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On formulation of nonlocal elasticity problems

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Abstract

Nonlocal elasticity models are tackled with a general formulation in terms of source and target fields belonging to dual Hilbert spaces. The analysis is declaredly focused on small movements, so that a geometrically linearised approximation is assumed to be feasible. A linear, symmetric and positive definite relation between dual fields, with the physical interpretation of stress and elastic states, is assumed for the local elastic law which is thus governed by a strictly convex, quadratic energy functional. Genesis and developments of most referenced theoretical models of nonlocal elasticity are then illustrated and commented upon. The purpose is to enlighten main assumptions, to detect comparative merits and limitations of the nonlocal models and to focus on still open problems. Integral convolutions with symmetric averaging kernels, according to both strain-driven and stress-driven perspectives, homogeneous and non-homogeneous elasticity models, together with stress gradient, strain gradient, peridynamic models and nonlocal interactions between beams and elastic foundations, are included in the analysis.

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Notes

  1. The dual space \(\,{\mathcal {X}}'\,\) of a normed linear space \(\,\mathcal {X}\,\) is composed of the continuous linear functionals \(\,f:\mathcal {X}\mapsto \mathfrak {R}\,\). Their values are denoted by means of the duality pairing \(\,\langle f,\mathbf{v }\rangle \,\) with \(\,\mathbf{v }\in \mathcal {X}\,\). In this context, continuity is equivalent to boundedness: \(\,|\langle f,\mathbf{v }\rangle |\le c\,\Vert \mathbf{v }\Vert _{\mathcal {X}}\,\).

  2. A Hilbert space is Banach with norm fulfilling the parallelogram law and therefore derivable from a symmetric, positive definite bilinear form [39, 42].

  3. Hu-Washizu-Fraeijs de Veubeke functional [57,58,59].

  4. Hellinger-Prange-Reissner functional [61,62,63].

  5. In non-finite dimensional Hilbert spaces, positive definiteness is to be replaced by the stronger assumption of coerciveness. Existence proofs in local elasticity are based on closedness of the kinematic operator [55].

  6. The displacement functional \(\,\varXi :\mathcal {L}\mapsto \mathfrak {R}\,\) is customarily named total potential energy, with an improper and misleading terminology, since no potential of the load \(\,\ell \,\) is required to exist and the term \(\,\langle \ell ,{\delta \mathbf{v }}\rangle \,\) is a virtual work.

  7. Here elastically ineffective means that no interaction with the elastic constraint is activated.

  8. In this paper the symbol \(\,\delta \,\) has no special meaning by itself. Adopted as a prefix, it denotes test fields belonging to linear test spaces.

  9. This result is a direct corollary of Volterra’s theorem [65] usually improperly attributed and quoted in literature as Poincaré lemma [66].

  10. The choice of the compact manifold \(\,{\varvec{\varOmega }}\,\) is a challenging point in the theory of nonlocal elasticity since it plays a basic role in the whole treatment.

  11. Strictly speaking, \(\,\mathcal {H}\,\) is a tensor bundle over the manifold \(\,{\varvec{\varOmega }}\,\) and \(\,{\mathcal {H}}'\,\) is the dual bundle, with the fiber \(\,{\mathcal {H}}'_{\mathbf{x }}\,\) dual to the fiber \(\,\mathcal {H}_{\mathbf{x }}\,\), for all \(\,{\mathbf{x }}\in {\varvec{\varOmega }}\,\).

  12. The limit property in Eq. (76) provides the correct formulation of the usual kernel normalising condition in which integration is improperly extended over a phantom reference unbounded domain \(\,{\varvec{\varOmega }}_\infty \,\), with the output equalled to \(\,s_{\mathbf{x }}\,\):

    $$\begin{aligned} \int _{{\varvec{\varOmega }}_\infty }^{}\varphi ({\mathbf{x }},\mathbf{y })\cdot {s_\mathbf{y }}\cdot \varvec{\mu }_\mathbf{y }=s_{\mathbf{x }}\,. \qquad (75) \end{aligned}$$

