Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
I don't have the time today, but both Saint Cyril and later Saint Severus discuss in detail issues like this and it would make a very insightful research project to compile respective references to this subject from these Oriental fathers. The short, gleaned version:
Human beings are not self-existing. Our nature is to be mortal (i.e., dependent upon the dynamic power of God) and our hypostasis reflects and manifests our inherently mortal nature in the tangible form of our human bodies. However, just because our hypostatic bodies are mortal, does not mean that the power of God does not interact to manifest our very existence (mechanically speaking). It could said to be the same mechanism then in the Incarnation. Just as the Divine Essence of God is the source of His power to create and sustain the entirety of the Universe for ourselves, Jesus Christ's own perfectly natural and human Body depends fully upon His own inherent Divinity to even exist! His human Body is as dependent on God as is our own, however He is God and so He rightfully depends upon Himself. Does His finger nails then diminish like our own, yes, because that is the very nature and essence of the human form and body, to diminish physically over time. However, the laws of entropy teach us that nothing physical every truly disappears, and further the doctrine of the Resurrection teaches that we shall be reanimated, and the atoms and molecules of our bodies, be they evaporated and decomposed, will be brought back together like the Dry Bones of Ezekiel 37. So even if Jesus Christ's finger nails or hair seem to disappear, much like our own, the actual substance of them remains intact, albeit by the power of His Divinity. His finger nails remain God's finger nails, but since by nature and kenosis God united with ordinary finger nails, they preserve a sense of ordinary-ness. In the analogy of the iron and the heat, the iron of His human body is in no way changed by Its natural proximity to the Heat of His own Divinity. Our iron bodies require the outside heating element of His Divinity, but He Himself is Divine, and needs no such additions. However, through kenosis, His iron remains Iron, it is not affected by the Heat to change into something different.