Harvard University

1. Failure Analysis of a Fin Design for the Micro-Mechanical Fish By Michael Petralia December 12, 2006 Fin Design by Kyla Grigg Wood MicroRobotics Laboratory Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

2. Design of the fin The final design will look like a fish fin, but to prototype the mechanical system, square elements are being used. Goal Prototype Design Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

3. Design of the fin Shape Memory Alloy (SMA) Silicone Rubber Glass Fiber Laminas Carbon Fiber Tendon Brass Plate Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

4. Forces acting on the fin FSMA FTendon FRubber FSMA Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

5. Forces acting on the fin Case 1: Maximum Deflection The SMA’s provide a maximum pull of, FSMA = 1.47 N FSMA Assuming the rubber stretches at most 1 mm, θmin = 6.5o θmin The horizontal force from the SMA to the rubber will be, Frubber = 2FSMAsin(θmin) = 0.333 N Assuming this load is equally distributed over the right edge of the fin, Frubber / A = 0.2641 N/mm2 Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

6. Forces acting on the fin Case 2: Maximum Force The maximum angle is, FSMA θmax = 22o Though this would mean the fin has not moved, let’s take this as our worst case scenario. The maximum horizontal force from the SMA will be, θmax Fmax = Frubber = 1.10 N Again, assuming this load is equally distributed over the right edge of the fin, Frubber / A = 0.8740 N/mm2 Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

7. Forces acting on the fin Force from the Tendon The force in the tendon is the force necessary to keep the fin in place during swimming. It will be the horizontal force from the SMAs for a given angle. Because the deflection will be relatively small, it should be safe to assume the force in the tendon will act parallel to the surface of the fin. Although it would mean there was no deflection of the fin, let’s use the maximum horizontal force provided by the SMA for analysis. Ftendon = 1.10 N Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

8. Properties of glass fiber According to Chou* unidirectional lamina can be treated as a homogeneous, orthotropic continuum. Additionally, for circular cross-section fibers randomly distributed in the unidirectional lamina, the lamina can be considered transversely isotropic. The result is that we only have five independent material constants: E11, E22, v12, G12, v23 These values of these constants were provided by the manufacturer: *Chou, Microstructural Design of Fiber Composites (1992) +This value was not provided by the manufacturer, but it was necessary to assume a value in order to please ABAQUS. Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

9. Orientation of the glass fiber + = Crossed Fibers Horizontal Fibers Vertical Fibers Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

10. List of Analyses The following situations were analyzed for the cases of maximum deflection and maximum forces. To look at the worst case scenario, the tendon force was included in each analysis. One Layer: Horizontal fiber direction Vertical fiber direction Two Layers: Horizontal fiber direction Vertical fiber direction Crossed fiber direction Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

11. A Note on the Results The maximum tensile and compressive stresses and strains in the 1 and 2 directions were compared to the maximum allowable values as provided by the manufacturer. The orientation of the fibers was considered, and the graphs are based on the stresses and strains with respect to the fiber orientation, not with respect to the fin orientation. Because of this, in the cross fiber analyses, the front and back square were considered separately. Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

22. Vertical Fibers - Max Deflection Deformation Scale Factor =100 σt,max = 58.5 MPa σc,max = 65.3 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

23. Horizontal Fibers - Max Deflection σt,max = 53.2 MPa σc,max = 52.3 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

24. Vertical Fibers - Max Deflection σt,max = 25.5 MPa σc,max = 32.6 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

25. Horizontal Fibers - Max Deflection σt,max = 26.7 MPa σc,max = 26.1 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

26. Crossed Fibers - Max Deflection Front Square: σt,max = 27.5 MPa σc,max = 33.1 MPa Back Square: σt,max = 38.9 MPa σc,max = 48.2 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

27. Vertical Fibers - Max Deflection σt,max = 25.5 MPa σc,max = 32.6 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

28. Vertical Fibers - Max Force Deformation Scale Factor =100 σt,max = 175.2 MPa σc,max = 62.7 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

29. Horizontal Fibers - Max Forces σt,max = 60.2 MPa σc,max = 59.3 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

30. Vertical Fibers - Max Force σt,max = 45.5 MPa σc,max = 30.7 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

31. Horizontal Fibers - Max Force σt,max = 28.6 MPa σc,max = 28.4 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

32. Crossed Fibers - Max Force Front Square: σt,max = 31.6 MPa σc,max = 37.7 MPa Back Square: σt,max = 37.5 MPa σc,max = 44.8 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

33. Vertical Fibers - Max Force σt,max = 45.5 MPa σc,max = 30.7 MPa Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

34. Table of Maximum Principal Stresses and Strains Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

35. Conclusions • As was previously thought, it is necessary to employ two layers of glass-fiber. • The crossed-fiber configuration provides lower strains in the the 2 direction, but the horizontal stresses will approach the limit of the material in the vertical-fiber square. • Despite the higher strains, the 2 layer, horizontal-fiber configuration does not experience any stresses or strains greater than 50% of the maximum allowable value. • It is recommended that the fin be constructed using two horizontal-fiber squares. Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo

36. Questions? Harvard University Eng-Sci 240: Solid Mechanics Division of Engineering and Applied Sciences Professor Zhigang Suo