# Integration of finite displacement interface element in reference and current configurations

- 149 Downloads
- 3 Citations

## Abstract

In the present paper the non-linear behaviour of a solid body with embedded cohesive interfaces is examined in a finite displacements context. The principal target is the formulation of a two dimensional interface finite element which is referred to a local reference frame, defined by normal and tangential unit vectors to the interface middle surface. All the geometric operators, such as the interface elongation and the reference frame, are computed as function of the actual nodal displacements. The constitutive cohesive law is defined in terms of Helmholtz free energy for unit undeformed interface surface and, in order to obtain the same nodal force vector and stiffness matrix by the two integration schemes, the cohesive law in the deformed configuration is defined in terms of Cauchy traction, as a function of separation displacement and of interface elongation. Explicit expression of the nodal force vector is integrated either over the reference configuration or over the current configuration, which is shown to produce the same analytical finite element operators. No differences between the integration carried out in the reference and in the current configuration are shown, provided that elongation of the interface is taken in to account.

## Keywords

Finite displacement Cohesive interface Integration Reference configuration Current configuration## Notes

### Acknowledgements

The financal support of the Italian Ministry for University and Research (MIUR), under the Grant PRIN-2015, Project No. 2015LYYXA8, “Multiscale mechanical models for the design and optimization of microstructured smart materials and metamaterials” is gratefully acknowledged.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Allix O, Corigliano A (1999) Geometrical and interfacial non-linearities in the analysis of delamination in composites. Int J Solids Struct 36(15):2189–2216CrossRefzbMATHGoogle Scholar
- 2.Scimemi G, Giambanco G, Spada A (2014) The interphase model applied to the analysis of masonry structures. Comput Methods Appl Mech Eng 279:66–85. https://doi.org/10.1016/j.cma.2014.06.026 ADSMathSciNetCrossRefGoogle Scholar
- 3.Sacco E, Toti J (2010) Interface elements for the analysis of masonry structures. Int J Comput Methods Eng Sci Mech 11(6):354–373. https://doi.org/10.1080/15502287.2010.516793 CrossRefzbMATHGoogle Scholar
- 4.Borino G, Fratini L, Parrinello F (2009) Mode i failure modeling of friction stir welding joints. Int J Adv Manuf Technol 41(5–6):498–503. https://doi.org/10.1007/s00170-008-1498-1 CrossRefGoogle Scholar
- 5.Camacho G, Ortiz M (1996) Computational modelling of impact damage in brittle materials. Int J Solids Struct 33(20):2899–2938. https://doi.org/10.1016/0020-7683(95)00255-3 CrossRefzbMATHGoogle Scholar
- 6.Pandolfi A, Krysl P, Ortiz M (1999) Finite element simulation of ring expansion and fragmentation: the capturing of length and time scales through cohesive models of fracture. Int J Fract 95(1):279–297. https://doi.org/10.1023/A:1018672922734 CrossRefGoogle Scholar
- 7.Mosler J, Scheider I (2011) A thermodynamically and variationally consistent class of damage-type cohesive models. J Mech Phys Solids 59(8):1647–1668ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 8.Ottosen N, Ristinmaa M, Mosler J (2015) Fundamental physical principles and cohesive zone models at finite displacements—limitations and possibilities. Int J Solids Struct 53:70–79. https://doi.org/10.1016/j.ijsolstr.2014.10.020 CrossRefGoogle Scholar
- 9.Vossen B, Schreurs P, van der Sluis O, Geers M (2013) On the lack of rotational equilibrium in cohesive zone elements. Comput Methods Appl Mech Eng 254:146–153ADSCrossRefzbMATHGoogle Scholar
- 10.Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Meth Eng 44(9):1267–1282CrossRefzbMATHGoogle Scholar
- 11.Qiu Y, Crisfield M, Alfano G (2001) An interface element formulation for the simulation of delamination with buckling. Eng Fract Mech 68(16):1755–1776. https://doi.org/10.1016/S0013-7944(01)00052-2 CrossRefGoogle Scholar
- 12.van den Bosch M, Schreurs P, Geers M (2007) A cohesive zone model with a large displacement formulation accounting for interfacial fibrilation. Eur J Mech A Solids 26(1):1–19CrossRefzbMATHGoogle Scholar
- 13.Tvergaard V (1990) Effect of fibre debonding in a whisker-reinforced metal. Mater Sci Eng A 125(2):203–213CrossRefGoogle Scholar
- 14.Geubelle PH, Baylor JS (1998) Impact-induced delamination of composites: a 2d simulation. Compos B Eng 29(5):589–602CrossRefGoogle Scholar
- 15.van den Bosch M, Schreurs P, Geers M (2006) An improved description of the exponential xu and needleman cohesive zone law for mixed-mode decohesion. Eng Fract Mech 73(9):1220–1234CrossRefGoogle Scholar
- 16.Reinoso J, Paggi M (2014) A consistent interface element formulation for geometrical and material nonlinearities. Comput Mech 54(6):1569–1581. https://doi.org/10.1007/s00466-014-1077-2 MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Gilormini P, Diani J (2015) Testing some implementations of a cohesive-zone model at finite strain. Eng Fract Mech 148:97–109. https://doi.org/10.1016/j.engfracmech.2015.09.013 CrossRefGoogle Scholar
- 18.Park K, Paulino G, Roesler J (2009) A unified potential-based cohesive model of mixed-mode fracture. J Mech Phys Solids 57(6):891–908ADSCrossRefGoogle Scholar
- 19.Taylor O Zienkiewicz (2000) The finite element method, 5th edn. Butterworth-Heinemann Press, OxfordzbMATHGoogle Scholar
- 20.Parrinello F, Failla B, Borino G (2009) Cohesive–frictional interface constitutive model. Int J Solids Struct 46(13):2680–2692CrossRefzbMATHGoogle Scholar
- 21.Parrinello F, Marannano G, Borino G (2016) A thermodynamically consistent cohesive–frictional interface model for mixed mode delamination. Eng Fract Mech 153:61–79. https://doi.org/10.1016/j.engfracmech.2015.12.001 CrossRefGoogle Scholar
- 22.Gdoutos E (1993) Fracture mechanics. An introduction. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar