Feb 10, 1992 - Correction and Addition to "The Solid-State Flow of Polymineralic Rocks" by Mark R. Handy. CORRECTION. In the paper "The Solid-State Flow ...
JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 97, NO. B2, PAGES 1897-1899, FEBRUARY
10, 1992
Correctionand Addition to "The Solid-StateFlow of PolymineralicRocks" by Mark R. Handy CORRECTION
In the paper "The Solid-State Flow of Polymineralic Rocks"by Mark R. Handy (Journal of GeophysicalResearch, 95 (B6), 8647-8661, 1990), equations(8), (12), and (13) in section4.2 on p. 8657 are incorrectand shouldincludevolume proportionterms.They shouldread N
• Ipi O'i•i =O'rock • rock
i=l
(8)
where •i and /•i are the stresses and strain rates of the ith
constituentphasein a rock of viscousstrength•rock at a bulk strain rate erock and where •i is the volume proportion of the ith phase. amck=
nj+ 1 tpi 'œi[•iFi'l+expn[1]+cpj FjtJj
the volume proportionof quartz (kquartz is equal to the clast spacing-sizefactor a in equation(5). Varying the size of the
clastsat constantspacingis like varyingthe volume proportionand thickness of feldsparin the uniformlyspaced layersof the hypotheticalmylonite. The mechanical and geometric opposite of a perfectly planar foliation is an ordered configuration of perfectly spherical clasts. A face-centered cubic configuration of feldsparclastsis chosenhere to representan ideal, unstrained microstructure in which stress concentration in the quartz matrix is maximized for a given volume proportion of feldspar. The compositionaldependenceof maximum stress concentrationin the quartz matrix at the onsetof deformation of this unstrained aggregate is obtained by replacing a in
equation(5) for the real mylonitewith 1-[(4.24/•t)(1-tkquartz]l/3 (12)
for the hypotheticalunstrainedrock. Figure 1 showsthe compositional dependence of maximum stressconcentrationin the quartz matrix of a quartz-feldspar aggregatefor both the ideal unstrained(dashedcurve) and strained microstructures(solid curve). Stress concentrationis expressed as a multipleof the creepstressin a pure quartzlayer (equation(4)). Note that becausestressconcentrationin the viscousmatrix only becomessignificant at very small clast spacings(Figure 2a), the maximum stressconcentrationin Figure 1 only slightly overestimates the average stress
•rock= tPi•i[•iFi'l+ exp n[1] +tPJ•'i[•'iF'fl+ exp n]l](13) • rock
where n is the creep exponent with differing values in the mineral phasesi andj. F is expressedasAexp(-Q/RT), whereT is the absolutetemperature,R is the universalgas constant,Q is the activation energy of creep, and A is an empirically determined constant. The values of Q and A are materialdependentand differ for the two phases. The omission of appropriatevolume proportion terms in equations (14) and (15) in section 4.3 on p. 8658 also invalidates the normalized strength versus composition diagram in Figure 9a. The stress concentration diagrams presentedin Figures 1 and 2 below replace Figure 9a and are consistentwith the hypothesisin section 4.3 of the original paperregardingthe effect of foliation developmenton viscous rock strength.
concentration in the matrix for Ckquartz valuescorresponding to the range of measuredz/d ratios (undottedparts of curves in Figure 1). Although the averagestressconcentrationis always less than the maximum
stress concentration,
the relative
positionof the two curvesin Figure 1 remainsthe samefor any given two-phase composition. Before discussing the mechanicalsignificanceof the curvesin Figure 1, I emphasize that they are only quantitatively valid for quartz undergoing power law creep under greenschistfacies conditions (300400øC) at a constantbulk natural strain rate. This bulk strain
rateispoorlyconstrained andranges fromabout10-11to 10-13 s-1. These estimatesare based on the extrapolationof
ADDITION TO FOLIATION DEVELOPMENT IN ROCKS
Using equation(5) and the data in Figure 7 of the original paper, one can calculate the maximum stressconcentrationin the quartz matrix for two hypothetical, end-member clastmatrix microstructures as discussed in section 4.3. An ideal,
high-strain mylonitic microstructure comprises perfectly planar, uniformly spaced, alternating layers of quartz and feldspar oriented parallel to the shearing plane. The compositionaldependenceof maximum stressconcentration in the quartz matrix of a perfectly foliated greenschistfacies mylonite is estimatedby equating the distancebetween the centers of neighboring feldspar clasts in the real mylonite (Figure 7) with the shortestdistancebetween the centerplanes of adjacentfeldspar layers in the hypotheticalmylonite. Thus,
rates tend to increase the stress concentration in the matrix, whereas low bulk strain rates decrease the stress concentration.
