Aerogel material is a new-type insulation material. The material shows different behaviors under tensile and compress loads due to the complexity in internal structure. Taking the stitched aerogel material as the research object, the constitutive relationship of the aerogel material is obtained by the out-of-plane tensile and compression tests. Then the failure analysis under tensile and compressive loads are carried out. The results show that the out-of-plane compression of aerogel exhibits nonlinear behavior due to structural densification; the aerogel is almost unloaded when stretched out-of-plane; the failure mode of the stitched is mainly the tensile or compression fracture at both ends of the stitches.