메뉴 건너뛰기




Volumn 48, Issue 13, 2011, Pages 1962-1990

Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results

Author keywords

Generalised continuum; Gradient elasticity; Internal length scale; Singularity removal; Size effect; Wave dispersion

Indexed keywords

GENERALISED CONTINUUM; GRADIENT ELASTICITY; INTERNAL LENGTH SCALE; SINGULARITY REMOVAL; SIZE EFFECT; WAVE DISPERSION;

EID: 79955562493     PISSN: 00207683     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijsolstr.2011.03.006     Document Type: Article
Times cited : (761)

References (163)
  • 1
    • 0010940116 scopus 로고
    • Fundamental equations of the theory of elastic media with rotationally interacting particles
    • E. Aero, and E. Kuvshinskii Fundamental equations of the theory of elastic media with rotationally interacting particles Soviet Phys. - Solid State 2 1961 1272 1281
    • (1961) Soviet Phys. - Solid State , vol.2 , pp. 1272-1281
    • Aero, E.1    Kuvshinskii, E.2
  • 3
    • 0023174893 scopus 로고
    • The physics of plastic deformation
    • E. Aifantis The physics of plastic deformation Int. J. Plast. 3 1987 211 247
    • (1987) Int. J. Plast. , vol.3 , pp. 211-247
    • Aifantis, E.1
  • 4
    • 0026930049 scopus 로고
    • On the role of gradients in the localization of deformation and fracture
    • E. Aifantis On the role of gradients in the localization of deformation and fracture Int. J. Eng. Sci. 30 1992 1279 1299
    • (1992) Int. J. Eng. Sci. , vol.30 , pp. 1279-1299
    • Aifantis, E.1
  • 8
    • 0037343002 scopus 로고    scopus 로고
    • Update on a class of gradient theories
    • E. Aifantis Update on a class of gradient theories Mech. Mat. 35 2003 259 280
    • (2003) Mech. Mat. , vol.35 , pp. 259-280
    • Aifantis, E.1
  • 9
    • 54049158022 scopus 로고    scopus 로고
    • Exploring the applicability of gradient elasticity to certain micro/nano reliability problems
    • E. Aifantis Exploring the applicability of gradient elasticity to certain micro/nano reliability problems Microsyst. Technol. 15 2009 109 115
    • (2009) Microsyst. Technol. , vol.15 , pp. 109-115
    • Aifantis, E.1
  • 10
    • 71649104342 scopus 로고    scopus 로고
    • On scale invariance in anisotropic plasticity, gradient plasticity and gradient elasticity
    • E. Aifantis On scale invariance in anisotropic plasticity, gradient plasticity and gradient elasticity Int. J. Eng. Sci. 47 2009 1089 1099
    • (2009) Int. J. Eng. Sci. , vol.47 , pp. 1089-1099
    • Aifantis, E.1
  • 12
    • 0026473605 scopus 로고
    • On the structure of the mode III crack-tip in gradient elasticity
    • DOI 10.1016/0956-716X(92)90194-J
    • S. Altan, and E. Aifantis On the structure of the mode III crack-tip in gradient elasticity Scripta Metall. Mater. 26 1992 319 324 (Pubitemid 23642730)
    • (1992) Scripta metallurgica et materialia , vol.26 , Issue.2 , pp. 319-324
    • Altan, S.B.1    Aifantis, E.C.2
  • 14
    • 0036471631 scopus 로고    scopus 로고
    • Mixed finite element formulations of strain-gradient elasticity problems
    • DOI 10.1016/S0045-7825(01)00353-X, PII S004578250100353X
    • E. Amanatidou, and N. Aravas Mixed finite element formulations of strain-gradient elasticity problems Comput. Methods Appl. Mech. Eng. 191 2002 1723 1751 (Pubitemid 34172228)
    • (2002) Computer Methods in Applied Mechanics and Engineering , vol.191 , Issue.15-16 , pp. 1723-1751
    • Amanatidou, E.1    Aravas, N.2
  • 15
    • 37049014689 scopus 로고    scopus 로고
    • Continuous models for 2D discrete media valid for higher-frequency domain
    • DOI 10.1016/j.compstruc.2007.05.013, PII S0045794907001927
    • I. Andrianov, and J. Awrejcewicz Continuous models for 2D discrete media valid for higher-frequency domain Comput. Struct. 86 2008 140 144 (Pubitemid 350251541)
    • (2008) Computers and Structures , vol.86 , Issue.1-2 , pp. 140-144
    • Andrianov, I.V.1    Awrejcewicz, J.2
  • 16
    • 5644239856 scopus 로고    scopus 로고
    • Asymptotic approaches in mechanics: New parameters and procedures
    • I. Andrianov, J. Awrejcewicz, and R. Barantsev Asymptotic approaches in mechanics: new parameters and procedures ASME Appl. Mech. Rev. 56 2003 87 110
    • (2003) ASME Appl. Mech. Rev. , vol.56 , pp. 87-110
    • Andrianov, I.1    Awrejcewicz, J.2    Barantsev, R.3
  • 18
    • 78650065241 scopus 로고    scopus 로고
    • Wave propagation in periodic composites: Higher-order asymptotic analysis versus plane-wave expansions method
    • I. Andrianov, J. Awrejcewicz, V. Danishevskyy, and D. Weichert Wave propagation in periodic composites: Higher-order asymptotic analysis versus plane-wave expansions method ASME J. Comput. Nonlinear Dyn. 6 2010 011015
    • (2010) ASME J. Comput. Nonlinear Dyn. , vol.6 , pp. 011015
    • Andrianov, I.1    Awrejcewicz, J.2    Danishevskyy, V.3    Weichert, D.4
  • 20
    • 0036768154 scopus 로고    scopus 로고
    • Mixed finite elements for elasticity
    • DOI 10.1007/s002110100348
    • D. Arnold, and R. Winther Mixed finite elements for elasticity Numer. Math. 92 2002 401 419 (Pubitemid 36161594)
    • (2002) Numerische Mathematik , vol.92 , Issue.3 , pp. 401-419
    • Arnold, D.N.1    Winther, R.2
  • 21
    • 34250136581 scopus 로고
    • A family of higher-order mixed finite element methods for plane elasticity
    • D. Arnold, J. Douglas Jr.; and C. Gupta A family of higher-order mixed finite element methods for plane elasticity Numer. Math. 45 1984 1 22
    • (1984) Numer. Math. , vol.45 , pp. 1-22
    • Arnold, D.1    Douglas Jr., J.2    Gupta, C.3
  • 22
    • 0036822265 scopus 로고    scopus 로고
    • Numerical modeling of size effects with gradient elasticity - Formulation, meshless discretization and examples
    • DOI 10.1023/A:1022225526483
    • H. Askes, and E. Aifantis Numerical modeling of size effect with gradient elasticity - formulation, meshless discretization and examples Int. J. Fract. 117 2002 347 358 (Pubitemid 36360802)
    • (2002) International Journal of Fracture , vol.117 , Issue.4 , pp. 347-358
    • Askes, H.1    Alfantis, E.C.2
  • 23
    • 33746437180 scopus 로고    scopus 로고
    • Gradient elasticity theories in statics and dynamics - A unification of approaches
