Displacement Vector Field and Strain The Techniques Of Modern Structural Geology Volume 1:Strain Analysis Session 4:

INTRODUCTION The aim of this session is to widen our view by studying more general type of displacement and the various strains that arise from them. Many different type of coordinate transformations lead to a variety of patterns of displacement vectors joining the original and final positions of certain points in the body. The aim of this session is to widen our view by studying more general type of displacement and the various strains that arise from them. Many different type of coordinate transformations lead to a variety of patterns of displacement vectors joining the original and final positions of certain points in the body.

Displacement vector Displacement vector: The vector joining the positions of a particle in the undeformed configuration (original position) and deformed configuration (final positions) is called the Displacement vector.The vector has 2 components defined as u abs = x' –x and v abs = x' –x Parallel to the x and y coordinate axes.

Displacement Vector Field Displacement vector field is the mathematical description of all the variously oriented and various valued vectors in a displaced body. If the coordinate transformation equation are given by: x' = f 1 (x,y) and y' = f 2 (x,y) The vector components defining the displacement vector field are given by: u = f 1 (x,y) –x and v= f 2 (x,y) –x

Type of displacement 1) Body translation 2) Body rotation 3) Simple shear 4) Pure shear 5) General homogeneous rotational strain 6) General heterogeneous strain

Rotational and irrotational strain

: Homogeneous and heterogeneous strain

Types of homogeneous deformation of a cube, circle, and sphere

Body translation In Body translation the displacement vectors are all parallel and of constant length throughout the body such a displacement plan moves the body in space without internal deformation or rotation. The coordinate transformation equation are: x'= x +A y'= y +B A, B are constants and defining the components u (parallel to x) and v (parallel to y) of the uniform displacement vector.

vo v u uo Body translation

Body rotation vary in direction length Body rotation: The displacement vector vary in direction and length with initial coordinate position. The effect here is the rotation around the origin (0,0) without internal deformation, through an angle -ω (negative sign for clockwise sense).The coordinate transformation for this rotation through an angle -ω is: x'= cos ωx + sin ωy y'= -sin ωx + cos ωy

Simple shear parallel unequal length homogeneous rotational. Simple shear: This displacement has parallel displacement vector but with unequal length. The transformation leads to a homogeneous strain because the shear gradient is constant across the body. The strain is rotational. If lines originally parallel to y axis are deflected through an angle Ψ, the coordinate transformation equations are x'= x + y tan Ψ y'=y

Pure shear shortening parallel to the y axis stretching parallel to the x axis varying in length and orientation Pure shear: This displacement has led to a homogeneous shortening parallel to the y axis and a homogeneous stretching parallel to the x axis. The displacement vectors varying in length and orientation through the body. The strain product is homogeneous and irrotational. The coordinate transformation equation are: x'= kx

General homogeneous rotational strain vector field is quite complex displacement vector gradient constant square shaped parallelogram homogeneous rotational This displacement vector field is quite complex, the vectors varying in length and direction with initial position. The displacement vector gradient through the body is constant. The vector field looks complex but the shape changes are simple and each originally square shaped element is transformed into a parallelogram. So the deformation is homogeneous. Because the direction of principle axes of the strain differ from those of the originally perpendicular lines which become the strain ellipse axes, the strain is rotational. The coordinate transformations are given by: x'= ax+by y'= cx+dy

General heterogeneous strain complicated gradient varying heterogeneous folding This displacement vector field is complicated.The displacement vector gradient varying in length and direction with initial position from point to point in the body. The shape changes are not the same and originally square shaped element is transformed into a shape that is not parallelogram. So the deformation is heterogeneous. Generally any initial set of straight parallel lines before deformation becomes curved. This type of deformation occur in situation of folding.

Ways of Studying distortion set 1.Focus attention on small elements in the body. So a heterogeneously strained body can be envisaged as an agglomeration of a large number of elemental units with homogeneous strain. 2.Examine mathematically 2.Examine mathematically how strains at a point relate to the general coordinate transformation equations.

The movement direction concept a direction: a direction: a terminology derived from coordinates applicable to simple shear displacement like card deck model. The direction of a is defined as the slip direction. ab is the slip plane and c is normal to ab. a b c Slip plane

Reference schemes to measure displacement The terms active, stable and direction of transport are generally impossible to determine, because all depend upon a knowledge of the absolute positions of parts of the crust before the movements took place. The termination of the absolute displacement vector depends upon a knowledge of the initial and the final position of a point relative to a fixed coordinate frame with unmoved origin so we can never measure absolute displacement.

In our study of the earth’s crust we have moving anchors. In this situation it is only possible to say how one point been displaced relative to another. There are 2 main types of moving anchors that we use for geological problems. Longitude 1. The displacement of one point relative to some other point acting as a moving origin for a new coordinate frame. Thus we relate all displacement relative to the that of some conveniently chosen point in the body.(Position of Longitude on the earth’s surface) Moving anchors

u + u o = u abs V + v o = v abs

Difference in vector component (either relative or absolute) of any 2 points and provides a measure of crustal shortening (difference in u component) and relative uplift (difference in v component).

Moving anchors displacement around any single point relative to that point 2- The second moving anchors makes use of the displacement which have taken place around any single point relative to that point. If an observer were stationed within the rock mass and carried along with the deform body, he would observe quite different displacement of the material surrounding him from those observe by an outsider at some fixed position.

