27th Risø International Symposium on Materials Science, Polymer Composite Materials for Wind Power Turbines, 2006

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1 27th Risø International Symposium on Materials Science, Polymer Composite Materials for Wind Power Turbines, 2006 SIMULATION OF CRACK INITIATION AND PROPAGATION IN AN ADHESIVE LAYER USING A MESOMECHANICAL MODEL Kent Salomonsson and Tobias Andersson Mechanics of Materials, University of Skövde P.O. Box 408, SE Skövde, Sweden ABSTRACT A finite element model of a double cantilever beam specimen is developed. The adherends are modeled using plane strain elastic continuum elements. Furthermore, the adhesive is modeled using a mesomechanical modeling technique which allows for simulation of initiation and propagation of micro-cracks. This enables the modeling of the entire process of degradation and fracture of the adhesive layer. The purpose of the present study is to compare the stress-deformation behavior in an idealized peel loading to the behavior in a double cantilever beam (DCB) specimen where the adhesive layer is deformed wilt a slight gradient along the layer. Previously performed experiments and simulations of the RVE are used as a comparison to the simulated results. 1. INTRODUCTION To study the possibility to model a structural adhesive layer using a representative volume element (RVE), a comparison is made between the behavior of a RVE and a finite element model of an entire double cantilever beam specimen. Several researchers model mode I behavior using cohesive zone models (CZM) together with either beam or shell finite elements. The disadvantage to use these methods is that the results depend entirely on the chosen CZM. A recent paper by Salomonsson and Andersson (2006), demonstrate a method using a RVE at the meso level to model the structural behavior of an adhesive. It is suggested that the model be used as a foundation of the development of a constitutive behavior controlling the CZM. With a finite element analysis

2 Salomonsson and Andersson of a RVE, initiation, propagation and coalesce of micro-cracks is simulated and interpreted as damage evolution. A brief overview of the finite element formulation is given in chapter 2, followed by the numerical results in section 3. Finally some conclusions and remarks are given in section THE FINITE ELEMENT MODEL Traditional continuum finite elements are used to model the adherends. The first 4 mm of the adhesive layer is modeled using the mesomechanical RVE developed by Salomonsson and Andersson (2006). The rest of the adhesive layer is modeled using the 4-node cohesive elements COH2D4 in ABAQUS v with a cohesive law consistent with a simulated stress-deformation relation from only one RVE. The adhesive layer consists of two types of materials, an epoxy/thermoplastic blend and a mineral. To allow for initiation and growth of micro cracks in the mesomechanical model, COH2D4 elements are used to couple all continuum elements at their respective facets. The mixed mode behavior of the cohesive law for the interface elements is controlled by the Benzeggagh-Kenane (BK) fracture criterion (Benzeggagh and Kenane, 1996). This criterion is particularly useful when the fracture energies in mode II and mode III are the same, i.e.. The criterion is given by II η I II I G% Gc + ( Gc Gc ) = G III G % (1) c II II III where G% III I II = G + G, G% = G + G% and η = 1 is a material parameter. The superscripts I, II and III indicate modes I, II and III; index c indicates the fracture energy. In pure peel and pure shear, the traction-separation laws are given in Fig. 1. σ ( δ n ) σˆ 1 k σ τ ( δ t ) ˆ τ 1 k τ δ σn δ n δ τt δ t Fig 1. The peel and shear stresses as functions of the relative separations in peel and shear, respectively. A representative volume element (RVE), cf. Fig. 2, is used to model the adhesive layer. The thickness of the adhesive layer is 0.2 mm in the experiments and this thickness is kept in the RVE. From a computational point of view, the RVE should be kept as small as possible. Based on previous studies a RVE with a length of 0.8 mm and a height of 0.2 mm is large enough to adequately capture the fracture process (Salomonsson and Andersson, 2006).

3 Simulation of crack initiation and propagation in an adhesive layer using a mesomechanical model Fig 2. The dark areas are the mineral grains and the lighter area is the polymer blend. Figure 2 illustrates the complexity of the structure of the adhesive at the meso level. The dark finite elements in Fig. 2 are given constitutive properties of mineral and the lighter elements are given properties of the polymer blend. As explained earlier, interface elements couple all continuum elements at their respective boundaries and interface elements that lie within mineral regions are thus given properties corresponding to mineral clusters. In a similar manner, interface elements that lie in the polymer blend region are given polymer properties. At the boundary between mineral and polymer blend as well as between the RVE s, the interface elements are given polymer blend properties. A schematic setup of the simulation model is given in Fig. 3, where the shaded areas represent RVE s and the black region is where the COH2D4 cohesive elements are used. The load is applied by a controlled displacement,. Furthermore, the load point is fixed in the horizontal direction. These are the only applied boundary conditions. Fig 3. Schematic of the model. indicates the controlled displacement. The shaded regions correspond to RVE s and in the black region COH2D4 cohesive elements are used. 4. NUMERICAL RESULTS The adherends are modeled using plane strain continuum elements with Young s modulus 210 GPa and Poisson s ratio 0.3. In the RVE s there are two types of material models, one for the polymer blend region and one for the mineral regions. Continuum elements in the polymer blend are given elasto-plastic properties with Young s modulus 2 GPa, Poisson s ratio 0.35 and the hardening modulus 200 MPa together with the yield stress 55 MPa. For the mineral regions, the continuum elements are linear elastic with Young s modulus 55 GPa and Poisson s ration The interface elements within the polymer blend are given the following properties: 5 k σ = 868 GPa/m, = mm, Γ = 760J/m 2 5, k τ = 844GPa/m, = mm and δ σ s σ c Γ τ c = 2300 J/m 2. The interface elements within the mineral regions are given the following properties: k σ = 50 GPa/m, δ σ s = mm, Γ σ c = 760 J/m 2 4, k τ = 25 GPa/m, δ τ s = 410 mm and Γ τ c = 2300 J/m 2. δ τ s

