Stresses in Cu Thin Films and Ag/Ni Multilayers - Harvard School of ...

Chapter 1: Background 5
and the Hookean stresses are defined by
dw = σij dɛij. (1.5)
For the small strain assumption, the distinction between the undeformed and deformed state
is unnecessary.
Making the small strain approximation is tantamount to choosing a harmonic potential
(the dashed curve in Figure 1.1 (a)). The stress then varies linearly with strain, as shown
in Figure 1.1 (b), and we may write
σij = cijklɛkl (1.6a)
or ɛij = sijklσkl (1.6b)
depending on which is the independent variable. For the one-dimensional model depicted
in Figure 1.1, c1111 is independent of strain (Figure 1.1 (c)).
In addition to writing the stress in terms of strain, I can also write stress in terms of its
definition of force per unit area:
σij = Fi
A j
where the area vector points along the plane normal.
(1.7)
A material’s measured elastic constants may differ substantially from those calculated
from the curvature of the potential well. This may result from either point, line, areal, or
volume defects in the material. If the properties are time-dependent and reversible, they are
anelastic. The reversible motion of defects can cause an abnormally large strain, thereby
reducing the measured elastic constant. This phenomenon may or may not be measurable
depending on the time scale of the experiment since the defect kinetics have characteristic
time constants.