Strain Rate
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In materials science, strain rate is the change in
strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
(
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
) 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" or ...
without changing its volume (
shear rate In physics, shear rate is the rate at which a progressive shearing deformation is applied to some material. Simple shear The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary ...
). 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 is a ...
(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 objec ...
, 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 force ...
. 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 m ...
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 describe ...
Newtonian fluid A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
, in particular, the viscous stress is a
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
of the rate of strain, defined by two coefficients, one relating to the expansion rate (the
bulk viscosity Volume viscosity (also called bulk viscosity, or dilatational viscosity) is a material property relevant for characterizing fluid flow. Common symbols are \zeta, \mu', \mu_\mathrm, \kappa or \xi. It has dimensions (mass / (length × time)), and the ...
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 inte ...
coefficient). In solids, higher strain rates can often cause normally
ductile Ductility is a mechanical property commonly described as a material's amenability to drawing (e.g. into wire). In materials science, ductility is defined by the degree to which a material can sustain plastic deformation under tensile stres ...
materials to fail in a
brittle A material is brittle if, when subjected to stress, it fractures with little elastic deformation and without significant plastic deformation. Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Bre ...
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 r ...
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. F ...
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 \epsilon between the amount of stretching and the original length of the band: :\epsilon(t) = \frac where L_0 is the original length and L(t) its length at each time t. Then the strain rate will be : \dot (t) = \frac = \frac \left ( \frac \right ) = \frac \frac = \frac 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
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 referr ...
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 (a
Poiseuille flow The poiseuille (symbol Pl) has been proposed as a derived SI unit of dynamic viscosity, named after the French physicist Jean Léonard Marie Poiseuille (1797–1869). In practice the unit has never been widely accepted and most international s ...
). 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 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 X(y+d,t) - X(y,t) of a nearby layer, divided by the spacing d between the layers: :\epsilon(y,t) = \lim_ \frac = \frac(y,t) Therefore, the strain rate is :\dot \epsilon(y,t) = \left(\frac\frac\right)(y,t) = \left(\frac\frac\right)(y,t) = \frac(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
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
. 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 tenso ...
, 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 Map (mathematics), mapping V \to W between two vect ...
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 is a ...
of the medium changes when one moves by a small distance away from the point in a given direction. This
strain rate tensor In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be def ...
can be defined as the time derivative of the
strain tensor In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
, 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 gradi ...
(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 is a ...
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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
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'' (franchis ...
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; 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 A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
quantity (a number that does not depend on the choice of
measurement unit A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multi ...
s). 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 (\dot) method which can be used to derive
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
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, \tau. :\gamma = \frac = \tan(\theta) Therefore the unidirectional shear strain rate can be defined as: :\dot=\frac{dt}


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 may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
*
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 and ...
* Stretch ratio


References


External links


Bar Technology for High-Strain-Rate Material Properties
Classical mechanics Materials science Temporal rates