Myron G. Best. Igneous and metamorphic 2003 Blackwell Science

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Kinetic Paths and Fabric of Magmatic Rocks

177

Consider, second, the origin of ﬂow layering. Ridge-

and-valley proﬁles of velocity in ﬂowing magma can

also translate grains of almost any shape toward the

higher-velocity region (see Section 8.2.3). In three di-

mensions, if the velocity gradients are nonuniform in a

planar sense (Figure 7.40c), then planar ﬂow layering,

expressed by layers of greater and lesser concentrations

of crystals, can develop. Extruded, viscous magmas

commonly have a ﬂow layering (Figures 7.2, 7.4, and

7.41) that is expressed by varying planar concentra-

tions or sizes of crystals or of vesicles that may have

originated in this manner. Nelson (1981) suggests that

thermal feedback in viscous ﬂow can also play a role in

development of ﬂow layering in lavas. During ﬂow,

movement becomes localized in shear surfaces between

more static layers; kinetic energy in the shear zones is

transformed into thermal energy; the increased T low-

ers the viscosity, facilitating further ﬂow in these zones

and further increase in T (thermal feedback), reducing

volatile solubility, increasing diffusion rates, and pro-

moting crystal and bubble nucleation and growth.

Mingling of contrasting magmas in the volcanic con-

duit is another way of producing ﬂow layered lavas

(Figure 7.42).

Field mapping of anisotropic ﬂow markers in mag-

matic rock bodies can potentially provide valuable

information concerning patterns of ﬂow in the vis-

cous magma during emplacement (Paterson et al.,

1998). However, it is important to note that many

anisotropic fabrics involving oriented mineral grains

in plutons result not so much from magma ﬂow as

from the way in which the crystal-rich magma body

was deformed, changing its shape, in the late stages of

solidiﬁcation. For example, almost wholly crystallized

magma near a pluton margin can be squeezed by em-

placement (“ballooning”) of additional magma within

the pluton interior, causing ﬂattening of the margin

and rotation of inequant mineral grains into an orien-

tation more or less parallel to the pluton contact with

its wall rocks; this foliation can resemble that created

by magma ﬂow along the wall rock contact during ini-

tial intrusion.

Grain-Size-Graded Layering. An example of rhythmic

layers characterized by a gradation in grain size is seen

FLOW PATTERN

IN MAGMA

(velocity vector

envelope)

CRYSTAL

MARKERS

(initially

random)

RESULTING MAGMA-

FLOW FABRIC IN

ROCK

Linear (lineated)

Planar and linear

Planar (foliated)

(a)

(b)

(c)

7.40 Schematic anisotropic fabrics produced by magmatic ﬂow. Each fabric is a compound function of the pattern of laminar ﬂow and asso-

ciated velocity gradients in the solidifying magma and the nature of the crystal ﬂow markers in it.

in the Duke Island complex in southeastern Alaska

(Irvine, 1987). In this composite ultramaﬁc intrusion,

several structures (Figure 7.43) typical of sedimentary

rocks, including cross- and graded-bedding and angu-

lar unconformities, testify to the action of currents of

low-viscosity melt depositing suspended pyroxene and

olivine crystals of varying size, in much the same way as

streams of water deposit sand grains on top of larger

pebbles.

Problematic Compositional Layers. It is tempting to in-

fer a similar control on the origin of layers of contrast-

ing mineral or modal composition: that is, composi-

tional layering. If larger particles sink faster than

smaller particles in a melt under the inﬂuence of grav-

ity, then one would likewise expect that denser parti-

cles would sink faster than less dense particles.

Chromite grains should sink more rapidly than less

dense olivines of the same size. Many petrologists in-

terpret compositional layers in terms of such gravita-

tional sorting. However, in a study of compositional

layers in the ultramaﬁc Stillwater intrusion in Montana,

Jackson (1961) found that small chromites near the

base give way upwards into larger olivines. Because set-

tling velocity (given by Stokes’s law in Section 8.3.3) is

proportional to the ﬁrst power of the density contrast

178 Igneous and Metamorphic Petrology

7.41 Folded ﬂow layering in a rhyolite lava ﬂow. Folded layers were

formed where the ﬂowing lava experienced local changes

in the ﬂow velocity regime. (Photograph courtesy of Glenn

Embree.)

