M a ssa c huset t s I nst it ut e of T ec hnol ogy M at hemat ical M et hods for M at er ials S cient ist s and E ngineer s 3.016 Fall 2005 W . C raig C arter Department of Materials Science and Engineering Massachusetts Institute of Technology 77 Massachusetts Ave. Cambridge, MA 02139 P roblem Set 2: Due Friday Sept. 23, B efore 5P M: email to the TA. Individual Exercise I2-1 rKreyszig Mathematica  Computer Guide: problem 6.2, page 77 Individual Exercise I2-2 rKreyszig Mathematica  Computer Guide: problem 6.10, page 77 1�   r   Individual Exercise I2-3 rKreyszig Mathematica  Computer Guide: problem 6.12, page 78 Group Exercise G2-1 A crack in a thin elastic material gives a stress concentration when the material is loaded in “mode I” as illustrated: Illustration of crack in thin sheet being loaded in mode I. The displacements in the x- and the y- direction of each point in the material located at a distance r from the crack tip and at an a ngle θ as illustrated are given by: ux KI(1 + ν) r 5−3ν cos θ cos 3θ  1+ν  2 − 2  = u7−ν 2 − sin 3θ y 2E 2π 1+ν sin θ 2 where KI = y y σ∞√πa is called t he “stress intensity factor” and E is the “Young’s elastic modulus.” Assume that the Poisson’s ratio, ν, is 1/4. 1. Assuming that the crack is very sharp (i.e., very thin), plot the shapes of the crack if the material is loaded to σ∞ = 0.1, 0.25, and 0.5 E. y y The strains in a material indicate how far two points have separated depending on the orig-inal separation between the points. The units of strain are (Δlength)/(length); in other words, dimensionless. Because there are two coordinates, x and y, there are different kinds of strain: ǫxx The xx strain indicates the rate of separation in the x-direction with original separation in the x-direction: t his is the “x-stretch.” ǫy y The yy strain indicates the rate of separation in the y-direction with original separation in the y-direction: this is the “y-stretch.” ǫxy The xy strain indicates the rate of separation in the x-direction with orig inal separation in the y-direction: this is the “shear.” ǫy x The yx strain indicates the rate of separatio n in the y-direction with original separation in the x-direction: t his is the same as ǫxy. 2      The strains a r e calculated from the displacements as follows: 1 ∂uη ∂uζǫηζ = + 2 ∂ζ ∂η e.g., 1 ∂ux ∂uyǫxy = + 2 ∂y ∂x 2. Find an expression for the strains for the mode-I problem. The stresses in a material indicate how much force is applied across a plane, per unit area of plane. Stresses have the same units as pressure and a s Young’s modulus E. Because fo r ces can point in two independent directions and the planes can be oriented in two independent directions, there are different kinds of stress: σxx The xx stress is the force in the x-direction per unit ar ea of a plane with normal in the x-direction: this is the “x tensile stress” σy y The yy stress is the force in the y-direction per unit area of a plane with normal in the y-direction: this is the “y tensile stress” σxy The xy stress is the force in the x-direction per unit area of a plane with normal in the y-direction: this is the “shearing stress” σy x The yx stress is the force in the y-direction per unit area of a plane with normal in the x-direction: this is the same as σxy if the material is in elastic equilibrium. In an isotropic linear elastic material in a state of plane stress, the strains are linearly related to the stresses through the compliance matrix:      ǫxx 11 −ν 0 σxx  ǫy y  =  −ν 1 0  σy y  Eǫxy 0 0 2(1 + ν) σxy 3. Find the corresponding compliance matrix that linearly relates the stresses to the strains. For plane stress, the hydrostatic pressure is given by 2(σxx + σy y)/3. 4. Plot contours of constant hydrostatic pressure. 5. Plot contours of constant magnitude of shear stress. Group Exercise G2-2 In two dimensions there are a set of symmetry operations on points ~v: vx v~v = y 3        that can be represented by matrix operations on vectors: mxx mxy vxM~v = my x my y vy Among the possible symmetry operation are: Mirror Reflection across the x-axis −1 0 0 1 Mirror Reflection across the y-axis   1 0 0 −1 Rotation by θ about the origin cos θ − sin θ sin θ cos θ rUse these operations and modify the Mathematica  example in and illustrate an object that has: 1. Mirror symmetry across t he x-axis. 2. A 2-fold rotation symmetry and mirror symmetry across the y-axis. 3. A mirror plane rotated by 4 5◦ from the x-axis. 4 ps2_setup.nb available in the Assignments