    • DOI 10.1007/s10704-006-8375-4
    • H. Askes, and E. Aifantis Gradient elasticity theories in statics and dynamics - a unification of approaches Int. J. Fract. 139 2006 297 304 (Pubitemid 44123485)
    • (2006) International Journal of Fracture , vol.139 , Issue.2 , pp. 297-304
    • Askes, H.1    Aifantis, E.C.2
  • 24
    • 77954736367 scopus 로고    scopus 로고
    • Gradient elasticity and flexural wave dispersion in carbon nanotubes
    • H. Askes, and E. Aifantis Gradient elasticity and flexural wave dispersion in carbon nanotubes Phys. Rev. B 80 2009 195412
    • (2009) Phys. Rev. B , vol.80 , pp. 195412
    • Askes, H.1    Aifantis, E.2
  • 25
    • 67449093521 scopus 로고    scopus 로고
    • Non-singular stresses in gradient elasticity at bi-material interface with transverse crack
    • H. Askes, and I. Gitman Non-singular stresses in gradient elasticity at bi-material interface with transverse crack Int. J. Fract. 156 2009 217 222
    • (2009) Int. J. Fract. , vol.156 , pp. 217-222
    • Askes, H.1    Gitman, I.2
  • 26
    • 80053575956 scopus 로고    scopus 로고
    • Review and critique of the stress gradient elasticity theories of Eringen and Aifantis
    • H. Askes, and I. Gitman Review and critique of the stress gradient elasticity theories of Eringen and Aifantis G. Maugin, A. Metrikine, Mechanics of Generalized Continua 2010 Springer 203 210
    • (2010) Mechanics of Generalized Continua , pp. 203-210
    • Askes, H.1    Gitman, I.2
  • 28
    • 0012716579 scopus 로고    scopus 로고
    • One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 2: Static and dynamic response
    • H. Askes, and A. Metrikine One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 2: static and dynamic response Eur. J. Mech. A/Solids 21 2002 573 588
    • (2002) Eur. J. Mech. A/Solids , vol.21 , pp. 573-588
    • Askes, H.1    Metrikine, A.2
  • 29
    • 5644294661 scopus 로고    scopus 로고
    • Higher-order continua derived from discrete media: Continualisation aspects and boundary conditions
    • H. Askes, and A. Metrikine Higher-order continua derived from discrete media: continualisation aspects and boundary conditions Int. J. Solids Struct. 42 2005 187 202
    • (2005) Int. J. Solids Struct. , vol.42 , pp. 187-202
    • Askes, H.1    Metrikine, A.2
  • 30
    • 0036542176 scopus 로고    scopus 로고
    • A classification of higher-order strain-gradient models - Linear analysis
    • DOI 10.1007/s00419-002-0202-4
    • H. Askes, A. Suiker, and L. Sluys A classification of higher-order strain gradient models - linear analysis Arch. Appl. Mech. 72 2002 171 188 (Pubitemid 34600220)
    • (2002) Archive of Applied Mechanics , vol.72 , Issue.2-3 , pp. 171-188
    • Askes, H.1    Suiker, A.S.J.2    Sluys, L.J.3
  • 32
    • 57849131284 scopus 로고    scopus 로고
    • Four simplified gradient elasticity models for the simulation of dispersive wave propagation
    • H. Askes, A. Metrikine, A. Pichugin, and T. Bennett Four simplified gradient elasticity models for the simulation of dispersive wave propagation Phil. Mag. 88 2008 3415 3443
    • (2008) Phil. Mag. , vol.88 , pp. 3415-3443
    • Askes, H.1    Metrikine, A.2    Pichugin, A.3    Bennett, T.4
  • 33
    • 43049121709 scopus 로고    scopus 로고
    • Finite element analysis with staggered gradient elasticity
    • H. Askes, I. Morata, and E. Aifantis Finite element analysis with staggered gradient elasticity Comput. Struct. 86 2008 1266 1279
    • (2008) Comput. Struct. , vol.86 , pp. 1266-1279
    • Askes, H.1    Morata, I.2    Aifantis, E.3
  • 34
    • 43049092202 scopus 로고    scopus 로고
    • Element size and time step selection procedures for the numerical analysis of elasticity with higher-order inertia
    • H. Askes, B. Wang, and T. Bennett Element size and time step selection procedures for the numerical analysis of elasticity with higher-order inertia J. Sound Vibr. 314 2008 650 656
    • (2008) J. Sound Vibr. , vol.314 , pp. 650-656
    • Askes, H.1    Wang, B.2    Bennett, T.3
  • 35
    • 60449103092 scopus 로고    scopus 로고
    • Finite element modelling of wave dispersion with dynamically consistent gradient elasticity
    • T. Bennett, and H. Askes Finite element modelling of wave dispersion with dynamically consistent gradient elasticity Comput. Mech. 43 2009 815 825
    • (2009) Comput. Mech. , vol.43 , pp. 815-825
    • Bennett, T.1    Askes, H.2
  • 36
    • 42149188760 scopus 로고    scopus 로고
    • Elasticity theories with higher-order gradients of inertia and stiffness for the modelling of wave dispersion in laminates
    • T. Bennett, I. Gitman, and H. Askes Elasticity theories with higher-order gradients of inertia and stiffness for the modelling of wave dispersion in laminates Int. J. Fract. 148 2007 185 193
    • (2007) Int. J. Fract. , vol.148 , pp. 185-193
    • Bennett, T.1    Gitman, I.2    Askes, H.3
  • 37
    • 44749085982 scopus 로고    scopus 로고
    • The Arnold-Winther mixed FEM in linear elasticity. Part I: Implementation and numerical verification
    • C. Carstensen, D. Günther, J. Reininghaus, and J. Thiele The Arnold-Winther mixed FEM in linear elasticity. Part I: implementation and numerical verification Comput. Meth. Appl. Mech. Eng. 197 2008 3014 3023
    • (2008) Comput. Meth. Appl. Mech. Eng. , vol.197 , pp. 3014-3023
    • Carstensen, C.1    Günther, D.2    Reininghaus, J.3    Thiele, J.4
  • 38
    • 2442619615 scopus 로고
    • Mémoire sur les systèmes isotropes de points matériels
    • Gauthier-Villars (reprint 1908), Paris
    • re Série - Tome II. Gauthier-Villars (reprint 1908), Paris, pp. 351-386.
    • (1850) re Série - Tome II , pp. 351-386
    • Cauchy, A.1
  • 39
    • 79955564858 scopus 로고
    • Mémoire sur les vibrations d'un double système de molécules et de l'éther continu dans un corps cristallisé
    • Gauthier-Villars (reprint 1908), Paris
    • re Série - Tome II. Gauthier-Villars (reprint 1908), Paris, pp. 338-350.
    • (1850) re Série - Tome II , pp. 338-350
    • Cauchy, A.1
  • 40
    • 0009827517 scopus 로고
    • Note sur l'équilibre et les mouvements vibratoires des corps solides
    • Gauthier-Villars (reprint 1899), Paris
    • re Série - Tome XI. Gauthier-Villars (reprint 1899), Paris, pp. 341-346.
