GY403 Structural GeologyThe general equations of the Mohr Circle for strain

Strain EllipsoidA three-dimensional ellipsoid that describes the magnitude ofdilational and distortional strain

PAssume a perfect sphere before deformationPThree mutually perpendicular axes X, Y, and ZPX is maximum stretch (SX) and Z is minimum

stretch (SZ)PThere are unique directions corresponding to

values of SX and SZ, but an infinite number ofdirections corresponding to SY

X

YZ

StrainThe results of deformation via distortion and dilation

PHeterogeneous strain: strain ellipsoid varies frompoint-to-point in deformed body

PHomogenous strain: strain ellipsoid is equivalentfrom point-to-point in deformed body

PAlthough hetereogenous strain is the rule in realrocks, often portions of a deformed body behaveas homogenous with respect to strain

Homogeneous Strain “Ground Rules”Characteristics of homogenous strain

PStraight lines that exist in the non-rigid bodyremain straight after deformation

PLines that are parallel in the non-rigid bodyremain parallel after deformation

P In a special case of homogenous strain termed“Plane Strain”, volume and area are conserved

General Strain EquationsExtension (e), Stretch (S), and Quadratic Elongation (λ)

These equations measure linear strain :

lF -lO

lOe =

S =

lF

lOλ =

lF

lO

2

lF = final lengthlO = original length

lO = 5cm

lF = 12cmS = =

12cm5cm

= 2.4

ouch!

lF

lO

e = (S-1) = 2.4 - 1 = 1.4λ = S2 = (2.4)2 = 5.76

Rotational Strain Equationsquantifying angular shear (ψ) and shear strain (γ)

X

Z θd=-25

Strain ellipse

LL’

M’ θ=-35

ψL (perpendicular to L relative to M) = -40

ψ

deformation

γL = tan(ψL) = tan(-40) = -0.839

θ = angle between reference line (L) and maximum stretch (X)measured from X to A (clockwise=+; anticlockwise=-)

αL = θd - θ = (-25) - (-35) = +10angle of internal rotation

X’

M

M

Mohr Circle for StrainGeneral equations as a function of λX, λZ, and θd

λ’= λ’Z+λ’X -λ’Z-λ’X cos(2θd)2 2

γ λ’Z-λ’X sin(2θd)2

tan θd = tan θSX

SZ

α = θd - θ(internal rotation)

λ’ = 1λ

λX = quadratic elongation parallel to X axis of finite strain ellipse

λZ = quadratic elongation parallel to Z axis of finite strain ellipse

λ =

Mohr Circle for StrainGeometric relations between the finite strain ellipse andthe Mohr Circle for strain

λX’ 2.0

1.0

-1.0

A

X

Z

1.414

θd=+30

1.0

0.816

2θd=60

Strain ellipseA

λ’ λZ’λ’

λX = (Sx)2=(1.414)2=2.0

λX’= 1/λ= 1/2.0 = 0.5

λZ= (Sz)2 = (0.816)2=0.666

λZ’= 1/λ = 1/0.666 = 1.50

SX = (lF/lO)=(1.414/1.0)=1.414SZ = (lF/lO)=(0.816/1.0)=0.816

Mohr Circle for StrainReference lines in the undeformed and deformed state

abcde f gh

i

j

k

lmnopqrs

a b c d e f g hi

j

klmnopqrs

SX=1.936SZ=0.707

Mohr Circle Strain Relationships

Values of quadratic elongation (λ), shear strain (γ), original θ angle,angular shear (ψ), and angle of internal rotation (α) as a function ofθd

Line

abcdefghijklmnopqrs

θd

-90-80-70-60-50-40-30-20-100102030405060708090

λ

0.5000.5130.5560.6380.7791.0171.4282.1293.1343.7483.1342.1291.4281.0170.7790.6380.5560.5130.500

γ

-0.000-0.152-0.310-0.479-0.665-0.868-1.072-1.187-0.9290.0000.9291.1871.0720.8680.6650.4790.3100.1520.000

