Metal foams are materials of recent development and application that show interesting combinations of physical and mechanical properties. Many applications are envisaged for such materials, particularly in equipments of passive safety, because their high capacity of energy absorption under impact conditions. The damage analysis in metallic

foams is a complex problem and must be performed in a finite strain context. Considering that compression is the dominant loading in impact situations, in this work a finite deformation simulation including damage effects of a compression test on a cellular metal sample is shown. A comparison with experimental results is also presented

Creus, G. J.

Cunda, Luiz A. B. da

Oliveira, Branca F. de

Oschner, Andreas

English

Universiti Teknologi Malaysia Institutional Repository

FINITE ELEMENT SIMULATION OF COMPRESSION TESTS ONCELLULAR METALSBranca F. Oliveira∗Department of Design and Graphic Expression, Federal University of Rio Grande do SulAv. Osvaldo Aranha, 99, sala 408, 90035-190, Porto Alegre, RS, BrazilLuiz A. B. da CundaMaterials and Construction Department, Federal University of Rio GrandeAv. Itália, km 08, Campus Carreiros, 96201-900, Rio Grande, RS, BrazilGuillermo J. CreusCivil Engineering Department, Federal University of Rio Grande do SulAv. Osvaldo Aranha, 99, 3. andar, 90035-190, Porto Alegre, RS, BrazilAndreas ÖchsnerDepartment of Materials Engineering, Universiti Teknologi Malaysia81310 UTM Skudai, Johor, MalaysiaAbstract. Metal foams are materials of recent development and application that showinteresting combinations of physical and mechanical properties. Many applications areenvisaged for such materials, particularly in equipments of passive safety, because their highcapacity of energy absorption under impact conditions. The damage analysis in metallicfoams is a complex problem and must be performed in a finite strain context. Considering thatcompression is the dominant loading in impact situations, in this work a finite deformationsimulation including damage effects of a compression test on a cellular metal sample isshown. A comparison with experimental results is also presented.Keywords: Cellular metals, Metallic foams, Damage, Gurson model, Finite elements1. INTRODUCTIONThis work continues the research on the simulation of metallic foams behavior that waspresented in earlier papers (OLIVEIRA et al., 2006, CUNDA et al., 2005, ÖCHSNER eLAMPRECHT, 2003).We understand by metal foam a material composed by a metallic matrix with internalvoids (Fig. 1). Metallic foams are increasingly used by the automotive industry, particularlyas recourse of passive safety. In that case, they act to absorb impact energy.                                                          ∗ Corresponding author. tel: +55 51 3308 4261 E-mail address: branca@ufrgs.br (Branca F. Oliveira)Figure 1 – Aluminum foam with open cell (DUOCEL®) (ÖCHSNER e LAMPRECHT, 2003).Metallic foams show mechanical behavior and physical properties that strongly differfrom those of solid materials and show interesting combinations of properties. For example,high stiffness combined with low specific weight, or permeability to gas flow combined withhigh thermal conductivity, offering possibilities of use in aerospace and automotive industries.The schematic stress-strain curve for a metallic foam in compression (Fig. 2), shows a largearea in the plateau region corresponding to high energy absorption at constant stress.Figure 2 – Stress-strain curve for a metallic foam showing large capacity of energy absorptionat constant stress (ÖCHSNER et al., 2003).In order to employ these materials as a main component in protection structures, it isnecessary to have reliable numerical methods and constitutive relations, developed andvalidated using experimental procedures that take into account the variety of mechanical,geometrical and physical properties of the foam and the base material.2. GURSON DAMAGE MODELGurson damage model was developed to describe the mechanical effect of high plasticdeformations in ductile metals. The loss of resistance is governed by the porosity level.  The(isotropic) damage variable employed is the volumetric void fraction, represented by f anddefined by f = Vv / V, where Vv is the volume of voids in a representative small volume V,corrected for effects as stress concentration, etc.; f is defined at each point of the continuum.The presence of voids alters the elastoplastic constitutive relations. The equations usuallyemployed in computational damage analyses, the Gurson-Tvergaard model (GURSON, 1977,TVERGAARD, 1981), considers a yield surface defined by023 =−=Φ yijij ss σω (1)where212321 23cosh21⎥⎥⎦⎤⎢⎢⎣⎡ +⎟⎟⎠⎞⎜⎜⎝⎛−= fpfyασααω (2)s p pij ij ij ij ij= − =σ δ σ δ                      13 (3)21321           0.1          5.11 αααα ====Uf(4)and σij are the Cauchy stresses, σy the yield stress in simple tension.  