Deformation memory in the lithosphere: A comparison of damage-dependent

Strain localization in the lithosphere and the formation, evolution, and maintenance of resulting plate boundaries play a crucial role in plate tectonics and thermo-chemical mantle convection. Previously activated lithospheric deformation zones often appear to maintain a “memory” of weakening, leading to tectonic inheritance within plate reorganizations including the Wilson cycle. Different mechanisms have been proposed to explain such strain localization, but it remains unclear which operate on what spatio-temporal scales, and how to best incorporate them in large-scale mantle convection models. Here, we analyze two candidates, 1), grain-size sensitive rheology and, 2), damage-style parameterizations of yield stress which are sometimes used to approximate the former. Grain-size reduction due to dynamic recrystallization can drive localization in the ductile domain, and grain growth provides a time-dependent rheological hardening component potentially enabling the preservation of rheological heterogeneities. We compare the dynamic weakening and hardening effects as well as the timescales of strength evolution for a composite rheology including grain-size dynamics with a pseudo-plastic rheology including damage(or “strain”-) dependent weakening. We explore the implications of different proposed grain-size evolution laws, and test to which extent strain-dependent rheologies can mimic the weakening and hardening effects of the more complex micro-physical behavior. Such an analysis helps to better understand the parallels and differences between various strainlocalization modeling approaches used in different tectonics and geodynamics communities. More importantly, our results contribute to efforts to identify the key ingredients of strainlocalization and damage hysteresis within plate tectonics and how to represent those in planetary-scale modeling.


Introduction
The Earth's current mode of heat transport is by means of plate tectonics which, by definition, is characterized by relatively rigid plate interiors and narrow plate boundaries where deformation due to relative plate motions is localized. The lithosphere, i.e. the top, cold, strong thermo-chemical boundary layer of mantle convection, is thus broken up such that brittle or plastic processes reduce the effective strength of rocks (i.e. the viscosity in the case of fluid behavior) which would otherwise be huge if temperature-dependent creep were the only relevant deformation mechanism (e.g. Kohlstedt et al., 1995;Burov, 2011). For some aspects of convection models, such behavior can be approximated by "Byerlee" type visco-plasticity with a depth-or pressure-dependent yield stress (e.g. Moresi & Solomatov, 1998;Enns et al., 2005). However, the yield stresses that are needed to break a homogenous lithosphere in convection models are typically much lower than those expected from rock mechanics, and pure plasticity is on its own not progressively weakening and inherently without memory of deformation (e.g. Bercovici 2003;Tackley, 2000a).
It is likely that because of this lack of strain localization, visco-plastic rheologies in mantle convection models only yield approximately plate-like surface motions (e.g. Tackley, 2000b;van Heck & Tackley, 2008;Foley & Becker, 2009). The planform of surface motions seems to become more realistic when a low-viscosity asthenosphere (Tackley, 2000c;Richards et al., 2001;Höink et al., 2012), a strongly temperature-dependent viscosity (e.g. Coltice et al., 2017, 2019), a free surface and weak oceanic crust (Crameri et al., 2012), and/or the presence of strong continents (Coltice et al., 2012) is included within visco-plastic models. However, a velocity/strain-weakening or pseudo stick-slip, strain localizing rheology is still required to achieve appropriate levels of toroidal motion and hallmark features of plate tectonics such as 74 transform faults offsetting spreading centers (e.g. Bercovici, 1993Bercovici, , 1995Tackley, 2000c;Gerya, 75 2013; Bercovici et al, 2015). 