Kinematic Deformation: Trajectories, Acceleration & Tensors Explained

[mullzer]

Homework Statement

Let a and b be two given orthonormal vectors around a fixed point O. The motion of a continuum is defined by the following velocity field:

v(M) = \alpha \vec{a} (\vec{b} . \vec{OM}) \\ [\tex]<br /> <br /> where \alpha [\tex] is a known positive constant.&lt;br /&gt; &lt;br /&gt; 1. Choosing a marker, determine the lagrangian representation of motion, given that at time t= 0, the particle occupies the position X= ([X][/1], [X][/2], [X][/3]).&lt;br /&gt; &lt;br /&gt; Determine the trajectories, streamlines and streaklines of the motion. Calculate the acceleration field.&lt;br /&gt; &lt;br /&gt; 2. Determine the composants of the deformation gradient tensor &lt;b&gt;F&lt;/b&gt;, the expansion tensor &lt;b&gt;C&lt;/b&gt;, the derformation tensor &lt;b&gt;X&lt;/b&gt;, the strain rate tensor &lt;b&gt;D&lt;/b&gt; and the tensors &lt;b&gt;G &lt;/b&gt;= &lt;b&gt;[F][/-1]&lt;/b&gt; and &lt;b&gt;B&lt;/b&gt; = [&lt;b&gt;G][/T] G&lt;/b&gt;&lt;br /&gt; &lt;br /&gt; Calculate the eigenvalues and principle directions of &lt;b&gt;C, X&lt;/b&gt; and &lt;b&gt;B&lt;/b&gt;. What would happen to these if the value of \alpha t [\tex] was small? Interperate the case.&amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; &amp;lt;h2&amp;gt;Homework Equations&amp;lt;/h2&amp;gt;&amp;lt;br /&amp;gt; The Lagrangian stament of motion: \vec {x_{i}} (t) = \vec{\phi_{i}} (\vec {X}, i = 1, 2, 3\\&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; and \vec{v} (\vec {u}, t) = \frac{d \vec{x}}{dt} \\[\tex]&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;h2&amp;amp;gt;The Attempt at a Solution&amp;amp;lt;/h2&amp;amp;gt;&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; First of all, is it alright to substitute \vec{a} [\tex] and \vec{b} [\tex] with basis vectors \vec{e_4{1}} [\tex] and \vec{e_{2}} [\tex] which would correspond with \vec{x_{i}} [\tex] ?&amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;gt; 1. use the second relevant eq:&amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;gt; \vec{v} (\vec {u}, t) = \frac{d \vec{x}}{dt} \\[\tex]&amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;gt; with respect to x_{1}, x_{2} and x_{3}. According to some notes i have jotted down, this yields:&amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;gt; \frac{d \vec{x_{1}}}{dt} = \alpha x_{1} \\&amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; \frac{d \vec{x_{2}}}{dt} = -\alpha x_{2} \\&amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; \frac{d \vec{x_{3}}}{dt} = 0 \\ [\tex]&amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; which is the main source of my confusion. How come it is different for each of these? Is it do with the wedge product of a and b?&amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; After that I rearrange these eq&amp;amp;amp;amp;amp;amp;amp;amp;amp;#039;s and solve for x_{i} \ [\tex] using the intial condition of X= ([x=[X][/1], [X][/2], [X][/3]) at t = 0:&amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; x_{1} = C_{1}e^(\alphat) \ [\tex] etc; where C_{1} \ [\tex]becomes X_{1}\\ [\tex]&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; I believe that these eq&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#039;s are also the trajectories of the motion. &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; Deriving these further with respect to t gives the lagrangian speed and then the lagrangian acceleration:&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; \Gamma_{1} = \alpha^2 X_{1}e^(\alphat) \ [\tex] etc.&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; 2. To obtain &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;F&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;/b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;, I derived the eq. of motion x_{1} = C_{1}e^(\alphat) \ [\tex] with respect to x_{1}, x_{2} and x_{3} [\tex] and then the other eq&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#039;s of motion with respect to x_{1}, x_{2} and x_{3} [\tex]. These nine values were used to create a diagonal tensor with trace e^(\alpha t) + e^(-\alpha t) + 1. \\ [\tex]&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;C&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;/b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; is then equal to [&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;F&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;/b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;][/T]&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;F&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;/b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;X&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;/b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; = \frac {1}{2} (C-I). \\ &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; I know how to get eigenvaues by solving &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;C&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;/b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; - \lambda[\tex]&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;I&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;/b&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;.&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; Mainly, I would just like an explanation of how the original equation came about (providing the rest of what I have done is correct). If you could explain it as simply as possible as I am now to this field of mechanics. Thanks in advance!