7.42 Mingled rhyolite-andesite pumice block from the pyroclastic

ﬂow that created the Valley of Ten Thousand Smokes, Alaska,

in 1912. The crude ﬂow layering was created as the two

magma types mingled in the volcanic conduit beneath No-

varupta volcano. (Sample courtesy of Richard V. Beesley.)

between crystals and melt and to the square of the

grain size, the chromite-olivine layers are upside-down.

For these and other reasons some petrologists have re-

jected any control by gravity and sought other explana-

tions for compositional layers.

Rhythmic isomodal layers having a uniform modal

composition that differs from that of adjacent layers

(Figure 7.44) also have a problematic origin, as do

rhythmic layers having gradational modal proportions

from top to bottom (Figure 7.45b). Upward increasing

plagioclase and decreasing maﬁc minerals are common

in modally graded layers. Grain-size variations are gen-

erally absent but igneous lamination is common. Layers

may be of uniform thickness, ranging from centimeters

to meters, that extend for hundreds of meters and

much more (Figure 7.45a) or they may be curved and

lenticular, extending only several meters before disap-

pearing into isotropic rock (Figure 7.45b). Numerous

ﬁeld photographs in Parsons (1987) illustrate the vari-

ety of layering in many intrusions.

Kinetic Paths and Fabric of Magmatic Rocks

179

(a)

(b)

7.43 Sedimentarylike structures in the Duke Island ultramaﬁc com-

plex of southern Alaska. (a) Size-graded layers in which the

base is dominantly large clinopyroxene crystals and minor

smaller olivines; pyroxenes decrease in size upward. Pocket

knife for scale. (b) Angular unconformity in layers of olivine

clinopyroxenite.

7.44 “Inch-scale” layering in gabbro, Stillwater Complex, Mon-

tana. Incredibly uniform, rhythmic layers are mostly dark py-

roxene alternating with mostly lighter plagioclase. Hammer

for scale. (Photograph courtesy of A. R. McBirney.)

(a)

7.45 Rhythmic modal layering in the gabbroic Skaergaard intru-

sion, east Greenland. (a) Panoramic view of mountain face

about 300 m high showing lighter plagioclase-rich layers. (b)

Lenticular layers enriched in darker maﬁc minerals at base

grading upward into lighter-colored, plagioclase enriched

tops. (Photographs courtesy of G. M. Brown.)

(b)

180 Igneous and Metamorphic Petrology

An + melt

Melt

Di +

melt

7.46 Possible nucleation phenomena in the binary system

CaMgSi

-CaAl

near its eutectic point. Solid lines are

the liquidus surfaces in the binary system (Figure 5.4); dashed

lines show schematically the undercooling needed to achieve

optimal homogeneous nucleation of diopside and anorthite.

The metastable liquidus of diopside is the ﬁne-dashed line ex-

tending to the right of and below the eutectic point E. The

track of a cooling melt with delayed nucleation of diopside and

ﬁnally plagioclase is shown by the dotted line.

02mm

7.47 Comb layering in a diabase dike. Note curved, branching crys-

tal morphologic characteristics in lighter-colored pyroxene-

rich layers that alternate with olivine- and plagioclase-rich lay-

ers. (Photomicrograph of a sample provided by S. Agrell and

H. Drever courtesy of G. E. Lofgren. Reproduced with per-

mission from Lofgren GE, Donaldson CH. Curved branching

crystals and differentiation in comb layered rocks. Contrib.

Mineral. Petrol. 1975;49:309–319.)

The fact that the most widespread and best devel-

oped modal layering occurs in gabbroic intrusions and

in some silica-undersaturated syenitic ones suggests

that low melt viscosity is a controlling factor. Flow of

the magma, gravity sorting of grains, or both, may be

involved, possibly in concert with other processes. For

layered gabbroic intrusions, gravitational sorting and

settling of crystals of differing densities, possibly in

conjunction with convective movement of the magma

across the ﬂoor of the chamber, are controversial

possibilities. Assertions made decades ago that each

modal layer originates in intrusion of new magma can

generally be rejected. This requires an unlikely repeti-

tion of special events—in some intrusions there are

hundreds of layers in the rhythmic sequence—that

does not explain the unique internal properties of each

layer.

