Strain-rate tensor
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.
[3] The concept has implications in a variety of areas of physics and engineering, including magnetohydrodynamics, mining and water treatment.
[4][5][6] 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.
By performing dimensional analysis, the dimensions of velocity gradient can be determined.
is called the spin tensor and describes the rate of rotation.
[7] Sir Isaac Newton proposed that shear stress is directly proportional to the velocity gradient:[8]
Consider a material body, solid or fluid, that is flowing and/or moving in space.
where vi is the component of v parallel to axis i and ∂jf denotes the partial derivative of a function f with respect to the space coordinate xj.
Note that J is a function of p and t. In this coordinate system, the Taylor approximation for the velocity near p is
Applying this to the Jacobian matrix with symmetric and antisymmetric components E and R respectively:
This decomposition is independent of coordinate system, and so has physical significance.
The antisymmetric term R represents a rigid-like rotation of the fluid about the point p. Its angular velocity
A rigid rotation does not change the relative positions of the fluid elements, so the antisymmetric term R of the velocity gradient does not contribute to the rate of change of the deformation.
The symmetric term E (the rate-of-strain tensor) can be broken down further as the sum of a scalar times the unit tensor, that represents a gradual isotropic expansion or contraction; and a traceless symmetric tensor which represents a gradual shearing deformation, with no change in volume:[9]
This decomposition is independent of the choice of coordinate system, and is therefore physically significant.
The trace of the expansion rate tensor is the divergence of the velocity field:
which is the rate at which the volume of a fixed amount of fluid increases at that point.
The shear rate tensor is represented by a symmetric 3 × 3 matrix, and describes a flow that combines compression and expansion flows along three orthogonal axes, such that there is no change in volume.
This type of flow occurs, for example, when a rubber strip is stretched by pulling at the ends, or when honey falls from a spoon as a smooth unbroken stream.
For a two-dimensional flow, the divergence of v has only two terms and quantifies the change in area rather than volume.
The factor 1/3 in the expansion rate term should be replaced by 1/2 in that case.
The study of velocity gradients is useful in analysing path dependent materials and in the subsequent study of stresses and strains; e.g., Plastic deformation of metals.
[3] The near-wall velocity gradient of the unburned reactants flowing from a tube is a key parameter for characterising flame stability.
[5]: 1–3 The velocity gradient of a plasma can define conditions for the solutions to fundamental equations in magnetohydrodynamics.
[10] If the velocity difference between fluid layers at the centre of the pipe and at the sides of the pipe is sufficiently small, then the fluid flow is observed in the form of continuous layers.
