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European Journal of Mechanics A/Solids 22 (2003) 815835

Constitutive modeling of aluminum foam including fractureand statistical variation of density

A. Reyes , O.S. Hopperstad, T. Berstad, A.G. Hanssen, M. Langseth

Structural Impact Laboratory (SIMLab), Department of Structural Engineering, Norwegian University of Science and Technology,

Rich. Birkelands vei 1A, N-7491 Trondheim, Norway

Received 7 January 2003; accepted 19 August 2003

Abstract

An existing constitutive model applicable for aluminum foam was evaluated. The model was implemented in LS-DYNA, andseveral test cases were analyzed and compared to experimental data. The numerical analyses consisted of foam subjected to bothsimple and more complex loading conditions where fracture was of varying importance. Therefore, simple fracture criteria wereadded to the model. Additionally, the inhomogeneities in the foam were modeled by incorporating the possibility of statisticalvariation of the foam density. The implemented model is efficient and robust, and gives satisfactory results compared with theexperimental results. 2003 ditions scientifiques et mdicales Elsevier SAS. All rights reserved.

Keywords:Aluminum foam; Finite element method; Constitutive relations; Fracture; Inhomogeneous foam

Notation

yield function equivalent stressY yield stressp foam plateau stressR strain hardening variablee von Mises effective stressm mean stress shape factor equivalent plastic strain ratee von Mises effective plastic strain ratem volumetric plastic strain rate2, , material parameterseD foam engineering compaction strainC0, C1, n material constantsf foam densityf0 density of base materialD true compaction strain proportional stress path,e/

* Corresponding author.E-mail address:[email protected] (A. Reyes).

0997-7538/$ see front matter 2003 ditions scientifiques et mdicales Elsevier SAS. All rights reserved.doi:10.1016/j.euromechsol.2003.08.001

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816 A. Reyes et al. / European Journal of Mechanics A/Solids 22 (2003) 815835

proportional strain pathe/cr critical strain1 maximum principal stresscr critical stressH (x) H (x) = 1 ifx 0, H (x) = 0 ifx < 0p plastic coefficient of contractionv velocity field

T total duration of loadingdmax final displacementp hydrostatic pressurepmax maximum pressurewe total elastic energye elastic strain tensoreH,

e, dev elastic hydrostatic and deviatoric strain Poissons ratiot timeTE total efficiencyFmean mean forceFmax maximum forced displacement

l length of specimenE Youngs moduluswe total elastic energy

1. Introduction

Energy absorbers are often used in cars, trains, buses etc. to protect passengers and the structure during impact. Due toits excellent energy-absorbing capability, aluminum foam may be used in such devices. As vehicle design requires numericalsimulations by finite element programs such as LS-DYNA (Hallquist, 1998), it is important to have good constitutive modelsalso for foams. Different models for metallic foams are already available in LS-DYNA, but Hanssen et al. (2002) concludedthat these models were not able to predict the behavior of different experimental verification tests. One of the reasons for thediscrepancy between experimental and numerical results was the lack of a suitable fracture criterion. An illustration of an

aluminum foam cube after a uniaxial tension test is given in Fig. 1. As crushable foams are very brittle in tension and shear (DuBois, 1996), fracture should be included in a model for aluminum foam.As foam is a cellular material, several challenges exist in the material modeling of foam. Contrary to metals, which maintain

the same volume when loaded plastically, the volume changes for foams during loading due to internal buckling and collapseof cell walls. A material model for foam should therefore include the possibility of yielding under hydrostatic load conditions.Another important characteristic of aluminum foam, is the inhomogeneity of the pores, which are of different sizes and arenot distributed evenly, see Fig. 2. A few approaches to modeling the inhomogeneities of foam exist in the literature: Daxneret al. (1999) and Gradinger and Rammerstorfer (1999) have studied spring-mass models, while Meguid et al. (2002) modeled3D foam with shell elements and let the thickness of the elements vary with a Gaussian distribution.

Fig. 1. Example of tension test of aluminum foam, from Hanssen et al. (2002).

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A. Reyes et al. / European Journal of Mechanics A/Solids 22 (2003) 815835 817

Fig. 2. Example of aluminum foam cell structure, from Hanssen et al. (2002).

Several constitutive models for foams exist in the literature (Schreyer et al., 1994; Zhang et al., 1997; Ehlers, 1999;Deshpande and Fleck, 2000; Miller, 2000). Some of them are quite simple; others are more complicated with several materialparameters. There are also few or no recommendations on how to include the uneven distribution of pores and fracture in themodels. Therefore, incorporation of statistical variation of density and different fracture criteria has been one of the objectives

in the present study.The main purpose of the present paper was to evaluate an existing constitutive model applicable for aluminum foam.

