×

zbMATH — the first resource for mathematics

Geometrically nonlinear higher-gradient elasticity with energetic boundaries. (English) Zbl 1294.74014
Summary: The objective of this contribution is to formulate a geometrically nonlinear theory of higher-gradient elasticity accounting for boundary (surface and curve) energies. Surfaces and curves can significantly influence the overall response of a solid body. Such influences are becoming increasingly important when modeling the response of structures at the nanoscale. The behavior of the boundaries is well described by continuum theories that endow the surface and curve with their own energetic structures. Such theories often allow the boundary energy density to depend only on the superficial boundary deformation gradient. From a physical point of view though, it seems necessary to define the boundary deformation gradient as the evaluation of the deformation gradient at the boundary rather than its projection. This controversial issue is carefully studied and several conclusions are extracted from the rigorous mathematical framework presented.
In this manuscript the internal energy density of the bulk is a function of the deformation gradient and its first and second derivatives. The internal energy density of the surface is, consequently, a function of the deformation gradient at the surface and its first derivative. The internal energy density of a curve is, consequently, a function of the deformation gradient at the curve.
It is shown that in order to have a surface energy depending on the total (surface) deformation gradient, the bulk energy needs to be a function of at least the first derivative of the deformation gradient. Furthermore, in order to have a curve energy depending on the total (curve) deformation gradient, the bulk energy needs to be a function of at least the second derivative of the deformation gradient. Clearly, the surface elasticity theory of Gurtin and Murdoch is intrinsically limited since it is associated with the classical (first-order) continuum theory of elasticity in the bulk. In this sense this contribution shall be also understood as a higher-gradient surface elasticity theory.

MSC:
74B20 Nonlinear elasticity
53C80 Applications of global differential geometry to the sciences
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agiasofitou, E.; Lazar, M., Conservation and balance laws in linear elasticity of grade three, J. Elasticity, 94, 69-85, (2009) · Zbl 1159.74329
[2] Aifantis, E. C., On scale invariance in anisotropic plasticity gradient plasticity, and gradient elasticity, Int. J. Eng. Sci., 47, 11-12, 1089-1099, (2009) · Zbl 1213.74061
[3] Akgoz, B.; Civalek, O., Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, Int. J. Eng. Sci., 49, 11, 1268-1280, (2011) · Zbl 1423.74338
[4] Akgoz, B.; Civalek, O., Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory, Arch. Appl. Mech., 82, 423-443, (2012) · Zbl 1293.74252
[5] Altan, B. S.; Evensen, H. A.; Aifantis, E. C., Longitudinal vibrations of a beama gradient elasticity approach, Mech. Res. Commun., 23, 1, 35-40, (1996) · Zbl 0843.73048
[6] Altan, B. S.; Miskioglu, I.; Vilmann, C. R., Propagation of s-h waves in laminated compositesa gradient elasticity approach, J. Vib. Control, 9, 11, 1265-1283, (2003) · Zbl 1078.74596
[7] Andreussi, F.; Gurtin, M. E., On the wrinkling of a free surface, J. Appl. Phys., 48, 9, 3798-3799, (1977)
