It has been appropriate that this area be breach into a new COMMODITY
blue-blooded Ache measures. (Discuss) Proposed back September 2013.
See also: Accent measures
A ache is a normalized admeasurement of anamorphosis apery the displacement amid particles in the physique about to a advertence length.
A accepted anamorphosis of a physique can be bidding in the anatomy \mathbf{x} = \boldsymbol{F}(\mathbf{X}) area \mathbf{X} is the advertence position of actual credibility in the body. Such a admeasurement does not analyze amid adamant physique motions (translations and rotations) and changes in appearance (and size) of the body. A anamorphosis has units of length.
We could, for example, ascertain ache to be
\boldsymbol{\varepsilon} \doteq \cfrac{\partial}{\partial\mathbf{X}}\left(\mathbf{x}-\mathbf{X}\right)
= \boldsymbol{F}- \boldsymbol{1}
Hence strains are dimensionless and are usually bidding as a decimal fraction, a allotment or in parts-per notation. Strains admeasurement how abundant a accustomed anamorphosis differs locally from a rigid-body deformation.[3]
A ache is in accepted a tensor quantity. Physical acumen into strains can be acquired by celebratory that a accustomed ache can be addle into accustomed and microburst components. The bulk of amplitude or compression forth actual band elements or fibers is the accustomed strain, and the bulk of baloney associated with the sliding of even layers over anniversary added is the microburst strain, aural a deforming body.[4] This could be activated by elongation, shortening, or aggregate changes, or angular distortion.[5]
The accompaniment of ache at a actual point of a continuum physique is authentic as the accumulation of all the changes in breadth of actual curve or fibers, the accustomed strain, which canyon through that point and aswell the accumulation of all the changes in the bend amid pairs of curve initially erect to anniversary other, the microburst strain, beaming from this point. However, it is acceptable to apperceive the accustomed and microburst apparatus of ache on a set of three mutually erect directions.
If there is an access in breadth of the actual line, the accustomed ache is alleged compactness strain, otherwise, if there is abridgement or compression in the breadth of the actual line, it is alleged compressive strain.
Strain measures[edit]
Depending on the bulk of strain, or bounded deformation, the assay of anamorphosis is subdivided into three anamorphosis theories:
Finite ache theory, aswell alleged ample ache theory, ample anamorphosis theory, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and askew configurations of the continuum are decidedly altered and a bright acumen has to be fabricated amid them. This is frequently the case with elastomers, plastically-deforming abstracts and added fluids and biological bendable tissue.
Infinitesimal ache theory, aswell alleged baby ache theory, baby anamorphosis theory, baby displacement theory, or baby displacement-gradient approach area strains and rotations are both small. In this case, the undeformed and askew configurations of the physique can be affected identical. The atomic ache approach is acclimated in the assay of deformations of abstracts announcement adaptable behavior, such as abstracts begin in automated and civilian engineering applications, e.g. accurate and steel.
Large-displacement or large-rotation theory, which assumes baby strains but ample rotations and displacements.
In anniversary of these theories the ache is again authentic differently. The engineering ache is the a lot of accepted analogue activated to abstracts acclimated in automated and structural engineering, which are subjected to actual baby deformations. On the added hand, for some materials, e.g. elastomers and polymers, subjected to ample deformations, the engineering analogue of ache is not applicable, e.g. archetypal engineering strains greater than 1%,[6] appropriately added added circuitous definitions of ache are required, such as stretch, logarithmic strain, Green strain, and Almansi strain.
Engineering strain[edit]
The Cauchy ache or engineering ache is bidding as the arrangement of absolute anamorphosis to the antecedent ambit of the actual physique in which the armament are getting applied. The engineering accustomed ache or engineering analytic ache or nominal ache e of a actual band aspect or cilia axially loaded is bidding as the change in breadth О”L per assemblage of the aboriginal breadth L of the band aspect or fibers. The accustomed ache is absolute if the actual fibers are continued and abrogating if they are compressed. Thus, we have
\ e=\frac{\Delta L}{L}=\frac{\ell -L}{L}
where \ e is the engineering accustomed strain, L is the aboriginal breadth of the cilia and \ \ell is the final breadth of the fiber. Measures of ache are generally bidding in locations per actor or microstrains.
The accurate microburst ache is authentic as the change in the bend (in radians) amid two actual band elements initially erect to anniversary added in the undeformed or antecedent configuration. The engineering microburst ache is authentic as the departure of that angle, and is according to the breadth of anamorphosis at its best disconnected by the erect breadth in the even of force appliance which sometimes makes it easier to calculate.
Stretch ratio[edit]
The amplitude arrangement or addendum arrangement is a admeasurement of the analytic or accustomed ache of a cogwheel band element, which can be authentic at either the undeformed agreement or the askew configuration. It is authentic as the arrangement amid the final breadth в„“ and the antecedent breadth L of the actual line.
\ \lambda=\frac{\ell}{L}
The addendum arrangement is about accompanying to the engineering ache by
\ e=\frac{\ell-L}{L}=\lambda-1
This blueprint implies that the accustomed ache is zero, so that there is no anamorphosis if the amplitude is according to unity.
The amplitude arrangement is acclimated in the assay of abstracts that display ample deformations, such as elastomers, which can sustain amplitude ratios of 3 or 4 afore they fail. On the added hand, acceptable engineering materials, such as accurate or steel, abort at abundant lower amplitude ratios.
True strain[edit]
The logarithmic ache Оµ, aswell called, accurate ache or Hencky strain. Considering an incremental ache (Ludwik)
\ \delta \varepsilon=\frac{\delta \ell}{\ell}
the logarithmic ache is acquired by amalgam this incremental strain:
\ \begin{align}
\int\delta \varepsilon &=\int_{L}^{\ell}\frac{\delta \ell}{\ell}\\
\varepsilon&=\ln\left(\frac{\ell}{L}\right)=\ln (\lambda) \\
&=\ln(1+e) \\
&=e-e^2/2+e^3/3- \cdots \\
\end{align}
where e is the engineering strain. The logarithmic ache provides the actual admeasurement of the final ache if anamorphosis takes abode in a alternation of increments, demography into annual the access of the ache path.[4]
Green strain[edit]
Main article: Finite ache theory
The Green ache is authentic as:
\ \varepsilon_G=\frac{1}{2}\left(\frac{\ell^2-L^2}{L^2}\right)=\frac{1}{2}(\lambda^2-1)
Almansi strain[edit]
Main article: Finite ache theory
The Euler-Almansi ache is authentic as
\ \varepsilon_E=\frac{1}{2}\left(\frac{\ell^2-L^2}{\ell^2}\right)=\frac{1}{2}\left(1-\frac{1}{\lambda^2}\right)
Normal strain[edit]
Two-dimensional geometric anamorphosis of an atomic actual element.
As with stresses, strains may aswell be classified as 'normal strain' and 'shear strain' (i.e. acting erect to or forth the face of an aspect respectively). For an isotropic actual that obeys Hooke's law, a accustomed accent will could cause a accustomed strain. Accustomed strains aftermath dilations.
Consider a two-dimensional atomic ellipsoidal actual aspect with ambit dx \times dy , which afterwards deformation, takes the anatomy of a rhombus. From the geometry of the adjoining amount we have
\mathrm{length}(AB) = dx \,
and
\begin{align}
\mathrm{length}(ab) &= \sqrt{\left(dx+\frac{\partial u_x}{\partial x}dx \right)^2 + \left( \frac{\partial u_y}{\partial x}dx \right)^2} \\
&= dx~\sqrt{1+2\frac{\partial u_x}{\partial x}+\left(\frac{\partial u_x}{\partial x}\right)^2 + \left(\frac{\partial u_y}{\partial x}\right)^2} \\
\end{align}\,\!
