Skip to main content
SpringerLink
Log in

A geometrically nonlinear size-dependent hypothesis for porous functionally graded micro-plate

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

The static bending behavior of porous functionally graded (PFG) micro-plate under the geometrically nonlinear analysis is studied in this article. A small-scale nonlinear solution is established using the Von-Kármán hypothesis and the modified couple stress theory (MCST). To obtain the deflection of the plate, the Reddy higher-order plate theory coupled with isogeometric analysis (IGA) is utilized. The distribution of porosities is assumed to be even and uneven across the plate’s thickness and the effective material properties of porous functionally graded micro-plate are calculated using the refined rule-of-mixture hypothesis. The influence of power index, porosity parameter and material length scale parameter on the nonlinear behaviors of static bending of porous FGM micro-plates are also investigated using several numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: Theory and experiment. Acta Metall Mater 42(2):475–487

    Google Scholar 

  2. Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8):1477–1508

    MATH  Google Scholar 

  3. Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11(1):385–414

    MathSciNet  MATH  Google Scholar 

  4. Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16:51–78

    MathSciNet  MATH  Google Scholar 

  5. Koiter WT (1964) Couple stresses in the theory of elasticity, I and II Nederl. Akad Wetensch Proc Ser B67:17–44

    MathSciNet  MATH  Google Scholar 

  6. Sahmani S, Fattahi AM, Ahmed NA (2019) Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Eng Computers 35(4):1173–1189

    Google Scholar 

  7. Sahmani S, Fattahi AM, Ahmed NA (2020) Develop a refined truncated cubic lattice structure for nonlinear large-amplitude vibrations of micro/nano-beams made of nanoporous materials. Eng Computers 36(1):359–375

    Google Scholar 

  8. Yang, X, Sahmani S, Safaei B (2020) Postbuckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects. Eng Computers 1–16

  9. Wu Q, Chen H, Gao W (2019) Nonlocal strain gradient forced vibrations of FG-GPLRC nanocomposite microbeams. Eng Computers 1–12

  10. Mohammadimehr M, Emdadi M, Rousta Navi B (2020) Dynamic stability analysis of microcomposite annular sandwich plate with carbon nanotube reinforced composite facesheets based on modified strain gradient theory. J Sandwich Struct Materials 22(4):1199–1234

    Google Scholar 

  11. Farzam A, Hassani B (2019) Size-dependent analysis of FG microplates with temperature-dependent material properties using modified strain gradient theory and isogeometric approach. Compos B Eng 161:150–168

    Google Scholar 

  12. Li Q, Wu D, Gao W, Tin-Loi F, Liu Z, Cheng J (2019) Static bending and free vibration of organic solar cell resting on Winkler-Pasternak elastic foundation through the modified strain gradient theory. Eur J Mechan A/Solids 78:103852

    MathSciNet  MATH  Google Scholar 

  13. Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731–2743

    MATH  Google Scholar 

  14. Asghari M (2012) Geometrically nonlinear micro-plate formulation based on the modified couple stress theory. Int J Eng Sci 51:292–309

    MathSciNet  MATH  Google Scholar 

  15. Ghayesh MH, Farokhi H (2016) Coupled size-dependent behavior of shear deformable microplates. Acta Mech 227(3):757–775

    MathSciNet  MATH  Google Scholar 

  16. Thai HT, Vo TP (2013) A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory. Compos Struct 96:376–383

    Google Scholar 

  17. Reddy JN, Kim J (2012) A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos Struct 94(3):1128–1143

    Google Scholar 

  18. Shaat M, Mahmoud F, Alshorbagy A, Alieldin S, Meletis E (2014) Mechanics, Size-dependent analysis of functionally graded ultra-thin films. Struct Eng Mechan 44(4):431–448

    Google Scholar 

  19. Akgöz B, Civalek Ö (2012) Investigation of size effects on static response of single-walled carbon nanotubes based on strain gradient elasticity. Int J Comput Methods 9(02):1240032

