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

2. Concepts of stress and strainz 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 A B A* B* P

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 StrainFN
positive side of plane Q
plane Q Fs
area A
negative side of plane Q