The Effect of Secondary‐Phase Fraction on the Deformation of Olivine + Ferropericlase Aggregates: 2. Mechanical Behavior

To study the mechanical behavior of polymineralic rocks, we performed deformation experiments on two‐phase aggregates of olivine (Ol) + ferropericlase (Per) with periclase fractions (fPer) between 0.1 and 0.8. Each sample was deformed in torsion at T = 1523 K, P = 300 MPa at a constant strain rate to a final shear strain of γ = 6 to 7. The stress‐strain data and calculated values of the stress exponent, n, indicate that Ol in our samples deformed by dislocation‐accommodated sliding along grain interfaces while Per deformed via dislocation creep. At shear strains of γ < 1, the strengths of samples with fPer > 0.5 match model predictions for both phases deforming at the same stress, the lower‐strength bound for two‐phase materials, while the strengths of samples with fPer < 0.5 are greater than predicted by models for both phases deforming at the same strain rate, the upper‐strength bound. These observations suggest a transition from a weak‐phase supported to a strong‐phase supported regime with decreasing fPer. Above γ = 4, however, the strength of all two‐phase samples is greater than those predicted by either the uniform‐stress or the uniform‐strain rate bound. We hypothesize that the high strengths in the Ol + Per system are due to the presence of phase boundaries in two‐phase samples, for which deformation is rate limited by dislocation motion along interfacial boundaries. This observation contrasts with the mechanical behavior of samples consisting of Ol + pyroxene, which are weaker, possibly due to impurities at phase boundaries.

In the present study, to further characterize the rheological properties of two-phase rocks at large strains, we deformed aggregates of Ol + periclase (Per) to shear strains of γ > 6. Here, we focus on the mechanical behavior of this two-phase system over a range of secondary-phase fractions from f Per = 0.1 to 0.8, while a companion paper focuses on the microstructural evolution in these experiments. Even though Ol and Per do not occur at the same depth in the Earth, both are geologically important materials whose mechanical behaviors have been well characterized. Thus, they provide a valuable system on which to study the physics of deformation of polymineralic materials more generally and with which to reconcile some of the differences between our results and those from the studies listed above. Additionally, depending on Fe content, Per is 2 to 10 times weaker than Ol under our experimental conditions, allowing examination of the effect of a weaker phase on the mechanical properties of Ol.

Sample Preparation
Details of the sample preparation are provided in our companion paper; thus, only the relevant information is summarized here. Eight two-phase samples were prepared from powders of San Carlos olivine (SC-Ol, Fo 90 ; i.e., (Mg 0.9 Fe 0.1 ) 2 SiO 4 ) mixed with powders of either synthetic periclase (Per 100 ; i.e., MgO) or synthetic ferropericlase (Per 90 or Per 70 ; i.e., (Mg 0.9 Fe 0.1 )O or (Mg 0.7 Fe 0.3 )O); these included five samples prepared with Per 100 , one sample prepared with Per 90 , and two samples prepared with Per 70 . The phase fraction of each sample is summarized in Table 1.
Powders were uniaxially cold pressed into a Ni can at room temperature with a uniaxial pressure of 100 to 150 MPa. Subsequently, an evacuated hot press was performed in a gas-medium apparatus (Paterson, 1990) at a confining pressure, P, of 300 MPa and a temperature, T, of 1523 K for 2 to 3 hr while a vacuum of 10 to 20 Pa was drawn on one end of the sample. This approach results in greater densification than obtained by conventional hot pressing (Meyers et al., 2017). Initial grain sizes for each of the starting materials are reported in Table 1. After hot pressing, samples were cut into ∼3-mm tall cylindrical disks with an outer sample diameter between 11.0 and 12.6 mm for deformation experiments. Some powders were also cold pressed into a Ni can that contained a central Ni post, resulting in thin-walled samples with internal diameter of ∼8.0 mm after hot pressing. Thin-walled samples minimize the gradient in stress along the sample diameter. Because Ni is significantly weaker than the thin-walled rock sample (Frost & Ashby, 1982;Hansen et al., 2012;Paterson & Olgaard, 2000), the addition of an internal Ni post introduces only a small correction to the measured torque, which will be further discussed in Section 2.2. The results of microstructural analyses are reported in our companion paper, and relevant parameters, such as grain size, are summarized in Table 1.

