Elastic plastic self-consistent (EPSC) modeling of San Carlos olivine deformed in a D-DIA apparatus

D-DIA apparatus

The experiments described in this manuscript were conducted using the deformation DIA (D-DIA) apparatus (Durham et al. 2002; Wang et al. 2003; Weidner and Li 2006; Weidner et al. 2010) located at beamline 6BM-B at the Advanced Photon Source, Argonne National Laboratory, which utilizes a bending magnet that produces a white X‑ray beam. The sample assembly (Supplemental^[1] Fig. S1), based on the “sphere-in-seats design” (Durham et al. 2009); is described in detail in the supplementary section as well as in Kaboli et al. (2017). The sample consisted of a pulverized single crystal of San Carlos olivine in series with a fully dense Al₂O₃ “inner piston” (Coors AD998) all enclosed in a 25 μm thick Ni metal jacket. A W-Re thermocouple was incorporated into the upper piston. Pt foils (25 μm thick) were placed at the top and bottom of the olivine specimen and at the bottom of the inner piston to measure the length of both from radiographs taken during the experiment.

The experiment was compressed to ~6 GPa at room temperature and annealed at 1200 °C for 3 h and 50 min. For experimental samples of San Carlos olivine produced in this fashion, we generally obtain an aggregate with various grain sizes ranging from 1–50 μm. For this particular experiment, we infer from the grain size distribution in the sample after the experiment (see Supplemental^[1] Fig. S7) that the resulting initial grain size was around ~35 μm as is described in detail in the supplementary material. After annealing, the temperature was then lowered to the first experimental temperature. The combination of cell relaxation during annealing and thermal contraction on cooling reduces the experimental pressure from that observed during the initial compression considerably. X‑ray spectra were collected at this initial condition and then the D-DIA inner rams were advanced to deform the specimen while in situ diffraction observations were made. The motor speed for the D-DIA ram pumps was chosen to produce a strain rate of ~5 × 10^–6/s, a strain rate that would allow for good documentation of the low strain behavior of the sample. After several percent strain was achieved the motors for the inner D-DIA rams were stopped. The temperature was then raised to 1200 °C and the inner rams were retracted briefly at the rate of ~10^–5/s to relax any remaining stresses. The temperature was then changed to the next experimental temperature and the next sequence begun. This sequence of short deformation experiments and relaxation periods was repeated for the four temperature conditions reported here. A fifth and final deformation sequence was conducted, but since during data analysis we found that the stress state was not fully relaxed before the start of the final sequence, that data was discarded. No effort was made to adjust the experimental pressure beyond the automatic feedback system that keeps the oil pressure constant. Thus the pressure for each deformation sequence was somewhat different. Conditions for the deformation sequences and annealing times before each sequence are given in Table 1.

In situ X‑ray measurements

Radiographs of the sample and inner piston were taken at ~12 min intervals during deformation and the length of each was analyzed using Image-J (Schneider et al. 2012). Sample strain was calculated as

where l is the instantaneous sample length and l₀ is starting length of the sample, which was recorded at the pressure and temperature conditions of the experiment immediately before the D-DIA rams begun advancing for each deformation sequence. Sample strain measurements are not synchronous with the diffraction measurements; therefore, the sample strain associated with each diffraction measurement must be calculated. Since we typically observe some sluggishness in the system when deformation first begins, rather than calculating sample strain from a linear fit of all the sample strain vs. time data, we fit the data with a polynomial function (see supplementary section^[1]). This is particularly important for characterizing the slope of the stress-strain curve at the lowest strains. Quoted strain rates (Table 1) are for the portion of the experiment after the sample strain vs. time behavior becomes linear.

