Strain-rate tensor

By performing dimensional analysis, the dimensions of velocity gradient can be determined. The dimensions of velocity are ${\displaystyle {\mathsf {L^{1}T^{-1}}}}$, and the dimensions of distance are ${\displaystyle {\mathsf {L^{1}}}}$. Since the velocity gradient can be expressed as ${\displaystyle {\frac {\Delta {\text{velocity}}}{\Delta {\text{distance}}}}}$. Therefore, the velocity gradient has the same dimensions as this ratio, i.e., ${\displaystyle {\mathsf {T^{-1}}}}$.

In 3 dimensions, the gradient ${\displaystyle \nabla \mathbf {v} }$ of the velocity ${\displaystyle \mathbf {v} }$ is a second-order tensor which can be expressed as the matrix ${\displaystyle \mathbf {L} }$:

$${\displaystyle \mathbf {L} =\nabla \mathbf {v} ={\begin{bmatrix}{\frac {\partial v_{x}}{\partial x}}&{\frac {\partial v_{x}}{\partial y}}&{\frac {\partial v_{x}}{\partial z}}\\{\frac {\partial v_{y}}{\partial x}}&{\frac {\partial v_{y}}{\partial y}}&{{\frac {\partial v_{y}}{\partial z}}\ }\\{\frac {\partial v_{z}}{\partial x}}&{\frac {\partial v_{z}}{\partial y}}&{\frac {\partial v_{z}}{\partial z}}\end{bmatrix}}}$$

${\displaystyle \mathbf {L} }$ can be decomposed into the sum of a symmetric matrix ${\displaystyle {\textbf {E}}}$ and a skew-symmetric matrix ${\displaystyle {\textbf {W}}}$ as follows

$${\displaystyle {\begin{aligned}\mathbf {E} &={\frac {1}{2}}\left(\mathbf {L} +\mathbf {L} ^{\textsf {T}}\right)\\\mathbf {W} &={\frac {1}{2}}\left(\mathbf {L} -\mathbf {L} ^{\textsf {T}}\right)\end{aligned}}}$$

${\displaystyle {\textbf {E}}}$ is called the strain rate tensor and describes the rate of stretching and shearing. ${\displaystyle {\textbf {W}}}$ 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]

$${\displaystyle \tau =\mu {\frac {\partial u}{\partial y}}.}$$

The constant of proportionality, ${\displaystyle \mu }$, is called the dynamic viscosity.

Consider a material body, solid or fluid, that is flowing and/or moving in space. Let v be the velocity field within the body; that is, a smooth function from R³ × R such that v(p, t) is the macroscopic velocity of the material that is passing through the point p at time t.

The velocity v(p + r, t) at a point displaced from p by a small vector r can be written as a Taylor series:

$${\displaystyle \mathbf {v} (\mathbf {p} +\mathbf {r} ,t)=\mathbf {v} (\mathbf {p} ,t)+(\nabla \mathbf {v} )(\mathbf {p} ,t)(\mathbf {r} )+{\text{higher order terms}},}$$

where ∇v the gradient of the velocity field, understood as a linear map that takes a displacement vector r to the corresponding change in the velocity.

Total field v(p + r).

Constant part v(p).

Linear part (∇v)(p, t)(r).

Non-linear residual.

The velocity field v(p + r, t) of an arbitrary flow around a point p (red dot), at some instant t, and the terms of its first-order Taylor approximation about p. The third component of the velocity (out of the screen) is assumed to be zero everywhere.

In an arbitrary reference frame, ∇v is related to the Jacobian matrix of the field, namely in 3 dimensions it is the 3 × 3 matrix

$${\displaystyle \left(\nabla \mathbf {v} \right)^{\mathrm {T} }={\begin{bmatrix}\partial _{1}v_{1}&\partial _{2}v_{1}&\partial _{3}v_{1}\\\partial _{1}v_{2}&\partial _{2}v_{2}&\partial _{3}v_{2}\\\partial _{1}v_{3}&\partial _{2}v_{3}&\partial _{3}v_{3}\end{bmatrix}}=\mathbf {J} .}$$

where v_i 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 x_j. Note that J is a function of p and t.

In this coordinate system, the Taylor approximation for the velocity near p is

$${\displaystyle v_{i}(\mathbf {p} +\mathbf {r} ,t)=v_{i}(\mathbf {p} ,t)+\sum _{j}J_{ij}(\mathbf {p} ,t)r_{j}=v_{i}(\mathbf {p} ,t)+\sum _{j}\partial _{j}v_{i}(\mathbf {p} ,t)r_{j};}$$

or simply

$${\displaystyle \mathbf {v} (\mathbf {p} +\mathbf {r} ,t)=\mathbf {v} (\mathbf {p} ,t)+\mathbf {J} (\mathbf {p} ,t)\mathbf {r} }$$

if v and r are viewed as 3 × 1 matrices.