  13. The push forward of a vector field from \(\,{\varvec{\varOmega }}\,\) to \(\,\varvec{\xi }({\varvec{\varOmega }})\,\) is its image through the tangent map, i.e. if \(\,\mathbf{v }_{\mathbf{x }}\,\) is the velocity of a curve through \(\,{\mathbf{x }}\in {\varvec{\varOmega }}\,\), the push forward is the velocity of the pushed curve at \(\,\varvec{\xi }({\mathbf{x }})\in \varvec{\xi }({\varvec{\varOmega }})\,\):

    $$\begin{aligned} (\varvec{\xi }{\uparrow }\mathbf{v })_{\varvec{\xi }({\mathbf{x }})}=(T_{\mathbf{x }}\varvec{\xi })\cdot \mathbf{v }_{\mathbf{x }}\,,\quad \forall \,{\mathbf{x }}\in {\varvec{\varOmega }}\,. \end{aligned}$$
    (84)

    The push-forward of a scalar field is defined by invariance and all other tensor fields are pushed accordingly.

  14. The locality recovery considered in [50] includes also the condition of a vanishing energy residual, see also [48, Eq. (4)]. We do not comment on this modification of the first principle of thermodynamics but just observe that the additional term therein is rather a power residual.

  15. This proposal was set forth in [28, 29] for a pure strain-driven model.

  16. This fact is likely to motivate the choice in [1] where it is said: “For simplicity, we will restrict our attention to macroscopically homogeneous bodies”.

  17. After having independently envisaged this way of getting a symmetric kernel, the authors became aware of the fact that a similar trick, involving the non-uniform mass density of a rotating shaft, was adopted by Tricomi in [12, p.3, Eq. (4)].

  18. This assumption was there motivated by the statement that “first gradients are suppressed as this would lead, in general, to third order tensors that previous linear models of gradient elasticity do not usually consider.” We can see from Eq. (123) that it suffices to consider the first gradient as argument of the potential, to get the differential condition Eq. (126).

  19. A three-field formulation is not feasible, unless an explicit inverse of the nonlocal response operator is available.

  20. The nabla \(\,\nabla \,\) and the apex \(\,'\,\) both denote differentiation with respect to \(\,x\,\).

References

  1. Bazant ZP, Jirasek M (2002) Nonlocal integral formulation of plasticity and damage: survey of progress. J Eng Mech ASCE 128:1119–1149. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:11(1119)

    Article  Google Scholar 

  2. Rogula D (1965) Influence of spatial acoustic dispersion on dynamical properties of dislocations. Bull Acad Pol Sci Ser Sci Tech 13:337–385

    Google Scholar 

  3. Rogula D (1976) Nonlocal theories of material systems. Ossolineum, Wrocław

    MATH  Google Scholar 

  4. Rogula D (1982) Introduction to nonlocal theory of material media. In: Rogula D (ed) Nonlocal theory of material media, CISM courses and lectures. Springer, Wien, pp 125–222. https://doi.org/10.1007/978-3-7091-2890-9

    Chapter  Google Scholar 

  5. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703. https://doi.org/10.1063/1.332803

    Article  Google Scholar 

  6. Romano G, Barretta R, Diaco M (2017) On nonlocal integral models for elastic nano-beams. Int J Mech Sci 131–132:490–499

    Google Scholar 

  7. Romano G, Barretta R (2017) Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams. Compos B 114:184–188

    Google Scholar 

  8. Eringen AC (2002) Nonlocal continuum field theories. Springer Verlag, New York

    MATH  Google Scholar 

  9. Karličić D, Murmu T, Adhikari S, McCarthy M (2015) Non-local structural mechanics. Wiley, Hoboken. https://doi.org/10.1002/9781118572030

    Book  MATH  Google Scholar 

  10. Rafii-Tabar H, Ghavanloo E, Fazelzadeh SA (2016) Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys Rep 638:1–97

    MathSciNet  Google Scholar 

  11. Peddieson J, Buchanan GR, McNitt RP (2003) Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 41(3–5):305–312

    Google Scholar 

  12. Tricomi FG (1957) Integral equations. Reprinted by Dover Books on Mathematics, Interscience, New York, 1985

  13. Polyanin AD, Manzhirov AV (2008) Handbook of integral equations, 2nd edn. CRC Press, Boca Raton

    MATH  Google Scholar 

  14. Jirásek M, Rolshoven S (2003) Comparison of integral-type nonlocal plasticity models for strain-softening materials. Int J Eng Sci 41:1553–1602

    MathSciNet  MATH  Google Scholar 

  15. Benvenuti E, Simone A (2013) One-dimensional nonlocal and gradient elasticity: closed-form solution and size effect. Mech Res Comm 48:46–51

    Google Scholar 

  16. Romano G, Barretta R, Diaco M, Marotti de Sciarra F (2017) Constitutive boundary conditions and paradoxes in nonlocal elastic nano-beams. Int J Mech Sci 121:151–156

    Google Scholar 

  17. Romano G, Barretta R (2016) Comment on the paper “Exact solution of Eringen’s nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams” by Meral Tuna & Mesut Kirca. Int J Eng Sci 109:240–242

    MathSciNet  MATH  Google Scholar 

  18. Romano G, Barretta R (2017) Nonlocal elasticity in nanobeams: the stress-driven integral model. Int J Eng Sci 115:14–27

    MathSciNet  MATH  Google Scholar 

  19. Romano G, Luciano R, Barretta R, Diaco M (2018) Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours. Continuum Mech Thermodyn 30:641

    MathSciNet  MATH  Google Scholar 

  20. Romano G, Barretta R, Diaco M (2018) A geometric rationale for invariance, covariance and constitutive relations. Continuum Mech Thermodyn 30:175–194

    MathSciNet  MATH  Google Scholar 

  21. Eringen AC (1972) Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Sci 5:425–435

    MATH  Google Scholar 

  22. Eringen AC (1987) Theory of nonlocal elasticity and some applications. Res Mechanica 21:313–342

    Google Scholar 

  23. Wang Y, Zhu X, Dai H (2016) Exact solutions for the static bending of Euler-Bernoulli beams using Eringen two-phase local/nonlocal model. AIP Adv 6(8):085114. https://doi.org/10.1063/1.4961695

    Article  Google Scholar 

  24. Fernández-Sáez J, Zaera R (2017) Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory. Int J Eng Sci 119:232–248

    MathSciNet  MATH  Google Scholar 

  25. Zhu XW, Wang YB, Dai HH (2017) Buckling analysis of Euler-Bernoulli beams using Eringen’s two-phase nonlocal model. Int J Eng Sci 116:130–140

    MathSciNet  MATH  Google Scholar 

  26. Koutsoumaris CC, Eptaimeros KG, Tsamasphyros GJ (2017) A different approach to Eringen’s nonlocal integral stress model with applications for beams. Int J Solids Struct 112:222–238

    Google Scholar 

  27. Pijaudier-Cabot G, Bazant ZP (1987) Nonlocal damage theory. J Eng Mech 113:1512–1533

    MATH  Google Scholar 

  28. Polizzotto C (2002) Remarks on some aspects of nonlocal theories in solid mechanics. In: Proc. of the 6th Congress of Italian Society for Applied and Industrial Mathematics (SIMAI), Cagliari, Italy

  29. Borino G, Failla B, Parrinello F (2003) A symmetric nonlocal damage theory. Int J Solids Struct 40(13–14):3621–3645. https://doi.org/10.1016/S0020-7683(03)00144-6

    Article  MATH  Google Scholar 

  30. Khodabakhshi P, Reddy JN (2015) A unified integro-differential nonlocal model. Int J Eng Sci 95:60–75. https://doi.org/10.1016/j.ijengsci.2015.06.006

    Article  MathSciNet  MATH  Google Scholar 

  31. Fernández-Sáez J, Zaera R, Loya JA, Reddy JN (2016) Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci 99:107-1-16. https://doi.org/10.1016/j.ijengsci.2015.10.013

    Article  MathSciNet  MATH  Google Scholar 

  32. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2–8):288–307. https://doi.org/10.1016/j.ijengsci.2007.04.004

    Article  MATH  Google Scholar 

  33. Reddy JN, Srinivasa AR (2017) An overview of theories of Continuum mechanics with nonlocal elastic response and a general framework for conservative and dissipative systems. Appl Mech Rev 69(3):030802. https://doi.org/10.1115/1.4036723

    Article  Google Scholar 

  34. Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313. https://doi.org/10.1016/j.jmps.2015.02.001

    Article  MathSciNet  MATH  Google Scholar 

  35. Barretta R, Marotti de Sciarra F (2018) Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. Int J Eng Sci 130:187–198

    MathSciNet  MATH  Google Scholar 

  36. Barretta R, Marotti de Sciarra F (2019) Variational nonlocal gradient elasticity for nano-beams. Int J Eng Sci 143:73–91

    MathSciNet  MATH  Google Scholar 

  37. Abdollahi R, Boroomand B (2019) On using mesh-based and mesh-free methods in problems defined by Eringen’s non-local integral model: issues and remedies. Meccanica. https://doi.org/10.1007/s11012-019-01048-6

