Pandat Help Center

Name

Operation

Examples

J

The transient nucleation rate

Nν

and

Potential nucleation sites

Z

Zeldovich factor

β*

Atomic attachment rate

τ

Incubation time

ΔG*

Nucleation barrier energy

R*

Critical nucleation radius

ΔGvol

ΔGV+ΔGS

Volume energy change during nucleation.ΔGV is the chemical driving force per volume and ΔGS is the elastic strain energy

ΔGS

Elastic strain energy. The volume misfit ∆ and particle aspect ratio AR are given in kdb file

σeff

Effective interfacial energy

σαβ

The estimated interfacial energy based on GBB method

Deff

Effective diffusivity for multi-component alloys

Name

Options

Description

model

KWN, Fast-Acting(FA)

Refer to Figure 5.2 

morphology

Sphere; Cylinder

Refer to Figure 5.3. The aspect ratio AR and shape factors are set to be 1 automatically when “Sphere” is selected

nucleation

Modified_Homogeneous; Grain_Boundary; Grain_Edge; Grain_Corner; Dislocation

Refer to Table 11.5. Both homogeneous and heterogeneous nucleation can be considered by “Modified_Homogeneous”. In this case, the values of Nν and ∆G* must be manually adjusted through Nf, ∆Gv and ∆GS as discussed in Table 11.8.

growth

Simplified;

SFFK;

SFFK_Shape_Evolution

A constant value of The aspect ratio AR can be assigned for “Simplified” and “SFFK”. Choose “SFFK_Shape_Evolution” for shape evolution, which means AR varies during particle growth

phase_name

Each “PrecipitatePhase” has a “name” and “phase_name”. “phase_name” must be consistent with the name in tdb/pdb. The “phase_name” tag can be empty if “name” and “phase_name” are same.

Name

Options

Description

model

Grain_OneSize, Grain_MultiSize

Refer to Figure 5.2

morphology

Sphere

Only spherical approximation is considered

nucleation

N/A

Recystallization process is not included

growth

Grain_Simplified

The simplified grain growth model is implemented as shown in Equations Eq. 5.38 and Eq. 5.41. User’s grain growth model can be defined in KDB (see example Section User-defined growth rate model for grain coarsening)

Name

Unit

Description

Equation

Molar_Volume

m3/mole

Molar volume of matrix or precipitate phase

<Parameter type="Molar_Volume" value="6E-6" description="Molar Volume" />

Vm in Eq. 5.2 and Vα in Eq. 5.3

Grain_Size

m

The grain size of the matrix phase

<Parameter type="Grain_Size" value="1e-4" description="Grain size, default value = 1e-4m" />

D in Eq. 5.12

Dislocation_Density

m-2

The dislocation density in the matrix phase

<Parameter type="Dislocation_Density" value="1e13" description = "Dislocation density, Default value  =1.0e12/m^-2" />

ρi in Eq. 5.10

Grain_Aspect_Ratio

N/A

The aspect ratio for the matrix grain

<Parameter type="Grain_Aspect_Ratio" value="1.0" description="grain aspect ratio, default value = 1.0" />

A in Eq. 5.12

Contact_Angle

degree

Contact angle of nucleus on grain boundary, default value = 90 degree

in Eq. 5.8

Aspect_Ratio

N/A

The aspect ratio of the precipitate phase. The value of ARis evolving if “SFFK_Shape_Evolution” is chosen as growth model.

<Parameter type="A_R" value="1" description ="Initial aspect ratio" />

AR in Eq. 5.6

Interfacial_Energy

J/m2

Interfacial energy

<Parameter type="Interfacial_Energy" value="0.2" description ="Interfacial Energy" />

User keyword “IFE_CAC(*)” to get the calculated interfacial energy:

<Parameter type="Interfacial_Energy" value="IFE_CALC(*)" description="Interfacial Energy" />

in Eq. 5.2 and Eq. 5.13

Interfacial_Energy_L

J/m2

Interfacial energy in L direction

<Parameter type="Interfacial_Energy_L" value="0.05" description="Interfacial Energy in L direction" />

Used in “SFFK_Shape_Evolution” model

Antiphase_Boundary_Energy

J/m2

Antiphase boundary energy

in Eq. 5.34 and Eq.  5.36

Atomic_Spacing

m

Usually use lattice constant

<Parameter type="Atomic_Spacing" value="7.6E-10" description="Atomic Spacing" />

a in Eq. 5.4

Nucleation_Site_Parameter

N/A

Homogeneous: choose a value close to solute concentration;

