basics of elasticity - helsinki

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Geodynamics Lecture 5

Basics of elasticityLecturer: David Whipp [email protected]

!16.9.2014

1

mailto:[email protected]

Goals of this lecture

• Introduce linear elasticity

!

• Look at the applications of elasticity to several simple stress/strain scenarios

!

• Provide the background needed to consider flexure of the lithosphere

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Toward rheology

• At this point, we’ve talked about how to describe stress and strain and a few examples of measurements that can be made in rocks

• For the rest of this lecture and the following two, our focus will turn to the connections between stress and strain

• The relationship between stress and strain (or strain rate) are given by constitutive equations that describe how rock deforms when a given force is applied (also known as rheological laws)

• Today, we start with elasticity

3

What parts of the Earth are elastic?

• Essentially all rocks are elastic under the right conditions

• Elastic: Low temperature, low deviatoric stress

• Brittle (fracture): Low temperature, high deviatoric stress

• Lecture 11 (7.10)

• Viscous (flow): High temperature

• Lecture 12 (9.10)

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Flexure of the lithosphere in foreland basins

5

Tectonic loading and subsidence of intermontane basins: Wyoming foreland province

E. Sven Hagen, Mark W. Shuster Department of Geology and Geophysics, University of Wyoming, Laramie, Wyoming 82071

Kevin P. Furlong Department of Geosciences, Pennsylvania State University, University Park, Pennsylvania 16802

ABSTRACT Results of two-dimensional flexural modeling of the northern Bighorn and northern

Green River basins in the Wyoming foreland province suggest that these basins formed as flexures in response to loading by basin-margin uplifts and basin sedimentary sequences. The northern Bighorn Basin subsided due to loading by the Beartooth uplift along its western margin. The northern Green River Basin developed as a result of concurrent loading by the Wyoming thrust belt to the west and the Wind River uplift to the east. Tectonic loading from basement-involved uplifts played a major role in subsidence and sedimentation, as evidenced by isopach patterns within each basin. Lithospheric flexural rigidities of 1021 to 1022 newton metres (N-m) can adequately explain subsidence in both basins.

INTRODUCTION The Laramide basins of the Wyoming fore-

land province are deep, asymmetric, structural depressions bounded in part by large-scale basement-involved uplifts representing several kilometres of structural relief (Fig. 1). These basins contain thick sequences of Upper Cre-

BEARTOOTH MOUNTAINS

taceous and Tertiary strata whose isopach patterns reflect the tectonic history of basin subsidence and adjacent uplifts. With the ex-ception of Keefer's (1965) work, little has been done to analyze the nature of subsidence in Laramide-style basins. The term "Laramide-style basins" here refers to those formed by

basement-involved structural deformation of the Wyoming foreland province during the Lar-amide orogeny that occurred from Campanian to Eocene time.

In contrast to rifted/passive margin basins that evolve primarily from thermal effects, Jor-dan (1981) and Beaumont (1981) suggested that the Cretaceous foreland basin of North America formed in flexural response to tectonic and sediment loading on an elastic (Jordan) or viscoelastic (Beaumont) lithosphere. The spatial and in part temporal superposition of Laramide-style basins within the Wyoming foreland province may suggest a similar mech-anism of subsidence.

In this study we examine the hypothesis that Laramide intermontane basins develop within a flexural depression caused by the loading effects

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A. NORTHERN BIGHORN BASIN 10 km

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WIND RIVER MOUNTAINS

Tertiary S Mesozoic

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B. NORTHERN GREEN RIVER BASIN Modified from Royse, Warner, and Reese, 1975

0 10 20 30 km

Top of Upper Cretaceous Mesaverde Formation

Figure 1. Generalized cross sections and location map for northern Bighorn and northern Green River basins showing major extrabasinal load configurations (structural elements) used to generate "present-day" flexural profiles shown in Figure 3. Dark dashed lines in each cross section represent base of present load configuration (datum) within each basin. Geographic reference points correspond to Figure 3.

GEOLOGY, v. 13, p. 585-588, August 1985 585

on September 15, 2014geology.gsapubs.orgDownloaded from

of basement uplifts (tectonic loading) and sed-imentary infill. In particular, the flexural model is applied to the northern Bighorn and northern Green River basins of western Wyoming where we think the development of the basins is closely tied to basement tectonics. Figure 1 il-lustrates the structural cross sections that were used to analyze the effects of both tectonic and sediment loading on the development of Laramide-style basins.

