improving load distributions in cellular materials using stress trajectory topologies

169 Pages·2016·6.69 MB·English

by  Scott Breault

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Open Access Master's Theses
2012
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Scott Breault
University of Rhode Island, [email protected]
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Breault, Scott, "IMPROVING LOAD DISTRIBUTIONS IN CELLULAR MATERIALS USING STRESS
TRAJECTORY TOPOLOGIES" (2012). Open Access Master's Theses. Paper 100.
https://digitalcommons.uri.edu/theses/100
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IMPROVING LOAD DISTRIBUTIONS IN
CELLULAR MATERIALS USING STRESS
TRAJECTORY TOPOLOGIES
BY
SCOTT BREAULT

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER’S OF SCIENCE
IN
MECHANICAL ENGINEERING AND APPLIED MECHANICS

UNIVERSITY OF RHODE ISLAND
2012
MASTER OF SCIENCE THESIS

OF

SCOTT BREAULT

APPROVED:

Thesis Committee:

Major Professor____Martin H. Sadd__________________

____David G. Taggart _______________

____Richard J Vaccaro________________

____Nasser H Zawia _________________
DEAN OF THE GRADUATE SCHOOL

UNIVERSITY OF RHODE ISLAND
2012
ABSTRACT
The mechanical behavior of cellular topologies aligned with stress trajectories
(ST) has been investigated. Two-dimensional stress trajectory topologies were
generated for each of seven problems. The problems include: cantilever beam with
shear end loading; two simply supported beams, one using the elasticity solution and
an identical problem using the strength of materials solution; a disk under diametric
compression; and a plate with a central stress free hole under equal biaxial, uniaxial
and unequal biaxial loadings. The stress trajectory topologies were generated using
MATLAB and the problems were simulated using Abaqus, a commercial finite
element package. In each problem results from the stress trajectory topology were
compared to other alternate topologies that came from random or uniform generation
or from another problem.
The purpose of this study was to determine if stress trajectory cellular
topologies would reduce the maximum and average stresses in comparison to control
and uniform topologies. It is believed that a topology aligned with the stress
trajectories will carry and distribute the stresses better than a random topology which
often has localized areas of high stress. In the beam and disk problems, the topology
effect was investigated by comparing the stress trajectory topology to a uniform
topology. In the plate problems, the effect of element size in the ST topologies as well
as the effect of the topology was investigated. The effect of the size of the element
was investigated by comparing three different ST topologies of various densities. The
effect of the topology was investigated by comparing the medium stress trajectory
topology to two random topologies and the medium topologies of other plate
problems.
For all cases it was found that stress trajectory topologies are better than other
alternate topologies. The stress trajectory topology lowered the maximum and
average stresses in the beam and disk problems. In the plate problems it was found
that the size of the element has little effect as the maximum and average stresses were
about the same. In looking at the different topologies for the plate problems it was
found that the medium stress trajectory topology was the best of the five alternate
topologies. The maximum and average stresses were lower than the random
topologies and the stresses on the hole were much less than the other medium
topologies.
ACKNOWLEDGMENTS
I would like to thank my major professor, Dr. Martin H. Sadd, who has supported
me throughout my thesis with his patience and knowledge of the subject. Without him
this thesis would not have been completed or written. One simply could not wish for a
better major professor.
I would also like to thank Dr. David Taggart and Dr. Richard Vaccaro, who were
both gracious enough to take time out of their schedules to be on my thesis committee
and help with any questions I may have had. I would also like to thank Dr. Lee for
being the chair of my thesis defense.
Finally, I would like to thank my parents for supporting me throughout my
studies at the University of Rhode Island and this thesis. I would especially like to
thank my mother, Susan Breault, for helping me input the thousands of intersecting
points into Abaqus.
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TABLE OF CONTENTS
ABSTRACT .................................................................................................................. ii
ACKNOWLEDGMENTS .......................................................................................... iv
TABLE OF CONTENTS............................................................................................. v
LIST OF FIGURES .................................................................................................... ix
1. INTRODUCTION .................................................................................................. 1
1.1 Cellular Materials ................................................................................................. 1
1.1.1 Geometries of Cellular Materials ................................................................... 2
1.1.2 Properties of Cellular Materials ..................................................................... 4
1.2 Finite Element Analysis of Cellular Materials ..................................................... 8
1.3 Stress Trajectories, Michell Structures and Applications to Cellular Materials 12
1.3.1 Stress Trajectories ........................................................................................ 12
1.3.2 Michell Structures ........................................................................................ 16
1.3.3 Application to Cellular Materials ................................................................ 18
1.4 Purpose of this Study .......................................................................................... 18
1.4.1 Problems Studied ......................................................................................... 19
2. MODELING APPROACH ................................................................................. 23
2.1 Determining Stress Trajectories in MATLAB ................................................... 23
2.1.1 Setting up the M-file .................................................................................... 23
