In
materials science, strain rate is the change in
strain (
deformation) of a material with respect to time.
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
Sheep shearing is the process by which the woollen fleece of a sheep is cut off. The person who removes the sheep's wool is called a '' shearer''. Typically each adult sheep is shorn once each year (a sheep may be said to have been "shorn" o ...
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
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
(same speed and direction) and/or rotating with the same
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an object ...
, as if that part of the medium were a
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
.
The strain rate is a concept of materials science and
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...
that plays an essential role in the physics of
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s and deformable solids. In an
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
Newtonian fluid, in particular, the
viscous stress is a
linear function of the rate of strain, defined by two coefficients, one relating to the expansion rate (the
bulk viscosity coefficient) and one relating to the shear rate (the "ordinary"
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
coefficient). In solids, higher strain rates can often cause normally
ductile materials to fail in a
brittle manner.
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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
the strain rate is generally defined as the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the strain with respect to time. Its precise definition depends on how strain is measured.
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
between the amount of stretching and the original length of the band:
:
where
is the original length and
its length at each time
. Then the strain rate will be
:
where
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
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
ly 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
In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...
of a fluid between two solid plates that slide parallel to each other (a
Couette flow
In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes a shear stress on the fluid and induces flow. ...
) or inside a circular
pipe
Pipe(s), PIPE(S) or piping may refer to:
Objects
* Pipe (fluid conveyance), a hollow cylinder following certain dimension rules
** Piping, the use of pipes in industry
* Smoking pipe
** Tobacco pipe
* Half-pipe and quarter pipe, semi-circular ...
of constant
cross-section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
(a
Poiseuille flow). In those cases, the state of the material at some time
can be described by the displacement
of each layer, since an arbitrary starting time, as a function of its distance
from the fixed wall. Then the strain in each layer can be expressed as the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of the ratio between the current relative displacement
of a nearby layer, divided by the spacing
between the layers:
:
Therefore, the strain rate is
:
where
is the current linear speed of the material at distance
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
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
, a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
between vectors, that expresses how the relative
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
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
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
(derivative with respect to position) of the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of the material.
With a chosen
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
, the strain rate tensor can be represented by a
symmetric 3×3
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
of real numbers. The strain rate tensor typically varies with position and time within the material, and is therefore a (time-varying)
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
. It only describes the local rate of deformation to
first order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hi ...
; but that is generally sufficient for most purposes, even when the viscosity of the material is highly non-linear.
Units
The strain is the ratio of two lengths, so it is a
dimensionless quantity (a number that does not depend on the choice of
measurement units). Thus, strain rate is in units of inverse time (such as s
−1).
Strain rate testing
Materials can be tested using the so-called epsilon dot (
) method which can be used to derive
viscoelastic parameters through
lumped parameter analysis.
Shear strain rate
Similarly, the shear strain rate is the derivative with respect to time of the shear strain. Engineering shear strain can be defined as the angular displacement created by an applied shear stress,
.
:
Therefore the unidirectional shear strain rate can be defined as:
:
See also
*
Flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
*
Strain
*
Strain gauge
A strain gauge (also spelled strain gage) is a device used to measure strain on an object. Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports ...
*
Stress–strain curve
In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress ...
*
Stretch ratio
In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body.
A deformation can ...
References
External links
Bar Technology for High-Strain-Rate Material Properties
Classical mechanics
Materials science
Temporal rates