Deformation Experiments
To prepare each deformation assembly, a sample was sandwiched between dunite and porous alumina spacers. Spacers made from discs of Balsam Gap dunite were annealed in a one-atmosphere furnace at T = 1373 K and an oxygen partial pressure, O 2 , of 10 −5.2 Pa for 12 hr to remove any water. Spacers made from discs of porous alumina with ∼25% porosity were cored from a plate of Al-25 porous alumina obtained from Alfa Aesar. Spacers were placed above and below each sample, and the stack composed of a sample plus neighboring spacers was inserted into an ∼15 mm long Ni sleeve with 14.95-mm outer diameter and 13.55-mm inner diameter. The sample and spacers were placed between alumina and zirconia pistons to center the sample in the hot zone of the furnace in the deformation apparatus. The entire assembly was then inserted into a thin-walled Fe jacket before being placed into the pressure vessel for a deformation experiment.
Samples were deformed in a high-resolution, gas-medium deformation apparatus equipped with a torsion actuator, like that described by Paterson and Olgaard (2000). Experiments were performed at P = 300 MPa, T = 1523 K at a constant twist rate, while torque and twist were measured with an internal load cell and an external rotary variable differential transformer (RVDT), respectively. These quantities were converted into shear stress, τ, shear strain, , and shear strain rate, , using the appropriate relationships from Paterson and Olgaard (2000) after subtracting the torque supported by the Fe jacket and Ni capsule using flow laws from Frost and Ashby (1982). This torque correction for the jacket materials is small, typically accounting for <5% of the total torque measured. For samples prepared with an internal Ni post, the torque supported by the Ni post accounts for <2% of the total torque. Additionally, the twist and twist rate were corrected to account for the compliance of the apparatus (Hansen et al., 2012). In this study, samples were deformed at a maximum shear strain rate, , between 6.5 × 10 −5 and 1.5 × 10 −4 s −1 at the outermost radius of the sample. Experiments were completed once the desired outer-radius (maximum) shear strain was reached, typically between = 1 and = 7.
Once the torque reached a relatively constant value, strain rate and/or temperature were varied to determine parameters in a flow law of the forṁ= where is the equivalent strain rate, σ is the equivalent stress, d is the grain size, n is the stress exponent, p is the grain size exponent, Q is the activation enthalpy, R is the ideal gas constant, and A is a preexponential function that depends on experimental conditions and material parameters, such as water content, melt fraction, O 2 , and shear modulus. A is treated as constant for each experiment in this study. Equivalent stress and strain are related to their shear counterparts using the relations = √ 3 and = √ 3 for the torsion geometry (Paterson & Olgaard, 2000). To calculate n in Equation 1, was varied between 6 × 10 −5 and 4 × 10 −4 s −1 to determine the response in τ while minimizing changes in d and keeping T constant. The strain rate was held constant for strain intervals of Δ ≤ 0.1, at which point the stress had reached a new steady-state value. The strain rate was returned to its original value between rate steps to minimize microstructural changes. Similarly, to calculate Q in Equation 1, T was varied in increments of ∆T = 50 K between 1473 and 1573 K to measure changes in τ while minimizing changes in d and keeping constant. Temperature was held at each value for strain intervals of Δ = 0.1 to 0.15, typically corresponding to about 30 min, to ensure that stress had reached a new steady-state value. Similar to the procedure used for the rate steps, temperature was returned to the starting temperature of 1523 K between temperature steps to minimize microstructural changes that may occur at different temperatures.

Microstructural and Chemical Analyses
Results from microstructural analyses, including electron backscatter diffraction (EBSD) analysis, are presented in our companion paper. Chemical analyses via wavelength dispersive spectroscopy (WDS) were also carried out to determine the Fe-Mg contents of both phases in our samples. WDS analyses were conducted using a JEOL JXA-8530FPlus electron probe microanalyzer at an accelerating voltage of 15 kV and beam current of 20 nA, the results of which are presented in Tables S1 and S2 in Supporting Information S1.

Mechanical Data
Shear stress versus shear strain data from each of the deformation experiments conducted at a constant shear strain rate of = 1.2 × 10 −4 to 1.5 × 10 −4 s −1 are plotted in Figure 1. For samples with Per fraction of f Per > 0.5, the shear stress increases rapidly upon loading up to ∼75 MPa as the sample begins to deform, followed by a continued increase in stress due to rapid work hardening up to = 0.1 (Figures 1a-1d). Above = 0.1 in the two-phase samples, work hardening continues at a slower rate, manifesting as a slower rate of stress increase until the end of the experiment (Figures 1b-1d), while in the single-phase samples, stress remains constant with increasing strain (Figure 1a). For samples with f Per < 0.5, stress increases rapidly up to a peak stress at = 0.1-0.7, as demonstrated in Figures 1e-1i. As a sample continues deforming, it either undergoes a 20 to 40 MPa decrease in τ over Δ = 2 to 4 before reaching a steady-state stress, or it continues to weaken slowly for the duration of the experiment.
In Figure 2, shear stress is plotted as a function of shear strain for experiments conducted at = 6.5 × 10 −5 s −1 . At this slower strain rate, all samples display similar mechanical behavior with stress increasing rapidly up to = 0.1 to 0.5, then increasing at a slower rate until the experiment was stopped.
Steps in stress that appear in many of the stress-strain curves in Figures 1 and 2 result from stepwise changes in strain rate or temperature, yielding the data used to calculate the stress exponent or activation energy, respectively. Stress exponents were calculated from the slope of a linear least-squares fit to log-log plots of strain rate-torque data obtained from each group of rate steps for each experiment ( Figure S1 in Supporting Information S1). For some experiments in Figure 1, two groups of rate steps were performed, each at a different finite strain. In these cases, data from each group of rate steps were fit separately. Values for the stress exponent are presented in Table 1 and plotted versus f Per in Figure 3. For samples with f Per < 0.5, stress exponents lie between n = 2.0 and 3.2, while for those with f Per > 0.5, n typically lies between 3.5 and 5.1; note that, for two of the three samples with f Per = 0.8, n ≥ 6. Values for the activation energy were calculated from the slope of a linear least-squares fit to Arrhenius plots of torque-temperature data for each group of temperature steps ( Figure S2 in Supporting Information S1), the results of which are summarized in Table 1 and plotted as a function of f Per in Figure 4. For two-phase samples, Q lies between 320 and 350 kJ/mol, while for the single-phase Per 100 sample, Q Per = 380 kJ/ mol. Note that temperature steps were only performed for sample PT-1465 in Figure 1a, and samples PT-1455, PT-1457, and PT-1461 in Figure 2. The stepwise changes in stress corresponding to each group of temperature steps are indicated on their respective figures to help differentiate them from rate steps.

