Strain rate

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 ${\displaystyle ϵ}$ between the amount of stretching and the original length of the band:

${\displaystyle ϵ(t)=\frac{L(t)-L_0}{L_0}}$

where ${\displaystyle L_0}$ is the original length and ${\displaystyle L(t)}$ its length at each time ${\displaystyle t}$. For example, if the rubber band starts at original length of 1 cm and after stretching some time ${\displaystyle t}$ the length is 7 cm, the strain at that time will be 6 (a unitless number). Then the strain rate will be

${\displaystyle \dot{ϵ}(t)=\frac{dϵ}{dt}=\frac{d}{dt}\left(\frac{L(t)-L_0}{L_0}\right)=\frac{1}{L_0}\frac{dL(t)}{dt}=\frac{v(t)}{L_0}}$

where ${\displaystyle v(t)}$ is the speed at which the ends are moving away from each other. To continue the above example, assuming the speed at which the ends were moving away from each other was constant and the stretch occurred over 2 seconds, the strain rate would be 3 s⁻¹.

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 infinitesimally thin parallel layers sliding against each other as if they were rigid sheets, in the same direction, without changing their spacing. This description fits the laminar flow of a fluid between two solid plates that slide parallel to each other (a Couette flow) or inside a circular pipe of constant cross-section (a Poiseuille flow). In those cases, the state of the material at some time ${\displaystyle t}$ can be described by the displacement ${\displaystyle X(y,t)}$ of each layer, since an arbitrary starting time, as a function of its distance ${\displaystyle y}$ from the fixed wall. Then the strain in each layer can be expressed as the limit of the ratio between the current relative displacement ${\displaystyle X(y+d,t)-X(y,t)}$ of a nearby layer, divided by the spacing ${\displaystyle d}$ between the layers:

${\displaystyle ϵ(y,t)=\underset{d\to 0}{lim}\frac{X(y+d,t)-X(y,t)}{d}=\frac{\mathrm{\partial }X}{\mathrm{\partial }y}(y,t)}$

Therefore, the strain rate is

${\displaystyle \dot{ϵ}(y,t)=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }t}\frac{\mathrm{\partial }X}{\mathrm{\partial }y}\right)(y,t)=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }y}\frac{\mathrm{\partial }X}{\mathrm{\partial }t}\right)(y,t)=\frac{\mathrm{\partial }V}{\mathrm{\partial }y}(y,t)}$

where ${\displaystyle V(y,t)}$ is the current linear speed of the material at distance ${\displaystyle y}$ 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 vector. In such cases, the rate of deformation must be expressed by a tensor, a linear map between vectors, that expresses how the relative velocity of the medium changes when one moves by a small distance away from the point in a given direction. This strain rate tensor can be defined as the time derivative of the 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.

Strain rate testing

Materials can be tested using the so-called epsilon dot (${\displaystyle \dot{\epsilon }}$) method which can be used to derive viscoelastic parameters through lumped parameter analysis.

Sliding rate or shear strain rate

Similarly, the sliding rate, also called the deviatoric strain rate or shear strain rate is the derivative with respect to time of the shear strain. Engineering sliding strain can be defined as the angular displacement created by an applied shear stress, ${\displaystyle \tau }$.

${\displaystyle \gamma =\frac{w}{l}=\tan (\theta )}$

Therefore the unidirectional sliding strain rate can be defined as:

${\displaystyle \dot{\gamma }=\frac{d\gamma }{dt}}$