During his classic pioneering investigations of the

Skaergaard intrusion, Greenland, Wager (1959) real-

ized that different nucleation rates for different miner-

als precipitating from a melt might explain modally

graded layers that appear to have originated through

differential gravitative settling. Hawkes (1967; see also

Maaløe, 1978) invoked this mechanism for rhythmic

layers as thick as 100 m in the Freetown Complex of

Sierra Leone in Africa, where the ideal sequence from

the base upward in each layer is olivine  Fe-Ti oxides,

olivine  plagioclase, olivine  pyroxene  plagio-

clase, and plagioclase alone. Multiply saturated gab-

broic magmas can coprecipitate all of these minerals

under equilibrium conditions. However, these phases,

as listed, apparently have decreasing ease of nucleation,

as discussed in Section 6.5.2. This can affect the crys-

tallization history of the magma, possibly yielding a se-

quence of rhythmic modally graded layers.

Crystallization kinetics in a basaltic magma can be

modeled by the simple binary system CaMgSi

CaAl

, in which diopside and anorthite should

coprecipitate from a eutectic melt under equilibrium

conditions. The eutectic melt E in Figure 7.46 (see also

Figure 5.4) may not begin to crystallize diopside until

it has undercooled to D, where it experiences optimal

nucleation. As diopside crystals grow, the melt be-

comes warmer in an adiabatic model system, because

of release of latent heat, and richer in the anorthite

component. Hence, the melt moves to P, where it

nucleates anorthite rather than diopside. Growth of

anorthite causes the melt now to track back toward the

eutectic point E. Depending on possible upward ex-

pulsion of residual melt and gravity settling of crystals,

the product of this crystallization excursion would

be an upward increase in concentration of anorthite

and decrease in diopside within each modally graded

layer. Further cooling of the melt E could repeat the

process, possibly creating alternating pyroxene- and

plagioclase-rich layers like those in Figure 7.44.

Kinetic Paths and Fabric of Magmatic Rocks

181

7.48 Maﬁc inclusions, or enclaves, in granodiorite, Alta, Utah.

Note thin subparallel leucocratic aplite dikes above author’s

wife.

Comb Layering. Long, branching, skeletal to feathery,

subparallel crystals that are oriented perpendicular to a

planar boundary of some sort deﬁne comb layering

(Figure 7.47). Varieties of speciﬁc occurrences or min-

eral compositions include crescumulate texture, har-

risitic layering, spinifex texture, Willow Lake layering,

and orbicular fabric. Commonly, the comb layers are

separated from one another by more granular isotropic

textured rock. Apparently, these represent parts of the

melt crystallized at low undercooling or supersatura-

tion, whereas the comb layers are created by substan-

tial supersaturation (Donaldson, 1977).

7.10 INCLUSIONS

Maﬁc inclusions, or enclaves, have attracted consider-

able attention and spawned long-lasting controversy.

They are maﬁc aggregates embedded in a more felsic

host rock and are especially common in granodiorite,

quartz diorite, and tonalite (Figure 7.48). The minerals

are essentially the same in the inclusion and its host;

only their modal proportions differ. Inclusions are gen-

erally ﬁner-grained than the host; grain shapes and mu-

tual relations are variable. Inclusions are polygenetic

and can originate in at least ﬁve ways:

1. As xenoliths, fragments of foreign country rock

around the intrusion caught up into the magma.

Existing minerals in the fragments recrystallize and

equilibrate with the magma, forming new minerals

like those stable in it

2. As restite, pieces of the crystalline residue of the

magma-generating process in the lower crust car-

ried in the ascending magma

3. As cognate inclusions, or autoliths, partially to

wholly crystallized parts of the intrusion, perhaps

lifted away from the margin of the intrusion

4. As cumulophyric accumulations of maﬁc minerals

5. As blobs of partially to completely recrystallized,

mantle-derived basaltic magma mingled into the

more felsic host magma; the basaltic magma may

have provided the heat for felsic magma generation

In any of these possible origins, magmatic condi-

tions were not conducive to complete assimilation, or

digestion, of the maﬁc inclusion into the magma.

SUMMARY

An inﬁnite variety of kinetic paths during solidiﬁcation

of magmas create a virtually inﬁnite variety of rock fab-

rics. Several time-dependent factors dictate the kinetic

paths that are involved in crystallization (or the lack of

it), textural equilibration, exsolution of volatile ﬂuids,

fragmentation of magma, and development of aniso-

tropic fabrics. These factors include the rate of change

in the intensive parameters of the magma system and

the triad of transport processes in the magma—transfer

of heat, transfer of mass by atomic diffusion, and trans-

fer of momentum in viscous ﬂow (Figure 6.1). A dom-

inant role is played by widely ranging melt viscosity,

which impacts rates of diffusion, crystal and bubble

nucleation and growth, and magma ascent. Because of

the complex interactions of these kinetic factors, accu-

rate interpretations of the origin of fabrics must utilize

careful and critical observations integrated with exper-

imental and theoretical information.