Aluminum foam is typically applied as an energy-absorbing core material in metal columns or plates. Crash analyses canoften be of large scale where CPU times are high. Therefore, it was important to choose a simple model and that resultswithin reasonable accuracy was achieved. It is also favorable if the calibration of a material model is straightforward with fewmaterial parameters. Robustness and efficiency of the numerical implementation routine was key characteristics. Accordinglythe material model presented by Deshpande and Fleck (2000) has been implemented as a user subroutine in LS-DYNA. Themodel was chosen because of its simplicity, as it can be regarded as an extension of the von Mises yield criterion, where thehydrostatic stresses are incorporated in the equivalent stress. Additionally, statistical variation of the material parameters, i.e.,the foam density, was implemented, together with two simple fracture criteria. Numerical analyses of existing verification testson Hydro aluminum foam (Hanssen et al., 2002) were performed, and compared with the experimental results.

2. Constitutive model and numerical implementation

The yield function for a porous material should include a hydrostatic stress term because the cells of the foam collapse whencompressed, and due to the voids that exist in the foam, the volume changes (Gibson and Ashby, 1997). The continuum-basedisotropic constitutive model for crushable foams, proposed by Deshpande and Fleck (2000), which contains this feature, wasimplemented in the present project.

The yield function is defined by

= Y 0 (1)and the yield stress Ycan be expressed as

Y= p + R(), (2)whereR ()represents the strain hardening andis the equivalent strain. The equivalent stress,, is given by Deshpande andFleck (2000)

2 = 1[1 + (/3)2]

2e+ 22m

, (3)

whereeis the von Mises effective stress and m is the mean stress. The parameter defines the shape of the yield surface.The following definition of the parameter is used (Deshpande and Fleck, 2000):

2 =92

(1 2p)(1 + p) , (4)

wherep is the plastic coefficient of contraction.

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Fig. 3. The influence of on the yield surface.

Fig. 3 shows how the yield surface changes shape for different values of. If 2 = 4.5, there will be no lateral plasticdeformation resulting from uniaxial compression, which means that the measured engineering stress is identical to the true

stress. The values of 2

should be limited within the range of 0 2 4.5 to be physically admissible. The upper limitcorresponds to zero plastic coefficient of contraction, while the lower limit corresponds to the von Mises criterion.

Remark.The von Mises yield criterion can be deduced by equating the shear energy in a multi-axial stress state to that of a one-dimensional state of stress in pure tension at the yield stress, disregarding the effects of the volumetric deformation (Lemaitreand Chaboche, 1990). The elastic energy can be written as a sum of the shear energy and the energy of volumetric deformation.Plastic deformation of metals results from slip due to shear stresses; hence, the deduction of the von Mises yield criterion isbased on the shear energy. Foams, on the other hand, can yield under hydrostatic as well as under deviatoric loading conditions(Chen, 1998). It is accordingly possible to modify the von Mises yield criterion by considering the total elastic energy in amulti-axial stress state, and equating it to the elastic energy in a one-dimensional state of stress in uniaxial tension at the yieldstress. This leads to Eqs. (1) and (3). The mathematical deduction is described in detail in Appendix.

According to Gioux et al. (2000), the Deshpande and Fleck (2000) and Miller (2000) yield criteria can also be deduced fromthe mechanistic yield surface for ideal open-cell foams (Gibson et al., 1989) by accounting for cell wall curvature.

The plastic rate-of-deformation and the equivalent strain rate is defined by the associated flow rule (Lemaitre and Chaboche,1990; Deshpande and Fleck, 2000). Deshpande and Fleck (2000) showed that the equivalent plastic strain rate can be expressedexplicitly as

2 =

1 +

3

22e+

1

22m

, (5)

where the volumetric and von Mises effective plastic strain rates are in turn defined as (Deshpande and Fleck, 2000):

m = 2

1 + (/3)2m

, e =

1 + (/3)2

e

. (6)

The elastoplastic constitutive model has been implemented in the explicit finite element program LS-DYNA. The integrationalgorithm (or stress-update scheme) for the rate constitutive equations is based on the work of Aravas (1987) for pressure-

dependent materials. A detailed description of the integration algorithm can be found in Reyes (2003).

3. Statistical variation

Foam is a cellular material, and the pores are of different sizes and not distributed evenly, which leads to variation in thedensity. It was therefore attempted to model the variation in properties, without modeling the pore structure. One approachof including the variation of density in the model is by introducing statistical variation of the material properties. A generalcharacteristic of foam is that the material properties are functions of the foam density. If each element is given a different

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density, the material parameters can be calculated from the density, which will lead to a variation of the properties. A hardeningmodel where it is possible to include the variation of foam density, is suggested by Hanssen et al. (2002):

= p + eeD

+ 2 ln

1

1 (e/eD)

. (7)

Here,e is the engineering strain, while p, 2, , , and eDare material parameters. Eq. (7) was calibrated against uniaxialcompression tests, and used as input in the different models in LS-DYNA in Hanssen et al. (2002). Furthermore, the material

properties,p, 2, , and , can be expressed as functions of the foam density (Hanssen et al., 2002):p, 2, ,

1

= C0 + C1

f

f0

n, (8)

whereC0, C1, andn are constants. Because the plastic coefficient of contraction was assumed zero by Hanssen et al. (2002),eD, also called the compaction strain, could be expressed aseD = 1f/f0, wherefis the foam density andf0is the densityof the base material.