Despitetheseuncertaintiesin bulk strainrate and temperature, the asymmetricalshape of the curves in Figure 1 probably typifies the compositionaldependenceof stressconcentration in any two-phasematerial containinga matrix that undergoes dislocationcreepbetweenclastsof a rigid phase. Figure 1 indicatesthat the developmentof a foliation is associated with a marked decrease in the stress concentration
within the viscousquartzmatrix, particularlyin rocks with low to moderatevolume proportionsof quartz.Both pairs of stress concentration curves in Figure 1 are only valid for hypotheticalend-membergeometricaldistributionsof feldspar
Copyfight 1992 by the AmericanGeophysicalUnion. Papernumber91JB02203. 0148-0227/92/91
experimentalsteady state creep laws to flow stresses(about 100 MPa) obtainedfrom dynamically recrystallizedgrain size piezometryin pure quartzmylonitesat the samelocality as the quartz-feldspar myloniteusedin this study[Handy, 1986]. For any volume proportionof quartz less than 1, high bulk strain
JB-02203 $02.00 1897
1898
HANDY: CORRECTION
6
clastsin a quartz-feldsparaggregateand so are likely to bracket stressconcentrationversus compositioncurves for naturally deformed quartz-feldspar rock at similar strain rates and temperatures. In natural mylonite, the clastsin the foliational layersare often irregularlyshaped(Figures2b, 2c, and 6 in the original paper). Such geometrical irregularities probably
rock 10-11-10-13S-1 - 300-400øC
x
. .
ß ß
ß ß
increase
.
ß
'idea[
: idea[ ß unfoliafed
' fo[iafed ß
the
stress
concentration
in
the
matrix
to levels
somewhat above those predicted for a perfectly planar foliationin Figure 1. Similarly,feldsparclastsin undeformed graniticrock are nonspherically shapedand their distribution is not
as ordered
as in
the
ideal
face-centered
cubic
configurationassumedhere. The stressconcentrationin the matrix of a rock undergoingincipientdeformationis therefore somewhat less than that predicted by the curve for ideal, unfoliatedrock in Figure 1. The importantpoint made in the original paper and reiterated here is that strain-dependent developmentof a foliation decreasesthe stressin the weak viscous
o:2
0:6
matrix
and
so
also
decreases
the
amount
of
o18
deformationalenergy expendedon the entire rock for a given incrementof strain.This in turn leads to a weakeningof rock deforming at a given bulk strain rate, a phenomenontermed "foliation weakening"in experimentallydeformedtwo-phase Fig. 1. Maximumstressconcentration in quartzmatrixversusvolume aggregates[Le Hazif, 1978; Jordan, 1988]. proportion of quartzfor twohypothetical, end-member microstructures What factors influence foliation development during in a greenschist faciesquartz-feldpar rock.Solidline is for perfecfiy shearing? Some insight into this question is gained by planarfoliationparallel to the plane of shear(ideal high strain microstructure),whereas dashedline is for a face-centeredcubic observinghow stressconcentrationvaries locally within the configuration of spherical feldspar clasts (ideal unstrained microstructure of deformed rocks. Figure 2a is a plot of microstructure). Dottedpans of curvescorrespond to extrapolation of equation (5) showing that stress concentration in the quartzgrainsizedatato clastz/d valueslessthan1.13andprobably dynamically recrystallized quartz matrix varies nonlinearly significanfiy overestimate the actualstressconcentration in the quartz with distancebetween the feldspar clasts and with clast size.
()quar,z
matrix (see text for further explanation).Curves calculatedfrom
modification ofequation (5)using a stress-grain sizeexponent p of Threeclastspacing-size intervals aredistinguished in Figure 0.71 [Twiss,1977].