    • (1851) re Série - Tome XI , pp. 341-346
    • Cauchy, A.1
  • 41
    • 0029357270 scopus 로고
    • Second-gradient constitutive theory for granular material with random packing structure
    • C. Chang, and J. Gao Second-gradient constitutive theory for granular material with random packing structure Int. J. Solids Struct. 32 1995 2279 2293
    • (1995) Int. J. Solids Struct. , vol.32 , pp. 2279-2293
    • Chang, C.1    Gao, J.2
  • 42
    • 0000258643 scopus 로고    scopus 로고
    • A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales
    • W. Chen, and J. Fish A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales ASME J. Appl. Mech. 68 2001 153 161 (Pubitemid 33715880)
    • (2001) Journal of Applied Mechanics, Transactions ASME , vol.68 , Issue.2 , pp. 153-161
    • Chen, W.1    Fish, J.2
  • 43
    • 0034171519 scopus 로고    scopus 로고
    • Subdivision surfaces: A new paradigm for thin-shell finite-element analysis
    • F. Cirak, M. Ortiz, and P. Schröder Subdivision surfaces: a new paradigm for thin-shell finite-element analysis Int. J. Numer. Methods Eng. 47 2000 2039 2072
    • (2000) Int. J. Numer. Methods Eng. , vol.47 , pp. 2039-2072
    • Cirak, F.1    Ortiz, M.2    Schröder, P.3
  • 44
    • 0002436381 scopus 로고
    • A quasicontinuum approximation for solitons in an atomic chain
    • M. Collins A quasicontinuum approximation for solitons in an atomic chain Chem. Phys. Lett. 77 1981 342 347
    • (1981) Chem. Phys. Lett. , vol.77 , pp. 342-347
    • Collins, M.1
  • 45
    • 0032647139 scopus 로고    scopus 로고
    • Computational modelling of gradient-enhanced damage in quasi-brittle materials
    • C. Comi Computational modelling of gradient-enhanced damage in quasi-brittle materials Mech. Coh.-Frict. Mater. 4 1999 17 36
    • (1999) Mech. Coh.-Frict. Mater. , vol.4 , pp. 17-36
    • Comi, C.1
  • 47
    • 0026908533 scopus 로고
    • Gradient-dependent plasticity: Formulation and algorithmic aspects
    • R. de Borst, and H.-B. Mühlhaus Gradient-dependent plasticity: formulation and algorithmic aspects Int. J. Numer. Methods Eng. 35 1992 521 539
    • (1992) Int. J. Numer. Methods Eng. , vol.35 , pp. 521-539
    • De Borst, R.1    Mühlhaus, H.-B.2
  • 48
    • 0029516648 scopus 로고
    • On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials
    • R. de Borst, J. Pamin, R. Peerlings, and L. Sluys On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials Comput. Mech. 17 1995 130 141
    • (1995) Comput. Mech. , vol.17 , pp. 130-141
    • De Borst, R.1    Pamin, J.2    Peerlings, R.3    Sluys, L.4
  • 49
    • 0040061610 scopus 로고
    • Dispersive corrections to continuum elastic theory in cubic crystals
    • D. DiVicenzo Dispersive corrections to continuum elastic theory in cubic crystals Phys. Rev. B 34 1986 5450 5465
    • (1986) Phys. Rev. B , vol.34 , pp. 5450-5465
    • Divicenzo, D.1
  • 50
    • 33847735473 scopus 로고    scopus 로고
    • Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics
    • W. Duan, C. Wang, and Y. Zhang Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics J. Appl. Phys. 101 2007 024305
    • (2007) J. Appl. Phys. , vol.101 , pp. 024305
    • Duan, W.1    Wang, C.2    Zhang, Y.3
  • 51
    • 31444432791 scopus 로고    scopus 로고
    • Waves in microstructured materials and dispersion
    • DOI 10.1080/14786430500362769, PII N77386151228740
    • J. Engelbrecht, A. Berezovski, F. Pastrone, and M. Braun Waves in microstructured materials and dispersion Phil. Mag. 85 2005 4127 4141 (Pubitemid 43149126)
    • (2005) Philosophical Magazine , vol.85 , Issue.33-35 , pp. 4127-4141
    • Engelbrecht, J.1    Berezovski, A.2    Pastrone, F.3    Braun, M.4
  • 52
    • 0037025251 scopus 로고    scopus 로고
    • Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity
    • DOI 10.1016/S0045-7825(02)00286-4, PII S0045782502002864
    • G. Engel, K. Garikipati, T. Hughes, M. Larson, L. Mazzei, and R. Taylor Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity Comput. Methods Appl. Mech. Eng. 191 2002 3669 3750 (Pubitemid 34842374)
    • (2002) Computer Methods in Applied Mechanics and Engineering , vol.191 , Issue.34 , pp. 3669-3750
    • Engel, G.1    Garikipati, K.2    Hughes, T.J.R.3    Larson, M.G.4    Mazzei, L.5    Taylor, R.L.6
  • 53
    • 0020821706 scopus 로고
    • On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves
    • DOI 10.1063/1.332803
    • A. Eringen On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves J. Appl. Phys. 54 1983 4703 4710 (Pubitemid 14450262)
    • (1983) Journal of Applied Physics , vol.54 , Issue.9 , pp. 4703-4710
    • Eringen A.Cemal1
  • 55
    • 29644446052 scopus 로고    scopus 로고
    • Weak spatial dispersion and the unfolding of wave arrival singularities in the elastodynamic Greens functions of solids
    • A. Every Weak spatial dispersion and the unfolding of wave arrival singularities in the elastodynamic Greens functions of solids Phys. Rev. B 72 2005 104302
    • (2005) Phys. Rev. B , vol.72 , pp. 104302
    • Every, A.1
  • 56
    • 33751165740 scopus 로고    scopus 로고
    • Unfolding of wave-arrival singularities in the elastodynamic Greens functions of anisotropic solids under weak spatial dispersion
    • A. Every, J. Kaplunov, and G. Rogerson Unfolding of wave-arrival singularities in the elastodynamic Greens functions of anisotropic solids under weak spatial dispersion Phys. Rev. B 74 2006 184307
    • (2006) Phys. Rev. B , vol.74 , pp. 184307
    • Every, A.1    Kaplunov, J.2    Rogerson, G.3
  • 58
    • 0037198567 scopus 로고    scopus 로고
    • Non-local dispersive model for wave propagation in heterogeneous media: Multi-dimensional case
    • DOI 10.1002/nme.424
    • J. Fish, W. Chen, and G. Nagai Non-local dispersive model for wave propagation in heterogeneous media: multi-dimensional case Int. J. Numer. Methods Eng. 54 2002 347 363 (Pubitemid 34490081)
    • (2002) International Journal for Numerical Methods in Engineering , vol.54 , Issue.3 , pp. 347-363
    • Fish, J.1    Chen, W.2    Nagai, G.3
  • 59
    • 0037198565 scopus 로고    scopus 로고
    • Non-local dispersive model for wave propagation in heterogeneous media: One-dimensional case
    • DOI 10.1002/nme.423
    • J. Fish, W. Chen, and G. Nagai Non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case Int. J. Numer. Methods Eng. 54 2002 331 346 (Pubitemid 34468208)
    • (2002) International Journal for Numerical Methods in Engineering , vol.54 , Issue.3 , pp. 331-346
    • Fish, J.1    Chen, W.2    Nagai, G.3
  • 60
    • 0012379292 scopus 로고    scopus 로고
