Finite Element Modelling Of Core

Composite Structures 86 (2008) 227–232

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Finite element modelling of core thickness effects in aluminium foam/composite sandwich structures under flexural loading M. Styles *, P. Compston, S. Kalyanasundaram Department of Engineering, Faculty of Engineering and Information Technology, The Australian National University, Engineering Building, 32, North Road, Canberra, ACT 0200, Australia

a r t i c l e

i n f o

Available online 16 March 2008 Keywords: Aluminium foam Sandwich structure Finite element modelling Flexural loading Strain distribution

a b s t r a c t This paper models the flexural behaviour of a composite sandwich structure with an aluminium foam core using the finite element (FE) code LS-DYNA. Two core thicknesses, 5 and 20 mm, were investigated. The FE results were compared with results from previous experimental work that measured full-field strain directly from the sample during testing. The deformation and failure behaviour predicted by the FE model compared well with the behaviour observed experimentally. The strain predicted by the FE model also agreed reasonably well with the distribution and magnitude of strain obtained experimentally. However, the FE model predicted lower peak load, which is most likely due to a size effect exhibited by aluminium foam. A simple modification of the FE model input parameters for the foam core subsequently produced good agreement between the model and experimental results. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Metallic foams offer good stiffness and strength to weight ratios, high energy absorption, good sound damping, electromagnetic wave absorption, thermal insulation and non-combustibility [1,2]. This range of properties has increased the interest in metallic foams as an alternative to polymer foams as the core material in composite sandwich structures. To date, attention has focussed on the potential for significant impact energy absorption in structural applications. Cantwell et al. [3] and Reyes and Cantwell [4] reported relatively high energy absorption in low and high velocity impact testing of sandwich structures with aluminium foam cores and fibre-reinforced composite skins. A significant amount of the energy absorption was attributed to the ductile failure modes in the aluminium foam. Compston et al. [5] also found that an aluminium foam sandwich structure exhibits less localised damage and significant out-of-plane, ductile deformation under low velocity impact loading compared to a polymer foam counterpart, which exhibited catastrophic failure at the impact point. Characterisation of surface strain contours during post-impact loading also showed lower strain concentrations at the impact point in the aluminium foam sandwich structure, which suggests increased damage tolerance compared polymer foam sandwich structures [5]. While impact energy absorption and damage tolerance are desirable properties, the main in-service attribute of a sandwich structure is high bending stiffness with minimal increase in weight * Corresponding author. Tel.: +61 2 6125 3072; fax: +61 2 6125 0506. E-mail address: [email protected] (M. Styles). 0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.03.024

due to the low density core. Therefore, the quasi-static flexural behaviour of a sandwich structure that includes aluminium foam must be predictable when designing for future applications. However, the cellular nature of commercially available aluminium foam presents a significant problem when attempting to predict mechanical behaviour [6]. The properties can depend on the relative magnitude of the average cell size and the geometry of the specimen. The bulk material behaviour of aluminium foam has exhibited a distinct size effect [7,8], where the compressive and shear strength properties were found to reach a plateau level as the ratio of specimen size to cell size increased [7]. Chen and Fleck [9] also found constraints on the foam core from skin sheets also resulted in a size effect. Two approaches based on finite element methodology have been used to model the mechanical behaviour of complex cellular foam materials. The first replicates the detailed topology and structure of the foam using repeated unit cells imitating the geometry of individual cell faces. This method has been implemented with a number of geometries and crushing techniques [10–15]. A second approach models the foam as a continuum, using empirical data to generate a standard yield surface to reproduce the bulk properties of the material. A number of constitutive models have been developed that attempt to incorporate the various stages of deformation the metal foam can display [16–20]. Some of these constitutive models have been implemented as material models within FE packages and have been used in a range of modelling studies [21–23]. However, it is notable that current FE models were developed for metallic foams as filler materials within energy absorbers, where compression forces dominate. The suitability of these models for

