I do not know of a close relative that was part of the D-Day invasion, but my father-in-law was part of the forces that pushed inland.
He was involved in the Battle of St. Lo that began on July 3, 1944. I encourage you to Google this battle to appreciate the ferocity and utter devastation that occurred.
I don't know how many days it was into the battle when he was wounded. What we do know was that his platoon was hit by enemy artillery fire. Only he and one other member of his platoon survived.
My father-in-law woke up in a hospital in Great Britain 24 days later. For decades afterwards, shrapnel would work its way around in his body and have to be surgically removed.
Ultimately, his most deviating injury was what we now call PTSD. He lived a productive life, but struggled with alcoholism until his untimely death from a heart attack on Christmas Eve, 1978.