MODELING APPROACH We have analyzed the flexural response of

an elastic lithosphere to loading from mass ex-cess associated with basement uplift and sedi-mentary fill in the basins. The lithosphere is modeled as a loaded two-dimensional infinite elastic beam buoyed up by a fluid substratum following the approach of Hetenyi (1946, p. 10-22) and Jordan (1981). Although there is disagreement regarding elastic versus visco-elastic behavior of the lithosphere (cf. Karner et al., 1983; Beaumont, 1981), the duration of the loading events considered in this study (~ 10 m.y. for basement uplifts, >30 m.y. for basin fill) is large compared to the relaxation time of the lithosphere (Willet et al., 1985; Courtney and Beaumont, 1983). Hence, it is difficult to see the effects of viscoelastic relaxation within the lithosphere from the sedimentary record (Courtney and Beaumont, 1983). Thus, we assume that the lithosphere can be effectively modeled as behaving elastically.

The flexural behavior of an elastic plate is a function of two factors: lithospheric flexural ri-gidity (which depends on the mechanical prop-erties of the lithosphere) and the specific load distribution. Flexural rigidity controls the char-acteristic wavelength and relative amplitude of the flexure; thus, the response of the lithosphere to a load depends only on the flexural rigidity (for assumed mantle and crustal densities) (Fig. 2). To calculate the flexure, we determine the mass excess (load) acting on each unit length (10 km) of the profiles (Fig. 2). This mass ex-

Figure 2. Flexural responses for arbitrary line load using lithospheric flexural rigidities of 1021 (21), 1023 (23), and 1025 (25) newton-metres (N-m) and mantle density (p) of 3.4 g cm'3. Line load has height of 5 km, width of 10 km, and density of 2.5 g cm'3.

586

cess can result from either added materials (e.g., sediments in basin, overthrust blocks) or lateral density variations from basement uplifts jux-taposing higher density crustal rocks with lower density supracrustal rocks. Sedimentary thick-nesses and basin geometries provide the pri-mary geologic constraints for the models (Fig. 1). The distribution of loads is normalized with respect to density and then convolved with the flexural response from a unit load to give the overall flexure (Beaumont, 1981).

We have used two complementary ap-proaches to test our hypothesis of flexural de-velopment for Laramide-style basins. The first approach ("present-day modeling") examines the conditions under which the present-day configuration of loads (basement uplifts and basin fill) can generate the observed present-day geometry—i.e., determining appropriate values for flexural rigidity. The second ap-proach ("sediment thickness profiling" or STP) examines the volume and distribution of basin fill consistent with presumed initial load condi-tions (uneroded basement uplifts) and com-pares that with the observed and estimated extent of sedimentary units. Ideally, one would like to remove the effects of sedimentary load-ing (i.e., "backstrip" the basin) and compare the remaining subsidence with that predicted from initial loading conditions (Watts and Ryan, 1976). However, such an approach re-quires assumptions on the flexural rigidity op-erative at varying times in the basins' evolution and the source regions for the basin infill (i.e., determine to what degree sediments are derived from the uplifted basement blocks). These parameters are difficult to determine in Laramide-style basins. Our approach evaluates end-member cases, allowing us to bracket the mechanical properties of the lithosphere and test the reasonableness of our assumption of flexural development.

Present-day profiling determines the average mechanical conditions under which the basin evolves. Included in the present-day basin geometry are the effects of initial load geome-try, initial flexural rigidity, the effects of subse-quent loading, load redistribution, and unloading (erosion) under varying flexural pa-rameters. The degree to which present-day conditions are matched by this average me-chanical model provides constraints on the var-iability in lithospheric mechanical properties in time.

Sediment thickness profiling is a means of examining more extreme conditions of basin evolution. We assume that virtually all of the basin fill is derived externally to the basin-load system. Hence, we can evaluate the maximum amounts of basin fill possible with flexural evolution. These results are compared with observed basin fill and estimates of paleo-geometries for the basin. This aspect of the modeling provides constraints on the flexur-

al parameters of the lithosphere at the time of loading. We would expect that this modeling will overestimate the basin fill because we have used maximum values for thrust load (un-eroded) and have assumed that the basin is 100% filled. Lack of geologic constraints pre-cluded the use of time-dependent models, ne-cessitating accurate estimates of uplift and ero-sion rates for progressive load redistribution.

