3.2.1. Representation of stress tensor in Cartesian coordinate system

Try these first

1. 1.

   Define shear stress.

   Hint

   Think about how layers of material might slide over each other.

2. 2.

   What does the normal stress component indicate?

   Hint

   Consider how forces applied can stretch or compress materials.

3. 3.

   What is the primary use of the stress tensor?

   • To measure thermal expansion
   • To analyze internal forces in materials
   • To calculate geometric dimensions

   Hint

   Consider its purpose in structural analysis.

4. 4.

   True or False: Shear stress acts perpendicular to a surface.

   • True
   • False

   Hint

   Review the definitions of shear and normal stresses.

5. 5.

   Given a cuboid with σ₁₁ = 10 MPa, σ₂₂ = -5 MPa, τ₁₂ = 3 MPa, calculate the resultant stress vector on one of its faces.

   Hint

   Break it down into components for calculations.

6. 6.

   A new material has an initial stress tensor of σ = [[5, 2], [2, 3]] in an arbitrary orientation. If it rotates by 45 degrees, how do the stress components transform?

   Hint

   Think about using trigonometric identities to analyze the transformations.