[8] Anthoine, A., Effect of couple-stresses on the elastic bending of beams, Int. J. Solids Struc., 37, 7, 1003-1018, (2000) · Zbl 0978.74044
[9] Arnold, V. I., Mathematical methods of classical mechanics, (1989), Springer
[10] Askes, H.; Aifantis, E. C., Gradient elasticity in statics and dynamicsan overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solids Struc., 48, 13, 1962-1990, (2011)
[11] Bar On, B.; Altus, E.; Tadmor, E., Surface effects in non-uniform nanobeamscontinuum vs. atomistic modeling, Int. J. Solids Struc., 47, May (9), 1243-1252, (2010) · Zbl 1193.74006
[12] Benveniste, Y.; Miloh, T., Imperfect soft and stiff interfaces in two-dimensional elasticity, Mech. Mater., 33, 6, 309-323, (2001)
[13] Benvenuto, E., An introduction to the history of structural mechanicspart II: vaulted structures and elastic systems, (1991), Springer Verlag
[14] Bowen, R.M., Wang, C.C., 1976. Introduction to Vectors and Tensors: Linear and Multilinear Algebra, vol. 1. · Zbl 0329.53008
[15] Cammarata, R. C., Surface and interface stress effects in thin films, Prog. Surf. Sci., 46, 1, 1-38, (1994)
[16] Cammarata, R. C., Surface and interface stress effects on interfacial and nanostructured materials, Mater. Sci. Eng.A, 237, 2, 180-184, (1997)
[17] Cammarata, R. C., Generalized thermodynamics of surfaces with applications to small solid systems, Solid State Phys., 61, 1-75, (2009)
[18] Cammarata, R. C.; Sieradzki, K.; Spaepen, F., Simple model for interface stresses with application to misfit dislocation generation in epitaxial thin films, J. Appl. Phys., 87, 3, 1227-1234, (2000)
[19] Chambon, R.; Caillerie, D.; Hassan, N. E., One-dimensional localisation studied with a second grade model, Eur. J. Mech. A/Solids, 17, 4, 637-656, (1998) · Zbl 0936.74020
[20] Chang, C.; Shi, Q.; Liao, C., Elastic constants for granular materials modeled as first-order strain-gradient continua, Int. J. Solids Struc., 40, 21, 5565-5582, (2003) · Zbl 1059.74019
[21] Charlotte, M.; Truskinovsky, L., Linear elastic chain with a hyper-prestress, J. Mech. Phys. Solids, 50, 2, 217-251, (2002) · Zbl 1035.74005
[22] Charlotte, M.; Truskinovsky, L., Towards multi-scale continuum elasticity theory, Continuum Mech. Thermodyn., 20, 3, 133-161, (2008) · Zbl 1172.74006
[23] Chen, S.; Feng, B., Size effect in micro-scale cantilever beam bending, Acta Mech., 219, 291-307, (2011) · Zbl 1333.74057
[24] Chhapadia, P.; Mohammadi, P.; Sharma, P., Curvature-dependent surface energy and implications for nanostructures, J. Mech. Phys. Solids, 59, 10, 2103-2115, (2011) · Zbl 1270.74018
[25] Ciarlet, P. G., An introduction to differential geometry with applications to elasticity, (2005), Springer · Zbl 1100.53004
[26] Ciarletta, P.; Ambrosi, D.; Maugin, G., Mass transport in morphogenetic processesa second gradient theory for volumetric growth and material remodeling, J. Mech. Phys. Solids, 60, 3, 432-450, (2012) · Zbl 1244.74084
[27] Cihan, Tekoglu; Patrick, R. Onck, Size effects in two-dimensional Voronoi foamsa comparison between generalized continua and discrete models, J. Mech. Phys. Solids, 56, 12, 3541-3564, (2008) · Zbl 1171.74411
[28] Cosserat, E.; Cosserat, F., Note sur la théorie de l’action euclidienne, (1908), Gauthier-Villars Paris · JFM 39.0856.01
[29] Cosserat, E.; Cosserat, F., Sur la théorie des corps Déformables, (1909), Herman Paris · JFM 40.0862.02
