Wire Arc Additive Manufacturing of NiTi 4D Structures: Influence of Interlayer Delay

Abstract

Shape memory alloy structures for actuator and vibration damper applications may be manufactured using wire arc additive manufacturing (WAAM), which is one of the additive manufacturing technologies. Multilayer deposition causes heat accumulation during WAAM, which rises the preheat temperature of the previously created layer. This leads to process instabilities, which result in deviations from the desired dimensions and mechanical properties changes. During WAAM deposition of the wall structure, a systematic research is carried out by adjusting the interlayer delay from 10 to 30 s. When the delay period is increased from 10 to 30 s, the breadth decreases by 45% and the height increases by 33%. Grain refinement occurs when the interlayer delay duration is increased, resulting in better hardness, phase transformation temperature, compressive strength, and shape recovery behavior. This study shows how the interlayer delay affects the behavior of WAAM-built nickel–titanium alloy (NiTi) structures in a variety of applications.

Introduction

Shape memory alloy (SMA) has established itself as a popular material for valve actuators and micropumps in a range of micro-electro-mechanical systems (MEMS).^1,2 Nitinol, also known, as nickel–titanium alloy (NiTi), is the most frequently used SMA because of its superior thermomechanical performance, pseudoelasticity, and biocompatibility. NiTi is used in the development of sensors, structural components, actuators, and other devices in a wide range of engineering areas.³ Hot pressing, hot extrusion, metal injection molding, and mechanical alloying⁴ are the traditional manufacturing techniques for fabricating NiTi structures. Traditional techniques, on the contrary, are difficult and expensive to manufacture complex-shaped NiTi components.⁵

The use of additive manufacturing (AM) techniques to manufacture NiTi SMA components appears to be extremely attractive and promising in this respect. Owing to its high accuracy and capacity to create large and complex structures, laser additive manufacturing (LAM) is the most widely utilized AM technology for creating NiTi structures.^6,7 However, LAM suffers from a low build rate and high system cost.⁸

Wire arc additive manufacturing (WAAM) is a low-cost AM method that includes melting the feedstock wire with an electric arc source like gas metal arc welding (GMAW)/gas tungsten arc welding/plasma arc welding. Fabrication of NiTi structures by WAAM has several benefits, including a high build rate (4 kg/h), low equipment cost, good deposition efficiency, and low operating expenses (owing to wire's lower cost than pre-alloyed powders).^9–11 The recent trend indicates that the WAAM is gaining significant attention from industrial manufacturing owing to the above advantages and its ability to build large-sized components sustainably.² However, its ability to build components with precise geometry limits the procedure.^5,12 Issues like melt-pool instability, residual stress, distortions, and heat accumulation during the process are particularly important when constructing wall structures because the process works on the principle of depositing layer-by-layer.

As the wall structures are built by laying single tracks one over the other, excessive heat accumulation takes place during the process. The heat accumulation increases the preheat temperature of the previously deposited layer during the multilayer deposition. This causes instability in the process, resulting in a deviation from the required dimension.⁶ In addition, the accumulation of preheating temperature changes the cooling rate during the WAAM deposition, especially which results in differences in the metallurgical characteristics (like microstructure, grain morphology, dislocation density, etc.). Thus, the geometry and metallurgical characteristics of WAAM-built structures can be controlled by varying the temperature distribution during the process. Literature shows that researchers have attempted to vary the process parameters and process conditions to control the temperature distribution and thereby control built characteristics.^6,11

According to Wang et al.,¹² the distance between the trailing end and the center of the molten pool increased by 1.95 mm from the first to the fifth layer owing to increased heat accumulation during WAAM. Similar findings were previously published (Zhao et al.) during the thermal investigation of multilayer WAAM deposition.¹³ Zhou et al. developed a three-dimensional model to simulate arc formation and metal transfer behavior during WAAM.¹⁴ For both single-bead and overlap deposition, the distribution of thermal conductivities and molten pool properties were studied. Because of the lower net heat flow, the molten pool's high-temperature zone is smaller during overlapping deposition than during single-bead deposition.¹⁴

Although this modeling and experimental study has offered some useful information, the underlying processes of arc characteristics and metal transfer behavior related to heat accumulation remain elusive owing to the complexity of the WAAM process. The properties of WAAM-built structures can be controlled by controlling the interlayer temperature. This can be carried out by applying changes in specific dwell time or using forced cooling.⁶ Controlling the interlayer temperature can also aid in controlling the oxidation during the deposition especially during the processing of reactive materials.^7,11,15,16 Denlinger et al. found that the interlayer dwell length, which is linked to thermal characteristics of the material, has a significant influence on the residual stresses and distortion in as-fabricated nickel and titanium alloy components.¹⁷ Considering the wide applications (actuators, vibration damper, etc.) of WAAM-built SMAs for customized MEMS devices, it is necessary to build thin wall structures with tailored geometries and built characteristics.

There is no publicly available literature that explains the influence of interlayer delay on the mechanical properties, microstructure, and shape memory properties of WAAM-built NiTi-based SMAs. As a result, the objective of this research was to investigate how varied interlayer delay periods impact the microstructure, shape memory properties, and mechanical properties of a WAAM-built NiTi-based SMA.

Materials and Methods

A 1.2 mm diameter wire of Ni50.9Ti49.1 (Table 1) composition is used as the feedstock material. For the WAAM deposition, mild steel is used as the substrate plate. More information regarding the system and the process can be found elsewhere.^18,19 The WAAM deposition is carried out with different interlayer delays of 10, 20, and 30 s, respectively, for six subsequent layers. Excepting the change in the interlayer delay time, all other parameters were kept constant for all the six layers. More details of the process parameter selection and system are reported elsewhere.⁵ Scanning speeds, voltages, feed rates, argon gas flow rates, and standoff distances for deposition are 8.57 mm/s, 16.5 V, 5 m/min, 20 L/min, and 15 mm, respectively. The deposition was performed in an open chamber and no specific cooling system was used for the WAAM deposition process. The density, geometry, microstructural characteristics, mechanical properties, and actuation behavior of the produced samples are thoroughly examined.

Table 1.

Properties of Shape Memory Alloy (NiTi)¹⁹

Properties	Value
Melting point	1310°C
Density	6.45 g/cm3
Thermal conductivity	0.1 W/cm°C
Resistivity	76–82 ohm·cm
Ultimate tensile strength	754–960 MPa
Heat capacity	0.077 cal/g°C
Typical elongation to fracture	15.5%
Typical yield strength	500–560 MPa
Latent heat	5.78 cal/g
Elastic modulus	28–75 GPa
Poisson's ratio	0.3

NiTi, nickel–titanium alloy.

A one-color pyrometer Sensortherm METIS M318 (Sensortherm GmbH, Sulzbach, Germany) with a temperature range of 150°C to 1200°C and a spectral range of 1.65–2.1 m was used to measure the temperature on the surface of the previously deposited layer. Porosity distribution is measured using the X-ray tomography technique (Make and Model: ZEISS and Xradia 620). WAAM-built wall structures are scanned with Comet Blue Light LED 3D Scanner (Make and Model—ZESIS and COMET L3D) for obtaining the 3D image. The width and height are measured at various locations using a vernier caliper. The walls are sectioned transversely to the laying direction using a wire-cut electrical discharge machine. In addition, samples are prepared according to conventional metallographic procedures. For microstructural analysis, samples are polished and etched with Kroll's reagent solution.

Phase analysis is performed using an X-ray diffractometer (BRUKER-D8 Advance) with a step size of 0.02° and a dwell period of 0.5 s from 20° to 90°. The Scherrer equation¹⁸ is used to compute crystallite size, and JCPDS file number 00-035-1281 is utilized to calculate X-ray diffraction lattice parameters. A scanning electron microscope (SEM) with Energy Dispersive Spectroscopy (Make and Model: S-4800 Hitachi) is used for microstructural investigation and compositional mapping. Samples were heated and cooled within the temperature range of 100 − 110°C at a rate of 10°C/min. At 1.96 N load, a Vickers microhardness tester (Make and Model: WlterUhl-VMHT002) with a 10-s dwell period is utilized to measure the microhardness. As per ASTM E9 standard,^18,20 samples with dimensions of 6 mm length and 3 mm width are utilized for compression testing (Make and Model: INSTRON). To test the sensing and bending capabilities, a set-up based on Joule heating is used.

