# Modified Deshpande–Fleck Model considering a Variable Plasticity Poisson’s Ratio and a Variable Ellipticity

Authors: Weimin Zhuang; Jian Sun; Dongxuan Xie
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

It is controversial whether the plastic Poisson’s ratio and ellipticity are invariable in the Deshpande–Fleck model, which describes the constitutive behavior of foam materials. In this paper, the Deshpande–Fleck model is modified by a variable plasticity Poisson’s ratio and a variable ellipticity based on a nonassociated flow rule. The plasticity Poisson’s ratio is measured by in situ digital image correlation (DIC) technology during the uniaxial compression experiment of aluminum foam. The ellipticity is then determined through the experimental data of uniaxial compression and hydrostatic compression. Both the plastic Poisson’s ratio and the ellipticity vary with the strain. The Hanssen model fitted with uniaxial compression experimental data is used to provide the hardening law for the modified Deshpande–Fleck model. The modified Deshpande–Fleck model is realized by the subroutine VUMAT in ABAQUS/Explicit. There is good agreement between the experimental results and the prediction results of the modified Deshpande–Fleck model under multiaxial compression conditions. Furthermore, the influence mechanism of the plastic Poisson’s ratio and ellipticity on the modified Deshpande–Fleck model are investigated. The results prove that it is necessary to consider both the variable plastic Poisson’s ratio and the variable ellipticity in order to improve the prediction accuracy of the modified Deshpande–Fleck model.