Because of the possibility of including the density dependence of the material properties, the hardening model in Eq. (7) wasused in the implemented model with a few modifications. Since the strain-hardening curve has about the same shape in bothengineering and true stressstrain space (Hanssen et al., 2002), true strains were used in the hardening model. Furthermore, itis assumed that the hardening depends upon the equivalent strain only. Accordingly, the yield stress is expressed as

Y

=p

+R(

)

=p

+

D +

2 ln 1

1 (/D) , (9)

where, p, 2, , , andDare the material parameters. Fig. 4 shows how the different parameters affect the strain-hardeningcurve. Eq. (9) can be calibrated to results from uniaxial compression tests. The material parameters are considered functions ofthe density, and Eq. (8) is used in the implementation. However, a general expression for the compaction strain is needed. FromEqs. (5) and (6), and from the fact that the model gives proportional straining under proportional loading, the equivalent straincan be expressed as

=1 + (/3)2

|m|1 + (/3)2 2

=

1 + (/3)2

|m|1 (1 + (/3)2)()2

(10)

for the proportional stress paths defined by the dimensionless stress-state parameter = e/or by the strain-path parameter = e/. The relationship between and is obtained as

= e

= 11 + (/3)

2

e

= 11 + (/3)

2. (11)

Fig. 4. Strain hardening model.

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The stress-path parameter was used in the strain hardening relation by Deshpande and Fleck (2000) to account for theinfluence of the stress path on strain hardening.

Theoretically, the compaction strainD(i.e., the equivalent strain at compaction) can be defined as the strain limit at whichthe density of the foam equals the density of the base material, i.e., when m = ln(f/f0). The equivalent strain D thatcorresponds to the compaction strain depends on the load case, and for proportional straining it can be expressed as

D = 1

+(/3)2

1 + (/3)2 2ln f

f0

. (12)

Since uniaxial compression tests are used for calibration of Eq. (9), the compaction strain was here defined accordingly. Uniaxialcompression gives = 1, and leads to

D = 9 + 2

32 ln

f

f0

. (13)

As the material parameters all can be considered functions of the foam density, a statistical variation of the foam densitywas introduced in the constitutive model, i.e., the mean value and standard deviation must be given as input parameters. Inthe initialization of the analysis, each element is given a foam density with a normal (Gaussian) probability distribution. Thematerial parameters are calculated from the density for each element, using Eqs. (8) and (13). A function, Gasdev (1992),was used to generate random deviates with a normal (Gaussian) probability distribution. When the foam density is distributedrandomly, each element is given a random density in the beginning of the analysis, which means that one numerical simulation

will be different from the next. Hence, one analysis can be an extreme case that is not likely to happen. Several analysesshould therefore be carried out to make sure that the response is of a wanted probability.The material constants, C0, C1, andnshould be calibrated to Eq. (8) when using the option with statistical variation of the

parameters, as the material constants all depend on the foam density. However, this requires that a larger experimental databaseis available. Hanssen et al. (2002) also points out that the fit to Eq. (8) is best for the plateau stress while there is some errorfor the other parameters. Nevertheless, as the main goal is to introduce a variation in the material parameters, the use of Eq. (8)seems reasonable.

In the presented approach, the elements will have different densities and the densities will be distributed randomly. Anelement with high density can be next to a low-density element, and a fine element mesh leads to small areas with differentdensities. In reality, due to the pores, larger areas should maybe have the same density and standard deviation. This could be

Fig. 5. Compaction strain as a function of stress state.

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accomplished by dividing the element model into different parts and introducing different mean values and standard deviationsin the various parts.

The definition of the compaction strain in the implementation leads to some problems. By considering Eq. (12), one cansee that the equivalent strain at compaction, D, depends on the load case. Fig. 5 shows Eq. (12) plotted against the strain-pathparameter = /

1 + (/3)2, which varies from zero to unity. Zero represents a pure hydrostatic stress state, while unity

corresponds to pure shear. For pure shear, equals 1, and an infinite value of the compaction strain is obtained, which meansthat pure shear does not lead to compaction. The point of uniaxial compression is also plotted in the figure. In the presentproject, the compaction strain from uniaxial compression is used as a material constant, and as one can see from the figure, thepresent approach predicts a compaction strain that is too high for a pure hydrostatic stress state, while the compaction strain forpure shear is too low.

