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Erschienen in: Journal of Elasticity 1-2/2019

18.02.2019

Interactions in a Multi-scale Representation of Sparse Media: From Mechanics to Thermodynamics

verfasst von: Gianfranco Capriz, Paolo Maria Mariano

Erschienen in: Journal of Elasticity | Ausgabe 1-2/2019

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Abstract

We develop further a discrete-to-continuum approach to sparse media for which we do not grant the common axiom of permanent identification of the material elements. After refining our previous work on this topic, we focus our attention on the identification of stresses and self-actions at continuum scale in terms of the molecular interactions. Moreover, we introduce and discuss a notion of tensorial temperature by considering a grand canonical statistical ensemble constituted by molecules freely flowing in and out a control volume.

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Literatur
1.
Zurück zum Zitat Andersen, H.C.: Molecular dynamics simulations at constant pressure and/or temperature. J. Chem. Phys. 72, 2384–2393 (1980) ADSCrossRef Andersen, H.C.: Molecular dynamics simulations at constant pressure and/or temperature. J. Chem. Phys. 72, 2384–2393 (1980) ADSCrossRef
2.
Zurück zum Zitat Bacci, M., Mariano, P.M.: Protein dynamics: an approach based on the Cauchy-Born rule. Physica E 61, 69–82 (2014) ADSCrossRef Bacci, M., Mariano, P.M.: Protein dynamics: an approach based on the Cauchy-Born rule. Physica E 61, 69–82 (2014) ADSCrossRef
3.
Zurück zum Zitat Blumenfeld, R., Edwards, S.F.: On granular stress statistics: compactivity, angoricity, and some open issues. J. Phys. Chem. 113, 3981–3987 (2009) CrossRef Blumenfeld, R., Edwards, S.F.: On granular stress statistics: compactivity, angoricity, and some open issues. J. Phys. Chem. 113, 3981–3987 (2009) CrossRef
4.
Zurück zum Zitat Brocato, M., Capriz, G.: Clockwork, ephemeral and hybrid continua. Phys. Mesomech. 14, 124–144 (2011) Brocato, M., Capriz, G.: Clockwork, ephemeral and hybrid continua. Phys. Mesomech. 14, 124–144 (2011)
5.
Zurück zum Zitat Capriz, G.: Elementary preamble to a theory of granular gases. Rend. Mat. Univ. Padova 110, 179–198 (2003) MathSciNetMATH Capriz, G.: Elementary preamble to a theory of granular gases. Rend. Mat. Univ. Padova 110, 179–198 (2003) MathSciNetMATH
6.
Zurück zum Zitat Capriz, G.: Pseudofluids. In: Capriz, G., Mariano, P.M. (eds.) Material Substructures in Complex Bodies: From Atomic Level to Continuum, pp. 238–261. Elsevier, Amsterdam (2006) Capriz, G.: Pseudofluids. In: Capriz, G., Mariano, P.M. (eds.) Material Substructures in Complex Bodies: From Atomic Level to Continuum, pp. 238–261. Elsevier, Amsterdam (2006)
7.
8.
Zurück zum Zitat Capriz, G.: On ephemeral continua. Phys. Mesomech. 11, 285–298 (2008) CrossRef Capriz, G.: On ephemeral continua. Phys. Mesomech. 11, 285–298 (2008) CrossRef
9.
Zurück zum Zitat Capriz, G., Giovine, P.: Hypocontinua. In: Albers, B., Kuczma, M. (eds.) Continuum Media with Microstructure 2, pp. 23–43. Springer, Berlin (2016) CrossRef Capriz, G., Giovine, P.: Hypocontinua. In: Albers, B., Kuczma, M. (eds.) Continuum Media with Microstructure 2, pp. 23–43. Springer, Berlin (2016) CrossRef
10.
Zurück zum Zitat Capriz, G., Giovine, P.: Classes of ephemeral continua. Math. Methods Appl. Sci. 43, 1175–1196 (2017) MathSciNetMATH Capriz, G., Giovine, P.: Classes of ephemeral continua. Math. Methods Appl. Sci. 43, 1175–1196 (2017) MathSciNetMATH
11.
Zurück zum Zitat Capriz, G., Mariano, P.M.: Objective fluxes in a multiscale continuum description of sparse medium dynamics. Physica A 415, 354–365 (2014) ADSMathSciNetCrossRefMATH Capriz, G., Mariano, P.M.: Objective fluxes in a multiscale continuum description of sparse medium dynamics. Physica A 415, 354–365 (2014) ADSMathSciNetCrossRefMATH
12.
