The Proceedings of the Eighth International Conference on Creationism (2018)

system, and v f is the volume fraction of primary mineral (olivine in this study). Clearly, the rate of grain growth increases as the temperature increases, pressure decreases, and the volume fraction of the primary mineral increases. Grain size reduction is logically related to dynamic recrystallization since dynamic recrystallization results in grain refinement. Therefore, we created a new formulation for the grain size reduction rate as follows: (9) (10) where X drx is the dynamically recrystallized volume fraction, c x is the reduction rate constant, and ω is the steady-state grain size. Therefore, the total average grain size rate of change is the sum of the grain growth rate and the grain size reduction rate (11) The average grain size is used in the two hardening equations, Eqs. (1) and (2), as the model is stepped in time. D. Grain size-stress relation With changes in grain size during deformation, the degree of dislocation-grain boundary interactions and the diffusion rate also change. These mechanisms affect the rock strength. The relationship between the grain size and stress can be expressed as (12) where σ 0 is the reference stress, d 0 is the reference grain size, and z is the exponent. In order to describe the grain size effect on the stress, Eq. (12) has been incorporated into the two hardening equations, Eqs. (1) and (2). E. Texture effect and stress-state dependence Anisotropy can develop during deformation and plays a significant role in reducing the rock’s strength. The effect of anisotropy (i.e., plastic spin or torsional softening) therefore needs to be included in a mathematical description of mantle deformation. To treat this phenomenon, we employ the invariants of the deviatoric stress tensor in the same manner as Horstemeyer et al. (2000) used in their damage model. The second and third invariants of the deviatoric stress tensor are expressed (13) and (14) where J 2 and J 3 are the second and third invariants, respectively, of the deviatoric stress tensor , and the subscripts obey Einstein tensor notation. The hardening and dynamic recovery terms in the ISV isotropic and kinematic hardening equations are influenced by these two deviatoric stress invariants as follows: (15) (16) (17) (18) where h is the pressure and temperature dependent kinematic hardening term, C 9 is the material specific hardening rate constant, C a is the torsional constant that differentiates torsional stress from other stress states, C b is the tension/compression constant that differentiates tension and compression behavior, μ is the pressure and temperature dependent shear modulus, r d is the dynamic recovery in the kinematic hardening, C 7 is the material-specific recovery rate constant, C 8 is the temperature sensitivity constant of the dynamic recovery, C 21 is the pressure sensitive constant of the dynamic recovery, P is the hydrostatic pressure, T is the absolute temperature, H is the isotropic hardening term, C 15 is the material- specific isotropic hardening rate constant, R d is the dynamic recovery in the isotropic hardening, C 13 is the material specific dynamic recovery rate constant, C 14 is the temperature sensitivity constant, and C 24 is the pressure sensitivity constant. Note that hardening terms h and H and recovery terms r d and R d appear in the hardening rate equations, Eqs. (1) and (2). F. Model calibrations based on the experimental rheological data for olivine The procedure by which one obtains the material specific parameters and constants required by a mathematical model to describe a physical process is known as model “calibration.” For a calibration of the grain growth of olivine (or forsterite), we used the experimental data from Hiraga et al. (2010) and Tasaka and Hiraga (2013). Since their experiments were performed at temperatures comparable to those in the upper mantle, it is appropriate to use their data to calibrate the model temperature dependence. However, since the applied pressures in their experiments were very low compared to actual mantle pressures, their experimental results are not appropriate to calibrate the pressure dependence of grain growth kinetics. Hence, the pressure dependent parameter for the grain growth (i.e., activation volume) was simply set to a reasonable value, 3.0 x 10 -6 , which is close to an average among the other mantle minerals: 3.8 x 10 -6 for garnets (Yamazaki et al. 2010; Paladino and Maguire 1970), 0.2 x 10 -6 for perovskites (Yamazaki et al. 1996; Wang et al. 1999), and 4.4 x10 -6 for ferropericlase (Tsujino and Nishihara 2009; 2010). The calibrated model and the experimental data are shown in Fig. 2. Next, the grain size effect on the rock’s strength also requires calibration. Based on laboratory data for olivine (Hansen et al. 2012; Karato et al. 1980; Van der Wal et al. 1993), the relationship between the grain size and the strength of olivine could be analyzed by using Eq. (12). The best calibrated exponent z from these experimental data was 0.8. The comparison between the experimental data and the model is shown in Fig. 3. With these calibrated constants and parameters, the stress-strain behavior at various temperatures, pressures, and strain rates was finally calibrated by using the ISV constitutive equations. Using Cho et al. ◀ Strength-reducing mechanisms in mantle rock during the Flood ▶ 2018 ICC 712