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Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity

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Abstract

In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the strain-gradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first- and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.

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REFERENCES

  1. F. dell’Isola, P. Seppecher, J. J. Alibert, T. Lekszycki, R. Grygoruk, M. Pawlikowski, D. Steigmann, I. Giorgio, U. Andreaus, E. Turco, M. Gołaszewski, N. Rizzi, C. Boutin, V. A. Eremeyev, A. Misra, et al., ‘‘Pantographic metamaterials: An example of mathematically driven design and of its technological challenges,’’ Continuum Mech. Thermodyn. 31, 851–884 (2019).

  2. F. dell’Isola, P. Seppecher, M. Spagnuolo, E. Barchiesi, F. Hild, T. Lekszycki, I. Giorgio, L. Placidi, U. Andreaus, M. Cuomo, S. R. Eugster, A. Pfaff, K. Hoschke, R. Langkemper, E. Turco, et al., ‘‘Advances in pantographic structures: Design, manufacturing, models, experiments and image analyses,’’ Continuum Mech. Thermodyn. 31, 1231–1282 (2019).

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. R. D. Mindlin and N. N. Eshel, ‘‘On first strain-gradient theories in linear elasticity,’’ Int. J. Solids Struct. 4, 109–124 (1968).

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. G. A. Maugin, ‘‘Generalized continuum mechanics: Various paths,’’ in Continuum Mechanics Through the Twentieth Century: A Concise Historical (Perspective Springer, Dordrecht, 2013), pp. 223–241.

    Book  Google Scholar 

  7. G. A. Maugin, Non-Classical Continuum Mechanics: A Dictionary (Springer, Singapore, 2017).

    Book  Google Scholar 

  8. N. Auffray, F. dell’Isola, V. A. Eremeyev, A. Madeo, and G. Rosi, ‘‘Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids,’’ Math. Mech. Solids 20, 375–417 (2015).

    Article  MathSciNet  Google Scholar 

  9. F. dell’Isola, U. Andreaus, and L. Placidi, ‘‘At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola,’’ Math. Mech. Solids 20, 887–928 (2015).

    Article  MathSciNet  Google Scholar 

  10. F. dell’Isola, A. Della Corte, and I. Giorgio, ‘‘Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives,’’ Math. Mech. Solids 22, 852–872 (2016).

    Article  MathSciNet  Google Scholar 

  11. J. G. Simmonds, A Brief on Tensor Analysis, 2nd ed. (Springer, New Yourk, 1994).

  12. V. A. Eremeyev, M. J. Cloud, and L. P. Lebedev, Applications of Tensor Analysis in Continuum Mechanics (World Scientific, New Jersey, 2018).

    Book  Google Scholar 

  13. T. J. Healey and S. Krömer, ‘‘Injective weak solutions in second-gradient nonlinear elasticity,’’ ESAIM: Control, Optim. Calculus Variat. 15, 863–871 (2009).

    Article  Google Scholar 

  14. A. Mareno and T. J. Healey, ‘‘Global continuation in second-gradient nonlinear elasticity,’’ SIAM J. Math. Anal. 38, 103–115 (2006).

    Article  MathSciNet  Google Scholar 

  15. V. A. Eremeyev, F. dell’Isola, C. Boutin, and D. Steigmann, ‘‘Linear pantographic sheets: Existence and uniqueness of weak solutions,’’ J. Elasticity 132, 175–196 (2018).

    Article  MathSciNet  Google Scholar 

  16. V. A. Eremeyev, F. S. Alzahrani, A. Cazzani, F. dell’Isola, T. Hayat, E. Turco, and V. Konopińska-Zmysłowska, ‘‘On existence and uniqueness of weak solutions for linear pantographic beam lattices models,’’ Continuum Mech. Thermodyn. 31, 1843–1861 (2019).

    Article  MathSciNet  Google Scholar 

  17. S. M. Nikol’skii, ‘‘On imbedding, continuation and approximation theorems for differentiable functions of several variables,’’ Russ. Math. Surv. 16 (5), 55–104 (1961).

