The interaction of deformation and metamorphic reactions

doi:10.1144/SP332.12 2010; v. 332; p. 189-223 Geological Society, London, Special Publications

Bruce E. Hobbs, Alison Ord, Maria Iole Spalla, Guido Gosso and Michele Zucali

The interaction of deformation and metamorphic reactions

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The interaction of deformation and metamorphic reactions

BRUCE E. HOBBS1,2*, ALISON ORD1,2, MARIA IOLE SPALLA3, GUIDO GOSSO3 &

MICHELE ZUCALI3

1CSIRO Exploration and Mining, PO Box, 1130 Bentley, Western Australia 6120, Australia2School of Earth and Geographical Sciences, The University of Western Australia, Perth,

Western Australia, Australia3Dipartimento di Scienze della Terra, Universita degli Studi di Milano, Via Mangiagalli 34,

20133 Milano, Italy

*Corresponding author (e-mail: [email protected])

Abstract: Feedback relations between deformation and metamorphic mineral reactions, derivedusing the principles of non-equilibrium thermodynamics, indicate that mineral reactions progressto completion in high-strain areas, driven by energy dissipated from inelastic deformation. Theseprocesses, in common with other time-dependent geological processes, lead to both strain, andstrain-rate, hardening/softening in rate-dependent materials. In particular, strain-rate softeningleads to the formation of shear zones, folds and boudins by non-Biot mechanisms. Strain-softeningalone does not produce folding or boudinage and results in low-strain shear zones; strain-rate soft-ening is necessary to produce realistic strains and structures. Reaction–mechanical feedbackrelations operating at the scale of 10–100 m produce structures similar to those that arise fromthermal–mechanical feedback relations at coarser (kilometre) scales and reaction–diffusion–mechanical feedback relations at finer (millimetre) scales. The dominance of specific processesat various length scales but the development of similar structures by all coupled processes leadsto scale invariance. The concept of non-equilibrium mineral stability diagrams is introduced. Inprinciple, deformation influences the position of mineral stability fields relative to equilibriumstability fields; the effect is negligible for the quartz! coesite reaction but may be importantfor others. Application of these results to the development of structures and mineral reactions inthe Italian Alps is discussed.

Metamorphism is commonly associated withdeformation and regional thermal gradients, to-gether with chemical reactions, all of which arenon-equilibrium situations. In many instances,another non-equilibrium process, namely fluidflow (either of melt or fluids comprised of H2O,CO2 and so on), accompanies the metamorphicprocess and is intimately coupled to deformation,infiltration, heat advection and chemical reactions.This paper explores the influence of energy dissi-pated by such processes upon: (i) the developmentof meso-scale structures commonly observed indeformed metamorphic rocks; (ii) the kinetics ofmetamorphic reactions; and (iii) the thermodynamicstability of metamorphic assemblages.

To focus the discussion we concentrate on meta-morphic rocks undergoing deformation and chemi-cal reactions with no large-scale mass infiltration(i.e. metasomatism) so that fluid flow is consideredinsignificant and also any regional thermal gradientsare neglected. As such, the energy dissipated duringdeformation and metamorphism consists of fourparts: (i) that due to mechanical processes; this

comprises dissipation arising from inelastic defor-mation and from the work done in forming chemicalcomponents in the deforming–reacting system bymineral reactions; (ii) that arising from the diffusiveflux of chemical components across gradients inboth chemical potentials and local gradients intemperature; (iii) that arising from chemical reac-tions; and (iv) that arising from thermal conduction.In the absence of metasomatism at the outcrop scale,many effects arising from mass flux and thermaltransport turn out to be unimportant. An importantingredient of the interaction between these dissipa-tive processes is that of spatial scale, so thatvarious processes dominate at different lengthscales. Thermal feedback effects dominate atscales larger than the outcrop scale, and mass trans-fer feedback effects dominate at the micro-scale.The dominance of these different processes atspecific length scales and the development ofsimilar geological structures at all scales by differ-ent coupled processes are the two principles at theheart of the scale invariance of structures observedin geology. This separation of spatial scales also

From: SPALLA, M. I., MAROTTA, A. M. & GOSSO, G. (eds) Advances in Interpretation of Geological Processes:Refinement of Multi-scale Data and Integration in Numerical Modelling. Geological Society, London, SpecialPublications, 332, 189–223. DOI: 10.1144/SP332.12 0305-8719/10/$15.00 # The Geological Society of London 2010.

greatly simplifies any study of the coupling effects.We include a brief consideration of the influence ofmetasomatism in the Discussion section.

In many metamorphic rocks there is evidence ofnon-equilibrium in the form of partially reactedmineral assemblages. An example is the MonteMucrone–Monte Mars region in the Italian Alps(Sesia–Lanzo Zone: Zucali et al. 2002) whereigneous relicts are preserved in a subductioncomplex; here in rocks immediately adjacent toeach other, the metamorphic reactions reach com-pletion only in highly deformed shear zones whichare ubiquitous. In such rocks the progress ofmineral reactions correlates with the amount ofstrain and not necessarily with the maximumpressure and temperature (P, T ) conditionsreached. The important questions are: (i) Whatroles do mineral reactions play in promoting defor-mation and influencing the style of deformation,particularly the development of shear zones, foldsand boudinage? (ii) What role does deformationplay in promoting metamorphic reactions? (iii) Inregions where the metamorphic reactions have notproceeded to completion and deformation accompa-nies the reactions, are estimates of pressure andtemperature conditions derived from equilibriumtheory relevant and/or accurate, and what influencedoes deformation have on the position of the mineralphase boundary in pressure–temperature spaceestablished by using equilibrium thermodynamics?The questions are summarized in Figure 1. Althoughthere is a wealth of literature published on theseissues, there is no discussion that integrates themwithin one framework. We attempt to do thiswithin the concepts of modern non-equilibriumthermodynamics.

The structure of the paper is as follows. We firstoutline the observations of partially completedmetamorphic reactions and the relationship to defor-mation observed in the Italian Alps. We then brieflyreview the development of non-equilibrium ther-modynamics as it is relevant to the coupling of geo-logical deformation and metamorphic reactions.Next we consider the influence of metamorphicreactions on constitutive behaviour, together with

the influence of mineral reactions on deformationwith particular emphasis on localization, foldingand boudinage. The concept of non-equilibriummineral phase diagrams is then introduced, and wediscuss the influence of deformation on mineralstability. Finally, we consider the implications ofthis non-equilibrium framework for metamorphicrocks in general, and the Italian Alps in particular.Terms are defined as they are introduced in thetext and in Table 1.

Examples of extent of reactions as a function

of degree of deformation from the Italian

Alps

Western Alps

Within the Monte Mucrone–Monte Mars area ofthe Sesia–Lanzo Zone (internal part of WesternItalian Alps, location 1 in Fig. 2) six periods ofdeformation and related metamorphism (commonlyassociated with incomplete mineral reactions) arerecorded (Zucali et al. 2002). The first and oldest(pre-D1) of these is a pre-Alpine high tempera-ture–low pressure (HT–LP) event, and five sub-sequent events are recorded during the Alpineevolution in which D1 corresponds to the P–T pro-grade path. D2 and D3 structures developed underTmax–PTmax conditions and are dated at approxi-mately 65 Ma, whereas D5 is earlier than 30 Ma.Events D2–D5 represent a progressive evolution inP–T conditions from approximately 2 GPa and550 8C for D2 to 0.4 GPa and 200 8C for D5, associ-ated with the exhumation after subduction of thisportion of Austroalpine continental crust (Fig. 3a).

The Monte Mucrone–Monte Mars is a singletectono-metamorphic unit (Spalla et al. 2005) inwhich the heterogeneous distribution of deforma-tion and metamorphic imprints can be representedin a map of deformation imprints (Fig. 3b) empha-sizing the metamorphic assemblages that predomi-nate in each region. The important point is that thedominant metamorphic imprint in a particularregion is not coincident with the P–T peak of thetectono-metamorphic unit but rather is the oneassociated with the most pervasive deformation ofeach region (Spalla et al. 2005). The dominant meta-morphic imprint is also better developed as theintensity of the most pervasive deformation eventincreases with a direct correlation at the micro-scalewith the progressive development of schistosity, asproposed by Bell & Rubenach (1983). The progresstowards completion of mineral reactions is distrib-uted in a patchy manner with lozenge-shaped,low-strain regions of very little mineral reactionembedded in an anatomosing network of high-strain shear zones that are apparently unrelated in

Fig. 1. The interplay between deformation andmetamorphic mineral reactions.

B. E. HOBBS ET AL.190

Table 1. Symbols used in this paper, units and typical values

Quantity Description Typical values,Units

A Affinity of mineral reaction J kg21

A Affinity vector in stress space J kg21

B Constant Pa sc Concentration of chemical species mol kg21

cp Specific heat at constant pressure 1450 J kg21 K21

Da Damkohler Number DimensionlessE Young’s modulus 4.5 � 109 PaGr Gruntfest Number Dimensionlessg Acceleration due to gravity 9.8 m s22

h Depth below surface of Earth mh Mechanical hardening coefficient Pahchemical Chemical hardening coefficient PaJ2 Second invariant of the deviatoric stress Pa2

JK Mass flux of K-th chemical component kg m22 s21

k Shear zone thickness mL Pre-exponential constant Pa2N s21

lthermal Thermal diffusion length scale mlchemical Chemical diffusion length scale mM Power-law dependence of chemical reaction rate on strain rate DimensionlessmK Concentration of K-th chemical component kg m23

N Stress power-law exponent DimensionlessPethermal Thermal Peclet Number DimensionlessP Pressure Mean stress Pap Power-law dependence of strain hardening DimensionlessQ Activation enthalpy J mol21

Qmechanical Activation enthalpy for mechanical deformation J mol21

Qchemical Activation enthalpy for mineral reaction J mol21

q Power-law dependence of strain-rate hardening DimensionlessR Gas constant 8.3143 J K21 mol21

s Specific entropy J kg21K21

s Rate of specific entropy production J kg21 K21 s21

T Absolute temperature KTc Critical temperature where thermal–mechanical feedback

becomes importantK

t Time su Specific internal energy J kg21

u Local material velocity vector m s21

xi Spatial co-ordinates mg Integrated shear strain Dimensionless1ij Strain tensor Dimensionless

1elasticij Elastic strain tensor Dimensionless

1viscousij Viscous strain tensor Dimensionless

_1ij Strain-rate tensor s21

_H@ Rate of heat production from latent heat production fromthe @-th mineral reaction

J kg21 s21

h Mechanical viscosity Pa shchemical Chemical viscosity Pa skthermal Thermal diffusivity 1026 m2 s21

kchemical Chemical diffusivity m2 s21

kij Diffusivity tensor m2 s21

l@ Dimensionless group that is an expression of dissipationof the @-th process

Dimensionless

mK Specific chemical potential of K-th, chemical component J kg21

v Specific volume m3 kg21

vK Specific volume of K-th chemical component m3 kg21

v0 Specific volume under hydrostatic stress m3 kg21

(Continued)

DEFORMATION AND METAMORPHIC REACTIONS 191

orientation to the axial planes of folds, and withinwhich the coeval mineral reactions have progressedto completion.

Central Alps

A belt of steep, thin slices of continental crustbounds the Periadriatic Lineament (internal partof Central Italian Alps, location 2 in Fig. 2) andcontains portions of Austroalpine crust as theLanguard–Tonale tectono-metamorphic unit whereAlpine and pre-Alpine structures have been dis-tinguished by their relationships with Permianintrusives (Gazzola et al. 2000). The pre-Alpinedeformations (D1 and D2) took place under amphi-bolite–granulite-facies conditions, and the mostpervasive S2 pre-Alpine fabric is marked bymineral assemblages peculiar to a HT–LP imprintof Permian age. The Alpine structures developedunder epidote–amphibolite-facies (D3) and greens-chist-facies conditions (D4) and are recorded duringthe progressive P–T evolution from approximately1.2 GPa and 500–600 8C, representing the Alpinemetamorphic climax, to �0.5 GPa and �350 8C,characterizing the retrograde exhumation path(Fig. 4a).

Structural mapping assisted by microstructuraland petrological analysis indicates that since thePermian the deformation history and associatedmetamorphic evolution has been spatially and tem-porally consistent throughout the Languard–Tonalepackage, which therefore represents a singletectono-metamorphic unit throughout its post-Permian evolution (Gosso et al. 2004; Spalla et al.2005; Salvi et al. 2010). Also in this case, as in

the Western Alps, the distribution of successivedeformation and metamorphic imprints is highlyheterogeneous. At the micro-scale (Fig. 4b), pro-gress to completion of mineral reactions is achievedin high-strain zones where the fabric is mylonitic;this is true for each rock type of the lithostratigra-phical association comprising metapelites, metain-trusives, marbles and metabasics (Zucali 2001).Figure 4b displays the progressive decrease ofthe modal amount of pre-Alpine minerals from thepoorly deformed (coronitic, Fig. 4b1, b2) to thehighly deformed (mylonitic, Fig. 4b4–b6) metape-lites, where pre-Alpine relicts are less than 10% involume. In this case again, different dominant meta-morphic imprints coexist in spatially adjacentvolumes of a single tectono-metamorphic unit andthey do not reflect the P–T peak reached withinthe tectono-metamorphic unit, but correspond tothe metamorphic conditions associated with themost pervasive fabric of each region. An importantobservation is that the microstructures of lessdeformed domains (Fig. 4b1–b3) in many casessuggest that grain-size reduction resulting from thenucleation of new phases is not sufficient to localizegranular-scale deformation.

Examples of natural synmetamorphic shear

zones, folds and boudins in contrasted

thermal environments from the Italian Alps

Shear zones

Many Alpine shear zones display V and Y shapes inaddition to the expected X shapes where conjugateshear zones intersect (Fig. 5).

Table 1. Continued

Quantity Description Typical values,Units

j Extent of chemical reaction or other process Dimensionlessj Rate of chemical reaction s21

r Mass density 2750 kg m23

sij Cauchy Stress tensor Pa

s 0ij Deviatoric stress tensor Pas Stress vector in stress space PaF Specific dissipation function J kg21 s21

Fmechanical Specific mechanical dissipation J kg21 s21

Fchemical Specific chemical dissipation J kg21 s21

Fdiffusive Specific diffusive dissipation J kg21 s21

Fthermal Specific thermal dissipation J kg21 s21

w Thermal-hardening parameter Pa K21

x Thermal efficiency DimensionlessC Specific free energy of the system J kg21

v Strain-rate-hardening parameter Pa svchemical Chemical strain-rate-hardening parameter Pa s@ Number of mineral reactions or processes Dimensionless


Fig. 2. Tectonic map of the Alps with the location of the natural examples described in the text (numbered black stars). Legend: 1, Southalpine basement; 2, Austroalpinebasement; 3, Penninic basement; 4, Helvetic–Dauphinois–Provencal basement; 5, Tertiary intrusive stocks.

DE

FO

RM

AT

ION

AN

DM

ET

AM

OR

PH

ICR

EA

CT

ION

S193

In the Monte Mucrone area (location 1 in Fig. 2)Y-shaped shear zones localize (Fig. 5a1) in the eclo-gitized metagranitoids during the early deformationperiod under eclogite-facies conditions. Startingfrom the Permian igneous assemblage (plagioclase,quartz, biotite, K-feldspar and accessory minerals)

the new eclogite-facies minerals ( jadeite-rich clino-pyroxene, zoisite, phengite, garnet, K-feldspar andquartz) developed both in poorly deformed grani-toids as coronas (Fig. 5a2) or as fine-grained aggre-gates or trails in the mylonitic bands (Fig. 5a3). Inthe weakly deformed rocks the modal amount of

Fig. 3. (a) P–T–d– t path inferred from rocks of Monte Mucrone–Monte Mars area (simplified after Zucali et al.2002). Pre-D1, D1, D2þD3, D4 and D5 represent P–T conditions estimated with respect to the successive deformationperiods (d). Pre-Alpine and Alpine P–T– t–d evolutions are compared with Vi (stable geotherm after Thompson &England 1984). (b) Map of deformation imprints, simplified after Zucali et al. (2002) with domains recording thesame relative timing of superposed structures. The areal extent of every new planar synmetamorphic fabric (dominantfabric) has been estimated and reported in the legend. Close-up displays the petrographic–structural map betweenMonte Mucrone and Monte Mars (Zucali et al. 2002). Legend: 1, metagranitoids with coronitic fabrics; 2, micaschistsand paragneisses; 3, metagranitoids showing mainly mylonitic fabrics, with minor tectonites; 4, eclogites.