Heterogeneous: choose a value close to nucleation density when “Modified_Homogeneous” option is chosen for nucleation model. Otherwise, use the model automatically estimate the nucleation density and default value of 1.0 can be used. Such an example is given in Section: Another Example for Heterogeneous Nucleation

Nf in Eq. 5.7 and Eq. 5.11

Driving_Force_Factor

N/A

A factor adjusting chemical driving force obtained by thermodynamic calculation

A pre-factor applied to ΔGV in Eq. 5.2

Strain_Energy

The elastic strain energy per volume of precipitate offsetting the calculated value by equation Eq. 5.5.

Volume_Misfit

N/A

The volume misfit

Δ in Eq. 5.6

Kinetic_Parameter_Factor

N/A

A factor adjusting kinetic parameter obtained by thermodynamic and mobility calculation

A pre-factor applied to adjust K in equation Eq.  5.14

Effective_Diffusivity_Factor

N/A

A factor adjusting effective diffusivity for nucleation obtained by mobility calculation

A pre-factor applied to adjust Deff  in Eq. 5.4

Steady_State_Nucleation_Rate

N/A

0: transient nucleation rate;

1: steady state nucleation rate;

in Eq. 5.1

Name

Unit

Description

Equation

Interfacial_Energy

J/m2

High angle grain boundary energy

γ in equations Eq. 5.40, Eq. 5.41 and Eq. 5.42

Grain_Boundary_Width

m

Grain boundary width. A suggested value is equal to twice the atomic radius.

δ in Eq. 5.39

Grain_Boundary_Mobility_Factor

N/A

A pre-factor to adjust grain boundary mobility.

A2 in Eq. 5.39

Zener_Drag_Force_Factor

N/A

A factor for Zener drag force, default is 0 for not considering Zener pinning effect.

β in Eq. 5.41

Name

Unit

Description

Equation

Grain_Boundary_Mobility

m/(s·J/mol)

User-defined grain boundary mobility. It can be an expression, for example: 1e-18/2.86e-10

<VariableTable name="Variables replacing built-in variables"
Parameter type="Grain_Boundary_Mobility" value="KP(@Grain)/2.86e-10" description="Grain_Boundary_Mobility" />

KP(@Fcc) ( in Eq. 5.39 is the effective atomic mobility for multi-component alloys calculated from mobility database.

Zener_Drag_Force

J/mol

User-defined drag force for Zener pinning effect. It can be an expression, for example: -1.5*7.1E-6*0.75*0.15/(0.3 * 1e-6), which is "-factor*Vm*Sigma*fv/size"

Growth_Rate

m/s

User-defined grain growth rate. It can be an expression, for example: KP(@Grain)/2.86e-10*2*0.75*7.1e-6/s(@Grain) with KP(@Grain) is the effective mobility and s(@Grain) is the mean grain size

Name

Unit

Description

Equation

Shear_Modulus

Pa

The shear modulus of the matrix phase

μ in Eq. 5.6 and Eq. 5.36

Burgers_Vector

m

The Burgers vector of the matrix phase

b in Eq. 5.36

Taylor_Factor

N/A

The Taylor factor of the matrix phase

M in Eq. 5.37

Solution_Strengthening_Factor

N/A

scaling factor of alloying element for solution strengthening

aj in Eq. 5.31

Strength_Parameter

N/A

Strengthening parameter due to precipitation hardening

kP in Eq. 5.27

Shearing_Critical_Radius

m

Critical radius shifting from shearing to looping mechanism

RC in Eq. 5.27

Intrinsic_Strength

MPa

The baseline contribution including lattice resistance, work-hardening and grain boundaries hardening.

σ0 in Eq. 5.30

Hardness_Factor

N/A

The yield strength in MPa can be converted to hardness in VPN based on Eq. 5.31

A in Eq. 5.32

Hardness_Constant

VPN

The yield strength in MPa can be converted to hardness in VPN based on Eq. 5.31

B in Eq. 5.32

Name

Unit

Description

Equation

f_WH

N/A

Work hardening coefficient for dislocation density evolution model

fw in Eq. 5.44

f_DRV

N/A

Dynamic recovery coefficient for dislocation density evolution model

fv in Eq. 5.45

f_SRV

N/A

Static recovery coefficient for dislocation density evolution model

f_RX

N/A

Recrystallization coefficient for dislocation density evolution model

fx in Eq. 5.46

Avrami_Exponent

N/A

Exponent for JMAK

n in Eq. 5.49

time_half

s

The time for 50% RX for JMAK

t0.5 in Eq. 5.49

Nucleation_Site_Parameter

N/A

Parameter for potential nucleation sites for the Fast-Acting nucleation model

Nf  in Eq. 5.57

Kinetic_Parameter_Factor

N/A

Kinetic parameter factor for the Fast-Acting growth model, which is a pre-factor to adjust the effective self-diffusivity Deff

Pre-factor of Deff in Eq. 5.60

Name

Description

Dislocation_Density_Rate

User-defined dislocation density rate to replace the built-in K-M model defined by Eq. 5.43

Nucleation_Rate

User-defined nucleation rate to replace the built-in model defined by Eq. 5.58

Nucleation_Barrier_Energy

User-defined Nucleation barrier energy to replace the built-in model defined by Eq. 5.56

Critical_Grain_Size_RX

User-defined Critical nucleus size for RX to replace the built-in model defined by Eq. 5.55

Dislocation_Mean_Free_Path

User-defined Dislocation mean free path for critical density to replace the built-in model defined by Eq. 5.54

Growth_Rate

User-defined Growth rate to replace the built-in RX growth model defined by Eq. 5.59

Name

Unit (SI)

Comments

time

second

Time

T

K

Temperature

vft

Total Transformed Volume Fraction: where vfp is the transformed volume fraction of phase

x(comp), w(comp)

Overall alloy composition

Name

Unit (SI)

Comments

s(@phase)

m

Average size/radius of equivalent sphere particles or grain

D(@phase)

m

Diameter of cylinder

L(@phase)

m

Length/Height of cylinder

A_R(@phase)

m

Aspect ratio of cylinder

nd(@phase)

#m-3

Number density

nr(@phase)

m-3sec-1

Nucleation rate

vf(@phase)

Volume fraction of specified phase

x(comp@phase), w(comp@ phase)

Instant composition of the matrix or precipitate phases

IFE_CALC(@phase)

J/m2

Model calculated interfacial energy

dgm(@phase)

J/mole

Nucleation driving force of phase(s)

vf_range(@phase,lb,ub)

The volume fraction for different particle groups defined by a size range [lb, ub] such as primary, secondary and tertiary in Ni-based super alloys, for example vf_range(@L12_FCC,0.5e-8, 0.5e-7)

s_range(@phase,lb,ub)

average size for different particle groups defined by a size range [lb, ub], for example s_range(@L12_FCC,0.5e-8, 0.5e-7)

Grain_Size

m

Mean grain size for the deformed matrix grains and the recrystallized grains

dislocation_density(@*)

1/m2

Dislocation density in the deformed matrix grains or the recrystallized grains. Example: dislocation_density(@Grain)

Name

Unit

Comments

time

s

The PSDs are saved for the user-specified times; the PSD for the last time step is automatically saved. Using time = t to get the PSD for time “t”.

psd_id

N/A

The PSD consists of a certain number of cells (size classes); psd_id gets the cell id.

psd_s(@phase)

m

The characteristic size of a precipitate phase or grain for each cell.

psd_nd(@phase)

#m-3

The number density of a precipitate phase or grain for each cell.

psd_gr(@phase)

m/sec

The growth rate of a precipitate phase or grain for each cell.

psd_ns(@phase)

Normalized size of the cell

psd_nnd(@phase)

Normalized number density of the cell 

psd_df(@phase)

The distribution function: with being the cell width

psd_cvf(@phase)

Cumulative volume fraction of phase(s). Example: psd_cvf(@L12_FCC).

Name

Unit (SI)

Comments

sigma_y

MPa

Overall yield strength. Example: sigma_y

Hv

vpn

Overall microhardness. Example: hv

sigma_i

MPa

Intrinsic yield strength. Example: sigma_i.

sigma_ss

MPa

Yield strength due to solution strengthening. Example: sigma_ss.

sigma_p(@*)

MPa

Yield strength due to precipitation hardening. Example: sigma_p(@Mg5Si6).

sigma_m(@Grain) 

MPa

Yield strength due to dislocation interactions in grain structure.

Name

Comments

_K

Boltzmann constant

_PI

Archimedes' constant.

_R

Molar gas constant.

_NA

Avogadro constant.

_E

Natural Logarithmic Base.