 

[mullzer]

This time with better notation (hopefully):

1. Homework Statement

Let a and b be two given orthonormal vectors around a fixed point O. The motion of a continuum is defined by the following velocity field:

v(M) = \alpha \vec{a} (\vec{b} . \vec{OM})

where \alpha is a known positive constant.

1. Choosing a marker, determine the lagrangian representation of motion, given that at time t= 0, the particle occupies the position X= (X_{1},X_{2},X_{3}).

Determine the trajectories, streamlines and streaklines of the motion. Calculate the acceleration field.

2. Determine the composants of the deformation gradient tensor F, the expansion tensor C, the derformation tensor X, the strain rate tensor D and the tensors G =F^{-1} and B = G^{T} G

Calculate the eigenvalues and principle directions of C, X and B. What would happen to these if the value of \alpha t was small? Interperate the case.



2. Homework Equations
The Lagrangian stament of motion: \vec {x_{i}} (t) = \vec{\phi_{i}} (\vec {X},t), i = 1, 2, 3

and \vec{v} (\vec {u}, t) = \frac{d \vec{x}}{dt}


3. The Attempt at a Solution

First of all, is it alright to substitute \vec{a} and \vec{b} with basis vectors \vec{ e_1} and \vec{e_2} [/tex] which would correspond with \vec{x_{i}} ?

1. use the second relevant eq:

\vec{v} (\vec {u}, t) = \frac{d \vec{x}}{dt}

with respect to x_{1}, x_{2} and x_{3}. According to some notes i have jotted down, this yields:

\frac{d \vec{x_{1}}}{dt} = \alpha x_{1}

\frac{d \vec{x_{2}}}{dt} = -\alpha x_{2}

\frac{d \vec{x_{3}}}{dt} = 0

which is the main source of my confusion. I don't know how to get this solution How come it is different for each of these? Is it do with the wedge product of a and b?

After that I rearrange these eq's and solve for x_{i} \ using the intial condition of

x= (X_{1},X_{2},X_{3}) at t = 0:

x_1 = C_{1}e^{\alpha t} etc;

where C_{1} becomes X_{1}

I believe that these eq's are also the trajectories of the motion.

Deriving these further with respect to t gives the lagrangian speed and then the lagrangian acceleration:

\Gamma_{1} = \alpha^2 X_{1}e^{\alpha t} \ etc.

2. To obtain F, I derived the eq. of motion x_{1} = C_{1}e^(\alpha t) \ with respect to x_{1}, x_{2} and x_{3} and then the other eq's of motion with respect to x_{1}, x_{2} and x_{3}. These nine values were used to create a diagonal tensor with trace e^{\alpha t} + e^{-\alpha t} + 1.

C is then equal to [F][/T]F

X = \frac {1}{2} (C-I).

I know how to get eigenvaues by solving C - \lambdaI.

Am I right in saying that if the value of \alpha t was small, there would be little or no deformation because these are directly proportional to it?

Mainly, I would just like an explanation of how the original equation came about (providing the rest of what I have done is correct). If you could explain it as simply as possible as I am now to this field of mechanics. I am also wondering how to interperate the trafectories etc into rough sketches.

Thanks in advance!