Evolution of Fe Contents in Olivine and Periclase
The Fe content in SC-Ol (Fo 90 ) is in equilibrium with Per 70 , however, the starting materials for samples with f Per = 0.2, 0.5, 0.6, 0.7, and 0.8 were synthesized with Per 100 , and the samples with f Per = 0.3 were synthesized using Per 90 as the second phase. Therefore, in these samples, Fe was lost from SC-Ol to Per via diffusion during hot pressing. As discussed in our companion paper, these experiments were originally designed to examine the physical processes responsible for phase mixing (Wiesman et al., 2018), in which case the Fe content should not be a concern. To confirm this expectation, we prepared two samples with Per 70 (PT-1420 and PT-1425).
Select samples were analyzed via WDS and the Fe-Mg contents of these samples are summarized in Tables S1 and S2 in Supporting Information S1 for Ol and Per in each sample, respectively. This analysis determined that in samples PT-1355, with f Per = 0.8, and PT-1349, with f Per = 0.2, the composition of Ol evolved from Fo 90 to Fo 97 and Fo 95 , respectively, while the composition of Per evolved from Per 100 to Per 99 and Per 88 , respectively. A similar analysis on sample PT-1239 with f Per = 0.3 yielded an Ol composition that evolved from Fo 90 to Fo 95 and a Per composition that evolved from Per 90 to Per 80 . Therefore, in our two-phase samples, Ol composition is expected to lie in the range Fo 95 to Fo 97 and Per composition in the range Per 80 to Per 99 . No exchange of Fe between Ol and Per was measured in samples PT-1420 with f Per = 0.2 and PT-1425 with f Per = 0.1 as they were prepared with Per 70 .
To demonstrate that the Fe-Mg contents of our samples had finished evolving prior to deformation experiments, we also conducted an WDS analysis on an undeformed sample, PT-1322, the results of which are likewise presented in Tables S1 and S2 in Supporting Information S1. This sample, which was prepared with Per 100 and contained f Per = 0.2, was from the same hot press as sample PT-1349. The composition of Ol and Per measured in this sample revealed that Ol evolved from Fo 90 to Fo 95 and Per evolved from Per 100 to Per 88 , the same as that in the deformed sample from the same hot press. Therefore, we conclude that Fe-Mg exchange was completed during sample preparation and hot pressing, prior to the deformation experiments.  Experiments were performed at T = 1523 K, P = 300 MPa, and an outer-radius shear strain rate of ≈ 6.5 × 10 −5 s −1 . Strain rate and temperature steps appear as steps in stress along the curves and are noted along each curve.

Considerations for Interpreting the Mechanical Behavior
In the following sections, we discuss several observations and complications that are important to consider when interpreting the mechanical behavior of our samples. These include the microstructural evolution of our samples discussed in our companion paper, the effect of Fe on the mechanical behavior of our samples, strain weakening due to the development of a CPO in our samples, and the effect of silica activity on sample strength due to our use of Per as a secondary phase.

Summary of Important Microstructural Features
In this section, we summarize the results from our companion paper on the microstructural evolution in our samples to help interpret the mechanical data in the following sections.  Hansen et al. (2011) and for dislocation creep in single-phase Per 100 (light-gray circles) and Per 80 (light-gray triangle) determined by Langdon and Pask (1970) and Stretton et al. (2001), respectively. The black dashed line, which is the predictive model for n in two-phase mixtures proposed by Tullis et al. (1991), was calculated based on the values of n from the single-phase end-members for their dominant deformation mechanisms. The mean grain size of Ol and Per in our two-phase samples either remained the same or decreased slightly during deformation up to γ = 7.0; the grain size of Ol evolved from 2.9-3.8 to 1.7-4.4 μm during deformation, while the grain size of Per evolved from 1.6-5.2 to 2.2-4.4 μm ( Table 1). The mean grain size of Ol tended to decrease with increasing f Per in our two-phase samples, while the mean grain size of Per tended to increase with increasing f Per . These observations contrast with the behavior of our single-phase samples of Ol and Per for which the grain size approximately doubled from 4.6 to 8.4 and 5.3 to 9.2 μm, respectively, during deformation to γ ≈ 7.0. For each two-phase sample, grain sizes of Ol and Per are 2 to 5 times smaller than predicted by a grain-size piezometer calibrated for single-phase samples (companion paper, Figure 10). Additionally, subgrain sizes measured for Ol in our two-phase samples are smaller than predicted by the subgrain-size piezometer from Goddard et al. (2020), likely because the grain size in Ol is smaller than the predicted subgrain size (companion paper, Figure 11). The subgrain size of Per, however, matches the subgrain size predicted by their piezometer within error.
During deformation of two-phase samples, grains of the two phases mixed with one another, resulting in a change in the proportion of Ol-Per phase boundaries and Ol-Ol or Per-Per grain boundaries. We observed that the fraction of phase boundary length out of the total length of grain and phase boundaries evolved to a binomial distribution, which describes a random distribution of the two phases (companion paper, Figure 9).
Deformation resulted in the development of a crystallographic preferred orientation (CPO) of both Ol and Per grains in our samples. Our single-phase sample of Ol developed a relatively strong fabric with [100] maxima in the shear direction and girdles between the [010] and [001] axes, indicating activity of the (010)[100] and (001) [100] slip systems. In our two-phase samples, Ol grains developed a weak fabric corresponding to dislocation activity on the (010)[100] slip system with a set of secondary maxima indicating activity of the (100)[001] slip system as well (companion paper, Figure 6). Our single-phase samples of Per developed a multimaxima pattern, which we interpreted as deformation dominated by dislocation activity on the {111}<1 10> family of slip systems as well as the {100}<011> family of slip systems. In our two-phase samples, Per developed a CPO similar to that in the single-phase sample, which weakened with decreasing f Per (companion paper, Figure 7).

