Abstract
Problems involving coupled multiple space and time scales offer a real challenge for conventional frameworks of either particle or continuum mechanics. In this paper, four cases studies (shear band formation in bulk metallic glasses, spallation resulting from stress wave, interaction between a probe tip and sample, the simulation of nanoindentation with molecular statistical thermodynamics) are provided to illustrate the three levels of trans-scale problems (problems due to various physical mechanisms at macro-level, problems due to micro-structural evolution at macro/micro-level, problems due to the coupling of atoms/molecules and a finite size body at micro/nano-level) and their formulations. Accordingly, non-equilibrium statistical mechanics, coupled trans-scale equations and simultaneous solutions, and trans-scale algorithms based on atomic/molecular interaction are suggested as the three possible modes of trans-scale mechanics.
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References
Tsien S.H. (1962). Physical Mechanics (in Chinese). Science Press, Beijing
Marder M. and Fineberg J. (1996). How things break. Phys. Today 499: 24–29
Conner R.D., Johnson W.L., Paton N.E. and Nix W.D. (2003). Shear bands and cracking of metallic glass plates in bending. J. Appl. Phys. 94: 904–911
Shen L.T., Zhao S.D., Bai Y.L. and Luo L.M. (1992). Experimental study on the criteria and mechanism of spallation in an Al alloy. Int. J. Impact Eng. 12: 9–19
Weinan E. and Engquist B. (2003). The heterogeneous multiscale methods. Commun. Math. Sci. 1: 87–132
Galileo G. (2001). Dialogue Concerning Two New Sciences. William Andrew Publishing, New York
Schlichting H. (1968). Boundary Layer Theory. McGraw-Hill, New York
Hall E.O. (1951). The deformation and aging of mild steel. Proc. Phys. Soc. Lond. Sect. B 64: 747–753
Petch N.J. (1953). The cleavage strength of polystrals. J. Iron Steel Inst. 173: 25–28
Glimm J. and Sharp D.H. (1997). Multiscale science: A challenge for the twenty-first century. SIAM News 30: 1–7
Sih, G.C.: Mesomechanics 2000 Role of Mechanics for Development of Science and Technology. In: Sih, G.C. (ed.) Tsinghua University Press, Beijing (2000)
Spaepen F. and Turnbull D. (1974). A mechanism for the flow and fracture of metallic glasses. Scripta Metall. 8: 563–568
Spaepen F. (1977). A microscopic mechanism for steady state inhomogeneous flow in metallic glasses. Acta Metall. 25: 407–415
Falk, M.L., Shi, Y.: Strain localization in a molecular-dynamics model of a metallic glass. In: Egami, T., Greer, A.L., Inoue, A., Ranganathan, S. (eds.). Supercooled Liquids, Glass Transition, Bulk Metallic Glasses. Mat. Res. Soc. Proc. vol. 754, pp. 1–6. Pittsburgh (2003)
Wang G.H., Pan H., Ke F.J., Xia M.F. and Bai Y.L. (2008). Study of mechanical properties of amorphous copper with molecular dynamics simulation. Chin. Phys. B 17: 1–5
Steif P.S., Spaepen F. and Hutchinson J.W. (1982). Strain localization in amorphous metals. Acta Metall. 30: 447–455
Huang R., Suo Z.G., Prevost J.H. and Nix W.D. (2002). Inhomogeneous deformation in metallic glasses. J. Mech. Phys. Solids. 50: 1011–1027
Dai L.H., Yan M., Liu L.F. and Bai Y.L. (2005). Adiabatic shear banding instability in bulk metallic glasses. Appl. Phys. Lett. 87: 141916
Bruck H.A., Rosakis A.J. and Johnson W.L. (1996). The dynamic compressive behavior of beryllium bearing bulk metallic glasses. J. Mater. Res. 11: 503–511
Lewandowski J.J. and Greer A.L. (2006). Temperature rise at shear bands in metallic glasses. Nat. Mater. 5: 15–18
Masumoto T. and Maddin R. (1971). Mechanical properties of palladium-20 atomic % silicon alloy quenched from the liqiud state. Acta Metall. 19: 725–741
