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Local integral equation method for viscoelastic Reissner–Mindlin plates

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Abstract

A meshless local Petrov-Galerkin (MLPG) method is applied to solve static and dynamic bending problems of linear viscoelastic plates described by the Reissner–Mindlin theory. To this end, the correspondence principle is applied. A weak formulation for the set of governing equations in the Reissner–Mindlin theory with a unit test function is transformed into local integral equations on local subdomains in the mean surface of the plate. Nodal points are randomly spread on the mean surface of the plate and each node is surrounded by a circular subdomain to which local integral equations are applied. A meshless approximation based on the moving least-squares (MLS) method is employed in the numerical implementation.

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References

  1. Atluri SN (2004) The meshless method, (MLPG) for domain & BIE discretizations. Tech Science Press, Forsyth

    MATH  Google Scholar 

  2. Atluri SN, Shen S (2002) The meshless local Petrov-Galerkin (MLPG) method. Tech Science Press, Forsyth

    MATH  Google Scholar 

  3. Atluri SN, Sladek J, Sladek V, Zhu T (2000) The local boundary integral equation (LBIE) and its meshless implementation for linear elasticity. Comput Mech 25:180–198

    Article  MATH  Google Scholar 

  4. Balas J, Sladek J, Sladek V (1989) Stress analysis by boundary element methods. Elsevier, Amsterdam

    MATH  Google Scholar 

  5. Belytschko T, Krogauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47

    Article  MATH  Google Scholar 

  6. Beskos DE (1991) Dynamic analysis of plates. In: Beskos DE (ed) Boundary element analysis of plates and shells. Springer, Berlin, pp 35–92

    Google Scholar 

  7. Bezine G (1980) A mixed boundary integral-finite element approach to plate vibration problems. Mech Res Commun 7:141–150

    Article  MATH  Google Scholar 

  8. Christensen RM (1971) Theory of viscoelasticity. Academia, New York

    Google Scholar 

  9. Dolbow J, Belytschko T (1999) Volumetric locking in the element free Galerkin method. Int J Numer Methods Eng 46:925–942

    Article  MATH  MathSciNet  Google Scholar 

  10. Donning BM, Liu KM (1998) Meshless methods for shear-deformable beams and plates. Comput Methods Appl Mech Eng 152:47–71

    Article  MATH  Google Scholar 

  11. Duflot M, Nguyen-Dang N (2002) Dual analysis by a meshless method. Comm Numer Methods Eng 18:621–631

    Article  MATH  MathSciNet  Google Scholar 

  12. Gaul L, Schanz M (1994) A viscoelastic boundary element formulation in time domain. Arch Mech 46:583–594

    Google Scholar 

  13. Jaswon MA, Maiti M (1968) An integral equation formulation of plate bending problems. J Eng Math 2:83–93

    Article  MATH  Google Scholar 

  14. Jin ZH, Paulino GH (2002) A viscoelastic functionally graded strip containing a crack subjected to in-plane loading. Eng Fract Mech 69:1769–1790

    Article  Google Scholar 

  15. Krysl P, Belytschko T (1996) Analysis of thin plates by the element-free Galerkin method. Comput Mech 17:26–35

    Article  MathSciNet  Google Scholar 

  16. Kusama T, Mitsui Y (1982) Boundary element method applied to linear viscoelastic analysis. Appl Math Model 6:285–290

    Article  MATH  Google Scholar 

  17. Lancaster P, Salkauskas K (1981) Surfaces generated by moving least square methods. Math Comput 37:141–158

    Article  MATH  MathSciNet  Google Scholar 

  18. Lee SS, Westmann RA (1995) Application of high-order quadrature rules to time-domain boundary element analysis of viscoelasticity. Int J Numer Methods Eng 38: 607–629

    Article  MATH  Google Scholar 

  19. Liu GR (2003) Mesh free methods, moving beyond the finite element method. CRC, Boca Raton

    MATH  Google Scholar 

  20. Long SY, Atluri SN (2002) A meshless local Petrov Galerkin method for solving the bending problem of a thin plate. Comput Model Eng Sci 3:11–51

    Google Scholar 

  21. Melenk JM, Babuska I (1996) The partition of unity element method: basis theory and applications. Comput Methods Appl Mech Eng 139:289–314

    Article  MATH  MathSciNet  Google Scholar 

  22. Mesquita AD, Coda HB, Venturini WS (2001) Alternative time marching process for BEM and FEM viscoelastic analysis. Int J Numer Methods Eng 51:1157–1173

    Article  MATH  Google Scholar 

  23. Mikhailov SE (2002) Localized boundary-domain integral formulations for problems with variable coefficients. Eng Anal Boundary Elements 26:681–690

    Article  MATH  Google Scholar 

  24. Mindlin RD (1951) Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech ASME 18:31–38

    MATH  Google Scholar 

  25. Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method. Comput Mech 10:307–318