Fig. 4. (a) P–T– t–d path inferred for the Alpine-subducted metaintrusives and surrounding metapelites of theLanguard–Tonale tectono-metamorphic unit (redrawn after Gazzola et al. 2000; Gosso et al. 2004). D2, D3 and D4

represent P–T conditions estimated with respect to the successive deformation periods; pre-Alpine and AlpineP–T– t–d evolution compared with Vi (stable geotherm after Thompson & England 1984). (b) Alpine microstructurescharacterizing the IT–HP metamorphic re-equilibration in metapelites from the Languard–Tonaletectono-metamorphic unit. Weakly deformed domains. (b1) Coronas of Alpine garnet (Grt II) around pre-Alpine relictgarnet (Grt I); Alpine phengitic white mica (Wm) and ilmenite (Ilm) partly replace pre-Alpine biotite (Bt), and theplagioclase site is fully pseudomorphed by a thin aggregate of epidote and white mica. (b2) Pre-Alpine staurolite (St)totally replaced by a very thin aggregate of new Alpine minerals, easily detected with the SEM (photomicrograph b3);sillimanite, marking with biotite the pre-Alpine S2 foliation, is overgrown by small kyanite Alpine grains. (b3) BSEimage of Alpine chloritoid (Cld) and kyanite (Ky) micro-aggregate occurring in microsites such as that of pre-Alpinestaurolite in photomicrograph b2. Highly deformed domains. (b4) The S3 pervasive foliation is marked by an Alpinephengitic white mica (Wm) and chloritoid (Cld) shape-preferred orientation; garnet porphyroblasts (Grt) preserve thepre-Alpine compositions at the cores, which represent the only pre-Alpine relicts in these rocks. (b5) The Alpinemylonitic foliation is marked by ribbon-quartz, phengitic white mica and trails of new garnet (Grt II); pre-Alpine relictsexclusively consist of garnet porphyroclasts (Grt I). (b6) New bluish amphibole (Amp) underlines the Alpine myloniticfoliation, together with garnet (Grt), white mica and ribbon-quartz; pre-Alpine red biotite (Bt) is rarely preserved assmall relict grains.


the preserved igneous relict minerals can reach70% in volume, whereas in the high-strain domains,where the fabric is mylonitic, the HP Alpine mineralscan occupy more than 90% in volume. The paragen-esis observed in shear zones, or as coronas in the

weakly deformed rocks, and the related mineral com-positions indicate that phase transitions occurredunder consistent P–T values in eclogite-facies con-ditions (Fig. 3a) (Fruh-Green 1994; Rubbo et al.1999; Tropper et al. 1999; Zucali et al. 2002).

Fig. 5. Examples of shear zones in the continental crust of the northern Sesia–Lanzo Zone of the Western Alps,eclogitized during Alpine subduction: (a) metagranitoids from Monte Mucrone and (b) eclogitized amphibolites fromValsesia. (a1) Convergent early Alpine shear zones developed under eclogite-facies conditions in a metagranitoid (baseof the ENE slope of Monte Mucrone). In the shear zone (a3) the mylonitic fabric is associated with a widespreaddevelopment of Alpine eclogitic minerals, such as garnet (Grt), phengitic white mica (Wm), jadeitic clinopyroxene(Cpx) and zoisite/clinozoisite (Ep); rare K-feldspar porphyroclasts (Kfs) represent the igneous pre-Alpine relicts (herec. 3% in volume). In the weakly deformed domains (a2) the igneous texture, as well as the igneous minerals (Kfs, Qtzand Bt), are well preserved (c. 60% in volume) and Alpine eclogitic minerals (Grt, Wm, Cpx, Ep) are confined to thincoronas around igneous biotite (Bt) and to the plagioclase (ex-Pl) microsite. (b1) Relicts of pre-Alpine amphibolitesfrom middle Val Sesia (redrawn after Lardeaux & Spalla 1991). In the shear zone (b2) pre-Alpine relicts are preserved(small grains of brown-green hornblende) only as inclusions in the omphacite porphyroblasts; the fine-grained foliatedmatrix consists of omphacite, phengite, zoisite, blue-green amphibole and minor garnet that occur also withinporphyroblasts. Conversely, outside the shear zone (b3) the pre-Alpine granoblastic fabric and mineral assemblages(brown hornblende) are still preserved; the plagioclase microsite is fully pseudomorphosed by a thin aggregate ofzoisite, jadeite and white mica.


In the northern part of the Sesia–Lanzo Zone(middle Val Sesia, location 3 in Fig. 2) Y- andX-shaped shear zones developed in the pre-Alpinegranulites and amphibolites (Fig. 5b1) during theearly Alpine deformation periods under eclogite-facies conditions (Lardeaux & Spalla 1991). As inthe previous case, the modal amount of eclogiticAlpine minerals increases strongly from nearlyundeformed eclogitized amphibolites to myloniticeclogites (Fig. 5b2, b3). Also in this case, thermo-barometric estimates performed on the weakly (cor-onitic) and highly (mylonitic) deformed rocks yieldcoherent P–T estimates.

In the Languard–Tonale unit of the Central Alps(location 2 in Fig. 2) Alpine deformation, occurringunder epidote–amphibolite-facies conditions, loca-lizes in metre- to 10 m-scale Y-shaped shear zonesin Permian intrusives (Fig. 6a). Here, as pointedout in the two previous cases, a direct relationshipbetween deformation progress and reaction accom-plishment exists, and is indicated also by thematching of fabric gradients with metamorphic re-equilibration as evidenced by comparisons of quan-titative fabric data with compositional variations inamphiboles of metadiorites (Spalla & Zucali 2004).In the low-strain domains of metadiorites (Fig. 6b)the Alpine IT–HP minerals (blue-green hornblende,albite-rich plagioclase, phengitic white mica,garnet, Mg-rich chlorite, quartz, epidote and ilme-nite) occupy no more than 20% in volume withrespect to the igneous mineral relicts (brown

hornblende, plagioclase, biotite, quartz and acces-sory minerals). In the mylonitic metadiorites(Fig. 6c) the modal amount of Alpine IT–HP min-erals increases up to more than 80% in volume(Salvi et al. 2010). P–T estimates performed bothon weakly deformed rocks and on mylonites givevery close values (P ¼ 1.1 + 0.2 GPa andT ¼ 500–600 8C) for the syn-D3 mineral assem-blages (Gazzola et al. 2000).

In the Oetztal nappe (southern Venosta Valley,location 4 in Fig. 2) conjugate shear zones (Fig. 7a)are widespread in metagranitoids deformed underamphibolite-facies conditions during Alpine times(Spalla & Zucali 2004 and references therein). Asalready described for the previous examples, newAlpine minerals replace the pre-Alpine relicts thatin D2 shear zones mainly consist of partly recrystal-lized K-feldspar porphyroclasts (Fig. 7b, c). Mineralassemblages developing in D2 shear zones arecompatible with parageneses observed in normallyfoliated L–S tectonites and coronites. P–T esti-mates derived from metagranitoids and from themetapelitic country rocks indicate that D2 tookplace at 550–600 8C and 0.6–0.8 GPa.

Folds

Natural cases of extreme thinning of fold limbsare frequent at the 10 m to km scales in a widerange of lithological associations and metamorphic

Fig. 6. (a) Form surface map of a Permian diorite deformed and metamorphosed during Alpine subduction (Gazzolaet al. 2000; Gosso et al. 2004) at Monte Pagano, Central Alps. The metamorphic reactions developing during the earlyAlpine deformation (D3) reach completion in the shear zones (c) where the sole igneous relics are brown hornblende(Amp I) porphyroclasts representing approximately 30% of the mineral modal amount and wrapped by a myloniticfoliation contemporaneous with the growth of new amphibole (Amp II), white mica and garnet (Grt). In the weaklydeformed domains the magmatic fabric is preserved and the Alpine transformations are only incipient (b).


environments (greenschist, blueschist, eclogite toamphibolite–granulite) in the Alpine orogen.

An example at the scale of 100–1000 m comesfrom the southern termination of the SchneebergComplex, in the core of the Oetztal nappe (location4 in Fig. 2), which consists of marbles, micaschists,gneisses and amphibolites, giving rise to the largestmegastructure of the Schlingen Zone of the EasternAlps (Fig. 8a). This rock association forms a multi-layer unit several kilometres thick, which is foldedunder amphibolite- and successively greenschist-facies conditions during D2 and D3 deformationperiods of Alpine exhumation, respectively(Helbig & Schmidt 1978; Hoinkes et al. 1987; vanGool et al. 1987; Spalla 1990). Extreme thinningof limbs in isoclinal fold systems, alternativelymarked by micaschist layers within thick marbles(Fig. 8a close-up), or vice versa, is ubiquitous;amphibolite metamorphic conditions dominatedthe entire D2 deformation period.

In the Languard–Tonale unit of the Central Alps(location 2 in Fig. 2) coupled scales of shortening(from kilometres to centimetres) are manifest innatural fold systems traced by schistosity (S3) thatare overprinted by localized zones of intensedevelopment of a new foliation (S4) positioned asa conjugate axial-planar set (Fig. 8b). Strain accom-modation within grain-scale layering during S4

is assisted by the replacement of IT–HP to LT–LP assemblages in schists (garnet, white mica,+chloritoid,+kyanite,+tschermakitic amphibole,plagioclase and quartz to chlorite, albite, epidote,white mica, +biotite), metadiorites (tschermakiticamphibole, plagioclase, phengitic white mica,zoisite/clinozoisite, garnet, quartz, +Mg-richchlorite, +ilmenite to actinolitic amphibole, albite,white mica, epidote, chlorite, quartz, +biotite andtitanite) and metagranitoids (plagioclase, phengiticwhite mica, zoisite/clinozoisite, garnet, minor blue-green amphibole, quartz, ilmenite/titanite to chlor-ite, white mica, albite, quartz, epidote +biotite).In the high-strain D4 zones the metamorphic repla-cement of earlier mineral assemblages exceeds65% in volume (Salvi et al. 2010).

Boudins

In the axial part of the Alpine belt, synmetamorphicboudinage always displays coupling of the defor-mation gradient with progress of metamorphictransformations at boudin margins.

Layered basic granulites of the Valpelline Seriesof the Dent Blanche nappe (location 5 in Fig. 2) aregenerally boudinaged in the high-grade Sill–Grt–Bt-bearing paragneisses during HT–LP pre-Alpinedeformation. The Valpelline Series is mainlycomposed of metapelites, mafics and carbonaterocks, with a dominant metamorphic and structural

Fig. 7. (a) Conjugate shear zones at the map scale inthe Tschigat metagranitoids, east of Milchsee Scharte(Texel Gruppe, Oetztal nappe) above Merano, SouthTyrol (Spalla 1990, 1993), developed underamphibolite-facies conditions during Alpine times, astestified at the micro-scale by mineral assemblagesmarking the mylonitic foliation within the shear zones(b, plane polarized light; c, crossed polars, with a slightbirefringence compensation). Map legend: orange,metagranitoids; light brown, gneisses; violet, myloniticbands; green, amphibolites; dashed bands, shear zones;light yellow, red dots and blue and red fans,Quaternary deposits.


imprint at amphibolite- to granulite-facies con-ditions of pre-Alpine age (Nicot 1977; Gardienet al. 1994). Three deformation events are relatedto the pre-Alpine evolution and are characterizedby a complex (Roda & Zucali 2008 and referencestherein) evolution from granulite to amphiboliteconditions, predating the Alpine tectono-metamorphic evolution, which is heterogeneouslyrecorded. In the example of Figure 8c1, a low-pressure granulite-facies basic boudin, composedof Cpx–Opxþ Pl + Amp + Bt, formed during ahigher-pressure–lower-temperature amphibolite-facies stage that produced an AmpþGrtþ Cpx þPl + Bt-bearing metamorphic assemblage. Themineral reaction zone migrates from the rim tothe core of the basic boudin (Fig. 8c2) and mimics

the fabric gradient: plagioclase and garnet porphyr-oblast growth is the most prominent result of thismetamorphic transformation.

Similar features characterize also the boudinageof eclogites (Fig. 9a) within the Eclogitic Micas-chists Complex of the southern Sesia–Lanzo Zone(location 6 in Fig. 2). Here the early S1 foliation ispreserved together with the associated eclogiticassemblage (omphacite, epidote, garnet, quartz,rutile) at the core of the boudins, which are fromcentimetre to decimetre in size. At the boudinmargins a new planar fabric appears and intensifiestowards the boudin margins, marked by the syn-D2

blueschist mineral assemblage (glaucophane, clino-zoisite, quartz, garnet, +white mica and titanite)developed during the P-retrograde path of this

Fig. 8. Natural examples of folding and boudinage. (a) Extreme thinning of fold limbs in a marble and schist sequenceof the Schneeberg Complex SW termination (Schlingen zone, Oetztal nappe of the Eastern Alps, on SW face of CimaFiammante, Schnalstal). Upper image: map view (in Spalla 1990); legend: turquoise, marbles; yellow, micaschists.Lower image: close-up of a cross-section of the fold system (arrow) of a natural slope dominated by light-colouredmarbles (photograph width is 600 m long); slope orientation differs from that of the average profile plane of the foldsystem of +158. (b) Lozenge-shaped kilometre-scale LT–IP relict rocks (1 in the legend) carrying an internal foldsystem traced by a S3 foliation; it is dissected by conjugate sets of localized zones of newly generated S4 schistositywithin the same rock types (2 in the legend). Sequence of deformation imprints (3 in the legend) in the Languard–Tonale unit (Austroalpine domain of the Central Alps) showed the existence of two deformation periods duringpre-Alpine evolution and two others during Alpine convergence (D3 and D4; from Gosso et al. 2004). (c) Granulite- toamphibolite-facies transition synchronous with boudinage in metabasics from Valpelline Series (Dent-Blanche nappe,Western Austroalpine domain; Lac Mort locality in Valpelline). The boudin core consists of orthopyroxene,clinopyroxene, plagioclase, +amphibole + biotite, and displays granoblastic texture. Rims and boudin necks areconstituted by an extremely foliated garnet-bearing amphibolite characterized by large plagioclase and dark amphibolegrains that define the newly developing foliation (c2, close up of detail 2 in c1).


portion of the Sesia–Lanzo Zone (Pognante 1989;Spalla & Zulbati 2003). This assemblage of thestructural and mineralogical features seems to berelated to a mechanism for the formation of conju-gate shear zones, initially affecting an eclogitelayer (Fig. 9a3) and localizing the initiation ofits transformation into glaucophanites; succes-sively, a boudin-like structure is intensified withina newly formed eclogite–glaucophanite multilayer(Fig. 9a2).

