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The advent of additive manufacturing, i.e., 3D printing, has enabled the flexibility to realize complex shapes and structures, such as triply periodic minimal surface (TPMS) structures, which are desirable in many engineering applications due to their unique mechanics. Common applications include protective armor or structural reinforcement for military or civilian uses. In this report, three TPMS structures (gyroid, Schwarz diamond, and Schwarz primitive) were fabricated using a hyperelastic photocurable resin and vat photopolymerization (VPP) technique. Additional sets of the same structures were fabricated with geometrical porosity to ascertain the mechanical response of each porous structure as a function of different strain rates (quasi-static and low-velocity impact), i.e., the effect of higher surface area to volume ratio. The results showed that irrespective of geometry, including pores in the TPMS structures causes reduced stress and truncated strain levels achieved under quasi-static loading. Gyroid structures outperformed the other TPMS structures, resulting in higher deformation, irrespective of porosity level. Alternatively, the drop impact results indicated that adding porosity decreased the stress levels and extended the plateau region, achieving greater strains than neat resin structures. The effects of porosity and glass microballoon reinforcement were investigated under the same loading regimes for the gyroid structure. The response of the dual hybridized structures proved to increase the impact efficacy of the gyroid structures compared to all other variations investigated. The results of this paper indicate the potential of additively manufactured TPMS structures made of hyperelastic materials and decorated with stochastic pores for improved impact response.