    Article  MathSciNet  Google Scholar 

  38. Romano G, Barretta R, Diaco M (2014) The geometry of non-linear elasticity. Acta Mech 225(11):3199–3235

    MathSciNet  MATH  Google Scholar 

  39. Romano G (November 2014) Geometry & continuum mechanics. Short course in Innsbruck, 24–25. ISBN-10: 1503172198, http://wpage.unina.it/romano/lecture-notes/

  40. Romano G, Barretta R, Diaco M (2017) The notion of elastic state and application to nonlocal models. Proceedings AIMETA III: 1145–1156. http://wpage.unina.it/romano/selected-publications

  41. Romano G, Barretta R (2013) Geometric constitutive theory and frame invariance. Int J Non-Linear Mech 51:75–86

    Google Scholar 

  42. Yosida K (1980) Functional analysis. Springer-Verlag, New York

    MATH  Google Scholar 

  43. Peetre J (1961) Another approach to elliptic boundary problems. Commun Pure Appl Math 14:711–731

    MathSciNet  MATH  Google Scholar 

  44. Tartar L (1987) Sur un lemme d’équivalence utilisé en Analyse Numérique. Calcolo XXIV(II):129–140

    MathSciNet  MATH  Google Scholar 

  45. Romano G (2000) On the necessity of Korn’s inequality. In: O’ Donoghue PE, Flavin JN (Eds) Trends in applications of mathematics to mechanics, Elsevier, Paris, pp 166–173, ISBN: 2-84299-245-8, http://wpage.unina.it/romano

  46. Romano G (2014) Continuum mechanics on manifolds. Downloadable from http://wpage.unina.it/romano

  47. Fichera G (1972) Existence theorems in elasticity. In: Handbuch der Physik, Vol.VI/a, Springer-Verlag, Berlin

  48. Polizzotto C (2003) Unified thermodynamic framework for nonlocal/gradient continuum theories. Eur J Mech A/Solids 22:651–668

    MathSciNet  MATH  Google Scholar 

  49. Polizzotto C, Fuschi P, Pisano AA (2004) A strain-difference-based nonlocal elasticity model. Int J Solids Struct 41:2383–2401

    MATH  Google Scholar 

  50. Polizzotto C, Fuschi P, Pisano AA (2006) A nonhomogeneous nonlocal elasticity model. Eur J Mech A/Solids 25:308–333

    MathSciNet  MATH  Google Scholar 

  51. Mindlin RD (1964) Micro-structure in linear elasticity. Arch Rat Mech Anal 16:51–78

    MathSciNet  MATH  Google Scholar 

  52. Polizzotto C (2015) A unified variational framework for stress gradient and strain gradient elasticity theories. Eur J Mech A/Solids 49:430–440

    MathSciNet  MATH  Google Scholar 

  53. Polizzotto C (2016) A note on the higher order strain and stress tensors within deformation gradient elasticity theories: physical interpretations and comparisons. Int J Solids Struct 90:116–121

    Google Scholar 

  54. Romano G, Barretta R (2016) Micromorphic continua: non-redundant formulations. Continuum Mech Thermodyn 28(6):1659–1670

    MathSciNet  MATH  Google Scholar 

  55. Romano G, Rosati L, Diaco M (1999) Well-posedness of mixed formulations in elasticity. ZAMM 79(7):435–454

    MathSciNet  MATH  Google Scholar 

  56. Romano G, Marotti de Sciarra F, Diaco M. Well-posedness and numerical performances of the strain gap method. Int J Num Meth Eng (51) 283–306

  57. Hai-Chang Hu (1955) On some variational principles in the theory of elasticity and the theory of plasticity. Scienza Sinica 4:33–54

    MATH  Google Scholar 

  58. Washizu K (1955) On the variational principles of elasticity and plasticity. Aeroelastic Research Laboratory, MIT Tech Rep, MIT Cambridge, pp 25–18

  59. Fraeijs de Veubeke BM (1965) Displacement and equilibrium models. In: Zienkiewicz OC, Hollister G (eds) Stress analysis. Wiley, London, pp 145–197 reprinted in Int J Numer Meth Engrg 2001;52:287–342

