Skip to main content

Plates with Incompatible Prestrain

Abstract

We study effective elastic behavior of the incompatibly prestrained thin plates, where the prestrain is independent of thickness and uniform through the plate’s thickness h. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric G, and seek the limiting behavior as \({h \to 0}\). We first establish that when the energy per volume scales as the second power of h, the resulting \({\Gamma}\) -limit is a Kirchhoff-type bending theory. We then show the somewhat surprising result that there exist non-immersible metrics G for whom the infimum energy (per volume) scales smaller than h 2. This implies that the minimizing sequence of deformations carries nontrivial residual three-dimensional energy but it has zero bending energy as seen from the limit Kirchhoff theory perspective. Another implication is that other asymptotic scenarios are valid in appropriate smaller scaling regimes of energy. We characterize the metrics G with the above property, showing that the zero bending energy in the Kirchhoff limit occurs if and only if the Riemann curvatures R 1213, R 1223 and R 1212 of G vanish identically. We illustrate our findings with examples; of particular interest is an example where \({G_{2 \times 2}}\), the two-dimensional restriction of G, is flat but the plate still exhibits the energy scaling of the Föppl–von Kármán type. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for \({G = Id_{3} + \gamma n \otimes n}\) given in terms of the inhomogeneous unit director field distribution \({ n \in \mathbb{R}^3}\).

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Barker B., Lewicka M., Zumbrun K.: Existence and stability of viscoelastic shock profiles. Arch. Rational Mech. Anal. 200(2), 491–532 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Borisov J.F.: \({\mathcal{C}^{1,\alpha}}\)-isometric immersions of Riemannian spaces. Doklady 163, 869–871 (1965)

    MathSciNet  Google Scholar 

  3. 3.

    Chen Y.-C., Fried E.: Uniaxial nematic elastomers: constitutive framework and a simple application. Proc. R. Soc. A 462, 1295–1314 (2006)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Conti S., De Lellis C., Szekelyhidi L. Jr.: h-principle and rigidity for \({\mathcal{C}^{1,\alpha}}\) isometric embeddings. Nonlinear Partial Differ. Equ. Abel Sympos. 7, 83–116 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Friesecke G., James R., Mora M.G., Müller S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris. 336(8), 697–702 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

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

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Friesecke G., James R., Müller S.: A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180(2), 183–236 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Gromov M.: Partial Differential Relations. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  9. 9.

    Gurtin M., Fried E., Anand L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  10. 10.

    Han, Q., Hong, J.X.: Isometric embedding of Riemannian manifolds in Euclidean spaces. Math. Surv. Monogr., 130 American Mathematical Society, Providence, RI (2006)

  11. 11.

    Hornung, P., Lewicka, M., Pakzad, R.: Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells. J. Elast., 111(1) (2013)

  12. 12.

    Klein Y., Efrati E., Sharon E.: Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315, 1116–1120 (2007)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Kuiper, N.: On \({\mathcal{C}^1}\) isometric imbeddings i,ii. Proc. Konwl. Acad. Wet. Amsterdam A 58, 545–556, 683–689 (1955)

  14. 14.

    Le Dret H., Raoult A.: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pure. Appl. 74, 549–578 (1995)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Lewicka M., Pakzad R.: Scaling laws for non-Euclidean plates and the W 2,2 isometric immersions of Riemannian metrics. ESAIM: Control. Optim. Calc. Var. 17(4), 1158–1173 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Lewicka M., Mahadevan L., Pakzad R.: The Foppl-von Karman equations for plates with incompatible strains. Proc. R. Soc. A 467, 402–426 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Lewicka,M., Mahadevan, L., Pakzad,M.: The Monge-Ampere constraint: matching of isometries, density and regularity and elastic theories of shallow shells. Ann. l’Institut Henri Poincare Non Linear Anal. (to appear)

  18. 18.

    Lewicka M., Mora M.G., Pakzad R.: The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells. Arch. Rational Mech. Anal. (3) 200, 1023–1050 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Liu F.C.: A Lusin property of Sobelov functions. Indiana Univ. Math. J. 26, 645–651 (1977)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Modes C.D., Bhattacharya K., Warner M., Disclination-mediated thermo-optical response in nematic glass sheets. Phys. Rev. E 81 (2010)

  21. 21.

    Modes C.D., Bhattacharya K., Warner M.: Gaussian curvature from flat elastica sheets. Proc. R. Soc. A 467, 1121–1140 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Modes, C.D., Warner, M.: Blueprinting Nematic Glass: Systematically Constructing and Combining Active Points of Curvature for Emergent Morphology. 84, 021711-1–021711-7 (2011)

  23. 23.

    Nash J.C.: \({\mathcal{C}^1}\) isometric imbeddings. Ann. Math. 60, 383–396 (1954)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Warner M., Terentjev E.: Liquid Crystal Elastomers. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Kaushik Bhattacharya.

Additional information

Communicated by The Editors

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bhattacharya, K., Lewicka, M. & Schäffner, M. Plates with Incompatible Prestrain. Arch Rational Mech Anal 221, 143–181 (2016). https://doi.org/10.1007/s00205-015-0958-7

Download citation

Keywords

  • Fundamental Form
  • Ricci Curvature
  • Isometric Immersion
  • Riemann Curvature
  • Christoffel Symbol