Three-phase model for low-speed crushing (quasi-static loading) (1) Phase I. Buckling phase In the range of small deformation in the beginning of compression, the model AZD0156 price describing thin-shell deformation under a point force is applicable [37, 38]. Considering LY2835219 a buckyball with wall thickness h = 0.066 nm compressed by F with deformation of W (with the subscript number denoting the phase number sketched in Figure  3), the force-deflection relation should be expressed as [39]

(2) where the bending stiffness G = Ehc 2; the reduced wall thickness and ν is the Poisson’s ratio. The linear deformation behavior continues until it reaches the critical normalized strain W b1. Experimental results for bulk thin spherical shell show that the transition from the flattened to the buckled configuration occurs at a deformation close to twice

the thickness of the shell [40]; while W b1 here is about 4 h, indicating a larger buckling strain in nanoscale structure. Figure 3 Illustration of deformation phases during compression for C 720 . Dynamic loading and low-speed crushing share the same morphologies in phase I while they are different in Copanlisib phase II. Analytical models are based on the phases indicated above and below the dash line for low-speed crushing and impact loading, respectively. The nanostructure has higher resistance to buckle than its continuum counterpart and based on the fitting of MD simulation, a coefficient f * ≈ 2.95 should be expanded to Equation 2 as (3) It is reminded that this equation is only valid for C720 under low-speed (or quasi-static) crushing. (2) Phase II. Post-buckling phase As the compression continues, buckyball continues Thiamine-diphosphate kinase to deform. Once the

compressive strain reaches W b1, the flattened area becomes unstable and within a small region, the buckyball snaps through to a new configuration in order to minimize the strain energy of the deformation, shown in Figure  3. The ratio between the diameter and thickness of buckyball is quite large, i.e., D/h ≈ 36.5, and only a small portion of volume is involved thus the stretching energy is of secondary order contribution to the total strain energy. Hubbard and Stronge [41] developed a model to describe the post-buckling behavior of a thin spherical shell under compression based on Steele’s [42] model (4) where . This nonlinear deformation behavior extends until it reaches the densification critical normalized strain W b2. The value of W b2 could be fitted from the simulation data for C720 where W b2 ≈ 11h. The first force-drop phenomenon is obvious once the buckling occurs where the loading drops to nearly zero. Therefore, by applying the boundary condition of F 2(W 2) ≈ 0, Equation 4 maybe further modified as (5) (3) Phase III.