Principal strain rates are the eigenvalues of the symmetric rate-of-strain tensor. Positive values are local extension and negative values local compression. In incompressible flow , implying . Their eigenvectors describe the principal deformation directions.
Bi-axial extension has two positive principal strain rates and one negative rate in incompressible flow. It flattens a material volume toward a sheet while extending its two in-plane directions. A negative mean product of the three rates indicates amplitude-weighted preference for this geometry, but alone does not establish an unweighted event-frequency majority.
The shear rate is the scalar rate at which adjacent material layers acquire shear strain. In simple shear , its signed value is .
A simple shear flow has velocity in Cartesian coordinates. Its velocity gradient has one nonzero off-diagonal entry.
The spin tensor is the antisymmetric part of the velocity gradient, . It describes local rigid-body rotation.
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The strain-rate tensor is a mathematical object used in continuum mechanics to characterize the rate of deformation of a material over time. It quantifies how the shape of a material changes as it deforms, which is particularly important in the study of fluid dynamics, solid mechanics, and material science. Mathematically, the strain-rate tensor \( \dot{\epsilon} \) is a second-order symmetric tensor that describes the instantaneous rate of change of the strain in the material.