One sample with a cross-section of 1 × 1 mm² and thickness of 1 mm was used for electrical actuation (Fig. 8c). A laser displacement sensor, a data acquisition system (Agilent DAQ 34790A), a monitor, a sample holder, and a configurable power supply are included in the setup (RIGOL DP1308). The Arduino relay circuit controls the cooling and heating of the SMA structures during a 15-s duty cycle. Loads of 40 g, 4 V voltage, and 4 A current were used in this actuation analysis. The temperature range and the experimental conditions for the actuation studies are taken from one of the previously published articles.⁵

FIG. 8.

(a) Time versus displacement graph for the WAAM-built NiTi samples as a function of interlayer delay. (b) Recovery angle versus time graph, (c) image of the actuation sample, and (d) shape recovery during hot plate actuation with respect to temperature change.

Numerical Modeling

ABAQUS 6.20 software is used to perform 3D finite element simulations for thermal analysis of the WAAM process.

For the analysis, the following assumptions are taken into account.

• In the beginning, the boundary conditions are applied to the substrate material at ambient temperature (i.e., 298 K).

• For simplicity, the deposit's geometry is considered to stay constant throughout the process, and the surface is assumed to be completely flat.

• The effects of convection and radiation are taken into account.

• It is believed that deposition is isotropic and homogeneous.

Temperature-dependent material properties are assumed, and their values are based on published reports.^21–25 The heat conduction equation [refer to Eq. (1)]¹⁸ is used to compute the temperature distribution.

Free convection boundary conditions are established on the bottom and top surfaces of the base plate, as well as on the vertical surfaces of the wall. According to literature correlations,^1,26 the convection coefficients for the top surface of the base plate, the bottom surface, and the vertical surface of the wall are 8.5, 4, and 12 W/m² K, respectively. General radiation to the environment boundary condition is included, with the material emissivity set at 0.2.²²

The heat loss q_loss from the building structure's surface owing to convective heat transfer and radiation is taken into account. q_loss is calculated using the following equation.^1,25,26

The chosen mesh for the wall was made up of cubic pieces, as described in previous articles.²¹ In the bedplate, the mesh size is doubled in the X and Z axes to minimize computing time. Two models are used to compute the idle time provided during deposition: one for cooling and the other for heating (deposition) simulation. The heat source, which is used only in the heating model, is the only difference between the two models in terms of material characteristics, mesh topology, and boundary conditions. At the end of the heating procedure, the deposition of the current layer is simulated without any idle time. The beginning circumstances for the cooling phase are determined by the end state of this simulation, which includes the initial temperature field and the active/inactive state of the elements. The initial value of the (T_max) variable determines the first element activation state. After then, the cooling simulation begins, modeling the workpiece's thermal behavior during the idle time following the deposition of the current layer.

The defined idle time is then utilized as input for the new layer's heating simulation. This study's heat source model is a modified version of the double ellipsoid model,⁶ which has been adapted for the GMAW process. The heat distribution function is given in Equation (3):

The boundary conditions used for modeling heat sources are presented in Equations (4) and (5).

If Equation (5) is met, the words f_f,r are distribution factors with different values for the frontward and backward ellipsoids. The heat input of the welding process is divided into two power density contributions in the suggested heat source: the base metal (q_b) and the filler or deposited metal (q_w) [refer to Eq. (6)]:

If the condition in Equation (7) is met, the amount of power transmitted to both the base q_b and the filler metal q_w can be regulated, where η = 50%,^1,6,21–26

The temperature distribution results show that the measurement point is critical for heat distribution and correlations between energy input and actual temperatures. A molten pool's shape may vary owing to the substantial increase in temperature. A change in the shape of a molten pool might cause the geometric shape of the deposited layer to deteriorate. As a result, the created model will be beneficial in the future for the development of a feedback control system for WAAM process temperature management.

More details about the simulation parameter and other related details are reported in our previously published report.¹¹

Results and Discussion

Thermal analysis

Using finite element analysis, the influence of interlayer delay on the preheat temperature of the previously built layer during WAAM is studied. Figure 1a provides the average temperature distribution generated by the simulation. The impact of interlayer delay on the preheat temperature is given in Figure 1a for various values of interlayer delay determined by numerical simulation and pyrometer. The variations/differences are provided as error bars. The maximum variation in the preheat temperature between experimental values and simulation is <2%. A short interlayer delay time results in higher preheat temperature, which has a preheat effect on the deposition of the subsequence layers. Furthermore, as the deposition progresses from the bottom to the top layer, the preheat temperature for a given delay time rises. This is mostly owing to the substrate effect, which causes fast heat dissipation at the lower layers. As the deposition moves to the top layer, the heat accumulation increases, and the heat dissipation rate reduces.

