1.5 Lecture Notes 5 Quantities in Different Coordinate Systems How to express quantities in different coordinate systems? x 3 x 3 P Direction Cosines Axis φ 11 φ 3 φ 1 x x x x 3 11 1 13 x 1 3 x 3 31 3 33 Figure 5.1 Figure 5.1 Direction cosine i is cosine of angle φ i between primed axis i and unprimed axis. If x and xˆ represent unit vectors that are the axes of two coordinate systems with the ˆi same origin, they are related by the equation 3 3 x ˆ = x, ˆ x = x x i i i i i = 1 = 1 where i is the cosine of the angle between the primed axis x and the unprimed axis xˆ. For example, 1 is the cosine of the angle between xˆ1 and ˆx. i represents a 9- component matrix called the transformation matrix. Unlike the stress tensor, it is not symmetric ( i i ). ˆi
i = 11 1 31 1 3 13 3 33 In matix equations, the transformation law is written x1 = x x 3 11 1 31 1 3 13 x1 3 x 33 x3 The inverse transformation law is written xˆ i = xˆ i Consider the following transformation of coordinates: x x 3 o x 3, x 3 Figure 5.1a Figure 5.1a
The transformation matrix is i 3 = 1 6 3 The explicit transformation equations are xˆ = cos3 xˆ 1 xˆ = cos1 xˆ 1 + cos6 xˆ 1 + cos3 xˆ Since x and xˆ are both unit length, these equations are easy to verify from the picture. ˆi First-order tensors First-order tensors or vectors have two components in D coordinates and three components in 3D coordinates. They transform according to the same laws as coordinate axes because coordinate axes are themselves vectors. If u is a vector in the xˆ coordinate system and ui is a vector in the x coordinate system, then the following equations describe their transformation: ˆi u = u u i i i = u i Note that i is positive if the angle is measured counterclockwise from x to xˆ. It is negative if the angle is measured clockwise. ˆi
Second-order tensors The transformation law for second-order tensors like stress and strain is more complicated than the transformation law for first-order tensors. It may be derived as follows: 1. Begin with the vector transformation of traction T i to T k : T = T k ki i. Rewrite T k and Ti using Cauchy s formulas: T = n and T = n k kl l i i Substitute Cauchy s formulas into the original transformation equation: n = n kl l ki i 3. Transform the normal vector n to n l and substitute into the previous equation: n = n l l n = n kl l ki i l l 4. Cancel the n l term on each side and group the s: = kl ki l i
Note that changing the position of the last term changes the order of its subscripts. In vector notation, the equation is T = where the double under ~ denote second-rank tensors and the superscript T denotes the transpose of matrix. Mohr s Circle We explained before that an obect resting on a slope will slide down when the shear traction on the slope is greater than or equal to the product of the normal traction and the coefficient of friction. τ = f s n On a shallow slope, n is large and the obect will not slide. On a steep slow, τ is large and the obect will slide. For any plane with normal nˆ, we can calculate if the plane will fail if the stress tensor i at the interface between the obect and the slope is known. Calculate n and, τ as follows: Vector and Tensor Notation T = nˆ ˆ n = T n τ = T nˆ n Summation Notation T i = i n n = T i n i τ=t i T k n k n i
This method is straightforward but cumbersome. A different approach involves rotating the coordinate system such that is along nˆ. In this case n and τ are much easier to derive: = n τ = Mohr s circle may be derived in two or three dimensions. This lecture explains the derivation in two dimensions because it is more straightforward and the results are easier to graph and understand. The derivation assumes that, x, and x 3 are principle directions. Consider the following figure in which the x i coordinate system is rotated clockwise about the x 3 axis to x i : 1 11 θ x Figure 5. φ 11 φ x φ 1 Figure 5. The rotation matrix is: i = 11 1 31 1 3 13 cosθ = 3 sinθ 33 sinθ cosθ 1
