Skip to main content
SpringerLink
Log in

Mixed convolved Lagrange multiplier variational formulation for size-dependent elastodynamic couple stress response

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

To model the dynamic response of materials at the submicron scale, non-classical theories of continuum mechanics must be invoked to capture the often-expected size-dependent behavior, and the corresponding variational formulations must be established. In addressing this need, a novel mixed convolved action approach is developed here for consistent couple stress theory. This action uses mixed variables, including displacements and rotations, along with impulses of symmetric force stress and skew-symmetric couple stress. In addition, the skew-symmetric components of force stress appear as Lagrange multipliers to enforce the rotation–displacement constraint, thus allowing development of a C0 variational statement. Meanwhile, the temporal convolution operator is introduced to permit proper accounting for the variations to maintain compatibility with the specified initial conditions. The resulting action functional is energy preserving for well-posed elastodynamic couple stress problems and leads directly to an innovative time–space finite element method. A low-order version of this finite element method is then applied to several two-dimensional problems to test the validity of the formulation and numerical implementation and to bring out a few interesting features of the underlying skew-symmetric couple stress theory under impulsive loading, including the dispersive nature of wave propagation. Note, however, that the formulation is not restricted to impulsive loading and, in fact, can be applied to problems with general time- and space-dependent loads.

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
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Similar content being viewed by others

References

  1. Voigt, W.: Theoretische Studien über die Elastizitätsverhältnisse der Kristalle (Theoretical studies on the elasticity relationships of crystals). Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen 34, 3–100 (1887)

    Google Scholar 

  2. Capecchi, D., Ruta, G., Trovalusci, P.: From classical to Voigt’s molecular models in elasticity. Arch. Hist. Exact Sci. 64, 525–559 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cosserat, E., Cosserat, F.: Théorie des Corps Déformables (Theory of Deformable Bodies). A. Hermann et Fils, Paris (1909)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Eringen, A.C.: Nonlinear theory of simple micro-elastic solids. Int. J. Eng. Sci. 2, 189–203 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)

    MathSciNet  MATH  Google Scholar 

  7. Eringen, A.C.: Theory of micropolar elasticity. In: Liebowitz, H. (ed.) Fracture, vol. 2, pp. 662–729. Academic Press, New York (1968)

    Google Scholar 

  8. Nowacki, W., Olszak, W.: The Linear Theory of Micropolar Elasticity. International Centre for Mechanical Sciences. Springer, New York (1974)

    Book  MATH  Google Scholar 

  9. Chen, S., Wang, T.: Strain gradient theory with couple stress for crystalline solids. Eur. J. Mech. A Solids 20, 739–756 (2001)

    Article  MATH  Google Scholar 

  10. Kunin, I.: On foundations of the theory of elastic media with microstructure. Int. J. Eng. Sci. 22, 969–978 (1984)

    Article  MATH  Google Scholar 

  11. Eringen, A.C.: Microcontinuum Field Theory. Springer, New York (1999)

    Book  MATH  Google Scholar 

  12. Trovalusci, P.: Molecular approaches for multifield continua: origins and current developments. In: Sadowski, T., Trovalusci, P. (eds.) Multiscale Modeling of Complex Materials: Phenomenological, Theoretical and Computational Aspects. Springer, Vienna (2014)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  15. Koiter, W.T.: Couple stresses in the theory of elasticity, I and II. In: Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen. Series B. Physical Sciences, vol. 67, pp. 17–44 (1964)

  16. Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)

    Article  MATH  Google Scholar 

  17. Hadjesfandiari, A.R., Dargush, G.F.: Couple stress theory for solids. Int. J. Solids Struct. 48, 2496–2510 (2011)

    Article  Google Scholar 

  18. Reissner, E.: On a variational theorem in elasticity. J. Math. Phys. 29, 90–95 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu, H.C.: On some variational principles in the theory of elasticity and the theory of plasticity. Sci. Sin. 4, 33–54 (1955)

