When dealing with the evaluation of residual stresses and deformations, two
quantities of interest are the temperature and the displacement vector. They are
respectively denoted by 
and 
. In elasticity modeling, the deformation tensor 
is defined from the displacement vector 
as follows:
 | (26–1) |
Following Chambers [6], the residual stress tensor
is obtained as follows:
 | (26–2) |
where 
and 
are the bulk and shear relaxation moduli, respectively,
is the thermal strain, 
is the unit tensor, and 
is the integration variable. The residual stress tensor must also
satisfy the following momentum equation at all times:
 | (26–3) |
In a general form, the bulk and shear relaxation moduli are described by means of Prony series as follows:
 | (26–4) |
 | (26–5) |
where 
and 
are bulk elastic moduli, 
and 
are shear elastic moduli, and 
and 
are the relaxation times for each Prony component. In general,
these moduli are characterized by 
and 
modes, respectively, and by a steady value reached after
relaxation. In Equation 26–2,
is a reduced time variable, which is given by the integral
invoking a shift function suggested by Narayanaswamy [26]:
 | (26–6) |
where 
and 
are a reference temperature and the fictive temperature,
respectively. In Equation 26–6,
denotes the ratio of activation energy to the ideal gas constant,
while 
is a fractional parameter ranging between 0 and 1. Often, the
temperature is expressed in a non-absolute scale, and the relationship for
calculating 
is then written as
 | (26–7) |
where 
denotes the absolute zero temperature in the current scale (for
example, –273 when the temperature is given in Celsius).
The fictive temperature 
is related to the actual temperature by means of the following
equation:
 | (26–8) |
where 
is a structural relaxation modulus, and is given by
 | (26–9) |
where 
are the individual weights for each Prony component of the
structural relaxation function and 
are the corresponding relaxation times.
Finally, the thermal strain 
in glass is a function of both temperature and fictive temperature
histories. It is therefore obtained from an integral involving liquid and glassy
properties described by means of the corresponding expansion coefficients
and 
as follows:
 | (26–10) |
where the coefficients 
and 
can be constants, or up to third-order polynomial functions of the
temperature.