Materials Science Forum Vols. 638-642 (2010) pp 2761-2765 © (2010) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.638-642.2761
X-ray computed tomography based modelling of polymeric foams J.G.F. Wismans a, J.A.W. van Dommelen b, L.E. Govaert c and H.E.H. Meijer d Materials Technology Institute, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands a
[email protected],
[email protected],
[email protected], d
[email protected]
Keywords: computed tomography, microstructure, polymer foam, viscoplasticity.
Abstract. A hybrid numerical-experimental approach is used to characterize the macroscopic mechanical behaviour of polymer foams. The method is based on characterization of foams with Xray Computed Tomography and conversion of the data to Finite Element (FE) models. Results of FE analyses revealed that plasticity has a large influence on the mechanical response of these structures. Introduction The macroscopic constitutive behaviour of polymeric foams is the result of a subtle interplay between the intrinsic material behaviour of the polymeric material and the complex microstructure that is present in the foams. The macroscopic constitutive behaviour of polymeric foams is determined by (i) the intrinsic constitutive behaviour of the polymeric material and (ii) the complex microstructure. The intrinsic behaviour of glassy polymers is strain-rate dependent and should accurately describe large deformations, up to plastic yield, followed by strain softening and strain hardening [1-2]. Because of the 3D microstructure, local deformation mechanisms exist that determine the global behaviour. The macroscopic properties are directly dependent on the local microstructure and the deformation and localisation mechanisms that occur at this level. When loaded in compressions, foams show bending of cell walls in the elastic regime, followed by plastic deformation which results in a plateau region [3]. Since the number of experiments that are available to characterize the macroscopic behaviour of polymeric foams is limited, finite element (FE) analyses based on the actual structure of foams are conducted. High resolution X-ray Computed Tomography (CT) is used to non-destructively determine the actual microstructure of foams [4, 5] and to investigate the deformation mechanisms in-situ [6]. At first, the segmented 3D reconstruction of a closed cell foam is converted into a 2D mesh and used in FE analyses to determine the best combination between (i) model size, (ii) element type and (iii) element size where contact between cell walls is considered and the models are loaded up to large deformations. A non-linear elasto-viscoplastic material model, the Eindhoven Glassy Polymer (EGP) model [7], is then used in combination with a representative model to determine the influence of strain-rate and viscoplasticity. Subsequently, this approach is applied to a 3D structure of open-cell foam resulting in a 3D tetrahedron FE model. All analyses are performed to determine the mechanical response in compression.
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Modelling of polymer foam Microstructure. An X-ray CT machine (Nanotom system | GE Sensing and Inspection Technology) is used to characterize the structures of both the closed and open-cell foams. A resolution of 3.75 µm is used to accurately capture the microstructure within the projections. Thereafter, segmentation is applied so that the reconstruction to only remain with the microstructure of the polymer foam in 3D remains, see Fig. 1. In order to determine the influence of model size, element size and element type, randomly chosen slices are taken. From these slices, multiple models, all with different sizes, are randomly taken and converted into 2D FE models [8].
Figure 1: Conversion from the segmented 3D reconstruction to 2D models. Different model sizes are represented by black lines. Statistical analyses are performed on models with equal size, corresponding to different slices. The influence of the element type is also determined by creating FE models that are based on triangular and a combination of triangular and quadrilateral elements [8]. Furthermore, the effect of element size is investigated by increasing the element size from 3.75 to 7.5, 11.25 and 15 µm, where for all models, the volume fraction is kept the same. Since the models are loaded in the regime where contact between cell walls occurs, nodes of elements are not allowed to penetrate other elements. In addition, symmetry boundary conditions are applied to the top, bottom and left boundary such that nodes initially and during deformation remain on the boundary and are not allowed to penetrate this boundary. The right boundary is considered as e free boundary. For all non-linear elastic analyses the Hermann element formulation is used to prevent locking of the triangular elements. Constitutive model for glassy polymers. In this study, the influence of the real microstructure in combination with a viscoplastic material model is investigated. A non-linear elasto-viscoplastic constitutive model, referred to as the EGP model [7], is used to investigate the effect of intrinsic material behaviour and the applied strain rate on the mechanical response of these structures. In this study material properties of polycarbonate are taken [7]. To investigate the influence of intrinsic material behaviour, also a non-linear elastic material model is incorporated which corresponds to the initial stiffness of the EGP model. For these analyses the viscosity is increased, to eliminate the contribution of the dashpot in the model.