Donvan P.E. and Stobbs W.M. (1981). The structure of shear bands in metallic glass. Acta Metall. 29: 1419–1436
Pekarskaya E., Kim C.P. and Johnson W.L. (2001). In-situ transmission electron microscopy studies of shear bands in a bulk metallic glass based composite. J. Mater. Res. 16: 2513–2518
Li J., Wang Z.L. and Hufnagel T.C. (2002). Characterization of nanometer-scale defects in metallic glasses by quantitative high-resolution transmission electron microscopy. Phys. Rev. B 65: 144201
Liu, L.F.: Formation mechanism of shear band in bulk metallic glass (in Chinese). PhD Thesis, Institute of Mechanics, Chinese Academy of Sciences, Beijing (2006)
Dai, L.H., Bai, Y.L.: Basic mechanical behaviors and mechanics of shear bending in BMGs. Int. J. Impact Eng. (in press)
Conner R.D., Dandliker R.B., Scruggs V. and Johnson W.L. (2000). Dynamic deformation behavior of W-fiber/metallic-glass matrix composites. Int. J. Impact. Eng. 24: 435–444
Wang W.H., Dong C. and Shek C.H. (2004). Bulk metallic glasses. Mater. Sci. Eng. R 44: 45–89
Liu L.F., Dai L.H., Bai Y.L. and Eckert J. (2006). Characterization of rate-dependent shear behavior of Zr-based bulk metallic glass using shear-punch testing. J. Mater. Res. 21: 153–160
Bai Y.L., Ling Z., Luo L.M. and Ke F.J. (1992). Initial development of microdamage under impact loading., ASME Trans. J. Appl. Mech. 59: 622–627
Tuler F.R. and Butcher B.M. (1968). A criterion for the time dependence of dynamic fracture. Int. J. Fracture Mech. 4: 431–437
Shen L.T., Zhao S.D., Bai Y.L. and Luo L.M. (1992). Experimental study on the criteria and mechanism of spallation in an Al alloy. Int. J. Impact Eng. 12: 9–19
Meyers M.A. (1994). Dynamic Behaviour of Materials. Wiley, New York
Grady D.E. and Kipp M.E. (1993). Dynamic fracture and fragmentation. In: Asay, J.R. and Shahinpoor, M. (eds) High-Pressure Shock Compression of Solids, pp 265–332. Springer, New York
Davison L. and Stevens A.L. (1972). Continuum measures of spall damage. J. Appl. Phys. 43: 988–994
Han W.S., Xia M.F., Shen L.T. and Bai Y.L. (1997). Statistical formulation and experimental determination of growth rate of micrometre cracks under impact loading. Int. J. Solids Struct. 34: 2905–2925
Bai Y.L., Xia M.F., Ke F.J. and Li H.L. (2002). Closed trans-scale statistical microdamage mechanics. Acta Mech. Sin. 18: 1–17
Xia M.F., Han W.S., Ke F.J. and Bai Y.L. (1995). Statistical meso-scopic damage mechanics and damage evolution induced catastrophe (in Chinese). ADv. Mech. 25(1–40): 145–173
Wang H.Y., Bai Y.L., Xia M.F. and Ke F.J. (2004). Spallation analysis with a closed trans-scale formulation of damage evolution. Acta Mech. Sin. 20: 400–407
Wang H.Y., Bai Y.L., Xia M.F. and Ke F.J. (2006). Microdamage evolution, energy dissipation and its trans-scale effects on macroscopic failure. Mech. Mater. 38: 57–67
Bai Y.L., Han W.S. and Bai J. (1997). A statistical evolution equation of microdamage and its application. ASTM STP 1315: 150–162
Bai Y.L. and Wang H.Y. (2004). The significance of the intrinsic Deborah numbers in failure. In: Hong, Y.S. (eds) Advances in Applied Mechanics. Science Press, Beijing
Freund J., Halbritter J. and Horber J.K.H. (1999). How dry are dried samples? Water adsorption measured by STM. Microsc. Res. Tech. 44: 327–338
Liu N., Bai Y.L., Xia M.F. and Ke F.J. (2005). Combined effect of surface tension, gravity and van der Waals force induced by a non-contact probe tip on the shape of liquid surface. Chin. Phys. Lett. 22: 2012–2015
Israelachvili J.N. (1985). Intermolecular and Surface Forces. Academic Press, San Diego
Cortat F.P.A. and Miklavcic S.J. (2004). Using stable and unstable profiles to deduce deformation limits of the air-water interfaces. Langmuir 20: 3208–3220