    Google Scholar 

  60. Fichera G (1972) Existence theorems in elasticity. Handbuch der Physik, vol VI/a. Springer-Verlag, Berlin

    Google Scholar 

  61. Hellinger E (1914) Die allgemeinen Ansätze der Mechanik der Kontinua. Art. 30 in Encyclopädie der Mathematichen Wissenschaften, 4:654, F. Klein and C. Müller (eds.), Leibzig, Teubner

  62. Prange G (1919) Das Extremum der Formänderungsarbeit, TH Hannover 1916. Veröffentlicht als: Prange, Theorie des Balkens in der technischen Elastizitätslehre, Zeitschrift für Architektur- und Ingenieurwesen, Band 65, S. 83–96, 121–150

  63. Reissner E (1950) On a variational theorem in elasticity. J Math Phy 29:90–95

    MathSciNet  MATH  Google Scholar 

  64. Vainberg MM (1964) Variational methods for the study of nonlinear operators. Holden-Day Inc, San Francisco

    MATH  Google Scholar 

  65. Volterra V (1889) Delle variabili complesse negli iperspazii, Rend. Accad. dei Lincei, ser. IV, vol. V, Nota I, 158–165, Nota II, 291–299 = Opere Matematiche, Accad. Nazionale dei Lincei, Roma 1954;1:403–410, 411–419

  66. Samelson H (2001) Differential forms, the early days; or the Stories of Deahna’s Theorem and of Volterra’s theorem. The American Mathematical Monthly. Math Assoc Am 108(6):522–530. http://www.jstor.org/stable/2695706

  67. Aifantis EC (2011) On the gradient approach—relation to Eringen’s nonlocal theory. Int J Eng Science 49:1367–1377

    MathSciNet  Google Scholar 

  68. Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48(1):175–209

    MathSciNet  MATH  Google Scholar 

  69. Silling SA, Lehoucq R (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44:73–168

    Google Scholar 

  70. Nishawala V, Ostoja-Starzewski M (2017) Peristatic solutions for finite one- and two-dimensional systems. Math Mech Solids 22(8):1639–1653

    MathSciNet  MATH  Google Scholar 

  71. Polizzotto C (2001) Nonlocal elasticity and related variational principles. Int J Solids Struct 38:7359–7380

    MathSciNet  MATH  Google Scholar 

  72. Romano G, Barretta R, Diaco M (October 2018) Iterative methods for nonlocal elasticity problems. Continuum Mech Thermodyn, published on line 03

  73. Winkler E (1867) Die Lehre von der Elastizität und Festigkeit. Prag, H. Dominicus https://archive.org/details/bub_gb_25E5AAAAcAAJ/page/n5

  74. Zimmermann H (1888) Die Berechnung des Eisenbahnoberbaues. Ernst U. Korn, Berlin

    Google Scholar 

  75. Wieghardt K (1922) Über der Balken auf nachgiebiger Unterlage. Zeit Angew Math Mech (ZAMM) 2:165–186

    MATH  Google Scholar 

  76. Föppl A (1909) Vorlesungen über technische Mechanik, vol III. Festigkeitslehre. Leipzig u. Berlin

  77. Van Langendonck T (1962) Beams on deformable foundation. Mémoires AIPC 22:113–128

    Google Scholar 

  78. Sollazzo A (1966) Equilibrio della trave su suolo di Wieghardt. Tecnica Italiana 31(4):187–206

    Google Scholar 

  79. Ylinen A, Mikkola M (1967) A beam on a Wieghardt-type elastic foundation. Int J Solids Struct 3:617–633

    Google Scholar 

  80. Capurso M (1967) A generalization of Wieghardt soil for two dimensional foundation structures. Meccanica 2:49. https://doi.org/10.1007/BF02128154

    Article  Google Scholar 

  81. Essenburg F (1962) Shear deformation in beams on elastic foundations. J Appl Mech 29(Trans. ASME84):313. https://doi.org/10.1115/1.3640547

    Article  MATH  Google Scholar 

  82. Barretta R (2019) Nonlocal elastic foundations. Private communication

  83. Thai HT et al (2017) A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos Struct 177:196–219

    Google Scholar 

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  1. Department of Structures for Engineering and Architecture, University of Naples Federico II, Via Claudio 21, 80125, Naples, Italy

    Giovanni Romano & Marina Diaco

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Romano, G., Diaco, M. On formulation of nonlocal elasticity problems. Meccanica 56, 1303–1328 (2021). https://doi.org/10.1007/s11012-020-01183-5

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