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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