Wang H.Y., Hu M., Liu N., Xia M.F., Ke F.J. and Bai Y.L. (2007). Multi-scale analysis of AFM tip and surface interactions. Chem. Eng. Sci. 62: 3589–3594
Liu, N.: The distortion and instability of liquid surface induced by a sub-microscale probe. [Master’s Thesis], Institute of Mechanics, Chinese Academy of Sciences, Beijing (2005) (in Chinese)
Haile J.M. (1997). Molecular Dynamics Simulation. Wiley, New York
Oliver W.C. and Pharr G.M. (1992). An improve technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7: 1564–1583
Ma Q. and Clark D.R. (1995). Size-dependent hardness of silver single crystals. J. Mater. Res. 10: 853–863
McElhaney K.W., Vlassak J.J. and Nix W.D. (1998). Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments. J. Mater. Res. 13: 1300–1306
Liu Y. and Ngan A.H.W. (2001). Depth dependence of hardness in copper single crystals measured by nanoindentation. Script Mater. 44: 237–241
Wei Y.G., Wang X.Z., Wu X.L. and Bai Y.L. (2001). Theoretical and experimental researches of size effect in micro-indentation test. Sci. China, A 44: 74–82
Chen S.D. and Ke F.J. (2004). MD simulation of the effect of contact area and tip radius on nanoindentation. Sci. China Ser. G-Phys. Astron. 47: 101–112
Noreyan A., Amar J.G. and Marinescu I. (2005). Molecular dynamics simulations of nanoindentation beta-SiC with diamond indenter. Mater. Sci. Eng. B Solid State Mater. Adv. Technol. 117: 235–240
Dupuy L.M., Tadmor E.B., Miller R.E. and Phillips R. (2005). Finite-temperature quasicontinuum: Molecular dynamics without all the atoms. Phys. Rev. Lett. 95: 060202
Hu, M., Wang, H.Y., Bai, Y.L., Xia, M.F., Ke, F.J.: Cluster statistical thermodynamics—to efficiently calculate quasi-static deformation at finite temperature based on molecular potential. In: IUTAM Symposium on Mechanical Behavior and Micro-Mechanics of Nanostructured Materials, pp. 163–170, Beijing, China, June 27–30, 2005 (2007)
Wang, H.Y., Hu, M., Xia, M.F., Ke, F.J., Bai, Y.L.: Molecular/cluster statistical thermodynamics methods to simulate quasi-static deformations at finite temperature. Int. J. Solids Struct. (in press)
Hu, M.: Statistical quasicontinuum method at nano/micro-meter scales and its applications (in Chinese). Ph.D. Thesis, Institute of Mechanics, Chinese Academy of Sciences, Beijing (2006)
Born M. and Huang K. (1954). Dynamical Theory of Crystal Lattices. Oxford Press, Landon
Ellis D.E., Mundim K.C., Fuks D., Dorfman S. and Berner A. (2000). Modeling of copper–carbon solid solutions. Mater. Sci. Semiconductor Process. 3: 123–127
Sneddon I.N. (1965). The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3: 47–57
Kadanoff L.P. (2000). Statistical Physics (Statics, Dynamics and Renormalization). World Scientific, Singapore
Barenblatt, G.I.: Micromechanics of fracture, Theoretical and Applied Mechanics. Bodner S.R., Singer J., Solan A., Hashin J. (eds.) Elesevier, Amsterdam, pp. 25–52 (1992)
Einstein A. (1954). Physics and Reality, in Einstein, Ideas and Opinions. Crown, New York
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The project supported by the National Basic Research Program of China (2007CB814800), the National Natural Science Foundation of China (10432050, 10572139, 10721202, 10772012, 10772181, 90715001), and CAS Innovation Program (KJCX2-SW-L08, KJCX2-YW-M04).
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Bai, Y.L., Wang, H.Y., Xia, M.F. et al. Trans-scale mechanics: looking for the missing links between continuum and micro/nanoscopic reality. Acta Mech. Sin. 24, 111–126 (2008). https://doi.org/10.1007/s10409-008-0147-0
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DOI: https://doi.org/10.1007/s10409-008-0147-0