XXIV ICTAM, 21-26 August 2016, Montreal, Canada

MODELLING THE CRUSHING BEHAVIOR OF A CERAMIC BRITTLE FOAM

Omar Kraiem ∗1, Nicolas Schmitt1, Amine Neffati2, and Han Zhao1,3

1LMT-Cachan UMR 8535, ENS Cachan, CNRS, Universite Paris Saclay, Batiment Leonard de Vinci, 61 avenuedu President Wilson, 94230 Cachan,

2TN International, 1 Rue des Herons, 78180 Montigny-le-Bretonneux,3Sorbonne Universites, UPMC Univ Paris 06, UFR 919, 75005 Paris, France.,

Summary This paper presents some experimental results and numerical computations carried out to model the quasi-static crushing behaviorof an open-cell ceramic foam (OCCF). The aim was to identify a continuum model able to predict the mechanical response under complexcompression loading paths. Different multiaxial compression tests have been performed to characterize the 3D-response of the material,which is slightly anisotropic, strongly non-linear and dependent on the compaction level. It has been shown that the Deshpande-Fleck (DF)yield criterion which depends on both the mean stress and the von Mises effective stress fit the experimental data well. The original DFmodel developed for foams has been modified by introducing a dependence of the plastic Poisson’s ratio with the plastic volumetric strainto improve the prediction of the radial expansion. It has been implemented into the finite-element code LS-Dyna via an usermat subroutine.Results of a simulation of a lab-scale shock absorber subjected to compression are in close agreement with the experimental results andvalidate the model robustness.

INTRODUCTIONOpen-cell ceramic foams (OCCF) constitute a new promising lightweight materials combining interesting physical and

thermo-mechanical properties [1]. In particular their ability to accommodate large deformation when it is confined makesthem good candidates for energy absorption engineering applications. When subjected to an impact, a confined foam part issubjected to a multiaxial stress state, which is more complex than the uniaxial compression state. Consequently, there is aneed to characterize the behavior under multiaxial loadings to identify a mechanical model.

METHODSTo do that, an experimental campaign including multiaxial compression tests was conducted to reproduce the stresses

present during the crushing impact. In addition, SEM and X-ray tomographic observations were made. In this paper onlyquasi-static loadings are described. The experimental results were obtained on cylindrical and cubical specimens of OCCF(ρ = 280 g·cm−3). Uniaxial compression tests with and without lateral confinement on the cylindrical specimens wereperformed with universal hydraulic testing machine (Instron). An ex-situ quasi-static compression test was also performed onan cylinder. To investigate their failure mechanisms, 3D images of the progressive degradation of the sample are reconstructedusing X-ray micro-computed tomography analysis. The multiaxial loadings program consisted of three types of triaxial testssuch as hydrostatic compression, oedometric compression and proportional compression. These tests were conducted withCanto’s compaction device [4] installed in the triaxial testing machine ASTREE. This machine comprises six independentactuators paired up along the three perpendicular directions pushing the blocks to reduce the hole, each one sliding relativeto the others. During the triaxial tests, the imposed displacements on the cubical specimen in each direction were measuredby laser displacement sensors. Based on the experimental data a slightly modified Deshpande-Fleck model [2] has beenidentified. The modification of the model has been implemented as a user-defined material model subroutine in the finiteelement code LS-DYNA [3]. Then simulations by finite element method of crushing tests on a metallic confined OCCF parthave been carried out in order to validate this model.

RESULTS AND DISCUSSIONFor the multiaxial characterization, under hydrostatic compression loading (ISOU), it was observed that the compressive

strength in the foaming direction is higher than that on the two other directions indicating an isotropic transverse behavior ofthe ceramic foam. Common to most ductile foams, when lateral displacement was prevented by metallic confinement (OEDOconditions), OCCF showed a typical compressive response with the three distinctive regions: at first a short elastic rangewith a brittle failure, then a crushing plateau and finally a densification with an increase in stiffness and stress (Fig. 1(a)).The confining pressure remained constant after the first damage of the specimen and then it increased as the volumetricstrain increase. This result indicates also that the foam displays a non-constant plastic Poisson’s ratio. The non-proportionalcompaction tests (TRIAX) consisted to impose an axial displacement at a constant rate while keeping constant the lateral force.For different confining pressures, this loading path permitted to show that the load-carrying capacity of the OCCF increasessignificantly with the confining pressure. The analysis of the 3D macroscopic images and SEM micrographs has enabledtracking the mechanisms of deformation during the test (Fig. 1(b)). The images have shown that the non linear deformationof the OCCF are controlled by the appearance of crushing bands in the sample which resulted from the fragmentation of thefoam sample into parts becoming more and more smaller until they are transformed into powder at the end of the densification.These observations were also confirmed by a 2D microscopic observation at small fine scale using SEM. In order to estimatethe failure surface of the ceramic foam in the different multiaxial loading conditions, the experimental data from all the tests

∗Corresponding author. Email: kraiem@lmt.ens-cachan.fr

may be plotted on a σm − σe diagram where σe is the Von Mises effective stress and σm is the mean stress(Fig. 2(a)). Theobtained behavior are well described by the elliptic yield criterion in the compressive zone of the space σm − σe defined bythe Deshpande-Fleck proposal

Φ = σ2 − Y 2 =1

[1 + (α(εv)3 )2][σ2e + α(εv)

2σ2m] − Y 2 (1)

where the parameter α defines the shape of the yield surface. To account for a good description of the hardening of the materialduring the densification, the initial Deshpande-Fleck model was modified to better predict the radial inelastic expansion in theplastic domain by introducing a dependence of the variable shape factor of the yield surface on the volumic plastic strain. Toassess the model identification, the crushing of a small metallic confined OCCF part has been simulated by the finite elementmodel and the numerical results compared with the experimental data obtained during a crushing test under quasi-staticcondition. Fig.2(b) shows the comparison of the force versus displacement curves from both the testing and the simulation. Agood realistic description of the test has been obtained.

CONCLUSIONSIt is found that the presence of a crushing plateau at high level of stress under confined case permits to the open-cell ceramic

foam to absorb a significantly quantity of energy during an impact. Its non-linear mechanical behaviour is slightly transverselyisotropic but it can well described with the isotropic DF model under compressive loadings. Further investigation will berequired to model the transversely isotropically behavior of the OCCF and more specifically the strong tension-compressionasymmetry observed for this material before this model can be used in engineering studies.

(a) Stress-strain curves under uniaxial (OEDO) and hydrostaticcompression (ISOU) tests

(b) SEM and X-ray tomography observations of the failuredeformation mechanisms under confined uniaxial compression

Figure 1: Experimental results of the OCCF material

(a) Yield surface of the DF model (b) Comparison of experimental and numericalforce-displacement curves from inclined crushing test

Figure 2: Numerical results obtained with the DF modelReferences

[1] Michael Scheffler, Paolo Colombo: Cellular Ceramics: Structure, Manufacturing, Properties & Applications. Weinheim: Wiley-VCH, 2005.[2] V.S. Deshpande and N.A. Fleck: Isotropic constitutive models for metallic foams. Journal of the Mechanics and Physics of Solids 48:1253 - 1283, 2000.[3] J.O.Hallquist,Ls-dyna theoretical manual,1998[4] Rodrigo C.: Theoretical and experimental study of the compaction and sintering processes of polytetrafluoroethylene (PTFE). Joint PhD thesis, ENS

Cachan / USP, Cachan (France), 1983.