A two-scale computational homogenization approach for elastoplastic truss-based lattice structures

Abstract

The revolutionary advancements in metal additive manufacturing have enabled the production of alloy-based lattice structures with complex geometrical features and high resolutions. This has encouraged the development of nonlinear material models, including plasticity, damage, etc., for such materials. However, the prohibitive computational cost arising from the high number of degrees of freedom for engineering structures composed of lattice structures highlights the necessity of homogenization techniques, such as the two-scale computational homogenization method. In the present work, a two-scale homogenization approach with on-the-fly exchange of information is adopted to study the elastoplastic behavior of truss-based lattice structures. The macroscopic homogenized structure is represented by a two-dimensional continuum, while the underlying microscale lattices are modeled as a network of one-dimensional truss elements. This helps to significantly reduce the associated computational cost by reducing the microscopic degrees of freedom. The microscale trusses are assumed to exhibit an elastoplastic material behavior characterized by a combination of nonlinear exponential isotropic hardening and linear kinematic hardening. Through multiple numerical examples, the performance of the adopted homogenization approach is examined by comparing forces and displacements with direct numerical simulations of discrete structures for three types of stretching-dominated lattice topologies, including triangular, X-braced and X-Plus-braced unit cells. Furthermore, the principle of scale separation, which emphasizes the need for an adequate separation between the macroscopic and microscopic characteristic lengths, is investigated.

Figures (17)

• Figure 1: Schematic representation of the two-scale computational homogenization approach. At each integration point of the macroscale BVP, the macroscopic strain field $\bar{\bm{\varepsilon}}$ is transferred to the microscale, where it is used to solve another BVP for a unit cell composed of truss elements. The homogenized stress $\bar{\bm{\sigma}}$ and tangent $\bar{\bm{C}}$ are then transferred back to the macroscale.
• Figure 2: Unit cell topology and node numbering for three types of stretching-dominated lattice structures: a) triangular lattice, b) X-braced lattice and c) XP-braced lattice. Dashed lines show the the boundaries of the unit cell, and the translational periodic vectors are denoted by $\bm{a}_{1}$ and $\bm{a}_{2}$.
• Figure 3: Schematic representation of a truss element in the global and local coordinate systems (i.e., $x$-$y$ and $x^{e}$-$y^{e}$ coordinate systems, respectively). $u_{1x}^{e}$, $u_{1y}^{e}$, $u_{2x}^{e}$ and $u_{2y}^{e}$ denote the nodal displacements in the global coordinate system, while $u_{1}^{e}$ and $u_{2}^{e}$ show such displacements in the local coordinate system.
• Figure 4: Geometry and BCs for a double-clamped beam, which, due to symmetry, is depicted as a one-half model with a length of $L$ and a thickness of $a$. The left boundary is fixed in both the $x$ and $y$ directions; the right boundary is fixed only in the $x$ direction, and a vertical displacement of $u_y = 0.1a$ is applied to the right boundary.
• Figure 5: Force-displacement curves of the double-clamped beam with different numbers of elements through the thickness for the a) triangular lattice, b) X-braced lattice and c) XP-braced lattice.