α i are materialparameters. The parameter fU = 1/α1 is the maximum volumetric void fraction admissiblebefore rupture in the absence of pressure. Another possible interpretation for the α1 and α2parameters is that they work as multipliers acting on porosity f and pressure p, respectively.In Fig. 3, yield surfaces for different levels of void content are shown, in a plot ofnormalized deviatoric stress versus normalized pressure.Figure 3 – Yield surface for a porous material: influence of volumetric void fraction.It can be seen that the elastic domain depends on the hydrostatic pressure. When thevolumetric void fraction f decreases, decreases the influence of pressure, leading to a largerelastic domain.  For f = 0, the model reduces to the von Mises model, which is independent ofpressure.  It should be noted here that in the absence of hydrostatic pressure, the coefficient ωreduces toUfff −=−= 11 1αω (5) The plastic strain rate tensor is given by,ijpijD σλ ∂Φ∂= & (6)and the equivalent plastic strain rate is defined bypijpijp DD32=ε& (7)The basic mechanisms of damage evolution are nucleation, growth and coalescence ofvoids. Nucleation occurs mainly due to material defects, in the presence of tension. Growthoccurs when the voids (preexistent or nucleated) change their size according to the volumechange in the continuum.Coalescence is related to the fast rupture process that occurs after that the volumetric voidfraction reaches a limit, indicated by fC. Coalescence consists in the union of neighbor voidsdue to the rupture of a ligament.The equations that govern damage evolution are modeled in a simplified form as follows.First, it is assumed that total void rate is given by⎩⎨⎧>≤+=CcCgnffffffff                                  &&&& (8)where nf&  is the void nucleation rate, gf&  is the void growth rate and cf&  is the void coalescencerate. Thus, as long as f is smaller than a characteristic value fC, only nucleation and growthdevelop. Above fC, only coalescence takes place.The nucleation rate is proportional to the rate of equivalent plastic strain& ( ) &f An p p= ε ε (9)  For A p( )ε  Chu and Needleman (CHU e NEEDLEMAN, 1980) propose the statisticaldistribution ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛ −−=221exp2)(NNpNNpssfA εεπε (10)  where fN, is the nucleation void volumetric fraction, εN is the plastic strain value fornucleation and sN is the standard deviation for the distribution. Sometimes it is assumed that  nucleation takes place only in tension (ABAQUS, 1992, CUNDA e CREUS, 1999), whatimplies that 0      if       0)( <= pA pε (11)Growth rate of voids is controlled by mass conservation through the expressionpiig Dff )1( −=& (12)  Voids increase or decrease their volume according to the volume variation in the continuum.Coalescence is governed (TVERGAARD, 1982) by the relation pCUcfff εε &&∆−= (13)3. MODELING CONSIDERATIONS FOR METTALIC FOAMSMetal foam properties can change along the production process, particularly according tothe cell being open or closed and the relative density of the foam in relation to the basematerial.Moreover, foams show (see Fig. 1) a fairly random structure. It is possible to determinethe exact geometry using CT-scan (computed tomography) and model it with a dense mesh offinite elements, as in Fig 4a. A different approach towards the understanding and modeling ofthese materials, adopted in this work, is an idealization as a cellular material with regularstructure, as in Fig. 4b. In this case, modeling a RVE (representative volume element, Fig. 4c)and still considering the existent symmetries, good results can be obtained with fewerelements. Then, specimens with regular structure may be tested to validate results anddetermine material parameters. These homogenization procedures have been used withsuccess in other areas of continuum mechanics.Figure 5 shows a specimen of a cellular metal made to represent a foam.