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A Geometric Theory of Nonlinear Morphoelastic Shells

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Abstract

Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms, are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.

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Notes

  1. Other examples of evolving material metrics in mechanics have been introduced in (Ozakin and Yavari 2010; Yavari and Goriely 2012a, b, 2013a, 2015, 2013b, 2014; Sadik and Yavari 2015).

  2. cf. (2.7) where the notation \({\bar{\varvec{G}}}_{{\mathcal {H}}}\) was introduced.

  3. The Lie derivative along the vector field \(\varvec{{\mathcal {V}}}\) is defined as \({\varvec{L}}_{\varvec{{\mathcal {V}}}}\varvec{{\mathcal {V}}}^\parallel = \left. \frac{\hbox {d}}{\hbox {d}t}\right| _{t=s} \left[ \left( \varphi _t \circ \varphi _s^{-1}\right) ^*\varvec{{\mathcal {V}}}^\parallel \right] \), where \(\varphi _t \circ \varphi _s^{-1}\) is the flow of \(\varvec{{\mathcal {V}}}\).

  4. Note that since the vector \({\varvec{U}}\) is tangent to \({\mathcal {H}}\) at \(\varphi (X,t)\), the vectors \(\left[ \varvec{{\mathcal {V}}}, {\varvec{U}} \right] ={\varvec{L}}_{\varvec{{\mathcal {V}}}}{\varvec{U}}\) and \(\tilde{\nabla }_{{\varvec{U}}} {\varvec{n}}\) are tangent to \({\mathcal {H}}\) as well.

  5. Note that (4.5) can also be obtained from (4.2) and (4.1) by writing

    $$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t} \int _{\varphi _t(\mathcal {U})}\varrho \hbox {d}s =\int _{\varphi _t(\mathcal {U})}s_m\hbox {d}s. \end{aligned}$$

  6. Since the Lagrangian density is a scalar, it depends on the metrics \({\varvec{G}}\) and \({\tilde{\varvec{g}}}\).

  7. For fixed X and t, we let \(\varphi _{\epsilon ,t}(X):=\varphi _{\epsilon }(X,t).\)

  8. We denote by \(F^{-A}{}_a\) the components of \({\varvec{F}}^{-1}\), the inverse of \({\varvec{F}}\). See Appendix 7 for the details of the derivation of (4.13) and (4.14) following (4.7).

  9. Recall the Piola identity \(\left( JF^{-A}{}_a\right) _{|A}=0\).

  10. We define the convected manifold to be the material manifold \({\mathcal {H}}\) equipped with the right Cauchy–Green deformation tensor \({\varvec{C}}\).

  11. The components of \({\varvec{C}}^{-1}\), the inverse of \({\varvec{C}}\), are denoted by \(C^{-AB}\).

  12. For details on the derivation of the Saint Venant–Kirchhoff shell model, see (Fox et al. 1993; Le Dret and Raoult 1993; Lods and Miara 1995; Miara 1998; Lods and Miara 1998; Friesecke et al. 2002a, b, 2003).

  13. Following (4.17b), there are three equilibrium equations

    figure c

    Because of the symmetry of the problem and the isotropy of the material, the stresses take the form (5.9). This implies that Eq. (5.10a) are trivially satisfied and the terms containing derivatives in (5.10b) vanish. Therefore, we are left with Eq. (5.11) as the only non-trivial equilibrium equation.

  14. When \(\omega =0\), the first fundamental form \({\varvec{G}}\) is not a dynamical variable anymore, and hence, the kinetic equation (5.20a) should be discarded.

  15. Recall that r depends on K as can be seen in (5.16), (5.17), or (5.18) depending on the value of the discriminant \(\Delta \).

  16. We let for example \(\omega _A=ZK_A(R,t)\) for \(A=R,\Theta \) in (5.24).

  17. Note that \(\left( \Sigma ^{AB}+C^{-AC}\Theta _{CD}\Lambda ^{DB}\right) _{||B} +C^{-AC}\Theta _{CD} \Lambda ^{DB}{}_{||B} =\left( \Sigma ^{AB}+2C^{-AC}\Theta _{CD}\Lambda ^{DB}\right) _{||B} -\left( C^{-AC}\Theta _{CD}\right) _{||B}\Lambda ^{DB}\). Also, we have \(\text {J} = \frac{r}{R}\). Therefore, the convected stress and couple-stress tensors read \({\varvec{\Sigma }}=\frac{R}{r}{\varvec{S}}\) and \({\varvec{\Lambda }}=\frac{R}{r}{\varvec{M}}\).