X‑ray diffraction data analysis

Diffraction data for both the sample and the inner piston were taken at 6 min intervals throughout each deformation sequence. The experimental setup had 10 energy-dispersive detectors, but our data analysis procedure relies primarily on three of the detectors, the two detectors (at ψ = 0° and 180° in Supplemental^[1] Fig. S2) that are positioned to record diffraction coming from planes nearly normal to the compression axis and one detector (at ψ = 90° in Supplemental^[1] Fig. S2) that measures diffraction coming from planes that are nearly parallel to the compression axis (the transverse direction). The other detectors should produce lattice strains that are intermediate between these two end-members and confirmation of this is used as a check on data quality. Further details regarding the data analysis procedure are contained in the supplementary material. Lattice strain (ε^hkl) is calculated for each diffraction peak as follows:

where d₀^hkl is the lattice spacing measured by a given detector immediately before the beginning of deformation for each sequence.

To interpret the diffraction measurements, lattice strain vs. sample strain curves for the experiments are then compared with simulated diffraction data generated with an EPSC model (Tome and Oliver 2002). The single-crystal elastic constants used in each model were calculated for the appropriate experimental temperature and pressure from constants given in (Isaak 1992; Anderson and Isaak 1995; Abramson et al. 1997; Liu and Li 2006) and are listed in the supplementary material. Typically for olivine, we model the eight commonly observed slip systems in olivine as well as three unidirectional slip systems to simulate the formation of kink bands (Burnley 2015; Kaboli et al. 2017). For this study, we also used an additional isotropic deformation mechanism that will be discussed below. The EPSC model uses a Voce hardening law to describe the evolution of the critical resolved shear stress (τ) for each slip system with shear strain (Γ) as follows:

where τ₀ is the initial critical resolved shear stress and τ₁, ϕ₀, and ϕ₁ are hardening parameters (Turner and Tome 1994; Tome and Oliver 2002). The values of τ₀, τ₁,ϕ₀, and ϕ₁ used in each model are listed in Table 2.

Application of EPSC models

Two of the observations above have important implications for developing an EPSC model that will fit the diffraction data. The first is that a deformation mechanism that has a very low critical resolved shear stress (τ₀) is required in order for deformation to deviate from elastic behavior so early in the deformation experiment. In addition, this mechanism cannot accommodate very much strain or else the entire aggregate would yield completely. The second important observation is that because the lattice strains for the individual reflections remain close to each other, whatever this mechanism is, it does not differentiate between any of the measured grain populations. All of the known slip systems for olivine as well as kink band formation produce dispersion between the olivine lattice strains (Burnley 2015). Thus a new deformation mechanism that affects all grain orientations to the same degree is required to keep the lattice strains from deviating from each other.

Although the exact nature of this new deformation mechanism has not been determined, we can simulate its behavior with a “fake” slip system in the EPSC model to improve the overall fit of the models. To do this, we created a deformation mechanism for which the Schmid factor is close to 0.5 for each grain. This “slip system” consisted of planes belonging to four rhombic prisms ({021},{101},{120},{301}) and two rhombic dipyramids ({111},{231}) with various slip directions (full details are found in the supplementary material). This system produced the observed lack of dispersion between the measured lattice strains. The slope of the lattice strain vs. sample strain curves is adjusted using the work hardening parameters. Results of this slip system operating alone are illustrated in Supplemental^[1] Figure S5. Once the low strain portion of the lattice strain vs. sample stain curves were successfully modeled then the slip systems typical of olivine as well as the model for kink band formation (Burnley 2015) were applied to produce the observed yielding and dispersion of the lattice strains. The inability of the models to reproduce the behavior of the (122) reflection in the second deformation sequence is probably due to issues with properly identifying the initial peak position at the start of that deformation sequence. Table 2 gives the parameters that we used to produce the model fits shown in Figure 2.

Figure 2

Lattice strain vs. sample strain data (symbols) for the four deformation sequences. The lines show the self-consistent models calculated to match the data. The slip system activity is plotted below each. The slip systems included in each group are listed in Table 2. The uncertainty in lattice strain is ±0.001, which is illustrated by an error bar placed to the right side of each deformation sequence.