(a)                                                   (b)                                                  (c)Figure 4 – Meshes used to represent a cellular metal (ÖCHSNER e LAMPRECHT, 2003).Figure 5 – Cube of cellular metal.3.1. Application of boundary conditionsOne important aspect to take into account when modeling cells as those described aboveare the boundary conditions, that must represent the average behavior of the foam. When avertical displacement is applied to the top side (see Fig. 6) the vertical faces must continuevertical while the distance between them may change.This behavior can be computationally imposed using multiple point constraints (MPC).The used FE code (ABAQUS) offers the possibility to realize such a boundary conditionwhere all nodes on a certain surface have the same x-displacement: uxi = … = uxj . The effectof differing boundary conditions on the deformation is shown in Fig. 6Figure 6 – Results of different boundary conditions applied to the right face (Öchsnerand Lamprecht, 2003).4. SIMULATION OF A COMPRESSION TEST ON A CUBE OF CELLULARMETALA cube of a cellular metal as shown in Fig. 5 is considered in the following. The cubecontains 3 mm diameter holes with distance of 4 mm among them. The presence of holessimulates a cellular metal with relative density of 0.2712.a)                                                                                                   b)c)                                                                                                               d)Figure 7 – Meshes used to study the cube. a)unit cell  b)2-cells  c)5-cells  d)7-cells.The matrix material properties are E = 72.7 GPa, ν = 0.34, initial yield value2500 =yσ MPa and final yield value 410=∞yσ MPa. Hardening is taken into account by therelation ])exp(1)[( 00 Pyyyy kεσσσσ −−−+= ∞ , with 25=k . Gurson model parametersconsidered are 5% of initial porosity, α1 = 1.5 and α2 = 1.0.Load is applied as a vertical displacement on the top surface of the cube.The objective of this analysis is to compare numerical results with experimental resultsavailable from a cellular metal cube compression tests, and to check if MPC boundarycondition works properly in compression as in tension situations (OLIVEIRA et al., 2006,ÖCHSNER e LAMPRECHT, 2003).Five models are used to simulate the problem, beginning with a single cell as shown inFig. 7a, and varying the number of cells in each direction and the boundary conditions. Thesingle cell model of Fig. 7a is studied with and without MPC boundary condition (see Fig. 6).With 2 cells in each direction the model called 2-cells (Fig. 7b) is obtained, representing acube with a hole. Employing 5 or 7 cells in each direction the models called 5-cells (Fig. 7c)and 7-cells (Fig. 7d) are obtained.Figure 8 shows a comparison among results for the five models used in the analysis. Itcan be seen that a unit cell (Fig. 7a) with MPC overestimates the stiffness of the model. UsingMPC, due the nature of the restriction (see section 3.1), the walls remain vertical. This effectis more significant after 2.5% macroscopic strain, when buckling occurs in the walls betweenthe holes (see Fig. 9). Figure 10 shows the von Mises stress distribution for the unit cell withMPC at an applied macroscopic strain of 22.73%.Figure 8 – Stress strain plots for the five models used in the analyses.010203040500 5 10 15 20 25 30Macroscopic Strain (%)Macroscopic Stress (MPa)Unit cell - MPC7 cells model5 cells modelUnit cell2 cells modelExperimentThe 2-cells model (Fig. 7b) and a unit cell without MPC that represents an eighth of the2-cells model, underestimate the stiffness of the cube because in this case all the walls sufferstrong buckling effects, that in the actual behavior of the real specimen (Fig. 5) is restricted(at this loading level) to the slenderer external walls. Figures 11 and 12 show the von Misesstress distribution for the unit cell without MPC and for the 2-cells model for an appliedmacroscopic strain of 25%, respectively.Figure 9 – Picture showing buckling process at the walls between holes.Figure 10 – Distribution of von Mises stress (MPa) for a unit cell with MPC for an appliedmacroscopic strain of 22.726%.Figure 11 – Distribution of von Mises stress (MPa) for a unit cell without MPC for an appliedmacroscopic strain of 25%.Figure 12 – Distribution of von Mises stress (MPa) for the 2-cells model for an appliedmacroscopic strain of 25%.Figure 13 – Distribution of von Mises stress (MPa) for the 7-cells model for an appliedmacroscopic strain of 7.143%.Figure 14 – Distribution of von Mises stress (MPa) for the 5-cells model for an appliedmacroscopic strain of 10%.It can be observed in Fig. 8 that for macroscopic strains larger than 12% there is adivergence in the unit cell and the 2-cells model plots, that can be explained by the fact thatthe 2-cells model present a shear strain global effect (see Fig. 12) not found in the one eighthmodel (unit cell without MPC).For the 7-cells and the 5-cells models, that represent more appropriately the effect of thebuckling at the walls between holes, numerical results are closer to experimental results. The7-cells model results are obtained only until an applied strain of 7.143% due time limitations.The mesh for this model has 253820 elements. The 5-cells model with a 92500 elements meshpresent a similar problem and so, the results are obtained until an applied strain of 10% (seeFig. 