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Size effects in two-dimensional layered materials modeled by couple stress elasticity

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Abstract

In the present study, the effect of material microstructure on the mechanical response of a two-dimensional elastic layer perfectly bonded to a substrate is examined under surface loadings. In the current model, the substrate is treated as an elastic half plane as opposed to a rigid base, and this enables its applications in practical cases when the modulus of the layer (e.g., the coating material) and substrate (e.g., the coated surface) are comparable. The material microstructure is modeled using the generalized continuum theory of couple stress elasticity. The boundary value problems are formulated in terms of the displacement field and solved in an analytical manner via the Fourier transform and stiffness matrix method. The results demonstrate the capability of the present continuum theory to efficiently model the size-dependency of the response of the material when the external and internal length scales are comparable. Furthermore, the results indicated that the material mismatch and substrate stiffness play a crucial role in the predicted elastic field. Specifically, the study also addresses significant discrepancy of the response for the case of a layer resting on a rigid substrate.

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References

  1. Ratner M A, Ratner D. Nanotechnology: A Gentle Introduction to the Next Big Idea. New Jersey: Prentice Hall, 2003

    MATH  Google Scholar 

  2. Booker R, Boysen E. Nanotechnology for Dummies. New Jersey: Wiley, 2005

    Google Scholar 

  3. Park H J, Xu T, Lee J Y, Ledbetter A, Guo L J. Photonic color filters integrated with organic solar cells for energy harvesting. ACS Nano, 2011, 5(9): 7055–7060

    Article  Google Scholar 

  4. Yang C, Ji C, Shen W, Lee K T, Zhang Y, Liu X, Guo L J. Compact multilayer film structures for ultrabroadband omnidirectional, and efficient absorption. ACS Photonics, 2016, 3(4): 590–596

    Article  Google Scholar 

  5. Bisheh H, Wu N. Wave propagation characteristics in a piezoelectric coupled laminated composite cylindrical shell by considering transverse shear effects and rotary inertia. Composite Structures, 2018, 191: 123–144

    Article  Google Scholar 

  6. Bisheh H, Wu N. On dispersion relations in smart laminated fiber-reinforced composite membranes considering different piezoelectric coupling effects. Journal of Low Frequency Noise, Vibration and Active Control, 2019, 38(2): 487–509

    Article  Google Scholar 

  7. Bisheh H, Wu N, Hui D. Polarization effects on wave propagation characteristics of piezoelectric coupled laminated fiber-reinforced composite cylindrical shells. International Journal of Mechanical Sciences, 2019, 161–162: 105028

    Article  Google Scholar 

  8. Yang Y T, Ekinci K L, Huang X M H, Schiavone L M, Roukes M L, Zorman C A, Mehregany M. Monocrystalline silicon carbide nanoelectromechanical systems. Applied Physics Letters, 2001, 78(2): 162–164

    Article  Google Scholar 

  9. Liao F, Girshick S L, Mook W M, Gerberich W W, Zachariah M R. Superhard nanocrystalline silicon carbide films. Applied Physics Letters, 2005, 86(17): 171913

    Article  Google Scholar 

  10. Peng B, Locascio M, Zapol P, Li S, Mielke S L, Schatz G C, Espinosa H D. Measurements of near-ultimate strength for multi-walled carbon nanotubes and irradiation-induced crosslinking improvements. Nature Nanotechnology, 2008, 3(10): 626–631

    Article  Google Scholar 

  11. Qian D, Liu W K, Zheng Q. Concurrent quantum/continuum coupling analysis of nanostructures. Computer Methods in Applied Mechanics and Engineering, 2008, 197(41–42): 3291–3323

    Article  MathSciNet  MATH  Google Scholar 

  12. Peter W H, Dehoff R R, Blau P J, Yamamoto Y, Chen W, Sabau A S, Klarner A D, Novatnak D, Lherbier L, DelCorsio G, Aprigliano L, Van Hoozier C, Moffett J. Application of Wear-Resistant, Nano Composite Coatings Produced from Iron-Based Glassy Powders. 2013

  13. Bisheh H K, Wu N. Analysis of wave propagation characteristics in piezoelectric cylindrical composite shells reinforced with carbon nanotubes. International Journal of Mechanical Sciences, 2018, 145: 200–220