FIG. 1.

(a) Temperature distribution profile observed in the WAAM samples according to the model, and CT scan images of the WAAM-built NiTi walls structures as a function of interlayer delay periods of (b) 10 s, (c) 20 s and (d) 30 s. CT, computed tomography; NiTi, nickel–titanium alloy; WAAM, wire arc additive manufacturing.

Geometrical analysis

Figure 2b–d provides the 3D scan images of the WAAM structures deposited at different interlayer delay times. The measured values of overall dimensional change (in terms of area) after six-layer deposition are given in Figure 2e. When the interlayer delay is increased from 10 to 20 s, the wall height decreases by 23%. For instance, when the interlayer delay is increased from 20 to 30 s, the wall height reduces by 34%. With an increase in interlayer delay from 10 to 20 s and 20 to 30 s, the reduction in wall width is observed to be 33% and 47%, respectively. The decrease in wall height as the interlayer delay increases might be attributed to a fall in the preheat temperature on the previously constructed layer's surface, resulting in a decrease in deposition efficiency. On the contrary, the decrease in wall width with an increase in interlayer delay is mostly attributable to the decrease in the melt pool size and subsequent melt outward flow owing to a decrease in preheat temperature. The viscosity of the molten metal decreases as the deposition progresses from lower to higher layers owing to a rise in preheat temperature (according to the negative connection between temperature and material viscosity), causing an increase in the wall width at a higher layer. Thus, at higher layers, the outward flow of the melt pool will be higher, which leads to larger widths. Cracks and depression are also observed at the lower interlayer delay owing to the heat accumulation effect and residual stresses. “Humping” and “undercutting” can be caused by fluid movement inside the molten pool (as seen in Fig. 8).²⁷ The Marangoni effect, also known as the thermocapillary effect, is a phenomenon in which fluid motion is regulated by variations in surface tension.²⁸ Because of the arc's oscillation frequency, ripples can be seen throughout the tracks,²⁹ as given in Figure 3b.

FIG. 2.

(a) WAAM deposited walls with different thermal delay periods and, laser scanning images of deposited walls with the interlayer delay periods of (b) 10 s, (c) 20 s and (d) 30 s, and effect of interlayer delay on the wall (e) area of the WAAM-built NiTi-based walls.

FIG. 3.

Scanning electron microscopy images of the WAAM-built NiTi samples with an interlayer delay of (a) 10 s, (b) 20 s, and (c) 30 s.

Density measurement and metallurgical characterization

The porosity of the NiTi wall structures built with different interlayer delay periods is analyzed using computed tomography. Figure 1b–d presents the 3D image of the sample indicating the presence of porosity. The average size of the pores is in the range of 304, 150, and 30 μm for walls built with 10, 20, and 30-s interlayer delay, respectively. The variation in the size of the pores can be attributed to the increase in the temperature at the lower interlayer delay period that can lead to cracking in addition to the usual porosity. The density of the wall structures is 93%, 97%, and 99.8% for walls built with the interlayer delay of 10, 20, and 30 s, respectively.

It can be observed that when the interlayer delay increases, the density of the wall structures rises. This is owing to the significant drop in melt pool temperature (increase in interlayer delay reduces the peak melt pool temperature). The reduction in the melt pool temperature suppresses the vaporization of the material, which leads to a reduction in the porosity with an increase in interlayer delay. Furthermore, a reduction in melt pool turbulences combined with a rise in interlayer delay improves its density. The variation in porosity is partly a function of melt pool size and the variable degassing behavior during WAAM, according to the literature.¹⁷ Larger weld pools combined with short welding periods result in different-sized solidification pores. The density is lower than the other counterparts because a large melt pool size is observed at the lower interlayer delay. The existence of much fewer hydrogen entrapment sites, such as grain boundaries, and the release of hydrogen into the environment by arc forces, might explain the lower porosity in the 30-s delay sample.³⁰

Figure 3a–c presents the microstructural images obtained using SEM for the WAAM-built NiTi walls. It shows that walls built with higher interlayer delay show finer grains in comparison with lower interlayer delay samples.^6,30 This is mainly owing to the higher solidification rate and subsequent formation of fine grain structure for walls built with higher interlayer delay. Figure 4a provides the schematics of grain growth for the different interlayer delay periods. Impressively, slower solidification in 10 s resulted in larger grains (54% larger in size) than walls built with 30-s delay. In addition, dendrites also serve as nucleation sites for pores during solidification (as given in Fig. 4b). The higher porosity in the sample with a 10-s interlayer delay can be attributed to the existence of a larger grain boundary area and the availability of appropriate interdendritic spaces.