4 Salomonsson and Andersson An interesting aspect, observed in experiments, is that micro cracks initiate some distance from the crack tip typically mm. Later on in the process, the micro cracks coalesce to form macroscopic cracks; one which opens the original crack tip and some others that propagate through the adhesive. A similar process is observed in the simulation. Figure 4 illustrates an initiation of a micro crack. Furthermore, it is seen that other micro cracks initiate around the larger ones. It seems that micro cracks initiate in and around mineral areas, as is also observed from scanning electron microscope (SEM) images, cf. Salomonsson and Andersson (2006). Fig. 4. The simulated DCB-specimen crack tip. The white regions in the blue region correspond to micro cracks. As in the experimental analyses, the energy release rate is derived by measuring the reaction force, F, and the rotation, θ, at the loading point (Andersson and Stigh, 2004). The energy release rate is given by J I 2 F θ = (1) b where b is the width of the specimen. Using an inverse method, the stress-deformation relation is derived from the numerical results by differentiating JI with respect to the crack opening displacement w. The result of the simulation is given in Fig. 6 together with a corresponding experimental curve of the stress-deformation relation. A curve representing the results from a simulation of only one RVE is also given in Fig. 6 for comparison. 25 (MPa) Meso RVE Experiments w (mm) Fig. 6. The stress-deformation relation from the simulation of the DCB-experiment (dotted curve), the result from a simulation of only one RVE (dashed curve) and the corresponding experimental curve (solid line).

5 Simulation of crack initiation and propagation in an adhesive layer using a mesomechanical model Exactly the same RVE s are used in the DCB-model as is used for the RVE simulation. An interesting difference is observed when comparing them. The curvature at the degradation part of the curves is opposite to each other. This bump is discussed in Salomonsson and Andersson (2006) where it is argued that the undamaged adhesive in front of the process zone in the DCB-specimen could cause the difference between the behavior of the RVE and the experiment. Furthermore, in the RVE simulation, the upper boundary of the RVE is controlled to have the same vertical displacement. This is not the case when analyzing the DCB-specimen; here the RVE s are loaded with the curvature as well as a separation of the adherends. 5. CONCLUSIONS AND FURTHER REMARKS A finite element model of a DCB-specimen with the possibility of micro-crack initiation and propagation is presented. Continuum finite elements are used in combination with cohesive elements. The adhesive structure is modeled based on scanning electron microscopy (SEM) images. In order to limit the size of the finite element model, only five RVE s are used in the beginning of the adhesive layer. The remaining layer is modeled using cohesive elements with an implemented constitutive law based on the stress-deformation relation from a single RVE. Micro-cracks are observed to initiate in and around the mineral regions. Later on in the process, these cracks coalesce to form macroscopic cracks. An inverse method is used to derive the stress-deformation relation for the simulated DCBspecimen. The results revile that the shape of the DCB simulation curve is in accordance with the experimental curve, but the parameters need to be calibrated better. It is seen that a RVE can capture almost the entire stress-deformation curve observed in experiments, but the shape of the curve is not similar. The RVE simulation comes from a simulation where the same constitutive parameters were used in both peel and shear. This is the reason in which the fit to the experimental curve is not fitted better, cf. Salomonsson and Andersson (2006). Further work on the calibration of the cohesive parameters will have to be done. REFERENCES Andersson, T., Stigh, U., (2004). The stress-elongation relation for an adhesive layer loaded in peel using equilibrium of energetic forces. International Journal of Solids and Structures 41, Benzeggagh, M. L., Kenane, M., (1996). Measurement of Mixed-Mode Delamination Fracture Toughness of Unidirectional Glass/Epoxy Composites with Mixed-Mode Bending Apparatus, Composites Science and Technology, vol. 56, pp , Mishnaevsky Jr, L. L, Schmauder, S., (2001). Applied Mechanics Reviews 54, no 1. Needleman, A., A continuum model for void nucleation by inclusion debonding. Journal of Applied Mechanics., vol. 54,

6 Salomonsson and Andersson Salomonsson, K., Andersson, T., (2006). Modeling and parameter calibration of an adhesive layer at the meso level. Submitted for publication. Xu, X.P., Needleman, A., (1994). Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids 42,