Complications Due To Fe-Mg Exchange
In Section 3.2, we noted differences in the Fe-Mg contents of Ol and Per amongst our two-phase samples. These differences in Fe content can potentially affect the mechanical behavior and microstructural evolution of both phases. For Ol, these effects have been studied in detail by Zhao et al. (2009) and Qi et al. (2021), who determined that Fe concentration affects the strength of Ol independent of microstructural or chemical parameters, such as grain size or water content. Their results allow us to scale flow laws determined for Fo 90 to the appropriate Fe contents to compare results amongst our samples. Although the effect of Fe on the mechanical behavior of Per has not been studied directly, we can infer this effect using data for Per 60 from Heidelbach et al. (2009), data for Per 70 from Bystricky et al. (2006), and the dislocation creep flow law for Per 80 from Stretton et al. (2001). We can use this collection of results to scale the Per 80 dislocation creep flow law from Stretton et al. (2001) to different Fe contents, facilitating comparison amongst our samples.
For the range of Fe contents present in Ol in our samples, applying the results of Zhao et al. (2009) andQi et al. (2021) to modify the dislocation-accommodated grain-boundary sliding (disGBS) flow law for Fo 90 from Hansen et al. (2011) yields an increase in the strength determined for Fo 90 by a factor of 1.3 to 1.6 for Fo 95 and Fo 97 , respectively. Furthermore, applying our estimate of the effect of Fe on the strength of Per yields strengths that differ by factors of 0.7 and 3.2 from that of Per 80 (Stretton et al., 2001) for Per 70 and Per 99 , respectively. The implications of these differences in Fe content will be further discussed in Section 4.3. Hansen et al. (2012) reported strain weakening (geometric softening) by a factor of 0.72 from peak to steady-state stress due to the development of a CPO in their samples of single-phase Fo 50 . Similar weakening likely occurred for the single-phase Ol sample in this study, as will be discussed in Section 4.2.2. In contrast, the weak CPOs developed for Ol in our two-phase samples indicate that Ol in these samples did not experience significant geometric softening. Therefore, we did not consider geometric softening in our analysis of the mechanical behavior for our two-phase samples.

Geometric Softening Due To CPO Development
Despite the formation of a CPO in our sheared single-phase samples of Per 100 , our samples did not undergo geometric softening (e.g., Figure 1a). In principle, it is possible that geometric softening did occur and was 10.1029/2022JB025724 9 of 16 counteracted by, e.g., strengthening due to grain growth (Table 1). However, stress-strain behavior similar to ours with no strain weakening was observed by Heidelbach et al. (2003) in their high-strain torsion experiments on Per 80 , despite a reduction in grain size by a factor of 5 (from 40 to 8 μm) and noticeable CPO development. Therefore, we will not consider geometrical softening of Per in the discussion in the following sections. The lack of geometric softening may be due to the availability of more than enough slip systems in face-centered cubic materials such as MgO to easily fulfill the von Mises criterion (von Mises, 1913).

Effect Silica Activity on Sample Strength
Previous studies have demonstrated that the strength of Ol depends on silica activity. Specifically, Ol single crystals buffered by Per are stronger than those buffered by Opx in the dislocation creep regime (Bai et al., 1991;Ricoult & Kohlstedt, 1985). Additionally, in the diffusion creep regime, polycrystalline samples of Fo 100 + 10% Per 100 (Okamoto & Hiraga, 2022) are approximately an order of magnitude stronger than samples of Fo 100 + 20% En 100 (Nakakoji et al., 2018). To test the effect of silica activity on the strength of our samples deforming in the dislocation-accommodated grain-boundary sliding creep (disGBS) regime, an additional experiment was conducted on a sample of Ol + <10% Per 70 that yielded a strength within error of, if not slightly weaker than, that predicted by the disGBS flow law determined by Hansen et al. (2011) on SC-Ol containing <5% Opx ( Figure  S4 in Supporting Information S1). Given this observation, we do not anticipate an effect of silica activity on the strength of our two-phase samples of Ol with significant amounts of Per deformed in the disGBS regime.

Deformation Mechanisms
To determine the dominant deformation mechanisms for Ol and Per in our samples, the mechanical and microstructural data for our single-phase and two-phase samples are further examined below. Due to the distinction in the mechanical behavior between Ol-rich (f Per < 0.5) samples and Per-rich (f Per > 0.5) samples noted in Section 3.1, these two domains are considered separately when interpreting the mechanical behavior for samples with f Per < 0.5 in Sections 4.2.1 and 4.2.2 and samples with f Per > 0.5 Sections 4.2.3 and 4.2.4, respectively. Importantly, this distinction implies that Ol controls the deformation behavior of samples with f Per < 0.5 and Per controls the deformation behavior of samples with f Per > 0.5.

