Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Computational Mechanics
Erschienen in: Computational Mechanics 3/2014

01.09.2014 | Original Paper

A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology

verfasst von: A. Javili, A. McBride, P. Steinmann, B. D. Reddy

Erschienen in: Computational Mechanics | Ausgabe 3/2014

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

A curvilinear-coordinate-based finite element methodology is presented as a basis for a straightforward computational implementation of the theory of surface elasticity that mimics the underlying mathematical and geometrical concepts. An efficient formulation is obtained by adopting the same methodology for both the bulk and the surface. The key steps to evaluate the hyperelastic constitutive relations at the level of the quadrature point in a finite element scheme using this unified approach are provided. The methodology is illustrated through selected numerical examples.
Vorheriger Artikel Extended particle difference method for moving boundary problems
Nächster Artikel A hysteretic multiscale formulation for nonlinear dynamic analysis of composite materials

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren
Fußnoten
2
Note that we distinguish between the material, spatial and natural configurations. A line element \(\text{ d }{\varvec{X}}\) in the material configuration is mapped to \(\text{ d }{\varvec{x}}\) in the spatial configuration via the linear map \({\varvec{F}}\) and to \(\text{ d }\varvec{\xi }\) in the natural (reference) configuration via \(\varvec{K}\), see Table 3. The material, spatial and natural configurations on the surface are defined in a near-identical fashion to the bulk, see Table 4.
 
3
The routine used, rsgene2D, produces a Gaussian height distribution with an exponential auto-covariance. The input parameters were 100 divisions, a surface length of 2, a root mean square height of 0.05, and an (isotropic) correlation length of \(0.25\).
 