[30] Daher, N.; Maugin, G. A., The method of virtual power in continuum mechanics application to media presenting singular surfaces and interfaces, Acta Mech., 60, 3-4, 217-240, (1986) · Zbl 0594.73004
[31] Danescu, A., Hyper-pre-stress vs. strain-gradient for surface relaxation in diamond-like structures, J. Mech. Phys. Solids, 60, 4, 623-642, (2012) · Zbl 1244.74019
[32] Danescu, A.; Sidoroff, F., Second gradient vs. surface energy in the asaro-grinfeld instability, J. Phys. IV France, 8, 79-86, (1998)
[33] Davydov, D.; Javili, A.; Steinmann, P., On molecular statics and surface-enhanced continuum modeling of nano-structures, Comput. Mater. Sci., 69, 510-519, (2013)
[34] dell’Isola, F.; Kosinski, W., Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modeling interphase layers, Arch. Mech., 45, 333-359, (1993) · Zbl 0815.73007
[35] dell’Isola, F.; Romano, A., On a general balance law for continua with an interface, Ric. Mat., 325-337, (1986) · Zbl 0629.76004
[36] dell’Isola, F.; Romano, A., On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface, Int. J. Eng. Sci., 25, 11-12, 1459-1468, (1987) · Zbl 0624.73001
[37] dell’Isola, F.; Seppecher, P., The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power, C. R. Acad. Sci. Ser. IIB - Mech.-Phys.-Astron., 7, 43-48, (1995) · Zbl 0844.73006
[38] dell’Isola, F.; Seppecher, P., Edge contact forces and quasi-balanced power, Meccanica, 32, 1, 33-52, (1997) · Zbl 0877.73055
[39] dell’Isola, F.; Seppecher, P.; Madeo, A., How contact interactions may depend on the shape of Cauchy cuts in nth gradient continuaapproach “à la D’alembert”, Z. für Angew. Math. Phys., 1-23, (2012)
[40] Dingreville, R.; Qu, J., Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films, J. Mech. Phys. Solids, 53, 8, 1827-1854, (2005) · Zbl 1120.74683
[41] Dingreville, R.; Qu, J., A semi-analytical method to compute surface elastic properties, Acta Materialia, 55, 1, 141-147, (2007)
[42] Dobovsek, I., Problem of a point defect, spatial regularization and intrinsic length scale in second gradient elasticity, Mater. Sci. Eng.A, 423, 92-96, (2006)
[43] Duan, H. L.; Karihaloo, B. L., Effective thermal conductivities of heterogeneous media containing multiple imperfectly bonded inclusions, Phys. Rev. B, 75, 064206, (2007)
[44] Duan, H. L.; Wang, J.; Huang, Z. P.; Karihalo, B. L., Eshelby formalism for nano-inhomogeneities, Proc. R Soc. A, 461, 2062, 3335-3353, (2005) · Zbl 1370.74068
[45] Duan, H. L.; Wang, J.; Huang, Z. P.; Karihaloo, B. L., Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress, J. Mech. Phys. Solids, 53, 7, 1574-1596, (2005) · Zbl 1120.74718
[46] Duan, H. L.; Wang, J.; Karihaloo, B. L., Theory of elasticity at the nanoscale, Adv. Appl. Mech., 42, 1-68, (2009)
[47] Exadaktylos, G., Gradient elasticity with surface energymode-i crack problem, Int. J. Solids Struc., 35, 5-6, 421-456, (1998) · Zbl 0930.74050
[48] Exadaktylos, G., Some basic half-plane problems of the cohesive elasticity theory with surface energy, Acta Mech., 133, 175-198, (1999) · Zbl 0922.73007
[49] Exadaktylos, G.; Vardoulakis, I., Surface instability in gradient elasticity with surface energy, Int. J. Solids Struc., 35, 18, 2251-2281, (1998) · Zbl 0935.74017
[50] Exadaktylos, G.; Vardoulakis, I.; Aifantis, E., Cracks in gradient elastic bodies with surface energy, Int. J. Fract., 79, 107-119, (1996) · Zbl 0919.73237
