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High cycle fatigue crack growth in Mode I of adhesive layers: modelling, simulation and experiments

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Abstract

The capability to predict high cycle fatigue properties of adhesive joints is important for cost-efficient and rapid product development in the modern automotive industry. Here, the adaptability of adhesives facilitates green technology through the widening of options of choosing and joining optimal materials. In the present paper a continuum damage mechanics model is developed based on the adhesive layer theory. In this theory, through-thickness averaged variables for the adhesive layer are used to characterise the deformation, damage and local loading on the adhesive layer. In FE-simulations, cohesive elements can thereby be used to model the adhesive layer. This simplifies simulations of large scale complex built-up structures. The model is adapted to experimental results for two very different adhesive systems; one relatively stiff rubber based adhesive and one soft polyurethane based adhesive. The model is able to reproduce the experimental results with good accuracy except for the early stage of crack propagation when the loads are relatively large. The model also predicts a threshold value for fatigue crack growth below which no crack growth occurs. The properties of the model are also compared with the properties of Paris’ law. The relations between the parameters of the continuum damage mechanics law and the parameters of Paris’ law are used to adapt the new law. It also shows that the properties of a joined structure influence the Paris’ law properties of the adhesive layer. Thus, the Paris’ law properties of an adhesive layer are not expected to be transferable to joints with adherends having different mechanical properties.

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Notes

  1. An aluminium sheet with the same bending stiffness as a steel sheet is about 40 % thicker but 50 % lighter.

  2. Formally, \(\sigma _{\mathrm{t}}\)(\(w_{\mathrm{t}}\)) is derived. However, we have already assumed that \(U\) is independent of \(x\). Thus, the relation \(\sigma \)(\(w\)) is independent of \(x\) and we have derived the cohesive law for all \(x\).

  3. A proof is given starting with Eq. (2.7). By neglecting the flexibility of the adhesive and modelling the free arms of the DCB-specimens as Euler–Bernoulli beams, Eq. (2.10) results. The angle \(\theta \) is then underestimated and Eq. (2.10) gives a too small value of \(J\).

  4. It can be noted that the integration points are located at the nodes.

  5. Data on Young’s modulus are according to the manufacturer.

Abbreviations

\(a\) :

Crack length (m)

\(a_{0}\) :

Initial crack length (m)

\(B\) :

Out-of-plane width of DCB-specimen (m)

\(C\) :

Compliance (m/N)

\(C_{\mathrm{P}}\) :

Constant in Paris’ law (\(\hbox {m}^{n+1}\)/\(\hbox {N}^{n}\))

\(D\) :

Damage (\(-\))

\(E\) :

Young’s modulus of the adherends (\(\hbox {N/m}^{2}\))

\(E_{\mathrm{a}}\) :

Young’s modulus of the adhesive (\(\hbox {N/m}^{2}\))

\(F\) :

Force applied to DCB-specimen (N)

\(h\) :

Thickness of adhesive layer (m)

\(J\) :

Energy release rate (\(\hbox {J/m}^{2}\))

\(J_{\mathrm{c}}\) :

Fracture energy (\(\hbox {J/m}^{2}\))

\(J_{\mathrm{cycle}}\) :

Measure of energy release rate during a load cycle (\(\hbox {J/m}^{2}\))

\(J_{\mathrm{f}}\) :

Final value of \(J_{\mathrm{cycle}}\) in an experiment (\(\hbox {J/m}^{2}\))

\(J_{\mathrm{th}}\) :

Threshold value of \(J_{\mathrm{cycle}}\) (\(\hbox {J/m}^{2}\))

\(K_{\mathrm{n}}\) :

Elastic stiffness of adhesive layer (\(\hbox {N/m}^{3}\))

\(L\) :

Length of adherend (m)

\(n\) :

Exponent in Paris’ law (\(-\))

\(M\) :

Bending moment applied to DCB-specimen (Nm)

\(N\) :

Load cycle (cycle)

\(p\) :

Constant in Berry’s law (1/\(\hbox {Nm}^{q-1}\))

\(q\) :

Exponent in Berry’s law (\(-\))

\(R_{\varepsilon }\) :

Load ratio (\(-\))

\(S\) :

Integration path (m)

T :

Traction vector (\(\hbox {N/m}^{2}\))

u :

Displacement vector (m)

\(U\) :

Strain energy density (\(\hbox {J/m}^{3}\))

\(w\) :

Separation of adherends along the adhesive layer (m)

\(w_{\mathrm{t}}\) :

\(w\) at the start of the adhesive layer (m)

\({\varDelta }\) :

Separation of loading points of DCB-specimen (m)

\(v_{\mathrm{a}}\) :

Poisson’s ratio of the adhesive (\(-\))

\(\alpha \) :

Damage growth proportionality parameter (1/cycle)

\(\beta \) :

Damage growth exponent (\(-\))

\(\sigma \)(\(w)\) :

Cohesive law of adhesive (\(\hbox {N/m}^{2}\))

\(\sigma _{0}\) :

Normalization parameter in damage evolution law (\(\hbox {N/m}^{2}\))

\(\sigma _{{\max }}\) :

Maximum stress in cohesive law, the strength (\(\hbox {N/m}^{2}\))

\(\sigma _{\mathrm{th}}\) :

Threshold stress (\(\hbox {N/m}^{2}\))

\(\sigma _{\mathrm{t}}\) :

Stress at tip of adhesive layer (\(\hbox {N/m}^{2}\))

\(\theta \) :

Rotation of loading point of DCB-specimen (\(-\))

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Acknowledgments

The authors would like to thank Dr Stephan Marzi for help with Abaqus, Dr. Anders Biel and Mr. Stefan Zomborcsevics for help with the experiments, Scania AB through LicEng Lars Gunnarsson for help with the SEM, DOW AG and Volvo AB for good collaboration with specimens and adhesives, and finally, the Knowledge Foundation for partly financing the study.

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Authors and Affiliations

  1. University of Skövde, 541 28, Skövde, Sweden

    Alexander Eklind, Tomas Walander, Thomas Carlberger & Ulf Stigh

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  1. Alexander Eklind
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Correspondence to Alexander Eklind.

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Eklind, A., Walander, T., Carlberger, T. et al. High cycle fatigue crack growth in Mode I of adhesive layers: modelling, simulation and experiments. Int J Fract 190, 125–146 (2014). https://doi.org/10.1007/s10704-014-9979-8

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