FIG. 4.

(a) Schematics illustrating the melt pool characteristics and their grain growth, and (b) scanning electron microscopy images showing the transition of grains from top to bottom for the WAAM-built NiTi walls with an interlayer delay of 10 s.

The liquid metal gets undercooled as the temperature difference from the bottom to the top of the molten pool decreases, causing the solidification front of the columnar grains to destabilize.²⁶ The columnar to equiaxed transition (seen in Fig. 4b) may occur when deposition progresses from bottom to top layers, as the temperature gradient decreases and the interface movement velocity increases.¹⁰ The SEM elemental mapping findings of WAAM-deposited tracks for samples with an interlayer delay of 10, 20, and 30 s are given in Figure 5. Distinct spectrums containing different Ti and Ni-rich areas are observed. Oxide layers are visible in samples generated with a 10 and 20-s interlayer delay, compared with a 30-s sample, owing to the lower cooling rate. At high temperatures, Ti may react with atmospheric oxygen and for walls built with 10 and 20-s interlayer delay, interlayer preheat temperature is high.

FIG. 5.

Elemental mapping of the WAAM-built NiTi samples as a function of interlayer delay (a) 10 s, (b) 20 s, and (c) 30 s.

Inner oxides are similarly multilayered, with complicated and nonuniform compositions. Oxides in the WAAM structure can cause structural and elemental discontinuity. The waviness of the oxidized surface causes arc length variation and, as a result, arc voltage fluctuation. Oxides have different electron emission capabilities than the matrix, and the waviness of the oxidized surface causes arc length variation and, as a result, arc voltage fluctuation. Because of their higher melting temperature and lower density than the matrix, oxides will float on the surface of the weld pool. When the weld pool encounters a large oxide island, it flows through it, changing the wetting boundary to an uneven curve until the island is completely passed.

The XRD patterns of NiTi deposited walls as a function of different interlayer delay periods is given in Figure 6a and b. The existence of the B2 phase is revealed by the high-intensity martensitic peak at 44.38° and 64.7° along the planes (020) and (200). The creation of Ni₄Ti₃ may be seen at 78.81°, which corresponds to (110). The change in the temperature history during deposition has a major impact on phase evolution, resulting in a distinct anisotropic microstructure. The two-way shape memory effect in NiTi is increased owing to the presence of Ni₄Ti₃. The Ni₄Ti₃ precipitates, in general, lead to the R phase transformation.³¹ The estimated value of crystallite size is 37 ± 10, 14 ± 6, and 7 ± 5 nm for walls built with 10, 20, and 30-s delay, respectively. The crystallite size reduces with an increase in interlayer delay, which is evident from the broadening of the peak. A shift in the peak position is also observed (refer to Fig. 6b) with variation in the interlayer delay, which is primarily owing to variations in the thermal strain as per the Braggs law.^10,19

FIG. 6.

X-ray diffraction patterns of the WAAM-built NiTi walls as a function of (a) different interlayer delay periods and (b) the plot showing the peak shift observed for the highest intense peaks, and DSC graphs for (c) 10 s, (d) 20 s and (e) 30 s. DSC, Differential Scanning Calorimetry.

Phase Transformation Results

Figure 6c–e provides the effect of the interlayer delay on the trend of Differential Scanning Calorimetry curves. The reason for this behavior is owing to the existence of high nickel in the matrix, which prevents martensitic transformation. By increasing interlayer delay, the intensity peaks of A → R transformation increased significantly compared with the lower interlayer delayer. The internal strain owing to the formation of Ni₄Ti₃ precipitates and high nickel content of the matrix inhibited the formation of M phase. Phase transformation temperatures are changed by alternation of the stress levels.

Mechanical Properties

The microhardness distribution of NiTi samples generated under various interlayer delay periods is given in Figure 7a along the vertical centerline of the cross-section. Higher hardness is observed for the bottom layer, owing to higher cooling rates and finer grain structure as a result of increased heat flow to the substrate. Because of the comparatively small grains at the higher interlayer delay, the average hardness values vary when the interlayer delay is raised from 10 to 30 s. The interlayer delay causes hardness variation owing to the development of distinct phases (Ni-rich phases). The hardness of WAAM-fabricated NiTi is largely regulated by solid solution and grain boundaries or dislocation distribution owing to some segregated components interacting between grain boundaries and edge dislocations.³²

FIG. 7.