Literatur
1.
Agrawal R, Peng B, Gdoutos EE, Espinosa HD (2008) Elasticity size effects in ZnO nanowires: a combined experimental–computational approach. Nano Lett 8(11):3668–3674CrossRef
2.
Altenbach H, Eremeyev V (2011) On the shell theory on the nanoscale with surface stresses. Int J Eng Sci 49(12):1294–1301CrossRefMathSciNet
3.
Bangerth W, Hartmann R, Kanschat G (2007) deal.II: A general purpose object oriented finite element library. ACM Trans Math Softw 33(4):24CrossRefMathSciNet
4.
5.
Benveniste Y, Berdichevsky O (2010) On two models of arbitrarily curved three-dimensional thin interphases in elasticity. Int J Solids Struct 47(1415):1899–1915CrossRefMATH
6.
Benveniste Y, Miloh T (2001) Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech Mater 33(6):309–323CrossRef
7.
8.
Cammarata RC (1994) Surface and interface stress effects in thin films. Prog Surf Sci 46(1):1–38
9.
Ciarlet PG (2005) An introduction to differential geometry with applications to elasticity. J Elast 78:1–215CrossRefMathSciNet
10.
Davydov D, Javili A, Steinmann P (2013) On molecular statics and surface-enhanced continuum modeling of nano-structures. Comput Mater Sci 69:510–519CrossRef
11.
12.
Dettmer W, Perić D (2006) A computational framework for free surface fluid flows accounting for surface tension. Comput Methods Appl Mech Eng 195(23–24):3038–3071CrossRefMATH
13.
Dingreville R, Qu J (2007) A semi-analytical method to compute surface elastic properties. Acta Mater 55(1):141–147CrossRef
14.
Dingreville R, Qu J, Cherkaoui M (2005) Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J Mech Phys Solids 53(8):1827–1854CrossRefMATHMathSciNet
15.
Duan H, Wang J, Karihaloo B (2009) Theory of elasticity at the nonoscale. Adv Appl Mech 42:1–68CrossRef
16.
Duan HL, Karihaloo BL (2007) Effective thermal conductivities of heterogeneous media containing multiple imperfectly bonded inclusions. Physi Rev B 75:64206CrossRef
17.
Duan HL, Wang J, Huang ZP, Karihalo BL (2005) Eshelby formalism for nano-inhomogeneities. Proc R Soc A 461(2062):3335–3353CrossRefMATH
18.
Fischer FD, Svoboda J (2010) Stresses in hollow nanoparticles. Sci Direct 47:2799–2805MATH
19.
Green AE, Zerna W (1968) Theoretical elasticity. Oxford University Press, OxfordMATH
20.
Gu ST, He Q-C (2011) Interfacial discontinuity relations for coupled multifield phenomena and their application to the modeling of thin interphases as imperfect interfaces. J Mech Phys Solids 59(7):1413–1426CrossRefMATHMathSciNet
21.
22.
He J, Lilley CM (2008) Surface effect on the elastic behavior of static bending nanowires. Nano Lett 8(7):1798–1802CrossRef
23.
Heltai L (2008) On the stability of the finite element immersed boundary method. Comput Struct 86(7–8):598–617CrossRef
24.
Herring C (1951) Some theorems on the free energies of crystal surfaces. Phys Rev 82(1):87–93CrossRefMATH
25.
Huang Z, Sun L (2007) Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis. Acta Mech 190:151–163CrossRefMATH
26.
Hung Z, Wang J (2006) A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mech 182:195–210CrossRef
27.
Itskov M (2007) Tensor algebra and tensor analysis for engineers. Springer, BerlinMATH
28.
Javili A, McBride A, Steinmann P (2013) Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface and curve structures at the nanoscale. A unifying review. Appl Mech Rev 65(1):010802CrossRef
29.
Javili A, McBride A, Steinmann P, Reddy BD (2012) Relationships between the admissible range of surface material parameters and stability of linearly elastic bodies. Philos Mag 92:3540–3563CrossRef
30.
Javili A, Steinmann P (2009) A finite element framework for continua with boundary energies. Part I: the two-dimensional case. Comput Methods Appl Mech Eng 198(27–29):2198–2208CrossRefMATHMathSciNet
31.
Javili A, Steinmann P (2010a) A finite element framework for continua with boundary energies. Part II: the three-dimensional case. Comput Methods Appl Mech Eng 199(9–12):755–765CrossRefMATHMathSciNet
32.
Javili A, Steinmann P (2010b) On thermomechanical solids with boundary structures. Int J Solids Struct 47(24):3245–3253CrossRefMATH
33.
Javili A, Steinmann P (2011) A finite element framework for continua with boundary energies. Part III: the thermomechanical case. Comput Methods Appl Mech Eng 200(21–22):1963–1977CrossRefMATHMathSciNet
34.
Kreyszig E (1991) Differential geometry. Dover Publications, New York
35.
Levitas VI (2013) Thermodynamically consistent phase field approach to phase transformations with interface stresses. Acta Mater 61:4305–4319CrossRef
36.
McBride A, Javili A (2013) An efficient finite element implementation for problems in surface elasticity.
37.
Miller RE, Shenoy VB (2000) Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11(3):139CrossRef
38.
Orowan E (1970) Surface energy and surface tension in solids and liquids. Proc R Soc 316:473–491CrossRef
39.
Park HS, Klein PA (2007) Surface Cauchy–Born analysis of surface stress effects on metallic nanowires. Phys Rev B 75(8):1–9CrossRef
40.
Park HS, Klein PA (2008) A Surface Cauchy–Born model for silicon nanostructures. Comput Methods Appl Mech Eng 197(41–42):3249–3260CrossRefMATHMathSciNet
41.
Park HS, Klein PA, Wagner GJ (2006) A surface Cauchy–Born model for nanoscale materials. Int J Numer Methods Eng 68(10):1072–1095CrossRefMATHMathSciNet
42.
Saksono PH, Perić D (2005) On finite element modelling of surface tension variational formulation and applications. Part I: quasistatic problems. Comput Mech 38(3):265–281CrossRef
43.
Scriven LE (1960) Dynamics of a fluid interface equation of motion for newtonian surface fluids. Chem Eng Sci 12(2):98–108CrossRef
44.
Scriven LE, Sternling CV (1960) The marangoni effects. Nature 187:186–188CrossRef
45.
Sharma P, Ganti S (2004) Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies. J Appl Mech 71:663–671CrossRefMATH
46.
Sharma P, Ganti S, Bhate N (2003) Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl Phys Lett 82(4):535–537
47.
Sharma P, Wheeler LT (2007) Size-dependent elastic state of ellipsoidal nano-inclusions incorporating surface/interface tension. J Appl Mech 74(3):447–454CrossRefMATHMathSciNet
48.
Shenoy VB (2005) Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys Rev B 71(9):1–11
49.
Shuttleworth R (1950) The surface tension of solids. Proc Phys Soc Sect A 63(5):444–457CrossRef
50.
Steigmann DJ (2009) A concise derivation of membrane theory from three-dimensional nonlinear elasticity. J Elast 97:97–101
51.
Sussmann C, Givoli D, Benveniste Y (2011) Combined asymptotic finite-element modeling of thin layers for scalar elliptic problems. Comput Methods Appl Mech Eng 200(4748):3255–3269CrossRefMATHMathSciNet
52.
53.
Wang Z-Q, Zhao Y-P, Huang Z-P (2010b) The effects of surface tension on the elastic properties of nano structures. Int J Eng Sci 48(2):140–150CrossRefMathSciNet
54.
Wei GW, Shouwen Y (2006) Finite element characterization of the size-dependent mechanical behaviour in nanosystems. Nanotechnology 17(4):1118–1122CrossRef
55.
Weissmüller J, Duan H-L, Farkas D (2010) Deformation of solids with nanoscale pores by the action of capillary forces. Acta Mater 58(1):1–13CrossRef
56.
Wriggers P (2008) Nonlinear finite element methods. Springer, BerlinMATH
57.
Yun G, Park HS (2009) Surface stress effects on the bending properties of fcc metal nanowires. Phys Rev B 79(19):32–35CrossRef
58.
Yvonnet J, Mitrushchenkov A, Chambaud G, He Q-C (2011) Finite element model of ionic nanowires with size-dependent mechanical properties determined by ab initio calculations. Comput Methods Appl Mech Eng 200(5–8):614–625CrossRefMATHMathSciNet
59.
Yvonnet J, Quang HL, He Q-C (2008) An XFEM level set approach to modelling surface–interface effects and computing the size-dependent effective properties of nanocomposites. Comput Mech 42(1):119–131CrossRefMATHMathSciNet
Metadaten
Titel
A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology
verfasst von
A. Javili
A. McBride
P. Steinmann
B. D. Reddy
Publikationsdatum
01.09.2014
Verlag
Springer Berlin Heidelberg
Erschienen in
Computational Mechanics / Ausgabe 3/2014
Print ISSN: 0178-7675
Elektronische ISSN: 1432-0924
DOI
https://doi.org/10.1007/s00466-014-1030-4

Weitere Artikel der Ausgabe 3/2014

Computational Mechanics 3/2014 Zur Ausgabe

Original Paper

Inverse material identification in coupled acoustic-structure interaction using a modified error in constitutive equation functional

Original Paper

New advances in the forced response computation of periodic structures using the wave finite element (WFE) method

Original Paper

Numerical modeling of corrosion pit propagation using the combined extended finite element and level set method

Original Paper

Entropy propagation analysis in stochastic structural dynamics: application to a beam with uncertain cross sectional area

Original Paper

A new hybrid smoothed FEM for static and free vibration analyses of Reissner–Mindlin Plates

Original Paper

A boundary element model for near surface contact stresses of rough surfaces

Neuer Inhalt

    Bildnachweise