References

  • Ambrosi, D., Guana, F.: Stress-modulated growth. Math. Mech. Solids 12(3), 319–342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Ambrosi, D., Ateshian, G., Arruda, E., Cowin, S., Dumais, J., Goriely, A., Holzapfel, G., Humphrey, J., Kemkemer, R., Kuhl, E., Olberding, J., Taber, L., Garikipati, K.: Perspectives on biological growth and remodeling. J. Mech. Phys. Solids 59(4), 863–883 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Angoshtari, A., Yavari, A.: Differential complexes in continuum mechanics. Arch. Ration. Mech. Anal. 216(1), 193–220 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Aron, H.: Das gleichgewicht und die bewegung einer unendlich dünnen, beliebig gekrümmten elastischen schale. J. Reine Angew. Math. 78, 136–174 (1874)

    MathSciNet  Google Scholar 

  • Amar, MBen, Goriely, A.: Growth and instability in elastic tissues. J. Mech. Phys. Solids 53(10), 2284–2319 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Bonnet, O.: Mémoire sur la théorie des surfaces applicables sur une surface donnée. J. l’École Polytech. 24, 209–230 (1865)

    Google Scholar 

  • Chien, W.-Z.: The intrinsic theory of thin shells and plates i. Q. Appl. Math. 1, 297–327 (1943)

    MathSciNet  Google Scholar 

  • Chladni, E.F.F.: Die Akustik. Breitkopf & Härtel, Wiesbaden (1830)

    Google Scholar 

  • Chuong, C., Fung, Y.: Residual stress in arteries. In Frontiers in Biomechanics, pp. 117–129. Springer, (1986)

  • Coddington, A., Levinson, N.: Theory of Ordinary Differential Equations. International series in pure and applied mathematics. Tata McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  • Cosserat, E., Cosserat, F.: Théories des Corps Déformables. Hermann, Paris (1909)

    MATH  Google Scholar 

  • Cox, D.: Galois Theory. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts.Wiley, (2012). ISBN 9781118218426

  • Delsanto, P.P., Guiot, C., Degiorgis, P.G., Condat, C.A., Mansury, Y., Deisboeck, T.S.: Growth model for multicellular tumor spheroids. Appl. Phys. Lett. 85(18), 4225–4227 (2004)

    Article  Google Scholar 

  • Dervaux, J., Ciarletta, P., Amar, MBen: Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the föppl-von kármán limit. J. Mech. Phys. Solids 57(3), 458–471 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New Jersey (1976)

    MATH  Google Scholar 

  • do Carmo, M.: Riemannian Geometry [translated by F. Flahetry from the 1988 Portuguese edition]. Mathematics: Theory & Applications. Birkhäuser Boston, (1992). ISBN 1584883553

  • Eckart, C.: The thermodynamics of irreversible processes. iv. the theory of elasticity and anelasticity. Phys. Rev. 73(4), 373 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  • Efrati, E., Sharon, E., Kupferman, R.: Elastic theory of unconstrained non-euclidean plates. J. Mech. Phys. Solids 57(4), 762–775 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Ericksen, J.L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1(1), 295–323 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  • Fox, D., Raoult, A., Simo, J.: A justification of nonlinear properly invariant plate theories. Arch. Ration. Mech. Anal. 124(2), 157–199 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55(11), 1461–1506 (2002a)

    Article  MathSciNet  MATH  Google Scholar 

  • Friesecke, G., Müller, S., James, R.D.: Rigorous derivation of nonlinear plate theory and geometric rigidity. Comptes Rendus Math. 334(2), 173–178 (2002b)

    Article  MathSciNet  MATH  Google Scholar 

  • Friesecke, G., James, R.D., Mora, M.G., Müller, S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by gamma-convergence. Comptes Rendus Math. 336(8), 697–702 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Fung, Y.: On the foundations of biomechanics. J. Appl. Mech. 50(4b), 1003–1009 (1983)

