54 A.R. Range et al. Fig. 7.8 Numerically simulated plate surface temperatures versus time in response to a 2 g excitation near the first resonant frequency for a representative plate of 85% solids loading with 0% additive content. Solid lines indicate insulated boundary conditions on all surfaces. Dotted lines indicate convective boundary conditions. Data are presented for the (a)mean and (b) maximum plate surface temperatures versus time where is the loss factor, !is the frequency of excitation, and 0 is the strain energy density given by 0 D E0 2 0 2.1 2/ (7.2) whereE0 is the real part of the dynamic modulus of the plate, is Poisson’s ratio, and 0 is the strain magnitude. The dynamic modulus was obtained using a system identification approach developed by Paripovic [2]. Specifically, the technique uses acceleration data from uniaxial compression tests to estimate stiffness and damping coefficients that are then used to identify dynamic mechanical properties. Transient thermal properties for each representative plate were determined using the transient plane source technique [12]. The dynamic modulus (E0), thermal conductivity (k), and thermal diffusivity (˛) were measured as 3.02 MPa, 0.35 W/(m-K), and 2:84 10 7 m2=s, respectively, for a representative plate of 85% solids loading with 0% additive content. The density of 1250 kg/m3 was taken as the average of the two 85-00 plate sample densities given in Table 7.2, and the structural loss factor was estimated as the inverse of the experimental quality factors for the two 85-00 plate samples given in Table 7.3. Thermal behavior for a representative plate of the minimum additive content formulation was simulated in response to the heat source in Eq. (7.1). Convective simulations assumed insulated clamped ends with convective boundary conditions on all other surfaces. Insulated boundary condition simulations assumed all of the surfaces to be perfectly insulated. The transient average and maximum simulated temperature response for a representative plate of 85% solids loading with 0% additive content over the 60 min time period is presented in Fig. 7.8a, b. Note that the assumption of perfectly insulated plate boundaries provides an upper bound of temperature increase for the experimental results presented above. As expected, the insulated surface assumption results in significantly greater temperature rise than the convective condition. Interestingly, the thermal simulation under-predicts the temperature increase in both mean and maximum surface temperature for the simulated 85-00 plate, as evident by the comparison of the first row of Table 7.4 with the final magnitude of the curves in Fig. 7.8a, b. This suggests the potential presence of particle-scale interactions that are not accounted for in the derivation of the model, such as friction between particles or de-bonding at the particle-binder interface [1, 13]. 7.4 Conclusions The thermal and mechanical responses of particulate composite, mock energetic plates, comprised of varying ratios of HTPB binder, sucrose particles, and spherical aluminum powder, under contact excitation have been presented. Clear resonant behavior was observed in each of the three formulation variations, differing solely based on the amounts of additive content.
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