Stress is a measure of the internal forces in a body between its constituent particlaes, as they resist separation, compression or sliding in response to externally applied forces. The mathematical definition of stress is defined as; the force exerted per unit area.
3.2. Stress on a plane
When an arbitary stress is applied to a plane, it can be resolved into three components, as seen in the figure below.
As it can be
seen above, when stress is applied to a plane, it can be expressed in terms
of one normal stress (the stress acting perpendicular to the plane), and
two shear stresses (the stress acting parallel to the plane).
As previously mentioned, simplification of the problem is required. The plane model works well when only considering a singular plane. This can be taken one step further to assume that the stress at a single point needs to be calculated, as the stress throughout a volume can vary. To calculate the stress at a point we take the simplification, that a point is in fact an infinitely small cube, and this is treated in the next section.
3.3. Stress at a point
If you treat a point as an infintiely small cube, it is obvious that a cube has six faces, or three pairs of planes as faces. It is only important to consider three faces, as the other three parallel faces are identical in natures.
As shown in
the diagram the stress on each face can be shown in terms of three seperate
stress vectors, and the stresses experienced can be expressed in nine stress
These nine stress vectors are usually expressed in a stress matrix such as in seen below, and is known as the Stress Tensor.
3.3.1. Mean Stress
The mean stress is simply the average of the three principal stresses.
Now that we can calculate the mean stress, we can break the stress tensor down into two components. The first part or isotropic component is the mean stress, and is responsible for the type of deformation mechanism, as well as dilation. The second component is the Deviatoric stress and is what actually causes distortion of the body. When considering the deviatoric stress, the maximum is always positive, representing compression, and the minimum is alway negative, representing tensional.
3.4. The Stress Ellipsoid
As with the
last example the cube, the stress ellipsoid is another, more acurate
way to study the state of stress at a point.
The Stress tensor, still applies, but the principal stresses now come into play.
In the large majority of cases, one principle stress is larger then the other two, and the remaining two also differ in magnitude. The maximum principal stress is usually called s1, the intermediate principal stress is usually called s2, and the minimum principal stress is usually called s3. When represented visually, you get the stress ellipsoid as shown below.
3.4.1. The kinds of stress
3.5. Mohr's Circle
Although the stress ellipsoid is good for visualising the orientation and relative magnitude of the principal stresses at a point, and for examining the relationship between stress vectors and the stress tensor, it is not easily used to show the relationship between the orientation of a plane and the magnitudes of the shearing and normal stresses on it. This is where the Mohr diagram comes into play.
The Mohr Circle
is used to calculate the shear and normal stresses on a plane. Using the
example below, the diagram shows a plane, upon which two forces are exerted,
F1 and F3. From the information given, using Trigonometry,
the calculation of the individual shear and normal stresses are possible
(although not covered in this report, the derivation is available here
- a Tutorial
on the Mohr Circle).
||F = Force
Fn = Normal Force
Fs = Shear Force
3.5.1. Setting up and interpreting a Mohr circle.
When a Mohr
circle is constructed, eventually it is possible to know that type of tension
or compression present by simply looking at the position of the circle.
Firstly a set of axes need to be drawn. The horizontal one will represent the normal stress, and the vertical one the shear stress.
A more precise way is to calculate the values of ss and sn.The following two equations achieve this.
After drawing the Mohr circle, a judgement about the type of stress experienced is able to be made. The diagram below shows these simple cases.