1. Continuum Mechanics (MTH487)
Lecture 7
Instructor
Dr. Junaid Anjum

2. Aims and Objectives Body and Surface forces Mass density Stress
Cauchy Stress principle
The Stress Tensor
Cauchy Stress formula

4. Stress Principles…
Body forces : forces acting on all volume element, and are distributed throughout the body. Gravity and Inertia are the best know examples. We designate body forces by the vector symbol “bi” (force per unit mass) or by the symbol “pi” (force per unit volume).
Surface forces : forces that act upon, and are distributed in some fashion over a surface element of the body, regardless of whether that element is part of the bounding surface or an arbitrary element of surface within the body. These are denoted by the vector symbol “fi” and have dimensions of force per unit area.

5. Stress Principles…
Mass density: Consider a material body B having a volume V enclosed by a surface S, and occupying a regular region Ro of the physical space. Let P be an interior point of the body located in the small element of volume whose mass is as indicated in the figure below.
Recall that mass is that property of a material body by virtue of which the body possesses inertia, that is, the opposition which the body offers to any change in its motion. We define the average density of this volume element by the ratio
and the density at point P by the limit of this ratio as the volume shrinks to the point P.

6. Stress Principles… Mass density
The units of density are kilograms per cubic meter (kg/m3). Notice that the two measures of body forces, bi, having units of Newton per kilogram (N/kg), and pi, having units of Newton per meter cubed (N/m3) are related through the density by the equation
In general the density is a scalar function of position and time
and thus may vary from point to point within the body.

7. Stress Principles… Cauchy Stress principle
Consider a homogenous, isotropic material body B. Let P be an interior point of B and imagine a plane surface S* passing through point P (sometimes referred to as cutting plane) so as to partition the body into two portions I & II (Fig. 1)
(Fig. 1)
(Fig. 2)

8. Stress Principles… Cauchy Stress principle
Consider a homogenous, isotropic material body B. Let P be an interior point of B and imagine a plane surface S* passing through point P (sometimes referred to as cutting plane) so as to partition the body into two portions I & II (Fig. 1)
The point P is in the small element of area of the cutting plane which is defined by the unit normal vector pointing in the direction from portion I to portion II as shown in Fig 2. The internal forces being transmitted across the cutting plane due to the action of plane II onto plane I result in a net force and net moment
(Fig. 2)
The Cauchy principle asserts that in
the limiting sense i.e.

9. Stress Principles… Cauchy Stress principle
The vector is called the stress vector. In above equations, we
assume that in the limit at P, the moment vector vanishes and there is no concentrated moment of couple stress as it is called.
Notice that in the symbol , the appearance of normal vector is important to emphasize that corresponding to a different cutting plane, there exists a different with a different associated stress tensor. The pair defines the state of stress at that point.

10. Stress Principles… Cauchy Stress principle
principle of linear momentum for portion I and portion II…
for the whole body
on surface element S*

11. Stress Principles…
Cauchy Stress principle
Portion I onto portion II

12. Stress Principles… The Stress Tensor
Cauchy stress principle associates a stress vector with each direction at point P. In particular, if we work in rectangular Cartesian reference frames at P, there exist vectors in each direction
In terms of their coordinate components,
or more compactly, using the summation convention
we will now work out the expression for components of stress tensor.

13. Stress Principles… Cauchy Stress formula
Consider the equilibrium of a small portion of body in the shape of a tetrahedron

14. Stress Principles… Cauchy Stress formula On the tetrahedron face ABC…
Consider the equilibrium of a small portion of body in the shape of a tetrahedron
On the tetrahedron face ABC…

15. Stress Principles… Cauchy Stress formula On the tetrahedron face ABC…
Consider the equilibrium of a small portion of body in the shape of a tetrahedron
On the tetrahedron face ABC…

16. Stress Principles… Cauchy Stress formula On the tetrahedron face PBC…
Consider the equilibrium of a small portion of body in the shape of a tetrahedron
On the tetrahedron face PBC…

17. Stress Principles… Cauchy Stress formula On the tetrahedron face PAC…
Consider the equilibrium of a small portion of body in the shape of a tetrahedron
On the tetrahedron face PAC…

18. Stress Principles… Cauchy Stress formula On the tetrahedron face PAB…
Consider the equilibrium of a small portion of body in the shape of a tetrahedron
On the tetrahedron face PAB…

19. Stress Principles… Cauchy Stress formula hence ….
Consider the equilibrium of a small portion of body in the shape of a tetrahedron
hence ….

20. Stress Principles… Cauchy Stress formula and the body force …
Consider the equilibrium of a small portion of body in the shape of a tetrahedron
and the body force …

21. Stress Principles… Cauchy Stress formula
Consider the equilibrium of a small portion of body in the shape of a tetrahedron
If dS is the area of the base , then the area of respective faces will be the projected are

22. Stress Principles… Cauchy Stress formula
Consider the equilibrium of a small portion of body in the shape of a tetrahedron
If dS is the area of the base , then the area of respective faces will be the projected are
equilibrium requires the vector sum of all forces acting on the tetrahedron to be zero, that is

23. Stress Principles…
Cauchy Stress formula
Cauchy Stress Formula

24. Aims and Objectives Body and Surface forces mass density Stress
The Stress Tensor
Cauchy Stress Principle
Cauchy Stress formula

25. Stress Principles…
Example: Let the components of the stress tensor at P be given in matrix form by
in units of mega-Pascals. Determine
The stress vector on the plane at P having the unit normal
The stress vector on a plane at P parallel to the plane ABC shown in the sketch.

26. Stress Principles…
Example: Let the components of the stress tensor at P be given in matrix form by
in units of mega-Pascals. Determine
The stress vector on the plane at P having the unit normal

28. Stress Principles…
Example: Let the components of the stress tensor at P be given in matrix form by
in units of mega-Pascals. Determine
(b) The stress vector on a plane at P parallel to the plane ABC shown in the sketch.

30. Stress Principles…
Example: Let the components of the stress tensor at P be given in matrix form by
in units of mega-Pascals. Determine
(b) The stress vector on a plane at P parallel to the plane ABC shown in the sketch.