2.1.2 Generating the Stress Trajectories ............................................................... 25
2.1.3 Solutions to Problems That Arose ............................................................... 26
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2.2 Establishing Finite Element Topologies Based on Stress Trajectories .............. 29
2.2.1 Curve Intersect ............................................................................................. 30
2.2.2 Finding the Intersecting points .................................................................... 31
2.3 Finite Element Analysis ..................................................................................... 32
2.3.1 Setting up the Problems ............................................................................... 32
2.3.2 Element Type ............................................................................................... 35
2.3.3 Stress Reporting ........................................................................................... 38
2.4 Verification Examples ........................................................................................ 39
2.4.1 Frame Example ............................................................................................ 39
2.4.2 Portal Example ............................................................................................. 41
2.4.3 Square Topology Simulation ....................................................................... 43
3. SIMULATION EXAMPLES AND RESULTS ................................................. 47
3.1 Examples Problems Studied ............................................................................... 47
3.2 Cantilever Beam Problem .................................................................................. 47
3.2.1 The Problem ................................................................................................. 47
3.2.2 Abaqus Simulations and Results ................................................................. 49
3.3 Simply Supported Beam Problems ..................................................................... 52
3.3.1 The Problem ................................................................................................. 52
3.3.2 Abaqus Simulations and Results ................................................................. 53
3.4 Disk Problem ...................................................................................................... 58
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3.4.1 The Problem ................................................................................................. 58
3.4.2 Abaqus Simulations and Results ................................................................. 60
3.5 Equal Biaxial Loaded Plate Problem .................................................................. 63
3.5.1 The Problem ................................................................................................. 63
3.5.2 Abaqus Simulations and Results ................................................................. 66
3.6 Uniaxial Loaded Plate Problem .......................................................................... 71
3.6.1 The Problem ................................................................................................. 71
3.6.2 Abaqus Simulations and Results ................................................................. 73
3.7 Unequal Biaxial Loaded Plate Problem ............................................................. 79
3.7.1 The Problem ................................................................................................. 79
3.7.2 Abaqus Simulations and Results ................................................................. 81
4. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ..................... 87
4.1 Summary ............................................................................................................ 87
4.2 Conclusions ........................................................................................................ 88
4.3 Recommendations .............................................................................................. 91
APPENDIX ................................................................................................................. 94
A. MATLAB Codes ................................................................................................. 94
BIBLIOGRAPHY .................................................................................................... 151
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LIST OF TABLES
TABLE PAGE
Table 2.1: List of Problems and Topologies ................................................................ 34
Table 2.2: Properties of Each Problem ........................................................................ 35
Table 2.3: Stresses in Each Member ............................................................................ 40
Table 2.4: Given Moments and Transverse Load ........................................................ 41
Table 2.5: Analytical vs Abaqus Stresses .................................................................... 42
Table 2.6: Analytical vs Abaqus Displacement ........................................................... 43
Table 2.7: Calculated Stresses in Vertical Elements .................................................... 44
Table 3.1: Stress Trajectory Topology vs Control Topology ...................................... 50
Table 3.2: Topological Data for Simply Supported Beams Examples ........................ 53
Table 3.3: Topological Data for Equal Biaxial Loaded Plate Examples ..................... 65
Table 3.4: Maximum, Minimum and Average Stresses for the Three Topologies ...... 67
Table 3.5: Maximum, Minimum and Average Stresses for the Five Topologies ........ 71
Table 3.6: Topological Data for Uniaxial Loaded Plate Examples ............................. 72
Table 3.7: Maximum, Minimum and Average Stresses for the Three Topologies ...... 75
Table 3.8: Maximum, Minimum and Average Stresses for the Five Topologies ........ 78
Table 3.9: Topological Data for Unequal Biaxial Loaded Plate Examples ................. 79
Table 3.10: Maximum, Minimum and Average Stresses for the Three Topologies .... 82
Table 3.11: Maximum, Minimum and Average Stresses for the Five Topologies ...... 86
Table 4.1: Maximum and Average Stress for the Each Problem ................................. 91

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