F s

1. Concepts of stress and strain P Stress at a point FN F positive side area A Plane Q Fs negative side

2. Concepts of stress and strain z y For the cube face in the –x direction, we have stresses: Stress tensor sxz s-x-x sxy s-x-y s-x-x sxx x

3. Concepts of stress and strain In general, we can define the stress vector acting on an arbitrary plane with direction cosines la1, la2, la3 referred to The x, y, z coordinate system. The arbitrary plane is defined in terms of its unit normal direction; Stress vector on an arbitrary plane

4. Concepts of stress and strain Transformation of stress components: suppose we have the stress components in one coordinate system (x, y, z) and we want to get the components in another (say rotated) coordinate system (x’, y’, z’):

5. Concepts of stress and strain Principal stresses and stress invariants For any general state of stress at a point P, there exists 3 mutually perpendicular planes on which the shear stresses Vanish. The resulting stresses on these planes are normal stresses and are called principal stresses (Eigen values) and The normal directions defining these planes are called principal directions (Eigen vectors). These principal directions can be referred to a coordinate system (x, y, z) in space.

6. Concepts of stress and strain The condition for a non trivial solution to exist is given by the so-called characteristic equation: This results in a cubic equation that can be solved for the 3 roots (sI, sII, sIII) corresponding to the 3 principal stresses. The principal directions can be solved by substituting separately each of the principal stresses into the set of linear equations and solving for the direction cosines subject to the condition that;

7. Concepts of stress and strain The cubic equation: where the I’s are invariant to the coordinate system:

8. Concepts of stress and strain Mean stress and deviator stress

9. Concepts of stress and strain Invariants of the stress deviator tensor:

10. Concepts of stress and strain Maximum shear stress: It can be shown using the method of Lagrange multipliers that the maximum shear stress is equal to one half the difference of the principal stresses and occur at 45 degree angles wrt the principal stresses.

11. Principal stress and direction example

12. Concepts of stress and strain Strain at a point (small displacements): 1-D strain B A P A* B*

13. Concepts of stress and strain More generally, the displacement, u, in the x-direction will also depend on the y and z coordinates of a point under a more general deformation, i.e., Similar expressions can be written for the displacement in the y-direction, v, and z-direction w.

14. Concepts of stress and strain The terms that look like Can Involve not only a deformation strain but also a rigid-body rotation. Pure shear Pure rotation with no shear simple shear

15. Concepts of stress and strain The 9 relative displacement components can be decomposed into a symmetric part defining the strain tensor and a rotation tensor, i.e., The strain tensor transforms the same way as the stress tensor and all 2nd order Symmetric tensors.

16. Concepts of stress and strain Very often, the shear strain is expressed as engineering shear strain, γ. However, be careful. since this form of shear stain does not transform the same way as a symmetric 2nd order tensor.

17. Concepts of stress and strain

18. Concepts of stress and strain

19. Concepts of stress and strain

20. Concepts of stress and strain

21. Concepts of Stress and Strain F FN positive side of plane Q Fs plane Q area A negative side of plane Q