This semi-inverse method is similar to that used in the so-called Saint-Venant problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. This semi-inverse method is similar to that used by Saint-Venant to solve the omonimus problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. Two examples are also presented. It is found that wedge forces are necessary to maintain the body in equilibrium and that these are not an artefact of the double application of the divergence theorem in the second-gradient material derivations.
GermainP.La méthode des puissances virtuelle sen mécanique des milieux continus Premiere partie: Theorie du second gradient. J Mécanique1973; 12(2): 235–274.
2.
GermainP.Sur l’application de la méthode des puissances virtuelles en mécanique des milieux continus. C R Acad Sci Paris Sér A-B1972; 274: A1051–A1055.
3.
GermainP.The method of virtual power in continuum mechanics Part 2: Microstructure. SIAM J Appl Math1973; 25(3): 556–575.
4.
ToupinRA.Elastic materials with couple-stresses. Arch Rat Mech Anal1962; 11: 385–414.
5.
ToupinRA.Theories of elasticity with couple-stresses. Arch Rat Mech Anal1964; 17: 85–112.
6.
GreenAERivlinRS.Simple force and stress multipoles. Arch Rat Mech Anal1964; 16: 325–353.
7.
GreenAERivlinRS.Multipolar continuum mechanics. Arch Rat Mech Anal1964; 17: 113–147.
8.
GreenAERivlinRS.Multipolar continuum mechanics: Functional theory. I. Proc R Soc Ser A1965; 284: 303–324.
9.
MindlinRD.Micro-structure in Linear Elasticity. New York: Department of Civil Engineering Columbia University New York, 1964.
10.
MindlinRD.Second gradient of strain and surface tension in linear elasticity. Int J Solids Struct1965; 1(4): 417–438.
11.
MindlinRDEshelNN.On first strain-gradient theories in linear elasticity. Int J Solids Struct1968; 4(1): 109–124.
12.
MindlinRDTierstenHF.Effects of couple stresses in linear elasticity. Arch Rat Mech Anal1962; 11: 415–448.
13.
MurdochAI.Symmetry considerations for materials of second grade. J Elast1979; 9: 43–50.
14.
PideriCSeppecherP.A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin Mech Thermodyn1997; 9(5): 241–257.
15.
SeppecherP.A numerical study of a moving contact line in Cahn–Hilliard theory. Internat J Eng Sci1996; 34: 977–992.
SunykRSteinmannP.On higher gradients in continuum-atomistic modelling. Int J Solids Struct2003; 40: 6877–6896.
18.
KronerE.Mechanics of Generalized Continua. Berlin: Springer, 1968.
19.
De FeliceGRizziN. Homogenization for materials with microstructure. Am Soc Mech Eng Pres Ves Pip Div1997; 369: 33–38.
20.
dell’IsolaFSeppecherP.Edge contact forces and quasi-balanced power. Meccanica1997; 32(1): 33–52.
21.
dell’IsolaFSeppecherP.The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. C R Mecanique1995; 321: 303–308.
22.
AifantisEC.On the role of gradients in the localization of deformation and fracture. Internat J Engrg Sci1992; 30(10): 1279–1299.
23.
LuongoA.On the amplitude modulation and localization phenomena in interactive buckling problems. Int J Solids Struct1991; 27(15): 1943–1954.
24.
LuongoA.Mode localization in dynamics and buckling of linear imperfect continuous structures. Nonlinear Dynam2001; 25(1): 133–156.
25.
LuongoADi EgidioA.Bifurcation equations through multiple-scales analysis for a continuous model of a planar beam. Nonlinear Dynam2005; 41: 171–190.
26.
TriantafyllidisNAifantisECA. Gradient approach to localization of deformation, I. Hyperelastic materials. J Elast1986; 16(3): 225–237.
27.
AndreausUChiaiaBPlacidiL.Soft-impact dynamics of deformable bodies. Contin Mech Thermodyn2013; 25: 375–398.
28.
CarcaterraACiappiE.Prediction of the compressible stage slamming force on rigid and elastic systems impacting over the water surface. Nonlinear Dynam2000; 21(2): 193–220.
29.
CarcaterraACiappiEIafratiACampanaEF.Shock spectral analysis of elastic systems impacting on the water surface. J Sound Vibr2000; 229(3): 579–605.
