Damage effects of adhesives in modern glass façades: a micro-mechanically motivated volumetric damage model for poro-hyperelastic materials

  • Michael DrassEmail author
  • Jens Schneider
  • Stefan Kolling


This paper presents a micro-mechanically motivated volumetric damage model accounting for cavitation effects in modern glass connections, e.g. laminated glass connections. The volumetric part of an arbitrary Helmholtz free energy function is equipped with an isotropic damage formulation. To develop a micro-mechanical damage model, the porous micro-structure of a transparent structural silicone adhesive is analyzed numerically applying hydrostatic loading conditions. Based on the structural responses of different types of cubic representative volume elements incorporating an initial void fraction, three damage parameters are fitted utilizing the LevenbergMarquard algorithm. The present volumetric damage model is implemented into ANSYS FE Code using a UserMat subroutine, where the algorithmic setting is described in detail in the present paper. To compare the structural responses of cubic equivalent homogeneous materials with representative volume elements, benchmark tests under hydrostatic loading are performed. The results indicate that the novel damage model accounts adequately for volumetric damage due to the cavitation effect. A special form of the pancake test is described briefly. The test allows for visualizing the cavitation effect during experimental testing. The experimental results of the pancake test are compared with numerical results, where the pancake test is simulated incorporating the micro-mechanical damage model. The micro-mechanically motivated scalar, internal damage variable is equipped with the obtained damage parameters from the structural response of the representative volume elements. The results show an adequate approximation of the experiment through the simulation. However, to optimize the results of the simulation, an optimization study on the damage parameters is conducted utilizing the Downhill-Simplex algorithm. Using the optimized damage parameters, the simulation of the pancake tests is further improved. Hence, it is shown that the novel micro-mechanically motivated volumetric damage model is excellently suited to represent the cavitation effect in poro-hyperelastic materials.


Laminated glass connections Structural silicone Cavitation Micro-mechanically motivated damage formulation ANSYS FE code UserMat subroutine 


  1. Ansarifar, A., Lim, B.: Reinforcement of silicone rubber with precipitated amorphous white silica nanofiller-effect of silica aggregates on the rubber properties. J. Rubber Res. 9(3), 140–158 (2006)Google Scholar
  2. Avril, S., Bonnet, M., Bretelle, A.S., Grediac, M., Hild, F., Ienny, P., Latourte, F., Lemosse, D., Pagano, S., Pagnacco, E.: Overview of identification methods of mechanical parameters based on full-field measurements. Exp. Mech. 48(4), 381–402 (2008). CrossRefGoogle Scholar
  3. Balzani, D.: Polyconvex Anisotropic Energies and Modeling of Damage Applied to Arterial Walls. VGE, Verlag Glückauf, Essen (2006)Google Scholar
  4. Biddis, E.A., Bogoch, E.R., Meguid, S.A.: Three-dimensional finite element analysis of prosthetic finger joint implants. Int. J. Mech. Mater. Des. 1(4), 317–328 (2004). CrossRefGoogle Scholar
  5. Chaves, E.W.: Notes on Continuum Mechanics. Springer, Berlin (2013)CrossRefGoogle Scholar
  6. Cheng, L., Guo, T.F.: Void interaction and coalescence in polymeric materials. Int. J. Solids Struct. 44(6), 1787–1808 (2007). CrossRefzbMATHGoogle Scholar
  7. Cho, J.R., Lee, H.W., Jeong, W.B., Jeong, K.M., Kim, K.W.: Finite element estimation of hysteretic loss and rolling resistance of 3-d patterned tire. Int. J. Mech. Mater. Des. 9(4), 355–366 (2013). CrossRefGoogle Scholar
  8. Christian Gasser, T.: An irreversible constitutive model for fibrous soft biological tissue: A 3-d microfiber approach with demonstrative application to abdominal aortic aneurysms. Acta Biomaterialia 7(6), 2457–2466 (2011). CrossRefGoogle Scholar
