3 032 Problem Set 3 Fall 2006 Due Start of lecture 9 29 06 1 In PS2 4 you considered a femur under a torsional load that induced a spiral fracture ap proximating the load as pure torsion Figure 1 Femur under pure torsional loading T a Draw a representative in plane element from the outermost surface of the femur diaph ysis The femur is idealized as an annular cylinder of uniform diameter subjected to a pure torque T Figure 1 Indicate all the normal and shear stresses i j acting on that material volume element expressed only as a function of T b Construct Mohr s circle of stress to identify the average normal stress avg in the bone c Draw the orientation of the representative material element indicate magnitude of ps and give the magnitudes of the principal normal stresses 1 and 2 assuming a critical torsional load F of 67 N for a moment arm of 2 m d Compare contrast this nding with what is observed for the principal stress state and maximum shear state for the same bone subjected to uniaxial compression with a mag nitude 11 1 60 MPa 2 Thin lms used in microelectronics can develop considerable internal strain when they are deposited onto a substrate of a di erent material The two materials may have di erent unit cell lattice parameters and or may have di erent coe cients of thermal expansion and the thin lm is forced to take on the lateral dimensions of the substrate to which it is adhered This strain can lead to substrate lm curvature Figure 2 The midplane of such thin lms is often approximated as being in a state of in plane normal strain in cylindrical coordinates r 0 rr o z where o is an isotropic extension of the substrate and the lm is the curvature of the lm substrate and z is the distance from the lm neutral axis its midplane 1 Figure 2 Top Substrate before deposition Bottom Substrate and thin lm after deposition the actual curvature may be in the opposite direction depending on the mismatch between lattice pa rameters and thermal expansion coe cients a In a scienti c programming language of your choice write a program that can be used to determine the normal strain x x and engineering shear strain x y on the thin lm material plane oriented at an angle with respect to your original coordinate system x y Your program must report i x x and engineering shear strain x y for a given ii principal strains 1 and 2 and the material volume element orientation pe iii maximum in plane shear strain x y max and the corresponding material volume element orientation pe max iv average normal strain avg Note Mathematica Matlab Maple C and Fortran90 are excellent choices to build this program Practice with Mathematica will bene t you in future lab experiment analyses so here s a good excuse to brush up available on MIT Server However a bruteforce execution in Excel is acceptable as a last resort In all cases you must provide the program and indicate the embedded equations with your problem set solution b Assume all the strain at the midplane of a Ge lm deposited on a Si substrate comes from a mismatch of the unit cell lattice parameters a of those two materials Compute that strain c Is the lm in a state of tensile or compressive strain and will the lm substrate system show positive smile or negative frown curvature d Assume the lm is 500 nm in thickness and that you have measured the radius of curva ture of the lm substrate system as 10 km the resolution of laser based instruments used to measure wafer curvature What is the in plane normal strain rr e This strain state is the principal strain state How do you know this 2 f Assume a Cartesian coordinate system is a fair approximation in the plane strain state of the lm i e rr 1 2 Use your program to determine the orientation and magnitude of maximum in plane shear strain in the Ge lm g If the principal strain state of a thicker piece of Ge were not in an equibiaxial strain state as described above but instead a random biaxial strain state e g 2 1 2 3 0 use Mohr s circle to determine the magnitude of maximum shear strain and the plane of the lm on which that shear strain would be 1 2 1 3 or 2 3 You can assume the 3 direction to be through the lm thickness 3 A beam of 2000 mm2 cross sectional area is composed of two pieces of wood glued together along a plane at an angle with the beam axis Figure 3 It has been found from previ ous experiments that the joint will fail at 20 MPa and 10 MPa normal and shear stresses respectively and the wood at 56 MPa and 28 MPa normal and shear stresses respectively Determine the maximum allowable axial load that the bar can carry with a safety factor of two and the corresponding value of the angle Figure 3 Wood pieces connected by glued joint 4 In 1654 the German scientist Otto von Guericke performed an experiment for Emperor Ferdinand III to demonstrate the nature of atmospheric pressure Guernicke connected two copper bowls with an outer diameter D of 14 in pumped out the air between them and tried to pull them apart with two teams of eight horses each Figure 4 The horses were unable to separate the bowls Image removed due to copyright restrictions Please see http chem ch huji ac il history guericke magdenburg experiment jpg Figure 4 Otto von Guericke s experiment with two connected copper bowls a vacuum pump and two teams of horses 3 a Assuming that Guernicke s pump was capable of removing 75 of the air from the as sembly and idealizing the connected bowls as a sphere with a wall thickness of 0 4 cm what was the maximum normal stress in the copper bowls before the horses started pulling b Using a failure stress F for copper in compression of 370 MPa how thin in theory could the copper bowls be made without failure If this value seems impractical ex plain why using the terminology and concepts of 3 032 c Assume that the bowls are not hemispheres but rather two identical shells of revolution of y 7 in sin x 14 in for x between 0 and 14 inches with axis of revolution x 7 inches The bowls are sealed at the joint with epoxy Figure 5 The wall thickness is 0 4 cm and the interior is 75 evacuated as before What is the actual shape of the stress distribution x and what is its average value Figure 5 Epoxy joint between two shells of revolution 4