# Three-dimensional model of interface between ice and aluminium

Mehran Riahi, Daniel Marceau, Jean Perron

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Mehran Riahi, Daniel Marceau, Jean Perron. Three-dimensional model of interface between ice and aluminium. 8e Colloque national en calcul des structures, CSMA, May 2007, Giens, France. hal-01507585

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# Three dimensional model of interface between ice and aluminium

Mehran Riahi† — Daniel Marceau† — Jean Perron‡

†CURAL – University research centre on aluminum

‡AMIL – Anti-icing Materials International Laboratory

University of Quebec at Chicoutimi, Chicoutimi (Qc), Canada G7H 2B1

Mehran_Matbou-Riahi@uqac.ca

Daniel_Marceau@uqac.ca

Jean_Perron@uqac.ca

**ABSTRACT.** In this paper, the mechanical behaviour of the interface between ice and substrate (Aluminium) under uniaxial tension were examined. Initially, a mathematical model allowing a good representation of assembling the experimental tests was developed. Then, a numerical model was developed (using ABAQUS software). In FEM analysing the various mechanisms and parameters characteristic of the delaminating and cracking of the ice were studied. To do so, the realistic behaviour of the ice, the geometry of the components as well as mechanisms occurring at the interface were considered and the brittle cracking method was considered to defining the brittle fracture of ice. Also the cohesive elements were used for detailed modeling of the de-icing and damage of adhesive layers at interfaces between bonded parts.

**RÉSUMÉ.** Dans ce papier le comportement mécanique de l'interface entre la glace et le substrat (aluminium) sous une tension uniaxiale a été examiné. Au commencement, un modèle mathématique permettant une bonne représentation des essais expérimentaux a été développée. Puis, un modèle numérique a été développé (en utilisant le code d'éléments finis ABAQUS) afin de simuler la structure du déglacage d'aluminium. Pour faire ainsi, le comportement réaliste de la glace, la géométrie des composants aussi bien que des mécanismes se produisant à l'interface ont été considérés et le comportement de la fissuration fragile a servit à définir la rupture fragile de la glace. En outre, les éléments cohésifs ont été employés pour modéliser la déformation et les dommages des couches adhésives aux interfaces entre la glace et l'aluminium.

**KEY WORDS:** interface, de-icing, the telemen ecohesiv, brittle cracking, finite element.

**MOTS-CLÉS :** interface, déglacage, les éléments cohésifs, rupture fragile, éléments finis.
# 1. Introduction

In the cold regions, ice causes many serious problems. For example, ice on aircraft wings and helicopter blades endanger its passengers. Icing of power lines and communication towers often cause their break and loss of power. Ice/material interfaces is a subject of great interest in such problems as icing of electrical transmission cables, highways and bridges, constructions, off-shore structures, aircrafts and helicopters. Therefore, an understanding of the mechanical properties of ice/material interfaces is essential to develop de-icing techniques. This research introduces an experimental investigation and a model finite element that predicts the initiation and growth of delamination of ice/aluminium using decohesion elements to model the interface between ice and aluminium (de-icing). Also contains a discussion of the brittle cracking constitutive low to define the brittle failure of ice.

# 2. Experimental tests

In this study, the fine granular ice deposited on the aluminium plate was fabricated by spraying very fine water mist at -10 °C at AMIL laboratory (see Figure 1a). For each different thickness of ice (2, 5 and 10 mm) three tensile tests were performed under a high strain rate about 10⁻⁴ s⁻¹ (brittle mode). The results of these tests show that for ice with 2 mm, the interfacial forces are larger than the cohesive forces of the ice. For this reason, when the sample is stressed, the failure occurs in the ice (see Figure 1b). In the case of 5 mm thickness of ice, it can be observed that, not only ice fracture (cohesive failure) but also ice removal (adhesive failure) is occurred. In this case, a transition is observed from cohesive to adhesive failure. Finally, for the case of 10 mm thickness of ice, ice de-bonds without breaking the ice sheet (cohesive break). Because in this case, the ice is stronger than the interface and we have ice removal without cracking the ice. The results obtained in test 2, 6 and 9 tests are presented in Table 1.

