Authentication remains a critical challenge in IIoT environments, particularly due to the resource constraints of edge devices. To address the limitations of traditional authentication methods (e.g., passwords or PKI) in resource-constrained IIoT devices, we propose a lightweight model that combines Physical Unclonable Functions (PUFs) and Blockchain, and validate it through a secure protocol design leveraging hardware uniqueness and blockchain immutability. The system incorporates smart contracts to manage the authentication process in a decentralised and efficient way. A theoretical validation is presented that shows how this combination overcomes the limitations of current methods, addressing key security and performance challenges in IIoT environments.