It has been shown recently that robust gradient models of elasticity offer an effective tool to resolve existing difficulties (e.g., elimination of singularities) or predict new experimentally observed phenomena (e.g., size effects) not captured by classical theory. The price that one pays for it, however, is the need to determine extra phenomenological coefficients and invent new boundary conditions associated with the gradient terms. The modest contribution of this note is to show that even the simplest possible gradient elasticity model may yield entirely different results for different higher-order boundary conditions used. This is demonstrated by considering the borehole problem, i.e., the determination of the state of stress and strain in an externally loaded or internally pressurized body containing a cylindrical hole. The standard practice of using variationally consistent boundary conditions leads to difficulties in physically interpreting them as well as in complex solution formulae without much physical insight. Alternative, mathematically less elegant arguments, as those employed here, lead to much neater formulas, usually consisting of a sum of the classical elasticity solution and an extra gradient term. These may easily be utilized to address engineering applications, ranging from processing/fabrication of metallic specimens at the micron and nanoscales to mining/drilling operations and tunneling excavation at geo-scales. The results are by no means conclusive, only suggestive of the various possibilities and choices that one can make at present. This, in itself, points to the pressing need of further work on the issue of extra boundary conditions with the aid of new novel experiments and accompanied multiscale simulations.
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Cook, G.: The yield point and initial stages of plastic strain in mild steel subjected to uniform and non-uniform stress distributions. Philos. Trans. R. Soc. A 230, 103–147 (1931)
Imamura, S., Stao, Y.: The size effect on yielding of a mild steel strip with a hole under tension. Trans. Jpn. Soc. Mech. Eng. Ser. A 52, 1440–1444 (1986). (in Japanese)
Taylor, M.B., Zbib, H.M., Khaleel, M.A.: Damage and size effect during superplastic deformation. Int. J. Plast. 18, 415–442 (2002)
Tsagrakis, I., Aifantis, E.C.: Recent developments in gradient plasticity-part I: formulation and size effects. ASME Trans. J. Eng. Mater. Technol. 124, 352–357 (2002)
Aifantis, E.C.: Higher order gradients and size effects. In: Carpinteri, A. (ed.) Size-Scale Effects in the Failure Mechanisms of Materials and Structures, pp. 231–241. E and FN Spon, London (1996)
Gao, X.L.: Analytical solution of a borehole problem using strain gradient plasticity. J. Eng. Mater. Technol. 124(3), 365–370 (2002)
Efremidis, G., Aifantis, E.C.: Gradient elasticity and size effect in borehole breakouts: an application of Ru-Aifantis theorem. J. Mech. Behav. Mater. 15(4–5), 279–289 (2004)
Efremidis, G., Rambert, G., Aifantis, E.C.: Gradient elasticity and size effect for a pressurized thick hollow cylinder. J. Mech. Behav. Mater. 15(3), 169–184 (2004)
Efremidis, G., Aifantis, E.C.: Gradient elasticity models for underground circular openings. In: Bazeos, N. et al. (eds.) Proceedings of 8th International Congress on Mechanics, Vol II, pp. 615–622. Patras (2007)
Efremidis, G., Pugno, N., Aifantis, E.C.: A proposition for a self-consistent gradient elasticity. J. Mech. Behav. Mater. 19(1), 15–29 (2009)
Efremidis, G.: On the theory of gradient elasto-plasticity and scale phenomena, Doctor of Philosophy (Ph.D.), Laboratory of Mechanics and Materials, Faculty of Engineering, Aristotle University of Thessaloniki (AUTh), Greece (2002); (in Greek)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)
Altan, B.S., Aifantis, E.C.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater. 8, 231–282 (1997)
Tsagrakis, I.: The role of gradients in elasticity and plasticity, Doctor of Philosophy (Ph.D.), Laboratory of Mechanics and Materials, Faculty of Engineering, Aristotle University of Thessaloniki (AUT), Greece (2001); (in greek)
Askes, H., Aifantis, E.C.: Numerical modeling of size effect with gradient elasticity. Part I: Formulation, meshless discretization and examples. Int. J. Fracture 117, 347–358 (2002). [or Askes, H., Aifantis, E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011)]
Lazar, M., Maugin, G.A., Aifantis, E.C.: On dislocations in a special class of generalized elasticity. Phys. Status Solidi B 242, 2365–2390 (2005)
Lurie, S., Volko-Bogorodsky, D., Leontive, A., Aifantis, E.C.: Eshelby’s inclusion problem in the gradient theory of elasticity: applications to composite materials. Int. J. Eng. Sci. 49, 1517–1525 (2011)
Lazopolulos, K.A., Alnefaie, N.H., Abu-Hamdeh, Aifantis, E.C.: The GRADELA plates and shells. In: Proceedings of the 10th Jubilee international Conference on Shell Structures, Theory and Applications (SSTA 2013), 16–18 October 2013, Gdansk, Poland, CRC Press/Balkema, Vol. 3, pp. 121–124 (2014)
Georgiadis, H.G.: The mode III crack problem in microstructured solids governed by dipolar gradient elasticity: static and dynamic analysis. J. Appl. Mech. 70(4), 517–530 (2003)
Aravas, N.: Plane-strain problems for a class of gradient elasticity models—a stress function approach. J. Elast. 104, 45–70 (2011)
Polizzotto, C.: Unified thermodynamic framework for nonlocal/gradient continuum theories. Eur. J. Mech. A Solids 22, 651–668 (2003). [or Polizzotto, C.: Non local elasticity and related variational principles. Int. J. Solids Struct. 38, 7359–7380 (2001)]
Altan, B.S., Aifantis, E.C.: On the structure of mode III crack tip in gradient elasticity. Scr. Met. Mater. 26, 319–324 (1992)
Ru, C.Q., Aifantis, E.C.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101, 59–68 (1993)
Aifantis, E.C.: Internal length gradient (ILG) material mechanics across scales and disciplines. Adv. Appl. Mech. 49, 1–110 (2016)
Aifantis, E.C.: Towards internal length chemomechanics. Rev. Adv. Mater. Sci. 48, 112–130 (2017)
Tsagrakis, I., Yasnikov, I.S., Aifantis, E.C.: Gradient elasticity for disclinated microcrystal. Mech. Res. Commun. https://doi.org/10.1016/j.mechrescom.2017.11.007
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Chen, H., Qi, C., Efremidis, G. et al. Gradient elasticity and size effect for the borehole problem. Acta Mech 229, 3305–3318 (2018). https://doi.org/10.1007/s00707-018-2109-3