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A great reference for its elastic properties including when isotropic approximations are appropriate is the paper by Hopcroft et al (M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s Modulus of Silicon?,”

Journal of Microelectromechanical Systems, vol. 19, pp. 229-238, 2010. link to researchgate https://www.researchgate.net/publication/224124507_What_is_the_Young%27s_Modulus_of_Silicon). Additionally Hopcroft’s course notes are useful http://micromachine.stanford.edu/~hopcroft/Publications/Hopcroft_E_Si_v1p1.pdf

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We begin with a very brief refresher of Euler-Bernoulli beam theory as described in one of my favorite texts \emph{Mechanical Vibration} by S.S. Rao (link to book). The governing equation for a vibrating beam is

For free vibration, and separation of variables is used to obtain two equations

and

where

The mode shape of the beam is

The boundary conditions of the beam provide four equations which are used to solve for the . Fixed boundary conditions result in and . Free boundary conditions result in zero moment and shear force at the ends and . Notice here that the free boundary conditions are two additional derivatives of the fixed boundary conditions. Pinned boundary conditions have zero displacement and zero moment at the ends and . Sliding restrained boundary conditions have zero slope and zero shear force at the ends , . Again these two boundary conditions are related by two differentiations.

For the fixed-fixed beam, substitution of the boundary conditions results in a four equations

Similarly for the free-free beam

Comparing the two equations, the only differences are that the first two columns of the free-free case are the negative of the first two columns of the fixed-fixed case. To solve for the non-trivial solution, one must set the determinant of the matrix equal to zero to obtain the characteristic equation. One property of the determinant is that . When we compare the transpose of the matrices we see that the first two rows of the free-free case and the fixed-fixed case differ only by a negative sign and can be made identical. Therefore, the characteristic equation will have the same solution for in each case and the natural frequencies are identical. When one goes back to solve for the mode shapes, it is quite clear that the negative sign results in different shapes.

The result of the characteristic equation is

The mode shape for the fixed-fixed beam is

where

The mode shape for the free-free beam is

where is the same as for the fixed-fixed case given above.

Below we plot the first mode shape as well as its derivatives for the fixed-fixed and free-free boundary conditions. The maximum displacement has been normalized to 1 and and and have also been set to 1 without loss of generality.

Another useful insight is to compare the strain and kinetic energy densities. Because the natural frequencies are the same, we know that the ratio of potential energy and strain energy are the same, but the energy density shows how the beams are different.

A similar procedure can also be followed to compare pinned and sliding boundary conditions. Furthermore, one also finds that this holds true for natural frequencies of plates.

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The shoelace formula gets its name from the arrangement of the coordinates and how they are combined to calculate the area. Arrange the *x-y* coordinates of the polygon in a *(n+1)x2* matrix where the order is determined by a counterclockwise pattern around the perimeter and the starting point is also repeated as the last row in the matrix.

Again, notice that the order is counterclockwise and that the first point is repeated in the last line of the matrix (there are *n+1* rows if the polygon has *n* points).

To calculate the area, we add when multiplying down and to the right and subtract pairs when multiplying down and to the left. That is

In the example polygon shown above, we have

Performing the calculation, we obtain

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Several command line options for ANSYS are hard-coded into this script. To generate the appropriate options and paths for your machine, set up your solve using the ANSYS launcher and then display the command that it would invoke from “Tools –> Display Command Line.”

I don’t show it here, but one can change the arguments to the APDL script by having the Python program edit either the body of the APDL script or its input files.

from subprocess import call import datetime import os def runAPDL(ansyscall,numprocessors,workingdir,scriptFilename): """ runs the APDL script: scriptFilename.inp located in the folder: workingdir using APDL executable invoked by: ansyscall using the number of processors in: numprocessors returns the number of Ansys errors encountered in the run """ inputFile = os.path.join(workingdir, scriptFilename+".inp") # make the output file be the input file plus timestamp outputFile = os.path.join(workingdir, scriptFilename+ '{:%Y%m%d%H%M%S}'.format(datetime.datetime.now())+ ".out") # keep the standard ansys jobname jobname = "file" callString = ("\"{}\" -p ansys -dis -mpi INTELMPI" " -np {} -dir \"{}\" -j \"{}\" -s read" " -b -i \"{}\" -o \"{}\"").format( ansyscall, numprocessors, workingdir, jobname, inputFile, outputFile) print("invoking ansys with: ",callString) call(callString,shell=False) # check output file for errors print("checking for errors") numerrors = "undetermined" try: searchfile = open(outputFile, "r") except: print("could not open",outputFile) else: for line in searchfile: if "NUMBER OF ERROR" in line: print(line) numerrors = int(line.split()[-1]) searchfile.close() return(numerrors) def main(): ansyscall = "C:\\Program Files\\ANSYS Inc\\v180\\ansys\\bin\\winx64\\MAPDL.exe" numprocessors = 8 workingdir = "G:\\scriptAnsDir" scriptFilename = "dymmyAnsysScript" nErr = runAPDL(ansyscall, numprocessors, workingdir, scriptFilename) print ("number of Ansys errors: ",nErr) if __name__ == '__main__': main()]]>

One topic that is particularly interesting and well written is the section on subcritical crack growth. Subcritical crack growth (also called stress corrosion) is the growth or extension of a crack over time with stress intensity factor* less than* the critical stress intensity factor . This is interesting because in classic linear elastic fracture mechanics (LEFM) cracks in brittle materials are typically viewed as stable (no growth under constant load) if the the stress intensity factor is less than the the critical value . The text only briefly mentions the underlying cause of subcritical crack growth in terms of chemical bonds breaking in the neighborhood of the crack tip.