GEOLOGIC SETTING Northern Bighorn Basin

The Beartooth Mountains are a major struc-tural element of the northern Bighorn Basin, having structural relief of about 7 km. The up-lift is adjacent to a deep asymmetric structural depression, and the basin axis is overthrust by Precambrian basement rock on the order of 3 km (Foose et al., 1961) to as much as 12 km (Bonini and Kinard, 1983). As the profile in Figure 1 illustrates, the Beartooth block proba-bly controlled both the structural configuration and post-Laramide (i.e., Fort Union Forma-tion) sedimentation of this region. Fault dips were assumed to be 45° from observed geology (Foose et al., 1961). Laramide deformation began in the Bighorn Basin area by late Cam-panian time. However, the intensity of orogenic activity and uplift of the Beartooth Range peaked in the Paleocene (Bown, 1980).

Northern Green River Basin The northern Green River Basin lies between

the Idaho-Wyoming thrust belt to the west, the Wind River Range to the east, and the Gros Ventre Range to the north and northeast (Fig. 1). The basin is asymmetric and deepens to the east; the basin axis is subparallel to the Wind River thrust fault. The area evolved from a retroarc foreland trough related to the develop-ment of the Idaho-Wyoming thrust belt during Late Jurassic and Cretaceous time (Jordan, 1981) to an ir.termontane basin during latest Cretaceous and early Tertiary time with the development of the Gros Ventre and Wind River uplifts. Structural studies of the over-thrust belt and Wind River thrust are presented by Royse et al. (1975), Dixon (1982), Berg (1961), Smithson et al. (1978), and Zawislak and Smithson (1981). The generalized cross-sectional geometry of the northern Green River Basin and its bounding structures is shown in Figure 1. A dip of 40° was assumed to repre-sent the Wind River thrust fault geometry (Za-wislak and Smithson, 1981).

FLEXURAL MODELS Present-day Profiles

Present-day profiling examines the deflection of a stratigraphic datum resulting from the present load configuration in a basin. Calcu-lated deflection profiles for a given flexural rigidity are compared to the present basin geometry. Flexural rigidities used in the models

GEOLOGY, August 1985

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Hagen et al., 1985

Flexure of the lithosphere Hawaii

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© Nature Publishing Group1985

Watts et al., 1985

© Nature Publishing Group1985

Inter-seismic strain accumulation

• Plate tectonic motions lead to the gradual build up of elastic strain near fault zones

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• Rather than being localized to the fault, this deformation is distributed over large areas (>10 km from the fault zones)

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• During an earthquake, slip on the faults leads to unrecoverable deformation and releases (some of) the stored elastic stress

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236˚ 237˚ 238˚ 239˚ 240˚ 241˚ 242˚ 243˚ 244˚ 245˚

33˚

34˚

35˚

36˚

37˚

38˚

39˚

236˚ 237˚ 238˚ 239˚ 240˚ 241˚ 242˚ 243˚ 244˚ 245˚

33˚

34˚

35˚

36˚

37˚

38˚

39˚

creeping

section

1906 rupture1857 rupture

PacificOcean

Figure S1. Shaded relief map of California. Pink lines denote sections of the San Andreasfault that ruptured in great earthquakes in 1857 and 1906. A red line denotes the southernpart of SAF that did not produce a major earthquake in historic times. Black wavy linesshow other geologically mapped faults. A white box outlines the study area shown in Figure 1in the main text.

4

© 2006 Nature Publishing Group

connecting the southern termination of the San Jacinto fault and theSuperstition Hills fault (see dashed red line in Fig. 1). In the case ofthe SAF, one might argue that the fault is dipping to the east26,27,based on the fact that seismicity occurs several kilometres off the faulttrace28. The hypothesized alternative locations of shear zones drivingthe interseismic deformation at the brittle–ductile transition aredenoted by dashed green lines in Fig. 2. Under these assumptions,it is possible to explain the data equally well without appealing tovariations in the effective shear modulus of the crust (see dashed redline in Fig. 2). In this case, the inferred slip rates are 25mmyr21 and19mmyr21 and the locking depths are 12 and 10 km for the SanAndreas and San Jacinto faults, respectively.The assumed position of the southern San Jacinto fault (Fig. 1) is

supported by a lineament of microseismicity28, yet the absence of anactive fault trace is puzzling. Partly, such absence might be explainedby alluvial burial from the ancient Lake Cahuilla29. However, it is notclear whether the corresponding fault segment remained quiescentover the 400 years since the lake retreat29, or if it could be a ‘blind’strike-slip fault19. For the SAF, the fault dip angle required by thehomogeneous model is ,608. A steeper fault implies a greaterrigidity contrast. Ultimately, the proposed interpretations of inter-seismic strain accumulation admit a robust observational test. Therigidity contrast model predicts that if future major earthquakesoccur on subvertical ruptures coincident with the mapped traces ofthe San Andreas and San Jacinto faults, the resulting coseismicdisplacement field should be essentially asymmetric, with displace-ments on the eastern side of a fault being at least two to three timeslarger than displacements on the western side of a fault. Ruptureshaving the proposed alternative fault geometries will be directevidence against large rigidity contrasts. These results suggest that,in general, information about coseismic deformation on a givenfault—such as asymmetries in the radiation pattern, and staticdisplacement fields18,27—may greatly reduce uncertainties in theinterpretation of the interseismic deformation data.Regardless of the details of fault geometry and mechanical proper-

ties of the ambient crust, results presented in this study lend supportto intermediate-term forecasts of a high probability of major earth-quakes on the southern SAF system2,7. Space geodetic data shown inFigs 1 and 2 clearly demonstrate that the southern SAF is accumu-lating significant elastic strain, as indicated by a broad area of high

gradients in the LOS velocity field on the eastern side of the fault.Although some creep may be occurring in the uppermost crust4, asevidenced by a steep gradient in the LOS velocity immediately to thewest of the surface trace of the SAF (Fig. 2), it does not prevent (and ifanything, enhances) a build-up of tectonic stress on the rest of thefault at seismogenic depths. I point out that the step-like increase inthe radar range across the SAFmay not be entirely due to right-lateralfault creep, and probably involves ground subsidence to the west ofthe fault. Without subsidence, the observed variations in the LOSvelocity would imply left-lateral deformation within the decorrelatedarea and immediately to the west of Salton Sea (at ,120–130 kmalong the profile A–A 0 , see Fig. 2), which is highly unlikely. Theinferred ground subsidence may be either man-made (for example,due to agricultural activities in the Coachella Valley), or of tectonicorigin (for example, indicating secular deepening of the SaltonTrough). Although it might be possible to discriminate betweenthe horizontal and vertical components of deformation using com-plementary InSAR data from ascending orbits, unfortunately nosuitable acquisitions are available from the ERS satellites.The current slip rate on the southern SAF of 25 ^ 3mmyr21

determined from the space geodetic data is in excellent agree-ment with some long-term geologic estimates (for example,25 ^ 4mmyr21; ref. 13), although other geologic studies maysuggest lower rates10. The discrepancy between different geologicdata sets might be due to along-fault variations in the mechanicalbehaviour of the uppermost crustal layer. For example, zones ofapparently low slip rates might represent substantial inelastic yield-ing of a shallow layer in the interseismic period, either in the formof creep4, or more distributed failure19. It is reasonable to assume thatthe geodetically determined slip rate remained relatively constant inthe recent geologic past, and, in particular, over the past severalhundred years. It follows that the relative seismic quiescence of thesouthern SAF over the past 250 years implies a slip deficit of 5.5–7m.This may be compared to palaeoseismological estimates of averagerecurrence times of large events on the southern SAF of 200–300years, and average coseismic displacements of 4–7m (refs 1, 2).Although simple time- and size-predictable earthquake models havebeen shown to be inadequate30–32, and the repeat interval betweenlarge earthquakes may vary significantly, it may be argued thatthe accumulated slip deficit cannot greatly exceed the maximum

Figure 2 |Average LOS velocities (grey dots) andGPS/EDMdata (colouredsymbols) projected onto the satellite LOS from the profile A–A 0 shown inFig. 1. Vertical bars denote the 2j errors of the pointmeasurements. Verticallines denote positions of the mapped fault traces (solid lines), andhypothetical positions of interseismic creep at the bottom of the brittle layer(dashed lines). Solid red line is a theoretical model of interseismic strain

accumulation due to a deep slip below themapped traces of the southern SanAndreas and San Jacinto faults in the presence of lateral variations in thecrustal rigidity. Dashed red line is a theoretical model of the same parameterfor the inferred alternative fault positions, assuming no lateral variations inthe rock rigidity.

