ThesisPDF Available

Abstract

This thesis focuses on the analysis of hexagonal lattices at different length scales. Practical influences (such as irregularity, viscoelasticity and vibration) are taken into consideration to comprehensively characterize the effective elastic properties of such lattices. Computationally efficient and physically insightful analytical formulae are developed for the effective elastic moduli of hexagonal lattices. Closed-form analytical formulae are derived to account for the effect of irregularities caused by spatially random variation of structural and materials attributes. The effective in-plane and out-of-plane elastic moduli are characterized following a probabilistic framework for randomly inhomogeneous and randomly homogeneous form of stochasticity. The effective elastic properties of hexagonal lattices are found to be considerably influenced by spatial irregularity. However, the effect of spatially random structural irregularity is more sensitive than the irregularity caused by spatial variation of material properties. As an application of the developed closed-form formulae, the free vibration analysis of a sandwich panel with spatially irregular honeycomb core is carried out. At room temperature many polymers are found to be near their glass temperature. Elastic moduli of honeycombs made of such materials are not constant, but changes in the time or frequency domain. To analyse such hexagonal lattices, the compound effect of irregularity and viscoelasticity is investigated in an analytical framework considering spatially correlated structural and material attributes. Effect of vibration on the elastic moduli of space-filled hexagonal lattices is investigated based on dynamic stiffness matrix of a beam element. Closed-form analytical formulae are derived for characterizing the frequency dependence of the effective elastic moduli of hexagonal lattices. Hexagonal structural forms are investigated at nano-scale to derive generalized closed-form analytical formulae for the elastic moduli of hexagonal multiplanar nano-structures and nano-heterostructures. The physics based high-fidelity analytical models are capable of obtaining the elastic properties in a computationally efficient manner for wide range of materials.
Mechanics of quasi-periodic lattices
College of Engineering
Swansea University
A dissertation
submitted to Swansea University
for the degree of Doctor of Philosophy
by
Tanmoy Mukhopadhyay
Swansea, United Kingdom
February, 2017
ii
To my Parents
Declaration
This work has not previously been accepted in substance for any degree and is not being concur-
rently submitted in candidature for any degree.
Signed...................................................
Date.....................................................
STATEMENT 1
This thesis is the result of my own investigation, except where otherwise stated. Where correction
services have been used, the extend and nature of the correction is clearly market in a footnote(s).
Signed...................................................
Date.....................................................
STATEMENT 2
I hereby give consent for my thesis, if accepted, to be available for photocopying and for
inter-library loan, and for the title and summary to be made available to outsider organizations.
Signed...................................................
Date.....................................................
v
Abstract
This thesis focuses on the analysis ofhexagonal lattices at different length scales. Practical influences (such
as irregularity, viscoelasticity and vibration) are taken into consideration to comprehensively characterize
the effective elastic properties of such lattices. Computationally efficient and physically insightful analytical
formulae are developed for the effective elastic moduli of hexagonal lattices.
Closed-form analytical formulae are derived to account for the effect of irregularities caused by spa-
tially random variation of structural and materials attributes. The effective in-plane and out-of-plane elastic
moduli are characterized following a probabilistic framework for randomly inhomogeneous and randomly
homogeneous form of stochasticity. The effective elastic properties of hexagonal lattices are found to be con-
siderably influenced by spatial irregularity. However, the effect of spatially random structural irregularity is
more sensitive than the irregularity caused by spatial variation of material properties. As an application of
the developed closed-form formulae, the free vibration analysis of a sandwich panel with spatially irregular
honeycomb core is carried out.
At room temperature many polymers are found to be near their glass temperature. Elastic moduli of
honeycombs made of such materials are not constant, but changes in the time or frequency domain. To
analyse such hexagonal lattices, the compound effect of irregularity and viscoelasticity is investigated in an
analytical framework considering spatially correlated structural and material attributes.
Effect of vibration on the elastic moduli of space-filled hexagonal lattices is investigated based on dy-
namic stiffness matrix of a beam element. Closed-form analytical formulae are derived for characterizing
the frequency dependence of the effective elastic moduli of hexagonal lattices.
Hexagonal structural forms are investigated at nano-scale to derive generalized closed-form analytical
formulae for the elastic moduli of hexagonal multiplanar nano-structures and nano-heterostructures. The
physics based high-fidelity analytical models are capable of obtaining the elastic properties in a computa-
tionally efficient manner for wide range of materials.
vii
Acknowledgements
I am grateful to the College of Engineering, Swansea University for awarding me the Zienkiewicz
scholarship to support this work and providing an environment propitious for my research.
