A versatile numerical approach for calculating the fracture toughness and R-curves of cellular materials

Abstract

We develop a numerical methodology for the calculation of mode-I R-curves of brittle and elastoplastic lattice materials, and unveil the impact of lattice topology, relative density and constituent material behavior on the toughening response of 2D isotropic lattices. The approach is based on finite element calculations of the J-integral on a single-edge-notch-bend (SENB) specimen, with individual bars modeled as beams having a linear elastic or a power-law elastoplastic constitutive behavior and a maximum strain-based damage model. Results for three 2D isotropic lattice topologies (triangular, hexagonal and kagome) and two constituent materials (representative of a brittle ceramic (silicon carbide) and a strain hardening elasto-plastic metal (titanium alloy)) are presented. We extract initial fracture toughness and R-curves for all lattices and show that (i) elastic brittle triangular lattices exhibit toughening (rising R-curve), and (ii) elasto-plastic triangular lattices display significant toughening, while elasto-plastic hexagonal lattices fail in a brittle manner. We show that the difference in such failure behavior can be * Corresponding Author. E-mail: Valdevit@uci.edu explained by the size of the plastic zone that grows upon crack propagation, and conclude that the nature of crack propagation in lattices (brittle vs ductile) depends both on the constituent material and the lattice architecture. While results are presented for 2D truss-lattices, the proposed approach can be easily applied to 3D truss and shell-lattices, as long as the crack tip lies within the empty space of a unit cell.exceed the relative density of the material, ̅ = / (Voigt bound). In the case of isotropic cellular materials, much tighter bounds exist, namely the Hashin-Shtrikman bound (Hashin and Shtrikman, 1963) and the Suquet-Ponte-Castaneda nonlinear bound (Castaneda and Debotton, 1992;Suquet, 1993), respectively. A number of nearly-isotropic topologies that achieve or approach the bounds have been identified, e.g., the plate-based closed cell architected material with cube-octet unit cell (Berger et al., 2017) and stochastic shell-based architected materials with spinodal topology (Hsieh et al., 2019).Conversely, the fracture toughness of cellular materials is theoretically unbounded, and has been much less investigated. The first systematic investigation dates back to 1984, when Maiti, Gibson, and Ashby derived the analytical expressions of mode I fracture toughness,, for both open and closed-cell brittle isotropic foams, by using dimensional analysis that relates the global stress intensity factor, K, to the local microscopic stress in the cell wall (Maiti et al., 1984); fracture occurs (K= ) when the maximum stress in the cell wall around the crack tip reaches the tensile strength, , of the constituent material. of open and closed-cell isotropic foams is found to scale with ̅ 1.5 and ̅ 2 , respectively (Maiti et al., 1984). Later, Gibson and Ashby showed that the of 2D elastic brittle hexagonal lattices scales with ̅ 2 (Gibson ...

The coupled strength and toughness of interconnected and interpenetrating multi-material gyroids

“…Figure 1b): (1) toughening due to the plasticity in the wake of the crack-a mechanism seen in continuum plastic solids and (2) crack tip bridging as seen in fiber and other composite materials. Previous work 21,25 on bending-dominated lattices suggests that the contribution to toughening from the former mechanism will be low due to their small plastic zone sizes. Predictions of the normalized J-R curves, viz., J ≡ J /(� D L) versus �a ≡ �a/L are included in Figure 5b, c for the 0 o orientation case for ε D = 0.1 and 0.4, respectively, across the range of ρ .…”

“…A large number of studies have been reported in the fracture of single-phase lattices. [19][20][21][22][23][24][25][26] Most of these are for twodimensional (2D) [19][20][21][22][23] lattices with very few investigations on 3D 24,26 lattices. The key finding of all these studies is that the mode-I fracture toughness K IC of a lattice with relative density ρ scales as K IC ∝ σ f ρ n √ ℓ , where σ f is the strength of the parent solid, ℓ is the strut length and n is a topology-dependent constant.…”

The fracture toughness of demi-regular lattices

“…The influence of topology on fracture toughness has been documented predominantly for planar lattices. Analytical [14][15][16][17][18], numerical [19][20][21][22][23][24][25] and experimental [26][27][28][29] studies have shown that the fracture toughness of an elastic-brittle lattice can be expressed as:…”

Stiff and strong, lightweight bi-material sandwich plate-lattices with enhanced energy absorption

“…For example, large number of unit cells are often required to investigate the crack propagations and fractures in cellular materials[54,55].…”