# 3. Model for the ice (brittle cracking theory)

The brittle cracking model is used for applications in which the material behaviour is dominated by tensile cracking and compressive failure is unimportant. In this approach it is assumed that a certain amount of energy G is absorbed by the formation of the unit area of the crack surface [Rots et al., 1989]. This fracture energy can be calculated from measuring the tensile stress (see Figure 2) as a function of the crack opening displacement such as $G_f^I = \int \sigma_f^I d u_n$.
<figure><img src="image_1.png" /><figcaption>a) Ice-beam sample before testing.</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>b) Ice-beam sample after testing.</figcaption></figure>

<caption>**Figure 1.** *Description of the tensile test.*</caption>

<table><caption>**Table 1.** *Tensile test data.*</caption><thead><tr><th>Test</th><th>Ice thickness (mm)</th><th>Strain (mm/mm)</th><th>Force (N)</th><th>Normal stress at bottom face in Al (MPa)</th></tr></thead><tbody><tr><td>2</td><td>1.97</td><td>0.000529</td><td>343.35</td><td>37.03</td></tr><tr><td>6</td><td>5.35</td><td>0.000503</td><td>329.20</td><td>35.21</td></tr><tr><td>9</td><td>10.09</td><td>0.000375</td><td>297.63</td><td>26.25</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Figure 2.** *Post-failure stress–fracture energy curve.*</figcaption></figure>

**4. Model for the interface (cohesive material theory)**

The cohesive material theory can be used for detailed modeling of the deformation and damage of adhesive layers at interfaces between bonded parts. Application includes finite-thickness, as well as zero thickness adhesive between two parts. In this approach a constitutive equation is used to connect the traction $\sigma$ to the relative displacement $\delta$, at the interface. The material response illustrates in Figure 3. Point 1 is related to a low tensile load in the linear elastic range. A high initial stiffness $K_P$ (penalty stiffness) is used to hold the top and bottom faces of the interface element together. Point 2 represents the onset of damage. In single-mode delamination, the traction at point 2 is equal to the corresponding interlaminar strength of the material $\sigma_C$. This value is denoted by $T$ for the interlaminar tensile strength in mode I and $S$ and $N$ for interlaminar shear strength in mode II and III. Therefore the onset displacements can be defined as (Rice, 1968):

$$ \delta_I^0 = T/K_P, \quad \delta_{II}^0 = S/K_P, \quad \delta_{III}^0 = N/K_P \quad [1] $$
The area of the triangle 0-2-3 shows the energy released at point 3. At point 4, the energy release rate reaches to the critical value. For any respective displacement larger than point 4, the interface is unable to transfer any tensile or shear loads and all the penalty stiffness revert to zero (point 5). This means that all the available interfacial fracture energy has been completely used at point 4. The area under the traction-relative displacement curves for Mode I, II or III is the relative critical energies released at failure $G_{IC}$, $G_{IIC}$, $G_{IIIC}$ respectively (Camanho et al., 2001).:

$$G_{IC} = \int_{0}^{\delta_{I}^{f}} \sigma_{z} d\delta, G_{IIC} = \int_{0}^{\delta_{II}^{f}} \tau_{xz} d\delta, G_{IIIC} = \int_{0}^{\delta_{III}^{f}} \tau_{yz} d\delta \quad [2]$$

<figure><img src="image_4.png" /><figcaption>Figure 3. Bilinear constitutive model (Camanho et al., 2001).</figcaption></figure>

## 5. Numerical simulation

Numerical finite simulations has been performed in ABAQUS/Explicit (version 6.5) using both linear and nonlinear approaches like geometric nonlinearity and constitutive brittle cracking model for ice. The basic ice properties and brittle cracking parameters are given in Table 2.

The ice beam and aluminium plate were modeled with 8-node linear brick, reduced integration elements. The interface between the ice and aluminium was modeled with cohesive elements with zero thickness that share nodes along the interface. The relevant cohesive material data are summarized in Table 3.

<table><caption>Table 2. Ice mechanical and brittle cracking properties used in the FE model.</caption><thead><tr><td>Ice properties</td><td>Values</td></tr></thead><tbody><tr><td>Density</td><td>897 (kg/m3)</td></tr><tr><td>Elasticity modulus</td><td>9000 (MPa)</td></tr><tr><td>Poisson ratio</td><td>0.31</td></tr><tr><td>Cracking failure stress</td><td>1.4 (MPa)</td></tr><tr><td>Mode I fracture energy</td><td>7.11E-4, 2e-3 (N/mm)</td></tr><tr><td>Post cracking shear behavior model</td><td>Power law with 1.0 power factor</td></tr></tbody></table>
<table><caption>**Table 3.** Cohesive material properties used in the FE model for Ice/Al interfaces.</caption><thead><tr><th>Abaqus option</th><th>Interface properties</th><th>Values</th></tr></thead><tbody><tr><td>Elastic type = traction</td><td>Cohesive layer modulus in all 3 directions, K<sub>nn</sub>, K<sub>ss</sub>, K<sub>tt</sub></td><td>1.0E6 (N/mm3)</td></tr><tr><td rowspan="3">Damage initiation, Criterion = quads</td><td>Ultimate strength in tensile, T</td><td>0.8 (N/mm2)</td></tr><tr><td>Ultimate strength in mode II, S</td><td>0.8 (N/mm2)</td></tr><tr><td>Ultimate strength in mode III, N</td><td>0.8 (N/mm2)</td></tr><tr><td rowspan="3">Damage Evolution, Type = energy</td><td>Normal mode fracture, G<sub>IC</sub></td><td>1.0 E-3 (N/mm)</td></tr><tr><td>Shear mode fracture, G<sub>II C</sub></td><td>2.0E-3 (N/mm)</td></tr><tr><td>Shear mode fracture, G<sub>III C</sub></td><td>2.0E-3 (N/mm)</td></tr></tbody></table>

## 6. Results and discussion

Table 4 shows a comparison between results obtained using the FEM models and the classical beam theory (the ice mass on the aluminium plate was assumed as a composite material). It can be observed that results from the beam theory are the same as results from linear numerical model. Also, the results from nonlinear models are approached to the experimental results. Differences between the linear FEM model and experimental results can be explained by the fact that the adherence and the cohesion between the ice and substrate as well as nonlinear behaviour of ice are not considered with linear FEM model. Therefore, as already pointed out, to obtain the good results in a better and more realistic model, the material nonlinear analyzing with the cohesive material approach was used. The results showed that both ABAQUS/Explicit brittle cracking model and cohesive material model can be used to model the de-icing phenomenon. As we expected, the simulation results show that for the model with 2 mm of ice, delamination was not observed, and for 5 mm of ice a little delamination was observed, while for 10 mm of ice the model exhibit progressive delamination. Table 5 shows that the values of shear stress from the model using brittle cracking and cohesive material is approximately the same value which was obtained from centrifuge ice adhesion reduction test in AMIL laboratory (0.47 ± 0.2 MPa) (Laforte et al, 2005). Figure 4 shows the shear stress distribution near the de-icing zone.

## 7. Conclusion

The good agreement between the experimental and numerical results indicates that the proposed approach can capture the onset of delamination in interface where the exact location of an initial crack may be difficult to determine a priori. It is concluded that the approach proposed here can be used in the failure prediction of ice-substrate structures when bending is the leading failure mechanism. Results from the finite element simulations correlated reasonably well with the analogous experimental measurements performed using ice/aluminium specimens.
<table><caption>**Table 4.** Comparative results along the monitored path.</caption><thead><tr><th rowspan="2">Thickness<br>s<br>(mm)</th><th rowspan="2">Force<br>(N)</th><th colspan="5">Normal stress σ<sub>11</sub> at bottom face in Al(MPa)</th></tr><tr><th>Exp.</th><th>The.</th><th>FEM<br>(linear)</th><th>FEM<br>(nl geom.)</th><th>FEM<br>(Brit./coh)</th></tr></thead><tbody><tr><td>2</td><td>343.35</td><td>37.03</td><td>43.27</td><td>43.37</td><td>33.19</td><td>34.90</td></tr><tr><td>5</td><td>329.20</td><td>35.21</td><td>32.44</td><td>32.47</td><td>30.46</td><td>34.13</td></tr><tr><td>10</td><td>297.63</td><td>26.25</td><td>22.27</td><td>22.34</td><td>22.00</td><td>24.36</td></tr></tbody></table>

<table><caption>**Table 5.** Longitudinal shear stress (S<sub>13</sub>).</caption><thead><tr><th rowspan="2">Thickness<br>(mm)</th><th rowspan="2">Force<br>(N)</th><th colspan="4">Longitudinal shear stress (S<sub>13</sub>) through the ice<br>(MPa)</th></tr><tr><th>Theoretical</th><th>FEM<br>(linear)</th><th>FEM<br>(nl geom.)</th><th>FEM<br>(Brit./coh)</th></tr></thead><tbody><tr><td>2</td><td>343.35</td><td>4.53</td><td>2.029</td><td>5.02</td><td>0.65</td></tr><tr><td>5</td><td>329.20</td><td>4.16</td><td>1.69</td><td>4.03</td><td>0.69</td></tr><tr><td>10</td><td>297.63</td><td>3.62</td><td>1.87</td><td>2.70</td><td>0.52</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Longitudinal shear stress for 10 mm of ice.</figcaption></figure>

## 8. References

Camanho, P. P., Dávila, C. G., and Ambur, D. R.; «Numerical simulation of delamination growth in composite materials», NASA-TP-211041 (2001).

Laforte C., Beisswenger A., «Icephobic Material Centrifuge Adhesion Test», 11 th International Workshop of Atmospheric Icing on Structure, Montréal, June 2005, 6 pp.

Rice J. R., *A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks*, Journal of Applied Mechanics, pp. 379-386 (1968).

Rots J.G., Blaauwendraad J., «Crack models for concrete: discrete or smeared? fixed, multi-directional or rotating?», Heron, **34** (1): 1-59 (1989).