8).Figures 13 and 14 show von Mises stress distribution for the 7-cells model at an appliedstrain of 7.143% and for the 5-cells model at an applied strain of 10%, respectively. It ispossible to observe there, the buckling of the external walls properly represented.The experimental curve in Fig. 8 indicates how complex is the real behavior. After 12,8%of the macroscopic strain there is a partial rupture with stiff decrease in load until a newequilibrium situation achieved by contact of the fragmented parts. This process repeats itselfuntil the final failure.5. FINAL COMMENTSThis paper presents some new results from a research in development (see OLIVEIRA etal., 2006, CUNDA et al., 2005).It is shown that MPC fails to represent the real behavior of the specimen in compressionfor large macroscopic strains (larger than 2.5% in the present case), mainly due to bucklingeffects that are important at this strain level. The use of more complex models gives a goodapproximation of experimental results, up to a point close to postcritical softening (12.8% inthe present case). To represent the real behavior at much larger deformations, the introductionof self contact effects should be needed. The research will continue looking for better modelsand damage parameters adjustment.AcknowledgementsWe thank CNPq, CAPES and GRICES for financial support though the CAPES/GRICESprogram (Proc. 127/05) and PROPESQ-UFRGS for continuous support of our researchproject.REFERENCESOLIVEIRA, B.F.; CUNDA, L.A.B.; ÖCHSNER, A.; CREUS, G.J.  “Gurson damage model:applications to case studies”, Proceedings of the XXVII Iberian Latin-AmericanCongress on Computational Methods in Engineering – CILAMCE 2006, Belém, Pará,Brazil, September 3 to 6, 2006.CUNDA, L.A.B.; ÖCHSNER, A.; CREUS, G.J.  “Damage in metallic foams”, Proceedings ofthe XXVI Iberian Latin-American Congress on Computational Methods in Engineering –CILAMCE 2005, Guarapari, Espírito Santo, Brazil, October  19 to 21, 2005.ÖCHSNER, A.; LAMPRECHT, K.  “On the uniaxial compression behavior of regular shapedcellular metals”, Mechanics Research Communications, Vol. 30, pp. 573-579, 2003.ÖCHSNER, A.; WINTER, W.; KUHN, G.  “On an elastic-plastic transition zone in cellularmetals” Archive of Applied Mechanics, Vol. 73, pp. 261-269, 2003.GURSON, A.L.  “Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media”, ASME Journal of EngineeringMaterials and Technology, Vol. 99, pp. 2-15, 1977.TVERGAARD, V.  “Influence of voids on shear band instabilities under plane strainconditions”,  International Journal of Fracture, Vol. 17, pp. 389-407, 1981.CHU, C.C.; NEEDLEMAN, A.  “Void nucleation effects in biaxially stretched sheets”, ASMEJournal of Engineering Materials and Technology, Vol. 102, pp. 249-256, 1980.ABAQUS, Theory Manual v. 5.2, Providence, USA: Hibbit, Karlsson & Sorensen, Inc., 1992.CUNDA, L.A.B.; CREUS, G.J.  “A note on damage analyses in process with nonmonotonicloading”, Computer Modeling and Simulation in Engineering, Vol. 4, pp. 300-303, 1999.TVERGAARD, V.  “Material failure by void coalescence in localized shear bands”,International Journal of Solids and Structures, Vol. 18, pp. 659-672, 1982.SIMULAÇÃO DE TESTES DE COMPRESSÃO EM METAISCELULARES POR ELEMENTOS FINITOSResumo. Espumas metálicas são uma nova, e ainda não perfeitamente caracterizada, classede materiais que apresentam interessantes combinações de propriedades físicas e mecânicas,como por exemplo alta rigidez combinada à peso específico muito baixo, ou permeabilidade àgases combinada com condutividade térmica alta, o que oferece amplas possibilidades deemprego, como por exemplo na indústria aeroespacial. Existem várias aplicações emdesenvolvimento para estes materiais, particularmente como segurança passiva devido a suaalta capacidade de absorção de energia em situações de impacto. A análise de dano nasespumas metálicas é um problema complexo que deve ser analisado no contexto de grandesdeformações. Considerando que a compressão é a carga dominante em situações de impacto,neste trabalho é mostrada a simulação de um teste de compressão em uma amostra de metalcelular incluindo efeitos de dano. São apresentadas comparações com dados experimentais.Palavras-Chave: Metais celulares, Espumas metálicas, Dano, Modelo de Gurson,, ElementosFinitos

Finite element simulation of compression tests on cellular metals

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Finite element simulation of compression tests on cellular metals

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