Glass forms by rapid undercooling of melts, bypass-

ing crystallization. All glasses are metastable at near-

atmospheric conditions and, over millions of years, hy-

drate and devitrify into more stable crystalline phases.

Grain size in a magmatic rock unmodiﬁed by tex-

tural equilibration, fragmentation, or other secondary

processes is proportional to the ratio of crystal growth

rate to nucleation rate. Aphanitic grain size reﬂects a

high rate of nucleation coupled with low growth rates

appropriate to rapid cooling of extruded magmas and

margins of shallow, small intrusions. Phaneritic grain

size is characteristic of deeply emplaced, slowly cooled

magmas that have low nucleation densities at small de-

grees of undercooling. The bimodal size distribution of

porphyritic aphanitic and vitrophyric rocks typically

reﬂects two different rates of cooling. However, a uni-

form cooling rate combined with contrasts in the nu-

cleation rates of different minerals, loss of volatile ﬂuid,

and inheritance of xenocrysts and restite crystals can

also be factors in the development of porphyritic tex-

ture.

Euhedral shapes of mineral grains—columnar, platy,

tabular, and so on—grown slowly without restriction in

slightly undercooled melts reﬂect their crystallographic

lattice, whereas restricted growth in highly crystalline

magmas creates subhedral to anhedral grains. Freely

grown crystals in melts are increasingly less compact

and have greater surface areas at increasing degrees

of undercooling, or rate of cooling, creating skeletal,

dendritic, feathery, and ﬁnally spherulitic morphologic

characteristics.

After their initial growth, crystals can adjust their

shapes and increase their sizes via Ostwald ripening,

which minimizes the surface free energy of the aggre-

gate. This textural equilibration is more likely to occur

in more slowly cooled magmas.

Unstable crystals are dissolved in a melt. Partial dis-

solution creates anhedral, embayed grains that may be

surrounded by a more stable reaction rim. Chemically

zoned solid solutions, exempliﬁed especially by plagio-

clase grains, are created in crystallizing magmas, where

rates of diffusion are so slow that the system cannot

equilibrate as intensive parameters change.

Exsolution of volatiles from melts in decompressing

and crystallizing magmas creates vesicular and vuggy

fabric in rocks.

Volcaniclasts are produced by autoclastic breakup

of solidifying margins of lava ﬂows and by explosive

pyroclastic fragmentation of magmas where they inter-

act with external surface water (hydromagmatic activ-

ity) and where volatiles exsolve from the melt as the

magma decompresses or crystallizes. Volcaniclasts are

classiﬁed according to whether they are juvenile pieces

of the magma or accidental rock and crystals; whether

they are crystals, lithic fragments, or vitroclasts of

quenched melt; and according to their size. Finer vol-

caniclasts (ash and lapilli) are produced in more ener-

getic explosive pyroclastic eruptions, whereas less ex-

plosive activity yields larger volcaniclasts (blocks and

bombs).

Deposits of vitroclasts that remain sufﬁciently hot

undergo compaction and welding, in addition to possi-

ble devitriﬁcation and vapor-phase crystallization.

Anisotropic fabric is geometrically planar, linear, or

both planar and linear and is expressed by preferred

orientation of inequant mineral grains and mineral ag-

gregates and by, for planar fabric, various types of

rhythmic textural and compositional layering. Layering

is polygenetic, and the origin of a particular sequence

can challenge the petrologist’s powers of observation

and ability to conceive viable physical-chemical mod-

els. Possible mechanisms include magma ﬂow, crystal

sedimentation in low-viscosity melts, and crystalliza-

tion controlled by differing ease of nucleation in differ-

ent minerals.

CRITICAL THINKING QUESTIONS

7.1 Contrast texture, structure, fabric, and ﬁeld re-

lations.

7.2 Describe the different fabrics of glassy rocks

and indicate their origin.

7.3 List all possible fabrics that can develop from a

granitic (rhyolitic) magma. Comment on these.

