Strain rates and displacement rates for two- and three-dimensional deformations

Basil Tikoff, University of Wisconsin-Madison
Ann L Everest, University of Wisconsin-Madison
Molly F Egan, University of Wisconsin-Madison
,
,
,
,
,

Abstract

The relation between strain rates (measured in units of 1/sec) and displacement rates (measured in units of m/sec) within shear zones is straight forward for simple shearing deformation. This contribution addresses this relation for cases of two- and three-dimensional deformation, that involve simultaneous coaxial and non-coaxial components. Strain rates are typically given by one-dimensional maximum elongation rate. We utilize effective strain rate, because: (1) it is proportional to the second invariant of the rate-of-stretching tensor (the second invariant describes deviation from a sphere for a second-rank, symmetric, 3x3 tensor); and (2) for simple shearing, it is exactly equivalent to the one-dimensional elongation rate. For coaxial deformations, the strain rate and the displacement rates cannot both be constant. Strain rates acting within a shear zone accumulate to finite strain over time, but coaxial deformation (e.g., two-dimensional pure shearing) accumulates finite strain quicker than simple shearing. Intermediate sub-simple shearing cases lie between these two endmembers. Maximum elongation rate is not a good measure for strain rates of three-dimensional deformations, as it overestimates strain rate for coaxial constricting and underestimates strain rate for coaxial flattening. Accumulations of finite strain for different strain rates can be calculated for any three-dimensional deformation (e.g., transpressing/transtensing). As with the two-dimensional case, coaxial deformations accumulates finite strain more quickly than simple shearing deformations for the same strain rate. In these three-dimensional cases, displacement rate should be based on the magnitude of the maximum converging (or diverging) flow apophysis (eigenvalues of the velocity gradient tensor), not on the maximum shortening (or elongating) rate; these quantities are neither parallel nor equivalent if there is any simple shear component.

For simple shearing, a displacement rate can be calculated from strain rate if a width is known. If there is any component of coaxial deformation, a displacement rate at the end of deformation can be calculated only if the width of the shear zone, type of deformation (e.g., transpression), angle of the oblique flow apophysis (or a kinematic vorticity), and the strain rate are known. Estimates of displacement rate through time can be modelled, if the timing of deformation is constrained. Finally, we apply these analyses to two separate cases: The transpressional western Idaho shear zone and the transtensional deformation in the Basin and Range province in the western US.

Session

Rates and dates