A novel metamaterial incorporating both auxeticity and thermal shrinkage

•A metamaterial incorporating the controllable coefficient of thermal expansion and auxeticity is proposed by introducing a bi-material double-square structure into the star-shaped cell.•The theoretical models based on the force method, exhibiting excellent consistency with the numerical results, are built to explore the mechanical and thermal responses of the metamaterial.•The desired properties of the proposed metamaterial can be customized by selecting the appropriate geometric and material parameters.•The metamaterial designed herein exhibits significant advantages in the auxeticity and thermal shrinkage under the identical relative density and dimension compared with the bi-material orthogonal-isotropic star-shaped metamaterial.