[51] Fatemi, J.; Van Keulen, F.; Onck, P., Generalized continuum theoriesapplication to stress analysis in bone, Meccanica, 37, 385-396, (2002) · Zbl 1141.74344
[52] Fernandes, R.; Chavant, C.; Chambon, R., A simplified second gradient model for dilatant materialstheory and numerical implementation, Int. J. Solids Struc., 45, 20, 5289-5307, (2008) · Zbl 1194.74027
[53] Fischer, F. D.; Svoboda, J., Stresses in hollow nanoparticles, Int. J. Solids Struc., 47, 20, 2799-2805, (2010) · Zbl 1196.74160
[54] 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, 3, 481-527, (2008)
[55] Fleck, N.; Hutchinson, J., Strain gradient plasticity, Adv. Appl. Mech., 33, 295-361, (1997) · Zbl 0894.73031
[56] Forest, S.; Barbe, F.; Cailletaud, G., Cosserat modeling of size effects in the mechanical behavior of polycrystals and multi-phase materials, Int. J. Solids Struc., 37, 46-47, 7105-7126, (2000) · Zbl 0998.74019
[57] Forest, S.; Cordero, N. M.; Busso, E. P., First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales, Comput. Mater. Sci., 50, 4, 1299-1304, (2011)
[58] Forest, S.; Francis, P.; Sab, K., Asymptotic analysis of heterogeneous Cosserat media, Int. J. Solids Struc., 38, 26-27, 4585-4608, (2001) · Zbl 1033.74038
[59] Forest, S.; Sievert, R., Elastoviscoplastic constitutive frameworks for generalized continua, Acta Mech., 160, 1-2, 71-111, (2003) · Zbl 1064.74009
[60] Fried, E.; Todres, R., Mind the gapthe shape of the free surface of a rubber-like material in proximity to a rigid contactor, J. Elasticity, 80, 97-151, (2005) · Zbl 1197.74013
[61] Ganghoffer, J.-F., Mechanical modeling of growth considering domain variation-part iivolumetric and surface growth involving eshelby tensors, J. Mech. Phys. Solids, 58, 9, 1434-1459, (2010) · Zbl 1429.74072
[62] Gao, X.-L.; Ma, H., Solution of eshelby’s inclusion problem with a bounded domain and Eshelby’s tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory, J. Mech. Phys. Solids, 58, 5, 779-797, (2010) · Zbl 1244.74023
[63] Gao, X.-L.; Park, S., Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem, Int. J. Solids Struc., 44, 22-23, 7486-7499, (2007) · Zbl 1166.74318
[64] Germain, P., Sur l’application de la méthode des puissances virtuelles en mécanique des milieux continus, C. R. Acad. Sci. Paris Série A-B, 274, A1051-A1055, (1972)
[65] Germain, P., The method of virtual power in continuum mechanics. part 2microstructure, SIAM J. Appl. Math., 25, 3, 556-575, (1973) · Zbl 0273.73061
[66] Gogosov, V. V.; Naletova, V. A.; Bin, C. Z.; Shaposhnikova, G. A., Conservation laws for the mass, momentum, and energy on a phase interface for true and excess surface parameters, Fluid Dyn., 18, 923-930, (1983) · Zbl 0545.76119
[67] Green, A.; Rivlin, R., Simple force and stress multipoles, Arch. Ration. Mech. Anal., 16, 325-353, (1964) · Zbl 0244.73005
[68] Green, A. E.; Rivlin, R. S., Multipolar continuum mechanics, Arch. Ration. Mech. Anal., 17, 113-147, (1964) · Zbl 0133.17604
[69] Green, A. E.; Rivlin, R. S., On Cauchy’s equations of motion, Z. Angew. Math. Phys. (ZAMP), 15, 290-292, (1964) · Zbl 0122.18403
[70] Green, A. E.; Rivlin, R. S., Multipolar continuum mechanicsfunctional theory. i, Proc. R. Soc. A, 284, 1398, 303-324, (1965)
[71] Gurtin, M. E.; Murdoch, A. I., A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57, 4, 291-323, (1975) · Zbl 0326.73001
[72] Gurtin, M. E.; Murdoch, A. I., Surface stress in solids, Int. J. Solids Struc., 14, 6, 43-440, (1978) · Zbl 0377.73001
