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
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
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