76 Strain localization is, of course, also observed in nature (e.g. Audet & Bürgmann, 2011;77 Montesi, 2013; Précigout & Almqvist, 2014) as well as in deformation experiments (e.g. 78 Hansen et al., 2012). In models, strain-localization has been explored for many 79 different processes, including but not limited to, thermal localization (e.g. Schubert 93 In the viscous regime, one important mechanism that has been suggested for localization 94 is grain-size evolution (GSE, e.g. Bercovici Bercovici et al., 2015;Foley, 2018). Diffusion creep viscosity is controlled by grain size, and 96 reduction of grain size due to dynamic recrystallization as well as a transition from dislocation 97 creep to diffusion creep dominated deformation can lead to localization (e.g. Braun et al., 1999; 98 Platt & Behr, 2011; Montesi, 2013). However, the physics and formulation of GSE, especially 99 for non-single-phase conditions (such as for a peridotite) and the effects of grain-growth limiting 100 Zener pinning, remain less well constrained (e.g. Bercovici Bercovici, 2017Bercovici, , 2018, and grain-size evolution laws remain expensive to implement in large-102 scale convection models (e.g. Barr and McKinnon, 2007;Dannberg et al., 2017;Foley and Rizo, 103 2017). Thus, a first order approximation of such microphysical behavior via a parameterized 104 weakening formulation could be helpful. 105 Damage or "strain" dependent rheologies can possibly provide such a simplification. 106 These are often motivated by dynamic weakening in the brittle/frictional regime where additional 107 weakening mechanisms, such as mineral transformations, serpentinization/mylonitization, 108 partial-melting assisted, flexural/bending weakening, or the coalescence of cracks occur. Such 109 mechanisms can result in a reduction of the effective yield stress (either due to a reduction of 110 cohesion, or reduction of the internal angle of friction) rather than viscosity, as in the case of ). With exceptions (e.g. Gerya,2013), one potential issue with many empirical formulations 120 is the lack of a recovery mechanism providing a time scale for a rheological memory, such as 121 would be expected, for example, for the growth of grain-sizes in GSE, or transformation of 122 minerals. This complicates the comparison of damage-dependent implementations to those based 123 on microphysical behavior such as grain-size evolution, and use of SDW models for long-term, 124 thermal convection models. 125 Given the promise of both GSE and SDW approaches and their respective advantages 126 and drawbacks in terms of physical realism and ease of implementation, we proceed to compare 127 different implementations to highlight their weakening and memory dependent healing behavior 128 using a range of simplified evolutionary deformation tests. We quantify the amplitude and time 129 scales of dynamic weakening and hardening for a pseudo-plastic rheology in combination with 130 "strain"or damage-dependent weakening (e.g. Tackley (1) 166 where T is the non-dimensional temperature, η0 a pre-exponential factor (here unity due 167 to non-dimensionalization) and η1 is the non-dimensional activation energy. 168 The yield and effective viscosity, ηy and ηeff, for a pseudo-plastic rheology can be defined 169 as (e.g. Tackley, 2000b, c): convenience, we will refer to this apparent strain variable γ as "strain" in the following. 201 The temporal evolution of the strain is defined by (e.g. Tackley The temperature-dependent healing rate is assumed to be an average of a possibly 212 constant and purely temperature-dependent healing rate (e.g. due to diffusion processes), which 213 can be described by half the inverse of the diffusion creep viscosity (e.g., Tackley, 2000b). 214 Temperature-dependent healing avoids infinite strain accumulation and leads to long-term strain 215 memory in the cold lithosphere and removal of damage within the hotter asthenosphere. The 216 apparent strain hardening mechanism mimics a reduction of the effective strain either by mixing 217 and stirring of the mantle with typical strain rates of the mantle or due to temperature-dependent 218 microphysical processes (e.g. diffusion or grain growth). For SDW, we always assume the strain 219 rate of eq. (6) to be the total strain rate. To allow for maximum weakening to be uniquely 220 described by the critical strain and maximum damage, and to avoid a time lag of strain hardening 221 once deformation ceases, we assume that no further damage accumulates once the critical strain 222 is reached. Thus, we assure that strain hardening initiates at the same time grain growth initiates. 223 The amplitude of weakening/hardening in the composite, grain-size sensitive rheology, is then 224 determined by the strain-weakening parameters, which control the rate and amplitude of the 225 strain weakening. 