30.
dell’IsolaFAndreausUPlacidiL.At the origins and in the vanguard of peridynamics, non-local and higher gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola. Math Mech Solids2015; 20: 887–928. DOI: 10.1177/1081286513509811.
dell’IsolaFAndreausUCazzaniCPeregoUPlacidiLRutaGScerratoD.On a debated principle of Lagrange’s analytical mechanics and on its multiple applications. In: AndreausUdell’IsolaFEspositoRForestSMaierGPeregoU (eds), The complete works of Gabrio Piola, vol. I. New York: Springer, 2014.
33.
dell’IsolaFAndreausUPlacidiLScerratoD.About the fundamental equations of the motion of bodies whatsoever, as considered following the natural their form and constitution. In: AndreausUdell’IsolaFEspositoRForestSMaierGPeregoU (eds), The complete works of Gabrio Piola, vol. I. New York: Springer, 2014.
34.
D’AgostinoMVGiorgioIGrecoLMadeoABoisseP.Continuum and discrete models for structures including (quasi-)inextensible elasticae with a view to the design and modeling of composite reinforcements. Int J Solids Struct2014; DOI: 10.1016/j.ijsolstr.2014.12.014.
35.
FerrettiMMadeoAdell’IsolaFBoisseP.Modelling the onset of shear boundary layers in fibrous composite reinforcements by second gradient theory. Z Angew Math Phys2014; 65(3): 587–612.
36.
MauginGA.The principle of virtual power: from eliminating metaphysical forces to providing an efficient modelling tool. Contin Mech Thermodyn2011; 25: 127–146.
37.
PlacidiLRosiGGiorgioIMadeoA.Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second gradient materials. Math Mech Solids2014; 19: 555–578.
38.
dell’IsolaFSteigmannD.A two-dimensional gradient-elasticity theory for woven fabrics. J Elasticity2015; 118(1): 113–125.
39.
CarcaterraAdell’IsolaFEspositoRPulvirentiM.Macroscopic description of microscopically strongly inhomogenous systems: A mathematical basis for the synthesis of higher gradients metamaterials. Arch Ration Mech Anal2015; 218(3): 1239–1262.
40.
dell’IsolaFLekszyckiTPawlikowskiMGrygorukRGrecoL.Designing a light fabric metamaterial being highly macroscopically tough under directional extension: First experimental evidence. Z Angew Math Phys. Epub ahead of print 17 July 2015. DOI: 10.1007/s00033-015-0556-4.
41.
AlibertJ-JSeppecherPdell’IsolaF.Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids2003; 8(1): 51–73.
42.
dell’IsolaFMadeoAPlacidiL.Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. Z Angew Math Mech2012; 92: 52–71.
43.
dell’IsolaFSciarraGVidoliS.Generalized Hooke’s law for isotropic second gradient materials. Proc R Soc Lond Ser A2009; 465(107): 2177–2196.
44.
Del VescovoDGiorgioI. Dynamic problems for metamaterials: review of existing models and ideas for further research. Internat J Engrg Sci2014; 80: 153–172.
45.
MadeoANeffPGhibaI-DPlacidiLRosiG.Band gaps in the relaxed linear micromorphic continuum. Z Angew Math Mech2015; 95: 880–887. DOI: 10.1002/zamm.201400036.
46.
NeffPGhibaI-DMadeoAPlacidiLRosiG.A unifying perspective: the relaxed linear micromorphic continuum. Contin Mech Thermodyn2014; 26: 639–681.
47.
ScerratoDGiorgioIMadeoALimamADarveF.A simple non-linear model for internal friction in modified concrete. Internat J Eng Sci2014; 80: 136–152.
48.
dell’IsolaFGouinHSeppecherP.Radius and surface tension of microscopic bubbles by second gradient theory. C R Acad Sci II Mec1995; 320: 211–216.
49.
dell’IsolaFRotoliG.Validity of Laplace formula and dependence of surface tension on curvature in second gradient fluids. Mech Res Commun1995; 22: 485–490.
50.
SteebHDiebelsS.Modeling thin films applying an extended continuum theory based on a scalar-valued order parameter - Part I: Isothermal case. Int J Solids Struct2004; 41: 5071–5085.
51.
AltenbachHEremeyevVALebedevLPRendònLA.Acceleration waves and ellipticity in thermoelastic micropolar media. Arch Appl Mech2010; 80(3): 217–227.
52.
EremeyevVA.Acceleration waves in micropolar elastic media. Dokl Phys2005; 50(4): 204–206.
53.
SoubestreJBoutinC.Non-local dynamic behavior of linear fiber reinforced materials. Mech Mater2012; 55: 16–32.
54.
Abd-allaANGalantucciLGiorgioIHamdanAMDel VescovoD.Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity. Contin Mech Thermodyn2014; DOI: 10.1007/s00161-014-0400-7.