The stress tensor in the x i coordinate system is transformed to in the x i coordinate system by the following equation: T = cosθ sinθ 1 cosθ sinθ = sinθ cosθ sinθ cosθ 1 33 1 1 cosθ sinθ cosθ sinθ = 1 sinθ cosθ sinθ cosθ 33 1 1 cos θ + sin θ (1 )sinθ cosθ = (1 )sinθ cosθ 1 sin θ + cos θ 33 Use the double-angle identities for sine and cosine to simplify the expressions for the normal stress 1 and the shear stress become in the new coordinate system: sin θ = sinθ cosθ cos θ = cos θ 1 = 1 sin θ 11 + 11 = n = 11 1 = τ = + 11 sin θ cos θ
The expression for the normal stress and shear stress can be shown graphically in shear space: τ Allowable tractions θ n τ Figure 5.3 This figure is called Mohr s circle. Mohr s circle plots in stress space. Admonton s law may also be plotted in stress space as a line with slope f s. When this line and Mohr s circle intersect, the criterion for failure across a plane is met. A note on signs. As derived, assumed >. Simplest case: consider θ = 45 o n v P x θ = 45 o P x x Figure 5.4 Figure 5.4
Consider several cases: τ 1 n 3 4 Figure 5.5 ) θ = 6 o 1) θ = 9 o ) θ = 18 o 3) θ = 7 o 4) θ = 3 o Back to the landslide: Consider a common experiment in soil mechanics or rock mechanics in which scientists apply a uniaxial stress to a cylindrical sample confined by a uniform stress.
x x 3 Figure 5.6 Figure 5.6 Continue increasing until failure. ˆn of failure plane typically at θ ; 3 o from. Seal Specimen Figure 5.7 Rubber acket Confining pressure Figure 5.7. Triaxial test deformation apparatus Figure 5.7
3 = 5 p.s.i. 55 5 35 3 = 4 35 3 = 1 p.s.i. 45 4 35 Stress: k.p.s.i. 3 5 15 1 Stress: k.p.s.i. 3 5 15 1 Stress: k.p.s.i. 3 5 15 1 5 5 5 1 3 4 5 Millistrains 1 3 4 5 Millistrains 1 3 4 5 6 7 Millistrains Figure 5.8. Stress-strain curves for Rand quartzite at various confining pressures. Figure 5.8 1) Shear fracture. ) Extension fracture. 3) Multiple shear fractures. Figure 5.9 4) Extension fracture produced by line loads. 5) Longitudinal splitting in uniaxial compression. Figure 5.9 Figures by MIT OCW.
9 8 7 165 Strain (%) 6 5 4 3 35 5 685 845 49 36 1 Figure 5.1 1 3 4 5-3 (bars) Figure 5.1. Stress-strain curves for Carrara marble at various confining pressures. The numbers on the curves are confining pressures in bars. Figure 5.1 15 Figure 5.11 Strain (%) 1 8 o C 5 o C 5 3 o C 5 o C 5 1 15-3 (k.bars) Figure 5.11. Stress-strain curves for granite at a confining pressure of 5 kilobars and various temperatures. Figure 5.11 Figures by MIT OCW.
Analyze this using a Mohr circle diagram: τ τ = µ n + n Figure 5.1 Figure 5.1 Mohr circle tangent to failure curve sample breaks. τ τ φ = tan -1 µ θ n Figure 5.13 φ n Compressive stress negative Compressive stress positive Figure 5.13
µ is the coefficient of friction. φ = tan 1 µ ; θ = 9 o φ Figure 5.14 θ = 9 o Figure 5.14 µ θ. 45.6 3 1. 3 lim µ Since most rocks have a coefficient of friction of about.6, the normal vector to the failure plane is typically 3 from the direction of the least compressive stress. Another way of saying this is that the failure plane is 3 from the direction of the most compressive stress. Styles of Faulting Faults are large-scale failure planes. Since the normals to failure planes are in the plane containing the least compressive and the most compressive stresses and are typically at 3 o from the direction of the least compressive stress, different styles of faulting can be used to infer the directions of 1,, and 3.
The following diagram shows the deviatoric stresses associated with thrust faults, normal faults, and strike-slip faults. Total Stress Figure 5.15 Deviatoric x β β y Or Total Deviatoric β Or Or "Left Lateral" "Right Lateral" Figure 5.15