What is a practical minimum thickness for concrete parts?
Depends on what the application is. For any sort of durability with art pieces, I wouldn't go less than 1/2". That said, my concrete balloon candle holders are very thin but hold up well enough to light handling.
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All concrete cracks! Concrete is usually chosen for its high compressive strength (resistance to crushing) but flexural strength (resistance bending) is considered ZERO by structural designers. Minimal thickness therefore depends upon the other narrowest dimension (Width) plus couple other factors:
a) will it be reinforced?
b) will the finished product be supported by a solid substrate (eg: floor tiles on a concrete slab)
Even un-reinforced cast-in-place concrete slabs are only as good as the ground underneath (substrate). EG: An un-reinforced concrete piece 2" wide x 4' long x 2" thick is going to break very easily under normal use. Increase the width to 2' and it becomes considerably stronger. Epoxy it to an existing solid substrate and it becomes almost unbreakable.