    Article  MathSciNet  Google Scholar 

  18. O. V. Besov, V. P. II’in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems (Wiley, New York, 1978), Vol. 1.

    Google Scholar 

  19. O. V. Besov, V. P. II’in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems (Wiley, New York, 1979), Vol. 2.

    Google Scholar 

  20. H. Triebel, Theory of Function Spaces III, Vol. 100 of Monographs in Mathematics (Birkhäuser, Basel, 2006).

  21. C. Boutin, F. dell’Isola, I. Giorgio, and L. Placidi, ‘‘Linear pantographic sheets: Asymptotic micro-macro models identification,’’ Math. Mech. Complex Syst. 5, 127–162 (2017).

    Article  MathSciNet  Google Scholar 

  22. L. Placidi, U. Andreaus, A. Della Corte, and T. Lekszycki, ‘‘Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients,’’ Zeitschr. Angew. Math. Phys. 66, 3699–3725 (2015).

    Article  MathSciNet  Google Scholar 

  23. L. Placidi, L. Greco, S. Bucci, E. Turco, N. L. Rizzi, ‘‘A second gradient formulation for a 2D fabric sheet with inextensible fibres,’’ Zeitschr. Angew. Math. Phys. 67 (5), 114 (2016).

    Article  MathSciNet  Google Scholar 

  24. S. Chandrasekhar, Liquid Crystals (Cambridge Univ. Press, Cambridge, 1977).

    Google Scholar 

  25. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Clarendon, Oxford, 1993).

    Google Scholar 

  26. P. Oswald and P. Pieranski, Smectic and Columnar Liquid Crystals, The Liquid Crystals Book Series (CRC, Boca Raton, 2006).

    Google Scholar 

  27. V. A. Eremeyev and F. dell’Isola, ‘‘A note on reduced strain gradient elasticity,’’ in Generalized Models and Non-classical Approaches in Complex Materials 1, Ed. by H. Altenbach, J. Pouget, M. Rousseau, B. Collet, and T. Michelitsch, Vol. 89 of Advanced Structured Materials (Springer, Cham, 2018), pp. 301–310.

  28. G. Fichera, ‘‘Existence theorems in elasticity,’’ in Handbuch der Physik, Ed. by S. Flügge (Springer, Berlin, 1972), Vol. VIa/2, pp. 347–389.

  29. P. G. Ciarlet, Mathematical Elasticity, Three-Dimensional Elasticity (North-Holland, Amsterdam, 1988), Vol. 1.

    MATH  Google Scholar 

  30. V. A. Eremeyev and L. P. Lebedev, ‘‘Existence of weak solutions in elasticity,’’ Math. Mech. Solids 18, 204–217 (2013).

    Article  MathSciNet  Google Scholar 

  31. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed. (Academic, Amsterdam, 2003).

    MATH  Google Scholar 

  32. L. P. Lebedev and I. I. Vorovich, Functional Analysis in Mechanics (Springer, New York, 2003).

    Book  Google Scholar 

  33. V. Maz’ya, Sobolev Spaces: With Applications to Elliptic Partial Differential Equations, Vol. 342 of Grundlehren der mathematischen Wissenschaften, 2nd ed. (Springer, Berlin, 2011).

  34. B. E Abali, W. H. Müller, and F. dell’Isola, ‘‘Theory and computation of higher gradient elasticity theories based on action principles,’’ Arch. Appl. Mech. 87, 1495–1510 (2017).

    Article  Google Scholar 

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Funding

Authors acknowledge the support of the Government of the Russian Federation (contract no. 14.Y26.31.0031).

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Authors and Affiliations

  1. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, 80-233, Gdańsk, Poland

    V. A. Eremeyev

  2. Research Institute for Mechanics, National Research Lobachevsky State University of Nizhny Novgorod, 603950, Nizhny Novgorod, Russia

    V. A. Eremeyev & F. dell’Isola

  3. International Research Center on Mathematics and Mechanics of Complex System (M&MOCS), Università dell’Aquila, l’Aquila, Italy

    F. dell’Isola

Authors
  1. V. A. Eremeyev
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  2. F. dell’Isola
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Correspondence to V. A. Eremeyev or F. dell’Isola.

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(Submitted by A. V. Lapin)

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Eremeyev, V.A., dell’Isola, F. Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity. Lobachevskii J Math 41, 1992–1998 (2020). https://doi.org/10.1134/S1995080220100078

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