    • On the stochastic interpretation of gradient-dependent constitutive equations
    • G. Frantziskonis, and E. Aifantis On the stochastic interpretation of gradient-dependent constitutive equations Eur. J. Mech. A/Solids 21 2002 589 596
    • (2002) Eur. J. Mech. A/Solids , vol.21 , pp. 589-596
    • Frantziskonis, G.1    Aifantis, E.2
  • 61
    • 0030107906 scopus 로고    scopus 로고
    • Damage, gradient of damage and principle of virtual power
    • DOI 10.1016/0020-7683(95)00074-7
    • M. Fremond, and B. Nedjar Damage, gradient of damage and principle of virtual power Int. J. Solids Struct. 33 1996 1083 1103 (Pubitemid 126403143)
    • (1996) International Journal of Solids and Structures , vol.33 , Issue.8 , pp. 1083-1103
    • Fremond, M.1    Nedjar, B.2
  • 62
    • 79955555764 scopus 로고    scopus 로고
    • On natural boundary conditions in linear 2nd-grade elasticity
    • F. Froiio, A. Zervos, and I. Vardoulakis On natural boundary conditions in linear 2nd-grade elasticity G. Maugin, A. Metrikine, Mechanics of Generalized Continua 2010 Springer 211 221
    • (2010) Mechanics of Generalized Continua , pp. 211-221
    • Froiio, F.1    Zervos, A.2    Vardoulakis, I.3
  • 63
    • 0141829271 scopus 로고    scopus 로고
    • The mode III crack problem in microstructured solids governed by dipolar gradient elasticity: Static and dynamic analysis
    • H. Georgiadis The mode III crack problem in microstructured solids governed by dipolar gradient elasticity: static and dynamic analysis ASME J. Appl. Mech. 70 2003 517 530
    • (2003) ASME J. Appl. Mech. , vol.70 , pp. 517-530
    • Georgiadis, H.1
  • 64
    • 36549027964 scopus 로고    scopus 로고
    • Problems of the flamant-boussinesq and kelvin type in dipolar gradient elasticity
    • DOI 10.1007/s10659-007-9129-x
    • H. Georgiadis, and D. Anagnostou Problems of the Flamant-Boussinesq and Kelvin type in dipolar gradient elasticity J. Elast. 90 2008 71 98 (Pubitemid 350178297)
    • (2008) Journal of Elasticity , vol.90 , Issue.1 , pp. 71-98
    • Georgiadis, H.G.1    Anagnostou, D.S.2
  • 65
    • 0000949779 scopus 로고    scopus 로고
    • Torsional surface waves in a gradient-elastic half-space
    • H. Georgiadis, I. Vardoulakis, and G. Lykotrafitis Torsional surface waves in a gradient-elastic half-space Wave Mot. 31 2000 333 348
    • (2000) Wave Mot. , vol.31 , pp. 333-348
    • Georgiadis, H.1    Vardoulakis, I.2    Lykotrafitis, G.3
  • 66
    • 29844447352 scopus 로고    scopus 로고
    • Representative Volume size as a macroscopic length scale parameter
    • Li, V.C. et al. (Ed.)
    • Gitman, I.; Askes, H.; Sluys, L.; 2004. Representative Volume size as a macroscopic length scale parameter. In: Li, V.C. et al. (Ed.), Fract. Mech. Concr. Struct. Ia-FraMCoS, pp. 483-489.
    • (2004) Fract. Mech. Concr. Struct. Ia-FraMCoS , pp. 483-489
    • Gitman, I.1    Askes, H.2    Sluys, L.3
  • 67
    • 29144461604 scopus 로고    scopus 로고
    • The representative volume size in static and dynamic micro-macro transitions
    • I. Gitman, H. Askes, and E. Aifantis The representative volume size in static and dynamic micro-macro transitions Int. J. Fract. 135 2005 L3 L9
    • (2005) Int. J. Fract. , vol.135
    • Gitman, I.1    Askes, H.2    Aifantis, E.3
  • 68
    • 29844442777 scopus 로고    scopus 로고
    • Quantification of stochastically stable representative volumes for random heterogeneous materials
    • DOI 10.1007/s00419-005-0411-8
    • I. Gitman, M. Gitman, and H. Askes Quantification of stochastically stable representative volumes for random heterogeneous materials Arch. Appl. Mech. 75 2006 79 92 (Pubitemid 43037091)
    • (2006) Archive of Applied Mechanics , vol.75 , Issue.2-3 , pp. 79-92
    • Gitman, I.M.1    Gitman, M.B.2    Askes, H.3
  • 69
    • 41649100457 scopus 로고    scopus 로고
    • Gradient elasticity with internal length and internal inertia based on the homogenisation of a representative volume element
    • I. Gitman, H. Askes, and E. Aifantis Gradient elasticity with internal length and internal inertia based on the homogenisation of a representative volume element J. Mech. Behav. Mater. 18 2007 1 16
    • (2007) J. Mech. Behav. Mater. , vol.18 , pp. 1-16
    • Gitman, I.1    Askes, H.2    Aifantis, E.3
  • 70
    • 34547494680 scopus 로고    scopus 로고
    • Representative volume: Existence and size determination
    • DOI 10.1016/j.engfracmech.2006.12.021, PII S0013794406004772
    • I. Gitman, H. Askes, and L. Sluys Representative volume: existence and size determination Eng. Fract. Mech. 74 2007 2518 2534 (Pubitemid 47177169)
    • (2007) Engineering Fracture Mechanics , vol.74 , Issue.16 , pp. 2518-2534
    • Gitman, I.M.1    Askes, H.2    Sluys, L.J.3
  • 71
    • 41649113212 scopus 로고    scopus 로고
    • Coupled-volume multi-scale modelling of quasibrittle material
    • I. Gitman, H. Askes, and L. Sluys Coupled-volume multi-scale modelling of quasibrittle material Eur. J. Mech. A/Solids 27 2008 302 327
    • (2008) Eur. J. Mech. A/Solids , vol.27 , pp. 302-327
    • Gitman, I.1    Askes, H.2    Sluys, L.3
  • 72
    • 77249142394 scopus 로고    scopus 로고
    • Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity
    • I. Gitman, H. Askes, E. Kuhl, and E. Aifantis Stress concentrations in fractured compact bone simulated with a special class of anisotropic gradient elasticity Int. J. Solids Struct. 47 2010 1099 1107
    • (2010) Int. J. Solids Struct. , vol.47 , pp. 1099-1107
    • Gitman, I.1    Askes, H.2    Kuhl, E.3    Aifantis, E.4
  • 73
  • 74
    • 0001652537 scopus 로고
    • Simple force and stress multipoles
    • A. Green, and R. Rivlin Simple force and stress multipoles Arch. Rat. Mech. Anal. 16 1964 325 353
    • (1964) Arch. Rat. Mech. Anal. , vol.16 , pp. 325-353
    • Green, A.1    Rivlin, R.2
  • 75
    • 0002333071 scopus 로고    scopus 로고
    • Nanoscopics of dislocations and disclinations in gradient elasticity
    • available via
    • M. Gutkin Nanoscopics of dislocations and disclinations in gradient elasticity Rev. Adv. Mater. Sci. 1 2000 27 60 available via
    • (2000) Rev. Adv. Mater. Sci. , vol.1 , pp. 27-60
    • Gutkin, M.1
  • 76
    • 0030393128 scopus 로고    scopus 로고
    • Screw dislocation in gradient elasticity
    • DOI 10.1016/1359-6462(96)00295-3, PII S1359646296002953
    • M. Gutkin, and E. Aifantis Screw dislocation in gradient elasticity Scripta Mater. 35 1996 1353 1358 (Pubitemid 126391736)
    • (1996) Scripta Materialia , vol.35 , Issue.11 , pp. 1353-1358
    • Gutkin, M.Y.1    Aifantis, E.C.2
  • 77
    • 0030869766 scopus 로고    scopus 로고
