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Cohesive zone modelling of nucleation, growth and coalesce of cavities

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Abstract

The stress-deformation relation i.e. cohesive law representing the fracture process in an almost incompressible adhesive tape is measured using the double cantilever beam specimen. As in many ductile materials, the fracture process of the tape involves nucleation, growth and coalesce of cavities. This process is studied carefully by exploiting the transparency of the used materials and the inherent stability of the specimen configuration. Utilising the path independence of the J-integral, the cohesive law is measured. The law is compared to the results of butt-joint tests. The law contains two stress peaks—the first is associated with nucleation of cavities at a stress level conforming to predictions of void nucleation in rubber elasticity. The second stress peak is associated with fracture of stretched walls between fully-grown cavities. After this second peak, a macroscopic crack is formed. The tape suffers at this stage an engineering strain of about 800%. A numerical analysis with the determined cohesive law recreates the global specimen behaviour.

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Notes

  1. Also known as pressure sensitive adhesive (PSA).

  2. Different names are used in the literature for this model. We suggest this notation to distinguish between models of material volumes, hence “layer”, and models of a prospective crack surface without a volume.

  3. Young’s modulus and Poisson’s ratio for PMMA are temperature dependent, cf. e.g. Mott et al. (2008).

  4. If we assume that the layer is incompressible and in a state of uniaxial strain, i.e. completely constrained by the adherends from deforming in the xz-plane, Hooke’s law for an isotropic material gives a hydrostatic state of stress for small strains, cf. e.g. Walander et al. (2016).

  5. Consider a local coordinate system at one wall with tangential and normal vectors t and n in the \(x-z\)-plane. The constraint by the adherents forces the strain in the t-direction to be very much smaller than the other strain components. The tape is almost incompressible, i.e. the strain in the n-direction is almost equal, but has opposite sign, as the strain component in the y-direction. Now, by setting the strain component in the t-direction equal to zero and the strain component in the n-direction equal to the negative of the strain in the y-direction, the strain state is revealed to be one in pure shear.

Abbreviations

a :

Distance between applied load and start of tape

b :

Width of the tape

h :

Height of the beam

l :

Length of the specimen

\(\mathbf{n}, n_{i}\) :

Unit normal vector and its components, respectively

\(p_{e}\) :

External pressure

t :

Thickness of the adhesive layer

\(\mathbf{u}, u_{i}\) :

Displacement vector and its components, respectively

w :

Normal deformation of the tape, DCB specimens

B :

Width of the adherends

E :

Young’s modulus

EI :

Bending stiffness of the adherends

F :

Applied force

J :

Energy release rate

\(J_{c}\) :

Fracture energy

M :

Applied moments

S :

Integration path

\(\mathbf{T}, T_{i}\) :

Traction vector and its components, respectively

U :

Strain energy density

\(\delta \) :

Normal deformation of the tape, butt-joints

\(\varepsilon _{y}\) :

Normal strain in the y-direction

\(\sigma \) :

Cohesive stress

\(\sigma _{n}\) :

Net stress in the material part of the layer

\(\sigma _{y}\) :

Normal stress in the y-direction

\(\vartheta _{\mathrm{n}}\) :

Fraction of area occupied by the material

\(\vartheta _\mathrm{o}\) :

Fraction of area occupied by open cavities

\(\vartheta _c \) :

Fraction of area occupied by closed cavities

v :

Poison’s ratio

\(\varDelta \) :

Displacement of the loading point

\(\theta \) :

Rotation at the loading point

References

  • Andersson T, Biel A (2006) On the effective constitutive properties of a thin adhesive layer loaded in peel. Int J Fract 141(1–2):227–246

    Article  Google Scholar 

  • Andersson T, Stigh U (2004) The stress–elongation relation for an adhesive layer loaded in peel using equilibrium of energetic forces. Int J Solids Struct 41(2):413–434

    Article  Google Scholar 

  • Ashby MF, Jones DRH (1980) Engineering materials: an introduction to their properties and applications. Pergamon Press, New York

    Google Scholar 

  • Ball JM (1982) Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos Trans R Soc Lond Ser A 306(1496):557–611

    Article  Google Scholar 

  • Biel A, Stigh U (2008) Effects of constitutive parameters on the accuracy of measured fracture energy using the DCB-specimen. Eng Fract Mech 75(10):2968–2983

    Article  Google Scholar 

  • Biel A, Stigh U (2010) Damage and plasticity in adhesive layer: an experimental study. Int J Fract 165(1):93–103

    Article  Google Scholar 

  • Biel A, Alfredsson KS, Carlberger T (2014) Adhesive tapes; cohesive laws for a soft layer. Proc Mater Sci 3:1389–1393

    Article  Google Scholar 

  • Broberg KB (1999) Cracks and fracture. Academic Press, Edinburgh

    Google Scholar 

  • Campilho RDSG, Banea MD, Neto JABP, da Silva LFM (2013) Modelling adhesive joints with cohesive zone models: effect of the cohesive law shape of the adhesive layer. Int J Adhes Adhes 44:48–56

