Cohesionless sample, deviator at 200 kPa

For a cohesionless soil ($c=0$), the failure circle touches a line through the origin.

Test 1: $\sigma_3 = 100$ kPa, $\sigma_d = 200$ kPa, so $\sigma_1 = 300$ kPa.

$$\sin\phi = \frac{\sigma_1-\sigma_3}{\sigma_1+\sigma_3} = \frac{200}{400} = 0.5 \;\Rightarrow\; \phi = 30^\circ$$

Test 2: $\sigma_3 = 200$ kPa.

$$\sigma_1 = \sigma_3\tan^2\left(45^\circ+\frac{30^\circ}{2}\right) = 200 \times 3 = 600\ \text{kPa}$$ $$\sigma_d = 600 - 200 = 400\ \text{kPa}$$

(The deviator stress is proportional to $\sigma_3$ for cohesionless soil, so doubling $\sigma_3$ doubles $\sigma_d$.)

Mohr circles and envelope

 tau        / tau = sigma tan30
  |       /
  |     /   ..-'''-.      circle 2: 200 to 600
  |   /  .-'.-'''-.  \
  | /  .'  /  c1   \  |   circle 1: 100 to 300
  +-+--+----+-+-----+---- sigma
   0  100  200 300  600

Answer: $\phi = 30^\circ$; deviator stress at failure for $\sigma_3 = 200$ kPa is 400 kPa.