Abstract
This contribution is concerned with the numerical implementation of boundary potential energies and the study of their impact on the deformations of thermomechanical solids. Although boundary effects can play a dominant role in material behavior, the common modelling in continuum mechanics takes exclusively the bulk into account, nevertheless, neglecting possible contributions from the boundary. In the approach of this contribution the boundary is equipped with its own thermodynamic life, i.e. we assume separate boundary energy, entropy and the like. Furthermore, the generalized local balance laws are given according to Javili and Steinmann (Int J Solids Struct, 47:3245–3253, 2010). Afterwards, derivations of a generalized weak formulation which is employed for the discretization and finite element implementation, including boundary potentials, are carried out completely based on a tensorial representation. Finally, numerical examples are presented to demonstrate the boundary effects due to the proposed thermohyperelastic material model.
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Notes
- 1.
In the current paper the discretization procedure for the bulk is skipped, for the sake of space.
References
Armero, F., Simo, J.C.: A new unconditionally stable fractional step method for non-linear coupled thermomechanical problems. Int. J. Numer. Methods Eng. 35, 737–766 (1992)
Dettmer, W., Peric, D.: A computational framework for free surface fluid flows accounting for surface tension. Comp. Methods Appl. Mech. Eng. 195, 3038–3071 (2006)
Dingreville, R., Qu, J.: A semi-analytical method to compute surface elastic properties. Acta Mater. 55, 141–147 (2007)
Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, 1–68 (2009)
Fischer, F.D., Waitz, T., Vollath, D., Simha, N.K.: On the role of surface energy and surface stress in phase-transforming nanoparticles. Prog. Mater. Sci. 53, 481–527 (2008)
Fischer, F.D., Simha, N.K., Svoboda, J.: Kinetics of diffusional phase transformation in multicomponent elastic-plastic materials. J. Eng. Mater. Technol. 125, 266–276 (2003)
Fried, E., Gurtin, M.E.: A unified treatment of evolving interfaces accounting for small deformation and atomic transport with emphasis on Grain-Boundaries and expitaxy. Adv. Appl. Mech. 40, 1–177 (2004)
Gibbs, J.W.: The scientific papers of JW Gibbs, vol. 1. Dover Publications, New York (1961)
Gurtin, M.E., Ian Murdoch, A.: A continuum theory of elastic material surfaces. Arch. Rational Mech. Anal. 57, 291–323 (1975)
Gurtin, M.E., Struther, A.: Multiphase thermomechanics with interfacial .structure 3. Evolving phase boundaries in the presence of bulk deformation. Arch. Rational Mech. Anal. 112, 97–160 (1990)
He, J., Lilley, C.M.: Surface effect on the elastic behavior of static bending nanowires. Nano Lett. 8, 1798–1802 (2008)
He, J., Lilley, C.M.: The finite element absolute nodal coordinate formulation incorporated with surface stress effect to model elastic bending nanowires in large deformation. Comput. Mech. 44, 395–403 (2009)
Javili, A., Steinmann, P.: A finite element framework for continua with boundary energies. Part I: the two-dimensional case. Comp. Methods Appl. Mech. Eng. 198, 2198–2208 (2009)
Javili, A., Steinmann, P.: A finite element framework for continua with boundary energies. Part II: the three-dimensional case. Comp. Methods Appl. Mech. Eng. 199, 755–765 (2010)
Javili, A., Steinmann, P.: On thermomechanical solids with boundary structures. Int. J. Solids Struct. 47, 3245–3253 (2010)
Johnson, W.C.: Superficial stress and strain at coherent interfaces. Acta Mater. 48, 433–444 (2000)
Kaptay, G.: Classification and general derivation of interfacial forces, acting on phases, situated in the bulk, or at the interface of other phases. J. Mater Sci. 40, 2125–2131 (2005)