  9. Cristiano, A., Marcellan, A., Long, R., Hui, C., Stolk, J., Creton, C.: An experimental investigation of fracture by cavitation of model elastomeric networks. J. Polym. Sci. B Polym. Phys. 48(13), 1409–1422 (2010). CrossRefGoogle Scholar
  10. Dal, H.: Approaches to the modeling of inelasticity and failure of rubberlike materials—theory and numerics. Ph.D. thesis, University of Dresden (2012)Google Scholar
  11. de Souza Neto, E.A., Perić, D., Owen, D.R.J.: A phenomenological three-dimensional rate-idependent continuum damage model for highly filled polymers: formulation and computational aspects. J. Mech. Phys. Solids 42(10), 1533–1550 (1994). CrossRefzbMATHGoogle Scholar
  12. Dimitrijevic, B.J., Hackl, K.: A method for gradient enhancement of continuum damage models. Technische Mechanik 28(1):43–52 (2008)
  13. Dorfmann, A., Fuller, K., Ogden, R.: Shear, compressive and dilatational response of rubberlike solids subject to cavitation damage. Int. J. Solids Struct. 39(7), 1845–1861 (2002). CrossRefzbMATHGoogle Scholar
  14. Dow Corning Europe SA (2017) On macroscopic effects of heterogeneity in elastoplastic media at finite strain. glasstec
  15. Drass, M., Schneider, J.: Constitutive modeling of transparent structural silicone adhesive—TSSA. In: Schröder J (ed) 14. Darmstädter Kunststofftage, vol 14 (2016a)Google Scholar
  16. Drass, M., Schneider, J.: On the mechanical behavior of Transparent Structural Silicone Adhesive (TSSA), CRC Press, book section 6. Material modelling, multi-scale modelling, composite materials, porous media, pp 446–451 (2016b). CrossRefGoogle Scholar
  17. Drass, M., Schneider, J., Kolling, S.: Novel volumetric helmholtz free energy function accounting for isotropic cavitation at finite strains. Mat. Design 138, 71–89 (2017a). CrossRefGoogle Scholar
  18. Drass. M., Schwind, G., Schneider, J., Kolling, S.: Adhesive connections in glass structures—part i: experiments and analytics on thin structural silicone. Glass Struct. Eng. (2017b). CrossRefGoogle Scholar
  19. Drass, M., Schwind, G., Schneider, J., Kolling, S.: Adhesive connections in glass structures—part ii: material parameter identification on thin structural silicone. Glass Struct. Eng. (2017c). CrossRefGoogle Scholar
  20. Flory, P.: Thermodynamic relations for high elastic materials. Trans. Faraday Soc. 57, 829–838 (1961)MathSciNetCrossRefGoogle Scholar
  21. Fond, C.: Cavitation criterion for rubber materials: a review of void-growth models. J. Polym. Sci. B Polym. Phys. 39(17), 2081–2096 (2001). CrossRefGoogle Scholar
  22. Gent, A.N.: Cavitation in rubber: a cautionary tale. Rubber Chem. Technol. 63(3), 49–53 (1990). CrossRefGoogle Scholar
  23. Gent, A.N., Lindley, P.: Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 249(1257), 195–205 (1959). CrossRefGoogle Scholar
  24. Gosse, J., Christensen, S.: Strain Invariant Failure Criteria for Polymers in Composite Materials, vol. 1184. American Institute of Aeronautics and Astronautics, Reston (2001). CrossRefGoogle Scholar
  25. Gunel, E., Basaran, C.: Stress whitening quantification of thermoformed mineral filled acrylics. J. Eng. Mater. Technol. 132(3), 031,002 (2010). CrossRefGoogle Scholar
  26. Henao, D., Mora-Corral, C., Xu, X.: A numerical study of void coalescence and fracture in nonlinear elasticity. Comput. Methods Appl. Mech. Eng. 303, 163–184 (2016). MathSciNetCrossRefGoogle Scholar
  27. Heyden, S., Conti, S., Ortiz, M.: A nonlocal model of fracture by crazing in polymers. Mech. Mater. 90, 131–139 (2015). CrossRefGoogle Scholar
  28. Holzapfel, G.A.: Nonlinear Solid Mechanics, vol. 24. Wiley, Chichester (2000)zbMATHGoogle Scholar
  29. Iman, R.L.: Latin Hypercube Sampling. Wiley, Hoboken (2008). CrossRefGoogle Scholar
  30. Iman, R.L., Conover, W.J.: A distribution-free approach to inducing rank correlation among input variables. Commun. Stat. Simul. Comput. 11(3), 311–334 (1982). CrossRefzbMATHGoogle Scholar
