Modeling the Cure Process in Thermosetting Polymers

The cure modeling capabilities:

are intended to model the curing process in adhesives and other polymer materials;
predict the degree of cure, volumetric heat generation, and shrinkage strain due to curing reactions;
allow you to specify a maximum value of the degree of cure;
are intended for use with existing elastic and viscoelastic behaviors that describe the mechanical response of the material as a function of the degree of cure; and
must be used with the fully coupled thermal-stress analysis (see Fully Coupled Thermal-Stress Analysis).

Alternatively, Abaqus offers another method of defining the cure model that uses dedicated cure modeling options and output variables (see Curing Processes in Polymers). This method is more general and is intended to supersede the capabilities described in this section.

This page discusses:

Reaction Kinetics
Cure Heat Generation
Cure Shrinkage Strain
Mechanical Response of the Material
Solution-Dependent State Variables
Example: Defining the Cure Modeling Capabilities
Elements
Procedures
Output
References

Reaction Kinetics

The degree of cure (or conversion) of a material is characterized by a normalized quantity, $α$ , with a value changing from 0 (corresponding to the uncured state; that is, no bonds) to 1 (corresponding to the fully cured material).

The cure reaction kinetics control the rate of conversion, $\dot{α}$ , as a function of $α$ and temperature. The Kamal equation provides a well accepted description of the cure reaction kinetics that is known to produce accurate results, particularly for epoxy resins. It is given by the following rate form:

\begin{matrix} (1) & \dot{α} = \sum_{i = 1}^{N} Z_{i} e^{- \frac{E_{i}}{R (θ - θ^{Z})}} {(b_{i} + α^{m_{i}}) (α_{\max} - α)}^{n_{i}}, \end{matrix}

where $N$ is the number of terms, $Z_{i}$ are rate constants, $E_{i}$ are activation energies, $m_{i}$ and $n_{i}$ are reaction constants, $θ^{Z}$ is the absolute zero on the temperature scale used, and $R$ is the universal gas constant. The material constants, $b_{i}$ , are introduced to allow a nonzero initial conversion rate by setting at least one of them to a small positive value. Alternatively, you can specify a nonzero initial conversion value, ${α |}_{t = 0}$ , using initial conditions (see Solution-Dependent State Variables). If you do not define $b_{i}$ or ${α |}_{t = 0}$ , the degree of cure (conversion) remains equal to zero throughout the analysis.

The constant, $α_{\max}$ , controls the maximum degree of cure. By default, $α_{\max} = 1$ . You can define a different value, as described in Maximum Conversion.

In addition to the Kamal equation, you can specify the rate of conversion, $\dot{α}$ , in a tabular format. This format allows you to express the rate as a function of conversion and, optionally, temperature and field variables:

\dot{α} = f (α, θ, F V s) .

Curing is an irreversible process; therefore, the value of the degree of cure that Abaqus computes never decreases. The value either increases or remains constant.

Input File Usage

Use the following options to define the conversion rate in the Kamal equation:

PHYSICAL CONSTANTS, UNIVERSAL GAS CONSTANT=R, ABSOLUTE ZERO=θ^Z
MATERIAL
PARAMETER TABLE, TYPE="ABQ_Cure_ReactionKinetics_Kamal"
 $Z_{1}$ ,  $E_{1}$ ,  $m_{1}$ ,  $n_{1}$ ,  $b_{1}$ 
 $Z_{2}$ ,  $E_{2}$ ,  $m_{2}$ ,  $n_{2}$ ,  $b_{2}$ 
…
 $Z_{N}$ ,  $E_{N}$ ,  $m_{N}$ ,  $n_{N}$ ,  $b_{N}$

Use the following options to define the conversion rate in tabular form:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_Cure_ReactionKinetics_Tabular", TEMPERATURE
 $\dot{α}$ ,  $α$ ,  $θ$

Maximum Conversion

In general, the degree of cure can reach a maximum value of 1 (corresponding to a fully cured material). However, at lower temperatures, the reaction might slow down considerably, and a fully cured state might not be reached (corresponding to $α_{\max} < 1$ ). You can specify the maximum conversion as a function of temperature. The default is $α_{\max} = 1$ .

Input File Usage

Use the following option to specify the maximum conversion:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_Cure_ReactionKinetics_MaxConversion"
 $α_{\max}$

Cure Heat Generation

Curing reactions are irreversible, exothermic processes that are activated by mixing or heating. The amount of heat released per unit volume per unit time is given by the relationship:

q = ρ_{0} Q \dot{α},

where $ρ_{0}$ is the density, $Q$ is the specific heat of the reaction, and $\dot{α} = \frac{d α}{d t}$ is the conversion rate.

