Glass Structures & Engineering

, Volume 3, Issue 2, pp 237–256 | Cite as

On cavitation in transparent structural silicone adhesive: TSSA

  • Michael Drass
  • Vladimir A. Kolupaev
  • Jens Schneider
  • Stefan Kolling
SI: Challenging Glass paper


Cavitation in rubber-like materials describes sudden void growth of an initially voided material under hydrostatic tension until the material fails. To study the cavitation effect numerically, classical cavitation criteria are coupled with a continuum damage formulation of a Neo-Hookean material. A cavitation criterion defines a failure surface in three-dimensional stress or strain space, which represents the onset of excessive void growth and therefore the strong degradation of the bulk modulus. To account for this special case of material softening, a novel continuum damage formulation at finite strains is presented, where the initially constant bulk modulus of a hyperelastic material is reduced after satisfying a cavitation criterion. Since this formulation leads to an abrupt damage initiation, additionally a continuously volumetric damage formulation is proposed and compared with it. Therefore, novel void growth criteria are developed, which describe the cavitation effect even under smallest volumetric strains. For numerical validation, a single element test is simulated under hydrostatic tension. Furthermore, pancake tests are numerically analysed. The results with regard on the chosen cavitation criterion and the abrupt/continuously damage formulation are compared with each other analysing TSSA.


TSSA Compressible hyperelasticity Cavitation criteria Continuum damage formulation 

List of symbols

\({\left( \bullet \right) _{{\mathrm{iso}}}}\)


\({\left( \bullet \right) _{{\mathrm{vol}}}}\)


tr\(\left( \bullet \right) \)

Trace of argument

\({\nabla _0}\left( \bullet \right) \)

Gradient of argument


Deformation gradient


Relative volume


Critical relative volume


Right Cauchy–Green tensor


Left Cauchy–Green tensor


Right material stretch tensor


Rotation tensor


Isochoric left Cauchy–Green tensor

\(\lambda _i\)

Principal stretches

\(\varepsilon _i^{{\mathrm{eng}}}\)

Principal engineering strain

\(I_{1,(\bullet )}\)

First invariant of its argument

\(I_{2,(\bullet )}\)

Second invariant of its argument

\(I_{3,(\bullet )}\)

Third invariant of its argument

\(\varvec{\sigma }\)

General Cauchy stress tensor

\(\varvec{\sigma }_{\mathrm{prin}}\)

Principal Cauchy stress tensor

\({\varepsilon _{\mathrm{vol}}}\)

Volume strain with engineering strains

\({\varepsilon _{\mathrm{cr,vol}}}\)

Critical volume strain for damage initiation

\({\varepsilon _{\mathrm{eqn,vol}}}\)

Equivalent volume strain with true strains


Hydrostatic pressure

\(\varPsi ( \bullet )\)

Helmholtz free energy

\(\xi \)

Internal scalar damage variable

\(\xi _{\mathrm{mod}}\)

Modified internal scalar damage variable

\(\mu \)

Initial shear modulus


Initial bulk modulus

\(\varvec{\Upphi }\)

Orthogonal matrix

\(I_{2,\sigma }'\)

Second invariant of stress deviator

\(I_{3,\sigma }'\)

Third invariant of stress deviator

\(\xi _1,\xi _2,\xi _3\)

Transformed coordinate system

\(\theta \)

Stress angle

\(\psi \)


tan\(\psi \)

Stress triaxiality

\(\alpha ,\beta ,\gamma \)