226 The amount of the "damage" D is assumed to depend linearly on the accumulated strain γ To test different weakening descriptions for SDW, we use three different formulations:   written as a sum of growth and reduction rates:  341 where Cg is a temperature and material dependent rate constant (see supporting 342 information S4). 343 The growth rate constant Cg, as defined in eq. (11), is controlled mainly by temperature 344 but also pressure, water content, and impurities (e.g. porosity, melt content, and secondary 345 phases). In addition, a calibration assuming a different GSE-model (e.g., as a piezometer; e.g.   For the pseudo-plastic rheology including SDW, we calculate the apparent strain γ (eq. 6-458 9) using the same ODE solver and a range of critical strains γcr (1, 5, 10) and healing time scales 459 B (10 -16 , 10 -14 , 10 -12 s -1 ). The accumulated strain γ(t) defines the amount of damage (eq. 8), and 460 hence weakening (eq. 9). We used SDW-I, SDW-II, and SDW-III, as defined above, to analyze 461 damage evolution. The apparent strain from eq. (6) is not the same as the total actual strain γtot 462 ( Figure 5), which is defined by the time integral of the second invariant of the total strain rate. 463 We show the temporal evolution of grain size, diffusion creep viscosity, deviatoric stress, 464 and logarithm of the ratio of the dislocation and diffusion creep strain rates for each GSE-model, 465 before comparing the weakening and hardening effects of different weakening mechanisms. which visco-plastic strain softening will be significant. For each temperature range, the transient behavior varies significantly between the 496 different models resulting in different viscosity variations. The strongest difference is observable 497 for Dan17 due to a significantly slower growth rate compared to Br99, Be09, or Ro11. A 498 reduction of grain size is seen at the first strain rate increase at every temperature, but grain 499 growth is negligible, even for the high temperature case. 500 In the following, we will focus on the differences between Br99, Be09, and Ro11. These 501 models are most similar at the intermediate temperature range (middle column in Figure 4). All 502 models show a grain-size reduction due to the increase of the total strain rate and growth 503 governed by the actual grain size and temperature (eqs. 11-13), except at the beginning of the last 504 stage of Br99. This is artificial, however, since the grains reach a size during grain growth in the 505 third stage similar to the steady-state grain size of the last stage. This shows a significant 506 difference between the GSE rates for the piezometer and the remaining models (cf. the diffusion 507 creep regime in Figure 3). Only minor differences are observable in grain-size reduction between 508 the wattmeter (Be09) and the thermodynamically self-consistent model (Ro11), both of which 509 adjust much faster than the piezometer (Br99). Regarding the growth rate, Ro11 is much faster 510 than Be09 and always reaches steady-state. This is even more pronounced at low temperatures 511 (due to the already small grains). 512 At the low temperature range (left column in Figure 4), viscosity and thus stress reaches a 513 maximum. Therefore, we obtain the smallest steady-state initial grain size (see supporting 514 information eqs. S24-S26), which also affects the growth rate (eq. 11). The combination of a 515 smaller steady-state grain size and a smaller growth rate (especially for Be09 and Ro11) results 516 in an overall smaller variation of the grain size. The smaller absolute and steady-state grain size 517 also favors a faster grain 'growth' for the piezometer (see eq. 12 and supporting information eqs. 518 S14-S16). However, grain growth in the low temperature range is significantly only for Ro11. 519 Steady-state grain size is never reached in the growth phase for the remaining GSE-models. 520 While grain growth remains faster for Br99 in comparison to Be09, grain-size reduction remains 521 smaller. 522 Overall, the most time-dependent GSE model is the piezometer (Br99) with the slowest 523 reduction and growth rate (besides Dan17). The wattmeter (Be09) has a much faster reduction 524 rate, but still a slower growth rate than the thermodynamically self-consistent model (Ro11) We compare three different SDW mechanisms with the GSE-models (Figures 5 and 6).