Zurück zum Zitat Capriz, G., Mariano, P.M.: Multi-scale kinetic description of granular clusters: invariance, balance, and temperature. Contin. Mech. Thermodyn. 30, 1323–1342 (2018) ADSMathSciNetCrossRef Capriz, G., Mariano, P.M.: Multi-scale kinetic description of granular clusters: invariance, balance, and temperature. Contin. Mech. Thermodyn. 30, 1323–1342 (2018) ADSMathSciNetCrossRef
13.
Zurück zum Zitat Capriz, G., Mazzini, G.: Invariance and balance in continuum mechanics. In: Nonlinear Analysis and Continuum Mechanics, pp. 27–35. Springer, New York (1998) CrossRef Capriz, G., Mazzini, G.: Invariance and balance in continuum mechanics. In: Nonlinear Analysis and Continuum Mechanics, pp. 27–35. Springer, New York (1998) CrossRef
14.
Zurück zum Zitat Capriz, G., et al. (eds.): Whence Boundary Conditions in Continuum Physics? Accademia Nazionale dei Lincei, Roma (2004) Capriz, G., et al. (eds.): Whence Boundary Conditions in Continuum Physics? Accademia Nazionale dei Lincei, Roma (2004)
17.
Zurück zum Zitat E, W., Huang, Z.: Matching conditions in atomistic-continuum modeling of materials. Phys. Rev. Lett. 87(1–4), 135501 (2001) ADSCrossRef E, W., Huang, Z.: Matching conditions in atomistic-continuum modeling of materials. Phys. Rev. Lett. 87(1–4), 135501 (2001) ADSCrossRef
18.
Zurück zum Zitat E, W., Ming, P.: Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007) MathSciNetCrossRefMATH E, W., Ming, P.: Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007) MathSciNetCrossRefMATH
19.
Zurück zum Zitat E, W., Ming, P.: Cauchy-Born rule and the stability of crystalline solids: dynamic problems. Acta Math. Appl. Sin. Engl. Ser. 529, 529–550 (2007) MathSciNetMATH E, W., Ming, P.: Cauchy-Born rule and the stability of crystalline solids: dynamic problems. Acta Math. Appl. Sin. Engl. Ser. 529, 529–550 (2007) MathSciNetMATH
20.
Zurück zum Zitat Ericksen, J.L.: On nonlinear elasticity theory of crystal defects. Int. J. Plast. 14, 9–24 (1998) CrossRefMATH Ericksen, J.L.: On nonlinear elasticity theory of crystal defects. Int. J. Plast. 14, 9–24 (1998) CrossRefMATH
21.
Zurück zum Zitat Goldhirsch, I.: Theory of rapid granular flows. In: Mehta, A. (ed.) Granular Physics, pp. 176–208. Cambridge University Press, Cambridge (2007) CrossRef Goldhirsch, I.: Theory of rapid granular flows. In: Mehta, A. (ed.) Granular Physics, pp. 176–208. Cambridge University Press, Cambridge (2007) CrossRef
23.
24.
Zurück zum Zitat Henkes, S., O’Hern, C.S., Chakraborty, B.: Entropy and temperature of a static granular assembly: an ab initio approach. Phys. Rev. Lett. 99, 038002 (2007) ADSCrossRef Henkes, S., O’Hern, C.S., Chakraborty, B.: Entropy and temperature of a static granular assembly: an ab initio approach. Phys. Rev. Lett. 99, 038002 (2007) ADSCrossRef
25.
Zurück zum Zitat Hoover, W.G.: Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A 31, 1695–1697 (1985) ADSCrossRef Hoover, W.G.: Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A 31, 1695–1697 (1985) ADSCrossRef
26.
Zurück zum Zitat Irving, J.H., Kirkwood, J.G.: The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950) ADSMathSciNetCrossRef Irving, J.H., Kirkwood, J.G.: The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950) ADSMathSciNetCrossRef
27.
Zurück zum Zitat Jenkins, J.T.: Boundary conditions for rapid granular flow: flat, frictional walls. J. Appl. Mech. 114, 120–127 (1992) CrossRefMATH Jenkins, J.T.: Boundary conditions for rapid granular flow: flat, frictional walls. J. Appl. Mech. 114, 120–127 (1992) CrossRefMATH
28.
Zurück zum Zitat Lehoucq, R.B., von Lilienfeld-Tonal, A.: Translation of Walter Noll’s “Derivation of the fundamental equations of continuum thermodynamics from statistical mechanics”. J. Elast. 100, 5–24 (2010) CrossRefMATH Lehoucq, R.B., von Lilienfeld-Tonal, A.: Translation of Walter Noll’s “Derivation of the fundamental equations of continuum thermodynamics from statistical mechanics”. J. Elast. 100, 5–24 (2010) CrossRefMATH
29.
