the direction of stress, respectively. \(\lambda\) and \(\mu\) are known as Lame’s parameters; \(\mu\) is also known as the shear modulus or modulus of rigidity.
Using the above equation, we can write normal stresses and normal strains : $\( \begin{aligned} \sigma_{xx} &= (\lambda + 2\mu)\varepsilon_{xx} + \lambda \varepsilon_{yy} + \lambda \varepsilon_{zz} \\ \sigma_{yy} &= \lambda \varepsilon_{xx} + (\lambda + 2\mu)\varepsilon_{yy} + \lambda \varepsilon_{zz} \\ \sigma_{zz} &= \lambda \varepsilon_{xx} + \lambda \varepsilon_{yy} + (\lambda + 2\mu)\varepsilon_{zz} \end{aligned}, \qquad \begin{aligned} \varepsilon_{xx} &= \frac{1}{E}\sigma_{xx} - \frac{\nu}{E}\sigma_{yy} - \frac{\nu}{E}\sigma_{zz} \\ \varepsilon_{yy} &= - \frac{\nu}{E}\sigma_{xx} + \frac{1}{E}\sigma_{yy} - \frac{\nu}{E}\sigma_{zz}\\ \varepsilon_{zz} &= - \frac{\nu}{E}\sigma_{xx} - \frac{\nu}{E}\sigma_{yy} + \frac{1}{E}\sigma_{zz} , \end{aligned} \)$
and shear strains and stresses as \(\sigma_{xy} = 2\mu\varepsilon_{xy} \) and so on, where, \(E = \frac{\mu(3\lambda + 2\mu)}{\lambda + \mu}\) and \(\nu = \frac{\lambda}{2(\lambda + \mu)}\) are known as Young’s modulus and Poisson’s ratio, respectively.
We can also write the above stress tensor using its principal directions, \(\sigma_1, \sigma_2, \sigma_3\), such that all the off-diagonal or shear components are zero.
The following figure illustrates the strains developed in a rock deformed under an applied stress:
Schematic diagram showing the triaxial stress-test equipment (left), modified from Zhao and Cai (2014), and the representative stress-strain relationship during such an experiment (Turcotte and Schubert, 2014).
Plane stress#
State of plane stress occurs when only one principal stress component is zero, e.g., \(\sigma_3\) = 0 and \(\sigma_2\), \(\sigma_1\) are non-zero. We can assume a plane-stress approximation for the Earth’s lithosphere such that the vertical stress is much smaller compared to the horizontal stresses.
Simple and pure shear#
Both simple and pure shear deformations are specific cases of plane stress state. In simple shear, the material only deforms through shearing such that off-diagonal shear strains are non-zero. An example of shear strain deformation is strike-slip faults such as the San Andreas fault. On the other hand, pure shear occurs when material only deforms across its axial directions such that the body is compresses in one direction and elongates in other.
Representation of pure and simple shear strain, from Jennings and Hambrey (2021).
Limits to linear elasticity in practice#
While linear elasticity provides a useful approximation for small deformations, it has some limitations when applied to real Earth rocks under large loads.
Non-linear behavior at higher stresses: beyond a critical stress (or yield point), strain no longer increases linearly with stress, and permanent (plastic) deformation may occur.
Material anisotropy: real materials often have different properties along different directions.
Micro-scale heterogeneity: small heterogeneity in the grains can localize deformation that deviates from the linear elastic behavior.
Time-dependent effects: material properties can change as it is deformed leading to strain-hardening (material is stronger as it is deformed) or strain-softening (material is stronger as it is deformed) effects.
The following figure illustrates the expected nonlinear deformation of a rock beyond its elastic limit:
Schematic diagram showing stress-strain curve for under brittle (left) and ductile (right) deformation.
Brittle deformation in the Earth forms faults, whereas ductile deformation results in folds.
When the magnitude of shear stress exceeds the local rock strength, failure occurs, leading to the brittle deformation of rocks generating earthquakes.
References#
Lecture notes by Robert L. Nowack for Introduction to Seismology (course EAS 557), Purdue University.
Turcotte, Donald L., and Gerald Schubert. Geodynamics. Cambridge university press, 2002.
