Structural Geology

Structural Geology Deformation And The concept of strain - Frédéric Flerit

San Andreas Big Bend San Andreas south SouthCalifornia Los Angeles

Courtesy of J.P. Petit, Montpellier Les Matelles

We need a tool to Describe the deformation process Rigid Body : motion Already understood A A A A A Initial final Rotation + Translation Deformation

Three level to understands the earth processes • Geometry - position • kinematics - displacements • Dynamics – forces,

The level of the kinematics • Rigid body motion: • translation • Rotation • 2) deformation

Deformation • Can be : • Continous / not continuous • () • Homogeneous / not homogeneous • (identical troughout the material)

Two type of elementary deformation 1) Change in length : longitudinal strain dL X L = Linit Lfinal exx = Lfinal - Linit / Linit = dL / L

Two type of elementary deformation 2) Change in angle : shear strain Y dLy X Lx exy = dLy / Lx

The Matrix notation • exx eyx • exy eyy ) ( e =

NOTE The strain matrix is symetric g = eyx = exy • exx g • g eyy ) ( e =

Deformation of the vector P ? exxg geyy Matrix: e = (< given) P = Px Py (given > ) P

The matrix product allow to resolve the components of the strain For a given direction P exxg gexx Px Py . e . P = That is Deformation of the vector P : ep = exxPx + gPy gPx + eyy Py Along x Along y

Exercices :deform the above square and circle using the following strainssupposed uniform • exx = 0.5 • eyy = 0.2 • g = 0.5 • g = -0.5 • exx = 0.5 and eyy=-0.5 • eyy = 0.2 and g = 0.2 • g = -0.2 and exx = 0.5 and eyy=-0.5

REMEMBER Displacement -Velocities To measure the rigid motion of the plates or of individual points we use the concepts of : Strain - Strain rates The math object associated is a vector To measure the deformation of the crust or of the lithosphere we use the concepts of : The math object associated is a matrix

Question Define the volume change associated with a strain Given : To be defined : DV/V = ? exxg gexx e =

Question for next course: Diagonalize the strain matrix (2D) exx g g eyy e = Given : e1 0 0 e2 e = The base : (V1, V2) To be defined :

San Andreas Big Bend San Andreas south SouthCalifornia Los Angeles

y The transformation Matrix D PA exx g g eyy C’ T = = ? g = 10-4 [1/yr] NA x A B’

The matrix of the deformation 0 - g 0 0 T = This matrix is not symetric so this is not a strain matrix Note : for the deformation on earth : g = 10-4 << 1

The strain matrix Deformation = strain + other deformation ??? 0-g/2 -g/2 0 0-g/2 g/2 0 0 - g 0 0 T = = + T = e + r

Simple shear = pure shear + rotation 0g 0 0 0g/2 g/2 0 0g/2 -g/2 0 T = = + Symetric = STRAIN AntiSymetric = ROTATION

Surface change : DS/S S’ • Simple Shear : 2) Pure Shear : S

Surface change : DS/S (continued) DS/S = e (if e<<1) e

Y y x X Let’s diagonalize e Same strain , different Coordinate System 0-g/2 - g/2 0 e1 0 0 e2 e = = A

y x In (A,x,y) the shear strain is maximum 0-g/2 - g/2 0 A

Y y X x e10 0 e2 In (A,X,Y) the longitudinal strain are maximum D’ AY = B’D’ : dir of max extension AX = A’C’ : dir of max shortening A B’ AY and AY are also called the principal direction And are perpendicular AY

Y y X x Eigenvectors : (X,Y) Calculation of the principal strain e1 ande2 Eigenvalues : (e1, ,e2 ) e10 0 e2 D’ B’

Y y X x Eigenvectors : (X,Y) e10 0 e2 Calculation of the principal strain e1 ande2 Eigenvalues : (e1, ,e2 ) D’ g/2 0 0 -g/2 A’ B’ Convention : lengthening < 0

AS geoscientist we would like to have a representation wherewe have at the same time : • Maximum shear and its direction, maximum lonitudinal strain e And its direction …

y x the mohr diagram It represents the state of strain (shear vs longitudinal) at a given point A, for all the coordinate system (A,xa,ya) g ya Xa Shear in dir. AXa a a ga A + e ea Shortening in direction AXa

Y y X x D’ We know the state of strain in (Ax,y) and in (A,X,Y) lets plot it g 0 0 -g e10 0 e2 B’

The Mohr circleof the San Andreas fault (pure shear) g Max shear g + Max lengthening Max shortening e + + g=e2 -g=e1 B’

y x The signification of angle in the mohr circle ya Xa g a A g a = P/4 + + e 2a + + g=e2 -g=e1 a = 0 a = P/2 B’ a = 3P/4

Properties of the mohr circle Symetric with respect to the axe of the extension-compression The diameter measures the max shear stress. The position of the centre of the cercle correspond to (e1+e2)/2 which is the Average strain and corespond to ½ of the Relative surface change.

San Andreas Big Bend San Andreas south SouthCaliforniaplot the directions-of shortening-of lenghtening-of sheardoes it make sense ? Los Angeles

Analysis of the strain pattern on a piece of Rock

Where are veins and solution surfaces

Surface solution or stylolite Fissures filled with calcite

The mechanism of creation of veins and surface solutions : Dissolution recristalisation + Lengthening - - + Shortening Stylolites are created by calcite removal Effect -> overall shortening Fissures are created and filled by precipitation of calcite Effect -> overall lengthening

Draw veins and solution surfaces (is the deformation Homogenous?) • 2 Define the principal direction of extension • 3 Define the principal direction of compression • 4 define the principal direction of shear strain • 5 conclude

1- Draw the veins • 2- Define the principal direction of extension • 3- Define the principal direction of compression • 4- Draw the solution surfaces • 5- Define the max. direction of shear strain

Evaluate the average lenghtening strain ?And comment

3D state of strain… e e e

IN 3D the strain matrix becomes in the princpal reference frame: z exxexyexz eyx eyy eyz ezxezyezz y x

IN 3D the strain matrix becomes in the princpal reference frame: z e100 0 e2 0 0 0 e2 y x

The mohr diagrams becomes g g z y e x e1 e2 e3 e100 0 e2 0 0 0 e2

define : g12 g13 g23 g g13 z g23 y g12 e x e1 e2 e3 e100 0 e2 0 0 0 e2

Exercices

z The circle becomes an ellipsoid define the coresponding strain matrixn ( convention for earth science shortening positive) y x Deformaed geometry

Structural Geology