Strain rate

Strain rate
Strain rate
In SI base units	s-1
Dimension

In

SI units of inverse second

, s⁻¹ (or its multiples).

The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change with time in the neighborhood of that point. It comprises both the rate at which the material is

expanding or shrinking (expansion rate), and also the rate at which it is being deformed by progressive shearing without changing its volume (shear rate). It is zero if these distances do not change, as happens when all particles in some region are moving with the same velocity (same speed and direction) and/or rotating with the same angular velocity, as if that part of the medium were a rigid body

.

The strain rate is a concept of materials science and

brittle manner.^[1]

Definition

The definition of strain rate was first introduced in 1867 by American metallurgist Jade LeCocq, who defined it as "the rate at which strain occurs. It is the time rate of change of strain." In physics the strain rate is generally defined as the derivative of the strain with respect to time. Its precise definition depends on how strain is measured.

The strain is the ratio of two lengths, so it is a

measurement units). Thus, strain rate has dimension of inverse time and units of inverse second

, s⁻¹ (or its multiples).

Simple deformations

In simple contexts, a single number may suffice to describe the strain, and therefore the strain rate. For example, when a long and uniform rubber band is gradually stretched by pulling at the ends, the strain can be defined as the ratio $\epsilon$ between the amount of stretching and the original length of the band:

\epsilon (t)={\frac {L(t)-L_{0}}{L_{0}}}

where $L_{0}$ is the original length and $L(t)$ its length at each time $t$ . Then the strain rate will be

{\dot {\epsilon }}(t)={\frac {d\epsilon }{dt}}={\frac {d}{dt}}\left({\frac {L(t)-L_{0}}{L_{0}}}\right)={\frac {1}{L_{0}}}{\frac {dL(t)}{dt}}={\frac {v(t)}{L_{0}}}

where $v(t)$ is the speed at which the ends are moving away from each other.

The strain rate can also be expressed by a single number when the material is being subjected to parallel shear without change of volume; namely, when the deformation can be described as a set of

Poiseuille flow

). In those cases, the state of the material at some time

t

can be described by the displacement

X(y,t)

of each layer, since an arbitrary starting time, as a function of its distance

y

from the fixed wall. Then the strain in each layer can be expressed as the limit of the ratio between the current relative displacement

X(y+d,t)-X(y,t)

of a nearby layer, divided by the spacing

d

between the layers:

\epsilon (y,t)=\lim _{d\rightarrow 0}{\frac {X(y+d,t)-X(y,t)}{d}}={\frac {\partial X}{\partial y}}(y,t)

Therefore, the strain rate is

{\dot {\epsilon }}(y,t)=\left({\frac {\partial }{\partial t}}{\frac {\partial X}{\partial y}}\right)(y,t)=\left({\frac {\partial }{\partial y}}{\frac {\partial X}{\partial t}}\right)(y,t)={\frac {\partial V}{\partial y}}(y,t)

where $V(y,t)$ is the current linear speed of the material at distance $y$ from the wall.

The strain-rate tensor

In more general situations, when the material is being deformed in various directions at different rates, the strain (and therefore the strain rate) around a point within a material cannot be expressed by a single number, or even by a single

strain tensor, or as the symmetric part of the gradient (derivative with respect to position) of the velocity

of the material.

With a chosen

coordinate system, the strain rate tensor can be represented by a symmetric 3×3 matrix of real numbers. The strain rate tensor typically varies with position and time within the material, and is therefore a (time-varying) tensor field. It only describes the local rate of deformation to first order

; but that is generally sufficient for most purposes, even when the viscosity of the material is highly non-linear.