The existence of trench art is well documented; poetry, paintings, sketches, carvings and trench magazines are all well known artefacts of the First World War. Why are there no {that I can find} obscene examples? We have graffiti in ancient Rome, hillside chalk figures and raunchy "what the butler saw" photographs from across history, yet a million men in the war to end all wars didn’t draw a single dick? Not one lewd cartoon of the local brothel madam and a raunchy officer? Given the close proximity of all the countries of the empire, the "pals" battalions and the intermingling of social classes in the trenches, you would expect to see some references to national stereotypes.... But no, for a bunch of men crammed together waiting to die, all the art seems very civilised..
Erotic trench art fails to survive in adequate amount or representation exactly for the reasons that erotic art made in other contexts to--it is very often destroyed along the way, either by the family members, the receiving institutions, or by prohibitions on researchers sharing them with the public. Regardless, there is a significant amount of it attested in written record, much of it following the same sorts of styles and variants that ocurred during the Civil War. For that you might be interested in this podcast we did on the topic. I also recommend Judith Ann Giesberg's Sex and the Civil War: Soldiers, Pornography, and the Making of American Morality.
I do have a couple of items here at the Kinsey that I pulled when I saw your question. Unfortunately I cannot share actual images of them, but they generally take the form of daggers carved in the shape of women's bodies, or lighters and medallions with women carved onto them. My favorite has a woman sitting on the head of George V. Another one is a soldier's kit with an erotic image in it. Here's the description:
Cartoons in response to women in the military [as telephone operators]. An officer facing a line of women soldiers who wear only skirts. Balloons over officer's head read "I said kit inspection."
Finally, I cant speak to the exact authenticity of these, but you might enjoy these two specimens that match items in our collections & are being sold online:
https://www.zacharymillerantiques.com/erotic-kissing-women-world-war-one-trench-art-shell/