Abstract The use of simple bar elements in nonlinear structural finite element formulations has the academic advantage of uncoupling element technology issues from the structural phenomena to be observed. In this work, we present a finite element setting for the formulation of different nonlinear material models applied to the transient analysis of trusses. While nonlinear elasticity is considered by studying a Hooke-like linear relationship between different pairs of nonlinear measures of stress and strain, hyperelasticity is formulated using Ogden’s model. Viscoelasticity is introduced using a generalized Kelvin rheological model to account for strain rate effects. The finite kinematics is set in a corotational total Lagrangian description where the virtual work is described using the Second Piola-Kirchhoff and the Green Lagrange measures. Although the derivation is omitted, the consistent tangent moduli are given for all these cases. Numerical problems involving simultaneously different truss models are studied and made available as benchmarks since little comparative data is found in literature.

Abstract Asymptotic Homogenization (AH) and the Extended Multiscale Finite Element Method (EMsFEM) are both procedures that allow working on a structural macroscale that incorporates the effect of averaged microscopic heterogeneities, thus resulting in computationally efficient strategies. EMsFEM works directly on coupled finite micro and macroscales using numerically built discrete interpolation functions. Periodic Truss Metamaterials (PTMMs) are cellular materials formed by the periodic repetition of a truss-like unit cell and engineeringly tailored to show a given macroscopic response. In this work we analyze the numerical behavior of selected PTMMs that were designed for extreme Poisson ratios using AH theory. As a first issue, we study macroscopic structures made of finite unit cells and verify how close their average behavior coincides with the material properties predicted by AH. For comparison, we solve the macroscopic plane stress associate problems that employ the elastic constitutive tensor obtained by AH. The second issue is concerned with the ability of EMsFEM to reproduce the structural behavior of the full macro-micro model. We employ two versions of the EMsFEM, adopting linear (LBC) and periodic (PBC) boundary conditions to build the numerical interpolation functions. The third and most important aspect discussed in this research concerns evaluation of the EMsFEM downscaled displacement fields. We observe that according to the layout of the AH designed unit cell, to the use of LBC or PBC and, depending on the boundary conditions present in the macroscopic problem, spurious downscaled displacements might occur. Such spurious displacements are due to excessive compliance of the corresponding unit cell and can be detected when building the numerical interpolation functions. We conclude that the layout optimization of PTMM using AH must be carefully interpreted and that EMsFEM is a good tool to detect a macroscopic excessively compliant response at an early design stage.

AbstractThe application of gradient-based methods to structural optimization problems usually requires the determination of displacement sensitivities with respect to design variables. In this regard, it is important to have at hand comprehensive methods for sensitivity analysis, which show stability, efficiency and accuracy. Particularly, application of the semi-analytical method for linear and nonlinear problems is generally a good trade-off between formulation simplicity and accuracy. In spite of that, semi-analytical methods are known to behave pathologically for shape design variables when the structure is subjected to rigid rotations. A large number of solutions for this problem have been presented in last years, although the formulation involved is generally not trivial, especially in the nonlinear case. A recent method, which adopts the semi-analytical approach and uses complex variables has rendered very promising results for all the aforementioned aspects: stability, efficiency and accuracy. Additionally, it is simple to codify. The present contribution is concerned with the application of this sensitivity analysis method to geometrically nonlinear truss problems. To this end, a finite element formulation is presented and displacement sensitivities are evaluated with respect to material and shape design variables. The results are compared to those obtained using the semi-analytical method with real variables and to global finite differences. An example demonstrates the potentiality of this new approach.