2a on the basis of microstructural observations' (1) at
xl] 1
2
3 N
//9
,
1.1
1.2
•
•
1.3
1./+ 1.5
a 1:6
Z/dofclasts
z
Fig. 2. (a) Stressconcentration in quartzmatrixversusaverage spacing to diameter ratio,z/d, of feldspar clastsin the greenschist faciesquartz-feldspar mylonite described in original paper,thethreeintervals of clastspacing to diameter ratios display distinctive microstructures andstress concentration characteristics (seetext);solidcurvein intervals 2 and3 derives froma bestfit of grainsizeandspacing measurements in thenaturalmylonite to anempirical stress concentration function (equations (3) and(5) in original paper); dotted curves in interval1 areinferred frommicrostinctures discussed in thetext; stress-grain sizeexponents p forhydrous quartzite usedin equation (5):0.68[Mercier etal., 1977],0.71[Twiss, 1977]and 1.11[Christieet al., 1980].(b) Schematic plotof stress concentration versusspacing to diameter ratioof clastsshowing theinferred change in localized stress concentration in thematrix(arrows) during theevolution of feldsparhie andquartzose layersfromaninitiallyundeformed granitic rockwithanisotropic distribution of feldspar grains(seetextfor explanation).
HANDY: CORRECTION
!899
moderateto high clast spacingto size ratios (1.43 < z/d < •o, interval 3 in Figure 2a), there is no significant stress concentrationin the quartz matrix except in the immediate vicinity of feldspar clasts [Prior et al., 1990]. To the extent that it is observed,localizedstressrise in the matrix appears to be controlledby viscousdrag aroundthe rigid clastsrather
clastsmay proceeduntil the Clastsize is approximately the samesize as the dynamicallyrecrystallized matrixgrainsand
[Masuda and Ando, 1988]; (2) at small to moderate clast
metamorphic fluid. Toriumi [1986] presents evidencein
the rigid feldspargrainscan no longerconcentratestressinto the viscousquartzmatrix.Both processes are associated with a
decrease in thespacing to sizeratioof theseclaststhatmaybe sufficiently large to allow the mutual overgrowth and than by the far field effects of neighboringfeldsparclasts cementationof the clastsin the presenceof a sytectonic,
spacingto size ratios(1.13 < z/d < 1.43, interval2 in Figure mylonitic metapelitic rockof eJongate aggregates comprising 2a), stressconcentration in the quartzmatrix variesstrongly syntectonically overgrown garnet clasts that are oriented
withtherelativespacing of feldspar clasts; (3) at extremely length-parallel to themyionitic foliation. Feldspar clasts may
low clastspacingto size ratios(1 < z/d < 1.13, interval 1 in Figure 2a), stress concentration in the viscous matrix is inferred to decrease from a maximum between intervals 1 to 2
to either a lesser finite value or to unity as the z/d ratio approachesunity. This inferencecomes from the observation that feldsparclastswithin foliationallayersare oftenin mutual contact (Figure 2b in original paper), suggestingthat hydrodynamic lubricationforcesin the viscousquartzmatrix breakdownat very smallclast spacings. The maximumstress concentration in the quartz matrix at the boundarybetween
intervals 1 and2 is limitedby oneor morefactors: High localized stress concentration in the matrix can induce either
continueto cluster in this way during shearinguntil a sufficientvolumeof weakquartzformscontiguous layersfor the volume-specificrate of strain energy dissipatedin the aggregate (equation (8)) to reach a mimimum value at the ambient temperature and bulk strain rate. Detailed
microsu'uctural andmineralogicalanalysesof grainboundaries in thefeldsparrich layersmay help to determinewhetherthese or
other
mechanisms
are
associated
with
foliation
developmentin mylonite.