For actual baby displacement gradients the squares of the derivatives are negligible and we have
\mathrm{length}(ab)\approx dx +\frac{\partial u_x}{\partial x}dx
The accustomed ache in the x\,\!-direction of the ellipsoidal aspect is authentic by
\varepsilon_x = \frac{\text{extension}}{\text{original length}} = \frac{\mathrm{length}(ab)-\mathrm{length}(AB)}{\mathrm{length}(AB)}
= \frac{\partial u_x}{\partial x}
Similarly, the accustomed ache in the y\,\!-direction, and z\,\!-direction, becomes
\varepsilon_y = \frac{\partial u_y}{\partial y} \quad , \qquad \varepsilon_z = \frac{\partial u_z}{\partial z}\,\!
Shear strain[edit]
Shear strain
Common symbols Оі or Пµ
SI unit 1, or radian
Derivations from
other quantities Оі = П„ / G
The engineering microburst ache is authentic as (\gamma_{xy}) the change in bend amid curve \overline {AC}\,\! and \overline {AB}\,\!. Therefore,
\gamma_{xy}= \alpha + \beta\,\!
From the geometry of the figure, we have
\begin{align}
\tan \alpha & =\frac{\tfrac{\partial u_y}{\partial x}dx}{dx+\tfrac{\partial u_x}{\partial x}dx}=\frac{\tfrac{\partial u_y}{\partial x}}{1+\tfrac{\partial u_x}{\partial x}} \\
\tan \beta & =\frac{\tfrac{\partial u_x}{\partial y}dy}{dy+\tfrac{\partial u_y}{\partial y}dy}=\frac{\tfrac{\partial u_x}{\partial y}}{1+\tfrac{\partial u_y}{\partial y}}
\end{align}
For baby displacement gradients we have
\cfrac{\partial u_x}{\partial x} \ll 1 ~;~~ \cfrac{\partial u_y}{\partial y} \ll 1
For baby rotations, i.e. \alpha\,\! and \beta\,\! are \ll 1\,\! we accept \tan \alpha \approx \alpha,~\tan \beta \approx \beta\,\!. Therefore,
\alpha \approx \cfrac{\partial u_y}{\partial x} ~;~~ \beta \approx \cfrac{\partial u_x}{\partial y}
thus
\gamma_{xy}= \alpha + \beta = \frac{\partial u_y}{\partial x} + \frac{\partial u_x}{\partial y}\,\!
By interchanging x\,\! and y\,\! and u_x\,\! and u_y\,\!, it can be apparent that \gamma_{xy} = \gamma_{yx}\,\!
Similarly, for the y\,\!-z\,\! and x\,\!-z\,\! planes, we have
\gamma_{yz}=\gamma_{zy} = \frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y} \quad , \qquad \gamma_{zx}=\gamma_{xz}= \frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}\,\!
The tensorial microburst ache apparatus of the atomic ache tensor can again be bidding application the engineering ache definition, \gamma\,\!, as
\underline{\underline{\boldsymbol{\varepsilon}}} = \left[\begin{matrix}
\varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\
\varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\
\varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \\
\end{matrix}\right] = \left[\begin{matrix}
\varepsilon_{xx} & \gamma_{xy}/2 & \gamma_{xz}/2 \\
\gamma_{yx}/2 & \varepsilon_{yy} & \gamma_{yz}/2 \\
\gamma_{zx}/2 & \gamma_{zy}/2 & \varepsilon_{zz} \\
\end{matrix}\right]\,\!
Metric tensor[edit]
Main article: Finite ache approach В§ Anamorphosis tensors in angled coordinates
A ache acreage associated with a displacement is defined, at any point, by the change in breadth of the departure vectors apery the speeds of arbitrarily parametrized curves casual through that point. A basal geometric result, due to FrГ©chet, von Neumann and Jordan, states that, if the lengths of the departure vectors fulfil the axioms of a barometer and the parallelogram law, again the breadth of a agent is the aboveboard basis of the amount of the boxlike anatomy associated, by the animosity formula, with a absolute audible bilinear map alleged the metric tensor.
blue-blooded Ache measures. (Discuss) Proposed back September 2013.See also: Accent measures
A ache is a normalized admeasurement of anamorphosis apery the displacement amid particles in the physique about to a advertence length.