LETTERS NATURE|Vol 441|22 June 2006

970

Fialko, 2006

Elasticity

8

�xx

= E"xx

!

• Stress is proportional to strain

• For 1-D normal stress

• If stress → 0, strain → 0 (recoverable)

𝐸 : Young’s modulus (1D)

E or 2µ

�n or �s

"n or "s

�n or �s

"n or "s

𝜇 : Shear modulus (1D)

�xx

= E"xx

Twiss and Moores, 2007

� / "

Normal stress Shear stress

Elasticity

9

�xx

= E"xx

!





�n or �s

"n or "s

�n or �s

"n or "s

𝐺 : Shear modulus (1D)


� / "

�xx

= E"xx

E or 2G


Elasticity

10

�xx

= E"xx

!





�n or �s

"n or "s

�n or �s

"n or "s



� / "

�xx

= E"xx

E or 2G


Elasticity

11

�xx

= E"xx

!





�n or �s

"n or "s

�n or �s

"n or "s


� / "

�xx

= E"xx


E or 2G


Linear elasticity in terms of stress

• For linear elasticity, stress is linearly proportional to strainwhere 𝜆 and 𝐺 are known as the Lamé parameters

• 𝐺 is also known as the modulus of rigidity or shear modulus

• Note, we assume the material properties are isotropic

12

�1 = (�+ 2G)"1 + �"2 + �"3

�2 = �"1 + (�+ 2G)"2 + �"3

�3 = �"1 + �"2 + (�+ 2G)"3

Linear elasticity in terms of stress

• For linear elasticity, stress is linearly proportional to strainwhere 𝜆 and 𝐺 are known as the Lamé parameters

• Principal strain 𝜀 produces a stress (𝜆 + 2𝐺)𝜀 in the same direction and stresses 𝜆𝜀 in the perpendicular directions

13

�1 = (�+ 2G)"1 + �"2 + �"3

�2 = �"1 + (�+ 2G)"2 + �"3

�3 = �"1 + �"2 + (�+ 2G)"3

Linear elasticity in terms of strain

• We can also formulate the equations of linear elasticity in terms of principal strainswhere 𝐸 is known as Young’s modulus and 𝜈 is Poisson’s ratio

• In this case, principal stress 𝜎 produces a strain of 𝜎/𝐸 in the direction it acts and strains of -(𝜈𝜎/𝐸) in the perpendicular directions

14

"1 =1

E�1 �

⌫

E�2 �

⌫

E�3

"2 = � ⌫

E�1 +

1

E�2 �

⌫

E�3

"3 = � ⌫

E�1 �

⌫

E�2 +

1

E�3

Material properties of common rock types

15

B.5 Properties of Rock 813

B.5 Properties of Rock

Density E G k α

kg m--3 1011 Pa 1011 Pa ν Wm--1 K--110--5 K--1

SedimentaryShale 2100–2700 0.1–0.7 0.1–0.3 0.1–0.2 1.2–3Sandstone 1900–2500 0.1–0.6 0.04–0.2 0.1–0.3 1.5–4.2 3Limestone 1600–2700 0.5–0.8 0.2–0.3 0.15–0.3 2–3.4 2.4Dolomite 2700–2850 0.5–0.9 0.2–6.4 0.1–0.4 3.2–5

MetamorphicGneiss 2600–2850 0.4–0.6 0.2–0.3 0.15–0.25 2.1–4.2Amphibole 2800–3150 0.5–1.0 0.4 2.1–3.8Marble 2670–2750 0.3–0.8 0.2–0.35 0.2–0.3 2.5–3

IgneousBasalt 2950 0.6–0.8 0.25–0.35 0.2–0.25 1.3–2.9Granite 2650 0.4–0.7 0.2–0.3 0.2–0.25 2.4–3.8 2.4Diabase 2900 0.8–1.1 0.3–0.45 0.25 2–4Gabbro 2950 0.6–1.0 0.2–0.35 0.15–0.2 1.9–4.0 1.6Diorite 2800 0.6–0.8 0.3–0.35 0.25–0.3 2.8–3.6Pyroxenite 3250 1.0 0.4 4.1–5Anorthosite 2640–2920 0.83 0.35 0.25 1.7–2.1Granodiorite 2700 0.7 0.3 0.25 2.0–3.5

