Modeling Metallurgical Phase Transformations

Material Physical Raw-Liquid-Solid (RLS) State

You can specify a temperature range to model physical state changes. During an additive manufacturing process, the raw (unfused) material is melted to a liquid state by an intense moving heat source, and then undergoes cooling and solidification when the heat source is removed. The solidified (fused) material can be remelted when the heat source passes by a nearby location. As shown in Figure 1, the transitions between raw, liquid, and solid states of a materal alloy are considered to be governed by temperature and occur within a temperature range bounded by the solidus and liquidus temperature of the alloy.

Input File Usage

Use the following options to specify the solidus and liquidus temperature for modeling material physical raw-liquid-solid (RLS) state transition:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_PHASE_TRANS_MeltingTemperature"
 $θ_{s o l i d u s}$ ,  $θ_{l i q u i d u s}$

Solid-Phase Transformation

The rapid heating and cooling events associated with additive manufacturing processes and slower heat treatment applications can induce metallurgical solid-phase transformations. You can define all possible solid phases and the set of rules that govern their transformations.

A solid-phase transformation can be written in the form

p_{1} + p_{2} + \dots + p_{n_{p}} \to c_{1} + c_{2} + \dots + c_{n_{c}},

where $n_{p}$ and $n_{c}$ are the number of parent and child phases, respectively; and $p_{i}$ and $c_{j}$ are the $i^{t h}$ parent and the $j^{t h}$ child phases, respectively.

The change of volume fraction of a parent phase during a transformation is assumed to be proportional to its current volume fraction. A similar assumption is made for child phases. With these assumptions, a transformation can be simplified as

p \to c,

consisting of one main parent phase and one main child phase with volume fractions of $f_{p} = \sum_{i = 1}^{n_{p}} f_{p_{i}}$ and $f_{c} = \sum_{j = 1}^{n_{c}} f_{c_{j}}$ , respectively.

A transformation usually occurs in a specific thermal condition. You can specify the conditions in terms of temperature rate and/or temperature range for a transformation. If both rate and range are specified, the transformation occurs when both conditions are satisfied.

Two types of transformation mechanisms are available: diffusional or martensitic (nondiffusional).

The following steps are required to define solid-phase transformations completely:

Define the information for all solid phases using a parameter table of type "ABQ_PHASE_TRANS_SolidPhases".
Define the information for all solid-phase transformations using a parameter table of type "ABQ_PHASE_TRANS_Transformations".
Define the kinetics of each transformation by either providing time-temperature-transformation (TTT) diagrams or by specifying model coefficients directly using dedicated parameter and property tables.

Input File Usage

Use the following options to define all possible solid phases:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_PHASE_TRANS_SolidPhases"
solid phase name, initial fraction upon solidification
…

Repeat the data line as often as necessary to define a list of solid phases.

Use the following options to define solid-phase transformations:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_PHASE_TRANS_Transformations"
transformation name and definition of parent-child relations, conditions, etc.
…

Repeat the data line as often as necessary to define a list of solid-phase transformations.

Diffusional Transformation

A diffusional transformation is a phase change that involves the long-range diffusion of atoms as a function of both time and temperature. The kinetics of diffusional transformations under isothermal conditions (constant temperature $θ$ ) can be described by the Johnson-Mehl-Avrami (JMA) model

f_{p} = f_{t o t} (1 - (1 - \exp [- k t^{n}]) (1 - f_{p, 0}^{e q})),

where $f_{t o t} = f_{p} + f_{c}$ is the total volume fraction of all parent and child phases participating in the transformation; $k$ and $n$ are the reaction rate constant and Avrami exponent, respectively; and $f_{p, 0}^{e q}$ is the equilibrium relative total volume fraction of the parent phases. The coefficients $k$ and $n$ can be temperature dependent, and they can be calibrated from a time-temperature-transformation (TTT) diagram. Figure 2 shows a representative TTT diagram illustrating a diffusional transformation (D) and a martensitic transformation (M). A transformation is usually represented by two curves in the TTT diagram: a start curve where the transformation begins (for example, $D_{s}, M_{s}$ in Figure 2) and a finish curve where the transformation completes (for example, $D_{f}, M_{f}$ in Figure 2). A curve in a TTT diagram represents the time (as a function of temperature) that a transformation has completed a certain proportion. For example, the start curve of the diffusional transformation, $D_{s}$ , can be represented by the time, $t_{s}^{D} (θ)$ , when the transformation begins and the corresponding relative total volume fraction of child phases, $f_{s}^{D}$ . Therefore, the JMA coefficients of a diffusional transformation can be calibrated from the TTT curves as

n (θ) = \frac{\ln (\ln (1 - f_{s}^{D}) / \ln (1 - f_{f}^{D}))}{\ln (t_{s}^{D} (θ) / t_{f}^{D} (θ))},

k (θ) = - \frac{\ln (1 - f_{s}^{D})}{{(t_{s}^{D} (θ))}^{n (θ)}} .