7.4 Thoroughly discuss the origin of the fabric of

the andesite in Figure 7.8.

7.5 Suppose there are no groundmass olivines in the

basalt in Figure 7.9. Propose two possible ori-

gins for the porphyritic fabric. What bearing

does your answer have on the order of crystal-

lization of olivine from the magma with respect

to the other constituent minerals?

7.6 In Figure 7.11, why might the spodumene crys-

tals have a preferred subvertical orientation,

rather than being random?

7.7 Propose an origin for the complex plagioclase in

Figure 7.19.

7.8 What contrasts between the fabrics of rhyolites

and basalts are dependent on the difference in

the viscosity of their melts?

7.9 Critically discuss factors controlling grain shape

in magmatic rocks. What evidence would allow

the different factors to be distinguished?

7.10 What are possible mechanisms for the origin of

igneous lamination? Do they differ from those

for trachytic texture? Discuss evidence for con-

trasting mechanisms.

7.11 How does ﬂow layering develop in extruded

lavas? What evidence could be used to decide

which process was involved in a particular rock?

7.12 How might oscillatory zoned plagioclases

formed by kinetic processes in the growth of

an individual crystal be distinguished from

zoned crystals formed by cyclic ﬂuctuations in

intensive variables throughout the magma

chamber?

7.13 How might reverse zoning develop in plagio-

clase?

7.14 Can a fused tuff and a welded tuff be distin-

guished in hand sample? In the ﬁeld? Discuss.

7.15 Examine the variety of phaneritic fabrics evi-

dent in several samples of granitic rock. Pol-

ished rock slabs are an excellent means for

study and can be examined on a “poor-man’s

ﬁeld trip” in a cemetery, a commercial monu-

ment or tile establishment, and facades on

buildings. Describe and sketch the variety of

fabrics, and discuss how it might be possible for

such a wide variety to originate between differ-

ent granitic intrusions and even within a single

intrusion (e.g. Clarke, 1992, Fig. 1.1).

182 Igneous and Metamorphic Petrology

UNDAMENTAL

UESTIONS

ONSIDERED IN

HIS

HAPTER

1. How do magmas and rocks respond to geologic

forces, formulated as states of stress?

2. What physical and compositional properties

of bodies of magma are most important in

controlling their dynamic behavior?

3. How do dynamic physical processes and heat

transfer interact in magmatic systems?

4. How do these physical-thermal interactions

inﬂuence the compositional evolution of magmas?

INTRODUCTION

Chapters 6 and 7 dealt mostly with atomic-scale chem-

ical dynamics and the resulting kinetic paths in solidi-

fying melts that control the evolution of magmatic fab-

ric. This chapter and the next two deal with larger-scale

physical and thermal dynamics of bodies of magma, in-

cluding their response to states of stress, and the ways

they ﬂow, cool, and convect. The basic driving force of

magma movement in the Earth is buoyancy. Acting

against the buoyant force is viscous resistance to ﬂow.

As viscosity is partly dependent on magma tempera-

ture, the thermal evolution of the magma system be-

comes important.

Crystallizing and convectively cooling magma bod-

ies evolve through time, unmixing into compositionally

contrasting parts by fractionation of crystals and melt.

Mixing of magmas of contrasting composition depends

on their contrasting densities and viscosities. Magmas

are contaminated by assimilation of wall rock if sufﬁ-

CHAPTER

Physical and

Thermal Dynamics

of Bodies of Magma

cient thermal energy and time are available. Hence,

chemical differentiation of magmas is intimately inter-

twined with their dynamic physical and thermal be-

havior.

8.1 STRESS AND DEFORMATION

8.1.1 Concepts of Stress

In buried rock and magma bodies, forces arise as a

result of the weight of overlying rock material and of

tectonism, mostly associated with lithospheric plates

pushing together or a plate being stretched apart.

Stress is the magnitude of a force, divided by the area

over which it is applied. Hence, both stress and pres-

sure have the same units of pascals or bars. When the

same force acts over a smaller area, a greater stress or

pressure results.

Suppose a body is acted upon by forces (Figure 8.1)

so that it is in a state of stress. On any arbitrarily ori-

ented planar surface through the body, there will be, in

the general case, one normal stress, , perpendicular to

the plane and two shear stresses, 

and 

, perpendic-

ular to one another and tangential (parallel) to the

plane. These three orthogonal (mutually perpendicu-

lar) stresses constitute the components of the total

stress on the plane. For a particular state of stress, the

magnitude of these three surface stresses varies with re-

spect to the orientation of the plane through the body.