The concept of local displacement is extremely important to the geologist it is closely connected with the concept of strain at a point. Example: a fossil embedded in a rock mass will often have its shape changed as a result of the local displacement. By measuring these changes it is possible to evaluate the strain at the point.

Rotational effect is our complication. It is not possible : 1. to know the initial position of any point and 2. to determine the origin of the initial coordinate frame. 3. to fix the initial direction of the x abs and y abs axes.(except where we have pale magnetic data). 4. To determine the correct orientation of the deformed mass of O ' relative to its original position. The direction of the x rel and y rel are chosen in any direction that is convenient. Complication of absolute and relative position of displaced rock mass

impossible 1. In geology it is impossible to measure the absolute displacement of any point Differentiate between active & passive elements in the crust Evaluate absolute body translation or absolute body rotation. possible 2. It is possible to Determine the relative displacement of any 2 points Compute value for crustal shortening and differential uplift between 2 point Evaluate the difference in body translation and body rotation between 2 points. possible 3. It is possible to Measure the changes in displacement vector in a local origin around a point. Define the state of strain around a point. Three important conclusion

The general coordinate transformation and strain principle strain e1,e2 θ') θ) The four linear coefficient of the coordinate transformation equation define a matrix known as the strain matrix and this matrix sets up distortions and rotations in the body. Because there are always four components to this matrix there are always four characteristic features of the strain state 2 are principle strain e1,e2.The third is the orientation of the strain ellipse axes (θ') and the fourth is the orientation of these lines before deformation (θ) related to the internal rotation component of the strain.(ω=θ'- θ)

 In order to determine the general form of a deformed ellipse centered at the origin given by: Ax 2 +2Bxy+Cy 2 =1 Using the eulerian form of the coordinate transformation equation

Which is the form of an ellipse centered at origin.  Two superposed homogeneous deformations combine to give a single homogeneous finite strain which may be analyzed using the strain ellipse concept.

 If we Transform the special ellipse given by: (a 2 +c 2 )x 2 +2(ab + cd) xy+(b 2 +d 2 ) y 2 =1 By Transforming the ellipse and simplifying the coefficient we find that the coefficient of x' 2, x' y', y 2 become 1,0,1 and the ellipse transform to : x' 2 + y' 2 =1 This is a equation of circle of a unit radius. reciprocal strain ellipse A ellipse that transform to a circle is known as reciprocal strain ellipse. Reciprocal strain ellipse Reciprocal strain ellipse

It has axial directions parallel to the lines in undeformed state which eventually become the axes of the finite strain ellipse and the value of this axes are the reciprocal value of those of the principle strains. reciprocal strain ellipse zero. It is possible for 2 finite strains to be superposed, if the first deformation is the reciprocal strain ellipse of the second deformation, the resulting strain become zero. Reciprocal strain ellipse Reciprocal strain ellipse

neutral strain surfaces  Many of neutral strain surfaces and neutral strain points arise by the build up strain during the early part of fold development which is subsequently removed during the later state of folding. aggregate of initially variably oriented elliptical objects conglomerate pebbles or oolites  When an aggregate of initially variably oriented elliptical objects such as conglomerate pebbles or oolites are subjected to deformation. It is not uncommon for certain of these objects to be initially shaped oriented like the reciprocal strain ellipse and in their deformed condition to take on a circular forms. examples Reciprocal strain ellipse

Question: Deform the circle with unit radius and centre (0,0) by successive transformations and then reverse the order of deformation? A. Simple shear (value γ)parallel to the x-axes followed by B. x' = X+γ y' =y B. Pure shear ( stretch value k parallel to x.shortening value 1/k parallel to y) using transformations x' = kx Commutative of strain

These strain ellipse are different. So the strains are not commutative.

The deformation matrix for simple shear followed by pure shear is derived by relating the fnal position of a point (x,y) to the original position (x,y) using 2 coordinate transformations. Since the deformation matrices are difficult, the strains are not identical.

Strain ellipse field and their geological significance Strain ellipse field and their geological significance

 e1>0, e2>o  Positive dilatation  Line of no finite longitudinal stain  All directions in the surface are stretched and develop boudin necks.  Field 1 can form during a single deformation of flattening type, or it can develop as the result of superposition of two or more phases of non-coaxial deformation.  Formation of complex boudinage with crossing boudin neck zones.  Formation of chocolate tablet boudinage. Field 1 :

Boudin necks If the strain arises during a Single deformation Irregular Boudin necks wider separation Sub perpendicular to the maximum extension direction.

Chocolate tablet Boudinage If the strain arises during a superposition of different deformations or different phases within one overall deformation often show periodic or sequential development. And the boudins often intersect at angles other than 90˚.

 e1>0, e2<o  Dominant shortening in one sector and dominant stretching in another.  Positive and negative dilatation  Competent layers involved in such a strain field will show sub perpendicular developments of boudins and buckle folds. Field 2:

 e1<0, e2<o  All directions in the surface show contraction.  Perpendicularly oriented fold axes.  Large negative dilatation Field 3:

 In one phase deformation : folds are highly irregular in shape(crumpled from all side). Fold axes and axial planes have many differing orientations. The intersections and margins of the folds will be unsystematic.  In two or more separate contractions: folds shows a more systematic interference geometry. Fold axes will be variable but ordered. The intersections and margins of the folds will be unsystematic.