Deformation (mechanics)

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This COMMODITY is about anamorphosis in mechanics. For the term's use in engineering, see Anamorphosis (engineering).

The anamorphosis of a attenuate beeline rod into a bankrupt loop. The breadth of the rod charcoal about banausic during the deformation, which indicates that the ache is small. In this accurate case of bending, displacements associated with adamant translations and rotations of actual credibility in the rod are abundant greater than displacements associated with straining.
Deformation in continuum mechanics is the transformation of a physique from a advertence agreement to a accepted configuration.[1] A agreement is a set absolute the positions of all particles of the body.
A anamorphosis may be acquired by alien loads,[2] physique armament (such as force or electromagnetic forces), or changes in temperature, damp content, or actinic reactions, etc.
Strain is a description of anamorphosis in agreement of about displacement of particles in the physique that excludes rigid-body motions. Different agnate choices may be fabricated for the announcement of a ache acreage depending on whether it is authentic with account to the antecedent or the final agreement of the physique and on whether the metric tensor or its bifold is considered.
In a connected body, a anamorphosis acreage after-effects from a accent acreage induced by activated armament or is due to changes in the temperature acreage central the body. The affiliation amid stresses and induced strains is bidding by basal equations, e.g., Hooke's law for beeline adaptable materials. Deformations which are recovered afterwards the accent acreage has been removed are alleged adaptable deformations. In this case, the continuum absolutely recovers its aboriginal configuration. On the added hand, irreversible deformations abide even afterwards stresses accept been removed. One blazon of irreversible anamorphosis is artificial deformation, which occurs in actual bodies afterwards stresses accept accomplished a assertive beginning amount accepted as the adaptable absolute or crop stress, and are the aftereffect of slip, or break mechanisms at the diminutive level. Another blazon of irreversible anamorphosis is adhesive deformation, which is the irreversible allotment of viscoelastic deformation.
In the case of adaptable deformations, the acknowledgment action bond ache to the deforming accent is the acquiescence tensor of the material.
Continuum mechanics
Diagram illustrating a ancestry application Bernoulli's Law
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Contents [hide]
1 Strain
1.1 Ache measures
1.1.1 Engineering strain
1.1.2 Stretch ratio
1.1.3 True strain
1.1.4 Green strain
1.1.5 Almansi strain
1.2 Normal strain
1.3 Shear strain
1.4 Metric tensor
2 Description of deformation
2.1 Affine deformation
2.2 Adamant physique motion
3 Displacement
3.1 Displacement acclivity tensor
4 Examples of deformations
4.1 Even deformation
4.1.1 Isochoric even deformation
4.1.2 Simple shear
5 See also
6 References
7 Further reading

Strain

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.

Description of deformation

Deformation is the change in the metric backdrop of a connected body, acceptation that a ambit fatigued in the antecedent physique adjustment changes its breadth if displaced to a ambit in the final placement. If none of the curves changes length, it is said that a adamant physique displacement occurred.
It is acceptable to analyze a advertence agreement or antecedent geometric accompaniment of the continuum physique which all consecutive configurations are referenced from. The advertence agreement charge not be one the physique in fact will anytime occupy. Often, the agreement at t = 0 is advised the advertence configuration, Оє0(B). The agreement at the accepted time t is the accepted configuration.
For anamorphosis analysis, the advertence agreement is articular as undeformed configuration, and the accepted agreement as askew configuration. Additionally, time is not advised if allegory deformation, appropriately the arrangement of configurations amid the undeformed and askew configurations are of no interest.
The apparatus Xi of the position agent X of a atom in the advertence configuration, taken with account to the advertence alike system, are alleged the actual or advertence coordinates. On the added hand, the apparatus xi of the position agent x of a atom in the askew configuration, taken with account to the spatial alike arrangement of reference, are alleged the spatial coordinates
There are two methods for analysing the anamorphosis of a continuum. One description is fabricated in agreement of the actual or referential coordinates, alleged actual description or Lagrangian description. A additional description is of anamorphosis is fabricated in agreement of the spatial coordinates it is alleged the spatial description or Eulerian description.
There is chain during anamorphosis of a continuum physique in the faculty that:
The actual credibility basic a bankrupt ambit at any burning will consistently anatomy a bankrupt ambit at any consecutive time.
The actual credibility basic a bankrupt apparent at any burning will consistently anatomy a bankrupt apparent at any consecutive time and the amount aural the bankrupt apparent will consistently abide within.
Affine deformation[edit]
A anamorphosis is alleged an affine anamorphosis if it can be declared by an affine transformation. Such a transformation is composed of a beeline transformation (such as rotation, shear, addendum and compression) and a adamant physique translation. Affine deformations are aswell alleged constant deformations.[7]
Therefore an affine anamorphosis has the form

\mathbf{x}(\mathbf{X},t) = \boldsymbol{F}(t)\cdot\mathbf{X} + \mathbf{c}(t)

where \mathbf{x} is the position of a point in the askew configuration, \mathbf{X} is the position in a advertence configuration, t is a time-like parameter, \boldsymbol{F} is the beeline agent and \mathbf{c} is the translation. In cast form, area the apparatus are with account to an orthonormal basis,

\begin{bmatrix} x_1(X_1,X_2,X_3,t) \\ x_2(X_1,X_2,X_3,t) \\ x_3(X_1,X_2,X_3,t) \end{bmatrix}
= \begin{bmatrix}
F_{11}(t) & F_{12}(t) & F_{13}(t) \\ F_{21}(t) & F_{22}(t) & F_{23}(t) \\ F_{31}(t) & F_{32}(t) & F_{33}(t)
\end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} +
\begin{bmatrix} c_1(t) \\ c_2(t) \\ c_3(t) \end{bmatrix}

The aloft anamorphosis becomes non-affine or inhomogeneous if \boldsymbol{F} = \boldsymbol{F}(\mathbf{X},t) or \mathbf{c} = \mathbf{c}(\mathbf{X},t).
Rigid physique motion[edit]
A adamant physique motion is a appropriate affine anamorphosis that does not absorb any shear, addendum or compression. The transformation cast \boldsymbol{F} is able erect in adjustment to acquiesce rotations but no reflections.
A adamant physique motion can be declared by

\mathbf{x}(\mathbf{X},t) = \boldsymbol{Q}(t)\cdot\mathbf{X} + \mathbf{c}(t)
where

\boldsymbol{Q}\cdot\boldsymbol{Q}^T = \boldsymbol{Q}^T \cdot \boldsymbol{Q} = \boldsymbol{\mathit{1}}

In cast form,

\begin{bmatrix} x_1(X_1,X_2,X_3,t) \\ x_2(X_1,X_2,X_3,t) \\ x_3(X_1,X_2,X_3,t) \end{bmatrix}
= \begin{bmatrix}
Q_{11}(t) & Q_{12}(t) & Q_{13}(t) \\ Q_{21}(t) & Q_{22}(t) & Q_{23}(t) \\ Q_{31}(t) & Q_{32}(t) & Q_{33}(t)
\end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} +
\begin{bmatrix} c_1(t) \\ c_2(t) \\ c_3(t) \end{bmatrix}

Displacement

A change in the agreement of a continuum physique after-effects in a displacement. The displacement of a physique has two components: a rigid-body displacement and a deformation. A rigid-body displacement consists of a accompanying adaptation and circling of the physique afterwards alteration its appearance or size. Anamorphosis implies the change in appearance and/or admeasurement of the physique from an antecedent or undeformed agreement \ \kappa_0(\mathcal B) to a accepted or askew agreement \ \kappa_t(\mathcal B) (Figure 1).
If afterwards a displacement of the continuum there is a about displacement amid particles, a anamorphosis has occurred. On the added hand, if afterwards displacement of the continuum the about displacement amid particles in the accepted agreement is zero, again there is no anamorphosis and a rigid-body displacement is said to accept occurred.
The agent abutting the positions of a atom P in the undeformed agreement and askew agreement is alleged the displacement agent u(X,t) = uiei in the Lagrangian description, or U(x,t) = UJEJ in the Eulerian description.
A displacement acreage is a agent acreage of all displacement vectors for all particles in the body, which relates the askew agreement with the undeformed configuration. It is acceptable to do the assay of anamorphosis or motion of a continuum physique in agreement of the displacement field. In general, the displacement acreage is bidding in agreement of the actual coordinates as
\ \mathbf u(\mathbf X,t) = \mathbf b(\mathbf X,t)+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J
or in agreement of the spatial coordinates as
\ \mathbf U(\mathbf x,t) = \mathbf b(\mathbf x,t)+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,
where О±Ji are the administration cosines amid the actual and spatial alike systems with assemblage vectors EJ and ei, respectively. Thus
\ \mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}
and the accord amid ui and UJ is again accustomed by
\ u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i
Knowing that
\ \mathbf e_i = \alpha_{iJ}\mathbf E_J
then
\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)
It is accepted to blanket the alike systems for the undeformed and askew configurations, which after-effects in b = 0, and the administration cosines become Kronecker deltas:
\ \mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}
Thus, we have
\ \mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J = x_i - X_i
or in agreement of the spatial coordinates as
\ \mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J =x_J - X_J
Displacement acclivity tensor[edit]
The fractional adverse of the displacement agent with account to the actual coordinates yields the actual displacement acclivity tensor \ \nabla_{\mathbf X}\mathbf u. Thus we have:

\begin{align}
\mathbf{u}(\mathbf{X},t) & = \mathbf{x}(\mathbf{X},t) - \mathbf{X} \\
\nabla_\mathbf{X}\mathbf{u} & = \nabla_\mathbf{X}\mathbf{x} - \mathbf{I} \\
\nabla_\mathbf{X}\mathbf{u} & = \mathbf{F} - \mathbf{I} \\
\end{align}
or \begin{align}
u_i & = x_i - \delta_{iJ} X_J = x_i - X_i\\
\frac{\partial u_i}{\partial X_K} & = \frac{\partial x_i}{\partial X_K}-\delta_{iK} \\
\end{align}
where \mathbf{F} is the anamorphosis acclivity tensor.
Similarly, the fractional adverse of the displacement agent with account to the spatial coordinates yields the spatial displacement acclivity tensor \ \nabla_{\mathbf x}\mathbf U. Thus we have,
\begin{align}
\mathbf U(\mathbf x,t) &= \mathbf x - \mathbf X(\mathbf x,t) \\
\nabla_{\mathbf x}\mathbf U &= \mathbf I - \nabla_{\mathbf x}\mathbf X \\
\nabla_{\mathbf x}\mathbf U &= \mathbf I -\mathbf F^{-1}\\
\end{align}
or \begin{align}
U_J& = \delta_{Ji}x_i-X_J =x_J - X_J\\
\frac{\partial U_J}{\partial x_k} &= \delta_{Jk}-\frac{\partial X_J}{\partial x_k}\\
\end{align}
Examples of deformations[edit]
Homogeneous (or affine) deformations are advantageous in elucidating the behavior of materials. Some constant deformations of absorption are
uniform extension
pure dilation
simple shear
pure shear
Plane deformations are aswell of interest, decidedly in the beginning context.
Plane deformation[edit]
A even deformation, aswell alleged even strain, is one area the anamorphosis is belted to one of the planes in the advertence configuration. If the anamorphosis is belted to the even declared by the base vectors \mathbf{e}_1, \mathbf{e}_2, the anamorphosis acclivity has the form

\boldsymbol{F} = F_{11}\mathbf{e}_1\otimes\mathbf{e}_1 + F_{12}\mathbf{e}_1\otimes\mathbf{e}_2 + F_{21}\mathbf{e}_2\otimes\mathbf{e}_1 + F_{22}\mathbf{e}_2\otimes\mathbf{e}_2 + \mathbf{e}_3\otimes\mathbf{e}_3

In cast form,

\boldsymbol{F} = \begin{bmatrix} F_{11} & F_{12} & 0 \\ F_{21} & F_{22} & 0 \\ 0 & 0 & 1 \end{bmatrix}

From the arctic atomization theorem, the anamorphosis gradient, up to a change of coordinates, can be addle into a amplitude and a rotation. Since all the anamorphosis is in a plane, we can write[7]

\boldsymbol{F} = \boldsymbol{R}\cdot\boldsymbol{U} =
\begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}
\begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & 1 \end{bmatrix}

where \theta is the bend of circling and \lambda_1,\lambda_2 are the arch stretches.
Isochoric even deformation[edit]
If the anamorphosis is isochoric (volume preserving) again \det(\boldsymbol{F}) = 1 and we have

F_{11} F_{22} - F_{12} F_{21} = 1

Alternatively,

\lambda_1\lambda_2 = 1

Simple shear[edit]
A simple microburst anamorphosis is authentic as an isochoric even anamorphosis in which there are a set of band elements with a accustomed advertence acclimatization that do not change breadth and acclimatization during the deformation.[7]
If \mathbf{e}_1 is the anchored advertence acclimatization in which band elements do not batter during the anamorphosis again \lambda_1 = 1 and \boldsymbol{F}\cdot\mathbf{e}_1 = \mathbf{e}_1. Therefore,

F_{11}\mathbf{e}_1 + F_{21}\mathbf{e}_2 = \mathbf{e}_1 \quad \implies \quad F_{11} = 1 ~;~~ F_{21} = 0

Since the anamorphosis is isochoric,

F_{11} F_{22} - F_{12} F_{21} = 1 \quad \implies \quad F_{22} = 1

Define \gamma := F_{12}\,. Then, the anamorphosis acclivity in simple microburst can be bidding as

\boldsymbol{F} = \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Now,

\boldsymbol{F}\cdot\mathbf{e}_2 = F_{12}\mathbf{e}_1 + F_{22}\mathbf{e}_2 = \gamma\mathbf{e}_1 + \mathbf{e}_2
\quad \implies \quad
\boldsymbol{F}\cdot(\mathbf{e}_2\otimes\mathbf{e}_2) = \gamma\mathbf{e}_1\otimes\mathbf{e}_2 + \mathbf{e}_2\otimes\mathbf{e}_2

Since \mathbf{e}_i\otimes\mathbf{e}_i = \boldsymbol{\mathit{1}} we can aswell address the anamorphosis acclivity as

\boldsymbol{F} = \boldsymbol{\mathit{1}} + \gamma\mathbf{e}_1\otimes\mathbf{e}_2