MantlePeridotite 3250 3–4.5 2.4Dunite 3000–3700 1.4–1.6 0.6–0.7 3.7–4.6

MiscellaneousIce 917 0.092 0.31–0.36 2.2 5

𝐸 : 0.1 - 1.6 ⨉ 1011 Pa (typical values: 10-100 GPa) 𝐺 : 0.04 - 6.4 ⨉ 1011 Pa 𝜈 : 0.1 - 0.4 (typical values 0.1 - 0.3)

Uniaxial stress

16

• Uniaxial stress occurs when only one of theprincipal stresses is nonzero (𝜎1 for this example)

• If 𝜎2 = 𝜎3 = 0, the equations for linear elasticity in terms of strain reduce to

• In this case, the equation above can be simplified further to yield Hooke’s law

"2 = "3 = � ⌫

E�1 = �⌫"1

�1 = E"1

Uniaxial stress

17

• We can also find the change in volume of the rock parcel or dilatation 𝛥 from the principal strains

� = "1 + "2 + "3 = "1(1� 2⌫)

Uniaxial stress

18

• We can also find the change in volume of the rock parcel or dilatation 𝛥 from the principal strains

� = "1 + "2 + "3 = "1(1� 2⌫)

What is Poisson’s ratio for an incompressible material?

Quartzite under uniaxial compression

19

http://www.controls-group.com

http://www.controls-group.com/

Quartzite under uniaxial compression

20

http://www.controls-group.com

What is happening here?

http://www.controls-group.com/

Stresses as a result of burial

• How does elastic stress change in sedimentary rocks as a result of burial?

!

• What stress/strain conditions are appropriate for this scenario?

21

Uniaxial strain

• Uniaxial strain occurs when only one component of the principal strains is nonzero (𝜀1 in this example)

• In this case, if we consider 𝜀2 = 𝜀3 = 0, the equations for linear elasticity reduce to

22

�2 = �3 =⌫

(1� ⌫)�1

�1 =(1� ⌫)E"1

(1 + ⌫)(1� 2⌫)

Uniaxial strain

• Let’s consider a parcel of rock initially at the surface that now has been buried by sediments of density 𝜌 to a depth ℎ

• In this case, we can assume 𝜎1 is vertical and equal to the weight of the overburden, 𝜎1 = 𝜌𝑔ℎ

• From the equations on the previous slide, we find

23

�2 = �3 =⌫

(1� ⌫)⇢gh

Uniaxial strain

• Let’s now consider the effects on the deviatoric stress, the principal stresses minus pressure 𝑝

!

• Which results in deviatoric stresses

24

p =1

3(�1 + �2 + �3) =

(1� ⌫)

3(1� ⌫)⇢gh

�01 = �1 � p =

2(1� 2⌫)

3(1� ⌫)⇢gh

�02 = �2 � p = �0

3 = �3 � p = � (1� 2⌫)

3(1� ⌫)⇢gh

Uniaxial strain

• Let’s now consider the effects on the deviatoric stress, the principal stresses minus pressure 𝑝

!

• Which results in deviatoric stresses

25

p =1

3(�1 + �2 + �3) =

(1� ⌫)

3(1� ⌫)⇢gh

�01 = �1 � p =

2(1� 2⌫)

3(1� ⌫)⇢gh

�02 = �2 � p = �0

3 = �3 � p = � (1� 2⌫)

3(1� ⌫)⇢ghUnder tension!

Pure shear and simple shear

• Plane stress occurs when only one principal stress is zero (𝜎3 = 0)

• Pure shear is a special case of plane stress

• Consider the case with 𝜎3 = 0 and 𝜎1 = -𝜎2

• In the case on the left with an angle 𝜃 = -45° between 𝜎1 and the 𝑥 axis, we find

• In this case, the equations for plane stress conditions yield

26

�xx

= �yy

= 0 and �xy

= �1

"1 =(1 + ⌫)

E�1 =

(1 + ⌫)

E�xy

= �"2


• Plane stress occurs when only one principal stress is zero (𝜎3 = 0)

• Pure shear is a special case of plane stress

• Consider the case with 𝜎3 = 0 and 𝜎1 = -𝜎2

• In the case on the left with an angle 𝜃 = -45° between 𝜎1 and the 𝑥 axis, we find

• In this case, the equations for plane stress conditions yield

27

�xx

= �yy

= 0 and �xy

= �1

"1 =(1 + ⌫)

E�1 =

(1 + ⌫)

E�xy

= �"2


• Similar to the stresses, with an angle 𝜃 = -45° between 𝜎1 and the 𝑥 axis, we find

• From the equation on the previous slide for we see

!