The general state of total stress in the body at a point,

which may be considered to be an inﬁnitesimally small

cube, is speciﬁed by these three surface stresses on

each of three orthogonal planes, parallel to the faces of

the cube, or nine stresses altogether. However, it can be

proved mathematically that it is always possible to ori-

ent a coordinate system so that no shear stresses exist

on the three orthogonal planes parallel to the faces of

the cube, reducing the number of stresses needed to

deﬁne the total state of stress to just three normal

stresses. These three normal principal stresses therefore

represent the total state of stress at a point in the body

and are designated with subscripts according to their

magnitude, 



. The specially oriented or-

thogonal planes on which only the principal stresses,

and no shear stresses, prevail are called principal

planes. Because shear stresses cannot be sustained on

the interface between the solid Earth and its atmo-

sphere, this more or less horizontal interface is a prin-

cipal plane and the other two principal planes perpen-

dicular to it must therefore be vertical. Hence, two

principal stresses near the surface of the Earth are

commonly considered to be horizontal and the third

vertical.

Within the Earth, compressive states of stress press

material together. Tensile stresses act in opposite di-

rections to pull material apart.

A special state of total stress in which principal

stresses are of equal magnitude in all directions, 







, is referred to as a hydrostatic state of stress,

because such states occur in bodies of water or other

liquids. This hydrostatic stress, also called pressure, is

approached at increasing depths in the Earth where

rocks have diminishing strength and ﬂow in the same

way as viscous liquids, especially on long geologic time

scales. This assumption is implicit in the formulation of

conﬁning pressure, P (Section 1.2), which is due to

gravitational loading, which is referred to throughout

this textbook as one of the three intensive thermody-

namic variables that deﬁnes states of equilibrium in

rock systems. This assumption was also made in calcu-

lating the geobaric gradient (equation 1.8).

Despite the obvious utility of the assumption of hy-

drostatic states of stress in rock systems, a nonhydro-

static state of stress is more typical and its effects on the

dynamics of magma bodies are one focus of this chap-

ter. In a nonhydrostatic state of stress the principal

stresses are unequal, 



 

8.1.2 Deformation

Deformation is the way a body responds to applied

stress. The character of the deformation depends on

the state of stress and the properties of the body.

Individual particles, such as mineral grains, within a

stressed rock body are displaced or changed in shape

or size until a new state of more stable equilibrium

between the imposed forces and the body is attained,

according to Le Chatelier’s principle. The body, or

parts of it, in its ﬁnal deformed state has experienced

one or more of the following components of defor-

mation:

1. Translation (movement, or displacement) from one

place to another with respect to Earth coordinates

(e.g., in a thrust sheet)

2. Rotation, as in the limbs of a fold

3. Distortion, or strain, which can consist of a change

in shape and/or volume

A nonhydrostatic state of stress can produce all of

these components in folding, faulting, and other types

of deformation of rocks in the Earth. On the other

hand, a hydrostatic state of stress can generally only

produce a change in volume; for example, a magnetite

grain is compressed into a smaller grain of the same

shape.

In rocks, strain can only be measured with respect

to a reference marker whose shape and/or size is

known in the initial, undeformed state. Reliable mark-

ers in rocks are difﬁcult to ﬁnd; a more or less ellip-

soidal basalt pillow (Figure 7.3a) is a possible marker,

but its exact shape and size are uncertain. Strain, , is

measured in terms of the change in some dimensional

or angular aspect of the ﬁnal deformed body relative to

the initial undeformed body. Examples of measures of

strain include volumetric strain

184 Igneous and Metamorphic Petrology

8.1 State of stress in a three-dimensional body. Force vectors (ar-

rows) acting on the body create the state of stress. On an arbi-

trarily oriented plane (shaded) through the body, forces acting

on the area of the plane can be resolved into mutually perpen-

dicular components of normal, , and shear stresses, 

and 

The normal stress, , is perpendicular to the plane, and the

two shear stresses, 

and 

, are parallel to the plane.

8.1 





(V 

)











and linear strain, either an elongation or a ﬂattening,

8.2 





(l 

)











where V

and l

represent the initial undeformed

state and V and l the ﬁnal deformed state. Note that

strain is a pure number without units, such as 0.45, or,

as a percentage change, 45%.

