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Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
Failure Analysis of a Fin Design for
the Micro-Mechanical Fish
By Michael Petralia
Wood MicroRobotics LaboratoryFin Design by Kyla Grigg
December 12, 2006
Design of the fin
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
Goal Prototype Design
The final design will look like a fish fin, but to prototype the mechanical system, square elements are being used.
Design of the fin
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
Brass Plate
Carbon Fiber Tendon
Glass Fiber Laminas
Silicone Rubber Shape Memory Alloy
(SMA)
Forces acting on the fin
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
FSMA
FTendonFRubber
FSMA
Forces acting on the fin
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
FSMA
FTendonFRubber
FSMA
θmin
FSMA
θmin = 6.5o Assuming the rubber stretches at most 1 mm,
The SMA’s provide a maximum pull of,
FSMA = 1.47 N
The horizontal force from the SMA to the rubber will be,
Frubber = 2FSMAsin(θmin) = 0.333 N
Case 1: Maximum Deflection
Assuming this load is equally distributed over the right edge of the fin,
Frubber / A = 0.2641 N/mm2
Forces acting on the fin
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
FSMA
FTendonFRubber
FSMA
FSMA
The maximum angle is,
θmax = 22o
Fmax = Frubber = 1.10 N
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,
Again, assuming this load is equally distributed over the right edge of the fin,
Frubber / A = 0.8740 N/mm2
Case 2: Maximum Force
θmax
Forces acting on the fin
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
FSMA
FTendonFRubber
FSMA
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.
Force from the Tendon
Ftendon = 1.10 N
Properties of glass fiber
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
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.
E11 (GPa) E22 (GPa) v12 G12 (GPa) v23+
50 7 0.33 5 0.33
Orientation of the glass fiber
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
+ =
Horizontal Fibers Vertical Fibers Crossed Fibers
List of Analyses
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
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
A Note on the Results
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
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.
Maximum value of σ1t
29 12 28 15 25 388
23 58 15 31 5
1400
0
200
400
600
800
1000
1200
1400
1600
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
2 Lay
ers C
rosse
d Fro
nt M
ax D
eflec
tion
2 Lay
ers C
rosse
d Bac
k Max
Defl
ectio
n
1 Lay
er Vert
ical M
ax F
orce
2 Lay
ers V
ertic
al M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Fro
nt M
ax F
orce
2 Lay
ers C
rosse
d Bac
k Max
For
ce
(MP
a)
Maximum value of σ1c
29 14 41 19 31 4 36 16 22 36 5
1225
54
0
200
400
600
800
1000
1200
1400
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
2 Lay
ers C
rosse
d Fro
nt M
ax D
eflec
tion
2 Lay
ers C
rosse
d Bac
k Max
Defl
ectio
n
1 Lay
er Vert
ical M
ax F
orce
2 Lay
ers V
ertic
al M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Fro
nt M
ax F
orce
2 Lay
ers C
rosse
d Bac
k Max
For
ce
(MP
a)
Maximum value of σ2t
47
2033
165
34
141
3730
155
31 35
0
20
40
60
80
100
120
140
160
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
2 Lay
ers C
rosse
d Fro
nt M
ax D
eflec
tion
2 Lay
ers C
rosse
d Bac
k Max
Defl
ectio
n
1 Lay
er Vert
ical M
ax F
orce
2 Lay
ers V
ertic
al M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Fro
nt M
ax F
orce
2 Lay
ers C
rosse
d Bac
k Max
For
ce
(MP
a)
Maximum value of σ2c
61
3137
196
46
59
2935
176
44
140
0
20
40
60
80
100
120
140
160
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
2 Lay
ers C
rosse
d Fro
nt M
ax D
eflec
tion
2 Lay
ers C
rosse
d Bac
k Max
Defl
ectio
n
1 Lay
er Vert
ical M
ax F
orce
2 Lay
ers V
ertic
al M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Fro
nt M
ax F
orce
2 Lay
ers C
rosse
d Bac
k Max
For
ce
(MP
a)
Maximum value of σ12
23
11
34
17
68
18
36
1813 15
63
0
10
20
30
40
50
60
70
80
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
1 Lay
er Vert
ical M
ax F
orce
2 Lay
ers V
ertic
al M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Max
Defl
ectio
n
2 Lay
ers C
rosse
d Max
For
ce
Ultimate
She
ar
(MP
a)
Maximum value of ε1t
0.00400.0017
0.0006 0.0003 0.0005 0.0005
0.0122
0.00110.0032
0.0003 0.0006 0.0006
0.0250
0.000
0.005
0.010
0.015
0.020
0.025
0.030
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
2 Lay
ers C
rosse
d Fro
nt M
ax D
eflec
tion
2 Lay
ers C
rosse
d Bac
k Max
Defl
ectio
n
1 Lay
er Vert