• If we recognise that the modulus of rigidity 𝐺 can be found fromthen we finally see the simple relationship

28

"xx

= "yy

= 0 and "xy

= "1

�xy

=E

1 + ⌫"xy

G =E

2(1 + ⌫)

�xy

= 2G"xy




!


29

"xx

= "yy

= 0 and "xy

= "1

�xy

=E

1 + ⌫"xy

G =E

2(1 + ⌫)

�xy

= 2G"xy




!


30

"xx

= "yy

= 0 and "xy

= "1

�xy

=E

1 + ⌫"xy

G =E

2(1 + ⌫)

�xy

= 2G"xy

Note this works for both pure and simple shear

Isotropic stress

• If all three principal stresses are equal, 𝜎1 = 𝜎2 = 𝜎3 ≡ 𝑝 and the state of stress is isotropic

• In this case, the principal strains are also equal, 𝜀1 = 𝜀2 = 𝜀3 = 1/3𝛥

• From the equations of elasticity in terms of stress we findwhere 𝐾 is the bulk modulus and its reciprocal 𝛽 is the compressibility

31

p =

✓3�+ 2G

3

◆� ⌘ K� ⌘ 1

��

Isotropic stress

• If all three principal stresses are equal, 𝜎1 = 𝜎2 = 𝜎3 ≡ 𝑝 and the state of stress is isotropic

• In this case, the principal strains are also equal, 𝜀1 = 𝜀2 = 𝜀3 = 1/3𝛥

• From the equations of elasticity in terms of stress we findwhere 𝐾 is the bulk modulus and its reciprocal 𝛽 is the compressibility

32

p =

✓3�+ 2G

3

◆� ⌘ K� ⌘ 1

��

Isotropic stress

• Any change in volume of rock must conserve mass

• For a parcel of rock with volume 𝑉 a change in volume 𝛿𝑉 will result in a change in density 𝛿𝜌, or

• In terms of dilatation of the rock parcel 𝛥 we can say assuming 𝛥 is small

• Thus, the density change as a function of pressure, compressibility and initial density is simply

33

�(⇢V ) = 0

��V

V= � =

�⇢

⇢

�⇢ = ⇢�p

Isotropic stress

• Any change in volume of rock must conserve mass

• For a parcel of rock with volume 𝑉 a change in volume 𝛿𝑉 will result in a change in density 𝛿𝜌, or

• In terms of dilatation of the rock parcel 𝛥 we can say assuming 𝛥 is small

• Thus, the density change as a function of pressure, compressibility and initial density is simply

34

�(⇢V ) = 0

��V

V= � =

�⇢

⇢

�⇢ = ⇢�p

Isotropic stress

• Finally, both the bulk modulus and compressibility can be found from the elastic properties of rock

35

K =1

�=

E

3(1� 2⌫)

Isotropic stress

• Finally, both the bulk modulus and compressibility can be found from the elastic properties of rock

!

• What does this suggest about rock (in)compressibility as a function of Poisson’s ratio 𝜈?

36

K =1

�=

E

3(1� 2⌫)

Recap

• Linear elasticity describes the linear relationship between stress and strain in rocks under low deviatoric stress and at low temperatures

!

• The elastic properties of rock, combined with assumptions about the stress acting on the rock parcel can be used to predict elastic deformation

!

• The elastic properties of rock suggest rocks are actually reasonably compressible (𝜈 = 0.1 - 0.3)

37

References

Fialko, Y. (2006). Interseismic strain accumulation and the earthquake potential on the southern San Andreas fault system. Nature, 441(7096), 968–971. doi:10.1038/nature04797

!Hagen, E. S., Shuster, M. W., & Furlong, K. P. (1985). Tectonic loading and subsidence of intermontane basins:

Wyoming foreland province. Geology, 13(8), 585. doi:10.1130/0091-7613(1985)13<585:TLASOI>2.0.CO;2 !Watts, AB, Brink, Ten, U. S., Buhl, P., & Brocher, T. M. (1985). A multichannel seismic study of lithospheric

flexure across the Hawaiian–Emperor seamount chain. Nature, 315(6015), 105–111.

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basics of elasticity - helsinki

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