[73] Gurtin, M. E.; Struthers, A., Multiphase thermomechanics with interfacial structure 3. evolving phase boundaries in the presence of bulk deformation, Arch. Ration. Mech. Anal., 112, 2, 97-160, (1990) · Zbl 0723.73018
[74] Gurtin, M. E.; Weissmüller, J.; Larché, F., A general theory of curved deformable interfaces in solids at equilibrium, Philos. Mag. A, 78, 5, 1093-1109, (1998)
[75] Haiss, W., Surface stress of Clean and adsorbate-covered solids, Rep. Prog. Phys., 64, 5, 591-648, (2001)
[76] Han, C.-S., Influence of the molecular structure on indentation size effect in polymers, Mater. Sci. Eng.A, 527, 3, 619-624, (2010)
[77] He, J.; Lilley, C. M., Surface effect on the elastic behavior of static bending nanowires, Nano Lett., 8, 7, 1798-1802, (2008)
[78] Huang, Y.; Zhang, L.; Guo, T.; Hwang, K.-C., Mixed mode near-tip fields for cracks in materials with strain-gradient effects, J. Mech. Phys. Solids, 45, 3, 439-465, (1997) · Zbl 1049.74521
[79] Huang, Z.; Sun, L., Size-dependent effective properties of a heterogeneous material with interface energy effectfrom finite deformation theory to infinitesimal strain analysis, Acta Mech., 190, 151-163, (2007) · Zbl 1117.74047
[80] Ibach, H., The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures, Surf. Sci. Rep., 29, 5-6, 195-263, (1997)
[81] Javili, A.; McBride, A.; Steinmann, P., Thermomechanics of solids with lower-dimensional energeticson the importance of surface, interface and curve structures at the nanoscale. a unifying review, Appl. Mech. Rev., 65, 1, 010802, (2013)
[82] Javili, A.; McBride, A.; Steinmann, P.; Reddy, B. D., Relationships between the admissible range of surface material parameters and stability of linearly elastic bodies, Philos. Mag., 92, 3540-3563, (2012)
[83] Javili, A.; Steinmann, P., A finite element framework for continua with boundary energies. part ithe two-dimensional case, Comput. Methods Appl. Mech. Eng., 198, 27-29, 2198-2208, (2009) · Zbl 1227.74075
[84] Javili, A.; Steinmann, P., A finite element framework for continua with boundary energies. part iithe three-dimensional case, Comput. Methods Appl. Mech. Eng., 199, 9-12, 755-765, (2010) · Zbl 1227.74074
[85] Javili, A.; Steinmann, P., On thermomechanical solids with boundary structures, Int. J. Solids Struc., 47, 24, 3245-3253, (2010) · Zbl 1203.74031
[86] Kahrobaiyan, M.; Asghari, M.; Rahaeifard, M.; Ahmadian, M., A nonlinear strain gradient beam formulation, Int. J. Eng. Sci., 49, 11, 1256-1267, (2011) · Zbl 1423.74487
[87] Kellogg, O. D., Foundations of potential theory, (1929), Springer Berlin · JFM 55.0282.01
[88] Kirchner, N.; Steinmann, P., A unifying treatise on variational principles for gradient and micromorphic continua, Philos. Mag., 85, 33-35, 3875-3895, (2005)
[89] Kirchner, N.; Steinmann, P., On the material setting of gradient hyperelasticity, Math. Mech. Solids, 12, 5, 559-580, (2007) · Zbl 1419.74071
[90] Kong, S.; Zhou, S.; Nie, Z.; Wang, K., Static and dynamic analysis of micro beams based on strain gradient elasticity theory, Int. J. Eng. Sci., 47, 4, 487-498, (2009) · Zbl 1213.74190
[91] Kreyszig, E., Differential geometry, (1991), Dover Publications
[92] Lam, D.; Yang, F.; Chong, A.; Wang, J.; Tong, P., Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids, 51, 8, 1477-1508, (2003) · Zbl 1077.74517