    Article  Google Scholar 

  • Carlberger T, Alfredsson KS, Stigh U (2008) FE-formulation of interphase elements for adhesive joints. Int J Comput Methods Eng Sci Mech 9(5):288–299

    Article  Google Scholar 

  • Carlberger T, Biel A, Stigh U (2009) Influence of temperature and strain rate on cohesive properties of a structural epoxy adhesive. Int J Fract 155(2):155–166

    Article  Google Scholar 

  • Carlberger T, Stigh U (2010) Influence of layer thickness on cohesive properties of an epoxy-based adhesive—an experimental study. J Adhes 86:814–833

    Article  Google Scholar 

  • Chiche A, Dollhofer J, Creton C (2005) Cavity growth in soft adhesives. Eur Phys J E 17(4):389–401

    Article  Google Scholar 

  • Christensen SF, Everland H, Hassager O, Almdal K (1998) Observations of peeling of a polyisobutylene-based pressure-sensitive adhesive. Int J Adhes Adhes 18:131–137

    Article  Google Scholar 

  • Creton C, Chiche A, Dollhofer J, Roos A, Hui CY, Muralidharan V (2005) Cavitation and fracture of soft adhesives. In: 11th international conference on fracture 2005, 7, pp 5468–5473

  • Gent AN, Lindley PB (1959) Internal rupture of bonded rubber cylinders in tension. Proc R Soc Lond Ser A 249:195–205

    Article  Google Scholar 

  • Gent AN, Park B (1984) Failure processes in elastomers at or near a rigid spherical inclusion. J Mater Sci 19:1947–1956

    Article  Google Scholar 

  • Good RJ, Gupta RK (1988) Rheological, interfacial and thermal control of polymer adhesion. 1. Isothermal theory. J Adhes 26(1):13–36

    Article  Google Scholar 

  • Hayashida S, Sugaya T, Kuramoto S, Sato C, Mihara A, Onuma T (2015) Impact strength of joints bonded with high-strength pressure-sensitive adhesive. Int J Adhes Adhes 56:61–72

    Article  Google Scholar 

  • Jousset P, Rachik M (2014) Implementation, identification and validation of an elasto-plastic-damage model for the finite element simulation of structural bonded joints. Int J Adhes Adhes 50:107–118

    Article  Google Scholar 

  • Kaelble DH, Reylek RS (1969) Peel adhesion. Rate dependence of micro fracture processes. J Adhes 1:124–135

    Article  Google Scholar 

  • Lefèvre V, Ravi-Chandar K, Lopez-Pamies O (2015) Cavitation in rubber: an elastic instability or a fracture phenomenon? Int J Fract 192(1):1–23

    Article  Google Scholar 

  • Lopez-Pamies O, Idiart MI, Nakamura T (2011) Cavitation in elastomeric solids: I—a defect-growth theory. J Mech Phys Sol 59(7):1464–1487

    Article  Google Scholar 

  • McClintock FA (1968) A criterion for ductile fracture by the growth of holes. J Appl Mech 35:363–371

    Article  Google Scholar 

  • May M, Hesebeck O, Marzi S, Boehme W, Lienhard J, Kilchert S, Brede M, Hiermaier S (2015) Rate dependent behavior of crash-optimized adhesives–experimental characterization, model development, and simulation. Eng Fract Mech 133:112–137

    Article  Google Scholar 

  • Marzi S, Biel A, Stigh U (2011) On experimental methods to investigate the effect of layer thickness on the fracture behavior of adhesively bonded joints. Int J Adhes Adhes 31(8):840–850

    Article  Google Scholar 

  • Mott PH, Dorgan JR, Roland CM (2008) The bulk modulus and poisson’s ratio of “incompressible” materials. J Sound Vib 312(4–5):572–575

    Article  Google Scholar 

  • Nase J, Creton C, Ramos O, Sonnenberg L, Yamaguchi T, Lindner A (2010) Measurement of the receding contact angle at the interface between a viscoelastic material and a rigid surface. Soft Matter 6(12):2685–2691

    Article  Google Scholar 

  • Niesiolowski F, Aubrey DW (1981) Stress distribution during peeling of adhesive tapes. J Adhes 13(1):87–98

    Article  Google Scholar 

  • Nilsson F (2005) A tentative method for determination of cohesive zone properties in soft materials. Int J Fract 136:133–142

    Article  Google Scholar 

  • Oberth AE, Bruenner RS (1965) Tear phenomena around solid inclusions in castable elastomers. Trans Soc Rheol 9(2):165–185

    Article  Google Scholar 

  • Olsson P, Stigh U (1989) On the determination of the constitutive properties of thin interphase layers—an exact inverse solution. Int J Fract 41(4):R71–76

    Article  Google Scholar 

  • Paris AJ, Paris PC (1988) Instantaneous evaluation of J and C*. Int J Fract 38(1):R19–R21

