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A two-dimensional flow that, at the highlighted point, has only a strain rate component, with no mean velocity or rotational component.

In continuum mechanics, the strain rate 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 defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the gradient (derivative with respect to position) of the flow velocity.

The strain rate tensor is a purely kinematic concept that describes the macroscopic 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, liquid or gas.

On the other hand, for any fluid except superfluids, 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 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.

Definition

Consider a material body, solid or fluid, that is flowing and/or moving in space. Let $v$ be the velocity field within the body; that is, a smooth function from $\mathbb {R} ^{3}\times \mathbb {R}$ such that $v(p,t)$ is the macroscopic velocity of the material that is passing through the point $p$ at time $t$.

The velocity $v(p+r,t)$ at a point displaced from $p$ by a small vector $r$ can be written as a Taylor series:

$v(p+r,t)=v(p,t)+(\nabla v)(p,t)(r)+{\text{(higher order terms)}},$

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In an arbitrary reference frame, $\nabla v$ is the Jacobian matrix of the field, namely the 3×3 matrix

$\nabla v=J={\begin{bmatrix}\displaystyle {\partial _{1}v_{1}}&\displaystyle {\partial _{2}v_{1}}&\displaystyle {\partial _{3}v_{1}}\\\displaystyle {\partial _{1}v_{2}}&\displaystyle {\partial _{2}v_{2}}&\displaystyle {\partial _{3}v_{2}}\\\displaystyle {\partial _{1}v_{3}}&\displaystyle {\partial _{2}v_{3}}&\displaystyle {\partial _{3}v_{3}}\end{bmatrix}}.$

where $v_{i}$ is the component of $v$ parallel to axis $i$ and $\partial _{j}f$ denotes the partial derivative of a function $f$ with respect to the space coordinate $x_{j}$. Note that $J$ is a function of $p$ and $t$.

In this coordinate system, the Taylor approximation for the velocity near $p$ is

$v_{i}(p+r,t)=v_{i}(p,t)+\sum _{j}J_{ij}(p,t)r_{j}=v_{i}(p,t)+\sum _{j}\partial _{j}v_{i}(p,t)r_{j};$

or simply $v(p+r,t)=v(p,t)+J(p,t)r$, if $v$ and $r$ are viewed as 3×1 matrices.

Symmetric and antisymmetric parts

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for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed

After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant

Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013

6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!

BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill

Any matrix can be decomposed into the sum of a symmetric matrix and an antisymmetric matrix. Applying this to the Jacobian matrix $J=\nabla v$ with symmetric and antisymmetric components $E$ and $R$ respectively:

$E={\frac {1}{2}}\left(J+J^{\mathsf {T}}\right)\quad \quad \quad R={\frac {1}{2}}\left(J-J^{\mathsf {T}}\right)$

That is,

$E_{ij}={\frac {1}{2}}(\partial _{j}v_{i}+\partial _{i}v_{j})\quad \quad \quad R_{ij}={\frac {1}{2}}(\partial _{j}v_{i}-\partial _{i}v_{j})$

This decomposition is independent of coordinate system, and so has physical significance. Then the velocity field may be approximated as

$v(p+r,t)\approx v(p,t)+E(p,t)(r)+R(p,t)(r),$

that is,

${\begin{array}{lcl}v_{i}(p+r,t)&=&v_{i}(p,t)+\sum _{j}E_{ij}(p,t)r_{j}+\sum _{j}R_{ij}(p,t)r_{j}\\~&=&v_{i}(p,t)+{\frac {1}{2}}\sum _{j}\left(\partial _{j}v_{i}(p,t)+\partial _{i}v_{j}(p,t)\right)r_{j}+{\frac {1}{2}}\sum _{j}\left(\partial _{j}v_{i}(p,t)-\partial _{i}v_{j}(p,t)\right)r_{j}\end{array}}$

The antisymmetric term $R$ represents a rigid-like rotation of the fluid about the point $p$. Its angular velocity $\omega$ is

$\omega ={\frac {1}{2}}\nabla \times v={\frac {1}{2}}{\begin{bmatrix}\partial _{2}v_{3}-\partial _{3}v_{2}\\\partial _{3}v_{1}-\partial _{1}v_{3}\\\partial _{1}v_{2}-\partial _{2}v_{1}\end{bmatrix}}.$

The product $\nabla \times v$ is called the rotational curl of the vector field. A rigid rotation does not change the relative positions of the fluid elements, so the antisymmetric term $R$ 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 $E$ term, which is the strain rate tensor.

Shear rate and compression rate

People are nonetheless rushing to buy properties in Singapore. A new residential condominium mission in Bedok had a long queue three days before its official launch and bought more than 350 units on the very first day. Nonetheless, there is a ‘development' that had emerged in 2011 that the majority have possibly overlooked.

In August, more than a hundred individuals across 16 states reportedly had been sickened with salmonella after consuming mangos In September, the Meals and Drug Administration detained mango imports from a Mexican packing house after the fruits had been linked to the sickenings Save Cash On Your Online Purchasing Make saving money when purchasing on-line a breeze with Bodge's favourite app, Invisible Hands. This helpful program does the price-checking analysis for you, and will warn you when a unique site has a better value - or let you realize that you have found the most effective deal. Watch Posted by Edison Foo September 29, 2013 HEARTH SALE!!! From only $9xx,000! We hope this listing will help launch your search for new properties in Singapore. It's unlawful to purchase ice cream after 6 pm, in Newark.

for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed

After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant

Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013

6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!

BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill The symmetric term $E$ of velocity gradient (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:

$E(p,t)(r)=D(p,t)(r)+S(p,t)(r).$

That is,

$E_{ij}=\underbrace {{\frac {1}{3}}(\sum _{k}\partial _{k}v_{k})\delta _{ij}} _{{\text{rate-of-expansion tensor}}D_{ij}}+\underbrace {\left({\frac {1}{2}}\left(\partial _{i}v_{j}+\partial _{j}v_{i}\right)-{\frac {1}{3}}(\sum _{k}\partial _{k}v_{k})\delta _{ij}\right)} _{{\text{rate-of-shear tensor}}S_{ij}},$

Here $\delta$ is the unit tensor, such that $\delta _{ij}$ is 1 if $i=j$ and 0 if $i\neq j$. This decomposition is independent of the choice of coordinate system, and is therefore physically significant.

The expansion rate tensor is 1/3 of the divergence of the velocity field:

$\nabla \cdot v=\partial _{1}v_{1}+\partial _{2}v_{2}+\partial _{3}v_{3};$

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 strip is stretched by pulling at the ends, or when honey falls from a spoon as a smooth unbroken stream.

For a two-dimensional flow, the divergence of $v$ 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 1/2 in that case.

See also

• Stress tensor

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.