    Article  Google Scholar 

  • Fung, Y.: What are the residual stresses doing in our blood vessels? Ann. Biomed. Eng. 19(3), 237–249 (1991)

    Article  MathSciNet  Google Scholar 

  • Fung, Y.-C.: Stress, strain, growth, and remodeling of living organisms. In Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids, pp. 469–482. Springer, (1995)

  • Fusi, L., Farina, A., Ambrosi, D.: Mathematical modeling of a solid-liquid mixture with mass exchange between constituents. Mathe. Mech. Solids 11(6), 575–595 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Geitmann, A., Ortega, J.K.: Mechanics and modeling of plant cell growth. Trends Plant Sci. 14(9), 467–478 (2009)

    Article  Google Scholar 

  • Germain, S.: Recherches sur la théorie des surfaces élastiques. Mme. Ve. Courcier, Paris (1821)

    Google Scholar 

  • Goriely, A., Amar, MBen: Differential growth and instability in elastic shells. Phys. Rev. Lett. 94(19), 198103 (2005)

    Article  Google Scholar 

  • Green, A., Zerna, W.: The equilibrium of thin elastic shells. Q. J. Mech. Appl. Math. 3(1), 9–22 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  • Han, H., Fung, Y.: Residual strains in porcine and canine trachea. J. Biomech. 24(5), 307–315 (1991)

    Article  Google Scholar 

  • Helmlinger, G., Netti, P.A., Lichtenbeld, H.C., Melder, R.J., Jain, R.K.: Solid stress inhibits the growth of multicellular tumor spheroids. Nat. Biotechnol. 15(8), 778–783 (1997)

    Article  Google Scholar 

  • Hicks, N.J.: Notes on differential geometry. Van Nostrand mathematical studies, no.3. Van Nostrand Reinhold Co., (1965). ISBN 9780442034108

  • Hori, K., Suzuki, M., Abe, I., Saito, S.: Increased tumor tissue pressure in association with the growth of rat tumors. Jpn. J. Cancer Res.: Gann 77(1), 65–73 (1986)

    Google Scholar 

  • Hsu, F.-H.: The influences of mechanical loads on the form of a growing elastic body. J. Biomech. 1(4), 303–311 (1968)

    Article  Google Scholar 

  • Humphrey, J.: Cardiovascular Solid Mechanics: Cells, Tissues, and Organs. Springer, (2002). ISBN 9780387951683

  • Ivey, T.A., Landsberg, J.M.: Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems. American Mathematical Society, Providence (2003)

    Book  MATH  Google Scholar 

  • Kadianakis, N., Travlopanos, F.: Kinematics of hypersurfaces in riemannian manifolds. J. Elast. 111(2), 223–243 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Kirchhoff, G.: Über das gleichgewicht und die bewegung einer elastischen scheibe. J. Reine Angew. Math. 40, 51–88 (1850)

    Article  MathSciNet  Google Scholar 

  • Koiter, W.T.: On the nonlinear theory of thin elastic shells. I- Introductory sections. II—Basic shell equations. III—Simplified shell equations. K. Ned. Akad. van Wet., Proc., Ser. B 69(1), 1–54 (1966)

    MathSciNet  Google Scholar 

  • Kondaurov, V., Nikitin, L.: Finite strains of viscoelastic muscle tissue. J. Appl. Math. Mech. 51(3), 346–353 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Kröner, E.: Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Arch. Ration. Mech. Anal. 4(1), 273–334 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  • Le Dret, H., Raoult, A.: Le modèle de membrane non linéaire comme limite variationnelle de l’élasticité non linéaire tridimensionnelle. Comptes Rendus l’Acad. Scie.Sér. 1, Math. 317(2), 221–226 (1993)

    Google Scholar 

  • Lods, V., Miara, B.: Analyse asymptotique des coques en flexion non linéairement élastiques. Comptes Rendus l’Acad. Scie.Sér. 1, Math. 321(8), 1097–1102 (1995)

    MathSciNet  Google Scholar 

  • Lods, V., Miara, B.: Nonlinearly elastic shell models: a formal asymptotic approach ii. the flexural model. Arch. Ration. Mech. Anal. 142(4), 355–374 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Love, A.E.H.: The small free vibrations and deformation of a thin elastic shell. Philos. Trans. R. Soc. Lond. A 179, 491–546 (1888)