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structural applications, especially within composite sandwich structure where flexural loading dominates, needs further investigation. The current study examines the ability of an FE model to predict the flexural behaviour of a composite sandwich structure with an aluminium foam core. Sample with core thickness of 5 and 20 mm were modelled in 4-point bending. The FE model utilises an existing LS-DYNA material model [24] developed for aluminium foam energy absorbers and based on the Deshpande–Fleck constitutive model [19]. Although this study is limited to quasi-static behaviour, the possible applications of this material system are likely to involve dynamic loading, making the use of the explicit FE code LS-DYNA appropriate. A composite damage material model that incorporates matrix cracking, compressive failure and fibre breakage is used for the sandwich skins. The results will be compared with load–displacement behaviour, failure modes and fullfield strain contours obtained from a previous experimental study of flexural behaviour of the same sandwich structures [25]. The strain contours in the previous study were obtained using an advanced optical 3D image correlation system, and showed the strain progression during the flexural loading. This strain data will be particularly useful for FE model validation. 2. Numerical implementation of the constitutive model Deshpande and Fleck [19] proposed a constitutive model for the plastic behaviour of metal foams including differential hardening effects from hydrostatic stress on the shape of the yield surface. The model is an extension of the von Mises yield criterion with the hydrostatic stresses incorporated in the equivalent stress term. This constitutive model was implemented by Reyes et al. [24] as a material model in LS-DYNA which was utilised in this study. The yield function is defined by ^Y 60 U¼r

ð1Þ

where, the yield stress Y can be expressed as Y ¼ rp þ Rð^eÞ

This constitutive model has been implemented as Mat_Deshpande_Fleck_Foam material model in LS-DYNA [27]. 3. Finite element model The results from the finite element model were obtained by using the commercial explicit code LS-DYNA version 971 on an SGI Altix UNIX platform. The 4-point bending loading conditions were replicated using four rollers of rigid shell elements with the material properties of tool steel. The sandwich beam was modelled with symmetry conditions along the width to reduce the computational time. The FE pre-processing package Hyperworks from Altair was used to develop the mesh and input deck for LS-DYNA. 3.1. Material model The material parameters for the aluminium foam were obtained by performing compressive tests on cubes of ALPORAS foam (30 mm sides). Fig. 1 illustrates the experimental stress strain curve and the curve fit used to obtain the material model parameters of plateau stress (rp), a2, c, and b. The values for different parameters are given in Table 1. The parameter Cfail is the value of the failure strain of foam and is used to remove failed elements during simulation. The damage progression in the composite skin was modelled using the composite material model Mat22 provided by LS-DYNA [27]. This is a model for orthotropic composites and can model matrix cracking, compressive failure and final failure due to fibre breakage. The mechanical properties for the composite skin were obtained from manufacturer’s data sheets for TwintexÒ. 3.2. Element and contact definitions The default eight-node brick element was used for the core with a one point reduced integration scheme and the LS-DYNA stiffnessbased hourglass control. Skin layers were modelled with shell elements using the Belytschko–Tsay formulation. The interface

ð2Þ

Here, Rð^eÞ is the strain hardening term and ^e is the equivalent strain [24]. Deshpande and Fleck [19] define the equivalent stress r ^ as r ^2  h

1 1 þ ða=3Þ

2

  i r2vm þ a2 r2m

ð3Þ

Here, the von Mises effective stress is rvm, rm is the mean stress and the shape of the yield surface is defined by the parameter a. The expression for a is given by a2 ¼

9 ð1  2tp Þ 2 ð1 þ tp Þ

ð4Þ

where, tp is the plastic coefficient of contraction. Details of the method used to implement this model as an integration algorithm within LS-DYNA are available elsewhere [26]. The material model requires several parameters to be obtained from compression testing. The expression for yield stress is given by 0 1 Y ¼ rp þ c

^e 1 B C þ a2 @  b A eD ^e 1  eD

ð5Þ

where rp, a2, c and b are material parameters obtained from a curve fit of the stress–strain data from uniaxial compression. The densification strain eD is determined from the density of the foam (qf) and bulk aluminium (qf0).   qf eD ¼  ln ð6Þ qf0

Fig. 1. Deshpande–Fleck yield surface curve fit of experimental compression data.