Symmetric and antisymmetric parts

The symmetric part E(p, t)(r) (strain rate) of the linear term of the example flow.

The antisymmetric part R(p, t)(r) (rotation) of the linear term.

Any matrix can be decomposed into the sum of a symmetric matrix and an antisymmetric matrix. Applying this to the Jacobian matrix with symmetric and antisymmetric components E and R respectively:

$${\displaystyle {\begin{aligned}\mathbf {E} &={\frac {1}{2}}\left(\mathbf {J} +\mathbf {J} ^{\textsf {T}}\right)&\mathbf {R} &={\frac {1}{2}}\left(\mathbf {J} -\mathbf {J} ^{\textsf {T}}\right)\\E_{ij}&={\frac {1}{2}}\left(\partial _{j}v_{i}+\partial _{i}v_{j}\right)&R_{ij}&={\frac {1}{2}}\left(\partial _{j}v_{i}-\partial _{i}v_{j}\right)\end{aligned}}}$$

This decomposition is independent of coordinate system, and so has physical significance. Then the velocity field may be approximated as

$${\displaystyle \mathbf {v} (\mathbf {p} +\mathbf {r} ,t)\approx \mathbf {v} (\mathbf {p} ,t)+\mathbf {E} (\mathbf {p} ,t)(\mathbf {r} )+\mathbf {R} (\mathbf {p} ,t)(\mathbf {r} ),}$$

that is,

$${\displaystyle {\begin{aligned}v_{i}(\mathbf {p} +\mathbf {r} ,t)&=v_{i}(\mathbf {p} ,t)+\sum _{j}E_{ij}(\mathbf {p} ,t)r_{j}+\sum _{j}R_{ij}(\mathbf {p} ,t)r_{j}\\&=v_{i}(\mathbf {p} ,t)+{\frac {1}{2}}\sum _{j}\left(\partial _{j}v_{i}(\mathbf {p} ,t)+\partial _{i}v_{j}(\mathbf {p} ,t)\right)r_{j}+{\frac {1}{2}}\sum _{j}\left(\partial _{j}v_{i}(\mathbf {p} ,t)-\partial _{i}v_{j}(\mathbf {p} ,t)\right)r_{j}\end{aligned}}}$$

The antisymmetric term R represents a rigid-like rotation of the fluid about the point p. Its angular velocity ${\displaystyle {\vec {\omega }}}$ is

$${\displaystyle {\vec {\omega }}={\frac {1}{2}}\nabla \times \mathbf {v} ={\frac {1}{2}}{\begin{bmatrix}\partial _{2}v_{3}-\partial _{3}v_{2}\\\partial _{3}v_{1}-\partial _{1}v_{3}\\\partial _{1}v_{2}-\partial _{2}v_{1}\end{bmatrix}}.}$$

The product ∇ × v is called the rotational curl of the vector field. 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 actual strain rate is therefore described by the symmetric E term, which is the strain rate tensor.

Shear rate and compression rate

The spherical part S(p, t)(r) (uniform expansion, or compression, rate) of the strain rate tensor E(p, t)(r).

The deviatoric part D(p, t)(r) (shear rate) of the strain rate tensor E(p, t)(r).

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]

$${\displaystyle \mathbf {E} (\mathbf {p} ,t)(\mathbf {r} )=\mathbf {S} (\mathbf {p} ,t)(\mathbf {r} )+\mathbf {D} (\mathbf {p} ,t)(\mathbf {r} ).}$$

That is,

$${\displaystyle E_{ij}=\underbrace {{\frac {1}{3}}\left(\sum _{k}\partial _{k}v_{k}\right)\delta _{ij}} _{{\text{rate-of-expansion tensor }}S_{ij}}+\underbrace {\overbrace {{\frac {1}{2}}\left(\partial _{i}v_{j}+\partial _{j}v_{i}\right)} ^{E_{ij}}-S_{ij}} _{{\text{rate-of-shear tensor }}D_{ij}},}$$

Here δ is the unit tensor, such that δ_ij is 1 if i = j and 0 if i ≠ j. 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:

$${\displaystyle \nabla \cdot \mathbf {v} =\partial _{1}v_{1}+\partial _{2}v_{2}+\partial _{3}v_{3};}$$

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.[4]

• Stress tensor (disambiguation)
• Finite strain theory § Time-derivative of the deformation gradient, the spatial and material velocity gradient from continuum mechanics