In Figure 8.2, a cubic volume of isotropic rock,

which has uniform properties in all directions, is sub-

jected to a nonhydrostatic state of stress, 

 

Consequently, it is ﬂattened perpendicular to 

(the

maximal compressive principal stress) and elongated

parallel to 

(the least compressive principal stress),

which is the direction of least work of deformation.

Note that a nonhydrostatically stressed body extends in

the directions of least and perhaps intermediate princi-

pal stress but not in the direction of the maximal com-

pressive principal stress, 

, as that would require the

maximum of work to be done on the surroundings. Na-

ture, being lazy and parsimonious, prefers the least-

work alternative.

8.1.3 Ideal Response to Stress

Stress and strain are mathematical concepts that apply

to any material. The response of real rocks and magmas

to applied stress can be complex and depends upon

many factors, as will be discussed in the next section.

But before dealing with real behavior we ﬁrst consider

ideal, or “end-member” elastic, plastic, and viscous be-

havior, which have simple relations between applied

stress and resulting strain.

Elastic behavior is the only ideal response to applied

stress that is recoverable, or reversible; the deformed

material returns instantaneously to its initial unde-

formed state when the stress is eliminated (Figure

8.3a). Rubber bands, springs, sheets of window glass,

and rocks under certain conditions behave elastically.

There is a linear, direct proportionality (Hooke’s law)

between stress and elastic strain,   E, where E,

Young’s modulus, is a property of the material. The

value of E is the slope of the stress-strain line in Figure

8.3a. As the applied stress increases, the material

reaches its elastic strength, 

, and the body ruptures

or breaks, losing cohesion and causing permanent de-

formation. If a pane of window glass is subjected to a

small applied stress, it bends slightly and reversibly, but

if it is stressed more to exceed its elastic strength, it

fractures into pieces.

Elastic strains of glass, minerals, and rocks are much

smaller than stretching of a rubber band. Small elastic

volumetric strains in geologic solids are a consequence

of their very small elastic compressibility. This prop-

erty, formally deﬁned in Section 8.3.1, is a measure of

how much a material body compacts under pressure;

even large pressure changes yield only small changes in

volume in most rocks, on the order of 1%.

Physical and Thermal Dynamics of Bodies of Magma

185

Extensional

fracture

Extensional fracturing

(cracking) and magma

injection

Elastically deformed

state in a nonhydro-

static stress field

Undeformed, initial

state

Tip

MAGMA

8.2 Deformation of a body under a nonhydrostatic state of stress, 



 

. (a) Undeformed body represented by a cubical stack of

smaller cubes. (b) Principal stresses of unequal magnitude cause a change in shape—a strain. Actual recoverable elastic strain produces

only a small percentage of strain, far less than is represented here for illustrative purposes. However, the relative amounts of strain are as

expected, that is, ﬂattening perpendicular to the maximal principal compressive stress, 

, and elongation, or extension, parallel to the

least principal compressive stress, 

. Strain parallel to 

can be either elongation or ﬂattening or nil. (c) Nonhydrostatic stresses may

exceed the brittle (essentially elastic) strength of a rock so that an extensional fracture forms perpendicular to the least principal com-

pressive stress, 

. These open cracks may be subsequently ﬁlled with magma or ﬂuid. The pressure exerted by the magma or ﬂuid itself

may be sufﬁcient to create an extensional fracture by hydraulic fracturing. Just beyond the injecting ﬂuid or magma that wedges apart

the walls of rock is a tip cavity where transient low pressure can suck out volatiles dissolved in the magma or pore ﬂuids lodged in the

wall rock.

occurs. Under some conditions, real minerals and

rocks behave more or less plastically, but the stress-

strain response is continuously curved and a yield

stress can only be approximately determined.

The ideal viscous response to applied stress was

discussed in Section 6.1 (see also Figure 6.1a) with ref-

erence to bodies of melt. Unlike elastic and plastic bod-

ies, a viscous body has no strength and deforms per-

manently by ﬂow under the smallest applied shear

stress. This property is typical of liquids. As shear stress

is applied, a proportional strain rate results (Figure

8.3c). Such linear ﬂow behavior is referred to as New-

tonian viscosity and the proportionality constant is the

viscosity, .

