Strain-rate tensor

In <a href="/facts/Continuum_mechanics/vI7LpFDk">continuum mechanics</a>, the strain-rate tensor or rate-of-strain tensor is a <a href="/facts/Physical_quantity/S8pWBEpK">physical quantity</a> that describes the <a href="/facts/Derivative/7Ito70Dh">rate of change</a> of the <a href="/facts/Strain_(mechanics)/xK7R59oI">strain</a> (i.e., the relative <a href="/facts/Deformation_(mechanics)/xK7R59oI">deformation</a>) of a material in the neighborhood of a certain point, at a certain moment of time. It can be defined as the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of the <a href="/facts/Strain_tensor/j4CJFdsU">strain tensor</a> with respect to time, or as the symmetric component of the <a href="/facts/Jacobian_matrix/0tNv6o8j">Jacobian matrix</a> (derivative with respect to position) of the <a href="/facts/Flow_velocity/amIT4Ht8">flow velocity</a>. In <a href="/facts/Fluid_mechanics/NIy4tRfE">fluid mechanics</a> it also can be described as the velocity gradient, a measure of how the <a href="/facts/Velocity_field/amIT4Ht8">velocity</a> of a fluid changes between different points within the fluid. Though the term can refer to a <a href="/facts/Velocity_profile/amIT4Ht8">velocity profile</a> (variation in velocity across layers of flow in a pipe), it is often used to mean the <a href="/facts/Spatial_gradient/2cmoCezd">gradient</a> of a flow's velocity with respect to its <a href="/facts/Coordinate_system/Mtd0q4SP">coordinates</a>. The concept has implications in a variety of areas of <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Engineering/RFVg8H3I">engineering</a>, including <a href="/facts/Magnetohydrodynamics/7mQbJPPD">magnetohydrodynamics</a>, mining and water treatment. 
The strain rate tensor is a purely <a href="/facts/Kinematics/o7mSyeJA">kinematic</a> concept that describes the <a href="/facts/Macroscopic/NLwPDtLe">macroscopic</a> motion of the material. Therefore, it does not depend on the nature of the material, or on the forces and stresses that may be acting on it; and it applies to any <a href="/facts/Continuum_mechanics/vI7LpFDk">continuous medium</a>, whether <a href="/facts/Solid/hj5UsXno">solid</a>, <a href="/facts/Liquid/t9GkLWzt">liquid</a> or <a href="/facts/Gas/hVVAoodV">gas</a>.
On the other hand, for any <a href="/facts/Fluid/tMQ46Tmz">fluid</a> except <a href="/facts/Superfluid/QuePDujN">superfluids</a>, any gradual change in its deformation (i.e. a non-zero strain rate tensor) gives rise to <a href="/facts/Viscosity/QLVfrdCz">viscous forces</a> in its interior, due to <a href="/facts/Friction/d0108Rpu">friction</a> between adjacent <a href="/facts/Fluid_element/yk70LICD">fluid elements</a>, that tend to oppose that change. At any point in the fluid, these stresses can be described by a <a href="/facts/Viscous_stress_tensor/ljeMKpDI">viscous stress tensor</a> that is, almost always, completely determined by the strain rate tensor and by certain intrinsic properties of the fluid at that point. Viscous stress also occur in solids, in addition to the elastic stress observed in static deformation; when it is too large to be ignored, the material is said to be <a href="/facts/Viscoelastic/pDhHgx2T">viscoelastic</a>.

Strain-rate tensor open-in-new

Strain-rate tensor