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## Body

## 1. Introduction

Metal foam has become an attractive material to automotive, military, architecture, and other industrial fields because of its lightweight and great ability of energy absorption [1–4]. In engineering, metal foam components are usually subjected to complex load conditions [5, 6], so it is very important to accurately describe the constitutive relationship of metal foam under multiaxial loads [7].Metal foam is a typical porous compressible material [8] that can still yield under hydrostatic pressure [9, 10]. Therefore, many scholars have proposed the yield surface of metal foam materials via the relationship between the von Mises equivalent stress σe and the average stress σm [1, 2, 11–13].The yield functions for metal foam are usually parabolic and elliptic. Gibson [1,2] proposed a parabolic yield function based on the ideal Gibson-Ashby geometric model, called the GAZT model. This model only describes the initial yield surface. It is suitable for open-cell aluminum foam with regular cell shapes [11–13]. Miller [14] proposed an improved incremental parabolic yield function based on the Drucker-Prager yield criterion. Miller’s model considered the tension and compression asymmetry of foam materials. The subsequent yield evolution of Miller’s model was controlled via the equivalent plastic strain. Chen and Lu [15] proposed a constitutive model suitable for plastic compressible materials based on the elastic complementary energy expression. The model introduced dimensionless characteristic stress and characteristic strain as independent variables, and the yield surface of the model was elliptical. Deshpande and Fleck [16] conducted a series of uniaxial and multiaxial compression experiments through Alporas and Duocel aluminum foam and proposed a self-similar isotropic hardening phenomenological foam constitutive (the D-F model). In this model, the yield surface is elliptical. Deshpande and Fleck also proposed a more complex differential reinforcement model [16], but its practicability is limited due to its overly complex enhancement law.The D-F model has been widely used in the research field of metal foam because it can approximately describe the mechanical response of metal foam under multiaxial loading. The accuracy of the D-F model for predicting the initial surface of metal foam has been proved [2, 11, 17, 18]. The D-F model has also been applied to various commercial finite element software, such as the crushable foam model with the isotropic hardening in ABAQUS [19] and the Mat154 material model in LS-DYNA [20].The elliptical yield surface of the D-F model is as follows [16]:(1)Φ=σ^−Y≤0,(2)σ^2=11+α/32σε2+α2σμ2,where Y is the uniaxial yield stress, σ^ is the equivalent stress, and α is the ellipticity of the elliptical yield surface. Deshpande and Fleck considered the ellipticity as a constant, and the associated flow rule was adopted.The plastic flow potential function of the D-F model can be written as(3)ϕ=σe2+φ2σm2,where φ represents the shape parameter of the plastic flow potential function. It is assumed that the strain increment is normal to the flow potential ϕ [16].(4)dεij=dλ∂ϕ∂σij,where dλ is the nonnegative plastic flow multiplier.The plastic Poisson’s ratioνp represents the compressibility of materials, which is defined as the negative ratio of the transverse logarithmic strain rate to the axial logarithmic strain rate [16], written as(5)νp=−ε˙22ε˙11.Combining equations (3) and (4) with (5) leads to(6)φ2=91−2νp21+νp.Deshpande and Fleck adopted the associated flow rule, so the plastic flow potential function has the same form as the yield function. Thus, the value ofφ is the same as the ellipticity α.The ellipticity and the plastic Poisson’s ratio, which influence the shape of the yield surface and the plastic flow potential, respectively, are not easy to measure. Tables1 and 2 illustrate investigations for the ellipticity and the plastic Poisson’s ratio, respectively, from which it can be seen that the views are very different. In most studies, the ellipticity was considered as a constant in subsequent yield [9, 22, 26, 27]. Deshpande and Fleck [16] found that the yield surface approximately evolves in the form of geometric self-similarity for uniaxial compression, while the yield surface extends along the axis of average stress σm for a primary loading path of hydrostatic compression. In fact, the ellipticity α is obviously affected by the compressibility of the material which varies with the plastic strain. Zhu et al. [7] assumed that the ellipticity varies with the equivalent strain but ignored the influence of the plastic Poisson’s ratio, which will bring errors [7]. The plastic Poisson’s ratio νp is also an important parameter in the D-F model. Different plastic Poisson’s ratios will lead to deviations in the prediction of the yield behavior of metal foams [28]. However, it is difficult to be measured by experiment [18]. Research on the plastic Poisson’s ratio is mostly based on mathematical methods and hypothetical methods, which to some extent leads to the scattered data of plastic Poisson’s ratio. Some literature [7, 11] noted the contradiction between the sensitivity of the D-F model to the plastic Poisson’s ratio and the lack of experimental data on the plastic Poisson’s ratio.Table 1
The investigation for the ellipticity of the D-F model.
EquationInvariableVariableAuthorsParameters involvedRef.α2=91−2νp/21+νp√—Deshpande et al.νp: the plastic Poisson’s ratio[16]α2=9σ02/9p02−σ02√—Deshpande et al.σ0： uniaxial compression yield stress[21]p0: hydrostatic compression yield stressα2=91−2ν/21+ν√—Vural et al.ν: the elastic Poisson’s ratio[22]α2=2.1496e−0.579ε¯+0.4314ε¯2−1.252—√Zhu et al.ε¯: equivalent strain[7]Table 2
The investigation for the plastic Poisson’s ratio of aluminum foam with different relative densities.
Relative densityPlastic Poisson’s ratioEquation of the plastic Poisson’s ratioAuthorsParameters involvedRef.0.10–0.250.17—Zhou et al.—[23]0.30–0.46—νp=0.278−0.409×εpSu et al.εp: the plastic strain[24]0.085—νp=0.01−0.20×εTFang et al.εT: the true axial plastic strain[25]νp=0.05e−εT−0.040.05, 0.10.17—Ruan et al.—[11]0.150.290.17, 0.20.300.05–0.200—Zhu et al.—[7]This study aims to propose a more accurate constitutive model for metal foam. In this paper, the plastic Poisson’s ratio and the ellipticity involved in the D-F foam model are calculated independently by experiments based on a nonassociated flow rule. Through the user-defined subroutine VUMAT in ABAQUS/Explicit, a modified D-F model is proposed by considering a variable plastic Poisson’s ratio and a variable ellipticity. Considering the influence of plastic Poisson’s ratio, a more accurate uniaxial compression stress-strain relationship of metal foam is employed. The modified D-F model is verified by the finite element simulation method, and the results show that the modified D-F model is more accurate. A control study has been performed to investigate the influence mechanism of ellipticity and plastic Poisson’s ratio on the modified D-F model.


## 2. Constitutive Models

### 2.1. The Deshpande–Fleck Model

The Deshpande–Fleck model assumes that metal foam exhibits similar mechanical properties under tension and compression conditions, which is contrary to the conclusion of other studies [14, 15]. As shown in Figure 1, the elliptical yield surface on the σe−σm stress plane is symmetric about the σe axis and evolves in a self-similar form. The Deshpande–Fleck model only considers the mechanical properties of metal foam under compression conditions, so this assumption can be accepted. The modified model proposed in this paper also only considers the compressive mechanical properties of foam metals.Figure 1
Yield surface for the Deshpande–Fleck foam model.Deshpande and Fleck [16] noted that when the equivalent plastic strain was taken as the independent variable, the prediction results of the constitutive model were more consistent with the experimental data. The definition of the equivalent plastic strain rate was given as σ^ε¯˙p=σijε˙ijp. σ^ is numerically equal to the true uniaxial yield stress Y associated with the current yield surface, and σij and ε˙ijp are the stress tensor and the plastic strain rate tensor, respectively, where j=1,2,3.Equations (1) and (2) are integrated to obtain(7)σe2+α2σm2−1+α32Y=0.Equation (7) contains only two parameters, the ellipticity α of the elliptic yield surface and the uniaxial yield stress Y. The plastic flow potential function of the D-F model is defined as shown in (3). The shape parameter φ of the plastic potential function in (3) is only related to the plastic Poisson’s ratio νp, as shown in (6). It is thus crucial to accurately measure the ellipticity α, the uniaxial yield stress Y, and the plastic Poisson’s ratio νp through experiments.


