In the analysis of stress, a state of stress at a point can be found by making an infinite number of cuts through the point from which, for each cut, the associated stress vector is known. Similarly, the state of strain at a point is defined as the totality of all the changes in length of lines (fibers) of the material which pass through the point and also the totality of all the changes in angle between pairs of lines radiating from this point.
However, it will be shown later that the change in length of any line of the material which passes through the point and the change in angle between any two lines radiating from this point can be calculated once the changes in length and angle for the three lines parallel to a set of mutually perpendicular coordinate axes through this point are known.
Figure 3.1 shows an infinitesimal line element OP at point O in a body in its unstrained original position with length equal to unity. After deformation, the element is displaced to the new position O ? P ?, as shown in the figure. Notice that for a very small length of the line element, and for smooth variation of the deformations in the neighborhood of point O, the displaced element O ? P ? remains straight. The relative displacement vector of...
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