ical M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers V
ertic
al M
ax F
orce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Fro
nt M
ax F
orce
2 Lay
ers C
rosse
d Bac
k Max
For
ce
Maximum value of ε1c
0.0042
0.00200.0008 0.0004 0.0006 0.0006
0.0052
0.00230.0011 0.0005 0.0007 0.0007
0.0220
0.000
0.005
0.010
0.015
0.020
0.025
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
2 Lay
ers C
rosse
d Fro
nt M
ax D
eflec
tion
2 Lay
ers C
rosse
d Bac
k Max
Defl
ectio
n
1 Lay
er Vert
ical M
ax F
orce
2 Lay
ers V
ertic
al M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Fro
nt M
ax F
orce
2 Lay
ers C
rosse
d Bac
k Max
For
ce
Maximum value of ε2t
0.00090.0004
0.0047
0.0024
0.0007 0.0007
0.0026
0.0007
0.0043
0.0022
0.0006 0.0006
0.0050
0.000
0.001
0.002
0.003
0.004
0.005
0.006
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
2 Lay
ers C
rosse
d Fro
nt M
ax D
eflec
tion
2 Lay
ers C
rosse
d Bac
k Max
Defl
ectio
n
1 Lay
er Vert
ical M
ax F
orce
2 Lay
ers V
ertic
al M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Fro
nt M
ax F
orce
2 Lay
ers C
rosse
d Bac
k Max
For
ce
Maximum value of ε2c
0.0012 0.0006
0.0054
0.00270.0009 0.0009 0.0012 0.0006
0.0051
0.00260.0009 0.0009
0.0200
0.000
0.005
0.010
0.015
0.020
0.025
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
2 Lay
ers C
rosse
d Fro
nt M
ax D
eflec
tion
2 Lay
ers C
rosse
d Bac
k Max
Defl
ectio
n
1 Lay
er Vert
ical M
ax F
orce
2 Lay
ers V
ertic
al M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Fro
nt M
ax F
orce
2 Lay
ers C
rosse
d Bac
k Max
For
ce
Maximum value of ε12
0.0045
0.0023
0.0068
0.0034
0.0136
0.0035
0.0071
0.00360.0026 0.0030
0.0130
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
1 Lay
er Vert
ical M
ax D
eflec
tion
2 Lay
ers V
ertic
al M
ax D
eflec
tion
1 Lay
er Hor
izonta
l Max
Defl
ectio
n
2 Lay
ers H
orizo
ntal M
ax D
eflec
tion
1 Lay
er Vert
ical M
ax F
orce
2 Lay
ers V
ertic
al M
ax F
orce
1 Lay
er Hor
izonta
l Max
For
ce
2 Lay
ers H
orizo
ntal M
ax F
orce
2 Lay
ers C
rosse
d Max
Defl
ectio
n
2 Lay
ers C
rosse
d Max
For
ce
Ultimate
She
ar
Vertical Fibers - Max Deflection
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
Deformation Scale Factor =100
σt,max = 58.5 MPa
σc,max = 65.3 MPa
Horizontal Fibers - Max Deflection
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
σt,max = 53.2 MPa
σc,max = 52.3 MPa
Vertical Fibers - Max Deflection
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
σt,max = 25.5 MPa
σc,max = 32.6 MPa
Horizontal Fibers - Max Deflection
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
σt,max = 26.7 MPa
σc,max = 26.1 MPa
Crossed Fibers - Max Deflection
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
Front Square:σt,max = 27.5 MPaσc,max = 33.1 MPa
Back Square:σt,max = 38.9 MPaσc,max = 48.2 MPa
Vertical Fibers - Max Deflection
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
σt,max = 25.5 MPa
σc,max = 32.6 MPa
Vertical Fibers - Max Force
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
Deformation Scale Factor =100
σt,max = 175.2 MPa
σc,max = 62.7 MPa
Horizontal Fibers - Max Forces
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
σt,max = 60.2 MPa
σc,max = 59.3 MPa
Vertical Fibers - Max Force
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
σt,max = 45.5 MPa
σc,max = 30.7 MPa
Horizontal Fibers - Max Force
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
σt,max = 28.6 MPa
σc,max = 28.4 MPa
Crossed Fibers - Max Force
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
Front Square:σt,max = 31.6 MPaσc,max = 37.7 MPa
Back Square:σt,max = 37.5 MPaσc,max = 44.8 MPa
Vertical Fibers - Max Force
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
σt,max = 45.5 MPa
σc,max = 30.7 MPa
Table of Maximum Principal Stresses and Strains
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
Number of Layers Fiber Direction Case
σt,max σc,max εt,max εc,max
1 Vertical 58.5 65.3 0.0044 0.0059
Horizontal 53.2 52.3 0.0064 0.0051
Vertical Max Deflection 25.5 32.6 0.0022 0.0063
2 Horizontal 26.9 26.1 0.0032 0.0067
Crossed - Front 27.5 33.1 0.0017 0.0027
Crossed - Back 38.9 48.2 0.0017 0.0027
1 Vertical 175.2 62.7 0.0133 0.0025
Horizontal 60.2 59.3 0.0063 0.0031
Vertical Max Force 45.5 30.7 0.0034 0.0033
2 Horizontal 28.6 28.4 0.0032 0.0018
Crossed - Front 31.6 37.7 0.0019 0.0016
Crossed - Back 37.5 44.8 0.0019 0.0016
Conclusions
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo
• 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.
Questions?
Harvard UniversityDivision of Engineering and Applied Sciences
Eng-Sci 240: Solid MechanicsProfessor Zhigang Suo