    MathSciNet  MATH  Google Scholar 

  20. Tsiatas GC (2009) A new Kirchhoff plate model based on a modified couple stress theory. Int J Solids Struct 46(13):2757–2764

    MATH  Google Scholar 

  21. Akgöz B, Civalek Ö (2013) Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory. Meccanica 48(4):863–873

    MathSciNet  MATH  Google Scholar 

  22. Tsiatas GC, Yiotis AJ (2015) Size effect on the static, dynamic and buckling analysis of orthotropic Kirchhoff-type skew micro-plates based on a modified couple stress theory: comparison with the nonlocal elasticity theory. Acta Mech 226(4):1267–1281

    MathSciNet  MATH  Google Scholar 

  23. Tran T, Thai CH, Nguyen-Xuan H (2018) Continua, A size-dependent functionally graded higher order plate analysis based on modified couple stress theory and moving Kriging meshfree method. Comp Mater Continua 57(3):447–483

    Google Scholar 

  24. Jrad H, Mars J, Wali M, Dammak F (2019) Geometrically nonlinear analysis of elastoplastic behavior of functionally graded shells. Eng Comp 35(3):833–847

    Google Scholar 

  25. Khatir S, Abdel Wahab M (2019) A computational approach for crack identification in plate structures using XFEM, XIGA, PSO and Jaya algorithm. Theor Appl Fracture Mechan 103:102240

    Google Scholar 

  26. Khatir S, Boutchicha D, Le Thanh C, Tran-Ngoc H, Nguyen TN, Abdel-Wahab M (2020) Improved ANN technique combined with Jaya algorithm for crack identification in plates using XIGA and experimental analysis. Theoret Appl Fract Mech 107:102554

    Google Scholar 

  27. Zenzen R, Khatir S, Belaidi I, Le Thanh C, Abdel Wahab M (2020) A modified transmissibility indicator and Artificial Neural Network for damage identification and quantification in laminated composite structures. Comput Struct 248:112497

    Google Scholar 

  28. Guo H, Zhuang X, Rabczuk T (2019) A deep collocation method for the bending analysis of Kirchhoff plate. Comput Mater Continua 59(2):433–456

    Google Scholar 

  29. Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh VM, Guo H, Hamdia K, Zhuang X, Rabczuk T (2020) An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Comput Methods Appl Mech Eng 362:112790

    MathSciNet  MATH  Google Scholar 

  30. Rabczuk T, Ren H, Zhuang H (2019) A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Comput Mater Continua 59:1

    Google Scholar 

  31. Thanh CL, Tran LV, Vu-Huu T, Nguyen-Xuan H, Abdel-Wahab M (2019) and Engineering, Size-dependent nonlinear analysis and damping responses of FG-CNTRC micro-plates. 353: 253–276

  32. Thanh C-L, Tran, LV, Vu-Huu T, Abdel-Wahab M (2019) The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis. Computer Methods Appl Mechan Eng

  33. Thanh CL, Tran LV, Bui TQ, Nguyen HX, Abdel-Wahab M (2019) Isogeometric analysis for size-dependent nonlinear thermal stability of porous FG microplates. Comp Struct

  34. Thanh C-L, Ferreira A, Wahab MA (2019) A refined size-dependent couple stress theory for laminated composite micro-plates using isogeometric analysis. Thin-Walled Struct 145:106427

    Google Scholar 

  35. Thanh CL, Vu-Huu T, Phung-Van P, Nguyen-Xuan H, Wahab MA (2018) Size-Dependent Analysis for FG-CNTRC Nanoplates Based on Refined Plate Theory and Modified Couple Stress. In: International Conference on Numerical Modelling in Engineering. Springer

  36. Thanh CL, Phung-Van P, Thai CH, Nguyen-Xuan H, Abdel Wahab M (2018) Isogeometric analysis of functionally graded carbon nanotube reinforced composite nanoplates using modified couple stress theory. Comp Struc 184:633–649

