Structural Relaxation in Glass

The structural relaxation model can predict the behavior of glass at temperatures near the glass transition temperature.

The structural relaxation model:

can be isotropic, orthotropic, or fully anisotropic; and
uses fictive temperature to describe the structure of glass.

This page discusses:

Structural Relaxation
Tool-Narayanaswamy-Moynihan Model
Elements
Procedures
Output
References

Structural Relaxation

At high temperatures glass behaves like a liquid, while at low temperatures glass behaves like a thermoelastic solid. In both temperature ranges the physical properties of glass depend only on the instantaneous value of the temperature. However, at temperatures in the vicinity of the glass transition temperature the behavior of glass is different. In this temperature range, the molecular structure of glass changes gradually with temperature and a noticeable delay is observed before the equilibrium state is reached. In this case the physical properties depend on both the temperature and the thermal history.

Tool-Narayanaswamy-Moynihan Model

In Abaqus/Standard the Tool-Narayanaswamy-Moynihan (TNM) model can be used to predict structural relaxation in glass. It uses the fictive temperature, $θ_{f}$ , originally proposed by Tool to describe the structure of the material.

In the TNM model the thermal strains are obtained from the following relation:

ϵ^{t h} = α_{g} (θ, f_{β}) (θ - θ^{0}) + [α_{l} (θ, f_{β}) - α_{g} (θ, f_{β})] (θ_{f} - θ^{0}) - α_{g} (θ^{I}, f_{β}^{I}) (θ^{I} - θ^{0}) - [α_{l} (θ^{I}, f_{β}^{I}) - α_{g} (θ^{I}, f_{β}^{I})] (θ_{f}^{I} - θ^{0})

where

$α_{g} (θ, f_{β})$: is the thermal expansion coefficient for the glassy state;
$α_{l} (θ, f_{β})$: is the thermal expansion coefficient for the liquid state;
$θ$: is the current temperature;
$θ^{I}$: is the initial temperature;
$θ_{f}^{I}$: is the initial fictive temperature;
$f_{β}$: are the current values of the predefined field variables;
$f_{β}^{I}$: are the initial values of the predefined field variables; and
$θ^{0}$: is the reference temperature for the thermal expansion coefficients.

For a special case when the thermal expansion coefficients are not temperature- or field variable-dependent, it can be shown that the thermal strain increment can be expressed as

Δ ϵ^{t h} = α_{g} Δ θ + (α_{l} - α_{g}) Δ θ_{f} .

In the TNM model we must specify the coefficients of thermal expansion for both glassy and liquid states.

Input File Usage

Use the following option to define the thermal expansion coefficient for the glassy state:

EXPANSION

Use the following option to define the thermal expansion coefficient for the liquid state:

EXPANSION, LIQUID

Defining the Reference Temperature

If the coefficients of thermal expansion are not functions of temperature or field variables, the value of the reference temperature, $θ^{0}$ , is not required. If $α_{g}$ or $α_{l}$ is a function of temperature or field variables, you can define $θ^{0}$ .

Input File Usage

Use the following option to define the reference temperature value:

EXPANSION, ZERO =  $θ^{0}$

Fictive Temperature

The fictive temperature is commonly expressed by introducing an equilibrium response function $M (t) :$

θ_{f} (t) = θ (t) - \int_{0}^{t} M (t - t') \frac{d θ}{d t'} d t' .

The response function,

M (t)

, is a decreasing function in time, such that

M (0) = 1

and

M (\infty) = 0

In the Tool-Narayanaswamy-Moynihan model the response function is expressed with a series of exponential functions:

M (t) = \sum_{i = 1}^{N} C_{i} e^{\frac{- t}{τ_{i}}}

where

$C_{i}$: is the material parameter for the $i^{t h}$ term of the response function;
$τ_{i}$: is the relaxation time for the $i^{t h}$ term of the response function; and
$N$: is the number of terms in the response function.

In addition, the material parameters $C_{i}$ must satisfy the relation $\sum_{i = 1}^{N} C_{i} = 1$ .

Combining the equations above, the expression for the fictive temperature has the form

θ_{f} = \sum_{i = 1}^{n} C_{i} θ_{f, i} (ξ (t)),

where

{θ_{f, i} = θ (t) - \int}_{0}^{t} e^{- \frac{ξ (t) - ξ (t')}{τ_{i}}} \frac{d θ}{d t'} d t',

and

ξ = \int_{0}^{t} \frac{1}{A (t')} d t'

is the reduced time, which is computed using the following shift function:

A = e^{\frac{E_{0}}{R} (\frac{χ}{θ - θ^{Z}} + \frac{1 - χ}{θ_{f} - θ^{Z}} - \frac{1}{θ_{R} - θ^{Z}})},

where

$E_{0}$: is the activation energy;
$R$: is the universal gas constant;
$θ_{R}$: is the reference temperature; and
$θ^{Z}$: is the absolute zero in the temperature scale used.

Computation of Fictive Temperature

The fictive temperature at time $t + Δ t$ is computed using the algorithm proposed by Markovsky and Soules:

θ_{f} (t + Δ t) = \sum_{i = 1}^{n} C_{i} θ_{f, i} (t + Δ t)

and

θ_{f, i} (t + Δ t) = \frac{τ_{i} θ_{f, i} (t) + θ (t + Δ t) \frac{d ξ}{d t} Δ t}{τ_{i} + \frac{d ξ}{d t} Δ t} .

Elements

The structural relaxation model can be used with any stress/displacement element in Abaqus/Standard.

Procedures

The model can be used with all stress/displacement procedure types. However, the time effects are taken into account only in quasi-static (see Quasi-Static Analysis), coupled temperature-displacement (see Fully Coupled Thermal-Stress Analysis), and direct-integration implicit dynamic (see Implicit Dynamic Analysis Using Direct Integration) analyses. In other stress/displacement procedure types the evolution of the fictive temperature is suppressed, so its value remains unchanged.

Output

In addition to the standard output identifiers available in Abaqus/Standard (Using Abaqus/Standard Output Variable Identifiers), the following variable has special meaning for the TNM model:

TFICT: Fictive temperature.

References

Markovsky, A., and T. F. Soules, “An Efficient and Stable Algorithm for Calculating Fictive Temperatures,” Journal of the American Ceramic Society, vol. 67, no. 4, pp. 56–57, 1984.
Narayanaswamy, O. S., “A Model of Structural Relaxation in Glass,” Journal of the American Ceramic Society, vol. 54, no. 10, pp. 491–498, 1971.
Tool, A. Q., “Relation Between Inelastic Deformation and Thermal Expansion of Glass in Its Annealing Range,” Journal of the American Ceramic Society, vol. 29, no. 9, pp. 240–253, 1946.