### 2.2. The Stress-Strain Relation under Uniaxial Compression

A plastic constitutive contains three parts. In addition to the yield criterion in (7) and flow rule in (3) and (4), it is also necessary to explain the law of hardening. The D-F model adopts the true stress-strain relationship in uniaxial compression as the hardening law. The elastic section of metal foam is short and negligible under a compressive load [29], so the ideal rigid plastic hardening(R-PH) model is often used to describe the constitutive behavior of metal foam under uniaxial compression. Hanssen [30] proposed an R-PH model to describe the uniaxial compression mechanical behavior of metal foam, in which the nominal stress-strain curve is expressed as(8)σn=σp+γεnεD+α2ln11−εn/εDβ.In (8), εn is the nominal strain, σp, γ, α2, and β are the material parameters, determined through experiments, and εD is the compaction strain, which is defined as(9)εD=1−ρfρf0,ρf is the density of the metal foam, and ρf0 is the density of the matrix metal. The relative density can be calculated according to ρ=ρf/ρf0. The elastic strain is assumed to be negligibly small under uniaxial compression conditions so the true axial total strain ε equals the true axial plastic strain εT. In addition, it should be noted that the tensile strain is generally considered to be positive, the compressive strain is negative, and the common relationship between the true and nominal strain is generally εT=ln1+εn. In this work, only compression conditions are involved, and the compressive strain is taken as a positive value for the convenience of reading, so the transformation formula of the true axial plastic strain and nominal strain can be rewritten as εT=−ln1−εn.The transformation of the axial true stress and the nominal stress needs to consider the expansion of the cross section during the compression process. According to (5), we can obtain the transverse true strain.(10)ε2=∫νpdεT.Then, the axial true stress can be expressed as(11)σT=σnA0A=σnl02l2,where A and l are the area and side length of a unit on the cross section, respectively. Combining (8), (10), and (11), the true stress-true strain relation can be expressed as(12)σT=σp+γ1−e−εT/εD+α2ln1/1−1−e−εT/εDβe∫νpdεT2.The equation of the plastic Poisson’s ratioνp regarding the true axial plastic strain εT will be discussed in Section 3.2. We take the optimized Hanssen model in (12) as the hardening law of the modified D-F model.


## 2.1. The Deshpande–Fleck Model

The Deshpande–Fleck model assumes that metal foam exhibits similar mechanical properties under tension and compression conditions, which is contrary to the conclusion of other studies [14, 15]. As shown in Figure 1, the elliptical yield surface on the σe−σm stress plane is symmetric about the σe axis and evolves in a self-similar form. The Deshpande–Fleck model only considers the mechanical properties of metal foam under compression conditions, so this assumption can be accepted. The modified model proposed in this paper also only considers the compressive mechanical properties of foam metals.Figure 1
Yield surface for the Deshpande–Fleck foam model.Deshpande and Fleck [16] noted that when the equivalent plastic strain was taken as the independent variable, the prediction results of the constitutive model were more consistent with the experimental data. The definition of the equivalent plastic strain rate was given as σ^ε¯˙p=σijε˙ijp. σ^ is numerically equal to the true uniaxial yield stress Y associated with the current yield surface, and σij and ε˙ijp are the stress tensor and the plastic strain rate tensor, respectively, where j=1,2,3.Equations (1) and (2) are integrated to obtain(7)σe2+α2σm2−1+α32Y=0.Equation (7) contains only two parameters, the ellipticity α of the elliptic yield surface and the uniaxial yield stress Y. The plastic flow potential function of the D-F model is defined as shown in (3). The shape parameter φ of the plastic potential function in (3) is only related to the plastic Poisson’s ratio νp, as shown in (6). It is thus crucial to accurately measure the ellipticity α, the uniaxial yield stress Y, and the plastic Poisson’s ratio νp through experiments.