Input File Usage

Use the following options to define the volumetric heat generation rate:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_Cure_HeatGeneration"
 $Q$ ,  $ρ_{0}$

Cure Shrinkage Strain

During the curing process, the material undergoes permanent shrinkage. The shrinkage is due to cross-linking because the formation of bonds moves the atoms closer together than in the unbonded state. The shrinkage and thermal strains that develop during the curing process result in residual stresses that might cause warpage of the final product. Predicting residual stress distributions is often one of the main reasons for performing numerical simulations. You can model the thermal strain using Abaqus capabilities (see Thermal Expansion). The cure shrinkage strain is expressed in the following general rate form:

{\dot{ε}}^{s} = - γ \dot{α},

where $ε^{s}$ is the cure shrinkage strain, $γ$ is the shrinkage coefficient matrix, and $\dot{α}$ is the conversion rate. Abaqus supports four forms to specify cure shrinkage coefficients: volumetric, isotropic, orthotropic, and anisotropic. You can use the orthotropic and anisotropic forms only with materials where the material directions are defined with local orientations (see Orientations).

The cure shrinkage strains enter the constitutive model in the form of an eigenstrain. Similar to thermal strains, the cure shrinkage strains are subtracted from the total deformation to compute the mechanical stress response.

Volumetric Cure Shrinkage Strain

The volumetric cure shrinkage strain is computed from the following rate equation:

{\dot{ε}}^{s} = - \frac{1}{3} γ_{v o l} I \dot{α} (t) .

In this case you only need to specify one value, $γ_{v o l}$ , as a function of temperature and field variables.

Input File Usage

Use the following options to specify the volumetric shrinkage coefficient:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_Cure_ShrinkageCoeff_Vol"
 $γ_{v o l}$

Isotropic Cure Shrinkage Strain

The isotropic cure shrinkage strain is computed from the following rate equation:

{\dot{ε}}^{s} = - γ I \dot{α} (t) .

In this case you only need to specify one value, $γ$ , as a function of temperature and field variables.

Input File Usage

Use the following options to specify the isotropic shrinkage coefficient:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_Cure_ShrinkageCoeff_Iso"
 $γ$

Orthotropic Cure Shrinkage Strain

The orthotropic cure shrinkage strain is computed from the following rate equation:

(\begin{matrix} {\dot{ε}}_{11}^{s} & 0 & 0 \\ 0 & {\dot{ε}}_{22}^{s} & 0 \\ 0 & 0 & {\dot{ε}}_{33}^{s} \end{matrix}) = - (\begin{matrix} γ_{11} & 0 & 0 \\ 0 & γ_{22} & 0 \\ 0 & 0 & γ_{33} \end{matrix}) \dot{α} (t) .

In this case you specify the coefficients in the principal material directions as functions of temperature and field variables.

Input File Usage

Use the following options to specify the orthotropic shrinkage coefficients:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_Cure_ShrinkageCoeff_Ortho"
 $γ_{11}$ ,  $γ_{22}$ ,  $γ_{33}$

Anisotropic Cure Shrinkage Strain

The anisotropic cure shrinkage strain is computed from the following rate equation:

(\begin{matrix} {\dot{ε}}_{11}^{s} & {\dot{ε}}_{12}^{s} & {\dot{ε}}_{13}^{s} \\ {\dot{ε}}_{12}^{s} & {\dot{ε}}_{22}^{s} & {\dot{ε}}_{23}^{s} \\ {\dot{ε}}_{13}^{s} & {\dot{ε}}_{23}^{s} & {\dot{ε}}_{33}^{s} \end{matrix}) = - (\begin{matrix} γ_{11} & γ_{12} & γ_{13} \\ γ_{12} & γ_{22} & γ_{23} \\ γ_{13} & γ_{23} & γ_{33} \end{matrix}) \dot{α} (t) .

In this case you must specify all six components of the coefficient, $γ$ , as functions of temperature and field variables.

Input File Usage

Use the following options to specify the anisotropic shrinkage coefficients:

MATERIAL
PROPERTY TABLE, TYPE="ABQ_Cure_ShrinkageCoeff_Aniso"
 $γ_{11}$ ,  $γ_{22}$ ,  $γ_{33}$ ,  $γ_{12}$ ,  $γ_{13}$ ,  $γ_{23}$

Mechanical Response of the Material

The mechanical response of the material in the uncured state is typically viscoelastic and, in the fully cured state, is often viscoelastic. Therefore, the cure modeling capabilities are typically used in combination with the small-strain viscoelastic modeling capabilities already available in Abaqus.

Cure-Dependent Material Properties

The elastic and viscoelastic material properties can change considerably as the cross-linking progresses and the material transitions from the uncured to the cured state. In general, these material properties depend on the degree of cure. To obtain accurate results, you must account for this dependency. You can consider the dependency of material properties on the degree of cure in the model by associating the value of a field variable with the degree of cure and specifying the material properties as a function of this field variable (see Specifying Material Data as Functions of Solution-Dependent Variables). Hedegaard et al. (2021) describe a testing procedure for measuring viscoelastic properties as a function of temperature and conversion level.