θ

-90.0-86.3-82.4-78.1-73.0-66.5-57.7-44.9-25.80.025.844.957.766.573.078.182.486.390.0

ψ

-0.0-8.7-17.2-25.6-33.6-41.0-47.0-49.9-42.90.042.949.947.041.033.625.617.28.70.0

α

0.06.312.418.123.026.527.724.915.80.0-15.8-24.9-27.7-26.5-23.0-18.1-12.4-6.30.0

SX=1.936SZ=0.707

a b c d e f g hi

j

klmnopqrs

Strain Ellipse General EquationValues for quadratic elongation (λ) and shear strain (γ) as a functionof θd

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-90 -70 -50 -30 -10 10 30 50 70 90

a b c d e f g hi

j

kl m n o p q r s

d

Strain Ellipse General EquationValues for angular shear (ψ) and internal rotation (α) as a function of θd

-60

-40

-20

0

20

40

60

-100 -80 -60 -40 -20 0 20 40 60 80 100 θd

ab

cd

e f g hi

j

kl m n o

pq

rs

Internal Rotation(α) Angular Shear(ψ)

Example strain problemGiven a finite strain ellipse of SX=1.936 and SZ=0.707,find for direction θd=-20E values of S, λ, γ, ψ, and α

λX=(1.936)2 = 3.750; λZ = (0.707)2 = 0.500; λ’X=0.267; λ’Z=2.0

2.0+0.267 - 2.0 - 0.267 Cos(-40) = 1.133-(0.866)(0.766) = 0.4702 2

λ’=

λ = 1/λ’ = 1/0.470 = 2.128 ˆ S = (2.128)0.5 = 1.459

γ =2.0-0.267

2Sin(-40) λ = (0.866)(-0.643)(2.128) = -1.185

ψ = tan-1(γ) = tan-1(-1.185) = -49.8E

Tan(θd) = tan(θ)SZ

SX ˆTan(θ) = tan(θd)

SX

SZ ˆ θ = -44.9E

α = θd -θ = (-20) - (-44.9) = +24.9E

Application of Plane StrainDeformed oolids from the study of Cloos (1947)Assuming plane strain: no dilational component to strain, therefore, constantvolume applies:

VS= 4/3πr3 where r is the radius of the sphereVe= 4/3πabc where (a,b,c) are the ½ axial legths of the ellipsoidVs=Ve

4/3πr3 = 4/3πabcBecause of plane strain r = b ˆr2 = acr = (ac)0.5

Example: a=4.2mm; c=2.5mm; r=(4.2*2.5)0.5 = 3.3 ˆ Sx = 4.2/3.3 = 1.27

Application to Deformed Strain Markers

PMarkers may be originalspheres or ellipsoids

PPebbles, sand grains,reduction spots, ooids,fossils, etc.

PAssume homogenousstrain domain

Measuring Length/Width Ratios (R f)

PMeasure major and minor axis of eachstrain ellipse

PRf = (Major/minor) (yields a unitless ratio)Pφ = Angle from reference direction

(usually foliation or cleavage), positiveangles are clockwise, negativecounterclockwise

Ellipse Length Width Rf φ1 0.3066 0.1600 1.916 32.82 0.0969 0.0704 1.376 51.93 0.1221 0.0729 1.675 61.84 0.0660 0.0389 1.697 54.65 0.1735 0.1392 1.246 22.76 0.0825 0.0539 1.531 76.27 0.1770 0.1275 1.388 67.58 0.0736 0.0347 2.121 37.69 0.0937 0.0797 1.176 ‐0.310 0.1184 0.0457 2.591 10.6

Spreadsheet setup for Rf/ φ analysis

P Note: φ is measured relative to a chosen reference direction suchas foliation

Hyperbolic Net

P Used to plot strainmarkers that wereoriginally ellipsoidal

P Statistically the Rf ratioswill tend to fall along oneof the hyperbolic curves

Nφ=+30

Rf=1.6