Zurück zum Zitat Mariano, P.M.: Multifield theories in mechanics of solids. Adv. Appl. Mech. 38, 1–93 (2002) CrossRef Mariano, P.M.: Multifield theories in mechanics of solids. Adv. Appl. Mech. 38, 1–93 (2002) CrossRef
30.
Zurück zum Zitat Mariano, P.M.: Mechanics of material mutations. Adv. Appl. Mech. XX, 1–92 (2014) Mariano, P.M.: Mechanics of material mutations. Adv. Appl. Mech. XX, 1–92 (2014)
31.
Zurück zum Zitat Mariano, P.M., Stazi, F.L.: Strain localization in elastic microcracked bodies. Comput. Methods Appl. Mech. Eng. 190, 5657–5677 (2001) ADSCrossRefMATH Mariano, P.M., Stazi, F.L.: Strain localization in elastic microcracked bodies. Comput. Methods Appl. Mech. Eng. 190, 5657–5677 (2001) ADSCrossRefMATH
32.
Zurück zum Zitat Memarnahavandi, A., Larsson, F., Runesson, K.: A goal-oriented adaptive procedure for the quasi-continuum method with cluster approximation. Comput. Mech. 55, 617–642 (2015) MathSciNetCrossRefMATH Memarnahavandi, A., Larsson, F., Runesson, K.: A goal-oriented adaptive procedure for the quasi-continuum method with cluster approximation. Comput. Mech. 55, 617–642 (2015) MathSciNetCrossRefMATH
33.
Zurück zum Zitat Murdoch, A.I.: A corpuscular approach to continuum mechanics: basic considerations. Arch. Ration. Mech. Anal. 88, 291–321 (1985) MathSciNetCrossRefMATH Murdoch, A.I.: A corpuscular approach to continuum mechanics: basic considerations. Arch. Ration. Mech. Anal. 88, 291–321 (1985) MathSciNetCrossRefMATH
34.
Zurück zum Zitat Murdoch, A.I.: Physical Foundations of Continuum Mechanics. Cambridge University Press, Cambridge (2012) CrossRefMATH Murdoch, A.I.: Physical Foundations of Continuum Mechanics. Cambridge University Press, Cambridge (2012) CrossRefMATH
35.
Zurück zum Zitat Naudts, J., van der Straeten, E.: The grand canonical ensemble in generalized thermostatistics. J. Stat. Mech. Theory Exp. 2004, P12002 (2004) MathSciNetCrossRefMATH Naudts, J., van der Straeten, E.: The grand canonical ensemble in generalized thermostatistics. J. Stat. Mech. Theory Exp. 2004, P12002 (2004) MathSciNetCrossRefMATH
36.
Zurück zum Zitat Noll, W.: Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistichen Mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955) MATH Noll, W.: Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistichen Mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955) MATH
37.
Zurück zum Zitat Noll, W.: La Mécanique classique, basée sur une axiome d’objectivité. In: La Méthode Axiomatique dans les Mécaniques Classiques et Nouvelles, Colloque International, Paris, 1959, pp. 47–56. Gauthier-Villars, Paris (1963) Noll, W.: La Mécanique classique, basée sur une axiome d’objectivité. In: La Méthode Axiomatique dans les Mécaniques Classiques et Nouvelles, Colloque International, Paris, 1959, pp. 47–56. Gauthier-Villars, Paris (1963)
38.
Zurück zum Zitat Nosé, S.: A unified formulation of the constant temperature molecular-dynamics methods. J. Chem. Phys. 81, 511–519 (1984) ADSCrossRef Nosé, S.: A unified formulation of the constant temperature molecular-dynamics methods. J. Chem. Phys. 81, 511–519 (1984) ADSCrossRef
40.
Zurück zum Zitat Parrinello, M., Rahman, A.: Polymorphic transitions in single crystals: a new molecular dynamics method. J. Appl. Phys. 52, 7182–7190 (1981) ADSCrossRef Parrinello, M., Rahman, A.: Polymorphic transitions in single crystals: a new molecular dynamics method. J. Appl. Phys. 52, 7182–7190 (1981) ADSCrossRef
41.
Zurück zum Zitat Peters, M.H.: Completing Irving and Kirkwood’s molecular theory of transport processes: nonequilibrium entropy conservation. Ind. Eng. Chem. Res. 48, 166–171 (2009) CrossRef Peters, M.H.: Completing Irving and Kirkwood’s molecular theory of transport processes: nonequilibrium entropy conservation. Ind. Eng. Chem. Res. 48, 166–171 (2009) CrossRef
42.
