User:Eas4200c.f08.vqcrew.f/hw4

Homework 4

Elastic Theory

For proper understanding of the elastic theory it can help to go back to the Kinematic Assumptions within the road map laid out in previous lectures and stated in the HW 3 report. Recall that in these assumptions that $$\displaystyle \theta$$ is considered to remain constant with respect to 'x' for a uniform bar.

A) Kinematic Assumptions


 * (1) $$u_x (y,z) = \displaystyle \theta \displaystyle \psi (y,z)$$


 * (2) $$u_y (x,z) = -\displaystyle \theta x z$$


 * (3) $$u_z (x,y) = +\displaystyle \theta x y$$



These equations vary slightly from the equations used in the textbook by C.T. Sun, this is due to the use of a different coordinated system. To transform the textbook equations to the uniform notation equations used above it is advised to use cyclic permutation, as seen in figure ###. The basic relationship can be followed by the figure, but the arrows are used to represent the conversion between the x-axis of C.T. Sun becomes the y-axis in uniform notation and likewise for the other axis.

Other important relations can be generated from use of the Kinematic Assumptions and that is the relationship to certain strains and strain rate, giving the following results:


 * $$\displaystyle \epsilon_{xx} = \displaystyle \epsilon_{yy} = \displaystyle \epsilon_{zz} = \displaystyle \gamma_{yz} = 0$$

Proof

These results can easily be proven using the partial derivatives of equations (1), (2), (3) above:


 * $$\displaystyle \epsilon_{xx} = \frac{\partial u_{x} (y,z)}{\partial x} = \frac{\partial (\displaystyle \theta \displaystyle \psi (y,z))}{\partial x} = 0$$


 * $$\displaystyle \epsilon_{yy} = \frac{\partial u_y (x,y)}{\partial y} = \frac{\partial (-\displaystyle \theta x z)}{\partial y} = 0$$


 * $$\displaystyle \epsilon_{yy} = \frac{\partial u_y (x,y)}{\partial y} = \frac{\partial (+\displaystyle \theta x y)}{\partial z} = 0$$


 * $$\displaystyle \gamma_{yz} = \frac{\partial u_y (x,z)}{\partial z} + \frac{\partial u_z (x,y)}{\partial y} = -\theta x + \theta x = 0$$

Formal Derivation

Now looking at the stains in 3-D. There are a total of six strain components for a 3 dimensional object due to symmetry.


 * $$\mathbf{\displaystyle \epsilon} = \begin{bmatrix}

\epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\ \epsilon_{yx} & \epsilon_{yy} & \epsilon_{yz} \\ \epsilon_{zx} & \epsilon_{zy} & \epsilon_{zz} \end{bmatrix}$$

Now that we have a 3x3 matrix with 9 coefficients, we need to change the notation to that of indicial notation.


 * $$ x \Leftrightarrow 1, y \Leftrightarrow 2 , z \Leftrightarrow 3 $$

Now we can write the matrix in indicial notation.


 * $$\mathbf{\displaystyle \epsilon} = \begin{bmatrix}

\epsilon_{11} & \epsilon_{12} & \epsilon_{13} \\ \epsilon_{21} & \epsilon_{22} & \epsilon_{23} \\ \epsilon_{31} & \epsilon_{32} & \epsilon_{33} \end{bmatrix}$$ $$ = \displaystyle \epsilon_{ij}$$

$$\displaystyle \epsilon_{ij}$$ is known at tensorial notation. For this matrix i,j range from 1 to 3, where i is the row number in the matrix and j in the column designation.

Now that the matrix for $$\displaystyle \epsilon_{ij}$$ is formed, the symmetry of the matrix can now be calculated.


 * $$\displaystyle \epsilon_{ij} = \frac{1}{2} (\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})$$

The coordinates x,y,z can now be rewritten: $$ x \Leftrightarrow x_1, y \Leftrightarrow x_2 , z \Leftrightarrow x_3 $$


 * $$\epsilon_{11} = \displaystyle \epsilon_{xx} = \frac{1}{2} (\frac{\partial u_1}{\partial x_1} + \frac{\partial u_1}{\partial x_1})$$


 * $$\displaystyle \epsilon_{xx} = \frac{\partial u_1}{\partial x_1} \Rightarrow \frac{\partial u_x}{\partial x}$$

HW Symmetry $$\displaystyle \epsilon_{ij} = \epsilon_{ji}$$

The symmetry of the matrix can be seen by examining the relationship between $$\displaystyle \epsilon_{21}$$ and $$\displaystyle \epsilon_{21}$$


 * $$\epsilon_{12} = \displaystyle \epsilon_{xy} = \frac{1}{2} (\frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1})$$

Now look at $$\displaystyle \epsilon_{21}$$


 * $$\epsilon_{21} = \displaystyle \epsilon_{xy} = \frac{1}{2} (\frac{\partial u_2}{\partial x_1} + \frac{\partial u_1}{\partial x_2})$$

It can now clearly be seen that $$\displaystyle \epsilon_{12} = \displaystyle \epsilon_{21}$$

From the Symmetrical proof it can be the following relation can be seen:


 * $$\displaystyle \epsilon_{12} = \displaystyle \epsilon_{21} \Leftrightarrow \epsilon_{xy} = \epsilon_{yx}$$

This relationship provides the result that simplifies the $$\displaystyle \epsilon_{ij}$$ matrix from 9 varialbe to only 6 individual components.

Stress Tensor

The stress tensor can be looked at much like the the strain relations as a 3x3 matrix. Also by symmetry there will be only 6 independent components of $$\displaystyle \sigma$$ in 3-D.


 * $$\displaystyle \sigma = [\sigma]_{3x3}$$

Based of the relations between the stress and strain, several stresses and a shear strain can be found by kinematic assumptions.


 * $$\displaystyle \sigma_{xx} = \displaystyle \sigma_{yy} = \displaystyle \sigma_{zz} = \displaystyle \tau_{yz} = 0$$