3.    Stress

    3.1.    Definition

         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.

  ie. Stress = Force / Area

        However as stresses experienced in the earth are so large, stress is more commonly expressed in MegaPascals (MPa, 1MPa = 9.8692 atm).
        As stress is a measure of the internal forces in response to external forces, stress is often required over  a large volume. However as with most problems, to be able to accurately describe the stress, a simplification is required. As such, stress is usually treated in a model of small scale, such as a plane, cube or ellipsoid.

    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 vectors.
        These nine stress vectors are usually expressed in a stress matrix such as in seen below, and is known as the Stress Tensor.

        If the cube shown is in equilibrium (not rotating), then it follows that s21 =s12, s31=s13 and s32=s23. Thus there are only six independent components to the stress tensor, and this means that the stress tensor is a symmetric tensor.
        The cube can be orientated in such a way that the major stress acting on it, is normal to one of the planes, and also that no shear stresses are caused, only normal stresses. This causes the stress tensor to be reduced to
        In this case the stress vectors s11, s22, and s33, are collectively known as the principal stresses. Note also that the principal stresses are all normal stresses, and as such s1, s2,  and s3 act parallel to an axis, which is termed the principal axes. The Principal axes become more important in a later section; the Stress Ellipsoid.

        3.3.1.    Mean Stress

        The mean stress is simply the average of the three principal stresses.

        3.3.2.    Deviatoric Stress

        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

        If all three of the principal axes are of equal magnitude, then the ellipsoid simplifies to a sphere, and each of the infinite number of stress vectors are equal.
        This particular type of stress is termed Hydrostatic stress. Under these conditions, none of the infinite plans feel a shear stress, and the material may undergo volume and/or mineralogical changes, but no deformation occurs. This type of stress is commonly experienced by deeply buried rocks.
        Uniaxial Stress is where only one of the principal axis is non-zero.
        Biaxial stress is where two of the principal axes are non-zero.
        Triaxial Stress is where all three of the principal axes are non-zero.

    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.

        Secondly the center of the circle needs to be located. The center of the circle is simply the mean stress, and it represents the hyrodstatic component of the principal stresses. Next the radius of the circle is needed, and this is the deviatoric stress (See the diagram for the equation), which is the nonhydrostatic component and as mentioned before tends to produce the distortion. Also the diameter of the circle is the differential stress (the top half of the radius equation, or twice the radius), and the greater this value is, the greater the potential for distortion. From this information, plus one more item (the angle q is the angle between s1 and the normal to the plane or 90 degrees minus the angle between s1 and the plane), the shear stress and normal stress pcomponents of the original stress can be calculated (see below).
only half of the circle is shown here
        Once all this information is known, a line is plotted at an angle of 2q in the same direction as it occurs in the diagram of forces (see below) from the center of the circle to the edge. Where the purple lines on the diagram touch the axes, it shows the values of ss and sn.

A more precise way is to calculate the values of ss and sn.The following two equations achieve this.

    Normal Stress                        Shear Stress

        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.


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