    Article  MATH  Google Scholar 

  • Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1892)

    MATH  Google Scholar 

  • Lu, J., Papadopoulos, P.: A covariant constitutive description of anisotropic non-linear elasticity. 51(2):204–217, (2000). ISSN 0044-2275

  • Lubarda, V., Hoger, A.: On the mechanics of solids with a growing mass. Int. J. Solids Struct. 39(18), 4627–4664 (2002)

    Article  MATH  Google Scholar 

  • Lubarda, V.A.: Constitutive theories based on the multiplicative decomposition of deformation gradient: thermoelasticity, elastoplasticity, and biomechanics. Appl. Mech. Rev. 57(2), 95–108 (2004)

    Article  Google Scholar 

  • Marsden, J., Hughes, T.: Mathematical Foundations of Elasticity, Dover Civil and Mechanical Engineering Series. Dover, London (1983)

    Google Scholar 

  • Marsden, J.E., Ratiu, T.: Introd. Mech. Symmetry. Springer, New York (1994)

    Book  Google Scholar 

  • Mathieu, E.: Mémoire sur le mouvement vibratoire des cloches. Gauthier-Villars (1882)

  • McMahon, J., Goriely, A., Tabor, M.: Nonlinear morphoelastic plates I: genesis of residual stress. Math. Mech. Solids 16(8), 812–832 (2011a)

    Article  MathSciNet  MATH  Google Scholar 

  • McMahon, J., Goriely, A., Tabor, M.: Nonlinear morphoelastic plates II: exodus to buckled states. Math. Mech. Solids 16(8), 833–871 (2011b)

    Article  MathSciNet  MATH  Google Scholar 

  • Miara, B.: Nonlinearly elastic shell models: a formal asymptotic approach i. the membrane model. Arch. Ration. Mech. Anal. 142(4), 331–353 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Naghdi, P.: Foundations of elastic shell theory. In: Sneddon, I.N., Hill, K. (eds.) Progress in Solid Mechanics, vol. 4, pp. 1–90. North Hollande Publishing Cy, Amsterdam (1963)

    Google Scholar 

  • Nishikawa, S.: Variational Problems in Geometry, volume 205 of Iwanami series in modern mathematics. American Mathematical Society, (2002). ISBN 9780821813560

  • Novozhilov, V.: The theory of thin shells [translated by P. G. Lowe from the 1951 Russian edition]. P. Noordhoff, (1964)

  • Olsson, T., Klarbring, A.: Residual stresses in soft tissue as a consequence of growth and remodeling: application to an arterial geometry. Eur. J. Mech.A/Solids 27(6), 959–974 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Omens, J.H., Fung, Y.-C.: Residual strain in rat left ventricle. Circ. Res. 66(1), 37–45 (1990)

    Article  Google Scholar 

  • Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51, 032902 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Pezzulla, M., Shillig, S.A., Nardinocchi, P., Holmes, D.P.: Morphing of geometric composites via residual swelling. Soft. Matter. 11, 5812–5820 (2015a)

    Article  Google Scholar 

  • Pezzulla, M., Smith, G.P., Nardinocchi, P., Holmes, D.P.: Geometry and mechanics of thin growing bilayers. pp. 1–5, (2015b). arXiv:1509.05259v2

  • Pollack, J.B., Hubickyj, O., Bodenheimer, P., Lissauer, J.J., Podolak, M., Greenzweig, Y.: Formation of the giant planets by concurrent accretion of solids and gas. Icarus 124(1), 62–85 (1996)

    Article  Google Scholar 

  • Polyanin, A., Zaitsev, V.: Handbook of Nonlinear Partial Differential Equations. CRC Press, (2004). ISBN 9780203489659

  • Rodriguez, E.K., Hoger, A., McCulloch, A.D.: Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27(4), 455–467 (1994)

    Article  Google Scholar 

  • Sadik, S., Yavari, A.: Geometric nonlinear thermoelasticity and the time evolution of thermal stresses. Math. Mech. Solids (2015). doi:10.1177/1081286515599458

  • Sadik, S., Yavari, A.: On the origins of the idea of the multiplicative decomposition of the deformation gradient. Math. Mech. Solids. (2016). doi:10.1177/1081286515612280,

