Question

At a certain depth in the crust, the maximum and minimum principal compressive stresses are 150 MPa and 75 MPa, respectively, which lead to normal faulting. If the average density of the crust is 2700 kg/m$^3$, the crustal depth of fracture initiation according to Anderson's theory of faulting is ................ km. (g = 10 m/s$^2$) [round off to one decimal place]

Hint:

In Anderson's faulting theory, vertical stress is lithostatic. Always use $\sigma_v = \rho g h$ to relate stress to crustal depth.

Answer 1

Step 1: Concept.
According to Anderson's theory, vertical stress ($\sigma_v$) corresponds to lithostatic stress: \[ \sigma_v = \rho g h \] where $\rho$ = density, $g$ = gravity, $h$ = depth.

Step 2: Identify given stresses.
Maximum principal stress $\sigma_1 = 150$ MPa, minimum principal stress $\sigma_3 = 75$ MPa. For normal faulting, $\sigma_1 = \sigma_v$. So, \[ \sigma_v = 150 \, \text{MPa} = 150 \times 10^6 \, \text{Pa} \]

Step 3: Calculate depth.
\[ \sigma_v = \rho g h \quad \Rightarrow \quad h = \frac{\sigma_v}{\rho g} \] \[ h = \frac{150 \times 10^6}{2700 \times 10} = \frac{150 \times 10^6}{27000} = 5555.5 \, \text{m} \]

Step 4: Convert to km.
\[ h = 5.6 \, \text{km (approx.)} \]

Final Answer: \[ \boxed{5.6 \, \text{km}} \]