The nine stress components are shown acting on the surfaces of this infinitesimal element in Figure 12-7 b and c . The components xx , yy , and zz are the normal stresses, so called because they act, respectively, in directions normal to the x , y, and z surfaces of the cube. The components xy and xz , for example, are shear stresses that act on the x face and whose directions of action are parallel to the y and z axes, respectively. The sign of any one of these components is defined as positive if the signs of its surface normal and its stress direction are the same, and as negative if they are different. Thus the components shown in Figure 12-7 b are all positive because they are acting on the positive faces of the cube and their directions are also positive. The components shown in Figure 12-7 c are all negative because they are acting on the positive faces of the cube and their directions are negative. This sign convention makes tensile normal stresses positive and compressive normal stresses negative.

For the 2-D case, only one face of the stress cube may be drawn. If the x and y directions are retained and z eliminated, we look normal to the xy plane of the cube of Figure 12-7 and see the stresses shown in Figure 12-8, acting on the unseen faces of the cube. The reader should confirm that the stress components shown in Figure 12-8 are all positive by the sign convention stated above.

Note that the definition of the dual subscript notation given above is consistent when applied to the normal stresses. For example, the normal stress xx acts on the x face and is also in the x direction. Since the subscripts are simply repeated for normal stresses, it is common to eliminate one of them and refer to the normal components simply as x , y , and z . Both subscripts are needed to define the shear-stress components and they will be retained. It can also be shown[2] that the stress tensor is symmetric, which means that



This reduces the number of stress components to be calculated.

12.7 STRAIN

Stress and strain are linearly related by Hooke’s law in the elastic region of most engineering materials as discussed in Chapter 2. Strain is also a second-order tensor and can be expressed for the 3-D case as

and for the 2-D case as

where   represents either a normal or a shear strain, the two being differentiated by their subscripts. We will also simplify the repeated subscripts for normal strains to x , y , and z for convenience while retaining the dual subscripts to identify shear strains. The same symmetric relationships shown for shear stress components in equation 12.3 also apply to the strain components.

12.8 PRINCIPAL STRESSES

The axis systems taken in Figures 12-7 and 12-8 are arbitrary and are usually chosen for convenience in computing the applied stresses. For any particular combination of applied stresses, there will be a continuous distribution of the stress field around any point analyzed. The normal and shear stresses at the point will vary with direction in any coordinate system chosen. There will always be planes on which the shear-stress components are zero. The normal stresses acting on these planes are called the principal stresses. The planes on which these principal stresses act are called the principal planes . The directions of the surface normals to the principal planes are called the principal axes , and the normal stresses acting in those directions are the principal normal stresses . There will also be another set of mutually perpendicular axes along which the shear stresses will be maximal. The principal shear stresses act on a set of planes that are at 45 ° angles to the planes of the principal normal stresses. The principal planes and principal stresses for the 2-D case of Figure 12-8 are shown in Figure 12-9.

FIGURE 12-9

Principal stresses on a two-dimensional stress element

Since, from an engineering standpoint, we are most concerned with designing our machine parts so that they will not fail, and since failure will occur if the stress at any point exceeds some safe value, we need to find the largest stresses (both normal and shear) that occur anywhere in the continuum of material that makes up our machine part. We may be less concerned with the directions of those stresses than with their magnitudes as long as the material can be considered to be at least macroscopically isotropic, thus having strength properties that are uniform in all directions. Most metals and many other engineering materials meet these criteria, although wood and composite materials are notable exceptions.

The expression relating the applied stresses to the principal stresses is



where   is the principal stress magnitude and nx , ny , and nz are the direction cosines of the unit vector n , which is normal to the principal plane:

For the solution of equation 12.5a to exist, the determinant of the coefficient matrix must be zero. Expanding this determinant and setting it to zero, we obtain

where

Equation 12.5c is a cubic polynomial in . The coefficients C 0, C 1, and C 2 are called the tensor invariants because they have the same values regardless of the initial choice of xyz axes in which the applied stresses were measured or calculated. The units of C 2 are psi (MPa), of C 1 psi2 (MPa2), and of C 0 psi3 (MPa3). The three principal (normal) stresses 1, 2, 3 are the three roots of this cubic polynomial The roots of this polynomial are always real[2] and are usually ordered such that 1 > 2 > 3. If needed, the directions of the principal stress vectors can be found by substituting each root of equation 12.5c into 12.5a and solving for nx , ny , and nz for each of the three principal stresses. The directions of the three principal stresses are mutually orthogonal.