Approximately, the eucalypti are not responsible for the damage.
The plant's vascular tissues, which help pump water from the roots to the leaves, travel directly underneath the damaged bark. The tissues' route is clearly so damaged that the plant is now suffering from drought damage.
What should you do? First, cut the wounds neatly. Remove all the dead bark at the edge of the wound until you reach the green, living bark. Cut the dead branches from the tree.
Once you have a clear idea of just how much of the tree is affected, you have two options:
1. If there is enough "live" tree left and it is clear that it will sprout again vigorously, then cut the crown of the willow back until the trunk:crown ratio is back to how it should be and the plant looks visibly better.
2. If the plant is beyond repair, it may be time to get rid of it.
But there remains the question of how this happened. If the stem wounds are exposed on the south side, the cause is likely to be frost cracking. The ideal time of year to plant a replacement tree would be in the late winter, when the temperatures are still low but the sun can warm the trunk up. Use tree trunk protection (e.g. a straw mat) to protect from the direct heat.