Examples on Calculation of Strains and Tractions, Principal Stresses and Directions — Lesson 3

This lesson covers the concept of stress and strain at a point in a body undergoing deformation. It explains the tensor representation of stress and strain, their components in 2 and 3 dimensions, and their symmetric nature. The lesson also discusses the difference between engineering shear strain and tensor shear strain. It further elaborates on the calculation of strains at a point in terms of displacement components and tractions on an arbitrary plane. The lesson concludes with the determination of principal planes and principal stresses, and the concept of stress invariants. For instance, it explains how to calculate the strain at a point where the displacements are given in millimeters.

Video Highlights

01:22 - Introduction to the strain tensor and its components
06:05 - Explanation of compact form of expanded expressions
10:26 - Calculation of strains strain components
22:22 - Determination of principal planes and principal stresses
39:36 - Explanation of stress invariants
48:42 - Explanation of why stress invariants are independent of coordinates

Key Takeaways

- Stress and strain at a point are represented by tensors with 9 components for 3 dimensions and 4 components for 2 dimensions.
- Engineering shear strain is twice the tensor shear strain.
- Strains at a point can be calculated in terms of displacement components.
- Traction components on an arbitrary plane can be calculated using the stress tensor and direction cosines.
- Principal planes and principal stresses are determined by finding the roots of a cubic equation involving stress invariants.
- Stress invariants are quantities that are independent of the coordinate system used.