Table 1 Material input parameters for foam model Mat154 qf (g/cm3)

E (GPa)

tp

a

c (MPa)

eD

a2 (MPa)

b

rp (MPa)

CFail

0.23

1.1

0.0

2.12

3.12

2.4629

0.368

4.47

1.35

0.2

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between the core and skin materials was replicated using a tied contact type with an offset. This contact definition used a soft constraint-based formulation. This contact formulation is recommended when the material constants of the surfaces in contact have large differences in elastic bulk moduli values. No failure criterion for the interface was implemented, to replicate the lack of any delamination observed experimentally. 3.3. Load application The load was applied through motion of the top load rollers. The bottom rollers were constrained to be stationary. A prescribed velocity in the z direction was applied to the load rollers to simulate the experimental conditions. To replicate quasi-static loading experienced during testing, the following velocity field was applied [24]:  p i p dmax h vðtÞ ¼ t ð7Þ 1  cos p2 T 2T Here, T is the total time of the loading and dmax is the maximum displacement of the load rollers. This velocity field produces an initial acceleration of zero, ensuring that the loading takes place gradually. For this study, the total loading time T was 750 ms and the maximum displacement was 30 mm. This velocity field ensured that quasi-static conditions were simulated in an explicit finite element formulation. 4. Results and discussion The deformation and failure of sandwich structure models having 20 mm and 5 mm aluminium foam cores was compared with the observed experimental behaviour [25], using both force–displacement curves and full-field strain distributions. 4.1. Failure behaviour of the sandwich structure with 20 mm core The general deformation shape and failure mechanisms of the 20 mm core model compared favourably with that observed in the physical testing. Fig. 2 illustrates the final deformed shapes after approximately 25 mm of crosshead displacement. The predominant deformation mechanisms observed in the tested sample were core crushing and indentation damage under the loading rollers. There was some minor skin failure in the form of slight fracture and wrinkling. A similar deformation shape was produced by the FE model. Core crushing was observed under the top rollers with little distortion of core elements elsewhere in the beam. There was some minor deformation of the skin elements following the core indentation but no significant skin wrinkling was observed. Fig. 3 depicts the load–displacement curves recorded during the physical testing and as produced by the numerical model. The general shape of the model curve matches the experiment with an initial linear elastic region followed by a decrease in slope up to a first peak load point. In the curve from the physical testing, this peak load point is followed by a plateau region. In this region the load level is reasonably constant with some small variation towards

Fig. 3. Comparison of the load–displacement curve from FE model with the curve from experimental work [25] for the sandwich structure with the 20 mm thick core.

the end of the test. This curve agrees with the deformation mechanisms observed; the initial peak corresponds to the first significant failure of foam cells followed by the progressive crushing and densification of the core. The fluctuation in the load magnitude may relate to the inconsistency in the cells; for example, as larger or weaker cells fail the load will drop considerably. The second part of the model curve also shows some small fluctuations throughout a semi-plateau region. The complete profile of the model curve matches well with the experimental curve. While the general shape of the curve produced by the model is in agreement with the experiment, there is a difference in the magnitude of the load. The peak load predicted by the model is 0.8 kN compared to the experimental value of 1.2 kN. This deformation behaviour has not been reported in other studies where the major emphasis of the work is on bulk compressive behaviour [22,28]. There are two likely reasons for the underestimation of the load by the model. The first is related to a size effect in the core material. Previous studies of metal foams have found a number of potentially significant size effects on material properties, with respect to the ratio of cell size to specimen size. In particular, Chen et al. [29] reported that shear response is sensitive to the thickness of the specimen, with a stronger response displayed by specimens of smaller thickness. Similar results have been discussed by Kesler et al. [6] as very important in considering sandwich panel design. The core thickness used in this investigation is less than the sample size used to generate the input parameters for the material model, and as such, a size effect may be involved. As discussed by Chen et al. [9] in an investigation of constrained deformation, the material model appears to be unable to predict the sample size effect on the strength. The inclusion of this effect is essential in developing an accurate model for sandwich structure applications. A second factor that may be contributing to the lower load prediction by

Fig. 2. Deformation in the sandwich structure with 20 mm thick core; (a) FE model and (b) observation from experimental work [25].