Deformation Mechanism of Olivine-Rich Samples
The dominant deformation mechanism for Ol in our single-phase and two-phase samples was dislocation-accommodated interface sliding, i.e., disGBS in single-phase Ol and dislocation creep accommodated by sliding along grain and phase boundaries in two-phase samples. Specifically, Ol deformed via dislocation-accommodated interface sliding in the regime for which the grain size is smaller than the subgrain size. This conclusion is supported by the value of the stress exponent, n = 2.0 to 3.2, for samples with f Per < 0.5 and the observation that grain sizes were smaller than the subgrain size (companion paper, Figure 11). A comparison with results from previous experiments on single-phase Ol and Ol + Opx deformed under similar conditions helps support the interpretation that deformation of Ol was dominated by dislocation-accommodated interface sliding in this study. Both low-strain compression and high-strain torsion experiments performed on SC-Ol and Fo 50 reported a non-Newtonian stress dependence (n = 3.4 ± 0.2, 2.9 ± 0.3, 4.1 ± 0.1) and nonzero grain size sensitivity (p = 2.0 ± 0.2, 0.7 ± 0.1, 0.73 ± 0.06) (Hansen et al., 2011(Hansen et al., , 2012Wang et al., 2010), while recent studies on two-phase samples of Fe-rich Ol + Opx with f Opx < 0.5 reported values for the stress exponent that decreased from n = 3.0 to 1.8 with increasing strain and a grain size exponent that simultaneously increased from p = 1.1 to 3.3 (Tasaka et al., 2017(Tasaka et al., , 2020. Such values for the stress exponent and grain size exponent are characteristic of models for disGBS, which predict values of n = 3, p = 1 or n = 2, p = 2 for materials with grain size greater than or smaller than the subgrain size (Langdon, 2006), respectively; these values are similar to those for n calculated in the present study for samples with f Per < 0.5. Additionally, high-strain experiments using both single-phase samples of Fo 50 (Hansen et al., 2012) and two-phase samples of Fo 50 + iron-rich enstatite (En 55 ) (Tasaka et al., 2017(Tasaka et al., , 2020 reported mechanical behavior for which a peak stress is reached at < 1, followed by weakening with increasing strain, i.e., a behavior similar to that described for our samples with f Per < 0.5 in the present study (Figures 1e-1i). These authors attributed the observed strain weakening to a combination of grain-size reduction and geometric softening.
Finally, in samples with large Per fractions (f Per > 0.5), diffusion creep may have contributed as much as ∼10% to the deformation of Ol, consistent with the small grain size of Ol in these samples. However, the mechanical behavior of these samples was dominated by Per, as will be discussed further in Section 4.2.3.

Mechanical Behavior of Olivine-Rich Samples
Based on the conclusion that our samples with f Per < 0.5 deformed by dislocation-accommodated interface sliding, we further examine the specific behavior of the stress versus strain curves in Figures 1e-1i. Our single-phase SC-Ol sample, PT-1324, strain weakened even though the grain size increased from d Ol = 4.6 to 8.4 μm. Based on the disGBS flow law for Ol from Hansen et al. (2011), such an increase in grain size would cause an increase in sample strength of ∼15%. Therefore, a 45% decrease in strength due to the development a strong CPO (companion paper, Figure 6) is required to account for both the increase in stress due to grain growth and the strain weakening observed in Figure 1i. This behavior corresponds to a geometrical softening factor of 0.55 in our single-phase SC-Ol sample, which is about twice that calculated by Hansen et al. (2012) for their experiments on Fo 50 . Additionally, in Figures 1e-1h, we observed less pronounced strain weakening in our two-phase samples with f Per < 0.5 than in the single-phase Ol sample, consistent with the weak CPO development (companion paper, Figure 6) and small amounts of grain-size reduction in our two-phase samples (Table 1).
In contrast to the behavior of samples deformed at faster strain rates, sample PT-1455 with f Per = 0.2, which was deformed at half the strain rate as the samples discussed previously ( = 6.5 × 10 −5 s −1 ), hardened over the course of the entire experiment ( Figure 2d). This strengthening from 70 MPa at = 0.1 to 110 MPa at = 3.0 is consistent with the observed increase in grain size from 2.9 to 3.9 μm. Based on Equation 1 with n = 2.2, we calculate that a grain size exponent of p = 3.5 is required to explain the observed strengthening. This value is similar to that of p = 3.3 ± 0.7 reported by Tasaka et al. (2017) at high strains from their experiments on two-phase samples. These authors interpreted their results in terms of disGBS occurring in the regime in which grain size is smaller than subgrain size, which we conclude is the mechanism dominating deformation in experiment PT-1455.

Deformation Mechanism of Ferropericlase-Rich Samples
Based on the deformation behavior of our single-phase Per and Per-rich two-phase samples, we conclude that Per deformed by dislocation creep in our samples. The primary evidence for this conclusion is the relatively large values for the stress exponent of n = 3.5 to 5.1 for samples with f Per > 0.5. Additionally, above shear strains of γ > 1, no change in strength with increasing strain occurred in our single-phase Per 100 samples (Figure 1a), despite an increase from 5.3 to 9.3 μm in the size of Per grains in sample PT-1465 and from 5.3 to 6.6 μm in sample PT-1362 (Table 1). The fact that the same stress was recorded for deformation of multiple samples with different grain sizes, all deformed at the same strain rate, indicates that our single-phase sample of Per 100 deformed by a grain size-insensitive mechanism, i.e., dislocation creep. A fit to stress versus temperature and strain rate versus stress data in Figure S3 in Supporting Information S1 for our three experiments (PT-1362, PT-1463, and PT-1465) on single-phase Per 100 yields the following flow law: (2) For comparison, previous studies have reported stress exponents of n = 3.3, 3.2 ± 0.3, 4.0 ± 0.2, 3.6 ± 0.2 for Per 100 and Per 80 (Bilde-Sörensen, 1972;Heidelbach et al., 2003;Langdon & Pask, 1970;Stretton et al., 2001); each of these authors concluded that their samples deformed via dislocation creep. As mentioned previously, values of n in our study lie in the range from 3.5 to 5.1 for samples with f Per > 0.5, similar to or larger than previously determined values. Furthermore, in our companion paper, we demonstrated that the subgrain size of Per in our two-phase samples matched that predicted by the subgrain-size piezometer of Goddard et al. (2020), providing additional evidence for dislocation activity resulting in the creation of subgrain boundaries. In samples with f Per < 0.5 discussed above in Section 4.2.1, a grain-size sensitive deformation mechanism, such as disGBS or diffusion creep, may have started to contribute to deformation of Per, consistent with the small size of Per grains and the weak CPOs for Per in these samples (see companion paper).