Kramer, D., Weissmüller, J.: A note on surface stress and surface tension and their interrelation via Shuttleworth’s equation and the Lippmann equation. Surf. Sci. 601, 3042–3051 (2007)
Leo, P.H., Sekerka, R.F.: The effect of surface stress on crystal-melt and crystal-crystal equilibrium. Acta Metall. 37, 3119–3138 (1989)
Miehe, C.: Entropic thermoelasticity at finite strains. Aspects of the formulation and numerical implementation. Comp. Methods Appl. Mech. Eng. 120, 243–269 (1995)
Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139 (2000)
Mitrushchenkov, A., Chambaud, G., Yvonnet, J., He, Q.C.: Towards an elastic model of wurtzite AlN nanowires. Nanotechnology 21, 255702 (2010)
Müller, P., Saul, A.: Elastic effects on surface physics. Surf. Sci. Rep. 54, 157–258 (2004)
Oden, J.T.: Finite Elements of Nonlinear Continua. Advanced Engineering Series. Mc Graw-Hill, New York (1972)
Olson, L., Kock, E.: A variational approach for modelling surface tension effects in inviscid fluids. Comput. Mech. 14(2), 140–153 (1994)
Park, H.S., Klein, P.A.: Surface Cauchy-born analysis of surface stress effects on metallic nanowires. Phys. Rev. B 75, 1–9 (2007)
Park, H.S., Klein, P.A.: A surface Cauchy-born model for silicon nanostructures. Comp. Methods Appl. Mech. Eng. 197, 3249–3260 (2008)
Park, H.S., Klein, P.A., Wagner, G.J.: A surface Cauchy-born model for nanoscale materials. Int. J. Numer. Methods Eng. 68, 1072–1095 (2006)
Rusanov, A.I.: Thermodynamics of solid surfaces. Surf. Sci. Rep. 23, 173–247 (1996)
Saksono, P.H., Peric, D.: On finite element modelling of surface tension: variational formulation and applications - part II: dynamic problems. Comput. Mech. 38, 251–263 (2006)
Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535–537 (2003)
Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B 71, 1–11 (2005)
Simha, N.K., Bhattacharya, K.: Kinetics of phase boundaries with edges and junctions. J. Mech. Phys. Solids 46, 2323–2359 (1998)
Steinmann, P.: On boundary potential energies in deformational and configurational mechanics. J. Mech. Phys. Solids 56, 772–800 (2008)
Steinmann, P., Häsner, O.: On material interfaces in thermomechanical solids. Arch. Appl. Mech. 75, 31–41 (2005)
Wang, B., She, H.: A geometrically nonlinear finite element model of nanomaterials with consideration of surface effect. Finite Elem. Anal. Des. 45, 463–467 (2009)
Wei, G., Shouwen, Y., Ganyun, H.: Finite element characterization of the size-dependent mechanical behaviour in nanosystems. Nanotechnology 17, 1118–1122 (2006)
Yang, F.: Effect of interfacial stresses on the elastic behavior of nanocomposite materials. J. Appl. Phys. 99, 054306 (2006)
Yun, G., Park, H.: A multiscale, finite deformation formulation for surface stress effects on the coupled thermomechanical behavior of nanomaterials. Comp. Methods Appl. Mech. Eng. 197, 3337–3350 (2008)
Yvonnet, J., Mitrushchenkov, A., Chambaud, G., He, Q.C.: Finite element model of ionic nanowires with size-dependent mechanical properties determined by ab initio calculations. Comp. Methods Appl. Mech. Eng. 200, 614–625 (2011)
Yvonnet, J., Quang, H.L., He, Q.C.: An XFEM level set approach to modelling surface/interface effects and computing the size-dependent effective properties of nanocomposites. Comput. Mater. Sci. 42, 119–131 (2008)
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Steinmann, P., Javili, A. (2013). Computational Thermomechanics with Boundary Structures. In: Cocks, A., Wang, J. (eds) IUTAM Symposium on Surface Effects in the Mechanics of Nanomaterials and Heterostructures. IUTAM Bookseries (closed), vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4911-5_16
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