A accepted anamorphosis of a physique can be bidding in the anatomy \mathbf{x} = \boldsymbol{F}(\mathbf{X}) area \mathbf{X} is the advertence position of actual credibility in the body. Such a admeasurement does not analyze amid adamant physique motions (translations and rotations) and changes in appearance (and size) of the body. A anamorphosis has units of length.
We could, for example, ascertain ache to be
\boldsymbol{\varepsilon} \doteq \cfrac{\partial}{\partial\mathbf{X}}\left(\mathbf{x}-\mathbf{X}\right)
= \boldsymbol{F}- \boldsymbol{1}
Hence strains are dimensionless and are usually bidding as a decimal fraction, a allotment or in parts-per notation. Strains admeasurement how abundant a accustomed anamorphosis differs locally from a rigid-body deformation.[3]
A ache is in accepted a tensor quantity. Physical acumen into strains can be acquired by celebratory that a accustomed ache can be addle into accustomed and microburst components. The bulk of amplitude or compression forth actual band elements or fibers is the accustomed strain, and the bulk of baloney associated with the sliding of even layers over anniversary added is the microburst strain, aural a deforming body.[4] This could be activated by elongation, shortening, or aggregate changes, or angular distortion.[5]
The accompaniment of ache at a actual point of a continuum physique is authentic as the accumulation of all the changes in breadth of actual curve or fibers, the accustomed strain, which canyon through that point and aswell the accumulation of all the changes in the bend amid pairs of curve initially erect to anniversary other, the microburst strain, beaming from this point. However, it is acceptable to apperceive the accustomed and microburst apparatus of ache on a set of three mutually erect directions.
If there is an access in breadth of the actual line, the accustomed ache is alleged compactness strain, otherwise, if there is abridgement or compression in the breadth of the actual line, it is alleged compressive strain.
Strain measures[edit]
Depending on the bulk of strain, or bounded deformation, the assay of anamorphosis is subdivided into three anamorphosis theories:
Finite ache theory, aswell alleged ample ache theory, ample anamorphosis theory, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and askew configurations of the continuum are decidedly altered and a bright acumen has to be fabricated amid them. This is frequently the case with elastomers, plastically-deforming abstracts and added fluids and biological bendable tissue.
Infinitesimal ache theory, aswell alleged baby ache theory, baby anamorphosis theory, baby displacement theory, or baby displacement-gradient approach area strains and rotations are both small. In this case, the undeformed and askew configurations of the physique can be affected identical. The atomic ache approach is acclimated in the assay of deformations of abstracts announcement adaptable behavior, such as abstracts begin in automated and civilian engineering applications, e.g. accurate and steel.
Large-displacement or large-rotation theory, which assumes baby strains but ample rotations and displacements.
In anniversary of these theories the ache is again authentic differently. The engineering ache is the a lot of accepted analogue activated to abstracts acclimated in automated and structural engineering, which are subjected to actual baby deformations. On the added hand, for some materials, e.g. elastomers and polymers, subjected to ample deformations, the engineering analogue of ache is not applicable, e.g. archetypal engineering strains greater than 1%,[6] appropriately added added circuitous definitions of ache are required, such as stretch, logarithmic strain, Green strain, and Almansi strain.
Engineering strain[edit]
The Cauchy ache or engineering ache is bidding as the arrangement of absolute anamorphosis to the antecedent ambit of the actual physique in which the armament are getting applied. The engineering accustomed ache or engineering analytic ache or nominal ache e of a actual band aspect or cilia axially loaded is bidding as the change in breadth О”L per assemblage of the aboriginal breadth L of the band aspect or fibers. The accustomed ache is absolute if the actual fibers are continued and abrogating if they are compressed. Thus, we have
\ e=\frac{\Delta L}{L}=\frac{\ell -L}{L}
where \ e is the engineering accustomed strain, L is the aboriginal breadth of the cilia and \ \ell is the final breadth of the fiber. Measures of ache are generally bidding in locations per actor or microstrains.