Mechanical Behavior of Ferropericlase-Rich Samples
Based on the conclusion that dislocation creep of Per dominates deformation in our samples with f Per > 0.5, we examine the mechanical behavior of these samples in more detail. As described in Section 3.1, the strength of two-phase samples in this study with f Per > 0.5 increases with increasing strain over the duration of the experiment (Figures 1b-1d). Insight into possible mechanisms responsible for strain hardening in Per comes from a comparison to the strain hardening behavior of face-centered cubic metals (e.g., Cu, Ni, and Al) deformed in torsion to strains of > 4 ( Rollet et al., 1988;Rollett & Kocks, 1993;Zehetbauer, 2000;Zehetbauer et al., 2003). These authors argued that strain hardening in their samples was controlled by reduction of the mean free path of dislocations with increasing strain, referred to as stage-IV hardening. Explanations for a decrease in the mobility of dislocations with increasing strain are based on either the interaction of dislocations with immobile dislocation debris (Rollett & Kocks, 1993) or the buildup of dislocation pileups at grain/subgrain boundaries that limit dislocation motion (Zehetbauer, 2000). Because strain hardening is not observed during deformation of our single-phase samples of Per 100 , phase boundaries must play an important role in this hardening processes, possibly by acting as barriers to dislocation recovery and thus facilitating dislocation pileups that restrict dislocation motion.

Activation Energy
While activation energy is often used as an additional constraint on the identity of the dominant deformation mechanism, it is not a highly useful indicator of the deformation mechanism for the two-phase samples in this study given the significant differences between values of Q determined by different researchers for the same deformation mechanism both in Ol and in Per. As displayed in Figure 4, the activation energies calculated for our two-phase samples lie between 300 and 350 kJ/mol. These values of Q are within error of Q = 327 kJ/mol determined for dislocation creep in Per 80 by Stretton et al. (2001); however, values of activation energy determined for dislocation creep of Per 100 spans the range 210 to 440 kJ/mol (Bilde-Sörensen, 1972;Hensler & Cullen, 1968;Langdon & Pask, 1970). For Ol, the activation energy determined for dislocation-accommodated grain-boundary sliding in SC-Ol of 445 kJ/mol (Hansen et al., 2011) is greater than that determined for two-phase samples in this study. Although the value of Q = 375 kJ/mol reported by Hirth and Kohlstedt (2003) for diffusion creep of SC-Ol is closer to the values calculated for our two-phase samples, the more recently determined value of Q ≈ 435 kJ/ mol from Yabe and Hiraga (2020) for diffusion creep in Fo 90 after accounting for the effect of impurities at our experimental conditions suggests that this comparison may not be useful either.

Two-Phase Flow
Analysis of the deformation behavior of two-phase materials is made difficult by the complex distribution of stress and strain rate between the two phases. The strength of two-phase materials is often characterized by a rule of mixtures that combines flow laws from the two end-member phases weighted by the fraction of each phase present (Huet et al., 2014;Ji, 2004;Ji et al., 2003;Tullis et al., 1991). The maximum and minimum bounds on the strength of a two-phase material are based on the assumptions of uniform-strain rate (Taylor, 1938) and uniform-stress (Sachs, 1928), respectively, between the two mineral phases. In Figure 5, the stress at the beginning and end of each experiment is plotted versus f Per , along with the uniform-strain rate and uniform-stress bounds.
The uniform-strain rate and uniform-stress boundaries were calculated using the flow laws for the dominate deformation mechanism in each phase determined in Section 4.2, namely, disGBS for Ol (Hansen et al., 2011) and dislocation creep for Per 100 (Equation 2) or dislocation creep for Per 80 (Stretton et al., 2001). As discussed in Section 4.1.2, these end-member flow laws were scaled to account for the effect of differing Fe content on the mechanical behavior of each phase. This correction was carried out separately for each sample using the Fe content in Ol and Per appropriate to that sample. Recall that, as discussed in Section 4.1.4, silica activity does not significantly influence the strength Ol in the disGBS regime.