Miller (2000) avoids the described problem by introducing the strain hardening as a product of the stress, dependent on theequivalent strain, and an amplification factor, dependent on the volumetric strain. The amplification factor can be unity untildensification, and can then increase rapidly after densification. Another solution would be to introduce Das a function of in the strain-hardening rule, i.e.,D = D(). It follows that the strain hardening variableR then would become a function ofboth equivalent strain and strain path:

R = R, . (14)This is similar to what was proposed by Deshpande and Fleck (2000) to account for the stress-path dependency of the strain

hardening observed in the experiments. This solution has not been considered in the present paper.

4. Fracture

Aluminum foam exhibits quite different behavior in tension than in compression. Foams show elastic, plastic and compactionphases in compression, while in uniaxial tension, they display an elastic deformation phase, followed by fracture. They alsotend to deform elastically under shear, followed by fracture. However, in hydrostatic compression, the stressstrain curve hasapproximately the same shape as for uniaxial compression (Du Bois, 1996). Uniaxial tensile tests performed by Hanssenet al. (2002) showed that the tensile failure stress for the foams in the present study is approximately equal to the initialplateau stress in compression, which is opposed to the extensive ductility in compression. In contrast, Olurin et al. (2000) andSugimura et al. (1997) found that the ultimate tensile strength was somewhat higher than the compressive stresses for the foamsthey investigated.

Fracture is often modeled in FEM-analyses by removing (eroding) elements when they reach a critical value of strain. Theplastic volumetric strain,m, is positive in tension and negative in compression. Hence, one possible fracture criterion can be

to use the volumetric strain as a measure for when an element should be eroded, i.e., when a critical, positive value of thevolumetric strain is reached, the element is eroded:

ifm cr erosion of element, (15)where cr is the critical strain. Eq. (15) was implemented in the constitutive model, and the criterion will be referred to ascriterion No. 1 in the following.

In addition, a second fracture criterion was evaluated. When the volumetric strain is used in the fracture criterion, onlyhydrostatic deformation is included. It is however, natural to assume that also deviatoric deformation can cause fracture.Therefore, a fracture criterion where the principal stress is used for evaluation of fracture was also implemented. The criterionis based on erosion of elements when the maximum principal stress reaches a critical value:

if1 cr erosion of element, (16)wherecris the critical stress. As the tensile failure stress was approximately equal to the initial plateau stress in compression(Hanssen et al., 2002), p can be used as a critical value of the principal stress, i.e., cr = p. However, because of spuriousnoise produced by contact forces and elastic stress waves, which are initiated when an element is eroded, the stress levels in theelements can at times be higher than the critical stress, although these should not necessarily cause fracture. To avoid prematureerosion of elements, an energy-based criterion was established from Eq. (16), motivated by a fracture criterion for metals dueto Cockcroft and Latham (1968). The fracture criterion reads:

if

0

H (1 cr)1 d W erosion of element, (17)

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where, H(x)is defined as

H(x) =

1 ifx 0,

0 ifx < 0.(18)

Eq. (17) was implemented in the constitutive model, and the criterion will be referred to as criterion No. 2 in the following.Be aware that the critical value ofWmay be problem dependent and may thus have to be selected based on experience orvalidation studies using experimental data.

The erosion of an element for which the fracture criterion is fulfilled is accomplished in LS-DYNA by setting all componentsin the stress tensor to zero. The mass is still contained in the connected nodes, while the internal energy of the element is includedin the energy balance. When many connecting elements are eroded, some of the related nodes will no longer be connected tothe rest of the structure.

5. Model verification

Several verification tests have been carried out by Hanssen et al. (2002), and have been compared to existing materialmodels in LS-DYNA. The experimental program included tests with various loading conditions, such as uniaxial compression(Cal 1) and hydrostatic compression (Cal 4), here called calibration tests. Three additional test cases, indentation testscalled Mval 1, diagonal loading called Mval 2, and two-stage perpendicular loading called Mval 3, were also performedto test the foam models under more general loading conditions. These tests will be referred to as material validation tests.

The experimental verification program is shown in Fig. 6. In the present project, numerical analyses of Cal 1, Cal 4,Mval 1, and Mval 2 were carried out with the implemented constitutive model. Analyses of foams with three densities,f= 0.17 g/cm3, f= 0.34 g/cm3, andf= 0.51 g/cm3, were performed.

5.1. Numerical modeling

The test specimens of Cal 1, Mval 1 and Mval 2 were all cubic with dimensions 70 70 70 mm3. Cal 4 hadcircular cross sections with diameter and length of 40 and 70 mm respectively. Due to symmetry, a quarter of the specimenwas modeled for Cal 1 and Mval 1, 1 /8 of the specimen was modeled for Cal 4 and half the specimen was modeledfor Mval 2. The different meshes are shown in Fig. 7. The default eight-node brick element of LS-DYNA (Hallquist, 1998)was applied with one point reduced integration scheme for Cal 1 and Cal 4. For Mval 1 and Mval 2 it was necessary

Fig. 6. Experimental verification program of Hanssen et al. (2002).

