Schmid Tensor / Resolved Shear Stress question

[muskie25]

Homework Statement

Calculate the magnitude of the resolved shear stress on the ## (112)## plane in the ##[11\bar{1}] ##direction in a bcc crystal.

Homework Equations

Tried to use:
##
t^{\alpha} = \frac{1}{2} \sigma_{ij} (m_{i} n_{j} + m_{i} n_{j})
##

where m is the direction and n is the plane.

The Attempt at a Solution

##
t^{\alpha} = \frac{1}{2} \left( \begin{array} (10 & -5 & 0 \\ -5 & 20 & 10 \\ 0 & 10 & 30 \end{array} \right ) \left( \begin{array} (-6 & 6 & 0 \end{array} \right ) = \left( \begin{array} (-90 & 150 & 60 \end{array} \right )
##

This does not sit right with me and I am not sure that I fully understand what is going on here. Can anyone help me figure out what I did wrong?

 

[KataKoniK]

The resolved shear stress on a specific plane and direction in a crystal is given by the formula:

$$\tau = \sigma_{ij} n_i m_j$$

Where $\sigma_{ij}$ is the stress tensor, $n_i$ is the normal vector to the plane, and $m_j$ is the direction vector. In this case, we are looking for the resolved shear stress on the (112) plane in the [11-1] direction.

The first step is to determine the normal vector to the (112) plane. This can be done by taking the cross product of two vectors that lie in the plane. One possible choice is the [1-1-2] and [1-1-1] vectors, which results in a normal vector of [1-1-1].

Next, we need to determine the direction vector in the [11-1] direction. This can be done by taking the cross product of two vectors that lie in this direction. One possible choice is the [1-1-1] and [1-1-2] vectors, which results in a direction vector of [1-1-1].

Now, we can plug these values into the formula to get:

$$\tau = \sigma_{ij} n_i m_j = \left( \begin{array}{ccc} 10 & -5 & 0 \\ -5 & 20 & 10 \\ 0 & 10 & 30 \end{array} \right ) \left( \begin{array}{c} 1 \\ -1 \\ -1 \end{array}\right) \left( \begin{array}{c} 1 \\ -1 \\ -1 \end{array}\right) = \left( \begin{array}{c} 0 \\ 0 \\ -40 \end{array}\right)$$

So, the magnitude of the resolved shear stress on the (112) plane in the [11-1] direction is 40.