time ξ. An important consequence is that theCorrespondence Principle (see, e.g., Gurtin and Sternberg [36]) is no longer useful. This principle, given non-time-varying boundary conditions, allows obtaining the solution for a linear viscoelasticity boundary value problem from the problem’s solution in linear elasticity, i.e., the solution of the problem with the linear elastic constitutive law. It can be seen through the examination of the field equations that the farthest point the Correspondence Principle would be valid is a temperature field that varies in time but homogeneous in space. For example, sufficiently slow cooling of a hot thermorheologically simple viscoelastic body may be approximated like this (see, e.g., Scherer and Rekhson [76]). Since the thermal tempering problem is certainly not in this class, further elaboration on this solution method is not necessary. Actually, semi- analytical solutions to thermoviscoelastic problems with non-uniform temperature fields are limited (e.g., Muki and Sternberg [62]) only to a few cases with simple geometries and often under simplifying assumptions such as elastic (instantaneous) bulk response. The thermoviscoelastic tempering solution for a uniformly cooled infinite plate by Lee at al. [50]
is in this group detailed in the next section.
the same way it is done in thermoelasticity (e.g., Boley and Weiner [15]). Hence
σkk=K0(εkk(x, t)−3εθ(x, t)) (2.19)
whereK0 = 3K, and K is the elastic bulk modulus.
This equation, along with equations (2.16), (2.17) and an appropriate shift functiona(T) define the constitutive behavior. The solution of this particular boundary value problem is not reproduced here since it is one of the problems that are considered for the thermal tempering of bulk metallic glasses in this thesis. The Finite Element Method (FEM) is generically employed throughout this study to solve the boundary value problems specifying constitutive laws also since problems other than the infinite plate case are too complicated for semi-analytic solutions and require numerical solutions4.
The instant freezing theory (e.g., Aggarwala and Saibel [3]) can be deduced from the viscoelastic theory by assigning an instant freezing temperature such that the material becomes a non-viscous fluid above it and an elastic solid below it. Therefore, glass transition is artificially shrank to a point. Under certain operating conditions, it is shown to yield good accuracy for the final residual stresses, though obviously, it does not comprise the physics to predict the correct evolution of stresses.
Now, let us worry about the validity of the assumed elastic bulk behavior. For silicate glasses, the partial justification behind this and another commonly used assumption for simplifying viscoelastic problems, K(t) G(t),∀t, is demonstrated in Figure 2.3 that has been adapted from Rekhson and Rekhson [72]. First, the bulk relaxation is seen to
4The solution of Lee et al. [50], too, requires numerical routines to tackle the coupled time integration of the constitutive law and the equilibrium equation which is not entirely trivial (this is the reason the term semi-analytic has been adopted for such solutions). Indeed, their solution method did not give enough accuracy resulting in erroneous results later corrected by Narayanaswamy and Gardon [65]
be more sluggish than the shear relaxation, and second, the equilibrium modulus K∞ is nonzero, unlike isochoric shear relaxation which is generally observed to continue until complete relaxation, i.e., G∞'0 andG0 = 2µ. Rekhson and Rekhson reported τid/τis= 6 and K0/K∞ = 3. Then, they calculated the uniaxial relaxation modulus E(t) and biaxial
K(t), G(t)
t(s)
Figure 2.3: Bulk relaxation modulus K(t) (curve 1) in comparison to shear relaxation modulusG(t) (curve 2) for silicate glass at 473◦C (adapted from Rekhson and Rekhson [72]).
relaxation modulus M(t) whose elastic counterparts are E and 1−νE , respectively, for the two cases simplified with the aforementioned assumptions and the exact case for which K(t)5 shown in Figure 2.3 is used. They observed that neither of the two assumptions yield particularly closer results to the exact solution and both seem reasonable for the problems of uniaxial and biaxial loading. The biaxial relaxation modulus is the viscoelastic function that relates in-plane stress to in-plane strain for the one-dimensional (i.e., fields (stress, temperature, etc.) vary only in the thickness direction) infinite plate problem. So, this study of Rekhson and Rekhson specifically targets the plate tempering problem with the treatment for biaxial modulus.
To draw conclusions for the present study, if metallic glass demonstrates similar charac-
5The bulk relaxation modulus curve here has been inverted from creep compliance measurements.
teristics, the elastic bulk response appears to be a good assumption. Unfortunately, there is lack of data from metallic glasses on this issue. For this reason, the present study limits itself to an elastic bulk deformation. In addition, the relatively low pressures employed in BMG casting may be expected to limit the inadequacy of ignoring bulk relaxation.
There is another significant shortcoming of the thermoviscoelastic theory that will be discussed starting in the next section, which led to the structural theory of tempering (Narayanaswamy [64]): The thermoviscoelastic theory makes use of the relaxation data for stabilized glass, which was shown to be indeed thermorheologically simple (see, e.g., Kurkjian [48]). Stabilized glass is a term in silicate glass terminology for liquid at metastable equilibrium. However, rapidly-cooled glass deviates from metastable equilibrium and mere temperature dependence does not describe the behavior well. Rather, it is required to account for the dependence of structure ontemperature history.