Homogenization of a unidirectional composite reinforced with two types of transtropic hollow fibers

A method for determining effective elastic constants of a composite unidirectionally reinforced with two types of transtropic hollow fibers is developed. Determining these characteristics is an integral step in the design of composite structures. The approach is based on analytical formulas for determining the elastic characteristics of a two-component composite with a transtropic matrix and hollow fiber. Hexagonal fiber lay-up with periodic reinforcement structure is considered. Double homogenization is used. The composite is conventionally divided into hexagonal regions of two types. The first is a hollow fiber of one material and the surrounding matrix. Similarly, the second one – with a hollow fiber of another material. In the first homogenization, elastic constants of the transtropic material of each of the two regions are determined. In the repeated homogenization, the region of the first type is taken as a “conditional” fiber, the region of the second type is taken as a “conditional” matrix. Effective elastic constants for a composite reinforced with two types of isotropic hollow fibers are calculated. The proposed method gives a good convergence of the results with calculations by known formulas. The maximum relative calculation error for the longitudinal elastic characteristics compared to known formulas does not exceed 0.05 %. The dependences of some effective elastic constants on the volume content of hollow fibers of various types are constructed. Using this approach, three-component composites can be modeled varying the materials of the matrix, hollow fibers and their volume content. This allows predicting the strength of such composites under certain deformations at the design stage

A closed-form orthotropic constitutive model for fused filament fabrication materials

We model two common fused filament fabrication mesostructures, square and hexagonal, using an orthotropic constitutive model and derive closed-form expressions for all nine effective elastic constants. The periodic void shapes are modeled using three and four point hypotrochoid curves with a single shape parameter that controls the sharpness of the points. Using the complex variable method of elasticity, we derive the in-plane elastic constants (Exx, Eyy, Gxy, nuxy) as well as out-of-plane antiplane shear constants (Gzx and Gzy). The remaining out-of-plane elastic constants (Ezz, nuzx, nuzy) are derived by directly solving the linearelasticity equations. We compare our results by conducting unit cell simulations on both mesostructures and at various porosity values. The simulations match the closed-form expressions exactly for Ezz, nuzx, and nuzy. For the remaining elastic constants, the simulation results match the closed-form expressions better for the square mesostructure than the hexagonal mesostructure. Differences between simulation and closed-form expressions are less than 10% for porosity values less than 6% (hexagonal mesostructure) and 10% (square mesostructure) for any of the nine elastic constants.

Stressed Cylinder Dispersion Curves Based on Effective Elastic Constants and SAFE Method

A Semi-Analytical Finite Element (SAFE) formulation is applied to determinethe dispersion curves in homogeneous and isotropic cylindrical waveguides subject touniaxial stress. Bulk waves are required for estimating the guided wave dispersion curvesand acoustoelasticity states a stress dependence of the ultrasound bulk velocities. Therefore,acoustoelasticity influences the wave field of the guided waves. Effective Elastic Constants(EEC) has emerged as a less complex alternative to deal with the acoustoelasticity; allowinga stressed material to be assumed as an unstressed material with EEC which considers thedisturbance linked to the presence of stress. In this approach the isotropic specimen subjectto load is studied by proposing an equivalent stress-free with a modified elasticity matrixwhich terms are the EEC. EEC provides an approximate stress-strain relation facilitating thedetermination of the dispersion curves using the well-studied numerical solution for the stressfreecases reducing the complexity of the numerical implementation. Therefore, a numericalmethod combining the SAFE and EEC is presented as a tool for the dispersion curve generationin stressed cylindrical specimens. The results of this methodology are verified by comparingthem with an approach previously reported in the literature based on SAFE including the fullstrain-displacement relation