I would like to thank my supervisor Prof. Sondipon Adhikari for his guidance and encour-
agement throughout the period of my research work in Swansea. I wish to thank my friends
and colleagues at the Zienkiewicz Centre for Computational Engineering, Swansea University for
a friendly and congenial work atmosphere in the laboratory. I would like to acknowledge sev-
eral sessions of intriguing discussions on the research with my colleagues and fellow researchers
around the globe, specially Mr. A. Mahata (Missouri University of Science and Technology), Dr.
A. Batou (University of Liverpool), Dr. D. Datta (Stanford University), Dr. S. Dey (Leibniz Insti-
tute of Polymer Research Dresden) and Dr. S. Chakraborty (University of Notre Dame). I am also
thankful to Prof. A. Chakrabarti (Indian Institute of TechnologyRoorkee) and Prof. R. Chowdhury
(Indian Institute of Technology Roorkee) for motivating me to pursue a PhD during the initial stage
of my research career.
I want to thank my parents and other family members for their inspiration and unconditional
support, in spite of being far away from me. Finally, I want to thank Susmita; without her constant
mental support and inspiration this work might not have come into this shape.
ix
9.3 Published works
(During the period of pursuing PhD)
Journal publications obtained directly from the thesis
1. Mukhopadhyay T., Adhikari S. (2016) Equivalent in-plane elastic properties of irregular
honeycombs: An analytical approach, International Journal of Solids and Structures, 91
169–184, Elsevier Publication (Chapter 2)
2. Mukhopadhyay T., Adhikari S. (2016) Effective in-plane elastic properties of auxetic hon-
eycombs with spatial irregularity, Mechanics of Materials, 95 204–222, Elsevier Publication
(Chapter 3)
3. Mukhopadhyay T., Adhikari S. (2016) Free vibration analysis of sandwich panels with
randomly irregular honeycomb core, Journal of Engineering Mechanics, 142 (11) 06016008,
ASCE Publication (Chapter 5)
4. Mukhopadhyay T., Adhikari S. (2017) Stochastic mechanics of metamaterials, Composite
Structures, 162 85–97, Elsevier Publication (Chapter 2)
5. Mukhopadhyay T., Mahata A., Adhikari S., Asle Zaeem M. (2017) Effectiveelastic proper-
ties of two dimensional multiplanar hexagonal nano-structures, 2D Materials, 4 025006, IOP
Publishing (Chapter 8)
6. Mukhopadhyay T., Adhikari S. (2017) Effective in-plane elastic moduli of quasi-random
spatially irregular hexagonal lattices, International Journal of Engineering Science, 119
142–179, Elsevier Publication (Chapter 4)
7. Mukhopadhyay T., Mahata A., Adhikari S., Asle Zaeem M., Effective mechanical proper-
ties of multilayer nano-heterostructures, Nature Scientific Reports, 7 15818, Springer Nature
Publication (Chapter 8)
8. Mukhopadhyay T., Adhikari S., Frequency domain homogenization for the viscoelastic
properties of spatially correlated quasi-periodic lattices, International Journal of Mechanical
Sciences, DOI: 10.1016/j.ijmecsci.2017.09.004, Elsevier Publication (Accepted) (Chapter 6)
9. Mukhopadhyay T., Mahata A., Adhikari S., Asle Zaeem M., Probing the shear modu-
lus of two-dimensional multiplanar nanostructures and heterostructures, Nanoscale, DOI:
10.1039/C7NR07261A, RSC Publication (Accepted) (Chapter 8)
10. Mukhopadhyay T., Adhikari S., Part of work based on Chapter 7 (Under review)
11. Mukhopadhyay T., Mahata A., Adhikari S., Asle Zaeem M., Part of work based on Chapter
8 (Under review)
12. Mukhopadhyay T., Adhikari S., Part of work based on Chapter 7 (Under review)
9.3. Published works 257
Journal publications which are not a part of the thesis
1. Mukhopadhyay T., Chakraborty S., Dey S., Adhikari S., Chowdhury R. (2017) A critical
assessment of Kriging model variants for high-fidelity uncertainty quantification in dynamics
of composite shells, Archives of Computational Methods in Engineering, 24(3) 495518,
Springer Publication
2. Dey S., Mukhopadhyay T., Adhikari S.(2017) Metamodel based high-fidelity stochastic
analysis of composite laminates: A concise review with critical comparative assessment,