  • Sanders Jr., J.L.: Nonlinear theories for thin shells. Technical report technical report no. 10, DTIC Document, (1961)

  • Silberberg, J.S., Barre, P.E., Prichard, S.S., Sniderman, A.D.: Impact of left ventricular hypertrophy on survival in end-stage renal disease. Kidney Int. 36(2), 286–290 (1989)

    Article  Google Scholar 

  • Skalak, R.: Growth as a finite displacement field. In: Proceedings of the IUTAM Symposium on Finite Elasticity, pp. 347–355. Springer, (1982)

  • Skalak, R., Dasgupta, G., Moss, M., Otten, E., Dullemeijer, P., Vilmann, H.: Analytical description of growth. J. Theor. Biol. 94(3), 555–577 (1982)

    Article  MathSciNet  Google Scholar 

  • Skalak, R., Zargaryan, S., Jain, R.K., Netti, P.A., Hoger, A.: Compatibility and the genesis of residual stress by volumetric growth. J. Math. Biol. 34(8), 889–914 (1996)

    Article  MATH  Google Scholar 

  • Stojanović, R., Djurić, S., Vujošević, L.: On finite thermal deformations. Arch. Mech. Stosow. 1(16), 103–108 (1964)

    MathSciNet  Google Scholar 

  • Synge, J.L., Chien, W.Z.: The intrinsic theory of elastic shells and plates. In: von Kármán anniv. vol., pp. 103–120. Cal. Inst. Tech., Pasadena, (1941)

  • Taber, L.A.: Biomechanics of growth, remodeling, and morphogenesis. Appl. Mech. Rev. 48(8), 487–545 (1995)

    Article  Google Scholar 

  • Takamizawa, K., Matsuda, T.: Kinematics for bodies undergoing residual stress and its applications to the left ventricle. J. Appl. Mech. 57(2), 321–329 (1990)

    Article  Google Scholar 

  • Verpoort, S.: The geometry of the second fundamental form: curvature properties and variational aspects. PhD thesis, Katholieke Universiteit Leuven, (2008)

  • Yavari, A.: A geometric theory of growth mechanics. J. Nonlinear Sci. 20(6), 781–830 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. 205(1), 59–118 (2012a)

    Article  MathSciNet  MATH  Google Scholar 

  • Yavari, A., Goriely, A.: Weyl geometry and the nonlinear mechanics of distributed point defects. Proc. R. Soc. A 468, 3902–3922 (2012b)

    Article  MathSciNet  Google Scholar 

  • Yavari, A., Goriely, A.: Nonlinear elastic inclusions in isotropic solids. Proc. R. Soc. A: Math., Phys. Eng. Sci. 469(2160), 20130415 (2013a)

    Article  MathSciNet  Google Scholar 

  • Yavari, A., Goriely, A.: Riemann–Cartan geometry of nonlinear disclination mechanics. Math. Mech. Solids 18(1), 91–102 (2013b)

    Article  MathSciNet  Google Scholar 

  • Yavari, A., Goriely, A.: The geometry of discombinations and its applications to semi-inverse problems in anelasticity. Proc. R. Soc. A 470, 20140403 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Yavari, A., Goriely, A.: The twist-fit problem: finite torsional and shear eigenstrains in nonlinear elastic solids. Proc. R. Soc. A 471, 20150596 (2015)

    Article  Google Scholar 

  • Yavari, A., Marsden, J.E., Ortiz, M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 042903 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

SS was supported by a Fulbright Grant. AG is a Wolfson/Royal Society Merit Award Holder and acknowledges support from a Reintegration Grant under EC Framework VII. We thank M.F. Shojaei for his help with some of the numerical examples. This research was partially supported by AFOSR – Grant No. FA9550-12-1-0290 and NSF—Grant No. CMMI 1042559 and CMMI 1130856.

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Souhayl Sadik & Arash Yavari

  2. Department of Civil and Environmental Engineering, The George Washington University, Washington, DC, USA

    Arzhang Angoshtari

  3. Mathematical Institute, University of Oxford, Oxford, UK

    Alain Goriely

  4. The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Arash Yavari

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  1. Souhayl Sadik
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Correspondence to Arash Yavari.