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Results 2D foam structure. Fig. 2 shows the deformation stages of a model with representative geometric properties and sufficiently fine discretization. The deformation mechanisms found are typical for foam structures, namely bending and buckling of cell struts [3]. The study on the 2D microstucture of the closed cell foams revealed that increasing the element size stiffened the mechanical response of the structures. Since using coarser elements results in the loss of detail, the remaining structure is thickened due to the equivalence in volume fraction. The most ideal discretization was found to be 3 to 4 elements over the thickness. Convergence to a equivalent solution for varying the model size was found for models with at least 8 cells in the width and height of the model. In contrast to the largest models, the smaller ones showed a larger spread in apparent properties where the mean response converged to the mean response of the largest models. For the latter model, boundary effects are reduced and significant cells are taken within the volume, resulting in a representative structure.
(a) ε = 0
(b) ε = 0.24
(c) ε = 0.48
Figure 2: Deformation of a 2D foam microstructure, using a non-linear elastic model, showing typical deformation mechanisms, e.g. bending and buckling of cell struts. The element type used did not have a significant influence on the solution, where it should be noticed that this was only verified with a non-linear elastic material model. When plasticity is introduced this could influence the numerical stability of the triangular elements significantly. The model shown in Fig. 2 is subsequently used in analyses incorporating the non-linear viscoplastic material model [7] to determine the effect of strain rate and intrinsic material properties. The mechanical responses of these analyses are presented in Fig. 3. The elastic response of the EPG model is given by the dashed line, where for this analysis the viscosity in the EGP was increased. Furthermore, the true strain rates applied ranged from 10-3-100 s-1 for the analyses performed with the EGP material model. The results from the EGP model reveal a clear deviation from the elastic response resulting in an earlier transition to the plateau region compared with the elastic response. Results show that plastic localization in the cell struts have a strong influence on the macroscopic response, where the strain rate has less influence. 3D foam structure. To capture the local deformation mechanisms and global mechanical responses of an open cell foam, a 3D model was subjected to this approach. The reconstructed data are segmented and automatically converted into a tetrahedron based FE model (see Fig. 4). Thereafter, a compression load was applied by a rigid plate. Furthermore, boundary conditions are applied analogous to the ones described in the 2D case. In the preliminary simulations, a non-linear elastic material model was incorporated. The deformed state is shown in Fig. 4, and was also observed in in-situ observations of compression on the same foam.
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(a)
(b) t0
(c) t1
(d) t2
Figure 3: (a) Influence of the intrinsic material behaviour and strain rate on the mechanical response of a 2D micro-structure of an open cell foam and (b)-(d) the corresponding deformed structures.
(a) (b) Figure 4: FE model (110.000 elements) of (a) an open cell polymer foam and (b) its deformed state.
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Conclusions and Recommendations A experimental-numerical approach has successfully been applied to polymer foams in order to characterize the mechanical response and to determine the influences of this approach. The results for a 2D foam showed that by using this approach, the influence of intrinsic material behaviour can be determined. Plasticity was found to have a large influence on the mechanical response of these foams. Furthermore, preliminary results showed that this approach is applicable to 3D structure. For future research, a non-linear viscoplastic material model will be adopted in 3D analyses. Acknowledgements This research is supported by the Dutch Technology Foundation STW, applied science division of NWO and Technology Program of the Ministry of Economic Affairs (under grant number 07345). References [1] H.G.H. van Melick, L.E. Govaert and H.E.H. Meijer, Polymer, Vol. 44 (2003), pp. 3579-3591. [2] T.A. Tervoort and L.E. Govaert, Journal of Rheology, Vol. 44, (2000), pp. 1263-1277. [3] L.J. Gibson, and M.F. Ashby, Cellular Solids, Structure and Properties, 2nd ed., Cambridge University Press: Cambridge (1999). [4] S.R. Stock, International Materials Reviews, Vol. 53 (2008), pp. 129-181. [5] J.G.F. Wismans, J.A.W. van Dommelen, L.E. Govaert, H.E.H. Meijer and B. van Rietbergen, Journal of Cellular Plastics, Vol. 45 (2009), pp. 157-179. [6] S. Youssef, E. Maire and R. Gaertner, Acta Materialia, Vol. 53 (2005), pp. 719-730. [7] E.T.J. Klompen, T.A.P. Engels, L.E. Govaert and H.E.H. Meijer, Macromolecules, Vol. 38 (2005), pp. 6997–7008. [8] D. Ulrich, B. van Rietbergen, H. Weinans, and P.J. Rüegsegger, Journal of Biomechanics, Vol. 31 (1998), pp. 1187–1192.