    Article  MATH  Google Scholar 

  26. Paris F, Leon SD (1986) Simply supported plates by the boundary integral equation method. Int J Numer Methods Eng 23:173–191

    Article  MATH  Google Scholar 

  27. Providakis CP, Beskos DE (1994) Dynamic analysis of elasto-plastic flexural plates by the D/BEM. Eng Anal Boundary Elements 14:75–80

    Article  Google Scholar 

  28. Providakis CP, Beskos DE (1999) Dynamic analysis of plates by boundary elements. Appl Mech Rev ASME 52:213–236

    Article  Google Scholar 

  29. Providakis CP, Beskos DE (2000) Inelastic transient dynamic analysis of Reissner–Mindlin plates by the D/BEM. Int J Numer Methods Eng 49:383–397

    Article  MATH  Google Scholar 

  30. Reddy JN (1997) Mechanics of laminated composite plates: theory and analysis. CRC, Boca Raton

    MATH  Google Scholar 

  31. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech ASME 12:A69-A77

    MathSciNet  Google Scholar 

  32. Schanz M, Antes H (1997) A new visco- and elastodynamic time domain boundary element formulation. Comput Mech 20:452–459

    Article  MATH  MathSciNet  Google Scholar 

  33. Sladek J, Sladek V (2004) Local boundary integral equation method for thin plate bending problems. Build Res J 52:89–120

    Google Scholar 

  34. Sladek J, Sladek V, Mang HA (2002) Meshless formulations for simply supported and clamped plate problems. Int J Numer Methods Eng 55:359–375

    Article  MATH  MathSciNet  Google Scholar 

  35. Sladek J, Sladek V, Mang HA (2003a) Meshless LBIE formulations for simply supported and clamped plates under dynamic load. Comput Struct 81:1643–1651

    Google Scholar 

  36. Sladek J, Sladek V, Zhang Ch (2003b) Application of meshless local Petrov-Galerkin (MLPG) method to elastodynamic problems in continuously non-homogeneous solids. Comput Model Eng Sci 4:637–648

    MATH  Google Scholar 

  37. Sladek J, Sladek V, Atluri SN (2004) Meshless local Petrov-Galerkin method in anisotropic elasticity. Comput Model Eng Sci 6:477–489

    MATH  MathSciNet  Google Scholar 

  38. Sladek J, Sladek V, Zhang Ch, Schanz M (2006) Meshless local Petrov-Galerkin method for continuously non-homogeneous viscoelastic solids. Comput Mech 37:279–289

    Article  MATH  Google Scholar 

  39. Soric J, Li Q, Jarak T, Atluri SN (2004) Meshless local Petrov-Galerkin (MLPG) formulation for analysis of thick plates. Comput Model Eng Sci 6:349–357

    MATH  Google Scholar 

  40. Stehfest H (1970) Algorithm 368: numerical inversion of Laplace transform. Comm Assoc Comput Mach 13:47–49

    Google Scholar 

  41. Tanaka M, Yamagiwa K, Miyazaki K, Udea T (1988) Free vibration analysis of elastic plate structures by boundary element method. Eng Anal Boundary Elements 5:182–188

    Article  Google Scholar 

  42. Van der Ween F (1982) Application of the boundary integral equation method to Reissner’s plate model. Int J Numer Methods Eng 18:1–10

    Article  Google Scholar 

  43. Wang J, Huang M (1991) Boundary element method for orthotropic thick plates. Acta Mech Sinica 7:258–266

    Article  Google Scholar 

  44. Wang J, Schweizerhof K (1996) Study on free vibration of moderately thick orthotropic laminated shallow shells by boundary-domain elements. Appl Math Model 20:579–584

    Article  MATH  Google Scholar 

  45. Wen PH, Aliabadi MH (2005) Boundary element frequency domain formulation for dynamic analysis of Mindlin plates. In: Aliabadi MH, Selvadurai APS, Tan CL (eds) Boundary element techniques. EC Ltd, UK, pp 35–42

    Google Scholar 

  46. Wen PH, Aliabadi MH, Zhang JZ (2005) Displacement discontinuity method for cracked Reissner plate analysis: static and dynamic. Int J Fract 135:95–116

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Construction and Architecture, Slovak Academy of Sciences, 845 03, Bratislava, Slovakia

    J. Sladek & V. Sladek

  2. Department of Civil Engineering, University of Siegen, 57068, Siegen, Germany

    Ch. Zhang

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  1. J. Sladek
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  2. V. Sladek
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  3. Ch. Zhang
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Correspondence to J. Sladek.

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Sladek, J., Sladek, V. & Zhang, C. Local integral equation method for viscoelastic Reissner–Mindlin plates. Comput Mech 41, 759–768 (2008). https://doi.org/10.1007/s00466-007-0169-7

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  • DOI: https://doi.org/10.1007/s00466-007-0169-7

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