Composite Structures, 171 227250, Elsevier Publication
3. Naskar S., Mukhopadhyay T., Sriramula S., Adhikari S. (2017) Stochastic natural frequency
analysis of damaged thin-walled laminated composite beams with uncertainty in microme-
chanical properties, Composite Structures, 160 312334
4. Metya S., Mukhopadhyay T., Adhikari S., Bhattacharya G. (2017) System Reliability
Analysis of Soil Slopes with General Slip Surfaces Using Multivariate Adaptive Regression
Splines, Computers and Geotechnics, 87 212228, Elsevier Publication
5. Mukhopadhyay T., Mahata A., Dey S., Adhikari S. (2016) Probabilistic analysis and design
of HCP nanowires: an efficient surrogate based molecular dynamics simulation approach,
Journal of Materials Science &Technology, 32(12) 13451351, Elsevier Publication
6. Kumar S., Mukhopadhyay T., Waseem S. A., Singh B., Iqbal M. A. (2016) Effect of platen
restraint on stress-strain behaviour of concrete under uniaxial compression: A comparative
study, Strength of Materials, 48(4) 592 602, Springer Publication
7. Dey S., Mukhopadhyay T., Sahu S. K., Adhikari S. (2016) Effect of cutout on stochastic
natural frequency of composite curved panels, Composites Part B: Engineering, 105, 188202,
Elsevier Publication
8. Mukhopadhyay T., Chowdhury R., Chakrabarti A. (2016) Structural damage identification:
A random sampling-high dimensional model representation approach, Advances in Structural
Engineering, 19(6) 908927, SAGE Publication
9. Dey S., Mukhopadhyay T., Spickenheuer A., Gohs U., Adhikari S. (2016) Uncertainty
quantification in natural frequency of composite plates - An Artificial neural network based
approach, Advanced Composites Letters, 25(2) 4348, Adcotec Publication
10. Dey S., Mukhopadhyay T., Khodaparast H. H., Adhikari S. (2016) Fuzzy uncertainty
propagation in composites using Gram-Schmidt polynomial chaos expansion, Applied Mathe-
matical Modelling, 40 (78) 44124428, Elsevier Publication
11. Mahata A., Mukhopadhyay T., Adhikari S. (2016) A polynomial chaos expansion
based molecular dynamics study for probabilistic strength analysis of nano-twinned copper,
Materials Research Express, 3 036501, IOP Publishing
12. Dey S., Naskar S., Mukhopadhyay T., Gohs U., Spickenheuer A., Bittrich L., Sriramula
S., Adhikari S., Heinrich G. (2016) Uncertain natural frequency analysis of composite plates
including effect of noise A polynomial neural network approach, Composite Structures, 143
130142, Elsevier Publication
13. Mukhopadhyay T., Naskar S., Dey S., Adhikari S. (2016) On quantifying the effect of
noise in surrogate based stochastic free vibration analysis of laminated composite shallow
shells, Composite Structures, 140 798805, Elsevier Publication
14. Dey S., Mukhopadhyay T., Spickenheuer A., Adhikari S., Heinrich G. (2016) Bottom up
surrogate based approach for stochastic frequency response analysis of laminated composite
plates, Composite Structures, 140 712727, Elsevier Publication
15. Dey S., Mukhopadhyay T., Khodaparast H. H., Adhikari S. (2016) A response surface
modelling approach for resonance driven reliability based optimization of composite shells,
Periodica Polytechnica - Civil Engineering, 60 (1) 103111, BUTE Publication
16. Dey S., Mukhopadhyay T., Khodaparast H. H., Kerfriden P., Adhikari S. (2015) Rotational
and ply-level uncertainty in response of composite shallow conical shells, Composite Struc-
tures, 131 594605, Elsevier Publication
17. Dey S., Mukhopadhyay T., Sahu S.K., Li G., Rabitz H., Adhikari S.(2015) Thermal
uncertainty quantification in frequency responses of laminated composite plates, Composites
Part B: Engineering, 80 186197, Elsevier Publication
18. Mukhopadhyay T., Dey T. K.,Chowdhury R., Chakrabarti A., Adhikari S. (2015)
Optimum design of FRP bridge deck: an efficient RS-HDMR based approach, Structural and
Multidisciplinary Optimization, 52 (3) 459-477, Springer Publication
19. Dey T.K., Mukhopadhyay T., Chakrabarti A., Sharma U.K.(2015) Efficient lightweight
design of FRP bridge deck, Proceedings of the Institution of Civil Engineers - Structures and