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Fig. 7. Element meshes of verification analyses.

to use a fully integrated S/R solid element. Hanssen et al. (2002) used the same elements (with only one integration point) forall analyses, but it was in the present case found that this gives unphysical results. The stiffness-based hourglass control #5 inLS-DYNA was used in order to avoid hourglassing. A prescribed velocity field in the upper nodes introduced the loading forCal 1, while the external nodes of Cal 4 was subjected to hydrostatic pressure. The load for Mval 1 and Mval 2 wasapplied through a rigid body given a prescribed velocity field.

The four tests were all loaded quasi-statically. To ensure quasi-static loading when using an explicit code, the prescribedvelocity field was given as

v(t) = 2

dmax

T

1 cos

2Tt

(19)

while the hydrostatic pressure was given as

p(t) = pmax

1 cos

2Tt

. (20)

Here,T is the total duration of the loading,dmaxis the final displacement andpmaxis the maximum pressure. When integrated

from t= 0 to t= T, Eq. (19) yields dmax, and when differentiated with respect to time, the initial acceleration equals zero.The pressure in Eq. (20) equals zero for t = 0 andpmaxfor t = T. The applied velocity and pressure fields should ensure thatthe loading takes place gradually and that unnecessary dynamics in the numerical solution are avoided (Ilstad, 1999). Analyseswith different values ofTwere carried out to check that the analyses really could be treated as quasi-static. The terminationtime,T, was 0.1 s for Cal 1, Mval 1, and Mval 2, while for Cal 4, T= 0.8, 0.3, and 0.2 s for f= 0.17, 0.34, and0.51 g/cm3, respectively.

Visual observations of the uniaxial compression tests indicated that the coefficient of contraction should be approximatelyzero (Hanssen et al., 2002). A zero plastic coefficient of compression gives = 9/2 2.12, which was used in the following.The hardening model (Eq. (9)) was calibrated to the uniaxial compression tests (Cal 1) performed by Hanssen et al. (2002),

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Table 1Material parameters for the analyses without statistical variation

f E D 2 p W

[g/cm3] [MPa] [MPa] [MPa] [MPa] [MPa]

0.17 377 1.87 2.77 93.5 5.79 1.15 0.080.34 1516 3.92 2.07 60.2 4.39 5.76 0.400.51 5562 5.37 1.67 66.9 2.99 14.82 1.04

Table 2Overview of analyses

[g/cm3] Cal 1 Cal 4 Mval 1 Mval 2 Type of analysis

0.17 3D 3D 3D, 2D 3D, 2D without fracture0.34 3D 3D 3D, 2D 3D, 2D

0.51 3D 3D 3D, 2D 3D, 2D

0.17 2D

with fracture criterion 10.34 2D 2D0.51 2D 3D 2D0.17 2D

with fracture criterion 20.34 2D 2D

0.51 2D 3D 2D0.17 2D

with statistical variation and fracture criterion 10.34 2D0.51 2D

0.17 2D

with statistical variation and fracture criterion 20.34 2D0.51 2D

and the compaction strain was calculated from Eq. (13). During testing, the foam cubes of the highest density (f= 0.51 g/cm3)fractured, which led to a smaller effective cross-sectional area, and hence the true stresses from these tests would be incorrect(Hanssen et al., 2002). Therefore, additional tests on shorter cubes (referred to as stocky column) were performed to find morerealistic stressstrain curves (Hanssen et al., 2002), and, consequently, the material parameters forf= 0.51 g/cm3 correspondto these tests, and not the ones that fractured. Analyses for f= 0.17 g/cm3, f= 0.34 g/cm3, andf= 0.51 g/cm3 have beenperformed, and the material parameters are given in Table 1. As the plastic coefficient of contraction was assumed zero, planestrain could be assumed. Therefore, both 2D and 3D analyses were carried out. An overview of the numerical analyses is givenin Table 2.

