Warren truss inspired hierarchical beams for three dimensional hierarchical truss lattice materials

Networks of beams are a subject of increasing interest to create architected materials with exceptional mechanical properties and low density. This paper investigates the mechanical properties of one dimensional (1D) hierarchical beams for the development of three dimensional (3D) truss lattice materials. These 1D hierarchical beams are constructed in two configurations by placing axial and inclined struts in single and double laced Warren truss patterns in each side of a beam with polygon cross section. Analytical and numerical analyses have been used to characterize their mechanical properties, including the elastic modulus, second moment of area, and shear stiffness of hierarchical beams drawn from a broad design space. Also, the failure limits of the beams with respect to parent material failure and various buckling modes are probed. Finally, the hierarchical beams have been implemented as the constituent members of Kelvin and octet lattices, and the elastic modulus and failure boundaries of the second-order hierarchical lattices are evaluated. The investigation reveals the competition between the elastic properties in the individual hierarchical beams based on different combinations of the design variables. The stiffness of the designs under compression and bending is found to be a function of the axial member size and cross sectional shape of the hierarchical beam. On the other hand, the shear stiffness of hierarchical beam designs is a function of the inclined member size and their inclination angle. It is demonstrated that incorporating hierarchy in the Kelvin and octet truss lattices can enhance the load bearing capacity of designs at low relative densities when compared to their hollow counterparts. Also, it is shown that second-order hierarchical stretching and bending-dominated lattices incorporating first-order hierarchical beams, can not only achieve but also surpass the strength and stiffness scaling relations established for first-order lattices. This becomes particularly noteworthy when considering bending-dominated lattices, as the hierarchy can drive their stiffness toward the boundaries, enabling them to outperform their equivalent stretching-dominated rivals.