Linear viscoelastic modeling and validation of functionally graded heterogeneous porous PLA structures with a C¹-continuous plate theory and novel homogenization

Cellular structures had been shown to provide improvements over their equivalent solid structures for energy-dissipation applications but on the expense of strength reduction. This limiting factor can be alleviated with functionally graded porous structures (FGPS) in addition to tailoring the strength-to-weight ratio as required for many structural applications. The FGPS considered in this work are plate-like structures with a through-thickness heterogeneity while kept homogeneous on-average for the planar cross sections. Plate-like structures of Polylactic Acid (PLA) are fabricated with a constrained foaming process utilizing a physical agent. The microstructure is controlled to provide a gradual change of pores' sizes and overall porosity as a function of the thickness coordinate. A numerical tool is developed to analyze those structures based on higher order displacement field theory including thickness stretching. The assumed displacement field satisfies the constraint on the consistency of the transverse shear strain energy. The numerical scheme is formulated in a finite element procedure with C¹ continuity across element boundaries which is a necessary condition for structural compatible deformation modes. The material model of PLA is assumed to obey linear viscoelasticity description with the hereditary integrals constitutive law. The stress relaxation function of PLA is obtained experimentally using DMAQ800 dynamic analyzer and then curve fitted to a two-terms Prony series which is used directly in the hereditary integrals. The formulation is fully discretized in space and time neglecting inertia effects (quasistatic). Homogenization is carried out through a novel model beased on the original Gibson-Ashby formula. A good agreement achieved between the numerical scheme and experimental findings and the work by other researchers.