[93] Lazar, M.; Maugin, G. A., Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity, Int. J. Eng. Sci., 43, 13-14, 1157-1184, (2005) · Zbl 1211.74040
[94] Lazar, M.; Maugin, G. A.; Aifantis, E. C., On dislocations in a special class of generalized elasticity, Phys. Status Solidi (b), 242, 12, 2365-2390, (2005)
[95] Lazar, M.; Maugin, G. A.; Aifantis, E. C., Dislocations in second strain gradient elasticity, Int. J. Solids Struc., 43, 6, 1787-1817, (2006) · Zbl 1120.74343
[96] Leo, P. H.; Sekerka, R. F., Overview no. 86the effect of surface stress on crystal-melt and crystal-crystal equilibrium, Acta Metall., 37, 12, 3119-3138, (1989)
[97] Li, S.; Miskioglu, I.; Altan, B., Solution to line loading of a semi-infinite solid in gradient elasticity, Int. J. Solids Struc., 41, 13, 3395-3410, (2004) · Zbl 1071.74006
[98] Li, W.; Duan, H.; Albe, K.; Weissmüller, J., Line stress of step edges at crystal surfaces, Surf. Sci., 605, 9-10, 947-957, (2011)
[99] Luongo, A.; Piccardo, G., Non-linear galloping of sagged cables in 12 internal resonance, J. Sound Vib., 214, 5, 915-940, (1998)
[100] Luongo, A.; Piccardo, G., Linear instability mechanisms for coupled translational galloping, J. Sound Vib., 288, 1027-1047, (2005)
[101] Luongo, A.; Romeo, F., Real wave vectors for dynamic analysis of periodic structures, J. Sound Vib., 279, 1-2, 309-325, (2005)
[102] Luongo, A.; Zulli, D.; Piccardo, G., Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables, J. Sound Vib., 315, 3, 375-393, (2008)
[103] Luongo, A.; Zulli, D.; Piccardo, G., On the effect of twist angle on nonlinear galloping of suspended cables, Comput. Struc., 87, 15-16, 1003-1014, (2009)
[104] Maranganti, R.; Sharma, P., A novel atomistic approach to determine strain-gradient elasticity constantstabulation and comparison for various metals, semiconductors, silica, polymers and the ir relevance for nanotechnologies, J. Mech. Phys. Solids, 55, 9, 1823-1852, (2007) · Zbl 1173.74003
[105] Mareno, A., Uniqueness of equilibrium solutions in second-order gradient nonlinear elasticity, J. Elasticity, 74, 99-107, (2004) · Zbl 1058.74021
[106] Marsden, J. E.; Hughes, T. J.R., Mathematical foundations of elasticity, (1994), Dover Publications
[107] Maugin, G. A., The method of virtual power in continuum mechanicsapplication to coupled fields, Acta Mech., 35, 1-70, (1980) · Zbl 0428.73095
[108] Miller, R. E.; Shenoy, V. B., Size-dependent elastic properties of nanosized structural elements, Nanotechnology, 11, 3, 139, (2000)
[109] Mindlin, R. D., Second gradient of strain and surface-tension in linear elasticity, Int. J. Solids Struc., 1, 4, 417-438, (1965)
[110] Moeckel, G. P., Thermodynamics of an interface, Arch. Ration. Mech. Anal., 57, 255-280, (1975) · Zbl 0338.73001
[111] Mühlich, U.; Zybell, L.; Kuna, M., Estimation of material properties for linear elastic strain gradient effective media, Eur. J. Mech. A/Solids, 31, 1, 117-130, (2012) · Zbl 1278.74145
[112] Müller, P.; Saul, A., Elastic effects on surface physics, Surf. Sci. Rep., 54, 5-8, 157-258, (2004)
[113] Murdoch, A. I., A thermodynamical theory of elastic material interfaces, Q. J. Mech. Appl. Math., 29, 3, 245-275, (1976) · Zbl 0398.73003
[114] Murdoch, A. I., Some fundamental aspects of surface modeling, J. Elasticity, 80, 1, 33-52, (2005) · Zbl 1089.74012
[115] Nikolov, S.; Han, C.-S.; Raabe, D., On the origin of size effects in small-strain elasticity of solid polymers, Int. J. Solids Struc., 44, 5, 1582-1592, (2007) · Zbl 1125.74007