    • Edge dislocation in gradient elasticity
    • PII S1359646296003521
    • M. Gutkin, and E. Aifantis Edge dislocation in gradient elasticity Scripta Mater. 36 1997 129 135 (Pubitemid 127415474)
    • (1997) Scripta Materialia , vol.36 , Issue.1 , pp. 129-135
    • Gutkin, M.Yu.1    Aifantis, E.C.2
  • 78
    • 0033524546 scopus 로고    scopus 로고
    • Dislocations in the theory of gradient elasticity
    • M. Gutkin, and E. Aifantis Dislocations in the theory of gradient elasticity Scripta Mater. 40 1999 559 566
    • (1999) Scripta Mater. , vol.40 , pp. 559-566
    • Gutkin, M.1    Aifantis, E.2
  • 79
    • 0028184669 scopus 로고
    • Discretization influence on regularization by two localization limiters
    • A. Huerta, and G. Pijaudier-Cabot Discretization influence on the regularization by two localization limiters ASCE J. Eng. Mech. 120 1994 1198 1218 (Pubitemid 24431608)
    • (1994) Journal of Engineering Mechanics - ASCE , vol.120 , Issue.6 , pp. 1198-1218
    • Huerta, A.1    Pijaudier-Cadot, G.2
  • 80
    • 0041027101 scopus 로고    scopus 로고
    • Kink dynamics in a lattice model with long-range interactions
    • DOI 10.1088/0305-4470/34/20/301, PII S0305447001187664
    • T. Ioannidou, J. Pouget, and E. Aifantis Kink dynamics in a lattice model with long-range interactions J. Phys. A - Math. Gen. 34 2001 4269 4280 (Pubitemid 33582225)
    • (2001) Journal of Physics A: Mathematical and General , vol.34 , Issue.20 , pp. 4269-4280
    • Ioannidou, T.1    Pouget, J.2    Aifantis, E.3
  • 81
    • 43449115098 scopus 로고    scopus 로고
    • Determination of the dispersive elastic constants of the cubic crystals Ge, Si, GaAs, and InSb
    • K. Jakata, and A. Every Determination of the dispersive elastic constants of the cubic crystals Ge, Si, GaAs, and InSb Phys. Rev. B. 77 2008 174301
    • (2008) Phys. Rev. B. , vol.77 , pp. 174301
    • Jakata, K.1    Every, A.2
  • 82
    • 0037644851 scopus 로고    scopus 로고
    • Determination of the size of the representative volume element for random composites: Statistical and numerical approach
    • DOI 10.1016/S0020-7683(03)00143-4
    • T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin Determination of the size of the representative volume element for random composites: statistical and numerical approach Int. J. Solids Struct. 40 2003 3647 3679 (Pubitemid 36659117)
    • (2003) International Journal of Solids and Structures , vol.40 , Issue.13-14 , pp. 3647-3679
    • Kanit, T.1    Forest, S.2    Galliet, I.3    Mounoury, V.4    Jeulin, D.5
  • 84
    • 34548757877 scopus 로고    scopus 로고
    • Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity
    • DOI 10.1016/j.cma.2007.07.006, PII S0045782507003015
    • G. Karlis, S. Tsinopoulos, D. Polyzos, and D. Beskos Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity Comput. Methods Appl. Mech. Eng. 196 2007 5092 5103 (Pubitemid 47423977)
    • (2007) Computer Methods in Applied Mechanics and Engineering , vol.196 , Issue.49-52 , pp. 5092-5103
    • Karlis, G.F.1    Tsinopoulos, S.V.2    Polyzos, D.3    Beskos, D.E.4
  • 88
    • 0037143110 scopus 로고    scopus 로고
    • Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme
    • DOI 10.1002/nme.541
    • V. Kouznetsova, M. Geers, and W. Brekelmans Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme Int. J. Numer. Methods Eng. 54 2002 1235 1260 (Pubitemid 34724730)
    • (2002) International Journal for Numerical Methods in Engineering , vol.54 , Issue.8 , pp. 1235-1260
    • Kouznetsova, V.1    Geers, M.G.D.2    Brekelmans, W.A.M.3
  • 89
    • 8644290775 scopus 로고    scopus 로고
    • Multi-scale second-order computational homogenization of multi-phase materials: A nested finite element solution strategy
    • DOI 10.1016/j.cma.2003.12.073, PII S0045782504002853, Advances in Computational Plasticity
    • V. Kouznetsova, M. Geers, and W. Brekelmans Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy Comput. Methods Appl. Mech. Eng. 193 2004 5525 5550 (Pubitemid 39499845)
    • (2004) Computer Methods in Applied Mechanics and Engineering , vol.193 , Issue.48-51 , pp. 5525-5550
    • Kouznetsova, V.G.1    Geers, M.G.D.2    Brekelmans, W.A.M.3
  • 90
    • 33644892364 scopus 로고    scopus 로고
    • Size of a representative volume element in a second-order computational homogenization framework
    • V. Kouznetsova, M. Geers, and W. Brekelmans Size of a representative volume element in a second-order computational homogenization framework Int. J. Multiscale Comput. Eng. 2 2004 575 598
    • (2004) Int. J. Multiscale Comput. Eng. , vol.2 , pp. 575-598
    • Kouznetsova, V.1    Geers, M.2    Brekelmans, W.3
  • 91
    • 0001463033 scopus 로고
    • On the physical reality of torque stresses in continuum mechanics
    • E. Kröner On the physical reality of torque stresses in continuum mechanics Int. J. Eng. Sci. 1 1963 261 278
    • (1963) Int. J. Eng. Sci. , vol.1 , pp. 261-278
    • Kröner, E.1
  • 92
    • 49949143801 scopus 로고
    • Elasticity theory of materials with long range cohesive forces
    • E. Kröner Elasticity theory of materials with long range cohesive forces Int. J. Sol. Struct. 3 1967 731 742
    • (1967) Int. J. Sol. Struct. , vol.3 , pp. 731-742
    • Kröner, E.1
  • 93
    • 0042620309 scopus 로고
    • Theory of elasticity with spatial dispersion. One-dimensional complex structure
    • I. Kunin Theory of elasticity with spatial dispersion. One-dimensional complex structure J. Appl. Math. Mech. 30 1966 1025 1034
    • (1966) J. Appl. Math. Mech. , vol.30 , pp. 1025-1034
    • Kunin, I.1
  • 95
    • 24644502714 scopus 로고    scopus 로고
    • Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity
    • DOI 10.1016/j.ijengsci.2005.01.006, PII S0020722505000881
    • M. Lazar, and G. Maugin Nonsingular stress and strain fields of dislocations and diclinations in first strain gradient elasticity Int. J. Eng. Sci. 43 2005 1157 1184 (Pubitemid 41285286)
    • (2005) International Journal of Engineering Science , vol.43 , Issue.13-14 , pp. 1157-1184
    • Lazar, M.1    Maugin, G.A.2
  • 96
    • 0015192864 scopus 로고
    • The relation between mathematical expectations of stress and strain tensors in elastic microheterogeneous media
    • V. Levin The relation between mathematical expectations of stress and strain tensors in elastic microheterogeneous media J. Appl. Math. Mech. 35 1971 694 701
    • (1971) J. Appl. Math. Mech. , vol.35 , pp. 694-701
    • Levin, V.1
  • 98
    • 34347372607 scopus 로고    scopus 로고