(a) Vickers hardness measurements observed for the WAAM-built NiTi samples observed as function of the sample position and (b) compression behavior of the WAAM-built NiTi walls as a function of interlayer delay.

With a larger interlayer delay, the deposit cools faster, resulting in more grain boundaries and dislocations, leading to higher microhardness values.³² The compression characteristics of walls deposited at various interlayer delays are given in Figure 7b. Because of the collective effects of oxidation behavior and grain size, increasing interlayer delay results in a substantial increase in compressive strength. The strength of 30-s samples is increased owing to grain refinement, decreased porosity, and the formation of a Ni-rich phase.^32,33

Shape Recovery Studies

The change in displacement over time is given in Figure 8a. For several cycles of heating and cooling at 30, 20, and 10-s delay, maximum deflections of 3.2, 2.2, and 1.68 mm, respectively, were determined. Such may be attributed to the variation in the stress pattern between the samples made as a function of different interlayer delays. The resulting stress field may have a negative impact on the detwinning process.^34,35 Different delay approaches can aid in the removal of the stress field's potential presence, as well as precipitate development, which can aid in the recovery of the observed shape.^34–37 For higher interlayer delays, the R phase and grain refinement will influence a rise in shape recovery behavior. The Ni-rich phase has a rhombohedral structure and is useful for altering the matrix Ni content and improving shape memory properties by raising critical stress for slip.^34–36

Figure 8b provides the bending angle change for temperature as a function of interlayer delay. Electrical actuation leads to free recovery (without load) and less heat loss as compared with increased recovery during hot plate actuation. When compared with near-equiatomic NiTi alloys, Ni-rich NiTi alloys with >50.5 at.% Ni (as given in Fig. 5) are capable of having a wider range of transformation temperatures and have higher thermomechanical cycle stability. It is especially vulnerable to intermediate aging temperatures of 300°C to 500°C because of the formation of coherent and semicoherent Ni₄Ti₃, affecting the shape memory properties.^31,38,39

Conclusions

The numerical modeling results present the temperature evolution during the deposition phase, the interlayer phase, and the cooling phase. With an increase in interlayer delay from 10 to 30 s, the density of the WAAM-constructed wall constructions increased. Cyclic high temperature (lower interlayer delay) exposure of the sample for longer time may have supported coalescence of small pores in lower interlayer temperature sample (higher interdelay sample). The existence of higher porosity in lower interdelay samples might be attributed to layer-by-layer floatation of pores and extended exposure of liquid metal to the air owing to slower solidification. Higher interlayer delay (30 s) helps additively built NiTi walls have a more appealing surface finish with less visible surface oxidation, better microstructure, improved hardness, and increased strength. Shape recovery increased with an increase in the interlayer delay owing to variation in stress pattern and grain size reduction that have been observed from XRD and SEM results.

The study paves a way to tailor the geometry, mechanical properties, microstructure, and shape recovery behavior of WAAM-built NiTi structures by varying the process conditions for various engineering applications.

Nomenclature

a_f,_r

ellipsoid x semi-axis (front or rear)

b

 ellipsoid y semi-axis

c

 ellipsoid z semi-axis (front)

q_f

 heat flux in the front ellipsoid

q_r

 heat flux in the rear ellipsoid

ν

 velocity

f_f

 heat fraction coefficient in the front ellipsoid

f_r

 heat fraction coefficient in the rear ellipsoid

η

 heat transfer efficiency

Q

 heat input

I

 applied current and

V

 voltage

f_f,_r

 ellipsoid distribution factor (front and rear)

q_w

 filler material

q_b

 base material

V _el

 volume of the element (depends on mesh size)

λ _act

 active material property

λ _quie

 inactive material property

γ

 activation variable

T

 current simulation time

ɛ

 surface emissivity

ρ

 density

k

 material conductivity

C _p

 specific heat capacity

A

 area

▿T

 temperature gradient

h

 heat transfer coefficient

σ

 Stefan's constant

T _room

 room temperature

Author Disclosure Statement

No competing financial interests exist.

Funding Information

European Regional Development Fund through ASTRA6-6.