    Article  Google Scholar 

  14. Bisheh H, Wu N. Wave propagation in smart laminated composite cylindrical shells reinforced with carbon nanotubes in hygrothermal environments. Composites. Part B, Engineering, 2019, 162: 219–241

    Article  Google Scholar 

  15. Bisheh H, Wu N. Wave propagation in piezoelectric cylindrical composite shells reinforced with angled and randomly oriented carbon nanotubes. Composites. Part B, Engineering, 2019, 160: 10–30

    Article  Google Scholar 

  16. Bisheh H, Rabczuk T, Wu N. Effects of nanotube agglomeration on wave dynamics of carbon nanotube-reinforced piezocomposite cylindrical shells. Composites. Part B, Engineering, 2020, 187: 107739

    Article  Google Scholar 

  17. Bisheh H, Wu N, Rabczuk T. Free vibration analysis of smart laminated carbon nanotube-reinforced composite cylindrical shells with various boundary conditions in hygrothermal environments. Thin-walled Structures, 2020, 149: 106500

    Article  Google Scholar 

  18. Wong E W, Sheehan P E, Lieber C M. Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotubes. Science, 1997, 277(5334): 1971–1975

    Article  Google Scholar 

  19. Pinyochotiwong Y, Rungamornrat J, Senjuntichai T. Rigid frictionless indentation on elastic half space with influence of surface stresses. International Journal of Engineering Science, 2013, 71: 15–35

    Article  MathSciNet  MATH  Google Scholar 

  20. Tadi Beni Y, Mehralian F, Razavi H. Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory. Composite Structures, 2015, 120: 65–78

    Article  Google Scholar 

  21. Gourgiotis P A, Zisis T. Two-dimensional indentation of micro-structured solids characterized by couple-stress elasticity. Journal of Strain Analysis for Engineering Design, 2016, 51(4): 318–331

    Article  Google Scholar 

  22. Song H, Ke L, Wang Y. Sliding frictional contact analysis of an elastic solid with couple stresses. International Journal of Mechanical Sciences, 2017, 133: 804–816

    Article  Google Scholar 

  23. Song H, Ke L, Wang Y, Yang J, Jiang H. Two-dimensional frictionless contact of a coated half-plane based on couple stress theory. International Journal of Applied Mechanics, 2018, 10(5): 1850049

    Article  Google Scholar 

  24. Mindlin R D, Tiersten H F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis, 1962, 11(1): 415–448

    Article  MathSciNet  MATH  Google Scholar 

  25. Mindlin R D. Influence of couple-stresses on stress concentrations. Experimental Mechanics, 1963, 3(1): 1–7

    Article  Google Scholar 

  26. Koiter W T. Couple-stresses in the theory of elasticity. Parts I and II. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series B: Physical Sciences, 1964, 67: 17–44

    MathSciNet  MATH  Google Scholar 

  27. Hadjesfandiari A R, Dargush G F. Couple stress theory for solids. International Journal of Solids and Structures, 2011, 48(18): 2496–2510

    Article  Google Scholar 

  28. Yang F, Chong A C M, Lam D C C, Tong P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 2002, 39(10): 2731–2743

    Article  MATH  Google Scholar 

  29. Arefi M. Static analysis of laminated piezo-magnetic size-dependent curved beam based on modified couple stress theory. Structural Engineering and Mechanics, 2019, 69(2): 145–153

    Google Scholar 

  30. Arefi M, Mohammad-Rezaei Bidgoli E, Rabczuk T. Effect of various characteristics of graphene nanoplatelets on thermal buckling behavior of FGRC micro plate based on MCST. European Journal of Mechanics. A, Solids, 2019, 77: 103802

    Article  MathSciNet  MATH  Google Scholar 

  31. Arefi M, Kiani M, Zenkour A M. Size-dependent free vibration analysis of a three-layered exponentially graded nano-/micro-plate with piezomagnetic face sheets resting on Pasternak’s foundation via MCST. Journal of Sandwich Structures & Materials, 2020, 22(1): 55–86