In a HT–HP-dominated metamorphic environ-ment, as for that of the Argentera–Mercantourmassif (Helvetic–Provencal domain of the external

SW Alps, location 7 in Fig. 2), a correlation maysimilarly be established between fabric intensifica-tion and metamorphic reaction progress. Variscaneclogites boudinaged within biotite-bearing meta-texite layers (Fig. 9b1) display marked texturaland grain-size variations from boudin core toboudin rims. The mineral assemblage of thepoorly amphibolitized eclogite in the boudin core(garnet, omphacite – totally replaced by diopsideþ plagioclase symplectite – hornblende, rutile,quartz and opaque minerals), where the texture isgranoblastic and weakly foliated with a centimetregrain size, is gradually replaced by the new HT

Fig. 9. Natural examples of boudinage. (a1) Field structural setting within the Eclogitic Micaschists Complex at AlpeMecio–Rocca Lunga, southern Sesia–Lanzo Zone (Western Austroalpine: Spalla & Zulbati 2003). Legend: lightorange, quartz-rich eclogitic micaschists; orange, glaucophane-rich micaschists; brown, metagranitoids; violet, albite–epidote–chlorite schists; light blue, glaucophanites; red, pre-Alpine relict amphibolites. Foliation trajectories: red, S1

eclogite-facies foliation; blue, S2 blueschist-facies foliation; green, S3 greenschists-facies foliation; double linetrajectories, mylonitic foliations; inferred trajectories are dashed. S1 syneclogitic foliation is preserved in the boudinsand is variably reoriented with respect to S2 of glaucophanites (a2). Amphibole-free eclogite boudins (a3) progressivelyisolated within intensely foliated (S2) glaucophanites representing their transformation products. (b) Variscanamphibolitized eclogite boudin in the metatexites of the Argentera–Mercantour massif (Helvetic–Dauphinois–Provencal domain of the external SW Alps). Strain gradient localized at the boudin margins (b1) corresponds to a modalmineral amount gradient with increase of amphibolite-facies minerals with respect to the eclogite-facies minerals, moreabundant at the boudin core (b2).


amphibolite-facies assemblage (hornblende II,plagioclase, quartz, opaque minerals) associatedwith a progressively stronger mineral layering andgrain-size reduction, down to mm size (Fig. 9b2).

Thus, there is a 1:1 correlation between both therelative and absolute intensity of a particular defor-mation and the tendency for a metamorphic reactioncorresponding to the P–T conditions of that defor-mation to proceed to completion. This observationis not unique and has been discussed by manyauthors including Beach (1973, 1976, 1980),Rubie (1983, 1990), Austrheim & Griffin (1985),Mork (1985), Koons et al. (1987), Austrheim (1990)and Brodie & Rutter (1990). Examples from exper-iments on the influence of deformation on phasetransitions and chemical reactions in mineralsystems are reported by Coe & Paterson (1969),Brodie & Rutter (1985) and Delle Piane et al.(2007), and in metal systems by Levitas et al.(1998a, b). The influence of mineral reactions onthe constitutive behaviour of materials has beenexplored by Shigematsu (1999), Burlini & Bruhn(2005) and de Ronde et al. (2005).

A common response to the observation thatdeformation influences the progress of metamorphicreactions is that the deformation influences thekinetics of the reaction perhaps by decreasing thegrain size or enhancing diffusion, particularly (butclearly not exclusively) that of water (Rubie 1983,1990; Vernon 2004). These processes are undoubt-edly important and are considered later, wherewe show that not only is there a direct influenceof deformation on the kinetics of a reaction, thereis also a small influence on the stability field ofthat reaction (at least at geological strain rates).More importantly, the feedback between the kin-etics of mineral reactions on the deformation isfundamental in influencing the strain hardening/softening, and particularly the strain-rate-hardening/softening behaviour, and hence controls the devel-opment of structures such as shear zones, foldsand boudinage that otherwise would not develop.In order to progress the discussion we proceedto consider some aspects of non-equilibriumthermodynamics.

Non-equilibrium thermodynamics and

metamorphic petrology

Non-equilibrium thermodynamics increasingly hasa very wide application in many fields of scienceand engineering, and has proved invaluable inexplaining aspects of the Earth’s climate (Paltridge1975, 1978, 2001), the oceanic general circulation(Shimokawa & Ozawa 2002), the fundamentalmechanical and chemical behaviour of materials(Ziegler 1983a, b; Kondepudi & Prigogine 1998;

Houlsby & Puzrin 2006), and fluid flow in deform-ing reactive porous media (Coussy 1995, 2004).

Apart from applications in metamorphic petrol-ogy in the 1970s and 1980s (Fisher 1970, 1973;Fisher & Lasaga 1981; Foster 1981; Joesten 1977)there has been relatively little application withinmetamorphic geology. There has been a growinginterest in non-equilibrium thermodynamics withrespect to the application of damage mechanics toseismology (Lyakhovsky & Ben Zion 1997; Main& Naylor 2008) and in structural geology/geodynamics (Lehner & Bataille 1984; Shimizu1992, 1995, 1997, 2001; Regenauer-Lieb & Yuen2003; Hobbs et al. 2007, 2008; Ricard & Bercovici2009; Hobbs & Ord 2010; Regenauer-Lieb et al.2009). In this paper we set out to discuss some appli-cations of non-equilibrium thermodynamics todeforming, reacting metamorphic systems at theoutcrop scale.

We make a distinction between classical equili-brium chemical thermodynamics, where minimiz-ation of the Gibbs Free Energy defines the stablestates (Gibbs 1906), and non-equilibrium thermo-dynamics, where either minimization or maximiza-tion of the entropy production rate defines theevolution of the system (Zeigler 1983a; Kondepudi& Prigogine 1998). It is also important to make adistinction between the area of study concernedwith the thermodynamics of elastic solids underthe influence of non-hydrostatic stress (with andwithout fluids) and that concerned with non-equilibrium thermodynamics. The isothermal defor-mation of elastic solids by non-hydrostatic stress isreversible. Thus, much of the varied literature thatis concerned with the chemical potential of astressed elastic solid (e.g. Kamb 1959, 1961;Green 1970, 1980; Paterson 1973) is a part of equi-librium chemical thermodynamics, as is emphasizedby McLellan (1980). We are concerned in this paperwith dissipative deformations and mineral reactionswhere the subject is a part of non-equilibriumthermodynamics.

The hesitation in applying non-equilibrium ther-modynamics to geological problems derives fromthe apparent lack of a set of guiding principles thatwould allow progress. In any system, whether atequilibrium or not, one can define a function, theGibbs Free Energy; care being taken to define mean-ingful thermodynamic state variables (Callen 1960;Kestin & Rice 1970; Shimizu 2001). This function isminimized at equilibrium, and so one can proceed todefine equilibrium assemblages of minerals as dis-cussed by many authors such as Kern & Weisbrod(1967). Another function, the entropy, is maximizedat equilibrium. For non-equilibrium systems, it hasnever been clear, until recently, that a similarguiding principle was available. In fact, two appar-ently opposing views seemed to emerge in the


literature. One was emphasized by Prigogine(1955), who claimed that in non-equilibrium sys-tems the rate of entropy production is minimized.This view was also adopted by Biot (1965a,1984). The other view is due to Zeigler (1983a),who claimed that the rate of entropy production ismaximized in non-equilibrium systems. This appar-ent paradox is resolved when one understands thatthe Prigogine principle holds for thermodynami-cally linear systems at steady state, whereas theZeigler principle is more general and holds forsystems that are not constrained to be at steadystate. The issue is discussed by Martyushev &Selezvev (2006). Both principles have a foundationin statistical mechanics (Dewar 2005), where it hasbeen shown that a state of maximum entropy pro-duction rate is the most probable state for largesystems not at equilibrium and not constrained tobe at steady state. This opens the way to describe theevolution of geological systems that are maintainedout of equilibrium by the continued supply of energyin the form of deformation, fluid flow, heat flow andchemical reactions (Hobbs & Ord 2010).

The incorporation of non-hydrostatic stress intoclassical equilibrium thermodynamics was dis-cussed by Gibbs (1906) for the situation where afluid is in contact with a stressed solid. Thisapproach was elaborated upon by Kamb (1959,1961), Green (1970, 1980), Fletcher (1973) andPaterson (1973) in particular; the interfacebetween the solid and fluid is particularly importantin such circumstances, and the emphasis is on gradi-ents in the chemical potential of the solid dissolvedin the fluid induced by elastic deformation of thesolid by variations in the normal stress across theinterface.

A number of other aspects of the thermodyn-amics of non-hydrostatically stressed solids (some-times in the absence of fluids) were considered byRamberg (1959), Bowen (1967), Bowen & Wiese(1969), Coe & Paterson (1969), Nye (1957),McLellan (1980), Truskinovskiy (1984), Bayly(1983, 1987, 1992) and Paterson (1995). Most ofthese discussions, although concerned with non-hydrostatic stresses, are in the realm of equilibriumthermodynamics where the entropy production rateis zero, and some are concerned with deriving theequilibrium thermodynamic properties (such asthermal expansion coefficient, specific heat, elasticmoduli) for crystals. The deformation of the solidis elastic so that the deformation is reversible andthere is no dissipation of energy. Although massdiffusion (a non-equilibrium process) is consideredin many of these publications, the approach isalways to consider what the configuration of thesystem is at equilibrium and the dissipation ofenergy associated with the diffusion process is notconsidered.

Authors who concentrated on gradients inchemical potential to drive diffusion in deformedand metamorphosed rocks include Korzhinskii(1959), Kamb (1959, 1961), Thompson (1959),Green (1970, 1980), Fisher (1973), Fletcher(1973), Paterson (1973), Foster (1981) and Lasaga(1998), but of these only Fisher, Lasaga and Fosterdeveloped approaches based on non-equilibriumthermodynamics. For instance, Fisher uses theprinciple of minimum entropy production rateto constrain the phenomenological coefficientsdescribing the diffusion process. All of the otherworkers adopt what is essentially an approachbased on equilibrium thermodynamics, and the con-cepts of ‘local equilibrium’ (Thompson 1959) or‘mosaic equilibrium’ (Korzhinskii 1959) are intro-duced to handle the need for gradients in chemicalpotential to exist. Dissipation of energy or entropyproduction arising from the diffusion process itselfis not considered. The same is true of workers inthe metals literature of the time (Larche & Cahn1985), although a notable exception is Kocks et al.(1975). Shimizu (1992, 1995, 1997, 2001) is oneof the few who have applied the principles ofnon-equilibrium thermodynamics in the geos-ciences to the behaviour of stressed solids coupledto diffusion driven by chemical potential gradientsgenerated by non-hydrostatic stress. Other authorsinclude Lehner & Bataille (1984) and Ghoussoub& Leroy (2001).

An approach sometimes considered in meta-morphic petrology when non-hydrostatic stressesare developed (Verhoogen 1951) is to accept thatthe ‘pressure’ relevant to phase equilibria studiesis actually the mean stress and so can differ fromthe lithostatic pressure, which is calculated as rrgh,where rr is the mean rock density, g is the accelera-tion due to gravity and h is the depth of the rock unitof interest below the surface of the Earth. The impli-cations of this are explored by Mancktelow (1993)and Stuwe & Sandiford (1994) amongst others.These considerations have no influence on themineral stability field derived using equilibriumthermodynamics. Essentially the identification ofthe mean stress as the thermodynamic pressure ina deforming metamorphic rock is equivalent to anoverestimation of the depth of burial correspondingto the metamorphic conditions if the thermodyn-amic pressure is equated with rrgh. Such an overes-timate can be significant, especially in drydeforming brittle materials where the mean stressis 3rrgh (Petrini & Podladchikov 2000). Otheraspects of the influence of non-hydrostatic stresson the stability of mineral assemblages, particularlymodifications of the phase rule, are discussed byKumazawa (1961, 1963) and Shimizu (2001).

The early historical development of non-equilibrium thermodynamics is given by Truesdell


(1969) and more recently by Maugin (1999). Animportant early contribution came from Duhem(1911), although many people worked in this areaprior to Duhem including well-known names suchas Kelvin, Thomson and Rayleigh. The chemicalapproach to non-equilibrium thermodynamics wasdeveloped by Prigogine (1955) and de Groot &Mazur (1969) based essentially on the approach ofGibbs (1906). The mechanics end of the spectrumwas developed by Coleman & Gurtin (1967),Truesdell (1969), Ziegler (1983a) and Biot (1984),and more recently by Coussy (1995, 2004),Collins & Houlsby (1997), Collins & Hilder(2002), Rajagopal & Srinivasa (2004) and Houlsby& Puzrin (2006). There have been attempts to inte-grate chemical and mechanical non-equilibriumapproaches by Coussy (1995, 2004), Levitas et al.(1998a, b) and Rambert et al. (2007). Biot in par-ticular published many papers that brought chem-istry and mechanics together (Biot 1984). Thisarray of publications sets a well-defined processfor studying the evolution and behaviour ofnon-equilibrium systems. We outline this processin the next section.

Coupling of deformation, chemical

reactions, diffusion and thermal transport

The general case

We present below a discussion of the couplingbetween deformation and mineral reactions.The aim is to arrive at a general expression thatdescribes constitutive behaviour, particularly strainand strain-rate softening. We consider general dissi-pative processes so the discussion does not involvepurely elastic materials, as is common in manydiscussions of related problems (e.g. Kamb 1961;McLellan 1980; Larche & Cahn 1985). Our discus-sion is restricted to elasto-plastic–viscous solids.

In discussing chemically reacting systems wherethe reaction has not proceeded to completion it isconvenient to use the state variable, j (called theextent of the reaction, with 0 � j � 1), and the con-jugate variable, A, the affinity of, or driving forcefor, the chemical reaction (Kondepudi & Prigogine1998). Some authors (e.g. Lasaga 1998) refer to jas the progress of the reaction. j is the rate of thereaction where the overdot indicates the materialderivative with respect to time, t. A is a linear func-tion of the difference in the sum of the chemicalpotentials of the reactants and products in the reac-tion (Kondepudi & Prigogine 1998).

Non-equilibrium thermodynamics attempts todescribe the evolution of systems not at equilibriumin terms of two potentials, the Helmholtz FreeEnergy (or, if pertinent to the problem, the Gibbs

Free Energy) and the dissipation function(Houlsby & Puzrin 2006). The Helmholtz FreeEnergy is useful if we want to describe the systemin terms of a thermodynamic state variable suchas the elastic strain, whereas the Gibbs FreeEnergy is useful if we want to use the conjugatestate variable, stress (Callen 1960; Houlsby &Puzrin 2006).

For instance, if we consider the specific Helm-holtz Free Energy, C, as the relevant function thenwe can choose the thermodynamic state variablesas the elastic strain, 1elastic

ij , the absolute temperature,T, and the extent, j, of a diffusive process or of amineral reaction (Kondepudi & Prigogine 1998).We assume that the system is closed, so that thereis no exchange of mass with the outside of thesystem and write:

C ¼ u� Ts ¼ C(1elasticij , T , j ) (1)

where u is the specific internal energy and s is thespecific entropy.

From equation (1) and following standardarguments in continuum thermodynamics (Callen1960; Coussy 2004) we obtain:

vosij ¼@C

@1elasticij

; s ¼ �@C

@T; A ¼ �

@C

@j(2)

where sij is the Cauchy stress.The kinetics of the reaction are given by (Coussy

1995):

_j ¼A

vohchemicalexp �

Qchemical

RT

� �(3)

where hchemical has the dimensions of viscosity,

Qchemical is the activation enthalpy for the chemicalreaction and R is the universal gas constant. One cansee from equation (3) that since A is a function ofchemical potential, it is in turn a function of non-hydrostatic stress (Shimizu 2001); the kinetics ofa chemical reaction are also influenced by anon-hydrostatic stress.

In order to proceed we need to couple tempera-ture to these processes and derive an expression(the Energy Equation) that describes the ways inwhich the various dissipative processes contributeto temperature changes. In order to achieve this(equation 9 later) we rewrite equation (1) with mK,the concentration of the K-th chemical species:

C ¼ C(1elasticij , T , mK): (4)


A combination of the First and Second Law ofThermodynamics may be written (Coussy 2004):

T _s ¼ F

¼ FmechanicalþFdiffusive

þFchemical

þFthermal� 0 (5)

where F is the dissipation function, and Fmechanical,Fdiffusive, Fchemical and Fthermal are the contributionsto the total dissipation from purely mechanical pro-cesses, chemical diffusion, chemical reactions andthermal diffusion, respectively. The mechanicaldissipation includes the introduction of the K-thchemical species into the deforming materialthrough a chemical reaction, whilst the diffusiveterm includes the mass flux, JK of the K-th chemicalspecies across gradients in both the chemical poten-tials, mK, and the temperature.