Acknowledgments.I would like to thank Martin Caseyfor voicingsuspicions that lead me to discovermy mistakeand to Dave OlgaardandHolgerSmenitzfor subsequent discussions. I shouldalso
fracturing[Mitra, 1978] or yieldingof the feldspargrains, mention thatStevenKirby,a reviewerof theoriginalpaper,wasfrom
dependingon the ambienttemperatureand local strain rate. Alternatively, the local stress-dependentdynamically recrystallizedgrain size of the quartz matrix becomessmall
the outsetskepticalof sections4.2 and 4.3. Finally, my sincere
apologies to anyonewhowastedtime tryingto test the validityof equation(8) in my paper.
enough(8= 10 mm or less) to inducea mechanismswitchfrom
dislocation creepto grainsize sensitivediffusioncreepin the matrix. The boundarybetweenintervals2 and 3 coincideswith clastspacingto size ratiosfor whichquartzmatrix grain sizes Christie, J.M., A. Ord, and P.S. Koch, Relationshipbetween recrystallized grainsizeandflow stressin experimentally deformed
fall within the rangeof quartzgrain sizesin pure quartzite mylonites(a > 0.3 in Figure7b of originalpaper). An interesting,albeit indirectconsequence of Figure2a is that the evolution of a mylonitic foliation is associatedwith complexstresshistorieson the scale of individual grains within the aggregate.This is shownschematicallyin Figure 2b for granite with initially homogeneous spatial and size distributions of feldsparandquartz.The formationof feldspar rich foliational layers may involve an initial localized increasein stressconcentrationfollowed by a reductionin stressconcentrationas feldsparclasts are pushedor grow together(leftward shift along the curve in Figure 2b). Even more complicated,cyclic stresshistoriesare conceivableif the
quartzite(abstract),Eos Trans.AGU, 61(17), 377, t980.
Handy,M.R., The structureand rheological evolutionof the Pogallo
faultzone:A deepcrustaldislocation in the soulhem Alpsof northwesternItaly, Ph.D. thesis, 327 pp., Univ. of Basel, Switzerland, June 1986.
Jordan,P., The rheologyof polymineralic rocks-An approach,Geol. Runtisch., 77(1), 285-294, 1988.
Le Hazif, R., Deformation plastiquedu systembiphaseFe-Ag,Acta Metall., $6, 247-257, 1978.
Masuda, T., andS. Ando,Viscous flowaround a rigidspherical body:A hydrodynamical approach,Tectonoph?•sics, 148, 337-346, 1988. Mercier, J.C., D.A. Anderson, and N.L. Carter, Stress in the
lithosphere: Inferences from steady-state flow of rocks,PureAppl. Geophys.,115, 199-226, 1977. Mitra, G., Ductile deformationzonesand mylonites:The mechanical processesinvolved in the deformation of crystalline basement
initialstressrisein thematrixleadsto repeated fracturingand rocks, Am. if. Sci.,278, 1057-1084, 1978. comminutionof the feldspar grains before they finally Prior, D.J., R.J. Knipeoand M.R. Handy, Estimatesof the ratesof microstructuralchangesin mylonites,in DeformationMechanisms, impinge.In contrast,the form•itionof contiguous quartzrich editedby R.J.KnipeandE.H. Rutter,Geol. layersappearsto involvea progressive reductionof stress RheologyandTectonics, Soc. Spec.Publ. London,54,309-319, 1990. concentration in thequartzmatrixasfeldsparclastsarepushed M., Mechanical segregation of garnetin synmetamorphic apart(rightwardshift alongthe curvein Figure2b). What is Toriumi, flow of pelitic schists,./. Pet.,27, 6, 1395-1408,1986. still poorly understoodis how thesedifferentlocalizedstresses Twiss, R.J., Theory and applicabilityof a recrystallizedgrainsize
withinthe aggregate act in concertto segregate materialwith paleopiezometer, Pure Appl. Geophys., 115, 227-244, 1977. contrasting mechanicalpropertiesinto layersparallel to the shearingplane (see discussionat end of section 4.3). Mark R. Handy,Geologisches Institut,Universit•tBern,3012 Bern, Possibly,a progressivechangeof the dominantdeformation Switzerland. mechanism in the fine grainedquartzmatrixfromdislocation creepto diffusioncreepreducesthe repellantforcesbetween (ReceivedMay 31, 1991; approaching feldsparclasts.Alternatively,fracturingof the acceptedAugust19, 1991.)