    Article  Google Scholar 

  32. Arefi M, Kiani M. Magneto-electro-mechanical bending analysis of three-layered exponentially graded microplate with piezomagnetic face-sheets resting on Pasternak’s foundation via MCST. Mechanics of Advanced Materials and Structures, 2020, 27(5): 383–395

    Article  Google Scholar 

  33. Muki R, Sternberg E. The influence of couple-stresses on singular stress concentrations in elastic solids. Journal of Applied Mathmatics and Physics, 1965, 16(5): 611–648

    MathSciNet  Google Scholar 

  34. Zisis T, Gourgiotis P A, Baxevanakis K P, Georgiadis H G. Some basic contact problems in couple stress elasticity. International Journal of Solids and Structures, 2014, 51(11–12): 2084–2095

    Article  Google Scholar 

  35. Karuriya A N, Bhandakkar T K. Plane strain indentation on finite thickness bonded layer in couple stress elasticity. International Journal of Solids and Structures, 2017, 108: 275–288

    Article  Google Scholar 

  36. Zisis T. Burmister’s problem extended to a microstructured layer. Journal of Mechanics of Materials and Structures, 2018, 13(2): 203–223

    Article  MathSciNet  Google Scholar 

  37. Wang Y, Shen H, Zhang X, Zhang B, Liu J, Li X. Semi-analytical study of microscopic two-dimensional partial slip contact problem within the framework of couple stress elasticity: Cylindrical indenter. International Journal of Solids and Structures, 2018, 138: 76–86

    Article  Google Scholar 

  38. Sneddon I N. Fourier Transforms. 1st ed. New York: McGraw-Hill, 1951

    MATH  Google Scholar 

  39. de Borst R. A generalisation of J2-flow theory for polar continua. Computer Methods in Applied Mechanics and Engineering, 1993, 103(3): 347–362

    Article  MATH  Google Scholar 

  40. Zhang T H, Huan Y. Nanoindentation and nanoscratch behaviors of DLC coatings on different steel substrates. Composites Science and Technology, 2005, 65(9): 1409–1413

    Article  Google Scholar 

  41. Chen S, Liu L, Wang T. Investigation of the mechanical properties of thin films by nanoindentation, considering the effects of thickness and different coating-substrate combinations. Surface and Coatings Technology, 2005, 191(1): 25–32

    Article  Google Scholar 

  42. Gourgiotis P A, Georgiadis H G. An approach based on distributed dislocations and disclinations for crack problems in couple-stress elasticity. International Journal of Solids and Structures, 2008, 45(21): 5521–5539

    Article  MATH  Google Scholar 

  43. Gourgiotis P A, Georgiadis H G. The problem of sharp notch in couple-stress elasticity. International Journal of Solids and Structures, 2011, 48(19): 2630–2641

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge support provided by the Thailand Research Fund (Grant No. RTA6280012). Furthermore, the first author gratefully acknowledges the financial support from the Graduate School and Faculty of Engineering, Chulalongkorn University, during her visit at Durham University.

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Authors and Affiliations

  1. Applied Mechanics and Structures Research Unit, Department of Civil Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, 10330, Thailand

    Wipavee Wongviboonsin & Jaroon Rungamornrat

  2. Department of Civil Engineering, University of Thessaly, Pedion Areos, Volos, GR-38334, Greece

    Panos A. Gourgiotis

  3. Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, Ho Chi Minh, 721400, Vietnam

    Chung Nguyen Van

  4. Department of Civil Engineering, Faculty of Engineering, Prince of Songkla University, Songkhla, 90112, Thailand

    Suchart Limkatanyu

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  1. Wipavee Wongviboonsin
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  2. Panos A. Gourgiotis
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  3. Chung Nguyen Van
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  4. Suchart Limkatanyu
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  5. Jaroon Rungamornrat
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Correspondence to Jaroon Rungamornrat.

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Wongviboonsin, W., Gourgiotis, P.A., Van, C.N. et al. Size effects in two-dimensional layered materials modeled by couple stress elasticity. Front. Struct. Civ. Eng. 15, 425–443 (2021). https://doi.org/10.1007/s11709-021-0707-y

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  • DOI: https://doi.org/10.1007/s11709-021-0707-y

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