Solution-Dependent State Variables

The cure modeling capability is available as a special-purpose material model that makes use of solution-dependent state variables (SDVs) for multiple purposes, such as defining state variables, initial values of state variables, and output (see Solution-Dependent State Variables). You must allocate at least three solution-dependent state variables. The meaning of the different SDVs is described in the table below.


SDV	Label	Description
1	ALPHA	Degree of cure or conversion. The value is between 0 and 1. The value is equal to 0 if the material is uncured and is equal to 1 if the material is fully cured.
2	ALPHAR	Conversion rate, $\dot{α}$ .
3	DALPHARDT	Derivative of the conversion rate with respect to temperature, $\frac{\partial \dot{α}}{\partial θ}$ .

Optionally, you can specify a nonzero initial value of the degree of cure to trigger the start of the curing reaction (see Equation 1).

Input File Usage

Use the following option to allocate space for solution-dependent state variables:

DEPVAR
3
1, ALPHA, “Degree of cure”
2, ALPHAR, “Degree of cure rate”
3, DALPHARDT, “Derivative of the rate of degree of cure with respect to temperature”

Use the following option to define initial conditions for the degree of cure:

INITIAL CONDITIONS, TYPE=SOLUTION

Example: Defining the Cure Modeling Capabilities

This example illustrates defining the cure modeling capabilities in combination with a typical viscoelastic material definition. The material properties are defined as a function of the degree of cure using field variable dependency. Field variable 1 is used for the purpose of the example.

HEADING
INCLUDE, INPUT=ABQ_Cure_types.inp
MATERIAL, NAME="ABQ_CURE_MATERIAL_matName"
DENSITY
Data lines to specify mass density
ELASTIC, DEPENDENCIES=1
Data lines to specify linear elastic parameters
VISCOELASTIC
Data lines to specify viscoelastic parameters
TRS, DEFINITION=TABULAR, DEPENDENCIES=1
Data line to specify logarithm of the shift function
EXPANSION
Data line to specify thermal expansion coefficients
EIGENSTRAIN, USER
PARAMETER TABLE, TYPE="ABQ_Cure_ReactionKinetics_Kamal"
Data line to specify the Kamal model parameters
PROPERTY TABLE, TYPE="ABQ_Cure_ShrinkageCoeff_Iso"
Data line to specify the isotropc shrinkage coefficients
CONDUCTIVITY
Data line to specify the thermal conductivity
SPECIFIC HEAT
Data lines to specify the specific heat
HEAT GENERATION
PARAMETER TABLE, TYPE="ABQ_Cure_HeatGeneration"
Data lines to define the volumetric heat generation rate
USER DEFINED FIELD, TYPE=SPECIFIED
1, SDV1
DEPVAR
3
1, ALPHA, “Degree of cure”
2, ALPHAR, “Degree of cure rate”
3, DALPHARDT, “Derivative of the rate of degree of cure with respect to temperature”
STEP
COUPLED TEMPERATURE-DISPLACEMENT
Data line to control incrementation and to specify the total time
END STEP

Elements

The cure model supports three-dimensional, plane strain, and axisymmetric continuum elements that have both displacements and temperatures as nodal variables (see Choosing the Appropriate Element for an Analysis Type).

Procedures

The cure model is supported only in fully coupled thermal-stress analyses (see Fully Coupled Thermal-Stress Analysis).

Output

In addition to the standard output identifiers available in Abaqus (Abaqus/Standard Output Variable Identifiers), the following variables have special meaning for the cure material model:

SDV1: Degree of cure (conversion), $α$ .

SDV2: Rate of degree of cure, $\dot{α}$ .

SDV3: Derivative of the rate of degree of cure with respect to temperature, $\frac{\partial \dot{α}}{\partial θ}$ .

EEIG: Cure shrinkage strain, $ε^{s}$ .

References

Hedegaard, A., E. Breedlove, S. Carpenter, V. Jusuf, C. Li , and D. Lindeman, Time-Temperature-Cure Superposition (TTCS) Methods for Determining Viscoelasticity of Structural Adhesives During Curing 44^th Meeting of the Adhesion Society, 2021.
Li, C., Y. Wang , and J. Mason, “The Effects of Curing History on Residual Stresses in Bone Cement During Hip Arthroplasty,” Journal of Biomedical Materials Research, vol. 70B, pp. 30–36, 2004.
Lindeman, D., S. Carpenter , and C. Li, Residual Stress Development During Curing of Structural Adhesives: Experimental Characterization and Modeling 44^th Meeting of the Adhesion Society, 2021.