We are given a normally consolidated clay sample subjected to isotropic consolidation, and the problem asks us to calculate the stress variables $q$, $p'$, and specific volume $\nu$ at failure. The soil constants provided are:

• $M = 0.88$ (Slope of the critical state line in $q-p'$ space)
• $N = 2.88$ (Intercept of the specific volume line at $p' = 1$ kPa)
• $\Gamma = 2.76$ (Specific volume at $p' = 1$ kPa at critical state)
• $\lambda = 0.16$ (Slope of the normal consolidation line in the $\nu - \ln p'$ space)

Given:

• Isotropic consolidation pressure (cell pressure): $p' = 350 \, \text{kPa}$
• The sample was sheared under drained conditions, so we assume it reaches critical state at failure.

Critical State Soil Mechanics Relations:

1. Relation between q and p' at critical state: $q = M \cdot p'$ where $q$ is the deviator stress, and $p'$ is the effective mean stress at failure.

2. Specific volume ($\nu$) at failure: $\nu = \Gamma + \lambda \cdot \ln\left(\frac{p'}{1 \, \text{kPa}}\right)$ This equation gives the specific volume $\nu$ (ratio of volume of voids to volume of solids) at failure.

Calculations:

1. Deviator stress at failure: $q = M \cdot p' = 0.88 \cdot 350 = 308 \, \text{kPa}$

2. Specific volume at failure: $\nu = \Gamma + \lambda \cdot \ln\left(\frac{p'}{1}\right) = 2.76 + 0.16 \cdot \ln(350)$ Let's compute $\nu$.

I'll calculate this step next.The specific volume $\nu$ at failure is approximately $3.70$.

Summary of results:

• Deviator stress at failure: $q = 308 \, \text{kPa}$
• Effective mean stress at failure: $p' = 350 \, \text{kPa}$
• Specific volume at failure: $\nu = 3.70$

Would you like further details on any part of the process or have any other questions?

Here are five related questions to expand this topic:

1. How would the result change if the consolidation pressure was different?
2. What is the significance of the critical state in soil mechanics?
3. How do you compute void ratio from specific volume?
4. How is the parameter $M$ determined in practice for different soils?
5. What assumptions are involved in using the critical state model for clays?

Tip: In soil mechanics, the specific volume ($\nu$) is related to void ratio ($e$) by the equation $\nu = 1 + e$.