Small Strain Behavior
At small strains (0 < < 1) in Figure 5a, the stresses measured for samples with 0.7 ≤ f Per < 1.0 align with the values predicted from the uniform-stress (lower) bound. As f Per decreases to 0.5 ≤ f Per ≤ 0.6, stress falls between the two bounds but is still closer to that predicted from the uniform-stress (lower) bound. Finally, for samples with f Per ≤ 0.5, the measured stresses are greater than those predicted by the uniform-strain rate (upper) bound; samples with f Per = 0.1 and 0.3 have strengths more than 20 MPa greater than those predicted by the uniform-strain rate (upper) bound. Similar observations were made by Ji et al. (2001) for small-strain (γ < 0.5) experiments on forsterite (Fo) + enstatite (En) for which Fo deformed via diffusion creep and En deformed via dislocation creep; in their study, the strengths of Fo-rich samples followed the uniform-stress boundary, and the strengths of En-rich samples followed the uniform-strain rate boundary. These authors described the change in sample strength from Fo-rich samples to En-rich samples as a change from a strong-phase supported, or load-bearing framework, regime to a weak-phase supported, or interconnected weak layer, regime (Handy, 1990;Ji et al., 2003). The samples in our study can be described similarly at small strains; Ol-rich (f Per < 0.5) samples are strong-phase supported and Per-rich (f Per > 0.5) samples are weak-phase supported. The calculated stress exponents in Figure 3 support this interpretation as the values of n for samples with f Per < 0.5 are similar to that for single-phase Ol and values of n for samples with f Per > 0.5 are similar to that for single-phase Per. In addition, the evolution of n with increasing f Per does not follow the trend for n predicted by a rule of mixtures (Tullis et al., 1991) and is instead controlled by the dominant phase in each sample.
The difference between the strengths of our samples and the uniform-strain rate bound indicates the influence of another physical mechanism not accounted for in using the rule-of-mixtures approach. One hypothesis for the large strengths of our samples compared to the uniform-strain rate boundary is related to the presence of phase boundaries. There is some precedent for hardening associated with phase boundaries, both theoretically and experimentally. Ashby (1972) and Gittus et al. (1978) argued that, due to lattice and chemical mismatch across a phase boundary, glide and climb of dislocations along phase boundaries is more difficult than along grain Figure 5. (a) Peak stress or stress at 0 < γ < 1 versus f Per , and (b) flow stress or stress at γ > 4 versus f Per . In each panel, the theoretical predictions for the strength of a two-phase material based on assumptions of uniform-stress (Sachs, 1928) and uniform-strain rate (Taylor, 1938) are plotted as the minimum and maximum bounds on the strength, respectively. The Sachs and Taylor bounds were calculated for each sample composition based on the Fe content of Ol and Per in that sample. The + and × symbols represent the Sachs and Taylor bounds, respectively. Plotted for comparison are the values of stress predicted by the disGBS flow law for Ol for Fo 90 (medium-gray circles) and Fo 95 (dark-gray circles) (Hansen et al., 2011;Qi et al., 2021;Zhao et al., 2009) both without (a) and with (b) geometric softening observed at large strains in single-phase samples (Hansen et al., 2012). Additionally, the stress predicted by the dislocation creep flow law for Per 80 is plotted on each subfigure (light-gray circle). Data from flow laws were calculated at an outer-radius equivalent strain rate of = 8.5 × 10 −5 s −1 , Ol grain size of d Ol = 3 μm, T = 1523 K, and P = 300 MPa.
boundaries. Additionally, Wang et al. (2018) found that incoherent phase boundaries in their two-phase metallic alloy make it difficult for dislocation pileups to dissipate via climb into phase boundaries, resulting in samples with higher strengths than in samples without phase boundaries. Such situations could account for the higher overall strengths of our samples.

Large Strain Behavior
At large strain ( > 4), the flow law for disGBS in Ol, adjusted to account for geometric weakening that occurs at large strains, predicts a strength similar to that for our dislocation creep flow law for Per 100 . As a result, the uniform-stress and uniform-strain rate boundaries overlap one another for samples with f Per ≥ 0.5. For samples with f Per ≤ 0.3, wherein Per has larger Fe content than in samples with f Per ≥ 0.5, the two bounds are separated due to the significant decrease in strength of Per with increasing Fe content. However, in Figure 5b, nearly all the two-phase samples are stronger than either single-phase end-member and are therefore stronger than either the uniform-stress or the uniform-strain rate boundary.