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the model is related to the Saint-Venant’s principle. The experimental flexural testing can be influenced by the concentrated loads of the rollers on the sample. These point loading conditions can lead to elevated stress values in the region around the points of loading or support, and thus can result in an amplified recorded load magnitude. In contrast, the compression testing for deriving the model input parameters, involves a relatively uniform stress distribution. It is suggested the combination of these issues of size effect and stress concentrations from point loads may have caused the difference in load magnitudes between the simulation and experimental results. The effect of some of the material model parameters was investigated in an attempt to match the experimental results more closely. The magnitude of the parameters used to describe the Deshpande–Fleck yield surface was increased and the resulting load–displacement curves are illustrated in Fig. 4. The parameters of plateau stress (rp), c and a2 were increased by factors of 1.5 and 2. The magnitude of the load–displacement curve increases accordingly, with the experimental curve most closely matched by the model with parameters increased by a factor of 1.5. The effectiveness of simply increasing these parameters supports the suggestion that the initial model underestimation of peak loads is, at least partially, related to the size effect. The effect of parameters in the skin material model was also investigated. The compressive, tensile and shear strength parameters were varied around the initial value without any significant effect on the behaviour of the model. Similarly the shear modulus parameter was found to have minimal effect on the model. The parameters relating to the longitudinal modulus were found to have the most effect on the magnitude of the curve. Fig. 5 illustrates load–displacement curves from models with Young’s modulus values having a very low value (5 GPa) or higher values (20 GPa) compared to manufacturer’s reported values of 15 GPa. As expected, these changes affect the initial slope of the curve and have only minimal effect on the overall magnitude of the curve. The minimal effect of varying the skin model properties suggests that the foam core material model dominates the overall behaviour of the sandwich structure for this particular geometry. 4.2. Strain distribution of the sandwich structure with 20 mm core A full-field strain distribution of the region of the sandwich structure between the load rollers was recording throughout the

Fig. 5. Load–displacement curves for the sandwich structure with the 20 mm thick core after modifying Young’s modulus for the composite skin in the FE model.

flexural testing. Fig. 6 provides a comparison of Von Mises strain contours between simulation and experiment at a crosshead displacement of 2.7 mm. This value of crosshead displacement corresponds to the initial peak load. The experimental results exhibit isolated regions of slightly higher strain dispersed throughout the sample, which can be associated with the cellular structure of the core. In the regions beneath the load rollers, small, more concentrated regions of high strain have appeared. On the right side there is a significant region of high strain in the centre of the thickness beneath the load roller. This is likely to be the site of a weak cell where initial crushing is beginning. The simulation results illustrate regions of increased strain directly under the load rollers. This region is also where the first cell failure and crushing was observed in the experiment. The remainder of the beam displays uniform regions of strain level unlike the dispersed higher strain regions seen in the experiment. This is a result of the use of the continuum material modelling method which does not include any variation in properties between elements. More importantly, the magnitude of the strain levels agrees well with the experimental strain values for most parts of the structure. Fig. 7 provides a comparison of Von Mises strain contours between simulation and experiment at a crosshead displacement of 10 mm. The simulation and experimental results indicate that the regions under the load rollers having concentrated high strain values. These regions correspond to the observed regions of core crushing. The maximum strain value in the model at this crosshead displacement was 0.436 which compares well with the maximum strain value of 0.47 observed in the experiment. The correlation of overall strain distribution between experimental and simulation results is very good. Therefore this study is the first of its kind to validate the constitutive model for sandwich foam structures through experimental observation for structures experiencing non-uniform strain fields. 4.3. Failure behaviour of the sandwich structure with 5 mm core

Fig. 4. Comparison of the load–displacement curves after modifying material parameters (plateau stress (rp), c and a2 magnitude) for the 20 mm thick aluminium foam core in the FE model.

The effect of reducing the core thickness on the model performance was investigated using a 5 mm core sample. Fig. 8 provides a comparison between simulation and experiment on the failure behaviour of this reduced core thickness. The main failure observed for this structure was skin wrinkling and fibre fracture with minor core cracking. There was no apparent crushing within the core structure. Instead, the structure exhibits plastic hinge type

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Fig. 6. Typical strain distribution at peak load (2.7 mm displacement) for the sandwich structure with 20 mm thick core; (a) FE model and (b) real-time experimental measurement [25].

