Skip to main content

Advertisement

SpringerLink
Log in

Modeling the Behavior of Heat-Shrinkable Thin Films

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

We describe an asymptotic model for the behavior of PET-like heat-shrinkable thin films that includes both membrane and bending energies when the thickness of the film is positive. We compare the model to Koiter’s shell model and to models in which a membrane energy or a bending energy are obtained by Γ-convergence techniques. We also provide computational results for various temperature distributions applied to the films.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acerbi, E., Buttazzo, G., Percivale, D.: A variational definition for the strain energy of an elastic string. J. Elast. 25, 137–148 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (1995)

    MATH  Google Scholar 

  3. Anzellotti, G., Baldo, S., Percivale, D.: Dimension reduction in variation problems, asymptotic development in Γ-convergence and thin structures in elasticity. Asympt. Anal. 9, 61–100 (1994)

    MATH  MathSciNet  Google Scholar 

  4. Bhattacharya, K., James, R.D.: A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 47(3), 531–576 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Bouchitté, G., Fonseca, I., Mascarenhas, M.L.: Bending moment in membrane theory. J. Elast. 73, 75–99 (2003)

    Article  MATH  Google Scholar 

  6. Braides, A.: Γ-convergence for Beginners. Oxford University Press, Oxford (2002)

    Book  MATH  Google Scholar 

  7. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)

    MATH  Google Scholar 

  8. Bělík, P., Luskin, M.: A computational model for the indentation and phase transformation of a martensitic thin film. J. Mech. Phys. Solids 50(9), 179–1815 (2002)

    Google Scholar 

  9. Bělík, P., Luskin, M.: A total-variation surface energy model for thin films of martensitic crystals. Interfaces Free Bound. 4, 71–88 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bělík, P., Luskin, M.: A computational model for martensitic thin films with compositional fluctuation. Math. Models Methods Appl. Sci. 14(11), 1585–1598 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bělík, P., Luskin, M.: Computational modeling of softening in a structural phase transformation. Multiscale Model. Simul. 3, 764–781 (2005)

    Article  MathSciNet  Google Scholar 

  12. Bělík, P., Luskin, M.: Computation of the training of a martensitic thin film. Calcolo 43, 197–215 (2006)

    Article  MathSciNet  Google Scholar 

  13. Bělík, P., Luskin, M.: The Γ-convergence of a sharp interface thin film model with non-convex elastic energy. SIAM J. Math. Anal. 38, 414–433 (2006)

    Article  MathSciNet  Google Scholar 

  14. Bělík, P., Brule, T., Luskin, M.: On the numerical modeling of deformations of pressurized martensitic thin films. M2AN Math. Model. Numer. Anal. 35(3), 525–548 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  16. Ciarlet, P.G.: An introduction to differential geometry with applications to elasticity. J. Elast. 78–79(1), 3–201 (2005)

    MathSciNet  Google Scholar 

  17. Dunavant, D.A.: High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int. J. Numer. Methods Eng. 21, 1129–1148 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  18. Findley, W.N., Lai, J.S., Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials. North Holland, Amsterdam (1976)

    MATH  Google Scholar 

  19. Fonseca, I., Francfort, G.: 3D-2D asymptotic analysis of an optimal design problem for thin films. J. Reine Angew. Math. 505, 173–202 (1998)

    MATH  MathSciNet  Google Scholar 

  20. Friesecke, G., James, R.D., Müller, S.: The Föppl–von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity. C.R. Acad. Sci. Paris, Sér. I 332, 1–6 (2002)

    Google Scholar 

  21. 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, 1461–1506 (2002)

    Article  MATH  Google Scholar 

  22. 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. C.R. Acad. Sci. Paris, Sér. I 336, 697–702 (2003)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    MATH  Google Scholar 

  25. Grabovsky, Y., Truskinovsky, L.: The flip side of buckling. Contin. Mech. Thermodyn. 19(3–4), 211–243 (2007)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  26. Gudmundson, P.: A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Gurtin, M.E.: Topics in Finite Elasticity. SIAM, Philadelphia (1981)

    Google Scholar 

  28. Gurtin, M.E.: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. Hilgers, M.G., Pipkin, A.C.: Bending energy of highly elastic membranes. Q. Appl. Math. 50(2), 389–400 (1992)