    Google Scholar 

  37. Shamekhi A, Nai MH (2005) Buckling analysis of circular FGM plate having variable thickness under uniform compression by mesh free method. Int J Comput Methods 2(03):327–340

    MATH  Google Scholar 

  38. Wu C-P, Li HY (2013) An RMVT-based finite rectangular prism method for the 3D analysis of sandwich FGM plates with various boundary conditions. Comp Mater Continua 34(1):27–62

    MathSciNet  Google Scholar 

  39. Ashrafi H, Shariyat M (2017) A Three-Dimensional Comparative Study of the Isoparametric Graded Boundary and Finite Element Methods for Nonhomogeneous FGM Plates with Eccentric Cutouts. Int J Comput Methods 14(01):1750006

    MathSciNet  MATH  Google Scholar 

  40. Hidayat MIP, Ariwahjoedi B, Parman S, Irawan S (2018) A meshfree approach for transient heat conduction analysis of nonlinear functionally graded materials. Int J Comput Methods 15(02):1850007

    MathSciNet  MATH  Google Scholar 

  41. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39):4135–4195

    MathSciNet  MATH  Google Scholar 

  42. Nguyen-Thanh N, Zhou K, Zhuang X, Areias P, Nguyen-Xuan H, Bazilevs Y, Rabczuk T (2017) Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Comput Methods Appl Mech Eng 316:1157–1178

    MathSciNet  MATH  Google Scholar 

  43. Nguyen-Thanh N, Valizadeh N, Nguyen MN, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T (2015) An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Comput Methods Appl Mech Eng 284:265–291

    MathSciNet  MATH  Google Scholar 

  44. Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wüchner R, Bletzinger KU, Bazilevs Y, Rabczuk T (2011) Rotation free isogeometric thin shell analysis using PHT-splines. Comput Methods Appl Mech Eng 200(47):3410–3424

    MathSciNet  MATH  Google Scholar 

  45. Wattanasakulpong N, Chaikittiratana A (2015) Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method. Meccanica 50(5):1331–1342

    MathSciNet  MATH  Google Scholar 

  46. Wattanasakulpong N, Ungbhakorn V (2014) Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aerosp Sci Technol 32(1):111–120

    Google Scholar 

  47. Nguyen NV, Nguyen HX, Lee S, Nguyen-Xuan H (2018) Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates. Adv Eng Softw 126:110–126

    Google Scholar 

  48. Levy S (1942) Square plate with clamped edges under normal pressure producing large deflections. Tech. Rep. 740, National Advisory Committee for Aeronautics

  49. Kim J, Reddy JN (2015) A general third-order theory of functionally graded plates with modified couple stress effect and the von Kármán nonlinearity: theory and finite element analysis. Acta Mech 226(9):2973–2998

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the financial support of VLIR-UOS TEAM Project, VN2017TEA454A103, ‘An innovative solution to protect Vietnamese coastal riverbanks from floods and erosion’, funded by the Flemish Government.

Author information

Authors and Affiliations

  1. Faculty of Civil Engineering, Ho Chi Minh City Open University, Ho Chi Minh City, Vietnam

    Cuong Le Thanh & Trong Nghia Nguyen

  2. Department of Electrical Energy, Metals, Mechanical Constructions, and Systems, Faculty of Engineering and Architecture, Ghent University, 9000, Gent, Belgium

    Truong Huu Vu & Samir Khatir

  3. Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Magd Abdel Wahab

  4. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Magd Abdel Wahab

Authors
  1. Cuong Le Thanh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Trong Nghia Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Truong Huu Vu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Samir Khatir
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Magd Abdel Wahab
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Magd Abdel Wahab.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Thanh, C.L., Nguyen, T.N., Vu, T.H. et al. A geometrically nonlinear size-dependent hypothesis for porous functionally graded micro-plate. Engineering with Computers 38 (Suppl 1), 449–460 (2022). https://doi.org/10.1007/s00366-020-01154-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01154-0

Keywords

Search

Navigation