SDW-II (VSS) a
Fast for γ cr < 5 Less effective ( (10)) than GSE Clear hardening effect only for B = 100 Less effective and slower than GSE

SDW-III (VSS) a
Fast for γ cr < 5 Somewhat more effective ( (1,000) For SDW-I, we assume deformation only takes place in the plastic regime, eq. (9), where 573 damage leads to a linear reduction of the yield stress or yield viscosity (top row in Figure 5). 574 Thus, the pseudo-plastic viscosity instantly changes with the strain rate.  laboratory grain-size evolution laws (Figure 1). However, to more widely, and perhaps more 733 realistically, explore the parameters, we analyzed the healing time of a lithospheric shear zone assuming a vertical deformation zone and an oceanic geotherm (Figure 8). For the geotherm we 735 assume a half-space cooling model for an oceanic lithosphere of 120 Ma age with constant 736 thermal parameters and a potential mantle temperature of 1315 °C. As an initial condition, we 737 assume a steady-state grain size at a background strain rate of 10 -15 s -1 within the shear zone that 738 might mimic a nearly rigid plate. We then calculate the grain-size reduction along a one-739 dimensional temperature profile for a sudden strain rate increase (up to 7 orders of magnitude), 740 say due to enhanced tectonic deformation (side stepping the issue of nucleation). A similar 741 analysis is feasible assuming constant stress conditions, but due to kinematic nature of our 742 previous experiments we focus on the constant strain-rate approach. Based on the weakening 743 behavior discussed above, we assume that the steady-state grain size responds instantly. 744 Assuming the strain-rate is reduced to its initial value after this deformation episode, we 745 calculated the time (th) for the effective viscosity and for the grain size to reach steady-state 746 again (within 1 and 0.1%, respectively) solving eq. (10) using a dry, composite rheology (Hirth 747 & Kohlstedt, 2003) and GSE of Ro11 and Be09 (Figures 8a and b). Additionally, we show the 748 time to reduce the accumulated damage by 95% (cf. supporting information eq. S9) for B = 10 -12

750
in press at JGR-SE 11/2020 The results show, that the healing time th does not depend on the actual grain size and is 760 thus independent of the amount of deformation; it is, however, strongly governed by 761 temperature. The same is true for the reduction of strain as defined in supporting information 762 eq. (S9). Within the shear zone, the healing times for grain growth are well matched by the strain reduction rates of the simplified description (Figures 8a and b)  pseudo-plastic rheology is unity (eq. 2) and falls into the range of an effective exponent for 805 diffusion creep deformation with steady-state grain size. 806 Considering instant weakening in GSE due to grain-size reduction, the steady-state 807 approximation is suitable to describe weakening. The weakening described by eq. (14), however, 808 only addresses the diffusion creep contribution due to a variation of the dislocation creep strain 809 rate. Considering that the dislocation creep strain rate decreases for a more dominant diffusion 810 creep deformation, weakening due to grain size reduction becomes less effective. Therefore, it is 811 only close to the transition between dislocation and diffusion creep where weakening due to 812 grain-size reduction can be approximated by plastic yielding. That said, a change in viscosity due 813 to grain-size reduction is almost of the same order of magnitude as a change in viscosity due to 814 plastic yielding. The choice of a small critical strain (e.g. γcr = 1) and a moderate damage 815 parameter (D max ~ 60-80 %) for a plastic strain softening rheology serves to best approximate the 816 weakening behavior expected from grain size reduction. strain-dependent weakening can model a healing behavior that is similar to the hardening due to 831 grain growth as described by a wattmeter. However, the healing behavior for a VSS rheology 832 fails to approximate grain growth strengthening. The rate, governed by B and η2, can be similar 833 to the grain growth rate, but VSS does not enable hardening larger than the 'undamaged' state. 834 This is, however, crucial for the composite dislocation-diffusion creep rheology expected for the 835 uppermost mantle, which is governed by grain size and temperature, respectively. On the other 836 hand, plastic strain softening and the associated yielding implementation do not only resemble 837 the amount of healing due to grain growth, but also its rate. 838 In particular, for the grain-size evolution models explored, the healing time scale B ~ 839 6•10 -11 -3•10 -10 s -1 and an activation energy of η2 ~ 30 -70 best approximate the time scales for 840 grain growth in a composite rheology. Therefore, the plastic strain softening rheology does 841 indeed enable a "realistic" hysteresis effect with a memory duration that is similar to that 842 expected for grain growth for lithospheric temperature conditions. This allows modeling the 843 formation, maintenance, and reactivation of lithospheric weak zones, but precludes further 844 weakening in the deeper mantle due to the higher temperatures and faster healing. Our results 845 help to identify the features and parameter ranges needed to represent grain-size evolution laws 846 and their associated rheologies with simplified approaches. Additional comparisons with laboratory and field observations using this simplified framework may serve to resolve 848 outstanding questions of plate tectonic strain localization.