Curing Processes in Polymers

Modeling Curing Processes

Curing processes are essential to the manufacturing of products that use thermosetting polymers (such as epoxy resins) to bond components. The use of epoxies and other cured structural adhesives is common in many industries. As a result of the curing reaction, chemical shrinkage strains and residual stresses develop, which can result in damage to the adherents or warpage of the bonded assembly. The Abaqus cure modeling capability enables you to analyze curing processes, including the reaction kinetics, heat generation, shrinkage strain development, and the evolution of mechanical properties. The model is based on the work of Lindeman et al. (2021) and Li et al. (2004). The mechanical response can include both elastic and viscoelastic effects. You can define the elastic, viscoelastic, and thermal expansion properties of the material as functions of the degree of cure and temperature.

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). Since curing is an irreversible process, the value of the degree of cure that Abaqus computes never decreases. The value either increases or remains constant. Abaqus provides four methods of specifying the cure kinetics: the Kamal equations, the Grindling equation, a conversation rate table, and a user-defined form using user subroutine UCURE.

Kamal model

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:

\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}},

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 Initial Conditions) or set at least one of the parameters,

m_{i}

, to zero. If you do not define

b_{i}

,

{α |}_{t = 0}

, or

m_{i} = 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.

Input File Usage

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

PHYSICAL CONSTANTS, ABSOLUTE ZERO=θ^Z
MATERIAL
CURE KINETICS, DEFINITION=KAMAL

Grindling model

The Kamal equation describes cure kinetics very well at cure temperatures higher than the glass transition temperature, where the reaction kinetics is chemically controlled. However, during isothermal curing processes the curing temperature can fall below the glass transition temperature, $θ_{g}$ , even if it was higher than $θ_{g}$ at the beginning of the curing process. This is because the glass transition temperature depends strongly on the degree of cure, and its value changes during the curing process. If the curing temperature falls below glass transition temperature, premature vitrification takes place and the material undergoes a transition from a rubbery to a glassy state, which slows the curing process rate considerably since the reaction kinetics becomes diffusion controlled. Although the Kamal model can account for this behavior by limiting the maximum value of conversion by specifying $α_{\max} < 1.0$ for temperatures below the glass transition temperature, the Grindling model can predict gradual transition from the chemically controlled to the diffusion-controlled reaction rates and produce accurate results in this case. The conversion rate for the Grindling model is given by:

\dot{α} = K_{1} {(α_{\max} - α)}^{n_{1}} + K_{e f f} {α^{m} (α_{\max} - α)}^{n_{2}}

where

n_{1}

,

n_{2}

, and

m

are material parameters,

K_{1}

is reaction rate, which is assumed to have the Arrhenius-type form:

K_{1} = A_{1} e^{- (\frac{E_{1}}{R (θ - θ^{Z})})},

where

A_{1}

is a material parameter and

E_{1}

is the activation energy.

The effective rate coefficient, $K_{e f f}$ , is used to switch between the chemically and the diffusion-controlled reaction, and it is given by

\frac{1}{K_{e f f}} = \frac{1}{K_{D}} + \frac{1}{K_{2}},

where

K_{2}

and

K_{D}

are the chemically and diffusion controlled rates, respectively. The rate coefficient,

K_{2}

, has the Arrhenius-type form similar to the rate coefficient

K_{1}

:

K_{2} = A_{2} e^{- (\frac{E_{2}}{R (θ - θ^{Z})})},

where

A_{2}

is a material parameter and

E_{2}

is the activation energy.

The diffusion-controlled rate coefficient, $K_{D}$ , is computed from the following relations depending on the value of the glass transition temperature:

\begin{matrix} If θ > (θ_{g} + Δ θ_{g}) : & K_{D} = K_{D}^{θ = θ_{g}} e^{\frac{c_{1} Δ θ_{g} (c_{2} + Δ θ_{g}) (θ - θ_{g}) {- c}_{1} c_{2} (θ_{g} - θ^{Z} + Δ θ_{g}) (θ_{g} - θ + Δ θ_{g})}{{(c_{2} + Δ θ_{g})}^{2} (θ - θ_{g})}} \\ If θ_{g} \leq θ \leq (θ_{g} + Δ θ_{g}) : & K_{D} = K_{D}^{θ = θ_{g}} e^{\frac{c_{1} (θ - θ_{g})}{c_{2} + θ - θ_{g}}} \\ If θ < θ_{g} : & K_{D} = K_{D}^{θ = θ_{g}} e^{- (\frac{c_{1}}{c_{2}}) (\frac{θ_{g} - θ^{Z}}{θ - θ^{Z}}) (θ_{g} - θ)}, \end{matrix}

where

c_{1}

,

c_{2}

, and

Δ θ_{g}

are material parameters and

K_{D}^{θ = θ_{g}}

is the diffusion-controlled rate coefficient at the temperature equal to the glass transition temperature.

Input File Usage

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

PHYSICAL CONSTANTS, ABSOLUTE ZERO=θ^Z
MATERIAL
CURE KINETICS, DEFINITION=GRINDLING

The diffusion-controlled rate, $K_{D}$ , strongly depends on the glass transition temperature, which in turn depends on the degree of cure and possibly other quantities, such as temperature. Therefore, the glass transition temperature must be specified when using the Grindling model. Abaqus provides two methods to define the glass transition temperature: the DiBenedetto model and the tabular form. The DiBenedetto model is widely used and defines $θ_{g}$ as a function of the degree of cure in the following form:

\frac{θ_{g} - θ_{g, 0}}{θ_{g, \infty} - θ_{g, 0}} = \frac{λ α}{1 - (1 - λ) α}

where

λ

is a material parameter (

λ > 0

),

θ_{g, 0}

is the glass transition temperature in the uncured state, and

θ_{g, \infty}

is the glass transition temperature in the fully cured state.