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Communicated by Paul Newton.

Appendix: Derivation of the Euler–Lagrange Equations

Appendix: Derivation of the Euler–Lagrange Equations

In this appendix, we work out in detail the derivation of the Euler–Lagrange equations first assuming that \(\delta {\varvec{G}}=\delta {\varvec{B}}={\varvec{0}}\). We substitute (4.8), (4.9), (4.10), (4.11), and (4.12) into (4.7) to obtain

$$\begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^a}{\delta \varphi ^a} +\frac{\partial {\mathcal {L}}}{\partial \varphi ^n}{\delta \varphi ^n} -\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\left( \delta \varphi ^a\beta ^b{}_a+\frac{\partial (\delta \varphi ^n)}{\partial x^a}g^{ab}\right) +\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}.\frac{\hbox {D}\delta \varphi }{\hbox {d}t}\nonumber \\&\quad +\,\frac{\partial {\mathcal {L}}}{\partial C_{AB}}\left( 2F^b{}_B{g}_{bc}\delta \varphi ^c{}_{|A} -2\delta \varphi ^nF^a{}_AF^b{}_B{\beta }_{ab} \right) +\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}\bigg ( F^a{}_AF^b{}_B{\beta }_{ab|c}\delta \varphi ^c\nonumber \\&\quad +\,2F^a{}_A{\beta }_{ac}\delta \varphi ^c{}_{|B}-\delta \varphi ^nF^a{}_AF^b{}_B{\beta }_{ac}{\beta }_{bd}g^{cd} +F^b{}_A\left( \frac{\partial \delta \varphi ^n}{\partial x^b}\right) _{|B} \bigg ) \bigg ]\nonumber \\&\quad \sqrt{\det {\varvec{G}}}\hbox {d}X^I\hbox {d}t =0. \end{aligned}$$
(7.1)

Hence, we have

$$\begin{aligned} \begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\bigg \{ \frac{\partial {\mathcal {L}}}{\partial \varphi ^a}{\delta \varphi ^a} +\frac{\partial {\mathcal {L}}}{\partial \varphi ^n}{\delta \varphi ^n} +\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}.\delta \varphi \right) \\&\quad -\,\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}\right) .\delta \varphi \\&\quad -\,\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\beta ^b{}_a\delta \varphi ^a -\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\delta \varphi ^ng^{ab}F^{-A}{}_a\right) _{|A}\\&\quad +\,\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _bg^{ab}F^{-A}{}_a\right) _{|A}\delta \varphi ^n\\&\quad +\,2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\delta \varphi ^a\right) _{|B} -2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\right) _{|B}\delta \varphi ^a \\&\quad -\,2\frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^a{}_AF^b{}_B{\beta }_{ab}\delta \varphi ^n\\&\quad +\,\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B{\beta }_{bc|a}\delta \varphi ^a +2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\delta \varphi ^a\right) _{|B}\\&\quad -\,2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\right) _{|B}\delta \varphi ^a\\&\quad -\,\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}} F^a{}_AF^b{}_B{\beta }_{ac}{\beta }_{bd}g^{cd}\delta \varphi ^n +\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\frac{\partial \delta \varphi ^n}{\partial x^b}\right) _{|B} \\&\quad -\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}\delta \varphi ^nF^{-D}{}_b\bigg ]_{|D}\\&\quad +\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}F^{-D}{}_b\bigg ]_{|D}\delta \varphi ^n \bigg \}\sqrt{\det {\varvec{G}}}\hbox {d}X^I\hbox {d}t=0 . \end{aligned} \end{aligned}$$
(7.2)