    • A novel atomistic approach to determine strain-gradient elasticity constants: Tabulation and comparison for various metals, semiconductors, silica, polymers and the (Ir) relevance for nanotechnologies
    • DOI 10.1016/j.jmps.2007.02.011, PII S0022509607000397
    • R. Maranganti, and P. Sharma A novel atomistic approach to determine strain-gradient elasticity constants: tabulation and comparison for various metals, semiconductors, silica, polymers and the (ir)relevance for nanotechnologies J. Mech. Phys. Sol. 55 2007 1823 1852 (Pubitemid 47021162)
    • (2007) Journal of the Mechanics and Physics of Solids , vol.55 , Issue.9 , pp. 1823-1852
    • Maranganti, R.1    Sharma, P.2
  • 100
    • 33747351012 scopus 로고    scopus 로고
    • On causality of the gradient elasticity models
    • A. Metrikine On causality of the gradient elasticity models J. Sound Vibr. 297 2006 727 742
    • (2006) J. Sound Vibr. , vol.297 , pp. 727-742
    • Metrikine, A.1
  • 101
    • 0012686118 scopus 로고    scopus 로고
    • One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 1: Generic formulation
    • A. Metrikine, and H. Askes One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 1: Generic formulation Eur. J. Mech. A/Solids 21 2002 555 572
    • (2002) Eur. J. Mech. A/Solids , vol.21 , pp. 555-572
    • Metrikine, A.1    Askes, H.2
  • 102
    • 33744999606 scopus 로고    scopus 로고
    • An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice
    • DOI 10.1080/14786430500197827, PII P7680553535544
    • A. Metrikine, and H. Askes An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice Phil. Mag. 86 2006 3259 3286 (Pubitemid 43862407)
    • (2006) Philosophical Magazine , vol.86 , Issue.21-22 , pp. 3259-3286
    • Metrikine, A.V.1    Askes, H.2
  • 103
    • 79955568809 scopus 로고    scopus 로고
    • On the uniqueness of the Lagrangian of gradient elastic continua
    • A. Metrikine, and J. Prokhorova On the uniqueness of the Lagrangian of gradient elastic continua G. Maugin, A. Metrikine, Mechanics of Generalized Continua 2010 Springer 149 160
    • (2010) Mechanics of Generalized Continua , pp. 149-160
    • Metrikine, A.1    Prokhorova, J.2
  • 104
    • 0040366845 scopus 로고
    • Micro-structure in linear elasticity
    • R. Mindlin Micro-structure in linear elasticity Arch. Rat. Mech. Anal. 16 1964 52 78
    • (1964) Arch. Rat. Mech. Anal. , vol.16 , pp. 52-78
    • Mindlin, R.1
  • 105
    • 0000095439 scopus 로고
    • Second gradient of strain and surface-tension in linear elasticity
    • R. Mindlin Second gradient of strain and surface-tension in linear elasticity Int. J. Solids Struct. 1 1965 417 438
    • (1965) Int. J. Solids Struct. , vol.1 , pp. 417-438
    • Mindlin, R.1
  • 106
    • 0012251847 scopus 로고
    • Theories of elastic continua and crystal lattice theories
    • R. Mindlin Theories of elastic continua and crystal lattice theories E. Kroner, IUTAM Symposium Mechanics of Generalized Continua 1968 Springer-Verlag Berlin 312 320
    • (1968) IUTAM Symposium Mechanics of Generalized Continua , pp. 312-320
    • Mindlin, R.1
  • 107
    • 0001357580 scopus 로고
    • On first strain-gradient theories in linear elasticity
    • R. Mindlin, and N. Eshel On first strain-gradient theories in linear elasticity Int. J. Solids Struct. 4 1968 109 124
    • (1968) Int. J. Solids Struct. , vol.4 , pp. 109-124
    • Mindlin, R.1    Eshel, N.2
  • 108
    • 33947128429 scopus 로고
    • Effects of couple-stresses in linear elasticity
    • R. Mindlin, and H. Tiersten Effects of couple-stresses in linear elasticity Arch. Rat. Mech. Anal. 11 1962 415 448
    • (1962) Arch. Rat. Mech. Anal. , vol.11 , pp. 415-448
    • Mindlin, R.1    Tiersten, H.2
  • 109
    • 33744663097 scopus 로고
    • Some properties of media defined by constitutive equations in implicit form
    • A. Morgan Some properties of media defined by constitutive equations in implicit form Int. J. Eng. Sci. 4 1966 155 178
    • (1966) Int. J. Eng. Sci. , vol.4 , pp. 155-178
    • Morgan, A.1
  • 110
    • 0030208823 scopus 로고    scopus 로고
    • Dispersion and wave propagation in discrete and continuous models for granular materials
    • H.-B. Mühlhaus, and F. Oka Dispersion and wave propagation in discrete and continuous models for granular materials Int. J. Solids Struct. 33 1996 2841 2858
    • (1996) Int. J. Solids Struct. , vol.33 , pp. 2841-2858
    • Mühlhaus, H.-B.1    Oka, F.2
  • 111
    • 34347405605 scopus 로고
    • Calculation of the coefficients describing the linear dependence of the stress tensor on the second order material gradients of the displacement gradients: Rare gas solids
    • A. Opie, and J. Grindlay Calculation of the coefficients describing the linear dependence of the stress tensor on the second order material gradients of the displacement gradients: rare gas solids J. Phys. C: Solid State Phys. 5 1972 3289 3295
    • (1972) J. Phys. C: Solid State Phys. , vol.5 , pp. 3289-3295
    • Opie, A.1    Grindlay, J.2
  • 112
    • 0036395220 scopus 로고    scopus 로고
    • Microstructural randomness versus representative volume element in thermomechanics
    • DOI 10.1115/1.1410366
    • M. Ostoja-Starzewski Microstructural randomness versus representative volume element in thermomechanics ASME J. Appl. Mech. 69 2002 25 35 (Pubitemid 35187176)
    • (2002) Journal of Applied Mechanics, Transactions ASME , vol.69 , Issue.1 , pp. 25-35
    • Ostoja-Starzewski, M.1
  • 114
    • 0041096950 scopus 로고
    • Fundamental equations of the theory of asymmetric elasticity
    • V. Pal'mov Fundamental equations of the theory of asymmetric elasticity J. Appl. Math. Mech. 28 1964 496 505
    • (1964) J. Appl. Math. Mech. , vol.28 , pp. 496-505
    • Pal'Mov, V.1
  • 116
    • 69249220198 scopus 로고    scopus 로고
    • Wave dispersion in gradient elastic solids and structures: A unified treatment
    • S. Papargyri-Beskou, D. Polyzos, and D. Beskos Wave dispersion in gradient elastic solids and structures: a unified treatment Int. J. Solids Struct. 46 2009 3751 3759
    • (2009) Int. J. Solids Struct. , vol.46 , pp. 3751-3759
    • Papargyri-Beskou, S.1    Polyzos, D.2    Beskos, D.3
  • 120
    • 41549131636 scopus 로고    scopus 로고
    • Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories
    • A. Pichugin, H. Askes, and A. Tyas Asymptotic equivalence of homogenisation procedures and fine-tuning of continuum theories J. Sound Vibr. 313 2008 858 874
    • (2008) J. Sound Vibr. , vol.313 , pp. 858-874
    • Pichugin, A.1    Askes, H.2    Tyas, A.3
  • 121
    • 0242490335 scopus 로고    scopus 로고
    • Gradient elasticity and nonstandard boundary conditions
