Question

The hydrostatic stress for the stress tensor provided below is .............. MPa (in integer). \[ \begin{bmatrix} 150 & 0 & 0
0 & -100 & 100
0 & 100 & 250 \end{bmatrix} \, \text{MPa} \]

Hint:

Hydrostatic stress is the mean of the three principal (normal) stresses. It represents the uniform pressure component in the stress state and is independent of shear stresses.

Answer 1

Step 1: Recall definition of hydrostatic stress.
Hydrostatic stress (also known as mean stress) is defined as the average of the normal stresses acting on a body. In tensor notation: \[ \sigma_{hydro} = \frac{\sigma_{xx} + \sigma_{yy} + \sigma_{zz}}{3} \] Only the diagonal terms (normal stresses) are considered. Shear stresses (off-diagonal terms like 100 MPa here) do not contribute to hydrostatic stress. Step 2: Identify the normal stress components.
From the given stress tensor: - Normal stress in x-direction: \(\sigma_{xx} = 150\) MPa - Normal stress in y-direction: \(\sigma_{yy} = -100\) MPa - Normal stress in z-direction: \(\sigma_{zz} = 250\) MPa Step 3: Substitute into the formula.
\[ \sigma_{hydro} = \frac{150 + (-100) + 250}{3} \] Step 4: Simplify.
\[ \sigma_{hydro} = \frac{300}{3} = 100 \, \text{MPa} \] \[ \boxed{100 \, \text{MPa}} \]