These dissipation functions are given by (Coussy2004):

Fmechanical¼

@C

@1elasticij

_1elasticij þ

@C

@T_T þ

@C

@mK

_mK (61)

Fdiffusive¼ �voJK� gradmK �

@mK

@TgradT

� �(62)

Fchemical¼ AK

_jK þ_H@ (63)

and

Fthermal¼ �kthermalcpr

2T : (64)

In equation (62) � represents the scalar product ofvectors; in equation (63) _H@ is the volume ratesupply of heat of reaction from the chemical reac-tion, @. kthermal is the thermal diffusivity and cp isthe specific heat at constant pressure given by:

cp ¼ T@2C

@T2(Nye 1957).

Using the expressions s ¼ �@C

@Tand

vomK ¼@C

@mK

, equation (61) becomes

Fmechanical¼ vosij _1

elasticij � s _T þ vomK _mK : (7)

If we assume that

_1totalij ¼ _1elastic

ij þ _1viscousij (8)

then, using equation (5), we can eventually arrive atthe Energy Equation that expresses the change intemperature arising from all of the dissipative

processes and, hence, does not include contributionsfrom elastic deformations:

cp_T ¼ xvosij _1

viscousij þ vomK _mK � ðF

diffusive

þFchemicalþFthermal

Þ (9)

where x is the Taylor–Quinney coefficient, and rep-resents the proportion of mechanical work arisingfrom dissipative deformation that is available toincrease the temperature or to drive diffusion,chemical reactions and structural adjustments suchas fracturing or grain-size reduction. At highstrains, where the energy arising from deformationis stored in crystal defects, x is generally in therange 0.85 � x � 1 (Taylor & Quinney 1934). Weassume in what follows that x ¼ 1.

Equation (9) is a critical equation and will beused when we come to examine the couplingbetween deformation and mineral reactions at thescale of 10–100 m.

Scale effects

In this paper we are concerned with the couplingbetween deformation and mineral reactions at thefield-outcrop scale (measured in tens to hundredsof metres), and we draw a distinction between thisscale and the regional scale measured in kilometresand the hand specimen–thin section scale measuredin fractions of a metre.

The length scale, l, associated with a particulardiffusion process is defined by:

lthermal,chemical � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(kthermal,chemical)=

p_1 (10)

where the superscripts thermal and chemical refer to thediffusion process with the diffusivity k. For thermaldiffusion a geologically realistic strain rate of10212 s21 leads to a length scale for the relatedprocess of approximately 2 km for a diffusivity ofkthermal ¼ 1026 m2 s21. At the same time a chemi-cal feedback process with a diffusivity of, say,kchemical ¼ 10218 m2 s21 has a typical millimetrelength scale. This vast separation of scales, as wellas providing a guide for the dominant scale of anylocalization, folding or boudinage phenomena,also provides an opportunity to simplify thenumerical treatment.

The models we consider in this paper are at theoutcrop scale and measured in tens to hundreds ofmetres. This means that with a thermal diffusivityof 1026 m2 s21, any heat produced by deformation,diffusion or chemically induced dissipation diffusesout of the system on timescales of 108–1010 s.These timescales are short compared to timescalesof, say, 1011 s (the time taken to reach 10%


shortening at a 10212 s21 strain rate), and so we con-sider that the deformation is isothermal. We alsosuppose that the surrounding rock does not heat upappreciably, so that the entire system remains iso-thermal. We also neglect the latent heat of anychemical reactions as expressed by equation (63),so that the chemical dissipation arises only fromthe term AKjK in equation (63). We consider theeffect of including the latent heat term in the Discus-sion section. This means that in the EnergyEquation, equation (9), T ¼ 0, rT ¼ 0 andFthermal ¼ 0. Thus, the Energy Equation duringthe chemical reactions reduces to:

voðxsij _1viscousij þm _mK Þ¼Fdiffusive

þFchemical: (11)

Similarly, for timescales as long as 1011 s andchemical diffusivities of 10218 m2 s21, materialdoes not diffuse more than about 6 � 1024 m.Thus, we propose that a negligible part of thedeformation-induced dissipation results in a diffu-sive flux at the scale we are considering, andhence mmK ¼ Fdiffusive ¼ 0. Such processes operateat a finer scale and are expressed in the form of theconstitutive equation used to describe the defor-mation. Thus, equation (11), at the scale of tens tohundreds of metres, reduces to:

x vosij _1viscousij ¼ Fchemical: (12)

Note that this chemical–mechanical formulationhas an interpretation in hyperplasticity (Houlsby &Puzrin 2006). We can think of equation (12) asexpressing a chemical ‘yield’ condition where thechemical reactions do not begin until the mechan-ical dissipation matches the chemical dissipation.If processes such as chemical reactions occur atthe same time as other deformation processes andcontribute to strain, then one way of considering thecoupled problem is to represent both the deforma-tion and the chemical reactions as different yield(or dissipation) surfaces. The coupled deforming–reacting system is then represented in stress spaceas a set of nested yield surfaces that change sizeand translate together according to the rules setout by Puzrin & Houlsby (2001) and Houlsby &Puzrin (2006). If we represent the chemical reactionas a von Mises yield surface (the chemical yieldsurface), then the radius of the yield surface is pro-portional to the affinity of the chemical reaction.

In this paper we follow this formalism and con-sider that the chemical yield surface is reachedwhen the mechanical dissipation is large enoughto be equal to the chemical dissipation, as indicatedin equation (12). This means that mechanicaldissipation drives the chemical reaction once the

products of the reactions are in their stability field.Chemical hardening or softening then dependsboth on the value of (s–A) and of A, where s andA are the stress and affinity vectors in stress space.

Once the chemical reaction is completed,the mechanical dissipation is available at a finescale to drive diffusion away from the reaction site(Regenauer-Lieb et al. 2009) and to generateincreases in temperature at a coarser scale (Hobbset al. 2008).

Compared to equation (9) we have nowdecoupled the thermal and chemical diffusionequations, and assumed that the thermal and chemi-cal diffusion processes operate on different length/time scales and there is no feedback between thescales. We have also assumed that the coarselength scale process loses heat such that the nettemperature change in the large-scale system iszero. We have also ignored the heat of the chemicalreaction and assumed that all of the mechanicaldissipation at the grain-scale is taken up in creatingnew phases or in driving chemical diffusion. Thissimplification means that the problem reduces toone identical in form to shear heating feedbackin mechanical systems; the latter is intrinsicallysimpler and is much better understood (Regenauer-Lieb & Yuen 2004).

We now have the following relationships expres-sing the coupling of deformation and chemicalreactions at the outcrop scale:

vosij _1viscous ¼ Fchemical

¼ A_j

¼ vohchemical _j

2exp

Qchemical

RT

� �: (13)

Coupling of deformation and mineral

reactions

First consider the situation where the viscous part ofthe strain is achieved (in a one-dimensional defor-mation) through power-law flow (specifically notat steady state), such that for a given deviatoricstress, s 0, and in the absence of chemical reactionsthe steady viscous strain rate, 1viscous, is given by:

_1viscous ¼ Ls 0N exp �Qmechanical

RT

� �(14)

where L is a material constant, N is the stress expo-nent and Qmechanical is the activation energy forsteady deformation. The deviatoric stress is givenby s 0 ¼ s 2 P, where s is the total stress and P isthe mean stress.

The mineral reaction can influence L by the gen-eration of new weaker or stronger phases and/or byreducing the grain size (Rubie 1983, 1990; Vernon


2004). The reaction can also influence Qmechanical

through the diffusion of species into the crystalstructures, as occurs in the hydrolytic weakeningeffect or in the case of the influence of water onthe kinetics of the coesite to quartz transition(Mosenfelder & Bohlen 1997). Qmechanical can alsobe substantially changed as mineral reactionschange the proportions of populations of differentminerals in the rock. Chemical species diffusedinto the crystal structure can also influence themobility of dislocations, and individual fine phasescan influence the migration of grain boundaries sothat the dominant mechanism of deformationchanges; thus influencing the value of N. Thus,chemical processes are capable of influencing L, Nand Qmechanical and so influence the strain rate fora given stress.

Conversely, the deformation can influence thechemical kinetics through a direct influence ofstress on Qchemical, as has been documented byauthors such as Hillig & Charles (1964), Lawn(1993), Schmalzried (1995) and Gilman (2003); anexample of this effect with respect to the influenceof stress on the incorporation of water into quartzis given by Zhu & Yip (2005). Deformation alsoinfluences hchemical through structural changes thatresult in the production of more reaction sitesarising from increased dislocation density orsmaller grain size. The affinity of the reaction alsodepends on stress through an influence on thechemical potentials of the products and reactants.

While the chemical reaction is in progress thereis, therefore, a dependence of some or all of L, N andQmechanical on the amount of strain, thus resulting inreaction-induced strain hardening or strain soften-ing. Thus, equation (14) can be rewritten for non-steady deformations as:

s ¼ L�10 _1

1N0 exp

Qmechanical0

RT

� �1 p (15)

where p is the strain-hardening exponent so thatp . 0 represents strain hardening and p , 0 rep-resents strain softening. The null subscripts referto values of the respective quantities at some refer-ence strain. This means that all of the strain harden-ing/softening is expressed in the term 1p. We admitthis may be simplistic but, failing systematic exper-imental data, it is a start. Experimental data onreaction-induced strain softening are rare but thelimited data (Brodie & Rutter 1985, 1990; Burlini& Bruhn 2005; de Ronde et al. 2005; Delle Pianeet al. 2007) suggest that, over a significant rangeof strain, the softening can be linear with strain;however, we assume that, overall, the dependencecan be represented by a power-law relation such asequation (15).

In many materials (Estrin & Kubin 1991), pro-cesses that evolve at the same time as deformationand that are coupled to the deformation result instrain-rate hardening (or softening), as well asstrain hardening (or softening). Thus, a materialdeforming solely by dislocation creep with diffusionplaying a negligible role needs to increase thedensity of mobile dislocations for an increase instrain rate, thus resulting in strain-rate hardening.Notice that this effect is greatest for Newtonianviscous materials (N ¼ 1 in eq. 14) and decreasesas N increases. The use of the terms strain-rate hard-ening and softening in this paper is meant to refer to

the sign of@s

@_1

� �1,T

. From equation (15) this is

L�1

Nexp

Qmechanical

RT

� �1p _1

1�NN

� �, which is always

positive for N � 1. In particular, a strictly Newto-nian material where the viscosity is a constant is astrain-rate hardening material, since an increase instrain rate always produces an increase in stress. Itis important to note that Smith (1977) referred tothe strain-rate response of power-law materialswith N . 1 as strain-rate softening because the

term1� N

Nis negative for N . 1, which means

that the tangent viscosity (Smith 1977) decreaseswith an increase in strain rate. This usage has beenfollowed by some workers in structural geology.We emphasize that for any power-law materialwith N � 1 the response to an increase in strain

rate is a hardening, sinceL�1

Nis always positive.

Thus, in our usage of the term, which coincideswith that of Poirier (1980) and of the general mech-anics community, the case N � 1 in equations (14)and (15) corresponds to strain-rate hardeningmaterial.

However, if time-dependent processes, such asthermal or mass diffusion, chemical reaction,melting, grain-size reduction or the developmentof a crystallographic preferred orientation, play adominant role and are coupled with the deformation,then an increase in strain rate can lead to hardeningor softening depending on the relative values of thetimescales associated with a strain-rate change andthe relaxation of the stress due to diffusion or theextent of the relevant process. Examples are soften-ing due to thermal diffusion coupled to a decrease inviscosity with temperature increase (Regenauer-Lieb & Yuen 2003) and various diffusive processesrelated to dislocation motion in some alloys (Estrin& Kubin 1991; Zaiser & Hahner 1997). The strain-rate hardening (softening) effect depends on theratio of the two timescales mentioned above andresembles the velocity weakening effect observedin frictional sliding (Ruina 1983). The formal


analogy between strain-rate softening in viscousmaterials and velocity weakening in frictionalsliding has been examined by Mesarovic (1995)and Kameyama (2003). Thus, there is always a tran-sient hardening effect associated with a rapidincrease in strain rate for a power-law materialwith N � 1, but with the elapse of time and the pro-gress of a chemical reaction or some other time-dependent process an ultimate weakening mayresult (Estrin & Kubin 1991). Strain-rate steppingexperiments in reacting–deforming materials wereconducted by Delle Piane et al. (2007), but a strain-rate softening effect arising from a mineral reactionwill be difficult to demonstrate at laboratory acces-sible strain rates.

The behaviour of strain-hardening (softening)and strain-rate-hardening (softening) materials isdiscussed by a number of workers including Estrin& Kubin (1991), Hahner (1995), Wang et al. (1997)and Zaiser & Hahner (1997). These authors definethree hardening (softening) parameters, namely:

h ¼@s

@1

� �_1,T

(161)

v ¼@s

@_1

� �1,T

or v ¼@s

@(ln _1)

� �1,T

(162)

w ¼@s

@T

� �1,_1

: (163)

If the timescale for the diffusive relaxationprocess associated with strain-rate softening isshort compared to the timescale for a strain-ratechange then a criterion for the onset of localizationin materials with an exponential dependence ofstress on strain rate is:

h� s

v, 0: (17)

Thus, localization can occur in these materials evenfor strain-hardening situations for h , s and v . 0,and for (h 2 s) . 0 and v , 0. For the constitutivelaw assumed by Estrin & Kubin (1991), strain andstrain-rate effects are coupled so that the compe-tition between the evolution of forest (dislocationsthat intersect the glide plane) and mobile dislocationdensities with continuing strain results in a situationwhere strain softening (h , 0) coupled with strain-rate softening (v , 0) does not arise.

An important outcome from these studies is thatfor materials that are both strain-softening andstrain-rate-hardening, deformation becomes loca-lized into individual zones that remained fixed inspace during the deformation. Localization phe-nomena similar to this have been investigatedby Mancktelow (2002). For strain-hardening,strain-rate-softening materials the competition

between strain hardening and strain-rate softeningresults in propagating (i.e. migrating) shear bands(Estrin & Kubin 1991; Wang et al. 1997). The soft-ening induced by thermal effects is identical to thatconsidered by Regenauer-Lieb & Yuen (2003) andHobbs et al. (2007, 2008).

The development of shear bands in strain-rate-sensitive materials appears to be different fromlocalization in strain-rate-insensitive materials(Needleman 1988). Localization in strain-rate-insensitive materials has been summarized inHobbs et al. (1990), where the localization processappears as a bifurcation in material behaviouronce some critical parameter such as amount ofstrain or strain-softening magnitude is exceeded.In strain-rate-sensitive materials the localization isan instability phenomenon that is quite sensitive toinitial imperfections such as variations in viscos-ity and does not involve bifurcation behaviour(Needleman 1988; Wang et al. 1997). To thisextent the theory for localization in strain-rate-sensitive materials has many features in commonwith the theory of folding proposed by Biot(1965a), where instability in the deformation isrelated to the growth of initial imperfections fromthe moment visco-plastic deformation begins.Needleman (1988) and Wang et al. (1997) showthat for quasi-static deformations the thickness ofshear zones in strain-rate-sensitive materials isrelated to the size of the initial imperfection.Viscous materials with N � 1 can never localizeunless at least strain softening exists (Anand et al.1987; Hobbs et al. 1990). Anand et al. (1987)analysed the onset of localization in viscoplasticstrain-rate hardening materials and showed thatlocalization arising from strain softening growsfastest just after initiation for weaker strain-ratehardening sensitivity. Another aspect arises fromthe intrinsic anisotropy of power-law viscousmaterials. Smith (1977) emphasized that viscousmaterials with N . 1 are, in effect, anisotropicbecause the viscosity measured in simple shearingis always less than the viscosity measured in pureshearing. Anisotropic viscous materials canbecome unstable (Biot 1965b; Hobbs et al. 2000)and form shear zones but the instability does notdevelop unless N is large (N . 10) which is unrea-listic for rocks. In the models of Mancktelow(2002) strain-rate hardening is always present asthe response of a power-law material (with nocoupled diffusive processes) to an increase instrain rate but the effect is greater for materialswith N ¼ 1 than for materials with N . 1. Thestrain softening invoked by Mancktelow (2002)competes with this strain-rate hardening so thatlocalization is better developed for N ¼ 3 ratherthan for N ¼ 1, in agreement with Anand et al.(1987). However, as pointed out by Mancktelow


(2002), the intensity of strain in these shear zonesalways remains small. This contrasts with the highintensity of strain developed in natural shear zones(Ramsay & Graham 1970) and in strain-rate soften-ing materials as demonstrated later.

The thermal–mechanical feedback process isworth further investigation. Fleitout & Froidevaux(1980) show that the effective viscosity within ashear zone that arises from thermal–mechanicalfeedback in a developing shear zone is given by:

heff � 8krcp

R T2c

Q_12k2: (18)

The effective viscosity is inversely proportional toboth the square of the strain rate and the square ofthe shear-zone thickness, k; equation (18) can bewritten more generally for constant k as:

heff � h0

_120

_12(19)

where h0 is the viscosity when the strain rate is _10.There is an additional feature that arises during thethermal–mechanical localization process. Theshear zone initially nucleates with a width k0 butcollapses in width as deformation proceeds untilthe conduction of heat away from the shear zonebalances that being generated by the shearing defor-mation. This results in a transient dependence of theviscosity on time, or equivalently, on strain. Thus,the evolution of the effective viscosity during defor-mation with thermal–mechanical feedback consistsof a transient strain hardening arising from thecollapse in shear-zone thickness and a strain-ratesoftening that is inversely proportional to the squareof the strain rate.

The general form of the constitutive equationconsidered by Estrin & Kubin (1991) can bewritten in incremental form as:

ds ¼ hd1þ vd(ln _1)þ wdT : (20)

It would appear that to consider the coupling betweenchemical reactions and deformation we need anincremental constitutive relation of the form:

ds ¼ hd1þ vd(ln _1)þ wdT þ hchemicaldj

þ vchemicald_j (21)

where hchemical and vchemical are the chemical strainand strain-rate hardening/softening parameters:

hchemical ¼@s

@j

� �1,_1,T ,_j

(221)

vchemical ¼@s

@_j

� �1,_1,T ,j

: (222)

Unfortunately, we have little knowledge of theprecise forms of these parameters and so we proceedin a somewhat simplistic manner below in order toexplore some scenarios of possible behaviour.

For a Newtonian viscous material, equation (13)can be rewritten as:

hmechanical(_1viscous)2 ¼ B_j2

(23)

or

hmechanical ¼ B_j

2

(_1viscous)2(24)

where B ¼ hchemical expQchemical

RT

� �. Equation (24)

is of similar form to equation (18). The differencebetween thermal–mechanical and reaction–mechanical coupling is that the strain rate has onlya small effect on the rate of heat flow and can beneglected. However, the strain rate can potentiallyhave large effects both on B and _j.

We need more theoretical and experimentalwork to define the precise form of equation (24),in particular the dependence of reaction rate onstrain and on strain rate. Neglecting thermaleffects, we for the moment write the general relationbetween the mechanical viscosity and the strain andstrain-rate softening as:

hmechanical ¼ h0(1viscous) p _10

_1viscous

� �q

(25)

where h0 is the mechanical viscosity at zero strainand a reference strain rate of _10, p is the strain-hardening exponent, and q is a parameter thatmeasures the dependence of viscosity on strain-rate; there is viscosity strain-rate hardening forq , 0 and viscosity strain-rate softening for q . 0.

The development of localized shear zones,

folding and boudinage

In order to gain some insight into the influence ofmineral reactions on the development of localizedzones of shear, and on folding and boudinage atthe outcrop scale, we explore a simple generic modelthat incorporates the issues discussed earlier. Forsimplicity, we assume that the reacting deformingmaterial can be represented at each instant as anelastic–viscous material with Newtonian viscositybut with reaction-induced strain and strain-ratehardening/softening. This behaviour is expressedas a dependence of the instantaneous viscosity onthe strain and on the strain rate as expressed by


equation (25). This follows the approach of authorssuch as Smith (1977) and Needleman (1988), wherethe instantaneous viscosity is expressed via adependence on strain and strain rate.

The constitutive law at each instant is theMaxwell constitutive relation:

_1ij ¼sij

hmechanicalþ

_sij

E(26)

where E is the elastic modulus.The governing equations are, first, the Continu-

ity Equation:

@r

@tþ rr � u ¼ 0 (27)

where u is the local material velocity vector. Sec-ondly, the momentum equation describes equili-brium of forces:

r �sij ¼ 0 (28)

where r �sij is the divergence of the Cauchy stresstensor. These equations are solved together withequations (25) and (26) using the finite-differencecode FLAC2D version 6.0 (ITASCA 2008).

We consider first a body of rock with no layeringbut with a random distribution of heterogeneities inthe viscosity (corresponding to initial variations inmineral composition). The viscosity is initially1021 Pa s and the standard deviation of the viscositydistribution is 1020 Pa s.

The model is shortened isothermally at a con-stant velocity parallel to x2 in Figure 10 correspond-ing to an initial strain rate of 10212 s21. The velocityparallel to x1 is varied with time to maintain anisochoric deformation. In Figure 10a, b, p fromequation (25) is 20.8 and q ¼ þ2, so that both vis-cosity strain softening and viscosity strain-rate soft-ening occur. This corresponds to a situation wherethe mineral reaction rate is independent of thestrain rate, but the reaction results in strain soften-ing. Shear zones are strongly developed, and inten-sify and decrease in number as the deformationproceeds; with increasing shortening the strainprogressively becomes focused in wider spacedshear zones.

In Figure 10c, d, p ¼ 0.5 and q ¼ þ2, corre-sponding to a situation where the mineral reactionresults in strain hardening and the reaction rate isindependent of the strain rate. In this case, theshear zones become more localized with increasingshortening. This resembles the development ofso-called propagating shear bands in rate-sensitivematerials (Estrin & Kubin 1991; Wang et al.1997) where the growth of an individual shear

band during progressive deformation is arresteddue to strain hardening, and the site of localizationmoves to a new area where strain-rate softeningdominates. This results in shear zones that localizeas the deformation proceeds up to a stage wherethe localization switches from one to anotherduring the deformation.

In both cases shear zones are strongly devel-oped with localized integrated strains, as measuredby g, the integral of the square root of thesecond invariant (I2) of the viscous strain rate

g ¼

ðt

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2(_1viscous

ij

r_1viscous

ij )dt

0@

1A. Values of g reach 5

in shear zones after 50% total shortening and 6 after70% shortening. In both strain-hardening and strain-softening situations the shear zones develop, inaddition to the expected X-shaped ‘conjugate’ con-figuration, a characteristic Y- or V-shaped geometryso that quite intense shear zones suddenly stop andare replaced by a diffuse pattern of deformation.This resembles shear-zone geometries seen in thefield (Figs 5 & 6).

Next, in Figure 11a–e we add a single layer ofslightly greater viscosity initially parallel to x1 andshorten the model parallel to x1. In all models theinitial viscosity contrast between the layer and theembedding material is 10. This viscosity contrastis too small to produce folding according to a Biot-type of mechanism (Biot 1965a; Hobbs et al. 2008).The model is shortened parallel to the initial orien-tation of the layer at 10212 s21 with p ¼ 20.8 andq ¼ þ2. No folds develop if only strain softeningis present, as the resultant shear zones developtotal accumulated strains too small to result insignificant inhomogeneous deformation. However,once strain-rate softening is incorporated, shearzones are strongly developed and any slight deflec-tion in the layer is amplified because of the influenceof strain-rate softening. The strain and, hence, thestrain rate is larger in the inner hinge of the incipientfold than in the outer hinge. Thus, the viscositydecreases in the inner hinge more than in the outerhinge (Fig. 11b). This generates a buckle thatis further amplified by the viscosity strain-ratesoftening. Thus, folds ultimately develop withwavelengths controlled by the distribution of imper-fections in the system such as the initial randomvariation of viscosity and the boundaries of themodel. The folds that ultimately form at high totalshortening have thinned limbs where shear zonescross the layer and thickened hinges in between.This is a non-Biot-type of folding mechanism thatis identical in principle to folds developed fromthermal–mechanical coupling at the kilometre scale(Hobbs et al. 2007, 2008) and folds developed bychemical reaction diffusion–mechanical coupling


at the centimetre scale (Regenauer-Lieb et al. 2009).We emphasize that in a Newtonian material, strainsoftening alone is not sufficient to generate folds;strain-rate softening is the important control. Theintensity of strain increases in the inner hinges.For reaction rates independent of strain rate theextent of the reaction is directly proportional tothe amount of viscous strain, so that Figure 11b–dare proxies for the extent of mineral reaction.

Figure 11f, g shows the same situation but withp ¼ 20.1 and q ¼ þ2. The shortening strains are50 and 70%, respectively. Note the thinning on thelimbs and boudinage in the hinge position.

Figure 12 shows the development of shear zonesand boudinage. In Figure 12a, p ¼ 20.8 andq ¼ þ2, and the deformation history comprisespure shortening parallel to x2. In Figure 12b, c,p ¼ 0.5 and q ¼ þ2, and the deformation historycomprises simultaneous pure shearing parallel tox2 and simple shearing parallel to x1.

Figure 12a (lower panel) shows the distributionof the total integrated strain (g) for the situationsshown where the reaction rate is independent ofthe strain rate. Again, these diagrams are equivalentto maps of j, the extent of the mineral reaction.Thus, one sees that the mineral reactions are moreadvanced in shear zones that form an anastomosingnetwork around the boudins, with reactions tendingto completion in the necks of boudins or on themargins of boudins particularly near necks. Again,the characteristic V- or Y-shaped geometry ofshear zones is developed resembling examplesseen in the field (Figs 5 & 6). Notice that for amaterial with N ¼ 1, with no coupled mineral reac-tions and with the initial viscosity ratio betweenlayers of 10, no boudins would develop (Smith1977; Schmalholz et al. 2008). Strain-rate softeningis essential.

Figure 12d shows part of a mutilayer sequenceshortened 80% parallel to x2. The extreme thinning

Fig. 10. Localization in unlayered Newtonian viscous strain-rate-softening materials (q ¼ þ2.0). Plots of thelogarithm of

ffiffiffiffiI2

p. (a) p ¼ 20.8; shortening 40%. (b) p ¼ 20.8; shortening 50%. (c) p ¼ 0.5; shortening 40%. (d)

p ¼ 0.5; shortening 50%.


of fold limbs is notable and resembles the extremethinning observed in the field (Fig. 8a). Althoughthis technically is a Class IC fold in the terminologyof Ramsay (1967), the intense thinning on the limbsis due to the development of a shear zone rather thanflattening as in the Ramsay model.

Influence of deformation on phase stability:

simple chemical reactions

The chemical potential of a chemical component is aquantity that measures the energy required to inserta unit quantity (usually expressed as 1 mole but hereexpressed as unit mass) of that component into asystem isothermally. This definition is relevantwhether the system is or is not at equilibrium and

whether the stress is hydrostatic or non-hydrostatic.Consider a system with K chemical components.We define the non-equilibrium specific chemicalpotential, mK, of the K-th component inserted intoa deforming, chemically reactive system as(Kondepudi & Prigogine 1998; Coussy 2004):

mK ¼ mK (s 0ij, P, T , jK ) (29)

where s 0ij is the deviatoric stress, P is the mean stressand jK is the extent of the chemical reaction thatproduces the K-th component. To be explicit, here

the mean stress is P ¼ �1

3skk where the Cauchy

stress, sij, is related to the deviatoric stressthrough s 0ij ¼ sij � skk=3:

Fig. 11. Folding and localization in Newtonian strain-softening, strain-rate-softening materials (p ¼ 20.8, q ¼ þ2.0).Layer viscosity initially 1021 Pa s. Embedding material initial viscosity 1020 Pa s. (a) Shortening 20%, passive markersshow deformation in embedding material. (b) Same as (a) but showing logarithm of current viscosity. Dark green is1021 Pa s, light green is 1019 Pa s, purple is 1017 Pa s. Viscosity is higher in the outer arc of the folded layer and low inthe inner arc. (c) Shortening 30%; logarithm of g. (d) Shortening 40%; logarithm of g. In both diagrams blue is g ¼ 3.5,green is g ¼ 0.2. (e) Deformed layer at 40% shortening. (f ) Single-layer fold with p ¼ 20.1 and q ¼ þ2; 50%shortening. (g) Same as (f ) but with 70% shortening.


Fig. 12. Deformation in multilayer materials. (a)–(c) boudinage in a three-layer sequence (d) Very strongly appressedfold. (a) Deformation is pure shearing; p ¼ 20.8, q ¼ þ2. Shortening is 70%. Plots of geometry and logarithms ofviscosity, strain rate and g, which is taken as a proxy for the extent of reaction. In the viscosity plot: dark green is1022 Pa s, yellow is 1020 Pa s, purple is 1018 Pa s. In the strain-rate plot, red is 10214 s21, blue is 10210 s21. In thestrain plot: blue is g ¼ 8, red is g ¼ ,1. (b) Boudinage developed with combined pure shearing (shortening 70%)parallel to x2 and simple shearing through 708 parallel to x1; p ¼ þ0.5, q ¼ þ2. (c) Map of log g for the geometryof (b): blue is g ¼ 8, red is g ¼ ,1. (d) p ¼ 20.8. q ¼ þ2; shortening 80%.


Even in a system not at equilibrium, at a phaseboundary where the reactants and the products of amineral reaction coexist, the difference in the sumof the chemical potentials of the phases on eitherside of the boundary is zero, as is also the differencein the affinities of the reactions involved. An impor-tant difference between the non-equilibrium and theclassical chemical potential is that the pressure forthe non-equilibrium situation is measured by themean stress. As the stress relaxes to hydrostaticand the chemical reactions proceed to completion,the non-equilibrium chemical potential evolves tobecome the classical chemical potential.

From equation (29):

dmK ¼@mK

@s 0ijds 0ij þ

@mK

@PdPþ

@mK

@TdT

þ@mK

@jKdjK (30)

or

dmK ¼ �v01K(elastic)ij ds 0ij þ vK dP� sK dT

þ AK djK (31)

where 1K(elastic)ij is the elastic strain of component K,

vK is the specific volume of component K, sK is thespecific entropy of component K, AK is the affinity ofthe reaction that produces K and v0 is the specificvolume of the reactant at hydrostatic stress. For adiscussion of the term v0 one should consultHoulsby & Puzrin (2006).

From equation (31) we obtain:

d(Dm) ¼ �D(v01elasticij ) ds 0ij þ Dv dP

� Ds dT þ DA dj (32)

where D(.) is the change in (.) during the chemicalreaction and deformation. That is, for the reactionquartz! coesite, Ds ¼ scoesite � squartz and so on.

At a phase boundary, d(Dm) ¼ 0 and DA ¼ 0hence:

dP

dT¼

Ds

DvþD(v01

elasticij )

Dv

ds 0ij

dT: (33)

Thus, at a phase boundary the classical Clausius–

Clapeyron slope at equilibrium,Ds

Dv, is modified in

the non-equilibrium case by a term that involvesthe elastic strain and specific volume contrastsbetween the two phases and the temperature deriv-ative of the deviatoric stress tensor. If the totalstrains arise from steady-state power-law creep of the

form involving s 0ij ¼ L�1=N _1ij(J2)1=(N�1) exp(Q=RT)then:

dP

dT¼

Ds

Dv�D(v01

elasticij )

Dv

Q

RT2L�1=N _1ij(J2)1=(N�1) exp(Q=RT) (34)

at a phase boundary. Hence, for a fixed strain rate,dP

dTapproaches the equilibrium Clausius–Clapeyron

slope at high temperatures, but at low temperaturesand/or fast strain rates the second term on the right-hand side of equation (34) can be of similar magni-tude to the classical Clausius–Clapeyron slope,Ds/Dv. As the strain rate decreases at constant

temperature,dP

dTapproaches the classical equili-

brium Clausius–Clapeyron slope.As the non-equilibrium phase boundary is

asymptotic to the equilibrium phase boundary asT! 1, the equation for the non-equilibriumphase boundary becomes through the integrationof equation (34):

P ¼ P0 þDs

DvT

þD(v01

elastic)

DvL�2=N _12=N exp

Q

NRT

� �

� expQ

NRT

� �� 1

� �(35)

where P0 is the intercept of the equilibriumphase boundary on the pressure axis for T ¼ 0 andD(v01

elastic) has been taken as D(v01elastic) ¼

s 0v

quartz0

Equartz�

vcoesite0

Ecoesite

� �and E is the Young’s

modulus. Note that for the quartz! coesitereaction, D(v01

elastic) is positive whereas Dv isnegative. The general behaviour of this boundaryis illustrated in Figure 13.

For the reaction quartz! coesite P0 ¼ 13 000

Pa,Ds

Dv¼ 14:8 � 106 Pa K21, v

quartz0 ¼ 3.76 �

1024 m3, vcoesite0 ¼ 3.41 � 1024 m3 and Dv ¼

22.7 � 1026 m3 mol21 (Kern & Weisbrod 1967).At atmospheric pressure the elastic moduli C11

and C33 of a-quartz are approximately 90 and80 GPa, respectively, whereas the equivalent elasticstiffnesses for coesite are about 150 and 200 GPa,respectively (Carpenter et al. 1998; Kimizukaet al. 2008). We take the elastic stiffness ofa-quartz to be about half of the equivalent moduliof coesite and so the elastic strain of quartz is abouttwice that of coesite for the same stress. We assume


Fig. 13. Non-equilibrium phase diagram. (a) Theoretical non-equilibrium pressure–temperature phase diagram withthe equilibrium phase boundary shown as a dashed line. The full line is the non-equilibrium phase boundary asymptotic


D1 ¼ 0.5s 0/E quartz, where E quartz is taken to be1011 Pa. If we also assume that the constitutiveparameters for coesite are the same as for quartzwe can write Q ¼ 135 KJ mol21, N ¼ 4 and L ¼1.33 � 10234 Pa2N s21. The resulting phase bound-ary for the quartz! coesite reaction is shown inFigure 13c for strain rates ranging from geologicalrealistic values of 10214 s21 to laboratory accessiblestrain rates of 1024 s21 and assuming Equartz ¼0.5Ecoesite. At geological strain rates the non-equilibrium phase boundary is within 0.1 MPa ofthe equilibrium boundary at temperatures above400 K and so the difference would not be detectablefor geological systems. However, examples forcoesite forming outside of its equilibrium stabilityfield during laboratory deformation experimentsare well known (Hobbs 1968; Green 1972), wherethe strain rates are 1026–1024 s21. This analysisseems to confirm that this is a metastable develop-ment of coesite, as discussed by Green (1972).

As the difference between the equilibrium andnon-equilibrium curves is strongly dependent onD1elastic there may be examples where the effect islarger than indicated for the quartz! coesite reac-tion, especially in reactions involving ‘hard’ phasesreacting to form ‘soft’ phases. Moreover, we haveassumed that the constitutive behaviour andparameters for the reactants are the same as for theproducts, and certainly this is not the case formore complicated reactions. Hence, the magnitudeof the difference between the equilibrium andnon-equilibrium behaviour in more complicatedreactions remains open, and would benefit fromdetailed, high-precision P–T investigations onassemblages in suitable structural environments.

Discussion

We have presented a model whereby mineral reac-tions occurring during metamorphism are coupledto the accompanying deformation. A two-way feed-back process operates whereby mineral reactionsinfluence deformation and deformation influencesmineral reactions. If the deformation enhances thereaction rate then the deformation is enhanced. Ifthe reaction softens the material either by strain-softening or strain-rate-softening mechanisms thendeformation is enhanced, which in turn furtherenhances the mineral reaction. An importantoutcome of employing non-equilibrium thermodyn-amics at the scale of tens to hundreds of metres is

that the thermal and chemical diffusion effects canbe decoupled from the mineral reaction effects sothat the energy dissipated by the inelastic deforma-tion is coupled only to the mineral reaction rate.Thermal diffusion effects dominate at larger scales(c. 1 km), whereas coupled mineral reactions–chemical diffusion effects dominate at smallerscales (c. 1 mm). This results in a strain-rate-softening effect at the scale of tens to hundreds ofmetres, in addition to any strain hardening or soften-ing. If the reaction rate is independent of the strainrate then the effective viscosity is inversely pro-portional to the square of the strain rate as it is inthermal–mechanical feedback situations (Fleitout& Froidevaux 1980). Unfortunately, we have noexperimental data on the dependence of reactionrates on strain rate and so we have explored (butnot presented) several simple situations where thereaction rate, j, is proportional to 1M. If M ¼ 0then the reaction rate is independent of the strainrate. For M ¼ 0.5 the stress becomes independentof the strain rate for a material that otherwisewould behave as a Newtonian material with noreaction coupling. For M ¼ 1 the viscosity is inde-pendent of strain rate and the material alwaysbehaves as a Newtonian material. These relationsare illustrated in Figure 14 where the structuralbehaviour of the material for various values of Mand q are also indicated.

At the scale of tens to hundreds of metres,although reaction-induced strain softening is com-monly quoted as a mechanism for generating shearzones, it appears from the modelling reported hereand by Mancktelow (2002) that strain softening,although capable of inducing localization in aviscous material, does not lead to the strain intensi-ties observed in natural examples such as thosestudied by Ramsay (1967) and Ramsay & Graham(1970) at least for values of N � 3. Part of thereason for this is that viscous materials with N � 1and with no coupling to time-dependent processesinherently exhibit strain-rate hardening, whichlargely offsets the strain softening (Anand et al.1987). The hardening is less for non-Newtonianmaterials with N . 1, so that localization is betterdeveloped in these power-law materials than inNewtonian materials (Mancktelow 2002). Even so,strain softening alone does not reproduce the strainintensities observed in natural shear zones(Mancktelow 2002).

In order to generate high-strain shear zones, non-linear strain-rate softening is necessary. Then three

Fig. 13. (Continued) to the equilibrium phase boundary at high temperatures. The non-equilibrium phase boundarymoves to the left with decreasing strain rate. The assemblage at point A is not stable under equilibrium conditions but isstable under non-equilibrium conditions. (b) Theoretical non-equilibrium phase boundaries at two different strain rates,_1I

ij (slow) and _1IIij (faster). The assemblage at point A is unstable at the slow strain rate but stable at the higher strain rate.

(c) Non-equilibrium phase boundary constructed for the quartz–coesite transition.


possibilities arise. For strain hardening togetherwith non-linear strain-rate softening shear zonesthat nucleate at a given site grow until the strainhardening arrests the growth. The localization thenshifts to an adjacent site. Such a shear-zonenetwork is to be expected in metamorphic terrainswhere the mineral reactions result in new phasesstronger than the original assemblage, such as intemperature-prograde metamorphic environments.Alternatively, non-linear strain softening accom-panied by strain-rate hardening leads to localizedshear zones that remain at the same site throughoutthe deformation but never progress to high-strainshear zones. Lastly, strain-rate softening accom-panied by strain softening leads to the formationof zones of intense localization that tend to evolveso that the strain is taken up in just a few wideshear zones. The result is relatively widely spacedshear zones with relatively high intensity of strain.Such a shear-zone pattern is to be expected in meta-morphic terrains where the mineral reactions resultin new phases weaker than the original assemblage.This is typical of areas undergoing temperature-retrograde metamorphism.

The development of localization in strain-rate-dependent materials is strongly sensitive to initialperturbations in mechanical properties or geometryso that shear-zone thickness is influenced by the sizeof initial departures from homogeneity (Needleman1988; Wang et al. 1997). This is particularly true iflayers of slightly different material are present.Slight departures from plane interfaces are ampli-fied because of differences in strain rate betweenthe outer and inner arcs of a slight buckle. Thestrain rate is marginally higher in the inner arc and

strain-rate softening induces a lower effective vis-cosity than in the outer arc. This amplifies the inci-pient buckle, and folds grow even if the initialviscosity contrast between the layers and the embed-ding material is small (�10). This folding mechan-ism is different from the classical Biot mechanism(Biot 1965a) but is identical to that which developsfrom thermal–mechanical and from chemicalreaction–diffusion–mechanical feedback processesat the kilometre and millimetre scales, respec-tively (Hobbs et al. 2007, 2008; Regenauer-Liebet al. 2009). The folds that develop from reaction–mechanical feedback are remarkably similar tonatural folds with thickened hinges and attenuatedlimbs, an irregular wavelength distribution andwavelength to thickness ratios of 4–7. Folds donot develop by this mechanism if strain softeningexists without strain-rate softening. However, bothstrain softening and strain hardening can lead tofold development when coupled with strain-ratesoftening.

Similarly, in strain-rate softening materials,boudinage is readily developed even for verysmall initial viscosity ratios (�10) between thelayer and the embedding material, and for materialsthat are Newtonian in the absence of coupledmineral reactions. This, again, is quite differentfrom the development of boudinage in power-lawmaterials with no strain-rate softening (even forsuch materials with strain softening) as discussedby Smith (1977) and Schmalholz et al. (2008).

Deformation not only influences the kinetics ofmineral reactions, it also influences the stability ofmineral phases. Thus, while a mineral reaction isin progress and viscous deformation is proceeding

Fig. 14. Behaviour of strain-rate hardening/softening viscous materials. M is the exponent that links reaction rateto strain rate: _j ¼ _1M . q is the exponent that describes strain-rate softening. q ¼ þ2 means that the reaction rate isindependent of the strain rate. (a) Changes in deformation behaviour with changes in M and q. (b) Changes inconstitutive behaviour with changes in M and q.


the field of stability of the minerals is shifted rela-tive to the equilibrium stability field. This non-equilibrium stability field relaxes to the equilibriumstability field as the mineral reaction proceeds tocompletion and as the deformation wanes. Theeffect is negligible at geological strain rates andtemperatures for the quartz! coesite reaction.The P–T estimates performed in natural cases pre-sented earlier also indicate no difference in P–Testimates from deformed and undeformed assem-blages. Nevertheless, the effect is present and mayinfluence nucleation to be preferentially favouredor inhibited in high-strain zones as is well documen-ted for martensitic transformations in deformedmetals (Ericksen 1998). Moreover, in the analysispresented here we have treated a very simple reac-tion, and we have assumed that the constitutivebehaviour is identical for the reactants and products.This is not the case in general and so the issue needsfurther attention.

Thus, in regions where mineral reactions havenot gone to completion the estimates of pressureand temperature obtained by assuming equilibriumare within the error of the physical conditions oper-ating in the non-equilibrium system except that therelevant measure of pressure defining equilibriumis the mean stress rather than the lithostatic pressure.This is true even with only elastic deformation if thestresses remain non-hydrostatic. It must be notedthat the difference, although small, corresponds toa value that may be in the medium error margin inbarometric estimates (e.g. Worley & Powell 2000)and so is amenable to high-precision P–T studies.

In the Italian Alps the shear zones that hostmineral reactions that have proceeded to completionare widely spaced with little evidence that localizeddeformation has migrated from one site to another.This would be interpreted as arising from strain soft-ening coupled to strain-rate softening. The lack ofobvious geometrical relation between the orien-tation of shear zones and coeval folds is to beexpected from the models presented here. Theoutcrop pattern of shear zones, the widespreadoccurrence of Y- and V-shaped shear zones, thewidespread occurrence of boudins, and the associ-ation of mineral reactions with the necks andmargins of boudins are all hallmarks of a terrainwhere deformation and chemical reactions arestrongly coupled to produce strain-rate-softeningbehaviour.

An important point to emphasize is that theresults depicted in Figures 10–12 are, in principle,scale invariant. Although we have developed thetheory here specifically for the outcrop-scale, thesame principles hold at all scales except thatdifferent processes lead to strain-rate softening atdifferent scales. Thus, thermal feedback processesdominate at the regional scale, chemical kinetic

feedback processes dominate at the outcrop scaleand chemical kinetic–diffusion processes dominateat the micro-scale. All of these processes producesimilar structures (shear zones, folds, boudinage)at the relevant scale and this leads to scale invar-iance of structures in deformed metamorphic rocks.

The reader will have noted that we have specifi-cally not included large-scale fluid and mass influx(metasomatism) into the deforming rock mass northe heat generated (or absorbed) during mineralreactions. The reason for this neglect is that theinclusion of such processes adds a significantadditional level of complexity that is best left forother papers. Our intent in this paper has been toemphasize that coupling between mechanical dissi-pation and that part of the chemical dissipationrepresented by the term AKjK in equation (63) issufficient to result in strain-rate softening and,hence, the development of instabilities in the defor-mation. In order to understand the complexity intro-duced by the inclusion of metasomatism and theheat of mineral reactions, it is convenient to recastthe Energy Equation, equation (9), in the form:

DT

Dt¼ kij

@2T

@xi@xj

þX@

l@ exp �E@

RT

� �(36)

where DT/Dt is the time material derivative of theabsolute temperature, T, kij is the thermal diffusivitytensor and xi is a spatial co-ordinate. l@ is a para-meter that is a measure of the dissipation for the@-th process; for thermal–mechanical, thermal–chemical and thermal–fluid-flow processes it isthe Gruntfest Number, the Damkoehler Numberand the Peclet Number, respectively (Law 2006;Veveakis et al. 2010). These dimensionless groupsare measures of the heat generated by deformation,chemical reactions and heat advection relative tothat generated by conduction. E@ is the activationenergy for the @-th process. Equation (36) is aspecial form of the reaction–diffusion equationand its form derives from an Arrhenius dependenceof the relevant geological processes upon tempera-ture. Coupled processes that obey equation (36)have been widely studied and the solutions ofmany of the equations involved are well known,especially in combustion physics (Law 2006). Inequation (36), if T is replaced by c, the concentra-tion of a mineral species, and the exponential depen-dence is replaced or augmented by non-linear termsinvolving c, then the result is the reaction–diffusionbehaviour discussed by Ortoleva (1994) for adiverse range of processes including metamorphicdifferentiation. All of these relations are specialforms of the Energy Equation, equation (9).

One outcome of this approach is that rate-dependent mechanical instabilities are inhibited by


endothermic reactions such as most anhydrous ordehydrating silicate reactions (including melting),and enhanced by exothermic reactions such ashydrating retrograde reactions and melt crystalliza-tion. The feedback relations also depend on thetimescale for the deformation with differenteffects at seismic strain rates as opposed to orogenicstrain rates. Veveakis et al. (2010) show that forseismic strain rates endothermic reactions (whichinclude most prograde reactions as well as grain-size reduction and melting) stabilize the defor-mation and produce a narrowing of the zone of loca-lized deformation within a developing wide shearzone. Thus, the development of ultramylonites andpseudotachylites within a wide mylonite zone isexplained by a coupling between these processes.The widening of the study of coupled processes inmetamorphic geology to include these other pro-cesses offers the opportunity to integrate many ofthe features that we see in deformed metamorphicrocks under the single umbrella of continuumthermodynamics.

Conclusions

We have examined the coupling between defor-mation and mineral reactions at the scale of tens tohundreds of metres, and developed a modelwhereby the coupling results in both strain harden-ing (or softening) and strain-rate hardening (or soft-ening). The precise forms of these relationshipsawait further theoretical and experimental develop-ments, but we have explored several realistic scen-arios in answer to the specific questions raised inthe beginning of the paper.