[116] Olsson, P. A.; Park, H. S., On the importance of surface elastic contributions to the flexural rigidity of nanowires, J. Mech. Phys. Solids, 60, 12, 2064-2083, (2012)
[117] Paolone, A.; Vasta, M.; Luongo, A., Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces inon-linear model and stability analysis, Int. J. Non-Linear Mech., 41, 4, 586-594, (2006)
[118] Park, H. S.; Klein, P. A., Surface Cauchy-Born analysis of surface stress effects on metallic nanowires, Phys. Rev. B, 75, 8, 1-9, (2007)
[119] Park, H. S.; Klein, P. A., A surface Cauchy-Born model for silicon nanostructures, Comput. Methods Appl. Mech. Eng., 197, 41-42, 3249-3260, (2008) · Zbl 1159.74312
[120] Park, H. S.; Klein, P. A.; Wagner, G. J., A surface Cauchy-Born model for nanoscale materials, Int. J. Numer. Methods Eng., 68, 10, 1072-1095, (2006) · Zbl 1128.74005
[121] Polizzotto, C., Gradient elasticity and nonstandard boundary conditions, Int. J. Solids Struc., 40, 26, 7399-7423, (2003) · Zbl 1063.74015
[122] Polyzos, D.; Fotiadis, D., Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models, Int. J. Solids Struc., 49, 3-4, 470-480, (2012)
[123] Pozrikidis, C., Mechanics of hexagonal atomic lattices, Int. J. Solids Struc., 45, 3-4, 732-745, (2008) · Zbl 1167.74322
[124] Rambert, G.; Grandidier, J.-C.; Aifantis, E. C., On the direct interactions between heat transfer, mass transport and chemical processes within gradient elasticity, Eur. J. Mech. A/Solids, 26, 1, 68-87, (2007) · Zbl 1111.74012
[125] Romeo, F.; Luongo, A., Vibration reduction in piecewise bi-coupled periodic structures, J. Sound Vib., 268, 3, 601-615, (2003)
[126] Rusanov, A.I., 1996. Thermodynamics of solid surfaces. Surf. Sci. Rep. 23, 173-247.
[127] Rusanov, A. I., Surface thermodynamics revisited, Surf. Sci. Rep., 58, 11-239, (2005)
[128] Seppecher, P., 1987. Etude d’une modélisation des zones capillaires fluides: Interfaces et lignes de contact. Thèse de l’Université Paris VI, Avril.
[129] Seppecher, P., 1989. Etude des conditions aux limites en théorie du second gradient: cas de la capillarité. C. R. Acad. Sci. Paris t. 309, Série II, 497-502.
[130] Seppecher, P., Alibert, J., dell’Isola, F., 2011. Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys.: Conf. Ser. 319 (1), 012018.
[131] Sharma, P.; Ganti, S., Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies, J. Appl. Mech., 71, 663-671, (2004) · Zbl 1111.74629
[132] Sharma, P.; Ganti, S.; Bhate, N., Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities, Appl. Phys. Lett., 82, 4, 535-537, (2003)
[133] Sharma, P.; Wheeler, L. T., Size-dependent elastic state of ellipsoidal nano-inclusions incorporating surface/interface tension, J. Appl. Mech., 74, 3, 447-454, (2007) · Zbl 1111.74630
[134] Shenoy, V. B., Atomistic calculations of elastic properties of metallic fcc crystal surfaces, Phys. Rev. B, 71, 9, 1-11, (2005)
[135] Shodja, H.; Tehranchi, A., A formulation for the characteristic lengths of fcc materials in first strain gradient elasticity via the sutton-Chen potential, Philos. Mag., 90, 14, 1893-1913, (2010)
[136] Shvartsburg, A. B.; Erokhin, N. S., Acoustic gradient barriers (exactly solvable models), Phys.-Usp., 54, 6, 605, (2011)
[137] Silhavy, M., The existence of the flux vector and the divergence theorem for general Cauchy fluxes, Arch. Ration. Mech. Anal., 90, 195-212, (1985) · Zbl 0593.73007