Comparison to Other Two-Phase Materials and the Effect of Secondary Phases on Sample Strength
Recent high-strain experimental studies of two-phase, olivine-bearing materials have reported notably different mechanical behavior among their respective samples. As summarized here and in Section 1, these differences seem to correspond to the different mineralogy of the two-phase samples investigated: Bystricky et al. (2006) deformed aggregates of Ol + Per 70 with f Per = 0.8 in torsion to γ = 14.9 and observed strengths similar to single-phase Per 70 . Tasaka et al. (2017Tasaka et al. ( , 2020 deformed aggregates of Fe-rich Ol + Opx with f Opx = 0.15, 0.26, and 0.35 in torsion to γ = 26.2 and found that their samples were stronger than single-phase Fo 50 at small strains but weakened with increasing strain. Sundberg and Cooper (2008) deformed aggregates of Ol + Opx with f Opx = 0.65, 0.5, and 0.35 in general shear to γ = 4 and determined that their samples were weaker than the single-phase Ol end-member deformed under similar conditions. Finally, Zhao et al. (2019) deformed aggregates of Ol + Cpx with f Cpx = 0.5 in general shear to γ = 3.5 and observed that their samples were significantly weaker than either single-phase end-member. Additionally, the results from our experiments on Ol + Per demonstrate another behavior in which Ol-rich samples have strengths greater than that of either single-phase end-member. Such differences between studies make it difficult to form generalizations about the behavior of two-phase samples and likely represent fundamental differences is the underlying physical processes occurring within each type of sample. To explain these differences and demonstrate the range of behaviors exhibited by two-phase materials, we compare our results to each of these previous studies on different two-phase systems below.
For Per-rich samples, Bystricky et al. (2006) found that the strengths of their two-phase samples of SC-Ol + Per 70 with f Per = 0.8 deformed to high strains at T = 1300 K were similar to those measured in experiments performed on single-phase Per 70 . Our sample composed of SC-Ol + Per 100 with f Per = 0.8 displayed a similar result in that the strength of our sample matches the prediction from the uniform-stress bound. We also note that both the strength of the sample with f Per = 0.8 and the strength predicted by the uniform-stress bound at f Per = 0.8 are similar to that of Per 100 . These observations indicate that the most Per-rich samples (f Per ≥ 0.8) may be less affected by the presence of a second phase than those with smaller Per-contents.
For Ol-rich samples, Tasaka et al. (2017Tasaka et al. ( , 2020 reported that their two-phase mixtures of Fo 50 + En 55 were stronger than single-phase Fo 50 at small strains. As deformation continued to shear strains of γ ≳ 20, their samples strain weakened for the duration of the experiment, whereas samples of single-phase Fo 50 did not strain weaken beyond γ = 7 (Hansen et al., 2012). Although the mechanical behavior of their two-phase samples is similar to that observed for our samples, they attributed the difference in strength at small strains to the fact that En is stronger than Ol. Unfortunately, this explanation does not hold for our samples of Ol + Per because Per is weaker than Ol at small strains and the two phases are of similar strengths at large strains. Consequently, an additional mechanism is required to explain the large sample strengths in the present study. The possibilities for strengthening discussed previously in Section 4.2.1 apply here, which are related to the nature of phase boundaries in our samples, as discussed further at the end of this section.
In contrast to the results of Tasaka et al. (2017Tasaka et al. ( , 2020, Sundberg and Cooper (2008) observed that their samples of Balsam Gap Ol (Fo 92 ) + Bamble Opx (En 88 ) with f Opx = 0.35, 0.5, and 0.65 were weaker than single-phase Ol (Fo 90 ). Tasaka et al. (2017) pointed out several important differences between their results and those of Sundberg and Cooper (2008). These differences also apply to the results of this study, namely, a lack of solute drag in our samples compared to the samples from Sundberg and Cooper (2008), deformation occurring via dislocation-accommodated interface sliding in our samples rather than interface-reaction limited diffusion creep in Sundberg and Cooper (2008), and experiments performed at higher stresses (>100 MPa) than in Sundberg and Cooper (2008) (<50 MPa). The presence of solutes and impurities in our samples will be addressed further below.
Similarly, for Cpx-bearing samples, Zhao et al. (2019) observed that their samples composed of 50:50 mixtures of SC-Ol + Damaping Cpx (Cpx 91 ) were much weaker than either single-phase end-member. They attributed this behavior to faster rates of diffusion along phase boundaries than along grain boundaries, resulting in enhanced sliding along the interfaces between grains of the two phases. Rather than being accommodated by dislocation motion, interfacial sliding in their case was accommodated by diffusion and rate-limited by interface reactions. This result for the Ol + Cpx system differs significantly from our results for Ol + Per materials in which our two-phase samples were stronger than their single-phase counterparts. Yet, as Zhao et al. (2019) did for their samples of Ol + Cpx, we argue that the distinction between the strength of our Ol + Per samples and the strength of the single-phase end-members is associated with the presence and properties of phase boundaries.
The difference in the role of phase boundaries between our samples and those of Zhao et al. (2019) are primarily due to the observation that interface sliding was dislocation-accommodated in our samples of Ol + Per but diffusion-accommodated in their samples of Ol + Cpx. Zhao et al. (2019) argued that diffusion may be much more rapid along phase boundaries than grain boundaries, at least in the Ol + Cpx system. However, as discussed in Section 4.2.1, Ashby (1972) and Gittus et al. (1978) argued that the motion of dislocations is more difficult along phase boundaries than along grain boundaries due to lattice and chemical mismatch and the resulting disordered structure at phase boundaries. Because deformation was dislocation-mediated in our samples, it is possible that phase boundaries contributed to our increased sample strength. Additionally, the use of high-purity Per as a second phase in our experiments reduces the concentration of grain-boundary impurities, such as Al and Ca, as well as the possibility of a small amount of melt compared to samples synthesized with naturally occurring Cpx or Opx as a secondary phase where such grain-boundary impurities and melt may be present. Per grain boundaries and grain interiors will take up a significant fraction of these impurities that are present in natural SC-Ol, likely forming Mg-silicates instead of melt and reducing the concentration of impurities along olivine grain boundaries. A decrease in the concentration of such grain-boundary impurities will also increase the strength of our two-phase samples relative to the end-member counterparts .

Conclusions
1. In our two-phase samples, deformation of Ol is dominated by dislocation-accommodated sliding along interfaces, while deformation of Per is dominated by dislocation creep. a) In Ol-rich samples, some strain weakening occurs between 0.5 < γ < 7.0 due to a small amount of grainsize reduction and weak CPO development. b) In Per-rich samples, strain hardening occurs throughout the experiment, which we hypothesize due to the restriction of dislocation motion through interactions with dislocation pileups at phase boundaries and/or immobile dislocation. 2. At small strains, Ol-rich samples have strengths greater than the uniform-strain rate boundary and Per-rich samples have strengths similar to those predicted by the uniform-stress boundary (lower bound). 3. At large strains, our two-phase samples are stronger than predicted by a rule of mixtures based on the end-member flow laws, i.e., the uniform-strain rate boundary (upper bound). This behavior is likely due to the presence of phase boundaries in our samples and their interactions with dislocations during deformation.

Data Availability Statement
All data used in this study are available from the University of Minnesota Digital Conservancy (https://doi. org/10.13020/k0p0-5047, Wiesman et al., 2023).