† The application of non-equilibrium thermodyn-amics at the scale of tens to hundreds of metresshows that mineral reactions induce both strainand strain-rate hardening (or softening) in rate-dependent materials. Strain softening alone instrain-rate hardening materials such as Newto-nian and power-law viscous materials with nocoupling produces shear zones, but the accumu-lated strain remains much smaller than isobserved in natural shear zones. Folds and bou-dinage do not form. Coupling to mineralreactions leads to strain-rate softening, and high-intensity strain localization now develops withstrain intensities similar to natural examples.This occurs no matter whether the mineral reac-tion results in strain hardening or strain softening.Moreover, folds and boudinage also develop inlayered materials with small (c. 10) viscosityratios between the layers and the embeddingmaterial. This is a non-Biot folding mechanismand is driven by strain-rate (and, hence, viscosity)differences that develop between the inner and

outer arcs of incipient buckles or betweenplanar and curved parts of incipient boudins.The folds that develop through this processresemble natural folds with strongly attenuatedlimbs. The ratios of wavelength to thickness arealso in the range of naturally observed ratios butdepend on the initial distribution of geometricalor material heterogeneities.

† Deformation influences mineral reactions atthe scale of tens to hundreds of metres wherethe energy dissipated by inelastic deformationenhances the mineral reaction rate. As indicatedby many authors (see Vernon 2004 for a discus-sion), this is equivalent to influencing the kin-etics. More theoretical and experimental work,coupled with detailed field and laboratory analy-sis, is required to develop better quantitativemodels for this influence, particularly the influ-ence (if any) of strain rate on mineral reactionrate. The result is a feedback relation resultingin both strain and strain-rate hardening (orsoftening).

† Although the concept of a non-equilibriummineral phase diagram can be developed, at geo-logical strain rates the non-equilibrium phaseboundary for the quartz! coesite reaction iswithin error of the equilibrium phase boundary.The departure depends strongly on the differ-ences in elastic strain and specific volumesbetween the reactants and the differences in con-stitutive behaviour between the reactants andthe products. At least for the quartz! coesitereaction the differences between the non-equilibrium and equilibrium phase boundariesdo not exceed approximately 0.1 Pa in the temp-erature range 600–1200 K at strain rates of10214–10212 s21. High-precision P and T esti-mates are needed, simultaneously on weaklyand highly deformed rocks (from coronites tomylonites), to explore the effect for more com-plicated assemblages. The non-equilibrium phasestability depends on the mean stress (P) insteadof the hydrostatic pressure and so this effectmay be important in some regions (Burg &Schmalholz 2008).

We thank K. Regenauer-Lieb for long and inspirationaldiscussions. C. Detournay is thanked for help with theremeshing routines in FLAC2D. R. Twiss and M. Brownare thanked for critical reviews that greatly improvedthe paper.

References

ANAND, L., KIM, K. H. & SHAWKI, T. G. 1987. Onset ofshear localization in viscoplastic solids. Journal of theMechanics and Physics of Solids, 35, 407–429.


AUSTRHEIM, H. 1990. The granulite–eclogite faciestransition: a comparison of experimental work andnatural occurrence in the Bergen Arcs, westernNorway. Lithos, 25, 163–169.

AUSTRHEIM, H. & GRIFFIN, W. L. 1985. Shear defor-mation and eclogite formation within granulite faciesanorthosites of the Bergen arcs Western Norway.Chemical Geology, 50, 267–281.

BAYLY, B. 1983. Chemical potential in viscous phasesunder nonhydrostatic stress. Philosophical MagazineA, 47, 39–44.

BAYLY, B. 1987. Nonhydrostatic thermodynamics indeforming rocks. Canadian Journal of Earth Sciences,24, 572–579.

BAYLY, B. 1992. Chemical Change in DeformingMaterials. Oxford University Press, New York.

BEACH, A. 1973. The mineralogy of high temperatureshear zones at Scourie NW Scotland. Journal ofPetrology, 14, 231–248.

BEACH, A. 1976. The interrelations of fluid transport,deformation, geochemistry and heat flow in earlyProterozoic shear zones in the Lewisian complex.Philosophical Transactions of the Royal Society,London, A280, 569–604.

BEACH, A. 1980. Retrogressive metamorphic processesin shear zones with special reference to the Lewisiancomplex. Journal of Structural Geology, 2, 257–263.

BELL, T. H. & RUBENACH, M. J. 1983. Sequentialporphyroblast growth and crenulation cleavagedevelopment during progressive deformation.Tectonophysics, 92, 171–194.

BIOT, M. A. 1965a. Mechanics of Incremental Defor-mations. Wiley, New York.

BIOT, M. A. 1965b. Theory of similar folding of the firstand second kind. Geological Society of AmericaBulletin, 76, 251–258.

BIOT, M. A. 1984. New variational-Lagrangian irrevers-ible thermodynamics with application to viscous flow,reaction–diffusion, and solid mechanics. Advances inApplied Mechanics, 24, 1–91.

BOWEN, R. M. 1967. Theory of Mixtures. In: ERINGEN,A. C. (ed.) Continuum Physics, Volume 3. AcademicPress, New York.

BOWEN, R. M. & WIESE, J. C. 1969. Diffusion in mixturesof elastic materials. International Journal of Engineer-ing Science, 7, 689–722.

BRODIE, K. H. & RUTTER, E. H. 1985. The role of transi-ently fine-grained reaction products in syntectonicmetamorphism natural and experimental examples.Canadian Journal of Earth Science, 24, 556–564.

BRODIE, K. H. & RUTTER, E. H. 1990. Mechanism ofreaction-enhanced deformability in minerals androcks. In: BARBER, D. J. & MEREDITH, P. G. (eds)Deformation Processes in Minerals, Ceramics andRocks. Unwin Hyman, London.

BURLINI, L. & BRUHN, D. 2005. High-strain zones:laboratory perspectives on strain softening duringductile deformation. In: BRUHN, D. & BURLINI, L.(eds) High-strain Zones: Structure and PhysicalProperties. Geological Society, London, SpecialPublications, 245, 1–24.

BURG, J.-P. & SCHMALHOLZ, S. M. 2008. Viscousheating allows thrusting to overcome crustal-scalebuckling: numerical investigation with application to

Himalayan syntaxes. Earth and Planetary ScienceLetters, 274, 189–203.

CALLEN, H. B. 1960. Thermodynamics: An Introductionto the Physical Theories of Equilibrium Thermostaticsand Irreversible Thermodynamics. Wiley, New York.

CARPENTER, M. A., SALJE, E. K. H., GRAEME-BARBER,A., WRUCK, B., DOVE, M. T. & KNIGHT, K. S. 1998.Calibration of excess thermodynamic properties andelastic constant variations associated with the a$ bphase transition in quartz. American Mineralogist,83, 2–22.

COE, R. S. & PATERSON, M. S. 1969. The a–b transitionin quartz: a coherent phase transition under nonhydro-static stress. Journal of Geophysical Research, 74,4921–4948.

COLEMAN, B. D. & GURTIN, M. E. 1967. Thermodyn-amics with internal state variables. Journal of Chemi-cal Physics, 47, 597–613.

COLLINS, I. F. & HILDER, T. 2002. A theoretical frame-work for constructing elastic/plastic constitutivemodels of triaxial tests. International Journal forNumerical and Analytical Methods in Geomechanics,26, 1313–1347.

COLLINS, I. F. & HOULSBY, G. T. 1997. Applicationsof thermomechanical principles to the modeling ofgeotechnical materials. Proceedings of the RoyalSociety, Series A, 453, 1975–2001.

COUSSY, O. 1995. Mechanics of Porous Continua. Wiley,Chichester.

COUSSY, O. 2004. Poromechanics. Wiley, Chichester.DE GROOT, S. R. & MAZUR, P. 1969. Non-equilibrium

Thermodynamics. North Holland, Amsterdam.DELLE PIANE, C., BURLINI, L. & GROBETY, B. 2007.

Reaction-induced strain localization: Torsion exper-iments on dolomite. Earth and Planetary ScienceLetters, 256, 36–46.

DE RONDE, A. A., STUNITZ, H., TULLIS, J. &HEILBRONNER, R. 2005. Reaction-induced weaken-ing of plagioclase–olivine composites. Tectonophy-sics, 409, 85–106.

DEWAR, R. C. 2005. Maximum entropy production andthe fluctuation theorem. Journal of Physics A: Mathe-matical and General, 38, L371–L381.

DUHEM, P. 1911. Traite d’energetique ou de Thermodyna-mique Generale. 2 vols. Gauthier-Villars, Paris.

ERICKSEN, J. L. 1998. Introduction to the Thermodyn-amics of Solids. Springer, New York.

ESTRIN, Y. & KUBIN, L. P. 1991. Plastic instabilities:phenomenology and theory. Materials Science andEngineering, A137, 125–134.

FISHER, G. W. 1970. The application of ionic equilibriato metamorphic differentiation: an example. Contri-butions to Mineralogy and Petrology, 29, 91–103.

FISHER, G. W. 1973. Non-equilibrium thermodynamicsas a model for diffusion-controlled metamorphic pro-cesses. American Journal of Science, 273, 897–924.

FISHER, G. W. & LASAGA, A. C. 1981. Irreversiblethermodynamics in petrology. American Reviews inMineralogy and Geochemistry, 8, 171–210.

FLEITOUT, L. & FROIDEVAUX, C. 1980. Thermal andmechanical evolution of shear zones. Journal ofStructural Geology, 2, 159–164.

FLETCHER, R. C. 1973. Propagation of a coherentinterface between two nonhydrostatically stressed


crystals. Journal of Geophysical Research, 78,7661–7666.

FOSTER, C. T. 1981. A thermodynamic model of mineralsegregation in the lower sillimanite zone near Range-ley, Maine. American Mineralogist, 66, 260–277.

FRUH-GREEN, G. L. 1994. Interdependence of deforma-tion, fluid infiltration and reaction progress recordedin eclogitic metagranitoids (Sesia Zone, WesternAlps). Journal of Metamorphic Geology, 12, 327–343.

GARDIEN, V., REUSSER, E. & MARQUER, D. 1994.Pre-Alpine metamorphic evolution of the gneissesfrom the Valpelline Series (Western Alps, Italy).Schweizerische Mineralogische PetrographischeMitteilungen, 74, 489–502.

GAZZOLA, D., GOSSO, G., PULCRANO, E. & SPALLA,M. I. 2000. Eo-Alpine HP metamorphism in thePermian intrusives from the steep belt of the centralAlps (Languard–Campo nappe and Tonale Series).Geodinamica Acta, 13, 149–167.

GHOUSSOUB, J. & LEROY, Y. M. 2001. Solid–fluid phasetransformation within grain boundaries during com-paction by pressure solution. Journal of the Mechanicsand Physics of Solids, 49, 2385–2430.

GIBBS, J. W. 1906. Collected Works of J. Willard Gibbs,Volume 1. Yale University Press, New Haven, CT.

GILMAN, J. J. 2003. Electronic Basis of the Strength ofMaterials. Cambridge University Press, Cambridge.

GOSSO, G., SALVI, F., SPALLA, M. I. & ZUCALI, M.2004. Map of deformation partitioning in the polyde-formed and polymetamorphic Austroalpine basementin Valtellina and Val Camonica, (Central Alps). In:PASQUARE, G., VENTURINI, C. & GROPPELLI, G.(eds) Mapping Geology in Italy. A.P.A.T., Rome,295–304.

GREEN, H. W., II. 1970. Diffusional flow in polycrystal-line materials. Journal of Applied Physics, 41,3899–3902.

GREEN, H. W., II. 1972. Metastable growth of coesite inhighly strained quartz. Journal of GeophysicalResearch, 77, 2478–2482.

GREEN, H. W., II. 1980. On the thermodynamics ofnonhydrostatically stressed solids. PhilosophicalMagazine, A41, 637–647.

HAHNER, P. 1995. A generalized criterion of plasticinstabilities and its application to creep damage andsuperplastic flow. Acta Metallurgica et Materialia,143, 4093–4100.

HELBIG, P. & SCHMIDT, D. 1978. Zur Tektonik undPetrogenese am W-Ende des Schneeberger Zuges(Ostalpen). Jahrbuch der Geologischen Bundesanstalt,121, 177–217.

HILLIG, W. B. & CHARLES, R. J. 1964. Chapter 17.Surfaces, stress-dependent surface reactions andstrength. In: ZACKAY, V. F. (ed.) High StrengthMaterials. Wiley, New York, 682–705.

HOBBS, B. E. 1968. Recrystallisation of single crystals ofquartz. Tectonophysics, 6, 353–401.

HOBBS, B. E. & ORD, A. 2009. The mechanics of grani-toid systems and maximum entropy production rates.Philosophical Transactions of the Royal Society ofLondon, A13, 368, 53–93.

HOBBS, B. E., MUHLHAUS, H.-B. & ORD, A.1990. Instability, softening and localization of defor-mation. In: KNIPE, R. J. & RUTTER, E. H. (eds)

Deformation Mechanisms, Rheology and Tectonics.Geological Society, London, Special Publications,54, 143–165.

HOBBS, B. E., MUHLHAUS, H.-B. & ORD, A. 2000. Theinfluence of chemical migration upon fold evolutionin multi-layered materials. Selbstorganisation. Jahr-buch fur Komplexitat in den Natur-, Sozial- undGeisttttteswissenschaften, 11, 229–252.

HOBBS, B. E., REGENAUER-LIEB, K. & ORD, A. 2007.Thermodynamics of folding in the middle to lowercrust. Geology, 35, 175–178.

HOBBS, B. E., REGENAUER-LIEB, K. & ORD, A. 2008.Folding with thermal–mechanical feed-back. Journalof Structural Geology, 30, 1572–1592.

HOINKES, G., FRANK, W., MAURACHER, J., PESCHEL,R., PURTSCHELLER, F. & TESSADRI, R. 1987. Petro-graphy of the Schneeberg Complex. In: FLUGEL,H. W. & FAUPL, P. (eds) Geodynamics of theEastern Alps. Deuticke, Wien, 190–199.

HOULSBY, G. T. & PUZRIN, A. M. 2006. Principles ofHyperplasticity. Springer, London.

ITASCA. 2008. Users Manual, FLAC6.0, Fast LagrangianAnalysis of Continua. ITASCA, Minneapolis, MN.

JOESTEN, R. 1977. Evolution of mineral assemblagezoning in diffusion metasomatism. Geochimica etCosmochimica Acta, 41, 649–670.

KAMB, W. B. 1959. Theory of preferred crystal orientationdeveloped by crystallisation under stress. Journal ofGeology, 67, 153–170.

KAMB, W. B. 1961. The thermodynamic theory ofnonhydrostatically stressed solids. Journal of Geophy-sical Research, 66, 259–271.

KAMEYAMA, M. 2003. Comparison between thermal-viscous coupling and frictional sliding. Tectonophy-sics, 376, 185–197.

KESTIN, J. & RICE, J. R. 1970. Paradoxes in the appli-cation of thermodynamics to strained solids. In:STUART, E. B., BRAINARD, A. J. & GAL-OR, B.(eds) A Critical Review of Thermodynamics. MonoBook Corp., Baltimore, MD, 275–298.

KERN, R. & WEISBROD, A. 1967. Thermodynamics forGeologists. Freeman, Cooper & Co.,San Francisco, CA.

KIMIZUKA, H., OGATA, S. & LI, J. 2008. Hydrostaticcompression and high-pressure elastic constants ofcoesite silica. Journal of Applied Physics, 103,053506-1–053506-4.

KOCKS, U. F., ARGON, A. S. & ASHBY, M. F. 1975.Thermodynamics and Kinetics of Slip. PergamonPress, Oxford.

KONDEPUDI, D. & PRIGOGINE, I. 1998. Modern Thermo-dynamics. Wiley, Chichester.

KOONS, P. O., RUBIE, D. C. & FRUEH-GREEN, G. 1987.The effects of disequilibrium and deformation on themineralogical evolution of quartz diorite during meta-morphism in the eclogite facies. Journal of Petrology,28, 679–700.

KORZHINSKII, D. S. 1959. Physicochemical Basis of theAnalysis of the Paragenesis of Minerals. ConsultantsBureau, New York.

KRAUSZ, A. S. & EYRING, H. 1975. DeformationKinetics. Wiley, New York.

KUMAZAWA, M. 1961. A note on the thermodynamictheory of nonhydrostatically stressed solids. Journalof Geophysical Research, 66, 3981–3984.