  31. Kachanov, L.: Introduction to Continuum Damage Mechanics, vol. 10. Springer, Berlin (2013)zbMATHGoogle Scholar
  32. Kachanov, L.M.: Time of the rupture process under creep conditions. Izv Akad Nauk SSR Otd Tech Nauk 8, 26–31 (1958)Google Scholar
  33. Khajehsaeid, H., Baghani, M., Naghdabadi, R.: Finite strain numerical analysis of elastomeric bushings under multi-axial loadings: a compressible visco-hyperelastic approach. Int. J. Mech. Mater. Des. 9(4), 385–399 (2013). CrossRefGoogle Scholar
  34. Kiziltoprak, N.: Development of a nano-mechanical-model to account for the cavitation-effect in rubber-like materials, Master Thesis, TU Darmstadt (2016)Google Scholar
  35. Kolling, S., Bois, P.A.D., Benson, D.J., Feng, W.W.: A tabulated formulation of hyperelasticity with rate effects and damage. Comput. Mech. 40(5), 885–899 (2007). CrossRefzbMATHGoogle Scholar
  36. Koplik, J., Needleman, A.: Void growth and coalescence in porous plastic solids. Int. J. Solids Struct. 24(8), 835–853 (1988). CrossRefGoogle Scholar
  37. Lemaitre, J., Desmorat, R.: Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures. Springer, Berlin (2005). CrossRefGoogle Scholar
  38. Leukart, M., Ramm, E.: A comparison of damage models formulated on different material scales. Comput. Mater. Sci. 28(34), 749–762 (2003). CrossRefGoogle Scholar
  39. Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2(2):164–168 (1944). MathSciNetCrossRefGoogle Scholar
  40. Li, W.: Damage models for soft tissues: a survey. J. Med. Biol. Eng. 36(3), 285–307 (2016). CrossRefGoogle Scholar
  41. Lopez-Pamies, O., Idiart, M.I., Nakamura, T.: Cavitation in elastomeric solids: I–A defect-growth theory. J. Mech. Phys. Solids 59(8), 1464–1487 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  42. Lopez-Pamies, O., Nakamura, T., Idiart, M.I.: Cavitation in elastomeric solids: Iionset-of-cavitation surfaces for neo-hookean materials. J. Mech. Phys. Solids 59(8), 1488–1505 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  43. Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963). MathSciNetCrossRefzbMATHGoogle Scholar
  44. Marsden, J.E., Hughes, T.J.: Mathematical Foundations of Elasticity. Courier Corporation, Chelmsford (1994)zbMATHGoogle Scholar
  45. Miehe, C.: Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int. J. Numer. Meth. Eng. 37(12), 19812004 (1994). MathSciNetCrossRefGoogle Scholar
  46. Miehe, C.: Discontinuous and continuous damage evolution in ogden-type large-strain elastic materials. Eur. J. Mech. A Solids 14(5):697–720 (1995).
  47. Miehe, C., Stein, E.: A canonical model of multiplicative elasto-plasticity formulation and aspects of the numerical implementation. Eur. J. Mech. A Solids 11, 25–43 (1992)Google Scholar
  48. Needleman, A.: Void growth in an elastic-plastic medium. J. Appl. Mech. 39(4), 964–970 (1972). CrossRefGoogle Scholar
  49. Nelder, J.A.: Inverse polynomials, a useful group of multi-factor response functions. Biometrics 22(1), 128–141 (1966). CrossRefGoogle Scholar
  50. Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308–313 (1965). Scholar
  51. Nguyen, N., Waas, A.M.: Nonlinear, finite deformation, finite element analysis. Zeitschrift für angewandte Mathematik und Physik 67(3), 35 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  52. Overend, M.: Optimising connections in structural glass. In: Proceedings of 2nd International conference on Glass in Buildings (2005)Google Scholar
  53. Pardoen, T., Hutchinson, J.W.: An extended model for void growth and coalescence. J. Mech. Phys. Solids 48(12), 2467–2512 (2000). CrossRefzbMATHGoogle Scholar
  54. Parisch, H.: Festkörper-kontinuumsmechanik. BG Teubner, Leipzig (2003)CrossRefGoogle Scholar
  55. Peters, S., Fuchs, A., Knippers, J., Behling, S.: Ganzglastreppe mit transparenten sgp-klebeverbindungen–konstruktion und statische berechnung. Stahlbau 76(3), 151–156 (2007). CrossRefGoogle Scholar
