Sucnakp Rugged Shield
While Sucnakp is a brand a lot of people don't know, they make some pretty awesome cases. The new Rugged Shield for the LG V40 looks good, fits great, and offers tons of protection unlike most other cases.
The Rugged Shield is all TPU, but they use multiple layers of TPU, raised edges, and air-gap technology for additional drop protection. That basically means there are small gaps of air in the corners so your phone won't break if you drop it.
And while some won't like the rugged and industrial design, those raised edges keep the corners safe, the cameras safe, and the screen raised up off any tables or surfaces even if you lay it face-down. This is a solid cheap case.