Zurück zum Zitat Pitteri, M.: Continuum equations of balance in classical statistical mechanics. Arch. Ration. Mech. Anal. 94, 291–305 (1986). Corrigendum published as “Correction of an error in my paper: “Continuum equations of balance in classical statistical mechanics” [Arch. Rational Mech. Anal. 94 (1986), no. 4, 291–305]”. Arch. Rational Mech. Anal. 100, 315–316 (1988) MathSciNetCrossRefMATH Pitteri, M.: Continuum equations of balance in classical statistical mechanics. Arch. Ration. Mech. Anal. 94, 291–305 (1986). Corrigendum published as “Correction of an error in my paper: “Continuum equations of balance in classical statistical mechanics” [Arch. Rational Mech. Anal. 94 (1986), no. 4, 291–305]”. Arch. Rational Mech. Anal. 100, 315–316 (1988) MathSciNetCrossRefMATH
43.
44.
Zurück zum Zitat Puckett, J.G., Daniels, K.E.: Equilibrating temperaturelike variables in jammed granular subsystems. Phys. Rev. Lett. 110, 058001 (2013) ADSCrossRef Puckett, J.G., Daniels, K.E.: Equilibrating temperaturelike variables in jammed granular subsystems. Phys. Rev. Lett. 110, 058001 (2013) ADSCrossRef
45.
Zurück zum Zitat Ray, J.R., Rahman, A.: Statistical ensemble and molecular dynamics studies of anisotropic solids. II. J. Chem. Phys. 82, 4243–4247 (1985) ADSCrossRef Ray, J.R., Rahman, A.: Statistical ensemble and molecular dynamics studies of anisotropic solids. II. J. Chem. Phys. 82, 4243–4247 (1985) ADSCrossRef
46.
Zurück zum Zitat Shenoy, V.B., Miller, R., Tadmor, E.B., Rodney, D., Phillips, R., Ortiz, M.: An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method. J. Mech. Phys. Solids 47, 611–647 (1999) ADSMathSciNetCrossRefMATH Shenoy, V.B., Miller, R., Tadmor, E.B., Rodney, D., Phillips, R., Ortiz, M.: An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method. J. Mech. Phys. Solids 47, 611–647 (1999) ADSMathSciNetCrossRefMATH
47.
48.
Zurück zum Zitat Tadmor, E.B., Miller, R.E.: Modeling Materials: Continuum, Atomistic and Multiscale Techniques. Cambridge University Press, Cambridge (2011) CrossRefMATH Tadmor, E.B., Miller, R.E.: Modeling Materials: Continuum, Atomistic and Multiscale Techniques. Cambridge University Press, Cambridge (2011) CrossRefMATH
49.
Zurück zum Zitat Truesdell, C.A., Muncaster, R.G.: Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas. Treated as a Branch of Rational Mechanics. Academic Press, New York (1980) Truesdell, C.A., Muncaster, R.G.: Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas. Treated as a Branch of Rational Mechanics. Academic Press, New York (1980)
50.
Zurück zum Zitat Ulz, M.H.: A multiscale molecular dynamics method for isothermal dynamic problems using the seamless heterogeneous multiscale method. Comput. Methods Appl. Mech. Eng. 295, 510–524 (2015) ADSMathSciNetCrossRef Ulz, M.H.: A multiscale molecular dynamics method for isothermal dynamic problems using the seamless heterogeneous multiscale method. Comput. Methods Appl. Mech. Eng. 295, 510–524 (2015) ADSMathSciNetCrossRef
51.
Zurück zum Zitat Wurm, P., Ulz, M.H.: A stochastic approximation approach to improve the convergence behavior of hierarchical atomistic-to-continuum multiscale models. J. Mech. Phys. Solids 95, 480–500 (2016) ADSMathSciNetCrossRef Wurm, P., Ulz, M.H.: A stochastic approximation approach to improve the convergence behavior of hierarchical atomistic-to-continuum multiscale models. J. Mech. Phys. Solids 95, 480–500 (2016) ADSMathSciNetCrossRef
52.
Zurück zum Zitat Xu, M., Gracie, R., Belytschko, T.: A continuum-to-atomistic bridging domain method for composite lattices. Int. J. Numer. Methods Eng. 81, 1635–1658 (2010) MathSciNetMATH Xu, M., Gracie, R., Belytschko, T.: A continuum-to-atomistic bridging domain method for composite lattices. Int. J. Numer. Methods Eng. 81, 1635–1658 (2010) MathSciNetMATH
Metadaten
Titel
Interactions in a Multi-scale Representation of Sparse Media: From Mechanics to Thermodynamics
verfasst von
Gianfranco Capriz
Paolo Maria Mariano
Publikationsdatum
18.02.2019
Verlag
Springer Netherlands
Erschienen in
Journal of Elasticity / Ausgabe 1-2/2019
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI
https://doi.org/10.1007/s10659-019-09727-x

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