Components of Stress and Strain

Suppose that a plane cut is made through a solid in equilibrium under the action of some forces (Fig. 3.12a). The distribution of force on the area A in the plane may be represented by an equivalent resultant force RA through point O (also in the plane) and a couple producing moment MA (Fig. 3.12b).
Three mutually perpendicular axes x, y, and z at point O are chosen such that axis x is normal to the plane and y and z are in the plane. RA can be resolved into components Rx, Ry, and Rz, and MA can be resolved into Mx, My, and Mz (Fig. 3.12c). Component Rx is called normal force. Ry and Rz are called shearing forces. Over area A, these forces produce an average normal stress Rx /A and average shear stresses Ry /A and Rz /A, respectively. If the area of interest is shrunk to an infinitesimally small area around point O, then the average stresses would approach limits, called stress components, Æ’x, vxy, and vxz, at point O. Thus, as indicated in Fig. 3.12d,

Because the moment MA and its corresponding components are all taken about point O, they are not producing any additional stress at this point.

If another plane is cut through O that is normal to the y axis, the area surrounding O in this plane will be subjected to a different resultant force and moment through O. If the area is made to approach zero, the stress components Æ’y, vyx, and vyz are obtained. Similarly, if a third plane cut is made through O, normal to the z direction, the stress components are Æ’z, vzx, vzy.
The normal-stress component is denoted by Æ’ and a single subscript, which indicates the direction of the axis normal to the plane. The shear-stress component is denoted by v and two subscripts. The first subscript indicates the direction of the normal to the plane, and the second subscript indicates the direction of the axis to which the component is parallel.
The state of stress at a point O is shown in Fig. 3.13 on a rectangular parallelepiped with length of sides Ax, Ay, and Ax. The parallelepiped is taken so small that the stresses can be

considered uniform and equal on parallel faces. The stress at the point can be expressed by the nine components shown. Some of these components, however, are related by equilibrium conditions:

Therefore, the actual state of stress has only six independent components.

A component of strain corresponds to each component of stress. Normal strains x, y, and z are the changes in unit length in the x, y, and z directions, respectively, when the deformations are small (for example, x is shown in Fig. 3.14a). Shear strains  xy,  zy, and zx are the decreases in the right angle between lines in the body at O parallel to the x and y, z and y, and z and x axes, respectively (for example,  xy is shown in Fig. 3.14b). Thus, similar to a state of stress, a state of strain has nine components, of which six are independent.

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