1. Continuum Mechanics (MTH487)
Lecture 8
Instructor
Dr. Junaid Anjum

2. Recap … Body and Surface forces mass density Stress The Stress Tensor
Cauchy Stress Principle
Cauchy Stress formula

3. Aims and Objectives some problems divergence theorem
local equilibrium equation
Stress Tensor symmetry

4. Problem
The stress tensor has components at point P as specified by the matrix below
Determine
the stress vector on the plane at P whose normal vector is
the magnitude of this stress vector,
the component of the stress vector in the direction of the normal,
the angle in degrees between the stress vector and the normal.

6. Problem
At a point P, the stress tensor relative to axes Ox1x2x3 has components given by the matrix below
Find the stress vectors , corresponding to the normal vectors
(ii) Show that the component of in the direction is equal to the component of in the direction

9. The local Equilibrium equation …
Consider a material body B having a volume V enclosed by a surface S. Let the body be subjected to surface traction and body forces as shown in the figure below.
As before we exclude concentrated body moments from consideration. Equilibrium requires that the summation of all the forces acting on the body be zero. This condition is expressed by the global (integral) equation representing the sum of the total surface and body forces acting on the body,
where is differential element of surface S and is that of volume.

10. The local Equilibrium equation …
The divergence theorem of Gauss establishes a relationship between the surface integral having as the integrand to the volume integral for which a coordinate derivative of is the integrand,
hence

11. Stress Tensor Symmetry …
In addition to the balance of forces, equilibrium requires that the summation of moments with respect to an arbitrary point must also be zero. Recall that moments is defined as the cross product of its position vector with the force. Therefore, taking the origin of coordinates as the centre for moments, and noting that is a vector for the typical elements of surface and volume, we express the balance of moments for the body as a whole by,

12. Stress Tensor Symmetry …

13. Problem
At a point P, the stress tensor relative to axes Ox1x2x3 has components On the area elements having the unit normal , the stress vector is , and on area element with normal the stress vector is Show that the component of in the direction is equal to the component of in the direction

15. Problem
With respect to axes Ox1x2x3 the stress state is given in terms of coordinates by the matrix
Determine
the body force components as functions of coordinates if the equilibrium equations are to be satisfied everywhere, and
the stress vector at point P(1,2,3) on the plane whose outward unit normal makes equal angles with the positive coordinate axes. .

18. Problem
Relative to the Cartesian axes Ox1x2x3 a stress field is given by the matrix
Show that the equilibrium equations are satisfied everywhere for zero body force.
Determine the stress vector at the point P(2,-1,6) of the plane whose equation is

21. Aims and Objectives divergence theorem local equilibrium equation
stress tensor symmetry