Deriving CRSS from EPSC

In the EPSC model, the CRSS and hardening constants are treated as fitting parameters. However, if the theory behind the model is correct and the modeling process takes all the deformation mechanisms into account, then the CRSS and hardening constants should also be related to the physical processes that they describe. We, therefore, compared the CRSS for the slip component of the EPSC models, with determination of the CRSS of [100] and [001] slip from previous work by Durinck et al. (2007) (Fig. 3). Durinck et al. (2007) compiled experimental data on the CRSS of olivine slip systems measured at low pressure in single-crystal studies and then parameterized the CRSS as a function of temperature. The dashed lines in Figure 3 show the range of CRSS as a function of temperature as indicated by the uncertainty in their parameterization. Some of our models required that different CRSS be used for different slip planes that have the same Burger’s vector; in this case, a weighted average was used in Figure 3. In the case of [100] slip at 440 °C, we found two EPSC models that were indistinguishable in terms of their fit to the experimental data, which had different CRSS (Table 2). This variation in CRSS is indicated by plotting a symbol for each model value and using a larger error bar. It is important to keep in mind that our CRSS values were determined at high pressure and that the CRSS for slip, especially along [100] should be somewhat higher (Durinck et al. 2005) than at low pressure. Differences in composition between forsterite and San Carlos olivine were ignored by (Durinck et al. 2007), but this small difference in chemistry has not been observed to have a large impact on the slip (Bollinger et al. 2012, 2015). Keeping in mind the pressure difference, the match between the CRSS for [001] slip from our models as compared to that from previous work is remarkable considering the difference in the experimental

Figure 3

CRSS of slip as a function of temperature for (a) [001] and (b) [100] slip. The data points (symbols) are derived from the CRSS listed in Table 2. The dashed lines indicate various parameterizations taken from (Durinck et al. 2007), based on the upper and lower bound of each parameter given by that study.

techniques used to determine the CRSS. It is interesting to note that the parameterization from (Durinck et al. 2007) gives a CRSS for [100] slip between 1550 and 2250 MPa at 440 °C that is not consistent with our models, which require some slip on [100]. However, it should be noted that this parameterization is based on only five experimental data points below 1000 °C, which offers little constraint at low temperature.

Inelastic behavior at low strain

While the isotropic slip system that we used in the EPSC models was useful to describe the physical phenomena that we observed, it is just a “hack”, and the input parameters (e.g., CRSS and hardening parameters) do not have a direct physical meaning. A more meaningful description of the phenomena is to calculate the apparent value of the Young’s modulus for each temperature and compare that to the Young’s modulus as derived from single-crystal elasticity (Fig. 4).

Figure 4

Plot of the apparent Young’s modulus from the initial portion of each deformation sequence compared with the Young’s modulus as calculated from the single-crystal elastic constants.

Although additional studies are required, we suggest that the physical process that is operating at low strain could be grain boundary sliding accommodated either elastically or by dislocation glide. The theory of elasticity of polycrystals with viscous grain boundaries was developed by (Zener 1941) who showed that the apparent elastic modulus reduction caused by relaxation on grain boundaries is a function of the viscosity of the grain boundaries, which is, in turn, a function of temperature. The decrease in the apparent Young’s modulus as a function of temperature that we observe is similar to that observed in metals (e.g., Ke 1947; ~20%) but of a greater magnitude. Displacement along grain boundaries in olivine aggregates has been directly observed in high-temperature deformation experiments (1200–1300 °C) (Maruyama and Hiraga 2017a, 2017b) and dislocation assisted grain boundary sliding is widely understood to be an important deformation Kaboli 3 process for high-temperature flow of olivine (Hirth and Kohlstedt 1995a, 1995b; Dimanov et al. 2011; Hansen et al. 2011; Hansen et al. 2012; Tielke et al. 2016). Elastically accommodated grain boundary sliding is thought to be an important process in the anelastic behavior of the mantle (Cooper 2002; Sundberg and Cooper 2010). At present, work on grain boundary sliding as a deformation mechanism in olivine aggregates has been confined to low pressure. Thus these observations may point to a means of using the D-DIA apparatus to study the effect of pressure on grain boundary sliding.