Fig. 7. Typical strain distribution at 10 mm displacement for the sandwich structure with 20 mm thick core; (a) FE model and (b) real-time experimental measurement [25].

Fig. 8. Deformation shape for the sandwich structure with 5 mm thick core; (a) FE model and (b) observation from experimental work [25].

deformation behaviour beneath each load roller. The simulation results for the overall deformation behaviour matches experimental results. The simulation results also exhibit some minor compression of the core directly beneath the load rollers, and some element rotation at each beam hinge. Fig. 9 illustrates a typical load–displacement curve for this structure. The curve recorded from the experiment shows initial linear elastic behaviour followed by a decrease in slope up to a maximum load magnitude. This is followed by a sharp drop in load before reaching a plateau. This progression agrees with the deformation mechanisms observed of skin wrinkling and fibre fracture. The curve produced by the simulation follows the general shape of the experimental curve closely though there are some differences in the load magnitude. The initial stiffness response of the structure has been overestimated by the model. In contrast, the peak load produced by the model is significantly lower than experimentally measured value. In this structure where the deformation appears dominated by skin failure mechanisms, the initial slope of the curve is highly dependent on the skin properties. An overestimated skin thickness is a possible cause for the model’s high initial stiffness. This model used the manufacturer’s nominal thickness for a single ply of consolidated Twintex. This nominal thickness may be greater than the effective thickness achieved by the metal foam sandwich manufacturing process used in this study, causing the overestimated stiffness. The underestimated load magnitude was also observed in the sandwich structure with 20 mm core and can be attributed to a combination of a core size effect and a Saint-Venant’s principle effect. The core size effect in constrained deformation as observed by Chen et al. [9] is likely to be highly significant to this thin core geometry. The load–displacement

Fig. 9. Load–displacement curves for the sandwich structure with 5 mm thick aluminium foam core; from the experimental work [25], the initial FE model, and after modifying material parameters (plateau stress (rp), c and a2 magnitude) for the aluminium foam core in the FE model.

response of the model can be increased to match the experiment data by increasing the core input parameters by a factor of 2. This is higher than the factor of 1.5 used with the 20 mm core,

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Fig. 10. Typical strain distribution at 10 mm displacement for the sandwich structure with 5 mm thickness; (a) FE model and (b) real-time experimental measurement [25].

indicating the importance of size effect in developing constitutive models for the foam material. 4.4. Strain distribution of the sandwich structure with 5 mm core Fig. 10 provides a comparison of Von Mises strain contours between simulation and experiment at a crosshead displacement of 10 mm. There are regions of higher strain beneath the load rollers corresponding to the regions where skin failure and core cracking were visually observed. There is a good agreement between simulation and experiments of the strain distribution of the structure. Overall, this study has illustrated that an existing constitutive material model for aluminium foam can be effectively utilised to model the behaviour of a complex sandwich structure with two different core thicknesses under flexural loading. The model underestimated the peak load magnitude for both thicknesses. However, the general deformation behaviour and load–displacement curve shapes were well matched. The discrepancy between the load magnitudes and its possible relationship to a core size effect needs to be further investigated. The strain distributions produced by the model were in good agreement with those recorded from the experiments. Future investigations are needed to further examine the behaviour of the model across a greater range of core and skin thicknesses. 5. Conclusion A finite element model of an aluminium foam composite sandwich structure undergoing 4-point flexural testing was produced using an existing foam material model based on the Deshpande– Fleck yield surface. The ability of the model to replicate the behaviour of structures for two different core thicknesses was investigated. The performance of the model was evaluated using experimental observations including data from an optical full-field strain distribution system. The damage progression and deformation of each of the models reflected the physical testing results although the load–displacement response was underestimated. This underestimation can be attributed to the non-inclusion of the size effect in the constitutive material model. The strain distributions produced by the models matched well with the experimental contours, providing a valuable alternative method of validating the deformation performance of the models. Further investigations and improvements are required to develop the model towards a useful FE design tool. References [1] Gibson LJ. Mechanical behavior of metallic foams. Ann Rev Mater Sci 2000;30:191–227. [2] Harte A-M, Fleck NA, Ashby MF. Sandwich panel design using aluminum alloy foam. Adv Eng Mater 2000;2(4):219–22.

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