We can rewrite (7.2) as

$$\begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\bigg \{ \frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}.\delta \varphi \right) +2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\delta \varphi ^a\right) _{|B}\nonumber \\&\quad +\,2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\delta \varphi ^a\right) _{|B}-\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}\delta \varphi ^nF^{-D}{}_b\bigg ]_{|D}\nonumber \\&\quad -\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\delta \varphi ^ng^{ab}F^{-A}{}_a\right) _{|A} +\,\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}\frac{\partial \delta \varphi ^n}{\partial X^A}\right) _{|B}\nonumber \\&\quad +\,\bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^a} -\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\beta ^b{}_a -\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }^a}\right) -2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\right) _{|B}\nonumber \\&\quad +\,\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B{\beta }_{bc|a} -2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\right) _{|B} \bigg ]\delta \varphi ^a \nonumber \\&\quad +\,\bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^n} +\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _bg^{ab}F^{-A}{}_a\right) _{|A}-\,\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }^n}\right) \nonumber \\&\quad -\,2\frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^a{}_AF^b{}_B{\beta }_{ab} -\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^a{}_AF^b{}_B{\beta }_{ac}{\beta }_{bd}g^{cd}\nonumber \\&\quad +\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}F^{-D}{}_b\bigg ]_{|D} \bigg ]\delta \varphi ^n \bigg \}\hbox {d}S\hbox {d}t=0. \end{aligned}$$
(7.3)

At \(t=t_1\), we assume that \(\varphi _{\epsilon ,t_1}=\varphi _{t_1}\) so that \(\delta \varphi _{t_1}=0\). Therefore, by integrating the first term in (7.3) in the time domain, we obtain only one term at \(t=t_0\) giving the initial condition on the velocity at \(t=t_0\). By applying Stokes’ theorem to the following five terms, if we denote by \(\varvec{{\mathsf {T}}}\) the outward in-plane vector field normal to the boundary curve \(\partial {\mathcal {H}}\), we obtain

$$\begin{aligned}&-\left. \int _{{\mathcal {H}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }}.\delta \varphi \hbox {d}S\right| _{t=t_0}+ \int _{t_0}^{t_1}\int _{\partial {\mathcal {H}}}\bigg \{ \left( 2\frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}+2\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\right) {\mathsf {T}}_{B}\delta \varphi ^a\nonumber \\&\quad -\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AC}}F^b{}_A\right) _{|C}F^{-B}{}_b+\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _bg^{ab}F^{-B}{}_a\bigg ]{\mathsf {T}}_{B}\delta \varphi ^n +\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}{\mathsf {T}}_{B}\frac{\partial \delta \varphi ^n}{\partial X^A}\bigg \}\hbox {d}L\hbox {d}t\nonumber \\&\quad \int _{t_0}^{t_1}\int _{{\mathcal {H}}}\bigg \{ \bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^a} -\left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _b\beta ^b{}_a -\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }^a}\right) -2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_A{g}_{ac}\right) _{|B}\nonumber \\&\quad +\,\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B{\beta }_{bc|a} -2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac}\right) _{|B} \bigg ]\delta \varphi ^a \nonumber \\&\quad +\bigg [ \frac{\partial {\mathcal {L}}}{\partial \varphi ^n}+\,\left( \left( \frac{\partial {\mathcal {L}}}{\partial \varvec{{\mathcal {N}}}}\right) _bg^{ab}F^{-A}{}_a\right) _{|A} -\,\frac{1}{\sqrt{\det {\varvec{G}}}}\frac{\hbox {d}}{\hbox {d}t}\left( \sqrt{\det {\varvec{G}}}\frac{\partial {\mathcal {L}}}{\partial \dot{\varphi }^n}\right) \nonumber \\&\quad -2\frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^c{}_AF^b{}_B{\beta }_{bc} -\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^a{}_AF^b{}_B{\beta }_{ac}{\beta }_{bd}g^{cd}\nonumber \\&\quad +\,\bigg [\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}F^{-D}{}_b\bigg ]_{|D} \bigg ]\delta \varphi ^n \bigg \} \hbox {d}S\hbox {d}t =0. \end{aligned}$$
(7.4)

By arbitrariness of \(\delta \varphi ^\parallel \), \(\delta \varphi ^n\), and \(d(\delta \varphi ^n)\), the Euler–Lagrange equations for shells (4.13) together with the initial and boundary conditions (4.14) follow from (7.4). Note that following Codazzi’s equation (2.3), we have \({\beta }_{bc|a}={\beta }_{ac|b}.\) Therefore \(\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B{\beta }_{bc|a}=\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_A{\beta }_{ac|B}.\)

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Sadik, S., Angoshtari, A., Goriely, A. et al. A Geometric Theory of Nonlinear Morphoelastic Shells. J Nonlinear Sci 26, 929–978 (2016). https://doi.org/10.1007/s00332-016-9294-9

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