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Generalization of strain-gradient theory to finite elastic deformation for isotropic materials

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Abstract

This paper concerns finite deformation in the strain-gradient continuum. In order to take account of the geometric nonlinearity, the original strain-gradient theory which is based on the infinitesimal strain tensor is rewritten given the Green–Lagrange strain tensor. Following introducing the generalized isotropic Saint Venant–Kirchhoff material model for the strain-gradient elasticity, the boundary value problem is investigated in not only the material configuration but also the spatial configuration building upon the principle of virtual work for a three-dimensional solid. By presenting one example, the convergence of the strain-gradient and classical theories is studied.

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References

  1. Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992)

    Article  MATH  Google Scholar 

  2. Aifantis, E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999)

    Article  Google Scholar 

  3. Aifantis, E.C.: Exploring the applicability of gradient elasticity to certain micro/nano reliability problems. Microsyst. Technol. 15, 109–115 (2009)

    Article  Google Scholar 

  4. Altan, B.S., Aifantis, E.C.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater. 8, 231–282 (1997)

    Article  Google Scholar 

  5. Bagni, C., Askes, H., Susmel, L.: Gradient elasticity: a transformative stress analysis tool to design notched components against uniaxial/multiaxial high-cycle fatigue. Fatigue Fract. Eng. Mater. Struct. 39, 1012–1029 (2016)

    Article  Google Scholar 

  6. Beheshti, A.: Large deformation analysis of strain-gradient elastic beams. Comput. Struct. 177, 162–175 (2016)

    Article  Google Scholar 

  7. Bertram, A.: Finite gradient elasticity and plasticity: a constitutive mechanical framework. Contin. Mech. Thermodyn. 27, 1039–1058 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Bertram, A., Forest, S.: The thermodynamics of gradient elastoplasticity. Contin. Mech. Thermodyn. 26, 269–286 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Dell’Isola, F., Sciarra, G., Vidoli, S.: Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. A 465, 2177–2196 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42, 475–487 (1994)

    Article  Google Scholar 

  11. Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)

    Article  MATH  Google Scholar 

  12. Lai, W.M., Rubin, D.H., Rubin, D., Krempl, E.: Introduction to Continuum Mechanics. Butterworth-Heinemann, Burlington (2010)

    MATH  Google Scholar 

  13. Lam, D.C.C., Yang, F., Chong, A.C.M.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)

    Article  ADS  MATH  Google Scholar 

  14. Lazar, M., Maugin, G.A.: Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lazar, M.: Irreducible decomposition of strain gradient tensor in isotropic strain gradient elasticity. ZAMM 96, 1291–1305 (2016)

    Article  ADS  Google Scholar 

  16. Mazière, M., Forest, S.: Strain gradient plasticity modeling and finite element simulation of Lüders band formation and propagation. Contin. Mech. Thermodyn. 27, 83–104 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solid. Struct. 4, 109–124 (1968)

    Article  MATH  Google Scholar 

  19. Papargyri-Beskou, S., Beskos, D.: Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates. Arch. Appl. Mech. 78, 625–635 (2008)

    Article  ADS  MATH  Google Scholar 

  20. Reddy, J.N.: An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, Oxford (2004)

    Book  MATH  Google Scholar 

  21. Sokolowski, M.: Theory of Couple-Stresses in Bodies with Constrained Rotations. In CISM Courses and Lectuers, vol. 26, Springer, Berlin (1970)

  22. Susmel, L., Askes, H., Bennett, T., Taylor, D.: Theory of critical distances versus gradient mechanics in modelling the transition from the short to long crack regime at the fatigue limit. Fatigue Fract. Eng. Mater. Struct. 36, 861–869 (2013)

    Article  Google Scholar 

  23. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ván, P., Berezovski, A., Papenfuss, C.: Thermodynamic approach to generalized continua. Contin. Mech. Thermodyn. 26, 403–420 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Wang, B., Zhao, J., Zhou, S.: A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur. J. Mech. Solids 29, 591–599 (2010)

    Article  ADS  Google Scholar 

  26. Wang, Z., Rudraraju, S., Garikipati, K.: A three dimensional field formulation, and isogeometric solutions to point and line defects using Toupin’s theory of gradient elasticity at finite strains. J. Mech. Phys. Solids 94, 336–361 (2016)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, Rasht, Iran

    Alireza Beheshti

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  1. Alireza Beheshti
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Correspondence to Alireza Beheshti.

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Communicated by Andreas Öchsner.

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Beheshti, A. Generalization of strain-gradient theory to finite elastic deformation for isotropic materials. Continuum Mech. Thermodyn. 29, 493–507 (2017). https://doi.org/10.1007/s00161-016-0542-x

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  • DOI: https://doi.org/10.1007/s00161-016-0542-x

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