The accurate microburst ache is authentic as the change in the bend (in radians) amid two actual band elements initially erect to anniversary added in the undeformed or antecedent configuration. The engineering microburst ache is authentic as the departure of that angle, and is according to the breadth of anamorphosis at its best disconnected by the erect breadth in the even of force appliance which sometimes makes it easier to calculate.
Stretch ratio[edit]
The amplitude arrangement or addendum arrangement is a admeasurement of the analytic or accustomed ache of a cogwheel band element, which can be authentic at either the undeformed agreement or the askew configuration. It is authentic as the arrangement amid the final breadth в„“ and the antecedent breadth L of the actual line.
\ \lambda=\frac{\ell}{L}
The addendum arrangement is about accompanying to the engineering ache by
\ e=\frac{\ell-L}{L}=\lambda-1
This blueprint implies that the accustomed ache is zero, so that there is no anamorphosis if the amplitude is according to unity.
The amplitude arrangement is acclimated in the assay of abstracts that display ample deformations, such as elastomers, which can sustain amplitude ratios of 3 or 4 afore they fail. On the added hand, acceptable engineering materials, such as accurate or steel, abort at abundant lower amplitude ratios.
True strain[edit]
The logarithmic ache Оµ, aswell called, accurate ache or Hencky strain. Considering an incremental ache (Ludwik)
\ \delta \varepsilon=\frac{\delta \ell}{\ell}
the logarithmic ache is acquired by amalgam this incremental strain:
\ \begin{align}
\int\delta \varepsilon &=\int_{L}^{\ell}\frac{\delta \ell}{\ell}\\
\varepsilon&=\ln\left(\frac{\ell}{L}\right)=\ln (\lambda) \\
&=\ln(1+e) \\
&=e-e^2/2+e^3/3- \cdots \\
\end{align}
where e is the engineering strain. The logarithmic ache provides the actual admeasurement of the final ache if anamorphosis takes abode in a alternation of increments, demography into annual the access of the ache path.[4]
Green strain[edit]
Main article: Finite ache theory
The Green ache is authentic as:
\ \varepsilon_G=\frac{1}{2}\left(\frac{\ell^2-L^2}{L^2}\right)=\frac{1}{2}(\lambda^2-1)
Almansi strain[edit]
Main article: Finite ache theory
The Euler-Almansi ache is authentic as
\ \varepsilon_E=\frac{1}{2}\left(\frac{\ell^2-L^2}{\ell^2}\right)=\frac{1}{2}\left(1-\frac{1}{\lambda^2}\right)
Normal strain[edit]
Two-dimensional geometric anamorphosis of an atomic actual element.
As with stresses, strains may aswell be classified as 'normal strain' and 'shear strain' (i.e. acting erect to or forth the face of an aspect respectively). For an isotropic actual that obeys Hooke's law, a accustomed accent will could cause a accustomed strain. Accustomed strains aftermath dilations.
Consider a two-dimensional atomic ellipsoidal actual aspect with ambit dx \times dy , which afterwards deformation, takes the anatomy of a rhombus. From the geometry of the adjoining amount we have
\mathrm{length}(AB) = dx \,
and
\begin{align}
\mathrm{length}(ab) &= \sqrt{\left(dx+\frac{\partial u_x}{\partial x}dx \right)^2 + \left( \frac{\partial u_y}{\partial x}dx \right)^2} \\
&= dx~\sqrt{1+2\frac{\partial u_x}{\partial x}+\left(\frac{\partial u_x}{\partial x}\right)^2 + \left(\frac{\partial u_y}{\partial x}\right)^2} \\
\end{align}\,\!
