1. Introduction
The term powder processing refers to compacting powders in dies and sintering the particles together heating to near melting. Generally, this process has been applied to wide variety of materials such as powdered metals, ceramics, glasses, graphites and diamond. Sintering refers to processing a compacted powdered (green) material, which is brittle, by directly heating it to 70-90% of melting temperature, for example, by placing it in a furnace, typically with three chambers:
(i) a burn-off chamber to the vaporize lubricants (used for easy pouring and compaction),
(ii) a high-temperature chamber to sinter, and
(iii) a cooling chamber to ramp down the temperature.
The binding occurs by small-scale mechanisms involving diffusion, plastic flow, recrystallization, grain growth and pore shrinkage. An oxygen-free environment is preferred to minimize oxides. Most importantly, sintering is a method that can be utilized to produce products with complex shapes that cannot be easily made with other processes. However, because powder processing is typically more expensive than other material processing involving full-blown melting, research is ongoing to improve the steps in the process. In this regard, electrically aided sintering techniques for heat delivery have become quite promising. In particular, electrically aided sintering, which uses the material’s inherent resistance to flowing current—resulting in Joule heating to bond the powder components has great promise because it produces desired materials without much post-processing. Furthermore, it has advantages over other methods, such as high purity of processed materials, in particular, because there are few steps during the fabrication approach.
(i) a burn-off chamber to the vaporize lubricants (used for easy pouring and compaction),
(ii) a high-temperature chamber to sinter, and
(iii) a cooling chamber to ramp down the temperature.
The binding occurs by small-scale mechanisms involving diffusion, plastic flow, recrystallization, grain growth and pore shrinkage. An oxygen-free environment is preferred to minimize oxides. Most importantly, sintering is a method that can be utilized to produce products with complex shapes that cannot be easily made with other processes. However, because powder processing is typically more expensive than other material processing involving full-blown melting, research is ongoing to improve the steps in the process. In this regard, electrically aided sintering techniques for heat delivery have become quite promising. In particular, electrically aided sintering, which uses the material’s inherent resistance to flowing current—resulting in Joule heating to bond the powder components has great promise because it produces desired materials without much post-processing. Furthermore, it has advantages over other methods, such as high purity of processed materials, in particular, because there are few steps during the fabrication approach.
2. Objectives:
In this project, we are simulate multi-physical analysis of electrically-aided sintering of compacted powdered materials. The objective of this project is to provide a framework to assemble a series of submodels for electrically-aided compaction, which can be collected into a single expression for the temperature evolution during the process. Each sub-model can be replaced with a different model if desired. We would analyze then analyze the deformation of the cube and the various material parameters like the stresses, densification parameter, temperature, thermal strain, plastic strain with respect to different time frames using the step time function.
In this project, we are simulate multi-physical analysis of electrically-aided sintering of compacted powdered materials. The objective of this project is to provide a framework to assemble a series of submodels for electrically-aided compaction, which can be collected into a single expression for the temperature evolution during the process. Each sub-model can be replaced with a different model if desired. We would analyze then analyze the deformation of the cube and the various material parameters like the stresses, densification parameter, temperature, thermal strain, plastic strain with respect to different time frames using the step time function.
3. Brief Procedure:
We also develop simple framework to investigate the densification and thermal strain changes in an electrically aided compaction process. We also utilize a simple elasto-thermal-plastic strain change deformation model to evaluate the temperature evolution building on Kirchoff St.Venant constitutive relation.
The key quantity of interest is the heat generated from an electrical field. The interconversions of various forms of energy (electromagnetic, thermal, etc) in a system are governed by the first law of thermodynamics,
Where 𝜌0 is the mass density in the reference configuration,
𝑤 is the stored energy per unit mass,
S is Second Piola-Kirchhoff stress,
E is the Green-Lagrange strain,
𝑞0 is heat flux in the reference configuration,
J is the Jacobian of the deformation gradient
Building on a classical Kirchhoff-St. Venant constitutive relation, we consider the following simple elasto-thermo-plastic decomposition:
where IE is the elasticity tensor ,
E is the Green-Lagrange strain,
Ep is a plastic strain parameter and
𝐸𝜃 is a thermal strain parameter
The temperature evolution rate is given by the equation,
In order to estimate the temperature parameters at different time intervals when the time step is given, we follow the explicit forward Euler time marching scheme which is given by
𝜃(𝑡+Δ𝑡)=𝜃(𝑡)+Δ𝑡( 𝜃̇ )
Similar approach is followed for 𝑑̇ 𝑎𝑛𝑑 𝐸𝑝̇
𝑑(𝑡+Δ𝑡)=𝑑(𝑡)+Δ𝑡( 𝑑̇ ),
𝐸𝑝(𝑡+Δ𝑡)=𝐸𝑝(𝑡)+Δ𝑡(𝐸𝑝̇ )
4. MATLAB Analysis:
a. Graphics of the deforming cube at different time intervals:
It can be observed that at time t=0, there is no deformation of the cube taking place (Z=0.001). However as the time progresses, the cube deformation takes place in the Z direction because of the electrical compaction of powdered material in that direction. It can be observed that the Z values for the cube are 0.000925 and 0.00085 corresponding to time intervals t=0.5 sec and t=1 sec respectively.
b. Stress values with respect to time:
It can be observed that as the time increases, the value of Cauchy stress also increases. The value of Cauchy stress increase from 0 Pa at time t=0 sec to 12×10^(9) Pa at time t=1sec. This is because with the increase in time, the dies come closer to each other and the compaction pressure increase. The powders are heated by an electric current, they are subjected to high electrical discharge, and compacted, simultaneously. As a result, the stress increases with time.
c. Norm of Deviatoric stress with respect to time
It can be observed that the deviatoric stress increases from 0Pa at time t=0sec to 4×10^(7) Pa within 0.1 sec, but reaches back to almost 0 by time t=1sec. Overall, the deviatoric stress remains around 0 within t=1 sec time interval. This implies that there is negligible/almost no shear force acting on the particles. Also, since the deviatoric stress remains almost zero, the dilatational stress/pressure increases with time but in a negative direction and reaches a value of -12×10^(9) Pa. The correlation between all the stress values can be easily observed. It can be observed that when compared with the Cauchy stress and dilatational stress, the deviatoric stress is negligible, implying minimal shear force.
d. Norm of Plastic Strain wrto time
It can be observed that with time, the plastic strain parameter increases from 0 at time t=0 sec and reaches a value of 0.04 at t=1sec. This implies that with the increase in the compaction of the electrified walls, the plastic strain of the particles increases.
e. Densification factor wrto time
It can be observed that the densification factor decreases from d=0.65 at time t=0sec to d=0 at time t=0.62sec. This implies that at around t =0.62 sec, powdered materials will be in their fully compacted stage because of the compaction by the electrified walls and it remains in this stage till time t=1sec. It can also be concluded that with higher densification (higher volume-fraction) in the fully compressed state, the better is the Joule heating induced bonding and less springback.
f. Temperature wrto time
It can be observed that with time the temperature increases from 0K at time t=0sec to 700K at around time t=0.62 sec. The temperature then starts decreasing and reaches a value of around 600K at time t=1sec. The temperature rises due to contact with the lower electrified wall. The compacting wall compresses the material to virtually 100% volume fraction and the temperature rises to a maximum due to contact with the upper and lower electrified walls by the time t=0.62sec. The temperature then starts to drop to around 600K at time t=1sec due to conduction with the surrounding walls. It can also be concluded that longer time in the fully compressed state, the better is the Joule-heating induced bonding and less springback.
g. Norm of thermal strain wrto time
It can be observed that with time the thermal strain increases from 0 at time t=0 sec to 8×10^(-3) at around t=0.62 sec. The thermal strain then starts decreasing and reaches a value of 6×10^(-3) at time t=1sec. Since the thermal strain with respect to time, is directly correlated with the temperature, we have similar graphs for temperature and thermal strain wrto time. This implies that as the temperature rises due to contact between the electrified walls, the thermal strain also increases and then starts decreasing due to conduction with surrounding walls
