Size-Dependent Responses of Timoshenko Beam Incorporating the Strain Gradient Theories of Elasticity

  • Sai SidhardhEmail author
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


This paper is concerned with the study of size effects over elastic response due to strain gradient elasticity (SGE). The general form of SGE with higher order gradients is simplified and either modified couple stress theory (MCST) or modified strain gradient theory (MSGT) models the size effects. An element-free Galerkin (EFG) model of the SGE response is obtained, and the algebraic governing equations of motion are derived here from the variational principles. Following validation, a comparison of the size effects exhibited by MCST and MSGT is carried out. The effect of each component of the higher gradients over the stiffness of the beam is also studied.


Strain gradient elasticity Couple stress Microstructure Meshfree methods 


  1. 1.
    Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4(1):109–124CrossRefGoogle Scholar
  2. 2.
    Yang FACM, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39(10):2731–2743CrossRefGoogle Scholar
  3. 3.
    Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8):1477–1508CrossRefGoogle Scholar
  4. 4.
    Akgoz B, Civalek O (2013) A size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int J Eng Sci 70:1–14MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ghiba I-D, Neff P, Madeo A, Munch I (2017) A variant of the linear isotropic indeterminate couple-stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and orthogonal boundary conditions. Math Mech Solids 22(6):1221–1266MathSciNetCrossRefGoogle Scholar
  6. 6.
    Paolucci S (2016) Continuum mechanics and thermodynamics of matter. Cambridge University PressGoogle Scholar
  7. 7.
    Belytschko T, Lu YY, Gu L (1994) Element-free galerkin methods. Int J Numer Meth Eng 37(2):229–256MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sidhardh S, Ray MC (2018) Element-free galerkin model of nano-beams considering strain gradient elasticity. Acta Mech 229(2765). Scholar
  9. 9.
    Liu GR (2009) Meshfree methods: moving beyond the finite element method. CRC PressGoogle Scholar
  10. 10.
    Ma HM, Gao X-L, Reddy JN (2008) A microstructure-dependent timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 56(12):3379–3391MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of TechnologyKharagpurIndia

Personalised recommendations