Casings and methods for manufacturing casings are provided. For example, a casing defining radial, axial, and circumferential directions is provided. The casing comprises an annular inner wall and an annular outer wall, each extending along the axial direction, with the outer wall radially spaced apart from the inner wall. The casing also comprises an auxetic structure extending from the inner wall to the outer wall and including a plurality of lattice elements. Each lattice element extends circumferentially and radially from the inner to the outer wall, and the lattice elements are axially spaced apart from one another. The auxetic structure may define at least one aperture for fluid flow from one portion to another of the auxetic structure and/or may be configured to vary the thermal characteristics of the casing along the axial direction. The casing may be integrally formed as a single monolithic component, e.g., by additive manufacturing.