Prefactors of void growth criteria


  1. Altenbach, H., Kolupaev, V.A.: Classical and non-classical failure criteria. In: Altenbach, H., Sadowski, T. (eds.) Failure and Damage Analysis of Advanced Materials, Springer, Wien, Heidelberg, International Centre for Mechanical Sciences CISM, Courses and Lectures, vol. 560, pp. 1–66 (2014)Google Scholar
  2. Altenbach, H., Bolchoun, A., Kolupaev, V.A.: Phenomenological yield and failure criteria. In: Altenbach, H., Öchsner, A. (eds.) Plasticity of Pressure-Sensitive Materials, pp. 49–152. Springer, Berlin (2014)CrossRefGoogle Scholar
  3. Alter, C., Kolling, S., Schneider, J.: An enhanced non-local failure criterion for laminated glass under low velocity impact. Int. J. Impact Eng. 109(Supplement C), 342–353 (2017). CrossRefGoogle Scholar
  4. Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 306(1496), 557–611 (1982). MathSciNetCrossRefzbMATHGoogle Scholar
  5. Busse, W.F.: Physics of rubber as related to the automobile. J. Appl. Phys. 9(7), 438–451 (1938). CrossRefGoogle Scholar
  6. Chaves, E.W.V.: Notes on Continuum Mechanics. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  7. Dispersyn, J., Hertelé, S., De Waele, W., Belis, J.: Assessment of hyperelastic material models for the application of adhesive point-fixings between glass and metal. Int. J. Adhes. Adhes. 77(Supplement C), 102–117 (2017). CrossRefGoogle Scholar
  8. Dow Corning Europe SA.: On macroscopic effects of heterogeneity in elastoplastic media at finite strain. glasstec (2017).
  9. Drass, M., Schneider, J., Kolling, S.: Damage effects of adhesives in modern glass façades: a micro-mechanically motivated volumetric damage model for poro-hyperelastic materials. Int. J. Mech. Mater. Des. (2017a). Google Scholar
  10. Drass, M., Schneider, J., Kolling, S.: Novel volumetric helmholtz free energy function accounting for isotropic cavitation at finite strains. Mater. Des. (2017b).
  11. 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. (2017c).
  12. 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. (2017d).
  13. Fahlbusch, N.C., Kolupaev, V.A., Becker, W.: Generalized Limit Surfaces—With an Example of Hard Foams, pp. 337–365. Springer, Berlin (2016)Google Scholar
  14. Flory, P.J.: Thermodynamic relations for high elastic materials. Trans. Faraday Soc. 57, 829–838 (1961). MathSciNetCrossRefGoogle Scholar
  15. Gent, A.N., Lindley, P.B.: Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 249(1257), 195–205 (1959). CrossRefGoogle Scholar
  16. Gosse, J., Christensen, S.: Strain invariant failure criteria for polymers in composite materials, vol 1184, American Institute of Aeronautics & Astronautics, book section collection of technical papers—AIAA/ASME/ASCE/AHS/ASC Structures, pp. 1–11 (2001).
  17. Hagl, A.: Development and test logics for structural silicone bonding design and sizing. Glass Struct. Eng. 1, 131–151 (2016)CrossRefGoogle Scholar
  18. Hamdi, A., Guessasma, S., Abdelaziz, M.N.: Fracture of elastomers by cavitation. Mater. Des. 53(Supplement C), 497–503 (2014).
  19. Hartmann, S., Neff, P.: Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Int. J. Solids Struct. 40(11), 2767–2791 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  20. Hohe, J., Becker, W.: An energetic homogenisation procedure for the elastic properties of general cellular sandwich cores. Compos. B Eng. 32(3), 185–197 (2001). CrossRefGoogle Scholar
  21. Hohe, J., Becker, W.: Effective mechanical behavior of hyperelastic honeycombs and two-dimensional model foams at finite strain. Int. J. Mech. Sci. 45(5), 891–913 (2003a). CrossRefzbMATHGoogle Scholar
  22. Hohe, J., Becker, W.: Geometrically nonlinear stress–strain behavior of hyperelastic solid foams. Comput. Mater. Sci. 28(3), 443–453 (2003b). CrossRefzbMATHGoogle Scholar
  23. Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering (2000). Wiley, West Sussex (2000)zbMATHGoogle Scholar
  24. Hou, H.S., Abeyaratne, R.: Cavitation in elastic and elastic–plastic solids. J. Mech. Phys. Solids 40(3), 571–592 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  25. James, H.M., Guth, E.: Theory of the elastic properties of rubber. J. Chem. Phys. 11(10), 455–481 (1943). CrossRefGoogle Scholar
  26. Kachanov, L.M.: Time of the rupture process under creep conditions. Izv. Akad. Nauk. SSR Otd. Tech. Nauk. 8, 26–31 (1958)Google Scholar
