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In
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...
, the strain-rate tensor or rate-of-strain tensor is a
physical quantity A physical quantity (or simply quantity) is a property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ''nu ...
that describes the rate of change of the strain (i.e., the relative deformation) 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 is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of the strain tensor 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. If this matrix is square, that is, if the number of variables equals the number of component ...
(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 plasma (physics), plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
it also can be described as the velocity gradient, a measure of how the
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
of a fluid changes between different points within the fluid. Though the term can refer to a velocity profile (variation in velocity across 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 f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
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 and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...
. The concept has implications in a variety of areas of
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
and
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
, including
magnetohydrodynamics In physics and engineering, magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single Continuum ...
, mining and water treatment. The strain rate tensor is a purely
kinematic In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with s ...
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 phenome ...
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 a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
,
liquid Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...
or gas. On the other hand, for any
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
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 vortex, vortices that continue to rotate indefinitely. Superfluidity occurs ...
s, any gradual change in its deformation (i.e. a non-zero strain rate tensor) gives rise to viscous forces 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. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
between adjacent fluid elements, that tend to oppose that change. At any point in the fluid, these stresses can be described by a viscous stress tensor 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 Viscosity, viscous and Elasticity (physics), elastic characteristics when undergoing deformation (engineering), deformation. Viscous mate ...
.


Dimensional analysis

By performing dimensional analysis, 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 associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
which can be expressed as the
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
\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 ...
\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 () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Enlightenment that followed. His book (''Mathe ...
proposed that
shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
is directly proportional to the velocity gradient: \tau = \mu\frac . The constant of proportionality, \mu, is called the
dynamic viscosity Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...
.


Formal definition

Consider a material body, solid or fluid, that is flowing and/or moving in space. Let be the velocity field within the body; that is, a smooth 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 phenome ...
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 ser ...
: \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 mapping V \to W between two vector spaces that p ...
that takes a displacement vector to the corresponding change in the velocity. In an arbitrary reference frame, 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. If this matrix is square, that is, if the number of variables equals the number of component ...
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 (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names ...
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). P ...
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 ...
and an antisymmetric matrix. 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
vorticity In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point an ...
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 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 rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
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. Types of polyisoprene ...
strip is stretched by pulling at the ends, or when
honey Honey is a sweet and viscous substance made by several species of 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 pl ...
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 of
metals A metal () is a material that, when polished or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. These properties are all associated with having electrons available at the Fermi level, as against no ...
. 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 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. 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. 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 Laminar flow () is the property of fluid particles in fluid dynamics to follow 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 m ...
. 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, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
between the layers.


See also

* Stress tensor (disambiguation) * , the spatial and material velocity gradient from continuum mechanics


References

{{reflist Continuum mechanics Rates Tensor physical quantities