Alternatively, you can use a tabular definition to specify $θ_{g}$ as a piecewise linear function of degree of cure, temperature, and field variables.

Input File Usage

Use the following option to specify the glass transition temperature using the DiBenedetto equation:

CURE GLASS TRANSITION TEMPERATURE, DEFINITION=DIBENEDETTO

Use the following option to specify the glass transition temperature in tabular form:

CURE GLASS TRANSITION TEMPERATURE, DEFINITION=TABULAR

Conversion rate in tabular form

You can also specify the rate of conversion, $\dot{α}$ , in 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) .

Input File Usage

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

MATERIAL
CURE KINETICS, DEFINITION=TABULAR

User-defined conversion rate

An alternative method to define the cure kinetics is to specify the degree of cure rate in the user subroutine UCURE. In addition to the degree of cure rate, you must also specify the values of the derivatives of the degree of cure rate with respect to the temperature and the degree of cure. Optionally, you can also specify the number of property values needed as data in the user subroutine as well as the number of solution-dependent variables (see About User Subroutines and Utilities).

Input File Usage

Use the following options to specify the degree of cure rate in user subroutine UCURE:

MATERIAL
CURE KINETICS, DEFINITION=USER, PROPERTIES=n

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:

CURE MAX CONVERSION

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
DENSITY
CURE HEAT GENERATION

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 of 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).

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 need to specify only a single coefficient, $γ_{v o l}$ , as a function of temperature and field variables.

Input File Usage

Use the following option to specify the volumetric shrinkage coefficient:

CURE SHRINKAGE, TYPE=VOL

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 need to specify only a single coefficient,

γ

, as a function of temperature and field variables.

Input File Usage

Use the following option to specify the isotropic shrinkage coefficient:

CURE SHRINKAGE, TYPE=ISO

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 option to specify the orthotropic shrinkage coefficients:

CURE SHRINKAGE, TYPE=ORTHO

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 shrinkage coefficient matrix, $γ_{i j}$ , as functions of temperature and field variables.

Input File Usage

Use the following option to specify the anisotropic shrinkage coefficients:

CURE SHRINKAGE, TYPE=ANISO

Example: Defining the Cure Properties of a Material

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 example.

HEADING
MATERIAL, NAME="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
CURE KINETICS, DEFINITION=KAMAL
Data line to specify the Kamal model parameters
CURE MAX CONVERSION
Data line to specify maximum value of the degree of cure
CURE SHRINKAGE, TYPE=ISO
Data line to specify the isotropic shrinkage coefficients
CURE HEAT GENERATION
Data lines to define the volumetric heat generation rate
CONDUCTIVITY
Data line to specify the thermal conductivity
SPECIFIC HEAT
Data lines to specify the specific heat
USER DEFINED FIELD, TYPE=SPECIFIED
1, DOC
STEP
COUPLED TEMPERATURE-DISPLACEMENT
Data line to control incrementation and to specify the total time
END STEP

Initial Conditions

You can specify a nonzero initial value of the degree of cure of the material. This approach is commonly used to define a small value to trigger the start of the curing reaction.

Input File Usage

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

INITIAL CONDITIONS, TYPE=CURE

Material Options

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.

Input File Usage

Use the following option to associate the degree of cure with a field variable:

USER DEFINED FIELD, TYPE=SPECIFIED
FV_number, DOC

Elements

You can use the cure modeling capabilities with any stress/displacement, coupled temperature-displacement, heat transfer, or coupled thermal-electrical elements in Abaqus (see Choosing the Appropriate Element for an Analysis Type). However, the cure shrinkage strain is ignored when elements without displacement degrees of freedom are used (that is, heat transfer and coupled thermal-electrical elements).

Procedures

The cure modeling capabilities are active in the following procedures:

Transient static analysis (Quasi-Static Analysis)
Implicit dynamic analysis (Implicit Dynamic Analysis Using Direct Integration)
Transient fully coupled temperature-displacement analysis unless no creep response (COUPLED TEMPERATURE-DISPLACEMENT, CREEP=NONE) is requested (Fully Coupled Thermal-Stress Analysis)
Transient fully coupled thermal-electrical-structural analysis unless no creep response (COUPLED TEMPERATURE-DISPLACEMENT, ELECTRICAL, CREEP=NONE) is requested (Fully Coupled Thermal-Stress Analysis)
Transient heat transfer analysis (Uncoupled Heat Transfer Analysis)
Transient coupled thermal-electrical analysis (Coupled Thermal-Electrical Analysis)

The capability is inactive during any other procedure, and the value of the degree of cure remains constant.

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:

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

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

DDOCRDTEMP: Derivative of the rate of degree of cure with respect to temperature.

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

References

Grindling, J., Simulation zur Verarbeitung von reaktiven Nonpost-cure-Epoxidharz-Systemen im Druckgelieren und konventionellen Vergiessen Dissertation, Shaker, Aachen, 2006.
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.