Theory of elasticity is useful in solving the problems in structural and continuum mechanics by the finite element method.
In theory of elasticity, usually right hand rule is used for selecting the coordinate system. Fig. shows various orientations of right hand rule of the coordinate systems.
In this Chapter orientation shown in Fig. (a) Is used for the explanation.
Fig. Shows a typical three dimensional element of size dx × dy × dz. Face abcd called as negative face of x and the face efgh as the positive face of x since the x value for face abcd is less than that for the face efgh. Similarly the face aehd is negative face of y and bfgc is positive face of y. Negative and positive faces of z are dhgc and aefb.
The direct stresses and shearing stresses acting on the negative faces are shown in the Fig. with suitable subscript. It may be noted that the first subscript of shearing stress is the plane and the second subscript is the direction. Thus the means shearing stress on the plane where x value is constant and y is the direction.
In a stressed body, the values of stresses change from face to face of an element. Hence on positive face the various stresses acting are shown in Fig. above with superscript ‘+’. All these forces are listed in table
Let the intensity of body forces acting on the element in x, y, z directions be X, Y and Z respectively as shown in Fig.
The intensity of body forces is uniform over entire body. Hence the total body force in x, y, z direction on the element shown are given by (i) X dx dy dz in x – direction (ii) Y dx dy dz in y – direction and (iii) Z dx dy dz in z – direction
Equation of equilibrium:
Strains: Corresponding to the six stress components given in equation, the state of strain at a point may be divided into six strain components as shown below
In equation, strains are expressed up to the accuracy of second order (quadratic) changes in displacements. These equations may be simplified to the first (linear) order accuracy only by dropping the second order changes terms. Then linear strain – displacement relation is given by: