Cohesive zone modelling of nucleation, growth and coalesce of cavities

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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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  1. 1.

    Also known as pressure sensitive adhesive (PSA).

  2. 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. 3.

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

  4. 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. 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.


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


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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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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) doi:10.1007/s10704-016-0168-9

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  • Experiment
  • Defect
  • Elastomer
  • Cavity
  • Post-bifurcation
  • Adhesive
  • Tape
  • Cohesive law