Strain rate
InSI base units	s−1
Dimension	${\displaystyle {\mathsf{T}}^{-1}}$

Inmechanics andmaterials science,strain rate is thetime derivative ofstrain of a material. Strain rate hasdimension ofinverse time andSI units ofinverse 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 isexpanding or shrinking (expansion rate), and also the rate at which it is being deformed by progressiveshearing 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 samevelocity (same speed and direction) and/or rotating with the sameangular velocity, as if that part of the medium were arigid body.

The strain rate is a concept of materials science andcontinuum mechanics that plays an essential role in the physics offluids and deformable solids. In anisotropicNewtonian fluid, in particular, theviscous stress is alinear function of the rate of strain, defined by two coefficients, one relating to the expansion rate (thebulk viscosity coefficient) and one relating to the shear rate (the "ordinary"viscosity coefficient). In solids, higher strain rates can often cause normallyductile materials to fail in abrittle manner.^[1]

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." Inphysics the strain rate is generally defined as thederivative 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 adimensionless quantity (a number that does not depend on the choice ofmeasurement units). Thus, strain rate has dimension of inverse time and units ofinverse second, s⁻¹ (or its multiples).

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${\displaystyle t}$. 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.

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 ofinfinitesimally thin parallel layers sliding against each other as if they were rigid sheets, in the same direction, without changing their spacing. This description fits thelaminar flow of a fluid between two solid plates that slide parallel to each other (aCouette flow) or inside a circularpipe of constantcross-section (aPoiseuille 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 thelimit 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.

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 singlevector. In such cases, the rate of deformation must be expressed by atensor, alinear map between vectors, that expresses how the relativevelocity of the medium changes when one moves by a small distance away from the point in a given direction. Thisstrain rate tensor can be defined as the time derivative of thestrain tensor, or as the symmetric part of thegradient (derivative with respect to position) of thevelocity of the material.

With a chosencoordinate system, the strain rate tensor can be represented by asymmetric 3×3matrix 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 tofirst order; but that is generally sufficient for most purposes, even when the viscosity of the material is highly non-linear.

Materials can be tested using the so-called epsilon dot (${\displaystyle \dot{\epsilon }}$) method^[2] which can be used to deriveviscoelastic parameters throughlumped parameter analysis.

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,${\displaystyle \tau }$.^[3]

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

Therefore the unidirectional sliding strain rate can be defined as:

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

• Flow velocity
• Strain
• Strain gauge
• Stress–strain curve
• Stretch ratio

• Bar Technology for High-Strain-Rate Material Properties

• Classical mechanics
• Materials science
• Temporal rates
• Viscoelasticity

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