Convexity and Quasiconvexity in a Cosserat Model for Fiber-Reinforced Elastic Solids
An inequality of the quasiconvexity type is derived in the context of a Cosserat model for fiber-reinforced elastic solids in which the Cosserat rotation field accounts for the kinematics of the embedded fibers, modeled as extensible, unshearable spatial rods. This is shown to be equivalent to the r...
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| Published in | Journal of elasticity Vol. 154; no. 1-4; pp. 555 - 567 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.11.2023
Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 0374-3535 1573-2681 |
| DOI | 10.1007/s10659-022-09963-8 |
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| Abstract | An inequality of the quasiconvexity type is derived in the context of a Cosserat model for fiber-reinforced elastic solids in which the Cosserat rotation field accounts for the kinematics of the embedded fibers, modeled as extensible, unshearable spatial rods. This is shown to be equivalent to the requirement that the fiber bend-twist strain vector associated with an energy minimizer be a point of convexity of the operative strain-energy function. The analysis requires careful consideration of constraints associated with the fact that fibers are convected by the continuum deformation field as material curves. |
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| AbstractList | An inequality of the quasiconvexity type is derived in the context of a Cosserat model for fiber-reinforced elastic solids in which the Cosserat rotation field accounts for the kinematics of the embedded fibers, modeled as extensible, unshearable spatial rods. This is shown to be equivalent to the requirement that the fiber bend-twist strain vector associated with an energy minimizer be a point of convexity of the operative strain-energy function. The analysis requires careful consideration of constraints associated with the fact that fibers are convected by the continuum deformation field as material curves. |
| Author | Steigmann, D. J. Shirani, M. |
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| Cites_doi | 10.1007/s10659-022-09934-z 10.1142/10959 10.1007/BF00948288 10.1007/BF00041892 10.1007/s11012-014-0042-7 10.1016/j.ijengsci.2015.08.001 10.1016/j.ijsolstr.2008.09.027 10.1016/j.ijnonlinmec.2012.04.007 10.1017/S0308210500004844 10.1007/978-3-662-03278-7 10.1007/s10659-021-09859-z 10.1093/qjmam/hbaa013 10.1002/sapm19755411 10.1007/BF00042633 10.1007/BF00379680 10.1007/BF00040817 10.3390/sym12071133 10.1007/BF00041227 10.1007/978-3-031-04548-6_13 |
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| Keywords | Quasiconvexity Cosserat elasticity 74A60 Fiber-reinforced materials 74A05 Convexity 74B20 74A30 |
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| References | ReissnerE.A further note on finite-strain force and moment stress elasticityZ. Angew. Math. Phys.19873866567310.1007/BF009482880626.73027 ShiraniM.SteigmannD.J.AltenbachH.Necessary conditions for energy minimizers in a Cosserat model of fiber-reinforced elastic solidsRecent Approaches in the Theory of Plates and Plate-Like Structures. Advanced Structured Materials2021SwitzerlandSpringer253265 DeseriL.OwenD.R.Stable disarrangement phases of elastic aggregates: a setting for the emergence of no-tension materials with nonlinear response in compressionMeccanica2014492907293210.1007/s11012-014-0042-71306.74009 EricksenJ.L.DickeyR.W.Loading devices and stability of equilibriumNonlinear Elasticity1975NYAcademic Press CosseratE.CosseratF.Théorie des Corps Déformables1909ParisHermann40.0862.02 McAvoyR.C.SteigmannD.J.Cosserat elasticity of helically wound cylindersJ. Elast.202210.1007/s10659-022-09934-z ShiraniM.SteigmannD.J.A Cosserat model of elastic solids reinforced by a family of curved and twisted fibersSymmetry202012113310.3390/sym12071133 ReissnerE.Note on the equations of finite-strain force and moment stress elasticityStud. Appl. Math.1975541844600910.1002/sapm197554110326.73006 ShiraniM.SteigmannD.J.GiorgioI.Quasiconvexity and rank-one convexity in Cosserat elasticity theoryTheoretical Analyses, Computations