5.2. Calibration tests

The two calibration tests were analyzed numerically with the material model without statistical variation or fracture. Thesenumerical models had quite coarse meshes due to calculation times. For Cal 1, one additional analysis of the stocky columnwith height 35 mm and density f= 0.51 g/cm3 was carried out. Force-displacement curves of Cal 1 are given in Fig. 8,while hydrostatic pressure is plotted versus the volume change for Cal 4 in Fig. 9. Cal 1 and Cal 4 deformed as expectedin the analyses. The force-displacement curve for the numerical analysis of Cal 1 lies very close to the experimental curve forbothf= 0.17 g/cm3 andf= 0.34 g/cm3. The analysis withf= 0.51 g/cm3 is not so near the experiment, but this is due tofracture in the experiments (Hanssen et al., 2002), as mentioned earlier. However, the results for the stocky column are good.Analyses including fracture were carried out of Cal 1 with f= 0.51 g/cm3, but fracture was not initiated. Additionally,analyses of Cal 4 where fracture was included were carried out, and as in the experiments, no fracture was initiated.The theoretical curve for pressure vs. volume change can be calculated for Cal 4 as follows: The values forandwhene = 0 ande = 0 were found from Eqs. (3) and (5), and were inserted in the equivalent stressstrain curve given in Eq. (9):

m =

1 + (/3)2

=

1 + (/3)2

Y () =

1 + (/3)2

Y

1 + (/3)2

m

. (21)

The volume change was calculated from the volumetric strain, whilemis the hydrostatic pressure. The theoretical curvesare plotted in Fig. 9. As one can see, the theoretical curves are in agreement with the results from the analyses. However, thetheoretical and numerical results are quite different from the experimental data. There are some uncertainties connected to the

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Fig. 8. Force-displacement plots for uniaxial compression test Cal 1, 3D-model.

Fig. 9. Hydrostatic compression tests, Cal 4, 3D-model.

experimental data, e.g., the initial stiffness from the hydrostatic tests was probably too low because of the test set-up (Hanssenet al., 2002). Nevertheless, the deviation between experimental and numerical results might imply that the plastic coefficient ofcontraction is not zero for the present experiments, as p >0 would have lifted the curves somewhat. Another speculation iswhether an associated flow rule is appropriate or not. Zhang et al. (1997) proposed a model for foam where the present yieldcriterion is used together with a non-associated flow rule.

5.3. Material validation tests

3D-models of the two material validation tests, Mval 1 and Mval 2 were analyzed numerically with the material model,without statistical variation or fracture. Due to calculation times, these models had quite coarse meshes. Therefore, in additionto the 3D-models, plane-strain models were generated, as it then was possible to use a much finer mesh. 2D-models of Mval 1

were then analyzed with the model including fracture and statistics. Mesh sensitivity studies were carried out, and the solutionconverged for a model of 2574 (Mval 1) and 1600 (Mval 2) elements. When using such a fine mesh, it was sufficient to useelements with one point reduced integration scheme. When statistical variation of the density was included, the density variedfrom one element to the other, and the model was not symmetric. Therefore, in addition to the half model where symmetry wasassumed, a full model was also analyzed.

The experimental program of Hanssen et al. (2002) included over 200 specimens, so the statistical variables, empirical meanand standard deviation were calculated from this database, and are given in Table 3. These values are based on a mean valueof the density, and do not really reflect the real variation of density in every specimen. No effort was made to measure the realvariation in each foam sample. The material parameters p, 2, , and 1/were all fitted to Eq. (8), and the different values

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Table 3Statistical variables from experimental database

0.17 g/cm3 0.34 g/cm3 0.51 g/cm3

Number of points 75 76 60Empirical mean [g/cm3] 0.172 0.314 0.515

Empirical variance [g

2/cm

6

] 0.000177 0.000507 0.000611Standard deviation [g/cm3] 0.0133 0.022517 0.02471

Table 4Material parameters for the analyses with statistical variation

p 2 1/ W [MPa] [MPa] [MPa] [MPa]

C0[MPa] 0 0 0.22 0 0C1[MPa] 590 140 320 40 41.3n 2.21 0.45 4.66 1.4 2.21

Fig. 10. Distribution of density for analysis with statistical variation, = 0.314 0.0225, half and full model.

ofC0, C1, andnare given in Table 4. An example of the variation of foam density in Mval 1 is shown in Fig. 10. The criticalvalue of the volumetric strain in fracture criterion 1 was chosen as 2 %, while the critical value Win fracture criterion 2 variedwith the foam density, and is shown in Table 1 and Table 4. In Table 4, Wis chosen as a function ofC1and n.

5.3.1. Analyses without fracture

Force-displacement curves from the experiments and analyses without fracture of Mval 1 and Mval 2 are given inFigs. 11 and 12. Pictures of the deformed meshes of Mval 1 and Mval 2 are given in Figs. 13 and 14. The analysis of

Mval 1 without fracture is of course different from the experiment, but the deformation behavior of Mval 2 seems quitephysical. The force-displacement curves of Mval 1 lie higher than the experiments, which is not surprising as the specimensin the experimental program fractured. The force-displacement curves of Mval 2 shows that the numerical model was able topredict the behavior of the experiments with good accuracy.