    • DOI 10.1016/j.ijsolstr.2003.06.001
    • C. Polizzotto Gradient elasticity and nonstandard boundary conditions Int. J. Solids Struct. 40 2003 7399 7423 (Pubitemid 37418353)
    • (2003) International Journal of Solids and Structures , vol.40 , Issue.26 , pp. 7399-7423
    • Polizzotto, C.1
  • 122
    • 0038710298 scopus 로고    scopus 로고
    • A boundary element method for solving 2-D and 3-D static gradient elastic problems Part I: Integral formulation
    • DOI 10.1016/S0045-7825(03)00289-5
    • D. Polyzos, K. Tsepoura, S. Tsinopoulos, and D. Beskos A boundary element method of solving 2-D and 3-D static gradient elastic problems Part I: Integral formulation Comput. Methods Appl. Mech. Eng. 192 2003 2845 2873 (Pubitemid 36882080)
    • (2003) Computer Methods in Applied Mechanics and Engineering , vol.192 , Issue.26-27 , pp. 2845-2873
    • Polyzos, D.1    Tsepoura, K.G.2    Tsinopoulos, S.V.3    Beskos, D.E.4
  • 123
    • 14044250125 scopus 로고    scopus 로고
    • Transient dynamic analysis of 3-D gradient elastic solids by BEM
    • DOI 10.1016/j.compstruc.2004.11.001, PII S0045794904004304
    • D. Polyzos, K. Tsepoura, and D. Beskos Transient dynamic analysis of 3-D gradient elastic solids by BEM Comput. Struct. 83 2005 783 792 (Pubitemid 40276592)
    • (2005) Computers and Structures , vol.83 , Issue.10-11 SPEC. ISS. , pp. 783-792
    • Polyzos, D.1    Tsepoura, K.G.2    Beskos, D.E.3
  • 124
    • 0000833634 scopus 로고
    • Lattice dynamics of grey tin and indium antomonide
    • D. Price, J. Rowe, and R. Nicklow Lattice dynamics of grey tin and indium antomonide Phys. Rev. B 3 1971 1268 1279
    • (1971) Phys. Rev. B , vol.3 , pp. 1268-1279
    • Price, D.1    Rowe, J.2    Nicklow, R.3
  • 125
    • 84867959680 scopus 로고    scopus 로고
    • On implicit constitutive theories
    • K. Rajagopal On implicit constitutive theories Appl. Math. 48 2003 279 319
    • (2003) Appl. Math. , vol.48 , pp. 279-319
    • Rajagopal, K.1
  • 128
    • 0002471276 scopus 로고
    • A simple approach to solve boundary-value problems in gradient elasticity
    • C. Ru, and E. Aifantis A simple approach to solve boundary-value problems in gradient elasticity Acta Mech. 101 1993 59 68
    • (1993) Acta Mech. , vol.101 , pp. 59-68
    • Ru, C.1    Aifantis, E.2
  • 129
    • 0342921375 scopus 로고
    • Continuum model of dispersion caused by an inherent material characteristic length
    • M. Rubin, P. Rosenau, and O. Gottlieb Continuum model of dispersion caused by an inherent material characteristic length J. Appl. Phys. 77 1995 4054 4063
    • (1995) J. Appl. Phys. , vol.77 , pp. 4054-4063
    • Rubin, M.1    Rosenau, P.2    Gottlieb, O.3
  • 130
    • 60849130961 scopus 로고    scopus 로고
    • A strain gradient generalized continuum approach for modelling elastic scale effects
    • C. Sansour, and S. Skatulla A strain gradient generalized continuum approach for modelling elastic scale effects Comput. Methods Appl. Mech. Eng. 198 2009 1401 1412
    • (2009) Comput. Methods Appl. Mech. Eng. , vol.198 , pp. 1401-1412
    • Sansour, C.1    Skatulla, S.2
  • 131
    • 1442334606 scopus 로고
    • Elastic wave propagation in a solid with microstructure
    • G. Savin, A. Lukashev, and E. Lysko Elastic wave propagation in a solid with microstructure Int. Appl. Mech. 6 1970 725 728
    • (1970) Int. Appl. Mech. , vol.6 , pp. 725-728
    • Savin, G.1    Lukashev, A.2    Lysko, E.3
  • 132
    • 0032710222 scopus 로고    scopus 로고
    • Finite elements for materials with strain gradient effects
    • J. Shu, W. King, and N. Fleck Finite elements for materials with strain gradient effects Int. J. Numer. Methods Eng. 44 1999 373 391
    • (1999) Int. J. Numer. Methods Eng. , vol.44 , pp. 373-391
    • Shu, J.1    King, W.2    Fleck, N.3
  • 135
    • 0034824728 scopus 로고    scopus 로고
    • Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient micro-polar constitutive theory
    • DOI 10.1007/BF01261670
    • A. Suiker, and R. de Borst Micro-mechanical modelling of granular material. Part 1: derivation of a second-gradient micro-polar constitutive theory Acta Mech. 149 2001 161 180 C.S.C. (Pubitemid 32864001)
    • (2001) Acta Mechanica , vol.149 , Issue.1-4 , pp. 161-180
    • Suiker, A.S.J.1    De Borst, R.2    Chang, C.S.3
  • 136
    • 0242285511 scopus 로고    scopus 로고
    • Analysis of materials with strain-gradient effects: A meshless local Petrov-Galerkin (MLPG) approach, with nodal displacements only
    • Z. Tang, S. Shen, and S. Atluri Analysis of materials with strain-gradient effects: a Meshless Local Petrov Galerkin (MLPG) approach, with nodel displacements only Comput. Model. Eng. Sci. 4 2003 177 196 (Pubitemid 37360208)
    • (2003) CMES - Computer Modeling in Engineering and Sciences , vol.4 , Issue.1 , pp. 177-196
    • Tang, Z.1    Shen, S.2    Atluri, S.N.3
  • 137
    • 27644439547 scopus 로고    scopus 로고
    • Size effects in the mechanical behavior of cellular materials
    • DOI 10.1007/s10853-005-5042-5
    • C. Tekoglu, and P. Onck Size effects in the mechanical behavior of cellular materials J. Mater. Sci. 40 2005 5911 5917 (Pubitemid 41563498)
    • (2005) Journal of Materials Science , vol.40 , Issue.22 , pp. 5911-5917
    • Tekoglu, C.1    Onck, P.R.2
  • 138
    • 0038746383 scopus 로고    scopus 로고
    • A two-dimensional finite element implementation of a special form of gradient elasticity
    • L. Tenek, and E. Aifantis A two-dimensional finite element implementation of a special form of gradient elasticity Comput. Model. Eng. Sci. 3 2002 731 741
    • (2002) Comput. Model. Eng. Sci. , vol.3 , pp. 731-741
    • Tenek, L.1    Aifantis, E.2
  • 139
    • 0343926464 scopus 로고
    • Plane-strain flow localization in tension blocks obeying strain-gradient-dependent constitutive equation
    • Y. Tomita, and T. Fujimoto Plane-strain flow localization in tension blocks obeying strain-gradient-dependent constitutive equation Mater. Sci. Res. Int. 1 1995 254 259
    • (1995) Mater. Sci. Res. Int. , vol.1 , pp. 254-259
    • Tomita, Y.1    Fujimoto, T.2
  • 140
    • 33947180779 scopus 로고
    • Elastic materials with couple-stresses
    • R. Toupin Elastic materials with couple-stresses Arch. Rat. Mech. Anal. 11 1962 385 414
    • (1962) Arch. Rat. Mech. Anal. , vol.11 , pp. 385-414
    • Toupin, R.1
  • 141
    • 34250940617 scopus 로고
    • Theories of elasticity with couple-stress
    • R. Toupin Theories of elasticity with couple-stress Arch. Rat. Mech. Anal. 17 1964 85 112
    • (1964) Arch. Rat. Mech. Anal. , vol.17 , pp. 85-112