    MATH  MathSciNet  Google Scholar 

  30. Hilgers, M.G., Pipkin, A.C.: Bending energy of highly elastic membranes II. Q. Appl. Math. 54(2), 307–316 (1996)

    MATH  MathSciNet  Google Scholar 

  31. Hornung, P.: A Γ-convergence result for thin martensitic films in linearized elasticity. SIAM J. Math. Anal. 40, 186–214 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kirchhoff, G.: Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math. 40, 51–88 (1850)

    MATH  Google Scholar 

  33. Koiter, W.T.: On the nonlinear theory of thin elastic shells. Proc. Kon. Ned. Akad. Wetensch. B69, 1–54 (1966)

    MathSciNet  Google Scholar 

  34. Le Dret, H., Raoult, A.: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 73, 549–578 (1995)

    MathSciNet  Google Scholar 

  35. Maso, G.D.: An Introduction to Γ-Convergence. Birkhäuser, Basel (1993)

    Google Scholar 

  36. Modica, L., Mortola, S.: Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Italiana A (5) 14(3), 526–529 (1977)

    MATH  MathSciNet  Google Scholar 

  37. Neff, P.: Local existence and uniqueness for a geometrically exact membrane-plate with viscoelastic transverse shear resistance. Math. Methods Appl. Sci. 28(9), 1031–1060 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Nicola, L., Xiang, Y., Vlassak, J.J., der Giessen, E.V., Needleman, A.: Plastic deformation of freestanding thin films: Experiments and modeling. J. Mech. Phys. Solids 54, 2089–2110 (2006)

    Article  MATH  ADS  Google Scholar 

  39. Osaki, K.: Constitutive equations and damping function for entangled polymers. Korea–Australia Rheol. J. 11(4), 287–291 (1999)

    Google Scholar 

  40. Oza, A., Vanderby, R., Lakes, R.S.: Interrelation of creep and relaxation for nonlinearly viscoelastic materials: application to ligament and metal. Rheol. Acta 42, 557–568 (2003)

    Article  Google Scholar 

  41. Pantz, O.: Une justification partielle du modèle de plaque en flexion par Γ-convergence. C.R. Acad. Sci. Paris, Sér. I 332, 587–592 (2001)

    MATH  ADS  MathSciNet  Google Scholar 

  42. Pantz, O.: On the justification of the nonlinear inextensional plate model. Arch. Ration. Mech. Anal. 167, 179–209 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  43. Polak, E.: Computational Methods in Optimization. Academic Press, New York (1971)

    Google Scholar 

  44. Shu, Y.C.: Heterogeneous thin films of martensitic materials. Arch. Ration. Mech. Anal. 153, 39–90 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  45. Sorvari, J., Malinen, M.: Determination of the relaxation modulus of a linearly viscoelastic material. Mech. Time-Depend. Mater. 10, 125–133 (2006)

    Article  ADS  Google Scholar 

  46. Steigmann, D.J.: Asymptotic finite-strain thin-plate theory for elastic solids. Comput. Math. Appl. 53, 287–295 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  47. Steigmann, D.J.: Thin-plate theory for large elastic deformations. Int. J. Non-Linear Mech. 42, 233–240 (2007)

    Article  ADS  Google Scholar 

  48. Zhdanov, G.S.: Experimental investigation of the temperature distribution in thin films heated by an electron beam. J. Eng. Phys. Thermophys. 21(6), 1557–1561 (1971)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, CB 93, Augsburg College, 2211 Riverside Ave, Minneapolis, MN, 55454, USA

    Pavel Bělík

  2. Structural Adhesives BTO, 3M Center Bldg 230-1G-06, Maplewood, MN, 55144, USA

    Bob Jennings

  3. Department of Mathematics, University of St. Thomas, 2115 Summit Avenue, St. Paul, MN, 55105, USA

    Mikhail M. Shvartsman

  4. Safety, Security and Protection Services Business Laboratory, 3M Center Bldg 201-2S-05, Maplewood, MN, 55144, USA

    Cristina U. Thomas

Authors
  1. Pavel Bělík
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bob Jennings
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mikhail M. Shvartsman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Cristina U. Thomas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavel Bělík.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bělík, P., Jennings, B., Shvartsman, M.M. et al. Modeling the Behavior of Heat-Shrinkable Thin Films. J Elasticity 95, 57–77 (2009). https://doi.org/10.1007/s10659-009-9194-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-009-9194-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation