In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the strain (i.e., the relative deformation) of a material in the neighborhood of a certain point, at a certain moment of time. It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the Jacobian matrix (derivative with respect to position) of the flow velocity. In fluid mechanics it also can be described as the velocity gradient, a measure of how the velocity of a fluid changes between different points within the fluid. Though the term can refer to a velocity profile (variation in velocity across layers of flow in a pipe), it is often used to mean the gradient of a flow's velocity with respect to its coordinates. The concept has implications in a variety of areas of physics and engineering, including magnetohydrodynamics, mining and water treatment. The strain rate tensor is a purely kinematic concept that describes the macroscopic 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 continuous medium, whether solid, liquid or gas. On the other hand, for any fluid except superfluids, any gradual change in its deformation (i.e. a non-zero strain rate tensor) gives rise to viscous forces in its interior, due to friction between adjacent fluid elements, that tend to oppose that change. At any point in the fluid, these stresses can be described by a viscous stress tensor 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 viscoelastic.
Dimensional analysis By performing dimensional analysis, the dimensions of velocity gradient can be determined. The dimensions of velocity are L 1 T − 1 {\displaystyle {\mathsf {L^{1}T^{-1}}}} , and the dimensions of distance are L 1 {\displaystyle {\mathsf {L^{1}}}} . Since the velocity gradient can be expressed as Δ velocity Δ distance {\displaystyle {\frac {\Delta {\text{velocity}}}{\Delta {\text{distance}}}}} . Therefore, the velocity gradient has the same dimensions as this ratio, i.e., T − 1 {\displaystyle {\mathsf {T^{-1}}}} .
In continuum mechanics In 3 dimensions, the gradient ∇ v {\displaystyle \nabla \mathbf {v} } of the velocity v {\displaystyle \mathbf {v} } is a second-order tensor which can be expressed as the matrix L {\displaystyle \mathbf {L} } :
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