Introduction to Elasticity/Body force potential

Body force potential

How do we find the body force potential? Before we proceed let us examine what conservative vector fields are.

Conservative vector fields

Work done in moving a particle from point A to point B in the field should be path independent.
The local potential at point P in the field is defined as the work done to move a particle from infinity to P.
For a vector field to be conservative

f_{2,1} - f_{1,2} = 0 ~;~~ f_{3,2} - f_{2,3} = 0 ~;~~ f_{1,3} - f_{3,1} = 0 \qquad \text{(28)}

\boldsymbol{\nabla}\times{\mathbf{f}} = 0 \qquad \text{(29)}

The field has to be irrotational.

Determining the body force potential

Suppose a body is rotating with an angular velocity $\dot{\theta}$ and an angular acceleration of $\ddot{\theta}$ . Then,

\text{(30)} \qquad \mathbf{a}_r = -\dot{\theta}^2 r \widehat{\mathbf{e}}_{r} ~;~~ \mathbf{a}_{\theta} = -\ddot{\theta} r \widehat{\mathbf{e}}_{\theta}

Let us assume that the $(r,\theta)$ coordinate system is oriented at an angle $\theta$ to the $(x_1,x_2)$ system. Then,

\text{(31)} \qquad \begin{bmatrix}a_1\\a_2\end{bmatrix} = \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix}a_r\\a_{\theta}\end{bmatrix}

or,

\text{(32)} \qquad \begin{bmatrix}a_1\\a_2\end{bmatrix} = \begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix}-\dot{\theta}^2r\\-\ddot{\theta}r\end{bmatrix}

or,

\begin{align} \text{(33)} \qquad a_1 & = -\dot{\theta}^2r\cos\theta + \ddot{\theta}r\sin\theta\\ \text{(34)} \qquad a_2 & = -\dot{\theta}^2r\sin\theta - \ddot{\theta}r\cos\theta \end{align}

or,

\begin{align} \text{(35)} \qquad a_1 & = -\dot{\theta}^2 x_1 + \ddot{\theta} x_2 \\ \text{(36)} \qquad a_2 & = -\dot{\theta}^2 x_2 - \ddot{\theta} x_1 \end{align}

If the origin is accelerating with an acceleration $\mathbf{a}_0$ (for example, due to gravity), we have,

\begin{align} \text{(37)} \qquad a_1 & = a_{01} -\dot{\theta}^2 x_1 + \ddot{\theta} x_2 \\ \text{(38)} \qquad a_2 & = a_{02} -\dot{\theta}^2 x_2 - \ddot{\theta} x_1 \end{align}

The body force field is given by

\begin{align} \text{(39)} \qquad f_1 & = -\rho\left(a_{01}-\dot{\theta}^2x_1+\ddot{\theta}x_2\right) \\ \text{(40)} \qquad f_2 & = -\rho\left(a_{02}-\dot{\theta}^2x_2-\ddot{\theta}x_1\right) \end{align}

For this vector body force field to be conservative, we require that,

f_{1,2} - f_{2,1} = 0 \Rightarrow 2\ddot{\theta} = 0

Hence, the field $\mathbf{f}$ is conservative only if the rotational acceleration is zero, i.e. = the rotational velocity is constant.=

\begin{align} \text{(41)} \qquad f_1 & = -\rho\left(a_{01}-\dot{\theta}^2x_1\right)\\ \text{(42)} \qquad f_2 & = -\rho\left(a_{02}-\dot{\theta}^2x_2\right) \end{align}

Now,

f_1 = - V_{,1} ~;~~ f_2 = - V_{,2}

Hence,

\begin{align} \text{(43)} \qquad V_{,1} & = \rho\left(a_{01}-\dot{\theta}^2x_1\right)\\ \text{(44)} \qquad V_{,2} & = \rho\left(a_{02}-\dot{\theta}^2x_2\right) \end{align}

Integrating equation (43),

\text{(45)} \qquad V = \rho\left(a_{01} x_1 -\dot{\theta}^2\cfrac{x_1^2}{2}\right) + h(x_2)

Hence,

(\text{46)} \qquad V_{,2} = h^{'}(x_2) = \rho\left(a_{02}-\dot{\theta}^2x_2\right)

Integrating,

\text{(47)} \qquad h(x_2) = \rho\left(a_{02} x_2 -\dot{\theta}^2\cfrac{x_2^2}{2}\right) + C

Without loss of generality, we can set $C = 0$ . Then,

\text{(48)} \qquad V = \rho\left(a_{01} x_1 -\dot{\theta}^2\cfrac{x_1^2}{2}\right) + \rho\left(a_{02} x_2 -\dot{\theta}^2\cfrac{x_2^2}{2}\right)

or,

\text{(49)} \qquad V = \rho\left[a_{01} x_1 + a_{02} x_2 - \cfrac{\dot{\theta}^2}{2} \left(x_1^2 + x_2^2\right)\right]

For a body loaded by gravity only, we can set $a_{01} = 0$ , $a_{02} = -g$ and $\dot{\theta} = 0$ , to get

\text{(50)} \qquad V = -\rho g x_2 \,

For a body loaded by rotational inertia only, we can set $a_{01} = 0$ , and $a_{02} = 0$ , and get

\text{(51)} \qquad V = -\cfrac{\rho\dot{\theta}^2}{2} \left(x_1^2 + x_2^2\right)

We can see that an Airy stress function + a body force potential of the form shown in equation (49) can be used to solve two-dimensional elasticity problems of plane stress/plane strain.

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