I created a destructible mesh from a cube that has the ground grass texture applied. The destruction works, but the chunks have the standard checkerboard texture from UE4 on the inside, i.e. the chunks are not textured as the original mesh on all sides. Is this some known bug?
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Open the destructible mesh - go to the skeletal mesh tab - there you can find “materials” -> add your materials there 
I see, there is Element 1 and Element 2, and 2 is for chunks. Now it works. Cheers!