KUMAZAWA, M. 1963. A fundamental thermodynamictheory on nonhydrostatic field and on the stability ofmineral orientation and phase equilibrium. Journal ofEarth Sciences, Nagoya University, 11, 145–217.

LARCHE, F. C. & CAHN, J. W. 1985. The interactions ofcomposition and stress in crystalline solids. ActaMetallurgica, 33, 331–357.

LARDEAUX, J. M. & SPALLA, M. I. 1991. From granulitesto eclogites in the Sesia zone (Italian Western Alps): arecord of the opening and closure of the Piedmontocean. Journal of Metamorphic Geology, 9, 35–59.

LASAGA, A. C. 1998. Kinetic Theory in the EarthSciences. Princeton University Press, Princeton, NJ.

LAW, C. K. 2006. Combustion Physics. CambridgeUniversity Press, Cambridge.

LAWN, B. 1993. Fracture of Brittle Solids. CambridgeUniversity Press, Cambridge.

LEVITAS, V. I., NESTERENKO, V. F. & MEYERS, M. A.1998a. Strain-induced structural changes and chemicalreactions – I. Thermomechanical and kinetic models.Acta Materialia, 46, 5929–5945.

LEVITAS, V. I., NESTERENKO, V. F. & MEYERS, M. A.1998b. Strain-induced structural changes and chemicalreactions – II. Modelling of reactions in shear band,Acta Materialia, 46, 5947–5963.

LEHNER, F. K. & BATAILLE, J. 1984. Non equilibriumthermodynamics of pressure solution. Pure andApplied Geophysics, 122, 53–85.

LYAKHOVSKY, V. & BEN-ZION, Y. 1997. Distributeddamage, faulting, and friction. Journal of GeophysicalResearch, 102, 27,635–27,649.

MAIN, I. G. & NAYLOR, M. 2008. Maximum entropyproduction and earthquake dynamics. GeophysicalResearch Letters, 35, L19311.

MANCKTELOW, N. S. 1993. Tectonic overpressure incompetent mafic layers and the development of iso-lated eclogites. Journal of Metamorphic Geology, 11,801–812.

MANCKTELOW, N. S. 2002. Finite-element modelling ofshear zone development in viscoelastic materials andits implications for localization of partial melting.Journal of Structural Geology, 24, 1045–1053.

MARTYUSHEV, L. M. & SELEZVEV, V. D. 2006.Maximum entropy production principle in physics,chemistry and biology. Physics Reports, 426, 1–45.

MAUGIN, G. A. 1999. The Thermomechanics of Non-linear Irreversible Behaviours. World Scientific,Singapore.

MCLELLAN, A. G. 1980. The Classical Thermodynamicsof Deformable Materials. Cambridge UniversityPress, Cambridge.

MESAROVIC, S. D. 1995. Dynamic strain aging and plasticinstabilities. Journal of the Mechanics and Physics ofSolids, 43, 671–700.

MORK, M. B. E. 1985. A gabbro to eclogite transition onFlemsoy, Sunnmore, Western Norway. ChemicalGeology, 50, 283–310.

MOSENFELDER, J. L. & BOHLEN, S. R. 1997. Kinetics ofcoesite to quartz transformation. Earth and PlanetaryScience Letters, 153, 133–147.

NEEDLEMAN, A. 1988. Material rate dependence andmesh sensitivity in localization problems. ComputerMethods in Applied Mechanics and Engineering, 67,69–85.

NICOT, E. 1977. Les roches meso et catazonales de la Val-pelline (nappe de la Dent Blanche; Alpes Italiennes).PhD thesis, Universite Pierre et Marie Curie, ParisVI, 211 pp.

NYE, J. F. 1957. Physical Properties of Crystals. OxfordUniversity Press, Oxford.

OGATA, S., KIMIZUKA, H. & LI, J. 2008. Hydrostaticcompression and high-pressure elastic constants ofcoesite silica. Journal of Applied Physics, 103,053506-1-4.

ORTOLEVA, P. 1994. Geochemical Self-organization.Oxford University Press, New York.

PALTRIDGE, G. W. 1975. Global dynamics and climate –a system of minimum entropy exchange. QuarterlyJournal of the Royal Meteorological Society, 101,475–484.

PALTRIDGE, G. W. 1978. The steady-state format ofglobal climate. Quarterly Journal of the Royal Meteor-ological Society, 104, 927–945.

PALTRIDGE, G. W. 2001. A physical basis for a maximumof thermodynamic dissipation of the climate system.Quarterly Journal of the Royal MeteorologicalSociety, 127, 305–313.

PATERSON, M. S. 1973. Nonhydrostatic thermodynamicsand its geological applications. Reviews in Geophysics,11, 355–389.

PATERSON, M. S. 1995. A theory for granular flow accom-modated by material transfer via an intergranular fluid.Tectonophysics, 245, 135–151.

PETRINI, K. & PODLADCHIKOV, Y. 2000. Lithosphericdepth–pressure relationship in compressive regionsof thickened crust. Journal of Metamorphic Geology,18, 67–77.

POGNANTE, U. 1989. Tectonic implications of lawsoniteformation in the Sesia zone (Western Alps). Tectono-physics, 162, 219–227.

POIRIER, J. P. 1980. Shear localization and shear instabil-ity in materials in the ductile field. Journal of Struc-tural Geology, 2, 135–142.

PRIGOGINE, I. 1955. Introduction to the Thermodynamicsof Irreversible Processes. Charles C. Thomas,Springfield, IL.

PUZRIN, A. M. & HOULSBY, G. T. 2001. Fundamentals ofkinematic hardening hyperplasticity. InternationalJournal of Solids and Structures, 38, 3771–3797.

RAJAGOPAL, K. R. & SRINIVASA, A. R. 2004. On thermo-mechanical retrictions of continua. Proceedings of theRoyal Society of London, A, 460, 631–651.

RAMBERG, H. 1959. The Gibbs’ free energy of crystalsunder anisotropic stress, a possible cause for preferredmineral orientation. Anais da Escola de Minas de OuroPreto, 32, Supplement, 1–12.

RAMBERT, G., GRANDIDIER, J.-C. & AIFANTIS, E. 2007.On the direct interactions between heat transfer, masstransport and chemical processes within gradient elas-ticity. European Journal of Mechanics, A/Solids, 26,68–87.

RAMSAY, J. 1967. Folding and Fracturing of Rocks.McGraw-Hill, New York.

RAMSAY, J. & GRAHAM, R. H. 1970. Strain variation inshear belts. Canadian Journal of Earth Sciences, 7,786–813.

REGENAUER-LIEB, K. & YUEN, D. A. 2003. Modelingshear zones in geological and planetary sciences:


solid– and fluid–thermal–mechanical approaches.Earth Science Reviews, 63, 295–349.

REGENAUER-LIEB, K. & YUEN, D. A. 2004. Positivefeedback of interacting ductile faults from couplingof equation of state, rheology and thermal mechanics.Physics of Earth and Planetary Interiors, 142,113–135.

REGENAUER-LIEB, K., HOBBS, B. E., ORD, A., GAEDE,O. & VERNON, R. H. 2009. Deformation with coupledchemical diffusion. Physics of the Earth and PlanetaryInteriors, 172, 43–54.

RICARD, Y. & BERCOVICI, D. 2009. A grained conti-nuum model of damage and coarsening. Journal ofGeophysical Research, 114, B01204; doi: 10.1029/2007JB005491.

RODA, M. & ZUCALI, M. 2008. Meso and microstructuralevolution of the Mont Morion metaintrusive complex(Dent-Blanche nappe, Austroalpine domain, Valpel-line, Western Italian Alps). Bollettino della SocietaGeologica Italiana, 127, 1–19.

RUBBO, M., BORGHI, A. & COMPAGNONI, R. 1999.Thermodynamic analysis of garnet growth zoning ineclogite-facies granodiorite from M. Mucrone, SesiaZone, Western Alps. Contributions to Mineralogyand Petrology, 137, 289–303.

RUBIE, D. C. 1983. Reaction-enhanced ductility: therole of solid–solid univariant reactions in deformationof the crust and mantle. Tectonophysics, 96, 331–352.

RUBIE, D. C. 1990. Role of kinetics in the formation andpreservation of eclogites. In: CARSWELL, D. A. (ed.)Eclogite Facies Rocks. Blackie, Glasgow, 111–140.

RUINA, A. 1983. Slip instability and state variablefriction laws. Journal of Geophysical Research, 88,10,359–10,370.

SALVI, F., SPALLA, M. I., ZUCALI, M. & GOSSO, G.2010. Three-dimensional evaluation of fabric evol-ution and metamorphic reaction progress in polycyclicand polymetamorphic terrains: a case from the CentralItalian Alps. In: SPALLA, M. I., MAROTTA, A. M. &GOSSO, G. (eds) Advances in Interpretation of Geo-logical Processes: Refinement of Multi-scale Dataand Integration in Numerical Modelling. GeologicalSociety, London, Special Publications, 332, 173–187.

SCHMALHOLZ, S. M., SCHMID, D. W. & FLETCHER,R. C. 2008. Evolution of pinch-and-swell structuresin a power-law layer. Journal of Structural Geology,30, 649–663.

SCHMALZRIED, H. 1995. Chemical Kinetics of Solids.VCH Verlagsgesellschaft, Weinheim.

SHIGEMATSU, N. 1999. Dynamic recrystallisation indeformed plagioclase during progressive shear defor-mation. Tectonophysics, 305, 437–452.

SHIMIZU, I. 1992. Nonhydrostatic and nonequilibriumthermodynamics of deformable materials. Journal ofGeophysical Research, 97, 4587–4597.

SHIMIZU, I. 1995. Kinetics of pressure solution creep inquartz: theoretical considerations. Tectonophysics,245, 121–134.

SHIMIZU, I. 1997. The non-equilibrium thermodynamicsof intracrystalline diffusion under non-hydrostaticstress. Philosophical Magazine A, 75, 1221–1235.

SHIMIZU, I. 2001. Chapter 3. Nonequilibrium thermo-dynamics of nonhydrostatically stressed solids.In: TEISSEYRE, R., MAJEWSKI, E. ET AL. (eds)

Earthquake Thermodynamics and Phase Transform-ations in the Earth’s Interior. Academic Press,New York, 81–102.

SHIMOKAWA, S. & OZAWA, H. 2002. On the thermodyn-amics of the oceanic general circulation: irreversibletransition to a state of higher rate of entropy pro-duction. Quarterly Journal of the Royal Meteorologi-cal Society, 128, 2115–2128.

SMITH, R. B. 1977. Formation of folds, boudinage andmullions in non-Newtonian materials. GeologicalSociety of America Bulletin, 88, 312–320.

SPALLA, M. I. 1990. Polyphased deformation dutinguplifting of metamorphic rocks: the example of thedeformational history of the Texel Gruppe (Central-Western Austroalpine Domain of the Italian EasternAlps). Memorie di Scienze Geologiche, Padova, 45,125–134.

SPALLA, M. I. 1993. Microstructural control on the PTpath construction in the metapelites from the Austroal-pine crust (Texel Gruppe, Eastern Alps). SchweizerMineralogische und Petrographische Mitteilungen,73, 259–275.

SPALLA, M. I. & ZUCALI, M. 2004. Deformation vs.metamorphic re-equilibration heterogeneities in poly-metamorphic rocks: a key to infer quality P–T–d–tpath. Periodico di Mineralogia, 73, 249–257.

SPALLA, M. I., ZUCALI, M., DI PAOLA, S. & GOSSO, G.2005. A critical assessment of the tectono-thermalmemory of rocks and definition of tectono-metamorphic units: evidence from fabric and degreeof metamorphic transformations. In: GAPAIS, D.,BRUN, J. P. & COBBOLD, P. R. (eds) DeformationMechanisms, Rheology and Tectonics: From Mineralsto the Lithosphere. Geological Society, London,Special Publications, 243, 227–247.

SPALLA, M. I. & ZULBATI, F. 2003. Structural and petro-graphic map of the Southern Sesia–Lanzo Zone (MonteSoglio–Rocca Canavese, Western Alps, Italy).Memorie di Scienze Geologiche, Padova, 55, 119–127.

STUWE, K. & SANDIFORD, M. 1994. Contribution ofdeviatoric stress to metamorphic P–T paths: anexample appropriate to low-P, high-T metamorphism.Journal of Metamorphic Geology, 12, 445–454.

TAYLOR, G. I. & QUINNEY, H. 1934. The latent energyremaining in a metal after cold working. Proceedingsof the Royal Society, of London, A, 143, 307–326.

THOMPSON, J. B., JR. 1959. Local equilibrium in metaso-matic processes. In: ABELSON, P. H. (ed.) Researchesin Geochemistry. Wiley, New York. 427–457.

THOMPSON, A. B. & ENGLAND, P. C. 1984. Pressure–Temperature–Time paths of regional metamorphismII. Their inference and interpretation using mineralassemblages in metamorphic rocks. Journal of Petrol-ogy, 25, 929–955.

TROPPER, P., ESSENE, E. J., SHARP, Z. D. & HUNZIKER,J. C. 1999. Application of K-feldspar–jadeite–quartzbarometry to eclogite facies metagranites and meta-pelites in the Sesia Lanzo Zone (Western Alps, Italy).Journal of Metamorphic Geology, 17, 195–209.

TRUESDELL, C. A. 1969. Rational Thermodynamics. 2ndedn. Springer, New York.

TRUSKINOVSKIY, L. M. 1984. The chemical potentialtensor. [English translation] Geochemistry Inter-national, 21, 22–36.


VAN GOOL, J. A. M., KEMME, M. M. J. & SCHREURS,G. M. M. F. 1987. Structural investigations along aE-W cross-section in the Southern Oetztal Alps. In:FLUGEL, H. W. & FAULP, P. (eds) Geodynamics ofthe Eastern Alps. Deuticke, Wien, 214–225.

VERHOOGEN, J. 1951. The chemical potential of a stressedsolid. Transactions of the American GeophysicalUnion, 32, 251–258.

VERNON, R. H. 2004. A Practical Guide to Rock Micro-structure. Cambridge University Press, Cambridge.

VEVEAKIS, E., ALEVIZOS, S. & VARDOULAKIS, I. 2010.The critical chemical capping of thermal runawayduring shear of frictional faults. Journal of the Mech-anics and Physics of Solids, in press.

WANG, W. M., SLUYS, L. J. & DE BORST, R. 1997.Viscoplasticity for instabilities due to strain softeningand strain-rate softening. International Journal forNumerical Methods in Engineering, 40, 3839–3664.

WORLEY, B. & POWELL, R. 2000. High-precision relativethermobarometry: theory and a worked example.Journal of Metamorphic Geology, 18, 91–101.

ZAISER, M. & HAHNER, P. 1997. A unified description ofstrain-rate softening instabilities. Materials Scienceand Engineering, A238, 399–406.

ZIEGLER, H. 1983a. An Introduction to Thermomecha-nics. 2nd edn. North-Holland, Amsterdam.

ZIEGLER, H. 1983b. Chemical reactions and the principleof maximal rate of entropy production. Journal ofApplied Mathematics and Physics, 34, 832–844.

ZHU, T., LI, J. & YIP, S. 2005. Stress-dependent molecularpathways of silica–water reaction. Journal of theMechanics and Physics of Solids, 53, 1597–1623.

ZUCALI, M. 2001. La correlazione nei terreni metamor-fici: due esempi dall’Austroalpino occidentale (ZonaSesia–Lanzo) e centrale (Falda Languard-Campo/Serie del Tonale). PhD thesis, University of Milano.

ZUCALI, M., SPALLA, I. M. & GOSSO, G. 2002. Strainpartitioning and fabric evolution as a correlation tool:the example of the Eclogitic Micaschists Complex inthe Sesia–Lanzo Zone (Monte Mucrone–MonteMars, Western Alps, Italy). Schweizer Mineralogischeund Petrographische Mitteilungen, 82, 429–454.


The interaction of deformation and metamorphic reactions

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