A New Phenomenon—Brittle to Ductile Transition

Most published literature analyzed the elastic modulus of silica aerogels by drawing inspiration from the cellular solids models. For example, Ashby and Gibson (1997) describe the open cellular foam model compressive modulus to follow power law dependence on the relative density as shown in Eq. (5.1) where C and /i are geometric constants that depend on the topological features and microstructure undergoing cell wall bending as the dominant deformation. 

Ashby and Gibson (1997) illustrated the model as a cubic array of interconnected beams as shown in Fig. 5.1 with an exponent of 2. Similarly for closed-cell foam, Ashby and Gibson approximated the relative modulus with an additional term accounted for internal gas pressure and membrane stresses on the faces as shown in Table 5.1. 

Mills and Zhu (1999) developed the closed-cell and opened-cell foams using the BBC lattice tetrakaidecahedral cell model were as shown in Table 5.1 where 0s is the volume fraction of solids in the cell edges (Zhu et al. 1997a, b Mills and Zhu 1999). Roberts and Garboczi (2001, 2002) investigated the modulus for both opened-cell and closed-ceU foam stracture of random models based on Voronoi tessellations and level-cut Gaussian random fields for a closed-cell tetrakaidecahedral model shown in Fig. 5.2. Lu et al. (1999) proposed macro-mechanic model to evaluate the modulus for low porosity. 

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A New Phenomenon— Brittle to Ductile Transition Density of Gelatin-Silica Aerogel... 

Although it was a simulation related to the cutting process, the brittle-ductile transition can be commonly evaluated by the critical depth of cut d<--Inamura et al. have developed a new simulation technique called renormalized molecular dynamics, which is able to deal with the dynamical phenomenon scaled from nanometer to micrometer. In the simulation, a defectless monocrystalline silicon was cut at the speed of 20 m/sec, the depth of cut of 1 pm, and in an absolute vacuum environment. 

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