DOC PREVIEW
MIT 3 032 - Problem Set 4

This preview shows page 1-2 out of 6 pages.

Save
View full document
Premium Document
Do you want full access? Go Premium and unlock all 6 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

3 032 Problem Set 4 Fall 2007 Due Start of Lecture 10 19 07 1 A microelectronic sensor is to be made of conductive wires deposited on a thin Si wafer During design considerations it was decided that only metals having cubic symmetry were to be used as fibers metallic wires in this composite sensor Another design criterion for the sensor is that the fibers of the sensor do not undergo any elastic true normal or shear strains 0 1 and 0 01 respectively when subjected to a given stress state We will assume plane stress since the device is very thin y 40 80 45 ij MPa 45 130 Si wafer Fiber direction x Metal wires fibers Figure 1 Orientation of composite sensor with respect to original loading conditions a The fibers in the sensor are subjected to an initial plane stress state given below with respect to the orientation seen in Fig 1 Determine the stress state on the fibers by resolving the original stress state onto an axis set aligned with the fibers instead of aligned with the applied stress state ij as shown in Fig 1 b Given the elastic compliance constants for various cubic crystals see Hosford Table 2 2 below calculate the complete strain tensor ij for the stress state you determined in part a for all of the metallic and organic materials from which we might make these fibers writing a small program would be ideal here Identify the metals that satisfy the design criterion that the fibers do not undergo elastic normal strains ii equal or greater than 0 1 or true shear strains ij equal to or greater than 0 01 Note The units of sij are given in TPa 1 1x10 12 Pa 1 Mechanical Behavior of Materials Material Cr Fe Mo Nb Ta W Ag Al Cu Ni Pb Pd Pt C Ge Si MgO MnO LiF KCl NaCl ZnS InP GaAs S11 S12 S44 3 10 0 46 10 10 7 56 2 90 6 50 6 89 2 45 22 46 15 82 15 25 7 75 94 57 13 63 7 34 1 10 9 80 7 67 4 05 7 19 11 65 26 00 22 80 18 77 16 48 11 72 2 78 0 816 2 23 2 57 0 69 9 48 5 73 6 39 2 98 43 56 5 95 3 08 1 51 2 68 2 14 0 94 2 52 3 43 2 85 4 66 7 24 5 94 3 65 8 59 8 21 35 44 12 11 06 22 22 03 35 34 13 23 8 05 67 11 13 94 13 07 1 92 15 00 12 54 6 60 12 66 15 71 158 6 78 62 21 65 21 74 16 82 Elastic compliances TPa 1 for various cubic crystals Table by MIT OpenCourseWare 2 2 Atomic interactions can be modeled using a variety of potential energy approximations One very common potential form is the Lennard Jones 6 12 potential 12 6 U r 4 r r where and are constants specific to a given material note they are NOT equivalent to stress and strain but this is the standard notation for the L J parameters The parameters and are related to the equilibrium bond length and the bond strength respectively Here r is the interatomic spacing given in units of Angstroms 1 Angstrom 10 nm 10 10 m and U r is given in units of eV atom A molecular dynamics simulation was performed by Zhang et al 1 to study the properties of Al thin films in which the authors proposed a Lennard Jones potential of the form above to model Al Al interactions The values used for the material parameters were 0 368 and 2 548 we ve rounded off the values in the paper for your pset a What are the assumed units of and in Zhang et al s potential for aluminum b Using the given material parameters and the form of the interatomic potential energy curve plot U r for aluminum from r 0 to 3 5 Angstroms in increments of 0 25 Angstroms Tips i We strongly suggest you use a scientific programming language such as Mathematica Matlab or Maple here to graph and take derivatives of U r as this will be a big help in prep for Lab 2 If you didn t take 3 016 or use this in another class now is your chance to learn e g by working with a classmate who did Regardless of what program 3 you use to do this you must include your own program commands and output in the pset of course not a duplicate of someone else s program ii Since r 0 will give you an error use a very low value for your starting point instead i e r 0 001 A iii In order to make your graph more readable you may want to adjust the scale of your axes so that it only emphasizes the points around the equilibrium interatomic spacing ro c Determine the equation for and graph the interatomic forces F as a function of interatomic separation r for Al over the same range of r used in part a indicating units of F r Also analytically and graphically determine the equilibrium interatomic spacing ro Mark this point on both the graphs produced in parts a and b d Compare this equilibrium interatomic spacing to the literature value of atomic radius for aluminum and from that comparison explain what you think Zhang et al assumed in choosing the constants and that made ro come out this way e Figure 2 below shows interatomic energy curves V r our U r for Mg that were calculated by Chavarria 2 squares and McMahan et al 3 dotted and solid curve These data are shown in arbitrary units or a u which is typical of computational experimental results that have funny units but r turns out to be expressed in 2 x Angstroms i e 9 a u 4 5 Angstroms Comparing these curves with that calculated in part a for Al explain whether you would expect magnesium to have a lower or higher elastic modulus than aluminum Is this confirmed by the literature values of elastic properties and physical properties of Al and Mg Hint One can compare the shape or curvature of the Al and Mg potential energy curves by normalizing energies by the minimum or binding energy Ub or U Ub and distances by ro or r ro Figure 2 Interatomic potential for magnesium as calculated by Chavarria 2 squares and McMahan et al 3 dotted and solid curves e Again considering the relationship between the Young s modulus and the equilibrium interatomic spacing and U r curvature what effect do you think temperature has on the measured Young s modulus Provide a conceptual explanation of your answer Hint Think about what happens to atoms inside of a material as you heat it up 4 3 Poly dimethylsiloxane is a crosslinked elastomer that is commonly used to build microscale devices via soft lithography borrowing from microelectronics processing …


View Full Document

MIT 3 032 - Problem Set 4

Download Problem Set 4
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Problem Set 4 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Problem Set 4 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?