29.14. Application: Principal Stress and Strain in 2D

Try these first

1. 1.

   What does the stress tensor represent in 2D solid mechanics?

   Hint

   Think about the components of the stress tensor.

2. 2.

   What are principal stresses and how are they determined?

   Hint

   Recall the characteristic polynomial.

3. 3.

   What do principal stresses represent in a material?

   • The average stress
   • The maximum and minimum normal stresses
   • Shear stresses only

   Hint

   Think about the eigenvalue definition.

4. 4.

   True or False: The principal planes can be determined from eigenvectors.

   • True
   • False

   Hint

   Reflect on the relationship between eigenvectors and angular orientation.

5. 5.

   Given the stress tensor σ = [4 2; 2 3], derive the principal stresses and describe their physical significance in a practical scenario.

   Hint

   Focus on the determinant calculation and ensure proper factorization.

6. 6.

   Evaluate the effect of increasing τ_xy on the eigenvalues derived from the stress tensor σ. How would this change the material's response?

   Hint

   Set up the matrix and recalculate eigenvalues based on modified shear stress.