    Google Scholar 

  • Plante J-S, Dubowsky S (2006) Large-scale failure modes of dielectric elastomer actuators. Int J Solids Struct 43(25–26):7727–7751

    Article  Google Scholar 

  • Pharr M, Sun J-Y, Suo Z (2012) Rupture of a highly stretchable acrylic dielectric elastomer. J Appl Phys 111:104114

    Article  Google Scholar 

  • Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by Notches and Cracks. J Appl Mech 35(2):379–386

    Article  Google Scholar 

  • Salomonsson K, Stigh U (2009) Influence of root curvature on the fracture energy of adhesive layers. Eng Fract Mech 76(13):2025–2038

    Article  Google Scholar 

  • Suo Z, Bao G, Fan B (1992) Delamination R-curve phenomena due to damage. J Mech Phys Sol 40(1):1–16

    Article  Google Scholar 

  • Steenbrink AC, van der Giessen E, Wu PD (1997) Void growth in glassy polymers. J Mech Phys Sol 45(3):405–437

    Article  Google Scholar 

  • Stigh U, Andersson T (2000) An experimental method to determine the complete stress-elongatioin relation for a structural adhesive layer loaded in peel. Fracture of Polymers, Composites and Adhesives, ESIS publication 27, pp 297–306

  • Stigh U, Alfredsson KS, Andersson T, Biel A, Carlberger T, Salomonsson K (2010) Some aspects of cohesive models and modelling with special application to strength of adhesive layers. Int J Fract 165(2):149–162

    Article  Google Scholar 

  • Stigh U, Biel A, Svensson D (2016) Cohesive zone modelling and the fracture process of a structural tape. Proc Struct Integr 2:235–244

    Article  Google Scholar 

  • Stringfellow R, Abeyaratne R (1989) Cavitation in an elastomer: comparison of theory with experiment. Mater Sci Eng A112:127–131

    Article  Google Scholar 

  • Sørensen BF (2010) Cohesive laws for assessment of materials failure: theory, experimental methods and application. Doctor of Technices thesis, Technical University of Denmark, Risø-R-1736

  • Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New York

    Google Scholar 

  • Toftegaard H, Rask M, Rasmussen S, Sørensen BF (2013) DCB fracture specimens with side notches. In: 6th International conference on composites testing and model identification, Department of Mechanical and Manufacturing Engineering, Aalborg University, Denmark, 2013, pp 75–76

  • Townsend BW, Ohanehi DC, Dillard DA, Austin SR, Salmon F, Gagnon DR (2011a) Characterizing acrylic foam pressure sensitive adhesive tapes for structural glazing applications—Part I: DMA and ramp-to-fail results. Int J Adhes Adhes 31(7):639–649

    Article  Google Scholar 

  • Townsend BW, Ohanehi DC, Dillard DA, Austin SR, Salmon F, Gagnon DR (2011b) Characterizing acrylic foam pressure sensitive adhesive tapes for structural glazing applications—Part II: Creep rupture results. Int J Adhes Adhes 31(7):650–659

    Article  Google Scholar 

  • Walander T, Biel A, Stigh U (2013) Temperature dependence of cohesive laws for an epoxy adhesive in Mode I and Mode II loading. Int J Fract 183(2):203–221

    Article  Google Scholar 

  • Walander T, Eklind A, Carlberger T, Stigh U, Rietz A (2016) Prediction of mixed-mode cohesive fatigue strength of adhesively bonded structure using Mode I data. Int J Adhes Adhes 66:15–25

    Article  Google Scholar 

  • Yang QD, Thouless MD (2001) Mixed-mode fracture analyses of plastically-deforming adhesive joints. Int J Fract 110:175–187

    Article  Google Scholar 

  • Zhao X, Suo Z (2008) Electrostriction in elastic dielectrics undergoing large deformation. J Appl Phys 104(12):123530

    Article  Google Scholar 

  • Zosel A (1998) The effect of fibrilation on the tack of pressure sensitive adhesives. Int J Adhes Adhes 18(4):265–271

    Article  Google Scholar 

Download references

Acknowledgements

Financial support was provided by the Åforsk foundation (Grant No. 14-312). The authors want to thank Stefan Zomborcsevics at University of Skövde for helping with manufacturing of the specimens and Dr. Roger Hagen at 3M for supply of adhesive tapes.

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Authors and Affiliations

  1. Department of Wind Energy, Technical University of Denmark, P.O. Box 49, Frederiksborgvej 399, 4000, Roskilde, Denmark

    A. Biel

  2. Engineering Science, University of Skövde, P.O. Box 408, Kaplansgatan 11, 541 28, Skövde, Sweden

    U. Stigh

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Correspondence to A. Biel.

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Biel, A., Stigh, U. Cohesive zone modelling of nucleation, growth and coalesce of cavities. Int J Fract 204, 159–174 (2017). https://doi.org/10.1007/s10704-016-0168-9

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