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}
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}
No comments:
Post a Comment