## 2.2. The Stress-Strain Relation under Uniaxial Compression

A plastic constitutive contains three parts. In addition to the yield criterion in (7) and flow rule in (3) and (4), it is also necessary to explain the law of hardening. The D-F model adopts the true stress-strain relationship in uniaxial compression as the hardening law. The elastic section of metal foam is short and negligible under a compressive load [29], so the ideal rigid plastic hardening(R-PH) model is often used to describe the constitutive behavior of metal foam under uniaxial compression. Hanssen [30] proposed an R-PH model to describe the uniaxial compression mechanical behavior of metal foam, in which the nominal stress-strain curve is expressed as(8)σn=σp+γεnεD+α2ln11−εn/εDβ.In (8), εn is the nominal strain, σp, γ, α2, and β are the material parameters, determined through experiments, and εD is the compaction strain, which is defined as(9)εD=1−ρfρf0,ρf is the density of the metal foam, and ρf0 is the density of the matrix metal. The relative density can be calculated according to ρ=ρf/ρf0. The elastic strain is assumed to be negligibly small under uniaxial compression conditions so the true axial total strain ε equals the true axial plastic strain εT. In addition, it should be noted that the tensile strain is generally considered to be positive, the compressive strain is negative, and the common relationship between the true and nominal strain is generally εT=ln1+εn. In this work, only compression conditions are involved, and the compressive strain is taken as a positive value for the convenience of reading, so the transformation formula of the true axial plastic strain and nominal strain can be rewritten as εT=−ln1−εn.The transformation of the axial true stress and the nominal stress needs to consider the expansion of the cross section during the compression process. According to (5), we can obtain the transverse true strain.(10)ε2=∫νpdεT.Then, the axial true stress can be expressed as(11)σT=σnA0A=σnl02l2,where A and l are the area and side length of a unit on the cross section, respectively. Combining (8), (10), and (11), the true stress-true strain relation can be expressed as(12)σT=σp+γ1−e−εT/εD+α2ln1/1−1−e−εT/εDβe∫νpdεT2.The equation of the plastic Poisson’s ratioνp regarding the true axial plastic strain εT will be discussed in Section 3.2. We take the optimized Hanssen model in (12) as the hardening law of the modified D-F model.


## 3. The Parameters in the Modified D-F Model

### 3.1. Fitting Parameters for the Hanssen Model

Quasi-static uniaxial compression experiments of closed-cell aluminum foam with a pure aluminum matrix were conducted. The relative densityρ values of the samples were measured to be 0.10, 0.15, and 0.20, and the sample size was 30 × 30 × 30 mm, as shown in Figure 2. To avoid errors caused by differences between individual samples, five samples were tested for each relative density. The average value of the stress-strain data of five samples with each relative density is taken as the effective value.Figure 2
Aluminum foam samples. (a)ρ=0.1. (b)ρ=0.15. (c)ρ=0.2.
(a)(b)(c)An MTS-810 hydraulic servo universal testing machine was used in the experiments. The whole experimental process is measured by the DIC system in real time, as shown in Figure3. DIC system has the advantages of no contact, a high measurement accuracy, full-field measurement, and strong anti-interference ability [31, 32].Figure 3
MTS-810 and DIC equipment.The loading head speed was set to 1.8 mm/min, corresponding to a quasi-static strain rate of 0.001s−1. When the axial loading reaches 24 mm, the nominal strain of the sample is 0.75, and the sample is essentially compacted. Figure 4 shows the compression process of three aluminum foams with different relative densities.Figure 4
Compression process of the aluminum foam samples. (a)ρ=0.1; (b)ρ=0.15; (c)ρ=0.2.
(a)(b)(c)The quasi-static stress-strain relationship obtained in the experiment was fitted using the Hanssen model through the CFtool module of MATLAB software. The discrete points in Figure5 are experimental effective values, and solid lines are fitting results. A good agreement is observed between the stress-strain responses obtained in the experiment and the fitting results, which proved the validity of the fitting. It should be noted that since the Hanssen model is an ideal rigid plastic hardening (R-PH) model, there is no elastic segment in the fitting curve.Figure 5
Uniaxial compression curve fitting of the Hanssen model.The parameters in (8) obtained by fitting are shown in Table 3. The material parameters involved in Table 3 are all related to the relative density ρ and can be expressed by (13) [30].(13)σp,γ,α2,1β=C0+C1ρfρf0n.Table 3
Parametersσp, γ, α2, and 1/β in equation (12).
ρσpMPaγα21/β0.101.5260.38112.360.13650.153.1821.56419.950.15030.207.9802.54034.810.2038The parameters in Table3 were substituted into (13), and the values of C0,C1,n were obtained by fitting with the relative density ρ, as shown in Table 4.Table 4
ParametersC0, C1, and n in equation (13).
σpMPaγα21/βC00000.1228C1750.289.07465.49.875n2.8292.1961.6213.0According to the data in Table4, we can obtain the uniaxial compressive stress-strain curves of aluminum foam samples with arbitrary relative density through (12) and (13).