and Experiments of Multiscale Materials2022SwitzerlandSpringer27328310.1007/978-3-031-04548-6_131496.74031 SteigmannD.J.Theory of elastic solids reinforced with fibers resistant to extension, flexure and twistInt. J. Non-Linear Mech.20124773474210.1016/j.ijnonlinmec.2012.04.007 NeffP.Existence of minimizers for a finite-strain micro-morphic elastic solidProc. R. Soc. Edinb. A2006136997101210.1017/S03082105000048441106.74010 HilgersM.G.PipkinA.C.Energy-minimizing deformations of elastic sheets with bending stiffnessJ. Elast.199331125139122097910.1007/BF000412270773.73051 DeseriL.OwenD.R.Stable disarrangement phases arising from expansion/contraction or from simple shearing of a model granular mediumInt. J. Eng. Sci.201596111130341118310.1016/j.ijengsci.2015.08.0011423.74203 TruesdellC.NollW.FlüggeS.The non-linear field theories of mechanicsHandbuch der Physik, Vol. III/31965BerlinSpringer SteigmannD.J.FaulknerM.E.Variational theory for spatial rodsJ. Elast.199333126125503810.1007/BF000426330801.73039 DillE.H.Kirchhoff’s theory of rodsArch. Hist. Exact Sci.199244123116600710.1007/BF003796800762.01012 ShiraniM.SteigmannD.J.Cosserat elasticity of lattice solidsJ. Elast.20221517388449644410.1007/s10659-021-09859-z1501.74007 ShiraniM.SteigmannD.J.NeffP.The Legendre-Hadamard condition in Cosserat elasticity theoryQ. J. Mech. Appl. Math.202073293303421285110.1093/qjmam/hbaa0131502.74015 BottemaO.RothB.Theoretical Kinematics19900747.70001 FosdickR.L.MacSithighG.P.Minimization in nonlinear elasticity theory for bodies reinforced with inextensible cordsJ. Elast.199126239289113547910.1007/BF000418920768.73097 FosdickR.L.MacSithighG.P.Minimization in incompressible nonlinear elasticity theoryJ. Elast.19861626730186016010.1007/BF000408170616.73026 AntmanS.S.Nonlinear Problems of Elasticity2005BerlinSpringer1098.74001 LandauL.D.LifshitzE.M.Theory of Elasticity19863OxfordPergamon0014.37504 PietraszkiewiczW.EremeyevV.A.On natural strain measures of the nonlinear micropolar continuumInt. J. Solids Struct.20094677478710.1016/j.ijsolstr.2008.09.0271215.74004 SteigmannD.J.DorfmannL.OgdenR.W.Effects of fiber bending and twisting resistance on the mechanics of fiber-reinforced elastomersNonlinear Mechanics of Soft Fibrous Tissues2015Wien and New YorkSpringer269305 GiaquintaM.HildebrandtS.Calculus of Variations I2004BerlinSpringer10.1007/978-3-662-03278-70853.49002 EremeyevV.A.CloudM.J.LebedevL.P.Applications of Tensor Analysis in Continuum Mechanics2018SingaporeWorld Scientific10.1142/109591471.74001 M. Shirani (9963_CR6) 2020; 12 O. Bottema (9963_CR24) 1990 L.D. Landau (9963_CR1) 1986 D.J. Steigmann (9963_CR4) 2012; 47 M. Giaquinta (9963_CR23) 2004 E. Cosserat (9963_CR9) 1909 D.J. Steigmann (9963_CR5) 2015 M. Shirani (9963_CR15) 2022 E.H. Dill (9963_CR2) 1992; 44 M. Shirani (9963_CR7) 2022; 151 R.L. Fosdick (9963_CR21) 1991; 26 E. Reissner (9963_CR12) 1987; 38 S.S. Antman (9963_CR3) 2005 M.G. Hilgers (9963_CR22) 1993; 31 L. Deseri (9963_CR25) 2014; 49 M. Shirani (9963_CR17) 2021 P. Neff (9963_CR14) 2006; 136 J.L. Ericksen (9963_CR19) 1975 R.L. Fosdick (9963_CR20) 1986; 16 R.C. McAvoy (9963_CR8) 2022 M. Shirani (9963_CR16) 2020; 73 D.J. Steigmann (9963_CR18) 1993; 33 V.A. Eremeyev (9963_CR27) 2018 L. Deseri (9963_CR26) 2015; 96 W. Pietraszkiewicz (9963_CR13) 2009; 46 C. Truesdell (9963_CR10) 1965 E. Reissner (9963_CR11) 1975; 54 |
| References_xml | – reference: CosseratE.CosseratF.Théorie des Corps Déformables1909ParisHermann40.0862.02 – reference: LandauL.D.LifshitzE.M.Theory of Elasticity19863OxfordPergamon0014.37504 – reference: ReissnerE.Note on the equations of finite-strain force and moment stress elasticityStud. Appl. Math.1975541844600910.1002/sapm197554110326.73006 – reference: HilgersM.G.PipkinA.C.Energy-minimizing deformations of elastic sheets with bending stiffnessJ. Elast.199331125139122097910.1007/BF000412270773.73051 – reference: EricksenJ.L.DickeyR.W.Loading devices and stability of equilibriumNonlinear Elasticity1975NYAcademic Press – reference: McAvoyR.C.SteigmannD.J.Cosserat elasticity of helically wound cylindersJ. Elast.202210.1007/s10659-022-09934-z – reference: SteigmannD.J.FaulknerM.E.Variational