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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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