  56. Rabotnov, Y.N.: On the equation of state of creep. Proc. Inst. Mech. Eng. Conf. Proc. 178(1), 2–117–2–122 (1963). CrossRefGoogle Scholar
  57. Santarsiero, M., Louter, C., Nussbaumer, A.: The mechanical behaviour of sentryglas ionomer and tssa silicon bulk materials at different temperatures and strain rates under uniaxial tensile stress state. Glass Struct. Eng. (2016). CrossRefGoogle Scholar
  58. Sasso, M., Chiappini, G., Rossi, M., Mancini, E., Cortese, L., Amodio, D.: Structural analysis of an elastomeric bellow seal in unsteady conditions: simulations and experiments. J. Mech. Mater. Des, Int (2016). CrossRefGoogle Scholar
  59. Schmidt, T., Balzani, D., Holzapfel, G.A.: Statistical approach for a continuum description of damage evolution in soft collagenous tissues. Comput. Methods Appl. Mech. Eng. 278, 41–61 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  60. Simo, J.: On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60(2), 153–173 (1987). CrossRefzbMATHGoogle Scholar
  61. Simo, J.C., Hughes, T.J.: Computational Inelasticity, vol. 7. Springer, Berlin (2006)zbMATHGoogle Scholar
  62. Simo, J.C., Ju, J.W.: Strain- and stress-based continuum damage models–i. Formulation. Int. J. Solids Struct. 23(7), 821–840 (1987). CrossRefzbMATHGoogle Scholar
  63. Sun, W., Chaikof, E.L., Levenston, M.E.: Numerical approximation of tangent moduli for finite element implementations of nonlinear hyperelastic material models. J. Biomech. Eng. 130(6), 061,003 (2008). CrossRefGoogle Scholar
  64. Sáez, P., Alastrué, V., Peña, E., Doblaré, M., Martí-nez, M.A.: Anisotropic microsphere-based approach to damage in soft fibered tissue. Biomech. Model. Mechanobiol. 11(5), 595–608 (2012). CrossRefGoogle Scholar
  65. Tanniru, M., Misra, R., Berbrand, K., Murphy, D.: The determining role of calcium carbonate on surface deformation during scratching of calcium carbonate-reinforced polyethylene composites. Mater. Sci. Eng. A 404(1), 208–220 (2005)CrossRefGoogle Scholar
  66. Tauheed, F., Sarangi, S.: Mullins effect on incompressible hyperelastic cylindrical tube in finite torsion. Int. J. Mech. Mater. Des. 8(4), 393–402 (2012). CrossRefGoogle Scholar
  67. Timmel, M., Kaliske, M., Kolling, S.: Modelling of microstructural void evolution with configurational forces. ZAMM - Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 89(8), 698–708 (2009). CrossRefzbMATHGoogle Scholar
  68. Tvergaard, V., Hutchinson, J.W.: Two mechanisms of ductile fracture: void by void growth versus multiple void interaction. Int. J. Solids Struct. 39(13–14), 3581–3597 (2002). CrossRefzbMATHGoogle Scholar
  69. Venkatesh Raja, K., Malayalamurthi, R.: Assessment on assorted hyper-elastic material models applied for large deformation soft finger contact problems. Int. J. Mech. Mater. Des. 7(4), 299 (2011). CrossRefGoogle Scholar
  70. Verron, E., Chagnon, G., Le Cam, J.: Hyperelasticity with Volumetric Damage. Constitutive Models for Rubber, vol. VI, pp. 279–284. Balkema, Rotterdam (2010)Google Scholar
  71. Voyiadjis, G.Z., Ju, J.W., Chaboche, J.L. (eds.): Continuum Damage Mechanics in Engineering Materials, book section 1–6, pp. 1–557. Elsevier, Amsterdam (1998)Google Scholar
  72. Waffenschmidt, T., Polindara, C., Menzel, A., Blanco, S.: A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials. Comput. Methods Appl. Mech. Eng. 268, 801–842 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  73. Xiao, H., Bruhns, O.T., Meyers, A.: Objective stress rates, path-dependence properties and non-integrability problems. Acta Mech. 176(3), 135–151 (2005). CrossRefzbMATHGoogle Scholar
  74. Zhang, W., Cai, Y.: Review of Damage Mechanics. Springer, Berlin (2010). CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Structural Mechanics and DesignTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Institute of Mechanics and MaterialsTechnische Hochschule MittelhessenGießenGermany

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