theory for spatial rodsJ. Elast.199333126125503810.1007/BF000426330801.73039 – reference: SteigmannD.J.Theory of elastic solids reinforced with fibers resistant to extension, flexure and twistInt. J. Non-Linear Mech.20124773474210.1016/j.ijnonlinmec.2012.04.007 – reference: DillE.H.Kirchhoff’s theory of rodsArch. Hist. Exact Sci.199244123116600710.1007/BF003796800762.01012 – reference: GiaquintaM.HildebrandtS.Calculus of Variations I2004BerlinSpringer10.1007/978-3-662-03278-70853.49002 – reference: ShiraniM.SteigmannD.J.NeffP.The Legendre-Hadamard condition in Cosserat elasticity theoryQ. J. Mech. Appl. Math.202073293303421285110.1093/qjmam/hbaa0131502.74015 – reference: NeffP.Existence of minimizers for a finite-strain micro-morphic elastic solidProc. R. Soc. Edinb. A2006136997101210.1017/S03082105000048441106.74010 – reference: FosdickR.L.MacSithighG.P.Minimization in incompressible nonlinear elasticity theoryJ. Elast.19861626730186016010.1007/BF000408170616.73026 – reference: FosdickR.L.MacSithighG.P.Minimization in nonlinear elasticity theory for bodies reinforced with inextensible cordsJ. Elast.199126239289113547910.1007/BF000418920768.73097 – reference: SteigmannD.J.DorfmannL.OgdenR.W.Effects of fiber bending and twisting resistance on the mechanics of fiber-reinforced elastomersNonlinear Mechanics of Soft Fibrous Tissues2015Wien and New YorkSpringer269305 – reference: ShiraniM.SteigmannD.J.Cosserat elasticity of lattice solidsJ. Elast.20221517388449644410.1007/s10659-021-09859-z1501.74007 – reference: ReissnerE.A further note on finite-strain force and moment stress elasticityZ. Angew. Math. Phys.19873866567310.1007/BF009482880626.73027 – reference: DeseriL.OwenD.R.Stable disarrangement phases of elastic aggregates: a setting for the emergence of no-tension materials with nonlinear response in compressionMeccanica2014492907293210.1007/s11012-014-0042-71306.74009 – reference: EremeyevV.A.CloudM.J.LebedevL.P.Applications of Tensor Analysis in Continuum Mechanics2018SingaporeWorld Scientific10.1142/109591471.74001 – reference: PietraszkiewiczW.EremeyevV.A.On natural strain measures of the nonlinear micropolar continuumInt. J. Solids Struct.20094677478710.1016/j.ijsolstr.2008.09.0271215.74004 – reference: ShiraniM.SteigmannD.J.AltenbachH.Necessary conditions for energy minimizers in a Cosserat model of fiber-reinforced elastic solidsRecent Approaches in the Theory of Plates and Plate-Like Structures. Advanced Structured Materials2021SwitzerlandSpringer253265 – reference: AntmanS.S.Nonlinear Problems of Elasticity2005BerlinSpringer1098.74001 – reference: DeseriL.OwenD.R.Stable disarrangement phases arising from expansion/contraction or from simple shearing of a model granular mediumInt. J. Eng. Sci.201596111130341118310.1016/j.ijengsci.2015.08.0011423.74203 – reference: BottemaO.RothB.Theoretical Kinematics19900747.70001 – reference: ShiraniM.SteigmannD.J.A Cosserat model of elastic solids reinforced by a family of curved and twisted fibersSymmetry202012113310.3390/sym12071133 – reference: TruesdellC.NollW.FlüggeS.The non-linear field theories of mechanicsHandbuch der Physik, Vol. III/31965BerlinSpringer – reference: ShiraniM.SteigmannD.J.GiorgioI.Quasiconvexity and rank-one convexity in Cosserat elasticity theoryTheoretical Analyses, Computations and Experiments of Multiscale Materials2022SwitzerlandSpringer27328310.1007/978-3-031-04548-6_131496.74031 – year: 2022 ident: 9963_CR8 publication-title: J. Elast. doi: 10.1007/s10659-022-09934-z – volume-title: Applications of Tensor Analysis in Continuum Mechanics year: 2018 ident: 9963_CR27 doi: 10.1142/10959 – volume: 38 start-page: 665 year: 1987 ident: 9963_CR12 publication-title: Z. Angew. Math. 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| SubjectTerms | Biomechanics Classical and Continuum Physics Classical Mechanics Convexity Engineering Fiber reinforced materials Kinematics Materials Science Mathematical Applications in the Physical Sciences Strain energy Theoretical and Applied Mechanics |
| Title | Convexity and Quasiconvexity in a Cosserat Model for Fiber-Reinforced Elastic Solids |
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