Additive manufacturing processes involve continuous temperature changes in rapid heating and cooling events. The kinetics of a diffusional transformation under general non-isothermal conditions is modeled by an extended Johnson-Mehl-Avrami (JMA) model that considers each time increment is a small isothermal step,

f_{p} (θ_{t} + Δ θ) = f_{t o t} (1 - (1 - \exp [- k {(ξ + Δ t)}^{n}]) (1 - f_{p, 0}^{e q})), i f f_{p} > f_{t o t} f_{p, 0}^{e q}

where $Δ t$ is the time increment size; $θ_{t}$ is the temperature at the end of the last increment; $Δ θ$ is the temperature change for the current time increment; and $ξ$ is the fictitious time that would provide an equivalent amount of transformed material, $f_{p} (θ_{t})$ , at a constant temperature $θ + Δ θ$ :

ξ = {[- \frac{1}{k} \ln (\frac{f_{p} (θ_{t}) - f_{t o t} f_{p, 0}^{e q}}{f_{t o t} (1 - f_{p, 0}^{e q})})]}^{\frac{1}{n}} .

If the diffusional transformation is reversible ( $p \leftrightarrow c$ ), the reverse transformation $c \to p$ can be described as

f_{p} (θ_{t} + Δ θ) = f_{t o t} (1 - \exp [- k {(ξ + Δ t)}^{n}]) f_{p, 0}^{e q}, i f f_{p} \leq f_{t o t} f_{p, 0}^{e q} .

You can define the kinetics of the transformation using either a TTT diagram or calibrated JMA model coefficients.

Input File Usage

Use the following options to define the kinetics of a diffusional transformation using a TTT diagram:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_PHASE_TRANS_Diffusional_TTT_Constants", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_TTT_Constants"
 $f_{s}$ ,  $f_{f}$ 
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_Diffusional_TTT_Diagram", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_TTT_Diagram", TEMPERATURE
 $t_{s} (θ)$ ,  $t_{f} (θ)$ ,  $θ$ 
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_Diffusional_ParentEquiFrac", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_ParentEquiFrac", TEMPERATURE
 $f_{p, 0}^{e q} (θ)$ ,  $θ$

Use the following options to define the kinetics of the diffusional transformation using JMA model coefficients:

MATERIAL
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_Diffusional_JMA_Coefficients", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_JMA_Coefficients", TEMPERATURE
 $k (θ)$ ,  $n (θ)$ ,  $θ$ 
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_Diffusional_ParentEquiFrac", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_ParentEquiFrac", TEMPERATURE
 $f_{p, 0}^{e q} (θ)$ ,  $θ$

NameOfTransformation is the name of the transformation defined in the parameter table of type ABQ_PHASE_TRANS_Transformations.

Martensitic Transformation

A martensitic transformation solely depends on temperature, as described by the Koistinen-Marburger (KM) model,

f_{c} (θ) = f_{c} (θ_{0}) + (f_{p} (θ_{0}) - f_{p r}) (1 - \exp [- γ (M_{s} - θ)]),

where $θ_{0}$ is the temperature at the beginning of the current cooling cycle, and the martensitic start temperature $M_{s}$ and the transformation coefficient $γ$ can be calibrated from TTT diagrams. Since a martensitic transformation is independent of time, this type of transformation is usually represented by at least two horizontal lines in the TTT diagram, as shown in Figure 2. For example, the start curve of the martensitic transformation can be represented by the temperature $M_{s}$ where the transformation begins and the corresponding volume fraction of the children phases $f_{s}^{M}$ . Thus, the transformation coefficient $γ$ can be computed as

γ = - \frac{\ln ((1 - f_{s}^{M}) / (1 - f_{f}^{M}))}{M_{f} - M_{s}} .

Abaqus uses a backward Euler integration of the differential form of the original KM model,

f_{c} (θ + Δ θ) = \frac{f_{c} (θ) - γ Δ θ (f_{p} (θ_{0}) - f_{p r} + f_{c} (θ_{0}))}{1 - γ Δ θ} .