The crack growth is given by

where is the crack growth velocity with respect to time, is the length of the crack, is time, is the stress intensity factor, is the critical stress intensity factor, and and are constants based on the particular material, temperature and environmental conditions. Another common parameter is which is given by

where is a constant related to the geometry, typically of order 1, with value for and infinite plate.

Zerodur (manufacturer Schott’s website) is one of the most commonly used ceramics or glass ceramics by precision engineers. Subcritical crack growth is an important consideration for Zerodur in the presence of water or water vapor. The term stress corrosion is often used in this case as many like to think of the water molecules attacking the bonds at the root of the crack.

“Fracture Toughness and Crack Growth of Zerodur” (*NASA Technical Memorandum 4185*) by Michael J. Viens, April 1990 (link to document) gives the stress corrosion constants to be and with 100% humidity. *The Properties of Optical Glass* edited by Hans Bach and Norbert Neuroth (google books link, amazon.com link) gives the value of for Zerodur. They also give the values of for Duran and soda-lime glass as 22.4 and 18.1, respectively. The Zerodur technical document by Schott TIE-33 (link) has an extensive discussion, examples, and list of fracture and related properties including stress corrosion.

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Consider as shown in the figure above a c-shaped iron core with a coil wrapped around one side. The iron core has a small air gap, into which a copper plate is inserted. If we drive a constant current through the coil, we induce a magnetic flux in the iron. The DC flux density across the air gap in our example is about 0.15 T. For a static (DC) field, the copper behaves just like air, not altering the magnetic field.

But when the current and therefore the magnetic field vary harmonically with time, an electromotive force is induced according to Faraday’s Law

(1)

where is the induced electric field, is an element of the boundary of the area through which the magnetic flux passes. We can obtain by integrating the flux density over the area according to

(2)

For our example, the flux density is nearly uniform over the area and drops off rapidly outside of , and hence is well approximated by . If varies harmonically with frequency , then the magnitude of and hence the magnitude of the EMF around any loop surrounding the area is .

The current density induced by the EMF on a circular loop of radius in a conductor is given by , where is the resistivity of the conductor. So the magnitude of the current density is approximately

(3)

Thus, for our example with =0.15 T and =(5 mm) and =1.7e-8 -m, we expect the flux density just outside the 5 mm square (at =3 mm) to be 2.8e6 A/m at 40 Hz. This agrees pretty well with the numerical results shown in the animation above, which was generated using Ansoft Maxwell from ANSYS.

This simple approximation works quite well at low frequencies. At higher frequencies, the EMF of the induced currents prevents the magnetic field from penetrating the conductor. The currents drop off rapidly with a skin depth characterized by

(4)

where is the magnetic permeability of the material. For copper at 40~Hz, we have a skin depth of about 10 mm, much larger then the 1 mm thickness of the plate in our example. We can therefore safely consider the induced current to be uniform through the thickness of the plate.

]]>As a refresher, the Reynolds number is the ratio of inertial to viscous forces and is given by

where is the fluid density, is the velocity, is a characteristic length such as a diameter, and is the viscosity. The Reynolds number can also be thought of as the ratio of the momentum diffusion rate to the viscous diffusion rate. At Reynolds numbers less than approximately 2000, the flow is laminar. For Reynolds numbers greater than approximately 4000, the flow is turbulent.The Navier-Stokes equations which govern fluid mechanics can be simplified greatly for very small Reynolds number ().

where is the velocity vector, is the pressure, and is the body force vector. It should be noticed that this is no longer explicitly a function of time because the terms have vanished.

The video lecture by G.I Taylor on Low Reynolds flow offers many good explanations. This video is part of a video lecture series featuring Ascher Shapiro and other renowned fluid mechanics experts. The video demonstrating reversible flow is quite simple and clear.

A classic paper by Edward M. Purcell (a Nobel prize winning physicist at Harvard) entitled “Life at Low Reynolds Number” is available online for free. In this paper, he helps explain some of the non-intuitive fluid behaviors. In particular, he develops the well-known “scallop theorem” for swimming at low Reynolds number. In short, a body composed of two links connected by a pivot such as a scallop shell cannot swim at low Reynolds number. The body must have at least three rigid links or have a flexible tail like body to create asymmetric motions. Figure 1.3 of the thesis by J. Hussong at TU Delft presents the different methods of asymmetry. These asymmetries were originally described in the paper by S.N. Khaderi et al (available here also from TU Deflt). Brennen at Caltech has an online textbook which has a section on low Reynolds number locomotion by a variety of mechanisms.

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