Closed-form orthotropic constitutive model for aligned square array mesostructure

Abstract

This work models the square array mesostructure found in fused filament fabrication parts using an orthotropic constitutive model and derives closed-form expressions for all nine effective elastic constants. The periodic void shape is modeled using a four point hypotrochoid curve with a single shape parameter that controls the sharpness of the points. In-plane elastic constants (Eyy, Ezz, vyz, Gyz) are derived by solving the 2D elasticity equations using the complex variable method of elasticity. The antiplane shear constants (Gxy and Gxz) are derived from a complex variable formulation of the antiplane shear equations. The remaining elastic constants (Exx, vxy, and vxz) are derived by directly solving the linearelasticity equations. We compare our results to unit cell finite element simulations. The simulations match the closed-form expressions exactly for Exx, vxy, and vxz. For the remaining elastic constants, the difference between closed-form model and simulation increases as porosity increases. Tensile testing specimens at various porosity values under 12% are fabricated to test six of the nine elastic constants. Experiments on Exx, vxy, and vxz agree well with closed-form predictions, differing by less than 2.6% for the tested specimen types. Experiments on Eyy, Ezz, and Gxy generally agree well, with larger differences at higher porosity values. The difference is less than 6% for all tested specimen types except for the Ezz specimen with a porosity of 0.113, which had a difference of 21%. Several reasons for the discrepancy are postulated, including non-ideal void shape and interfacial bond stiffness.