If I understand correctly, it is similar to the condition of line to ground fault in the electrical lines. When there is a line to ground fault say for phase a, the phase voltage at that line becomes zero, but the phase voltages of other phases (b and c) increases to line voltages.
You can look at the mathematical derivations for more understanding. But, intuitively you can understand it from the pictures I have attached.
You have actually a complex system here and not one easy to analyse because of the distributed current flows between the electrodes. You can certainly simulate a furnace - (don't forget to accurately and fully specify the impedance of the driving transformer which will significantly affect the results) and change the inter-electrode resistances and see what happens. However don't forget that what you simulate will NEVER be identical to what goes on in a real furnace so your results will be indicative only. Best wishes with your analysis!
I expect - as also said above - that an electric arc furnace used for this, is isolated from the grid by a transformer, that should be designed such to filter the disturbances from the grid.
First, the primary side of the transformer should be connected without connection to the neutral, meaning there are only 2 independent phases, not 3.
Second, there should be adequate stray inductance between primary and secondary, so that the voltage drops on the secondary side in case of 'short circuits' or any anomalies.
Third, there should be capacitors I dont know where and how to filter off the harmonics generated, so they are attenuated on the grid.