2 1. NATURE AND SOURCE OF RESIDUAL STRESSES
For horizontal equilibrium with zero force resultant, the area of the tensile (C) stress
region shaded in Figure 1.1a equals the sum of the areas of the compressive () stress regions.
e presence of the compressive stress regions also at the left and right sides of the photoelastic
measurement in Figure 1.1b illustrates that equilibrium is obeyed on all planes, giving zero force
resultants both horizontally and vertically. Similar zero in-plane force resultants occur with the
shear stresses. In this example, moment equilibrium is automatically obeyed because of the stress
symmetry, but it is also obeyed for the general non-symmetric case.
Because they exist locked-in within a material that has no external loads, residual stresses
seemingly have a “hidden” character. ey give few readily apparent indications of their presence
and so can easily be overlooked. ere is also sometimes a tendency among designers to shy away
from considering residual stresses because they can be difficult to measure, predict or analyze.
However, inclusion of residual stresses is very important for practical applications because they
add to the structural stresses due to the applied loads and can possibly make the total stress
much different than anticipated. Inclusion of residual stresses in engineering design analysis has
always been an important need, but it is increasingly urgent as modern structures become lighter
and less conservative.
1.2 ORIGIN OF RESIDUAL STRESSES
e combination of tensile and compressive stress regions within a material that characterize
residual stresses are caused by “misfits” among those regions. For example, the stresses shown
for the heat-tempered glass illustrated in Figure 1.1 occur as a result of dimensional differences
between the outer surface material and the interior material. During manufacture, the glass is
heated to become a liquid, formed into a flat shape and then cooled to become a solid glass
sheet. Figure 1.2 schematically illustrates the steps in the process. Figure 1.2a shows the glass as
a uniformly very hot liquid. For clarity, the cross-section is divided into three conceptual regions
comprising the two surfaces and the interior. e solidification step in Figure 1.2b is done by
rapidly cooling the surfaces using air jets. is causes the surfaces to solidify while the hotter
interior material is still liquid, allowing it to flow freely and adjust to the thermal shrinkage of
the cooler surface material. As the overall cooling progresses, the hotter interior material also
solidifies as shown in Figure 1.2c and forms a continuous solid with the cooler surface material.
Subsequently, all material cools to ambient temperature, as shown in Figure 1.2d. However,
during this final cooling, the interior cools through a larger temperature range than the outer
surfaces. Consequently, it shrinks more due to thermal contraction. us, to maintain material
continuity, compressive residual stresses are formed in the surfaces, balanced by tensile stresses
in the interior, as schematically shown in Figure 1.2e.
e thermally toughened glass example of Figures 1.1 and 1.2 illustrates a case where
residual stresses are deliberately created. However, material misfits and their associated residual
stresses are induced by almost all manufacturing methods, e.g., casting, grinding, forming and
welding. Indeed, the name residual stresses derives from their creation as a residue from such
1.2. ORIGIN OF RESIDUAL STRESSES 3
Figure 1.2: Residual stress creation in thermally toughened glass.
prior processing. Figure 1.3 shows an example of residual stresses within an optical lens. In this
example, insufficiently slow cooling during solidification of the lens material created significant
residual stresses in the final product, as revealed by crossed isoclinic lines seen when viewing in
plane polarized light. ese lines show the locations where the principal stress directions cor-
respond to the principal polarization directions. is method of observation in plane polarized
light is a common quality control step in lens manufacture. e response shown in Figure 1.3 is
very severe and would be unacceptable for a good quality lens.
e material misfits that are the source of residual stresses are called “inherent strains” or
“eigenstrains.” e measurement and identification of these strains provides a powerful method
for quantifying residual stresses. An idealized “stress-free” material without any interior misfits
could be a perfect single crystal of a metal or a perfectly uniform amorphous material created
through a quasi-infinitesimal cooling rate. However, all practical materials have some amount
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