Strain and the Small-Strain Tensor

Part 2, Part 2: Strain, Elasticity, and the Poroelastic Rock

Learning objectives

Define normal strain as fractional length change and shear strain as angle change between perpendicular lines
Assemble the small-strain tensor from displacement gradients, the deformation twin of the stress tensor
Read the volumetric strain as the sum of the normal strains, and show pure shear changes shape without volume
Calibrate the magnitudes: reservoir strains live around a millistrain, seismic strains around a microstrain

Deformation, Made Local

Stress is only half a mechanics; the other half is what the rock does. Displacements themselves are useless for that, a rigid block can translate a kilometer without deforming at all, so the meaningful object is how displacement varies from point to point. From those gradients, small-strain theory builds a tensor with exactly the structure you already know. Normal strains $\varepsilon_{xx}, \varepsilon_{yy}$ are fractional length changes along the axes, positive in contraction under our compression-positive convention. Shear strain $\varepsilon_{xy}$ xy is half the closing of the angle between two initially perpendicular lines; engineers often quote the whole angle, $\gamma_{xy} = 2\varepsilon_{xy}$ , and the factor of two between conventions has ambushed generations of careful people. Symmetric, six independent components, principal directions of its own: the stress tensor's deformation twin.

Deform the grid above. Normal strains stretch or squeeze the squares; shear strain shears them into parallelograms; and the readout tracks the one combination that will matter most in this course: the volumetric strain $\varepsilon_v = \varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz}$ varepsilonyy+varepsilonzz, the fractional volume change, the strain tensor's trace and, like every trace, an invariant. The pure-shear preset makes the point cleanly: the grid distorts visibly while its area holds to machine precision. Shape change and volume change are separate businesses, and the elastic moduli of the next section will price them separately.

How Big Is Big?

Strain's dimensionless numbers deserve calibration, because they are small in a way intuition mishandles. A reservoir drawn down by 10 MPa compacts by something near a millistrain, one part in a thousand, which on a hundred-meter interval is the ten centimeters of thinning that Part 10 will turn into subsidence bowls and sheared casings. A passing seismic wave carries a microstrain or less, which is why rocks can transmit it elastically forever. And the strains at which rocks stop being elastic at all, where Part 3 begins, sit near a few millistrain in brittle rock: the entire elastic life of a reservoir plays out inside a fraction of one percent of deformation. Small numbers, large consequences; the moduli that connect them to megapascals are next.

References

Jaeger, J. C., Cook, N. G. W., & Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics (4th ed.). Blackwell.
Fjaer, E., Holt, R. M., Horsrud, P., Raaen, A. M., & Risnes, R. (2008). Petroleum Related Rock Mechanics (2nd ed.). Elsevier.