    MATH  Google Scholar 

  20. Washizu, K.: On the variational principles of elasticity and plasticity. Technical Report, 25–18 MIT, Aeroelastic and Structures Research Laboratory, Cambridge, MA (1955)

  21. Washizu, K.: Variational Methods in Elasticity and Plasticity. Pergamon, New York (1968)

    MATH  Google Scholar 

  22. Cadzow, J.A.: Discrete calculus of variations. Int. J. Control 11, 393–407 (1970)

    Article  MATH  Google Scholar 

  23. Sivaselvan, M.V., Reinhorn, A.M.: Lagrangian approach to structural collapse simulation. J. Eng. Mech. ASCE. 132, 795–805 (2006)

    Article  Google Scholar 

  24. Sivaselvan, M.V., Lavan, O., Dargush, G.F., Kurino, H., Hyodo, Y., Fukuda, R., Sato, K., Apostolakis, G., Reinhorn, A.M.: Numerical collapse simulation of large-scale structural systems using an optimization-based algorithm. Eqk. Eng. Struct. Dyn. 38, 655–677 (2009)

    Article  Google Scholar 

  25. Lavan, O., Sivaselvan, M.V., Reinhorn, A.M., Dargush, G.F.: Progressive collapse analysis through strength degradation and fracture in the mixed Lagrangian formulation. Eqk. Eng. Struct. Dyn. 38, 1483–1504 (2009)

    Article  Google Scholar 

  26. Lavan, O.: Dynamic analysis of gap closing and contact in the mixed Lagrangian framework: toward progressive collapse prediction. J. Eng. Mech. ASCE 136, 979–986 (2010)

    Article  Google Scholar 

  27. Apostolakis, G., Dargush, G.F.: Mixed Lagrangian formulation for linear thermoelastic response of structures. J. Eng. Mech. ASCE 138, 508–518 (2012)

    Article  Google Scholar 

  28. Apostolakis, G., Dargush, G.F.: Mixed variational principles for dynamic response of thermoelastic and poroelastic continua. Int. J. Solids Struct. 50, 642–650 (2013)

    Article  Google Scholar 

  29. Apostolakis, G., Dargush, G.F.: Variational methods in irreversible thermoelasticity: Theoretical developments and minimum principles for the discrete form. Acta Mech. 224, 2065–2088 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Apostolakis, G., Dargush, G.F.: Mixed Lagrangian formalism for temperature-dependent dynamic thermoplasticity. J. Eng. Mech. ASCE 143, 04017094 (2017)

    Article  Google Scholar 

  31. Deng, G., Dargush, G.F.: Mixed Lagrangian formulation for size-dependent couple stress elastodynamic response. Acta Mech. 227, 3451–3473 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Deng, G., Dargush, G.F.: Mixed Lagrangian formulation for size-dependent couple stress elastodynamic and natural frequency analyses. Int. J. Numer. Methods Eng. 109, 809–836 (2017)

    Article  MathSciNet  Google Scholar 

  33. Gurtin, M.E.: Variational principles in the linear theory of viscoelasticity. Arch. Ration. Mech. Anal. 13, 179–191 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gurtin, M.E.: Variational principles for linear initial-value problems. Q. Appl. Math. 22, 252–256 (1964)

    Article  MATH  Google Scholar 

  35. Gurtin, M.E.: Variational principles for linear elastodynamics. Arch. Ration. Mech. Anal. 16, 34–50 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tonti, E.: On the variational formulation for linear initial value problems. Annali di Matematica XCV, 331–360 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tonti, E.: Inverse problem: its general solution. In: Rassias, G.M., Rassias, T.M. (eds.) Differential Geometry, Calculus of Variations and Their Applications. Marcel Dekker, New York (1985)

    Google Scholar 

  38. Oden, J.T., Reddy, J.N.: Variational Methods in Theoretical Mechanics. Springer-Verlag, Berlin (1983)

    Book  MATH  Google Scholar 

  39. Dargush, G.F., Kim, J.: Mixed convolved action. Phys. Rev. E 85, 066606 (2012)

    Article  Google Scholar 

  40. Dargush, G.F.: Mixed convolved action for classical and fractional-derivative dissipative dynamical systems. Phys. Rev. E 86, 066606 (2012)