The transformation occurs if the initial volume fraction of parent phases is higher than the retained amount of parent phases; namely, $f_{p} (θ_{0}) > f_{p r}$ .

You can define the kinetics of the transformation using either a TTT diagram or calibrated KM model coefficients.

Input File Usage

Use the following options to define the kinetics of the martensitic transformation using a TTT diagram:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_PHASE_TRANS_Martensitic_TTT_Constants", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_TTT_Constants"
 $f_{s}$ ,  $f_{f}$ 
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_Martensitic_TTT_Diagram", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_TTT_Diagram"
 $M_{s}$ ,  $M_{f}$ 
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_Martensitic_ParentRetainedFrac", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_ParentRetainedFrac"
 $f_{p r}$ ,  $f_{p} (θ_{0})$

Use the following options to define the kinetics of the martensitic transformation using KM model coefficients:

MATERIAL
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_Martensitic_KM_Coefficients", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_KM_Coefficients"
 $γ$ 
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_Martensitic_ParentRetainedFrac", 
LABEL="ABQ_PHASE_TRANS_Trans_NameOfTransformation_ParentRetainedFrac"
 $f_{p r}$ ,  $f_{p} (θ_{0})$

NameOfTransformation is the name of the transformation defined in the parameter table of type "ABQ_PHASE_TRANS_Transformations".

Grain Morphology

You can specify a solidification map for modeling grain morphology. As shown in Figure 3, given the magnitude of thermal gradient $G = | \frac{\partial θ}{\partial x} |$ and solidification rate $R = | \dot{θ} | / G$ , the grain morphology can be determined to be equiaxed, columnar, or a mixture of the two morphologies.

Input File Usage

Use the following options to define a solidification map to model the evolution of the grain morphology:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_PHASE_TRANS_SolidificationMap_Constants"
critical solidification rate  $R_{c r}$ , slope of the constant grain size line  $k_{d 0}$ 
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_SolidificationMap_Diagram", 
LABEL="ABQ_PHASE_TRANS_SolidificationMap_Diagram_Equiaxied"
 $\log_{10} (G_{e q u i a x e d})$ ,  $\log_{10} (R)$ 
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_SolidificationMap_Diagram", 
LABEL="ABQ_PHASE_TRANS_SolidificationMap_Diagram_Columnar"
 $\log_{10} (G_{c o l u m n a r})$ ,  $\log_{10} (R)$

Grain Size

The evolution of the grain size can be estimated using a parabolic growth law in the form:

d^{n} - d_{0}^{n} = k t \exp (- \frac{Q}{R (θ - θ^{Z})}),

where $d$ is the grain size at time $t$ ; $d_{0}$ is the initial grain size upon solidification; $k$ , $n$ , and $Q$ are material grain-growth properties; $R$ is the universal gas constant; and $θ^{Z}$ is the absolute zero.

Input File Usage

Use the following options to define the constants and coefficients of the grain-growth law:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_PHASE_TRANS_GrainSize_Definitions"
 $d_{0}$ , grain growth temperature condition,  $θ_{1}$ ,  $θ_{2}$ 
*PROPERTY TABLE, TYPE="ABQ_PHASE_TRANS_GrainSize_Coefficients"
 $k$ ,  $n$ ,  $Q$

Use the following option to define physical constants used in the grain-growth law:

*PHYSICAL CONSTANTS, ABSOLUTE ZERO= $θ^{Z}$ , UNIVERSAL GAS CONSTANT= $R$

Specifying Material Properties as Functions of Metallurgical State

You can specify material properties as functions of metallurgical state. For example, thermal conductivity and specific heat capacity can be different for raw and fused materials. This is achieved by introducing a series of field variables that characterize the metallurgical state and by defining the material properties as a function of those field variables (see Specifying Material Data as Functions of Temperature and Independent Field Variables). The metallurgical state includes material physical state, volume fraction ratio of solid phases, volume fraction of columnar grain, and grain size. The volume fraction ratio of solid phase $i$ ( $i > 1$ ) is defined as

r_{i} = \frac{f_{i}}{\sum_{k = 1}^{i} f_{k}} .

This definition ensures that the $n_{s} - 1$ volume fraction ratios are independent and that their values range from 0 to 1. For example, if a material has four solid phases and it is found that Young's modulus, $E$ , and Poisson's ratio, $ν$ , are different for different solid phases, the material data can be defined as


$E$	$ν$	$r_{2}$	$r_{3}$	$r_{4}$
$E_{1}$	$ν_{1}$	0	0	0
$E_{2}$	$ν_{2}$	1	0	0
$E_{3}$	$ν_{3}$	0	1	0
$E_{4}$	$ν_{4}$	0	0	1

where $E_{i}$ and $ν_{i}$ are Young's modulus and Poisson's ratio of solid phase $i$ , respectively.

