Modeling the Cure Process in Thermosetting Polymers

The degree of cure (or conversion) of a material is characterized by a normalized quantity, $\alpha$, 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, $\dot{\alpha }$, as a function of $\alpha$ 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:

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, ${\theta }^Z$ 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, ${\alpha |}_{t=0}$, using initial conditions (see Solution-Dependent State Variables). If you do not define $b_i$ or ${\alpha |}_{t=0}$, the conversion remains equal to zero throughout the analysis.

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

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

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.

PHYSICAL CONSTANTS, UNIVERSAL GAS CONSTANT=R, ABSOLUTE ZERO=θZ
MATERIAL
PARAMETER TABLE, TYPE="ABQ_Cure_ReactionKinetics_Kamal"
$Z_1$, $E_1$, $m_1$, $n_1$, $b_1$
$Z_2$, $E_2$, $m_2$, $n_2$, $b_2$
…
$Z_N$, $E_N$, $m_N$, $n_N$, $b_N$

MATERIAL
PROPERTY TABLE, TYPE="ABQ_Cure_ReactionKinetics_Tabular", TEMPERATURE
$\dot{\alpha }$, $\alpha$, $\theta$

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

where ${\rho }_0$ is the density, $Q$ is the specific heat of the reaction, and $\dot{\alpha }=\frac{d\alpha }{dt}$ is the conversion rate.

MATERIAL
PARAMETER TABLE, TYPE="ABQ_Cure_HeatGeneration"
$Q$, ${\rho }_0$

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:

where ${\mathbf{\epsilon }}^s$ is the cure shrinkage strain, $\mathbf{\gamma }$ is the shrinkage coefficient matrix, and $\dot{\alpha }$ 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.

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.

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.

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

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”

INITIAL CONDITIONS, TYPE=SOLUTION

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