5.3.2. Plane-strain analyses of Mval 1 including fracture

Pictures of the different analyses of Mval 1 with fracture criterion 1 are shown in Figs. 15 and 16. The foam begins torupture along the sides of the indenter, which is similar to the experiments. However, there are some visual differences betweenthe analyses and experiments. The fracture in the real tests continued to develop along the sides of the indenter, whereas in the

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Fig. 11. Force-displacement plots, indentation test, Mval 1, 3D-model.

Fig. 12. Force-displacement plots, diagonal loading Mval 2, 3D-model.

analyses, the fracture moves from the indenter towards the outer edge of the specimen. One can also see that there are minordifferences between the half and full model with statistical variation. The full model is clearly asymmetric, but still quite similarto the half model.

The corresponding force-displacement curves of Mval 1 are given in Fig. 17, together with the experimental results. Theanalyses with fracture criterion 1 clearly lowers the force levels compared to the analyses without fracture, but the forces arestill higher than the experimental results. One can also see from the figure that the analyses with statistical variation of thedensity lie somewhat lower than the analyses without. Additionally, the force levels fall somewhat before rising again at theend of the analyses with statistical variation. This behavior is similar to the experimental results. Fig. 18 depicts the differencebetween using a half and full model. There is obviously not much variation in response between the two models.

Fig. 19 shows the deformation behavior of the analyses with fracture criterion 2. As one can see, the deformation is very

similar to the deformation in the experiments. The rupture begins just beside the indenter, and continues along the indenter.At a later stage in the experiments, however, fracture also develops from the outer sides of the specimen, and grows towardsthe fracture along the indenter. The deformation modes of the analyses with and without statistical variation of density arevery similar. As one can see in Fig. 19, tear lines beneath the indenter, are developed along the perimeter of the indenter. Thisbehavior corresponds to the observations in a recent experimental study by Kumar et al. (2003) of indentation of the isotropicAlporas foam (Doyoyo and Wierzbicki, 2003).

The force-displacement curves of Mval 1 with fracture criterion 2 are given in Fig. 20. As one can see, the differencesbetween the analyses with and without statistical variation of density are very small, and there is a very good agreement betweennumerical and experimental results. Fracture criterion 2 seems to be more suitable for the indentation test than criterion 1.

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Fig. 13. Deformation behavior of Mval 1.

Fig. 14. Deformation behavior of Mval 2.

For the analyses where fracture occurred, the loss of energy due to erosion of elements was checked and found to be

negligible.

5.3.3. Plane strain analyses of Mval 2

Pictures of the deformation behavior of the plane strain analyses of Mval 2 are shown in Fig. 21, and it is very similarto the 3D-model and the experiments. Force-displacement curves of Mval 2 are given in Fig. 22, including the experimentalresults together with the results from the analyses. As one can see, both the 3D-model and the 2D-model are able to predictthe experimental results with good accuracy. Analyses of Mval 2 with fracture was carried out, and as in the experiments,fracture was not initiated.

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Fig. 15. Deformation behavior of Mval 1, plane strain model, with fracture criterion 1.

(a) (b)

Fig. 16. Deformation behavior of Mval 1, plane strain model, with fracture criterion 1 and statistical variation of density: (a) half model and(b) full model.

Fig. 17. Force-displacement plots for indentation test, Mval 1, including results from analyses with fracture criterion 1 and statistical variationof density, plane strain analyses.

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Fig. 18. Force-displacement plots for indentation test, Mval 1: analyses with fracture criterion 1 and statistical variation of density. Differencebetween half and full model, plane strain analyses.

Fig. 19. Deformation behavior of Mval 1, plane strain model, with fracture criterion 2.

Fig. 20. Force-displacement plots for indentation test, Mval 1: analyses with fracture criterion 2 and statistical variation of density, planestrain analyses.

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Fig. 21. Deformation behavior of Mval 2, plane strain model.

Fig. 22. Force-displacement plots, diagonal loading Mval 2, difference between 3D-model and plane strain model.

Fig. 23. Correlation between mean loads from numerical analysesand experiments.

Fig. 24. Effect ofon the stressstrain curve.

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Fig. 25. Results from parametric studies.

5.3.4. Mean loads

The mean load, Fmean, is defined as the energy absorption divided by the displacement and was determined for theexperiments and analyses of Cal 1, Mval 1 and Mval 2 at the displacement corresponding to the maximum value ofthe total efficiency which is defined as (Hanssen et al., 1999)

TE =Fmean

Fmax d

l. (22)

Here,Fmaxis the maximum force, dthe displacement, andlis the length of the specimen. Fig. 23 shows the correlation betweenthe mean forces from the various experiments and the analyses. As one can see, the analyses of Cal 1, Mval 1 with fracture

criterion 2, and Mval 2 gives very good results, while the other analyses of Mval 1 gives higher mean loads than found inthe experiments.