### 3.2. Modified Equation of the Plastic Poisson’s Ratio

The plastic Poisson’s ratioνp, defined as the negative value of the ratio of the transverse and axial logarithmic strain rates, is difficult to measure. Some studies propose very different values. The uniaxial compression experiments in Section 3.1 were measured by the DIC system. Figure 6 shows the measurement process of the aluminum foam sample with a relative density of 0.15, and the serial numbers ①, ②, and ③ in Figure 6 are correspond to the numbers marked in Figure 7. The acquisition frequency of the system is 2 Hz, and the transverse true strain εTr and the axial true strain εT are calculated according to the acquisition results. To avoid errors caused by the nonuniformity of the samples, the average value of the calculation results of the five samples of each relative density aluminum foam is taken as the final effective value.Figure 6
Axial and transverse nominal strain contour maps obtained via DIC technology.Figure 7
Relationship between the transverse and axial true strain.Figure7 shows the corresponding relationship between the true transverse strain and the true axial strain of aluminum foam materials with different relative densities measured via DIC technology. By deriving the curve in Figure 7 to the true axial strain, the relationship between the instantaneous plastic Poisson’s ratio of the material and the true axial strain can be obtained, as shown in Figure 8. According to Figure 8, the plastic Poisson’s ratio of the aluminum foam with a higher relative density is larger. The plastic Poisson’s ratio of each relative density aluminum foam shows a trend of first decreasing and then increasing with an increase in the true axial strain, where the higher the relative density, the earlier the transition occurred.Figure 8
Variations in the plastic Poisson’s ratio with the axial true strain.As the compression progresses, the space occupied by the internal pores of the aluminum foam gradually decreases, until the compression reaches the limit. For an incompressible entity, the plastic Poisson’s ratio is always 0.5 [23]. Since the excellent energy absorption performance of the aluminum foam is mainly reflected in the platform stress section of plastic deformation when the axial nominal strain reaches 0.75, the aluminum foam is considered to be essentially compacted, and the experiment is stopped. Therefore, only a slight increase of the plastic Poisson’s ratio is observed at the end of the curve in Figure 8.Considering the variation characteristics of the plastic Poisson’s ratioνp with the true axial strain εT in Figure 8, it is assumed that the fitting equation is a quadratic polynomial function.(14)νp=b1εT2+b2εT+b3.Figure8 shows that the fitting effect is good. The fitting values of the parameters at each relative density in (14) are then shown in Table 5.Table 5
Fitting results of the parameters in equation (14).
ρb1b2b30.100.1301−0.30680.19210.150.1272−0.26710.23020.200.1250−0.22580.2714The parametersb1, b2, and b3 in (14) are regarded as functions of the relative density ρ and fitted by (15), which only contains two parameters k and n, and the fitting result is shown in Table 6.(15)b1,b2,b3=k+nlnρ.Table 6
Fitting results of the parameters in equation (15).
b1b2b3k0.1135−0.04260.4468n−0.00720.11570.1113According to equations (14) and (15) and the parameter data in Table 6, the relationship between the plastic Poisson’s ratio with respect to the relative density and the true axial strain can be established.