[138] Silhavy, M., Cauchy’s stress theorem and tensor fields with divergences in L^{p}, Arch. Ration. Mech. Anal., 116, 223-255, (1991) · Zbl 0776.73003
[139] Silhavy, M., Equilibrium of phases with interfacial energya variational approach, J. Elasticity, 105, 271-303, (2011) · Zbl 1290.74032
[140] Simha, N.; Bhattacharya, K., Kinetics of phase boundaries with edges and junctions, J. Mech. Phys. Solids, 46, 2323-2359, (1998) · Zbl 1017.74051
[141] Simha, N. K.; Bhattacharya, K., Kinetics of phase boundaries with edges and junctions in a three-dimensional multi-phase body, J. Mech. Phys. Solids, 48, 12, 2619-2641, (2000) · Zbl 1005.74049
[142] Steigmann, D.J., Ogden, R.W., 1999. Elastic surface-substrate interactions. Proc. R. Soc. A 455 (1982), 437-474. · Zbl 0926.74016
[143] Steinmann, P., On boundary potential energies in deformational and configurational mechanics, J. Mech. Phys. Solids, 56, 3, 772-800, (2008) · Zbl 1149.74006
[144] Suiker, A. S.J.; Chang, C. S., Application of higher-order tensor theory for formulating enhanced continuum models, Acta Mech., 142, 223-234, (2000) · Zbl 0966.74007
[145] Sun, L.; Han, R. P.S.; Wang, J.; Lim, C. T., Modeling the size-dependent elastic properties of polymeric nanofibers, Nanotechnology, 19, 45, 455706, (2008)
[146] Sunyk, R.; Steinmann, P., On higher gradients in continuum-atomistic modeling, Int. J. Solids Struc., 40, 24, 6877-6896, (2003) · Zbl 1137.74311
[147] Temizer, İ.; Wriggers, P., An adaptive multiscale resolution strategy for the finite deformation analysis of microheterogeneous structures, Comput. Methods Appl. Mech. Eng., 200, 37-40, 2639-2661, (2011) · Zbl 1230.74157
[148] Tenek, L. T.; Aifantis, E., On some applications of gradient elasticity to composite materials, Compos. Struc., 53, 2, 189-197, (2001)
[149] Toupin, R. A., Elastic materials with couple-stresses, Arch. Ration. Mech. Anal., 11, 385-414, (1962) · Zbl 0112.16805
[150] Toupin, R. A., Theories of elasticity with couple-stress, Arch. Ration. Mech. Anal., 17, 85-112, (1964) · Zbl 0131.22001
[151] Tran, T.-H.; Monchiet, V.; Bonnet, G., A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media, Int. J. Solids Struc., 49, 5, 783-792, (2012)
[152] Tsepoura, K. G.; Pavlou, D. G., Solution of 3-d gradient elastic problems with surface energy via BEM, WSEAS Trans. Math., 5, 3, 314-321, (2006)
[153] Van, P.; Papenfuss, C., Thermodynamic consistency of third grade finite strain elasticity, Proc. Est. Acad. Sci., 57, 3, 132-141, (2008) · Zbl 1375.74005
[154] Vardoulakis, I.; Exadaktylos, G.; Aifantis, E., Gradient elasticity with surface energymode-iii crack problem, Int. J. Solids Struc., 33, 30, 4531-4559, (1996) · Zbl 0919.73237
[155] Weissmüller, J.; Duan, H.-L.; Farkas, D., Deformation of solids with nanoscale pores by the action of capillary forces, Acta Materialia, 58, 1-13, (2010)
[156] Yi, D.; Wang, T.; Xiao, Z., Strain gradient theory based on a new framework of non-local model, Acta Mech., 212, 51-67, (2010) · Zbl 1351.74051
[157] 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, Comput. Methods Appl. Mech. Eng., 200, 5-8, 614-625, (2011) · Zbl 1225.74112
[158] Zheng, Y.; Zhang, H.; Chen, Z.; Ye, H., Size and surface effects on the mechanical behavior of nanotubes in first gradient elasticity, Composites Part BEng., 43, 1, 27-32, (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.