A dedicated parameter table of type "ABQ_PHASE_TRANS_FieldVariables" is available to assign field variables to metallurgical state quantities.

Input File Usage

Use the following options to assign metallurgical state quantities to field variables used in the definition of material properties:

MATERIAL
PARAMETER TABLE, TYPE="ABQ_PHASE_TRANS_FieldVariables"
field variable index, metallurgical state quantitiy

Repeat the data line as often as necessary to assign the metallurgical state to field variables.

Solution-Dependent State Variables

The metallurgical phase transformation framework is available as a special-purpose modeling technique that makes use of solution-dependent state variables (SDVs) (see About User Subroutines and Utilities) for multiple purposes, such as defining initial values, output, and workspaces used for computation. You must allocate at least $3 + n_{s} + {3 n}_{t}$ solution-dependent state variables, where $n_{s}$ and $n_{t}$ are the number of sets of data defined in the parameter tables of type "ABQ_PHASE_TRANS_SolidPhases" and "ABQ_PHASE_TRANS_Transformations", respectively. You must ensure that those spaces are properly allocated and not overwritten by other user subroutines. The meaning of the different SDVs is described below.


SDV	Label	Description
1	RLS	Raw/liquid/solid state. This variable is varying between –1.0 to 1.0. The state is –1.0 if the material is raw, 0.0 if the material is liquid, and 1.0 if the material is solid. The state is between –1.0 and 0.0 if the material is a mixture of raw and liquid. The state is between 0.0 to 1.0 if the material is a mixture of liquid and solid.
2	FGRAINCOLUMNAR	Volume fraction of columnar grain. The variable is 0.0 if the grain is fully equiaxed, 1.0 if the grain is fully columnar, and between 0.0 and 1.0 if the grain is a mixture of equiaxed grain and columnar grain. The variable is –1.0 when the grain morphology is unavailable when the material is liquid.
3	GRAINSIZE	Grain size.
4	FPHASE_1	Volume fraction of solid phase 1.
5	FPHASE_2	Volume fraction of solid phase 2.
...	...	...
$3 + n_{s}$	FPHASE_N	Volume fraction of solid phase $n_{s}$ .

You can output SDV1 to SDV( $3 + n_{s}$ ). The ordering of volume fraction of solid phases starting from SDV4 follows the ordering defined in the parameter table of type "ABQ_PHASE_TRANS_SolidPhases". You can customize a more meaningful output label of SDVs based on the name of the solid phases. For example, if phase 1 is named "alpha", you can use FPHASE_ALPHA as the output label instead of FPHASE_1.

SDV( $3 + n_{s} + 1$ ) to SDV( $3 + n_{s} + 3 n_{t}$ ) are workspaces used for solid-phase transformations.

Input File Usage

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

DEPVAR
 $3 + n_{s} + {3 n}_{t}$ 
1, RLS, “raw/liquid/solid state”
2, FGRAINCOLUMNAR, “volume fraction of columnar grain”
3, GRAINSIZE, “grain size”
4, FPHASE_1, “volume fraction of solid phase 1”    
5, FPHASE_2, “volume fraction of solid phase 2”
…
 $3 + n_{s}$ , FPHASE_N, “volume fraction of solid phase  $n_{s}$ ”

Initial Conditions

You can specify initial values of material physical state, volume fraction of solid phases, volume fraction of columnar grain, and grain size by defining initial values of the corresponding solution-dependent state variables. For example, you can specify the initial value –1 for SDV1 to represent the initial raw material. See Defining the Initial Values of Solution-Dependent State Variables for more details.

References

Crespo, A., “Modelling of Heat Transfer and Phase Transformations in the Rapid Manufacturing of Titanium Components,” Convection and Conduction Heat Transfer, IntechOpen, 2011.
Kobryn, P. A., , and S. L. Semiatin, “Microstructure and Texture Evolution During Solidification Processing of Ti–6Al–4V,” Journal of Materials Processing Technology, vol. 135, no. 2-3, pp. 330–339, 2003.
Zhang, Q., J. Xie, Z. Gao, T. London, D. Griffiths , and V. Oancea, “A Metallurgical Phase Transformation Framework Applied to SLM Additive Manufacturing Processes,” Materials & Design, vol. 166, 2019.