5.3.5. Parametric study

A study was carried out on the effects of different parameters in the model. Mval 1 with density 0.34 g /cm3 was chosenas a test case, and was analyzed with the model including statistical variation of density and fracture criterion 1. The previousstudies showed that the force levels were somewhat higher than in the experiments for larger displacements. Analyses with noinitial hardening and initial softening were carried out by changing in Eq. (9). This was set to 0 and 1.97 MPa. The stressstrain curves for the different values ofare shown in Fig. 24. The results from the analyses are given in Fig. 25(a) and one can

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Fig. 26. Mean load from parametric study. Fig. 27. Mean load from statistical study.

see that the force levels are lowered by introducing no initial hardening and initial softening. Yet, the effects are not very large.The critical fracture strain was also changed; analyses with a critical volumetric strain of 0.2% and 1% were performed, and theresulting force-displacement curves are depicted in Fig. 25(b). There are some small effects of changing the critical strain, butthey seem insignificant.

As mentioned earlier, the standard deviation calculated from the experimental database of Hanssen et al. (2002), wascalculated based on mean densities of whole specimens and could not reflect the variation of density within the specimens.Therefore, one analysis with standard deviation of 20% of the mean value was also carried out, and the result is given inFig. 25(c). This actually led to a little higher force levels.

There might have been friction between the indenter and the foam in the experiments as no attempts were made to makethe surfaces friction-free. Consequently, this was also studied by including friction in one analysis. A friction factor of 0.2 wasused, and as one can see in Fig. 25(d), this increased the force levels somewhat, but the effect is not large.

The mean loads for the analyses in the parametric study were calculated at the displacement corresponding to the maximum

value of the total efficiency, and are shown in Fig. 26 together with the results from the other analyses of Mval 1 with density0.34 g/cm3. Most of the results are quite close to each other and lie higher than the experiment, while the analysis with fracturecriterion 2 is somewhat lower than the experiment.

When statistical variation of density is included in the model, the foam density will be distributed randomly in the specimen,which means that the numerical simulation with the same input will yield different results. Therefore, 50 identical simulationsof Mval 1 with density 0.34 g/cm3 were carried out to determine the scatter in the response due to the statistical variation ofthe density. The mean value and standard deviation had the values given in Table 3. The force-displacement curves of all thesimulations are shown in Fig. 25(e) which illustrates the upper and lower boundaries. The mean load was determined for all thesimulations, and the mean value was found to be 15.16 kN, with a standard deviation of 0.42 kN, which corresponds to 2.8% ofthe mean value. The mean loads are also plotted in Fig. 27 and one can see that the scatter in the results is not very large.

6. Concluding remarks

An existing constitutive model for foam, proposed by Deshpande and Fleck (2000), has been evaluated for aluminum foam.The foam model can be regarded as an extension of the von Mises yield criterion, where the hydrostatic stresses are incorporatedin the equivalent stress, as it is possible to derive the yield criterion by equating the total elastic energy in a multiaxial stressstate to that of a one-dimensional state of stress in pure tension at the yield stress. The model was implemented in LS-DYNA,and, in addition, two simple fracture criteria and statistical variation of foam density were implemented.

Analyses of four verification tests were carried out, and the results were compared to the experimental data. The uniaxialcompression tests were, as expected, predicted with good accuracy, and the hydrostatic compression tests gave satisfactoryresults as they predicted the theoretical results. The indentation test, Mval 1, with the strain-based fracture criterion gave

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load levels higher than the experiment, while the stress-based fracture criterion predicted the experimental response with goodaccuracy. The tests with diagonal loading, Mval 2, were also predicted with good accuracy. The study has shown that byincluding a simple fracture criterion, it was possible to predict the behavior of the quite complex indentation test. The effectof including statistical variation, however, was small in the test cases in the present study, although it should be noted that theeffect might be larger for other problems.

Acknowledgements

The authors would like to thank Hydro Automotive Structures for their generous support of the research project that formsthe basis for the present work.

Appendix

The total elastic energy can be written as (Lemaitre, 1996)

we =1 + 3E

2e+

9

2

(1 2)(1 + )

2m

.

Let2 = (9/2)(1 2)/(1 + ), and the following expression for total elastic energy can be obtained

we = 12E

1

1 + (/3)2

2e+ 22m

.

The elastic energy of a one-dimensional state of stress in uniaxial tension is (Lemaitre and Chaboche, 1990)

we = 12E

2.

For a one-dimensional state, plastic yielding will occur when the equivalent stress,= , so that=

2Ewe

which leads to the following:

2 = 2Ewe = 11 + (/3)2

2e+ 22m

.

This equals the equivalent stress proposed by Deshpande and Fleck (2000). It is clear that depends on Poissons ratio, ,but the elastic coefficient of contraction for foams is close to zero. In any case, most of the deformation is in fact plastic, so itwould be more convenient to use the plastic coefficient of contraction.

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