Buildings, 168 (10) 697 - 707, ICE Publication
20. Dey S., Mukhopadhyay T., Khodaparast H. H., Adhikari S. (2015) Stochastic natural fre-
quency of composite conical shells, Acta Mechanica, 226 (8) 2537-2553, Springer Publication
21. Mukhopadhyay T., Dey T. K.,Chowdhury R., Chakrabarti A.(2015) Structural damage
identification using response surface based multi-objective optimization: A comparative study,
Arabian Journal for Science and Engineering, 40 (4) 1027-1044, Springer Publicationn
9.3. Published works 259
22. Mukhopadhyay T., Dey T. K., Dey S., Chakrabarti A.(2015) Optimization of fiber
reinforced polymer web core bridge deck A hybrid approach, Structural Engineering
International, 25 (2) 173-183, IABSE Publication
23. Dey S., Mukhopadhyay T., Adhikari S.(2015) Stochastic free vibration analyses of
composite doubly curved shells - A Kriging model approach, Composites Part B: Engineering,
70 99112, Elsevier Publication
24. Dey S., Mukhopadhyay T., Adhikari S.(2015) Stochastic free vibration analysis of
angle-ply composite plates - A RS-HDMR approach, Composite Structures, 122 526536,
Elsevier Publication
25. Mukhopadhyay T., A multivariate adaptive regression splines based damage identification
methodology for web core composite bridges including the effect of noise, Journal of
Sandwich Structures &Materials, DOI: 10.1177/1099636216682533, SAGE Publication
(Accepted)
26. Bera A. K., Mukhopadhyay T., Mohan P. J., Dey T. K., A multi-attribute decision making
approach of mix design based on experimental soil characterization, Frontiers of Structural
and Civil Engineering, DOI: 10.1007/s11709-017-0425-7, Springer Publication (Accepted)
27. Dey S., Mukhopadhyay T., Naskar S., Dey T. K., Chalak H. D., Adhikari S., Probabilistic
characterization for dynamics and stability of laminated soft core sandwich plates, Journal
of Sandwich Structures &Materials, DOI: 10.1177/1099636217694229, SAGE Publication
(Accepted)
28. Dey S., Mukhopadhyay T., Sahu S. K., Adhikari S., Stochastic dynamic stability analysis
of composite curved panels subjected to non-uniform partial edge loading, European Journal
of Mechanics / A Solids, DOI: 10.1016/j.euromechsol.2017.09.005, Elsevier Publication
(Accepted)
29. Karsh P. K., Mukhopadhyay T., Dey S., Spatial vulnerability analysis for the first ply
failure strength of composite laminates including effect of delamination, Composite Structures,
DOI: 10.1016/j.compstruct.2017.09.078, Elsevier Publication (Accepted)
Book/ book chapter publications which are not a part of the thesis
Books
1. Dey S., Mukhopadhyay T., Adhikari S. (2018) Uncertainty quantification in laminated
composites: A meta-model based approach, CRC Press, Taylor &Francis Group
Book chapters
1. Naskar S., Mukhopadhyay T., Sriramula S. (2018) A comparative assessment of ANN
and PNN model for low-frequency stochastic free vibration analysis of composite plates,
Handbook of Probabilistic Models for Engineers and Scientists, Elsevier Publication
2. Karsh P. K., Mukhopadhyay T., Dey S. (2018) Fuzzy based frequency response function
analysis of functionally graded plates, Hierarchical Composite Materials, De Gruyter Publi-
cation
3. Metya S., Mukhopadhyay T., Adhikari S., Bhattacharya G. (2017) Efficient System
Reliability Analysis of Earth Slopes based on Support Vector Machine Regression Model,
Handbook of Neural Computation, Elsevier Publication
Conference publications obtained based on the thesis
1. Mukhopadhyay T., Adhikari S., Dynamics of harmonically excited irregular cellular meta-
materials, 8th International Conference on Metamaterials, Photonic Crystals and Plasmonics,
July, 2017, Incheon, Korea
2. Mukhopadhyay T., Adhikari S., Wave propagation in irregular honeycombs, Probabilistic
Mechanics &Reliability Conference 2016 (PMC 2016), May, 2016, Vanderbilt, USA
3. Mukhopadhyay T., Adhikari S., Mechanics of irregular honeycombs, Sixth International
Congress on Computational Mechanics and Simulation (ICCMS 2016), June, 2016, Mumbai,
India
4.Mukhopadhyay T., Batou A., Adhikari S., Stochastic analysis for in-plane elastic moduli of
irregular honeycombs with viscoelastic properties, 13th International Probabilistic Workshop
2015 (IPW 2015), November, 2015, Liverpool, United Kingdom
5.Mukhopadhyay T., Adhikari S., Prediction of equivalent elastic properties of irregular
cellular solids, 23rd UK Conference of the Association for Computational Mechanics in
Engineering (ACME 2015), April, 2015, Swansea University, Swansea, United Kingdom
6. Mukhopadhyay T., Adhikari S., Homogenization and ergodicity of random lattices-A
physics based approach, 1st International Conference on Uncertainty Quantification in Com-
putational Sciences and Engineering (UNCECOMP 2015), May, 2015, Crete Island, Greece
7. Mukhopadhyay T., Adhikari S. Free vibration analysis of sandwich panels including the
effect of irregularity in honeycomb core, 5th International Conference on Computational
Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015), May, 2015,
Crete Island, Greece
9.3. Published works 261
Conference publications which are not a part of the thesis
1. Naskar S., Mukhopadhyay T., Sriramula S., Non-probabilistic analysis of laminated
composites based on fuzzy uncertainty quantification, 20th International Conference on
Composite Structures (ICCS20), 4 - 7 September, 2017, Paris, France
2. Metya S., Bhattacharya G., Mukhopadhyay T., Adhikari S., Multivariate Adaptive Regres-
sion Splines for System Reliability Analysis of Slopes, Indian Geotechnical Conference 2017
(IGC 2017), 14 - 16 December, 2017, Guwahati, India
3. Karsh P. K. Mukhopadhyay T., Dey S., Stochastic natural frequency analysis of functionally
graded plates - A Polynomial Neural Network approach, 13th International Conference on
VibrationProblems (ICOVP-2017), 29th November - 2nd December, 2017, Guwahati, India
4. Maharshi K., Roy L., Mukhopadhyay T., Dey S., Stochastic dynamic behaviour of polymer
hydrodynamic journal bearing, 6th International Conference on Functional Electroceramics
and Polymers (ICEP-2017), February 20-22, 2017, Kharagpur, India
5. Dey S., Mukhopadhyay T., Adhikari S., A meta-law for functionally graded materials based
on low velocity impact parameters, 4th International Conference on Advances in Materials
and Materials Processing (ICAMMP-IV) on 5-7 November, 2016, Kharagpur, India.
6. Dey S., Mukhopadhyay T., Chakraborty S., Chowdhury R., Adhikari S., Karmakar A.,
Spickenheuer A., Stochastic natural frequency of composite plates using Kriging model, Sixth
International Congress on Computational Mechanics and Simulation (ICCMS 2016), June,
2016, Mumbai, India
7.Dey S., Mukhopadhyay T., Spickenheuer A., Gohs U., Adhikari S., Artificial neural network
based stochastic natural frequency analysis of composite plates, International Conference on
VibrationProblems (ICOVP-2015), December, 2015, Guwahati, India
8.Dey S., Mukhopadhyay T., Khodaparast H. H., Adhikari S. Reliability based optimization
of composite spherical shells, 23rd UK Conference of the Association for Computational
Mechanics in Engineering (ACME 2015), April, 2015, Swansea University, Swansea, United
Kingdom m
9. Dey S., Mukhopadhyay T., Khodaparast H. H., Adhikari S. Uncertainty quantification of
dynamic characteristics of composites A fuzzy approach, 1st International Conference on
Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2015),
May, 2015, Crete Island, Greecee
10.Dey S., Mukhopadhyay T., Adhikari S., Free vibration analysis of angle-ply composite
plates with uncertain properties, AIAA Science and Technology Forum and Exposition
2015 (SciTech2015): 17th AIAA Non-Deterministic Approaches Conference, January 2015,
Kissimmee, FL, USA.
11. Dey S., Mukhopadhyay T., Adhikari S., Transient response of delaminated torsion stiff
composite conical shell panel subjected to low velocity oblique impact, The Twelfth Inter-
national Conference on Computational Structures Technology (CST2014) on 2-5 September,
2014, Naples, Italy.
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