55.
Abd-allaANHamdanAMGiorgioIDel VescovoD.The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials. Arch Appl Mech2014; 84(9–11): 1229–1248.
56.
dell’IsolaFGuarascioMHutterK.A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle. Arch Appl Mech2000; 70: 323–337.
57.
ForestS.Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J Eng Mech2009; 135(3): 117–131.
58.
ForestSAifantisEC.Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. Int J Solids Struct2010; 47(25–26): 3367–3376.
59.
MisraAChangCS.Effective elastic moduli of heterogeneous granular solids. Int J Solids Struct1993; 30: 2547–2566.
60.
MisraAChingWY.Theoretical nonlinear response of complex single crystal under multi-axial tensile loading. Sci Rep2013; 3: 1488.
61.
MisraASinghV.Micromechanical model for viscoelastic-materials undergoing damage. Contin Mech Thermodyn2013; 25: 1–16.
62.
MisraAYangY.Micromechanical model for cohesive materials based upon pseudo-granular structure. Int J Solids Struct2010; 47: 2970–2981.
63.
RinaldiAKrajcinovicDPeraltaPLaiY-C.Lattice models of polycrystalline microstructures: A quantitative approach. Mech Mater2008; 40: 17–36.
64.
YangYChingWYMisraA.Higher-order continuum theory applied to fracture simulation of nano-scale intergranular glassy film. J Nanomech Micromech2011; 1: 60–71.
65.
YangYMisraA.Higher-order stress–strain theory for damage modeling implemented in an element-free Galerkin formulation. Comput Model Eng Sci2010; 64: 1–36.
66.
YangYMisraA.Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. Int J Solids Struct2012; 49: 2500–2514.
67.
RizziNLVaranoV.The effects of warping on the postbuckling behaviour of thin-walled structures. Thin Wall Struct2011; 49(9): 1091–1097.
68.
El DhabaARDeformation of an infinite, square rod of an elastic magnetizable material, subjected to an external magnetic field by a boundary integral method: a numerical approach. J Vib Control2014; DOI: 10.1177/1077546314563968.
69.
El DhabaARAbou-DinaMS.Deformation for a rectangle by a finite Fourier transform. J Comput Theor Nanos2015; 12: 31–37.
70.
El DhabaARAbou-DinaMS.Thermal stresses induced by a variable heat source in a rectangle and variable pressure at its boundary by finite Fourier transform. J Therm Stresses2015; 38(6): 677–700.
71.
El DhabaARGhalebAFAbou-DinaMS.Deformation of a long, current-carrying elastic cylinder of square cross-section. Numerical solution by boundary integrals. Arch Appl Mech2014; 84(9–11): 1393–1407.
72.
El DhabaARGhalebAFAbou-DinaMS.Deformation of an infinite, square cylinder of an elastic magnetizable material, subjected to an external magnetic field by a boundary integral method. A numerical approach. J Vib Control2015; accepted.
73.
El DhabaARGhalebAFAbou-DinaMS.Numerical treatment of a problem of plane, uncoupled linear thermo-elasticity for a square cylinder by a boundary integral method. J Comput Theor Nanos2015; 12: 1–15.
74.
El DhabaARGhalebAFPlacidiL. Deformation of an elastic magnetizable square rod due to a uniform electric current inside the rod and an external transverse magnetic field. Math Mech Solids; in press.
75.
CuomoMContrafattoL.Stress rate formulation for elastoplastic models with internal variables based on augmented Lagrangian regularisation. Int J Solids Struct2000; 37: 3935–3964.
76.
CuomoMVenturaG.Complementary energy approach to contact problems based on consistent augmented Lagrangian regularization. Math Comput Model1998; 28: 185–204.
77.
dell’IsolaFGouinHRotoliG.Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. Eur J Mech B Fluids1996; 15(4): 545–568.
78.
RosiGGiorgioIEremeyevVA.Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. Z Angew Math Mech2013; 93(12): 914–927.
79.
TurcoE.Identification of axial forces on statically indeterminate pin-jointed trusses by a nondestructive mechanical test. Open Civil Eng J2013; 7(1) 50–57.
80.
PlacidiLHutterK.Thermodynamics of polycrystalline materials treated by the theory of mixtures with continuous diversity. Contin Mech Thermodyn2006; 17: 409–451.
81.
SpivakM.A comprehensive introduction to differential geometry, Vol. I and II, 2nd edn.Wilmington, DE: Publish or Perish, Inc., 1979.