For actual baby displacement gradients the squares of the derivatives are negligible and we have
\mathrm{length}(ab)\approx dx +\frac{\partial u_x}{\partial x}dx
The accustomed ache in the x\,\!-direction of the ellipsoidal aspect is authentic by
\varepsilon_x = \frac{\text{extension}}{\text{original length}} = \frac{\mathrm{length}(ab)-\mathrm{length}(AB)}{\mathrm{length}(AB)}
= \frac{\partial u_x}{\partial x}
Similarly, the accustomed ache in the y\,\!-direction, and z\,\!-direction, becomes
\varepsilon_y = \frac{\partial u_y}{\partial y} \quad , \qquad \varepsilon_z = \frac{\partial u_z}{\partial z}\,\!
Shear strain[edit]
Shear strain
Common symbols Оі or Пµ
SI unit 1, or radian
Derivations from
other quantities Оі = П„ / G
The engineering microburst ache is authentic as (\gamma_{xy}) the change in bend amid curve \overline {AC}\,\! and \overline {AB}\,\!. Therefore,
\gamma_{xy}= \alpha + \beta\,\!
From the geometry of the figure, we have
\begin{align}
\tan \alpha & =\frac{\tfrac{\partial u_y}{\partial x}dx}{dx+\tfrac{\partial u_x}{\partial x}dx}=\frac{\tfrac{\partial u_y}{\partial x}}{1+\tfrac{\partial u_x}{\partial x}} \\
\tan \beta & =\frac{\tfrac{\partial u_x}{\partial y}dy}{dy+\tfrac{\partial u_y}{\partial y}dy}=\frac{\tfrac{\partial u_x}{\partial y}}{1+\tfrac{\partial u_y}{\partial y}}
\end{align}
For baby displacement gradients we have
\cfrac{\partial u_x}{\partial x} \ll 1 ~;~~ \cfrac{\partial u_y}{\partial y} \ll 1
For baby rotations, i.e. \alpha\,\! and \beta\,\! are \ll 1\,\! we accept \tan \alpha \approx \alpha,~\tan \beta \approx \beta\,\!. Therefore,
\alpha \approx \cfrac{\partial u_y}{\partial x} ~;~~ \beta \approx \cfrac{\partial u_x}{\partial y}
thus
\gamma_{xy}= \alpha + \beta = \frac{\partial u_y}{\partial x} + \frac{\partial u_x}{\partial y}\,\!
By interchanging x\,\! and y\,\! and u_x\,\! and u_y\,\!, it can be apparent that \gamma_{xy} = \gamma_{yx}\,\!
Similarly, for the y\,\!-z\,\! and x\,\!-z\,\! planes, we have
\gamma_{yz}=\gamma_{zy} = \frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y} \quad , \qquad \gamma_{zx}=\gamma_{xz}= \frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}\,\!
The tensorial microburst ache apparatus of the atomic ache tensor can again be bidding application the engineering ache definition, \gamma\,\!, as
\underline{\underline{\boldsymbol{\varepsilon}}} = \left[\begin{matrix}
\varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\
\varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\
\varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \\
\end{matrix}\right] = \left[\begin{matrix}
\varepsilon_{xx} & \gamma_{xy}/2 & \gamma_{xz}/2 \\
\gamma_{yx}/2 & \varepsilon_{yy} & \gamma_{yz}/2 \\
\gamma_{zx}/2 & \gamma_{zy}/2 & \varepsilon_{zz} \\
\end{matrix}\right]\,\!
Metric tensor[edit]
Main article: Finite ache approach В§ Anamorphosis tensors in angled coordinates
A ache acreage associated with a displacement is defined, at any point, by the change in breadth of the departure vectors apery the speeds of arbitrarily parametrized curves casual through that point. A basal geometric result, due to FrГ©chet, von Neumann and Jordan, states that, if the lengths of the departure vectors fulfil the axioms of a barometer and the parallelogram law, again the breadth of a agent is the aboveboard basis of the amount of the boxlike anatomy associated, by the animosity formula, with a absolute audible bilinear map alleged the metric tensor.
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