  27. 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
  28. Kraus, M.A., Schuster, M., Kuntsche, J., Siebert, G., Schneider, J.: Parameter identification methods for visco- and hyperelastic material models. Glass Struct. Eng. 2(2), 147–167 (2017). CrossRefGoogle Scholar
  29. Kuhn, W.: Über die gestalt fadenförmiger moleküle in lösungen. Kolloid-Zeitschrift 68(1), 2–15 (1934). CrossRefGoogle Scholar
  30. Kuhn, W.: Beziehungen zwischen molekülgröße, statistischer molekülgestalt und elastischen eigenschaften hochpolymerer stoffe. Kolloid-Zeitschrift 76(3), 258–271 (1936a). CrossRefGoogle Scholar
  31. Kuhn, W.: Gestalt und eigenschaften fadenförmiger moleküle in lösungen (und im elastisch festen zustande). Angew. Chem. 49(48), 858–862 (1936b). CrossRefGoogle Scholar
  32. Kuhn, W., Grün, F.: Beziehungen zwischen elastischen konstanten und dehnungsdoppelbrechung hochelastischer stoffe. Kolloid-Zeitschrift 101(3), 248–271 (1942). CrossRefGoogle Scholar
  33. Li, J., Mayau, D., Song, F.: A constitutive model for cavitation and cavity growth in rubber-like materials under arbitrary tri-axial loading. Int. J. Solids Struct. 44(18), 6080–6100 (2007). CrossRefzbMATHGoogle Scholar
  34. Lindsey, G.H.: Triaxial fracture studies. J. Appl. Phys. 38(12), 4843–4852 (1967). CrossRefGoogle Scholar
  35. 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 (2011a). MathSciNetCrossRefzbMATHGoogle Scholar
  36. Lopez-Pamies, O., Nakamura, T., Idiart, M.I.: Cavitation in elastomeric solids: II—onset-of-cavitation surfaces for neo-Hookean materials. J. Mech. Phys. Solids 59(8), 1488–1505 (2011b). MathSciNetCrossRefzbMATHGoogle Scholar
  37. Overend, M.: Optimising connections in structural glass. In: Proceedings of 2nd International conference on Glass in Buildings (2005)Google Scholar
  38. Poulain, X., Lefévre, V., Lopez-Pamies, O., Ravi-Chandar, K.: Damage in elastomers: nucleation and growth of cavities, micro-cracks, and macro-cracks. Int. J. Fract. 205(1), 1–21 (2017). CrossRefGoogle Scholar
  39. Pourmoghaddam, N., Schneider, J.: Finite-element analysis of residual stresses in tempered glass plates with holes or cut-outs. Glass Struct. Eng. (2017). Google Scholar
  40. Rühl, A., Kolling, S., Schneider, J.: Characterization and modeling of poly(methyl methacrylate) and thermoplastic polyurethane for the application in laminated setups. Mech. Mater. 113(Supplement C), 102–111 (2017). CrossRefGoogle Scholar
  41. Santarsiero, M., Louter, C., Nussbaumer, A.: Laminated connections for structural glass applications under shear loading at different temperatures and strain rates. Constr. Build. Mater. 128, 214–237 (2016a). CrossRefGoogle Scholar
  42. 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. 1(2), 395–415 (2016b). CrossRefGoogle Scholar
  43. Santarsiero, M., Louter, C., Nussbaumer, A.: Laminated connections under tensile load at different temperatures and strain rates. Int. J. Adhes. Adhes. 79(Supplement C), 23–49 (2017). CrossRefGoogle Scholar
  44. Sarkawi, S.S., Dierkes, W.K., Noordermeer, J.W.M.: Morphology of silica-reinforced natural rubber: the effect of silane coupling agent. Rubber Chem. Technol. 88(3), 359–372 (2015). CrossRefGoogle Scholar
  45. Sitte, S., Brasseur, M., Carbary, L., Wolf, A.: Preliminary evaluation of the mechanical properties and durability of transparent structural silicone adhesive (tssa) for point fixing in glazing. J. ASTM Int. 10(8), 1–27 (2011). Google Scholar
  46. Timmel, M., Kaliske, M., Kolling, S., Müller, R.: A micromechanical approach to simulate rubberlike materials with damage. Comput. Mater. Contin. 5, 161–172 (2007). zbMATHGoogle Scholar
  47. Timmel, M., Kaliske, M., Kolling, S.: Modelling of microstructural void evolution with configurational forces. ZAMM 89(8), 698–708 (2009). CrossRefzbMATHGoogle Scholar
  48. Treloar, L.: The Physics of Rubber Elasticity. Oxford University Press, Oxford (1975)zbMATHGoogle Scholar
  49. Yerzley, F.L.: Adhesion of neoprene to metal. Ind. Eng. Chem. 31(8), 950–956 (1939). CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Structural Mechanics and DesignTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Fraunhofer Institute for Structural Durability and System ReliabilityDarmstadtGermany
  3. 3.Institute of Mechanics and MaterialsTechnische Hochschule MittelhessenGiessenGermany

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