Strain rate tensor
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In
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 ...
, the strain-rate tensor or rate-of-strain tensor is a
physical quantity A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
that describes the rate of change of the
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 in the neighborhood of a certain point, at a certain moment of time. It can be 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 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 ...
with respect to time, or as the symmetric component of the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
(derivative with respect to position) of the
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 ...
. In
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
it also can be described as the velocity gradient, a measure of how 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 a fluid changes between different points within the fluid. Though the term can refer to the differences in velocity between layers of flow in a pipe, it is often used to mean 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 ...
of a flow's velocity with respect to its
coordinates 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 concept has implications in a variety of areas of
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 ...
and
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, including
magnetohydrodynamics Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, ...
, mining and water treatment. The strain rate tensor is a purely
kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
concept that describes the
macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenomena an ...
motion of the material. Therefore, it does not depend on the nature of the material, or on the forces and stresses that may be acting on it; and it applies to any continuous medium, whether
solid Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
,
liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
or
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
. On the other hand, for any
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 ...
except
superfluid Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
s, any gradual change in its deformation (i.e. a non-zero strain rate tensor) gives rise to
viscous forces 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 inter ...
in its interior, due to
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
between adjacent
fluid element In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constan ...
s, that tend to oppose that change. At any point in the fluid, these stresses can be described by a
viscous stress tensor The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point. The viscous stress ...
that is, almost always, completely determined by the strain rate tensor and by certain intrinsic properties of the fluid at that point. Viscous stress also occur in solids, in addition to the elastic stress observed in static deformation; when it is too large to be ignored, the material is said to be
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 ...
.


Dimensional analysis

By performing
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
, the dimensions of velocity gradient can be determined. The dimensions of velocity are \mathsf , and the dimensions of distance are \mathsf. Since the velocity gradient can be expressed as \frac. Therefore, the velocity gradient has the same dimensions as this ratio, i.e., \mathsf.


In continuum mechanics

In 3 dimensions, the gradient \nabla\mathbf of the velocity \mathbf is a second-order
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 ...
which can be expressed as the
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 ...
\mathbf: \mathbf = \nabla\mathbf = \begin & & \\ & & \\ & & \end \mathbf can be decomposed into the sum of a
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
\textbf and a
skew-symmetric matrix In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ ...
\textbf as follows \begin \mathbf &= \frac \left(\mathbf + \mathbf^\textsf\right) \\ \mathbf &= \frac \left(\mathbf - \mathbf^\textsf\right) \end \textbf is called the strain rate tensor and describes the rate of stretching and shearing. \textbf is called the spin tensor and describes the rate of rotation.


Relationship between shear stress and the velocity field

Sir Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the great ...
proposed that
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
is directly proportional to the velocity gradient: \tau = \mu\frac . The
constant of proportionality In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constant ...
, \mu, is called the
dynamic 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 inter ...
.


Formal definition

Consider a material body, solid or fluid, that is flowing and/or moving in space. Let be the velocity
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
within the body; that is, a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
function from such that is the
macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenomena an ...
velocity of the material that is passing through the point at time . The velocity at a point displaced from by a small vector can be written as a
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
: \mathbf(\mathbf + \mathbf, t) = \mathbf(\mathbf, t) + (\nabla \mathbf)(\mathbf, t)(\mathbf) + \text, where the gradient of the velocity field, understood as 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 ...
that takes a displacement vector to the corresponding change in the velocity. In an arbitrary
reference frame In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale are specified by a set of reference point ...
, is related to the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
of the field, namely in 3 dimensions it is the 3 × 3 matrix \left(\nabla \mathbf\right)^ = \begin \partial_1 v_1 & \partial_2 v_1 & \partial_3 v_1 \\ \partial_1 v_2 & \partial_2 v_2 & \partial_3 v_2 \\ \partial_1 v_3 & \partial_2 v_3 & \partial_3 v_3 \end = \mathbf. where is the component of parallel to
axis An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis * Axis (mathematics), a designator for a Cartesian-coordinat ...
and denotes the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
of a function with respect to the space coordinate . Note that is a function of and . In this coordinate system, the Taylor approximation for the velocity near is v_i(\mathbf + \mathbf, t) = v_i(\mathbf, t) + \sum_j J_(\mathbf, t) r_j = v_i(\mathbf, t) + \sum_j \partial_j v_i(\mathbf, t) r_j; or simply \mathbf(\mathbf + \mathbf, t) = \mathbf(\mathbf, t) + \mathbf(\mathbf, t) \mathbf if and are viewed as 3 × 1 matrices.


Symmetric and antisymmetric parts

Any matrix can be decomposed into the sum of a
symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
and an
antisymmetric matrix Antisymmetric or skew-symmetric may refer to: * Antisymmetry in linguistics * Antisymmetric relation in mathematics * Skew-symmetric graph * Self-complementary graph In mathematics, especially linear algebra, and in theoretical physics, the adject ...
. Applying this to the Jacobian matrix with symmetric and antisymmetric components and respectively: \begin \mathbf &= \frac\left(\mathbf + \mathbf^\textsf\right) & \mathbf &= \frac\left(\mathbf - \mathbf^\textsf\right) \\ E_ &= \frac\left(\partial_j v_i + \partial_i v_j\right) & R_ &= \frac\left(\partial_j v_i - \partial_i v_j\right) \end This decomposition is independent of coordinate system, and so has physical significance. Then the velocity field may be approximated as \mathbf(\mathbf + \mathbf, t) \approx \mathbf(\mathbf, t) + \mathbf(\mathbf, t)(\mathbf) + \mathbf(\mathbf, t)(\mathbf), that is, \begin v_i(\mathbf + \mathbf, t) &= v_i(\mathbf, t) + \sum_j E_(\mathbf, t) r_j + \sum_j R_(\mathbf, t) r_j \\ &= v_i(\mathbf, t) + \frac\sum_j \left(\partial_j v_i(\mathbf, t) + \partial_i v_j(\mathbf, t)\right)r_j + \frac\sum_j \left(\partial_j v_i(\mathbf, t) - \partial_i v_j(\mathbf, t)\right)r_j \end The antisymmetric term represents a rigid-like rotation of the fluid about the point . Its angular velocity is \vec = \frac \nabla \times \mathbf = \frac \begin \partial_2 v_3 - \partial_3 v_2 \\ \partial_3 v_1 - \partial_1 v_3 \\ \partial_1 v_2 - \partial_2 v_1 \end. The product is called the ''
rotational Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
curl'' of the vector field. A rigid rotation does not change the relative positions of the fluid elements, so the antisymmetric term of the velocity gradient does not contribute to the rate of change of the deformation. The actual strain rate is therefore described by the symmetric term, which is the strain rate tensor.


Shear rate and compression rate

The symmetric term (the rate-of-strain tensor) can be broken down further as the sum of a scalar times the unit tensor, that represents a gradual isotropic expansion or contraction; and a
traceless In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...
symmetric tensor which represents a gradual shearing deformation, with no change in volume: \mathbf(\mathbf, t)(\mathbf) = \mathbf(\mathbf, t)(\mathbf) + \mathbf(\mathbf, t)(\mathbf). That is, E_ = \underbrace_ + \underbrace_, Here is the unit tensor, such that is 1 if and 0 if . This decomposition is independent of the choice of coordinate system, and is therefore physically significant. The trace of the expansion rate tensor is the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of the velocity field: \nabla \cdot \mathbf = \partial_1 v_1 + \partial_2 v_2 + \partial_3 v_3; which is the rate at which the volume of a fixed amount of fluid increases at that point. The shear rate tensor is represented by a symmetric 3 × 3 matrix, and describes a flow that combines compression and expansion flows along three orthogonal axes, such that there is no change in volume. This type of flow occurs, for example, when a
rubber Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, and ...
strip is stretched by pulling at the ends, or when
honey Honey is a sweet and viscous substance made by several bees, the best-known of which are honey bees. Honey is made and stored to nourish bee colonies. Bees produce honey by gathering and then refining the sugary secretions of plants (primar ...
falls from a spoon as a smooth unbroken stream. For a two-dimensional flow, the divergence of has only two terms and quantifies the change in area rather than volume. The factor 1/3 in the expansion rate term should be replaced by in that case.


Examples

The study of velocity gradients is useful in analysing path dependent materials and in the subsequent study of stresses and strains; e.g.,
Plastic deformation In engineering, deformation refers to the change in size or shape of an object. ''Displacements'' are the ''absolute'' change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain ...
of
metals A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
. The near-wall velocity gradient of the unburned reactants flowing from a tube is a key parameter for characterising flame stability. The velocity gradient of a
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...
can define conditions for the solutions to fundamental equations in magnetohydrodynamics.


Fluid in a pipe

Consider the velocity field of a fluid flowing through a
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-circula ...
. The layer of fluid in contact with the pipe tends to be at rest with respect to the pipe. This is called the
no slip condition In fluid dynamics, the no-slip condition for viscous fluids assumes that at a solid boundary, the fluid will have zero velocity relative to the boundary. The fluid velocity at all fluid–solid boundaries is equal to that of the solid boundary. ...
. If the velocity difference between fluid layers at the centre of the pipe and at the sides of the pipe is sufficiently small, then the fluid flow is observed in the form of continuous layers. This type of flow is called
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 ...
. The
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 ...
difference between adjacent layers can be measured in terms of a velocity gradient, given by \Delta u / \Delta y. Where \Delta u is the difference in flow velocity between the two layers and \Delta y is the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between the layers.


See also

*
Stress tensor (disambiguation) Stress tensor may refer to: * Cauchy stress tensor, in classical physics * Stress deviator tensor, in classical physics * Piola–Kirchhoff stress tensor, in continuum mechanics * Viscous stress tensor, in continuum mechanics * Stress–energy ten ...
* , the spatial and material velocity gradient from continuum mechanics


References

{{reflist Continuum mechanics Rates Tensor physical quantities