2020 The Author(s). Published by IOP Publishing Ltd. Chemical vapor deposition is a popular technique for producing high-quality graphene sheets on a substrate. However, the cooling process causes the graphene sheet to experience a strain-induced, out-of-plane buckling. These wrinkles structures can have undesirable effects on the properties of the graphene sheet. We construct a pair of models to analyse the conformation structure of these wrinkles. An arch-shaped wrinkle is first modelled then expanded to incorporate self-adhesion between the wrinkle edges. Variational techniques are employed on both models to determine the optimal conformation for graphene supported on Cu and Ni substrates. We find these models predict a similar structure to experimental analysis of graphene wrinkles on these solid metal substrates.