8.2 RHEOLOGY OF ROCKS

AND MAGMAS

Real rocks and magmas generally deviate, commonly

substantially, from the three ideal modes of deforma-

tional behavior just discussed. The real deformational

behavior of rocks and magmas—their rheology—is a

combination of the ideally elastic, plastic, and viscous

end-member responses that are expressed either simul-

taneously or under particular conditions. For example,

basalt magmas near their liquidus T where few if any

crystals exist exhibit Newtonian viscous behavior,

whereas with increasing crystallinity the magma rheol-

ogy becomes non-Newtonian and the magma possesses

yield strength; its behavior is a compound of viscous

and plastic, or viscoplastic. Accordingly, the applied

stress must exceed a critical yield strength in order for

permanent viscous ﬂow to occur. For an applied shear

stress less than this critical yield strength no permanent

deformation occurs. Highly crystalline magmas near

their solidus temperatures can fracture more or less

elastically as wholly crystalline rock does. Bodies of

folded rock layers (Figure 8.4) exemplify compound

viscous-elastic behavior. They are viscoelastic. At high

T over long periods the compositional layers were

folded as if they had low viscosity. Yet if it had been

possible to strike the hot body with a hammer when the

folds were developing, it would have fractured into

pieces. Thus, rocks and magmas have a duality of be-

havior that differs with circumstances.

Viscoelastic behavior illustrates the importance of

the time rate of strain, /t, which has units of

reciprocal seconds (s

1

). A rock subjected to a large

strain rate, in which strain accumulates quickly in a

small time, such as during a blow with a hammer, re-

sults in an elastic response; the rock breaks into pieces.

On the other hand, slower strain rates allow a greater

amount of permanent but slowly developing viscous

ﬂow to occur. The viscoelastic mantle of the Earth

transmits elastic seismic waves at speeds measured

in kilometers per second (fast strain rate) yet also

186 Igneous and Metamorphic Petrology

In contrast to elastic behavior, ideal plastic response

involves a nonrecoverable, or irreversible, strain occur-

ring at a stress equal to a critical value known as the

yield strength, 

(Figure 8.3b). For applied stresses

less than the yield stress, only reversible elastic strain

ELASTIC

Stress σ

σ = Eε

Strain ε Stress ε Strain

Time

(a)

(b)

Stress σ

PLASTIC

Strain ε Strain ε Strain ε

Rock

Stress σ

Stress τ

Strain rate ε Stress Strain

τ = ηε

Time

8.3 Deﬁnitions and comparisons of three ideal types of response

to applied stress. For each type of ideal behavior, the relation

between stress and strain is shown in the left-hand diagram

and a mechanical analog is shown in the diagram on the right;

the middle diagrams show special attributes. (a) Elastic be-

havior in which stress is linearly proportional to strain. If the

stress exceeds the elastic limit, or elastic strength, 

, the ma-

terial breaks permanently. Another attribute of elastic be-

havior is that strain is instantaneous and nonpermanent (re-

versible). This is shown in the middle diagrams, where a par-

ticular stress, 

, is applied between time t

and t

resulting in

an immediate accompanying strain whose magnitude is /E.

After t

when the stress is eliminated there is no longer any

strain. Elastic behavior is modeled by a spring. (b) Plastic be-

havior in which no strain occurs until the applied stress

reaches the yield strength, 

, after which permanent strain

can accumulate indeﬁnitely. The mechanical model of plastic

deformation is a block resting on a table. An applied force can-

not move (deform) the block until the frictional resistance (the

yield strength) is overcome. Before the stress reaches the yield

strength in a material, some amount of reversible elastic strain

may result (second diagram); in this case, removal of the stress

after an excursion through elastic and then plastic behavior is

tracked by the dashed line. A typical response of a stressed

rock (third diagram) shows initial elasticlike behavior grading

continuously into more plasticlike behavior. The yield strength

can only be approximated and the plastic-like region shows

“strain hardening” in which increasing stress is required to in-

duce more strain. (c) Newtonian viscous behavior in which

the time rate of shear strain, d/dt 



, and shear stress, , are

linearly proportional. The proportionality constant is the coef-

ﬁcient of viscosity, . Any applied shear stress between time t

and t

results in an immediate, permanent strain that increases

until the stress is removed. After t

when the stress is elimi-

nated the strain that had accumulated at time t

remains per-

manently. A mechanical model of viscous response is the dash-

pot—a cylinder containing a viscous liquid, such as honey, in

which moves a loose-ﬁtting piston.