    • Toupin, R.1
  • 142
    • 0022891455 scopus 로고
    • A gradient approach to localization of deformation. I. Hyperelastic materials
    • N. Triantafyllidis, and E. Aifantis A gradient approach to localization of deformation. I. Hyperelastic materials J. Elast. 16 1986 225 237 (Pubitemid 17459967)
    • (1986) Journal of Elasticity , vol.16 , Issue.3 , pp. 225-237
    • Triantafyllidis, N.1    Aifantis Elias, C.2
  • 143
    • 0036814242 scopus 로고    scopus 로고
    • A boundary element method for solving 3D static gradient elastic problems with surface energy
    • DOI 10.1007/s00466-002-0348-5
    • K. Tsepoura, S. Papargyi-Beskou, and D. Polyzos A boundary element method for solving 3D static gradient elastic problems with surface energy Comput. Mech. 29 2002 361 381 (Pubitemid 35422277)
    • (2002) Computational Mechanics , vol.29 , Issue.4-5 , pp. 361-381
    • Tsepoara, K.G.1    Papargyri-Beskou, S.2    Polyzos, D.3
  • 144
    • 0037623750 scopus 로고    scopus 로고
    • A boundary element method for solving 2-D and 3-D static gradient elastic problems Part II: Numerical implementation
    • DOI 10.1016/S0045-7825(03)00290-1
    • K. Tsepoura, S. Tsinopoulos, D. Polyzos, and D. Beskos A boundary element method of solving 2-D and 3-D static gradient elastic problems Part II: numerical implementation Comput. Methods Appl. Mech. Eng. 192 2003 2875 2907 (Pubitemid 36882081)
    • (2003) Computer Methods in Applied Mechanics and Engineering , vol.192 , Issue.26-27 , pp. 2875-2907
    • Tsepoura, K.G.1    Tsinopoulos, S.V.2    Polyzos, D.3    Beskos, D.E.4
  • 145
    • 0002516837 scopus 로고
    • The asymptotic solution of gradient elasticity for mode-III
    • D. Unger, and E. Aifantis The asymptotic solution of gradient elasticity for mode-III Int. J. Fract. 71 1995 R27 R32
    • (1995) Int. J. Fract. , vol.71
    • Unger, D.1    Aifantis, E.2
  • 146
    • 0034320847 scopus 로고    scopus 로고
    • Strain gradient elasticity theory for antiplane shear cracks. Part I: Oscillatory displacements
    • D. Unger, and E. Aifantis Strain gradient elasticity theory for antiplane shear cracks. Part I: oscillatory displacements Theor. Appl. Fract. Mech. 34 2000 243 252
    • (2000) Theor. Appl. Fract. Mech. , vol.34 , pp. 243-252
    • Unger, D.1    Aifantis, E.2
  • 147
    • 0034324258 scopus 로고    scopus 로고
    • Strain gradient elasticity theory for antiplane shear cracks. Part II: Monotonic displacements
    • D. Unger, and E. Aifantis Strain gradient elasticity theory for antiplane shear cracks. Part II: monotonic displacements Theor. Appl. Fract. Mech. 34 2000 253 265
    • (2000) Theor. Appl. Fract. Mech. , vol.34 , pp. 253-265
    • Unger, D.1    Aifantis, E.2
  • 150
    • 0042119232 scopus 로고
    • A theory of elasticity with spatial distribution of matter. Three-dimensional complex structure
    • V. Vdovin, and I. Kunin A theory of elasticity with spatial distribution of matter. Three-dimensional complex structure J. Appl. Math. Mech. 30 1966 1272 1281
    • (1966) J. Appl. Math. Mech. , vol.30 , pp. 1272-1281
    • Vdovin, V.1    Kunin, I.2
  • 151
    • 36049060807 scopus 로고
    • Lattice dynamics of lithium hydride
    • J. Verble, J. Warren, and J. Yarnell Lattice dynamics of lithium hydride Phys. Rev. 168 1968 980 989
    • (1968) Phys. Rev. , vol.168 , pp. 980-989
    • Verble, J.1    Warren, J.2    Yarnell, J.3
  • 152
    • 0003389412 scopus 로고
    • Theoretische Studien über die Elasticitätsverhältnisse der Krystalle. I. Ableitung der Grundgleichungen aus der Annahme mit Polarität begabter Moleküle
    • W. Voigt Theoretische Studien über die Elasticitätsverhä ltnisse der Krystalle. I. Ableitung der Grundgleichungen aus der Annahme mit Polarität begabter Moleküle Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen 34 1887 3 52
    • (1887) Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen , vol.34 , pp. 3-52
    • Voigt, W.1
  • 153
    • 0003389412 scopus 로고
    • Theoretische Studien über die Elasticitätsverhältnisse der Krystalle. II. Untersuchung des elastischen Verhaltens eines Cylinders aus krystallinscher Substanz, auf dessen Mantelfläche keine Kräfte wirken, wenn die in seinem Innern wirkenden Spannungen längs der Cylinderaxe constant sind
    • W. Voigt Theoretische Studien über die Elasticitätsverhä ltnisse der Krystalle. II. Untersuchung des elastischen Verhaltens eines Cylinders aus krystallinscher Substanz, auf dessen Mantelfläche keine Kräfte wirken, wenn die in seinem Innern wirkenden Spannungen längs der Cylinderaxe constant sind Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen 34 1887 53 79
    • (1887) Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen , vol.34 , pp. 53-79
    • Voigt, W.1
  • 154
    • 0003389412 scopus 로고
    • Theoretische Studien über die Elasticitätsverhältnisse der Krystalle. III. Untersuchung des elastischen Verhaltens eines Cylinders aus krystallinscher Substanz, auf dessen Mantelfläche keine äussern Drucke wirken, wenn die in seinem Innern wirkenden Spannungen lineäre Functionen der Axenrichtung sind
    • W. Voigt Theoretische Studien über die Elasticitätsverhä ltnisse der Krystalle. III. Untersuchung des elastischen Verhaltens eines Cylinders aus krystallinscher Substanz, auf dessen Mantelfläche keine äussern Drucke wirken, wenn die in seinem Innern wirkenden Spannungen lineäre Functionen der Axenrichtung sind Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen 34 1887 80 100
    • (1887) Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen , vol.34 , pp. 80-100
    • Voigt, W.1
  • 155
    • 28544452524 scopus 로고    scopus 로고
    • Flexural wave propagation in single-walled carbon nanotubes
    • L. Wang, and H. Hu Flexural wave propagation in single-walled carbon nanotubes Phys. Rev. B 71 2005 195412
    • (2005) Phys. Rev. B , vol.71 , pp. 195412
    • Wang, L.1    Hu, H.2
  • 158
    • 21344461668 scopus 로고    scopus 로고
    • On using strain gradient theories in the analysis of cracks
    • J. Yang, and S. Guo On using strain gradient theories in the analysis of cracks Int. J. Fract. 133 2005 L19 L22
    • (2005) Int. J. Fract. , vol.133
    • Yang, J.1    Guo, S.2
  • 161
    • 38349192853 scopus 로고    scopus 로고
    • Finite elements for elasticity with microstructure and gradient elasticity
    • A. Zervos Finite elements for elasticity with microstructure and gradient elasticity Int. J. Numer. Methods Eng. 73 2008 564 595
    • (2008) Int. J. Numer. Methods Eng. , vol.73 , pp. 564-595
    • Zervos, A.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.