    Article  Google Scholar 

  41. Dargush, G.F., Darrall, B.T., Kim, J., Apostolakis, G.: Mixed convolved action principles in linear continuum dynamics. Acta Mech. 226, 4111–4137 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Dargush, G.F., Apostolakis, G., Darrall, B.T., Kim, J.: Mixed convolved action variational principles in heat diffusion. Int. J. Heat Mass Trans. 100, 790–799 (2016)

    Article  MATH  Google Scholar 

  43. Darrall, B.T., Dargush, G.F.: Variational principle and time-space finite element method for dynamic thermoelasticity based on mixed convolved action. Eur. J. Mech. A Solids 71, 351–364 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Darrall, B.T., Dargush, G.F.: Mixed convolved action variational methods for poroelasticity. J. Appl. Mech. 83, 091011-1-091011–12 (2016)

    Article  Google Scholar 

  45. Darrall, B.T., Dargush, G.F., Hadjesfandiari, A.R.: Size-dependent response in skew-symmetric couple stress planar elasticity. Acta Mech. 225, 195–212 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Simulia: Abaqus 3DEXPERIENCE R2019x documentation. Dassault Systèmes Simulia Corp. (2019)

  47. Dargush, G.F., Apostolakis, G., Hadjesfandiari, A.R.: Two- and three-dimensional size-dependent couple stress response using a displacement-based variational method. Eur. J. Mech. A Solids 88, 104268-1-104268–13 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  48. Deng, G., Dargush, G.F.: Mixed variational principle and finite element formulation for couple stress elastostatics. Int. J. Mech. Sci. 202, 106497 (2021)

    Article  Google Scholar 

  49. Mindlin, R.D.: Influence of couple-stresses on stress-concentrations. Exp. Mech. 3, 1–7 (1963)

    Article  Google Scholar 

  50. Hadjesfandiari, A.R., Dargush, G.F.: Boundary element formulation for plane problems in couple stress elasticity. Int. J. Numer. Methods Eng. 89, 618–636 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. Pedgaonkar, A., Darrall, B.T., Dargush, G.F.: Mixed displacement and couple stress finite element method for anisotropic centrosymmetric materials. Eur. J. Mech. A Solids 85, 104074-1-104074–17 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  52. Toupin, R.A.: A note on stress concentration around an elliptic hole in micropolar elasticity. Arch. Ration. Mech. Anal. 11, 385–414 (1962)

    Article  MathSciNet  Google Scholar 

  53. Basu, A.: A note on stress concentration around an elliptic hole in micropolar elasticity. J. Aust. Math. Soc. B 19, 289–293 (1976)

    Article  MATH  Google Scholar 

  54. Jasiuk, I., Ostoja-Starzewski, M.: Planar Cosserat elasticity of materials with holes and intrusions. Appl. Mech. Rev. 48, S11–S18 (1995)

    Article  MATH  Google Scholar 

  55. Huang, F.Y., Liang, K.Z.: Boundary element analysis of stress concentration in micropolar elastic plate. Int. J. Numer. Methods Eng. 40, 1611–1622 (1997)

    Article  Google Scholar 

  56. Tuna, M., Trovalusci, P.: Stress distribution around an elliptic hole in a plate with ‘implicit’ and ‘explicit’ non-local models. Compos. Struct. 256, 113003–113011 (2021)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, Buffalo, NY, 14260, USA

    Guoqiang Deng & Gary Dargush

Authors
  1. Guoqiang Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gary Dargush
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gary Dargush.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

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

Deng, G., Dargush, G. Mixed convolved Lagrange multiplier variational formulation for size-dependent elastodynamic couple stress response. Acta Mech 233, 1837–1863 (2022). https://doi.org/10.1007/s00707-022-03187-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-022-03187-6

Search

Navigation