GG313 Lecture 14 Coordinate Transformations Strain

Coordinate Transformation One use of matrices is in describing deformation of materials under stress. When forces act on materials, the materials deform. This deformation is called STRAIN, and it can be described by changes in position of a particle inside a body. This change in position can be modeled by a coordinate transformation that moves the particle from one point to another with respect to a fixed point. First, we need to define how to measure changes.

The red lines show the location vectors with respect to the origin, and the blue line shows the direction and magnitude of how the particle moved during deformation. The point A has components x 1, y 1, and B has components x 2, y 2. The length or magnitude of the vectors is (x i 2 +y i 2 ) 1/2, and the angle each vector makes with the x axis is defined by tan(  i )=(y i /x i ). The angle between the two vectors is  b -  a, and the length of the displacement vector (blue) is [(xb-xa) 2 +(yb-ya) 2 ] 1/2.

We can define a matrix describing the location of three points in the x-y plane and add a third column of ones: It is important to note that if the determinant of the above matrix =0, then the three points lie on a straight line.

What about 4 points? If the similar matrix with 4 points and three dimensions has a zero determinant, then the four points must lie on a plane: Let’s check the above statement using four points that lie in a plane with unit coordinate values. This is easy: choose two points x 1,y 1,z 1, and x 1,y 2,z 2 (a straight line parallel to the x axis) two others x 3,y 3,z 3, and x 3,y 4,z 4 (another line parallel to the z-axis), and see if the determinant of the above equation is zero.

In geological situations, such a verification could be important - where you detect a particular layer in four wells for example, and want to know if the structure is a plane. In the true geological case, however, there is always noise, and even if the structure is planar, the determinant will likely not be equal to zero. Try the same calculation as above, but move one of the points slightly off the plane - say by 0.1 unit. How much does that change the determinant? How large can the determinant get? What if the four points are as far from a plane as you can get? What is the value of the determinant then? - (Such a structure with four points forms a tetrahedron.)

Back to strain: Let’s go to two dimensions for simplicity of visualization. Consider a particle that is deformed with respect to some other point (at the origin). The initial location of the particle is x,y, and the new location is x’,y’. Then the displacement of the particle is given by u=x’-x and v=y’-y. If the displacement is only in the x-direction, then y’=y, and v=0. We define the shear strain in this case as , where  =tan(  ), the angle between the initial and final locations with respect to the origin, and u=  y, v=0.

The strain matrix has normal strain components along the diagonal and shear strains on the off-diagonals. The strain matrix above is called simple shear: show Matlab. If the determinant of the strain matrix has particular values, we can say something about the area change of the deformation: Determinant valueResult of deformation 0one dimension is lost 1The area doesn’t change >1The area increases >0<1The area decreases

Shear strains change angles of vectors, normal strains preserve angles but change lengths. Simple shear rotates the whole body. Pure shear does not rotate the body. Normal strains are related to compressional waves in solids and liquids. Shear strains are related to shear waves. NEXT CLASS: We will go over two examples of the uses of eignevectors and eigenvalues.