### 3.3. Modified Equation of Ellipticity

The ellipticity can be obtained by fitting the von Mises effective stressσe and the mean stress σm of the aluminum foam under different loading conditions through the ellipse standard equation. On the plane of σe-σm, the ellipticity can also be determined by any two asymmetric points on the elliptic yield surface, as shown in (16)α2=σe22−σe12σm12−σm22.The two points corresponding to the uniaxial compression and the hydrostatic compression conditions are the farthest on theσe−σm plane. The experimental data of uniaxial compression and hydrostatic compression are thus sufficient to accurately determine the ellipticity [7].The hydrostatic compression experiment of aluminum foam was conducted through 3D Voronoi and finite element simulation technology. 3D Voronoi technology has been proven to have a high accuracy in the simulation of metal foam materials [6, 29, 33, 34]. The commercial software LS-DYNA was used for the finite element calculation. The model size is 30 mm × 30 mm × 30 mm, as shown in Figure 9(a), and the cell wall thickness of the Voronoi model is uniform. The relative density of the model can be adjusted by changing the thickness of the cell wall. Finally, the relative densities of the aluminum foam samples are 0.1, 0.15, and 0.2, which are the same as the experimental samples. All the models are meshed with shell elements. To ensure the accuracy of the model, elements that were too small size and had sharp angles at the cell wall junction were deleted, and finally, there were approximately 282,200 S4R shell elements in each sample. The matrix material of the model is pure aluminum, and the rate-independent bilinear strengthening model is adopted. The specific parameters are shown in Table 7 [11], and the loading method is shown in Figure 9(b). Six rigid plates are set on the six surfaces of the aluminum foam cube sample. The two parallel rigid plates form a group. In each group, one rigid plate is fixed, and the other moves in opposite directions at a constant quasi-static velocity, and the velocity of each rigid plate can be adjusted independently. In this section, three groups of rigid plates are set to move at the same velocity to achieve quasi-static hydrostatic compression. A surface-to-surface contact was employed to simulate the contact between the rigid plate and the Voronoi model, and the coefficient of friction was set to zero. A single surface contact was employed to simulate the contact among cell walls, and the coefficient of friction was 0.02.Figure 9
Finite element model (a) and loading method (b) .
(a)(b)Table 7
Matrix material parameters.
DensityYoung’s modulus (GPa)Poisson’s ratioYield stress (MPa)Tangent modulus (MPa)2.7 g/cm3700.338030In order to prove the validity of the above mesoscopic finite element model, uniaxial simulation experiments were conducted with only two loading plates in theZ direction. Comparisons between the experimental results in Section 3.1 and the simulation results show a good agreement, as shown in Figure 10, which proves the effectiveness of the finite element model in simulating foam metal.Figure 10
Uniaxial compression results of experiments and simulations.Then, the hydrostatic compression simulation was conducted through the simultaneous loading of the six loading plates in Figure9(b). Five samples were calculated for each relative density, and the average value of the five stress-strain curves was taken as the effective value. The relationship between the hydrostatic compression stress and the real volumetric strain of the samples with three relative densities is shown in Figure 11.Figure 11
Hydrostatic pressure and the true volumetric strain diagram.Based on the uniaxial compression experiment in Section3.1 and the hydrostatic compression simulation experiment in this section, the Mises effective stress σe and the mean stress σm of three kinds of relative density aluminum foam samples are fitted using (16). Figure 12 shows the fitting results when the true equivalent plastic strain is 0.3, 0.6, 0.9, and 1.2. The abscissa and ordinate axes in Figure 12 are the normalized mean stress and normalized equivalent stress. σ0 is the uniaxial yield stress of samples, which is equal to σp. As the true equivalent plastic strain increases, the stress level rises, and the elliptical yield surface gradually expands.Figure 12
A schematic diagram of ellipse fittings for three relative density samples with the equivalent plastic strains of 0, 1.2, 1.7, and 2.0.The curve of the square of ellipticityα2 with respect to the true equivalent plastic strain is obtained from the fitting result of the elliptical yield surface, as shown in Figure 13. The square α2 of the ellipticity is hardly affected by the relative density. The α2 values of the three types of relative density aluminum foam samples are similar, and the variation trend is the same. With an increase in the true equivalent plastic strain, the ellipticity decreases first and then increases. According to the variation trend of α2 with the true equivalent plastic strain, the fitting equation is assumed to be composed of a cubic polynomial.(17)α2=d1ε¯Tp3+d2ε¯Tp2+d3ε¯Tp+d4.Figure 13
The relationship between ellipticity and the true equivalent plastic strain.The fitting results ared1=−0.5746, d2=1.567, d3=−1.435, and d4=0.6665. The fitting curve of (17) is shown in Figure 13, and the fitting result is good.


## 3.1. Fitting Parameters for the Hanssen Model

Quasi-static uniaxial compression experiments of closed-cell aluminum foam with a pure aluminum matrix were conducted. The relative densityρ values of the samples were measured to be 0.10, 0.15, and 0.20, and the sample size was 30 × 30 × 30 mm, as shown in Figure 2. To avoid errors caused by differences between individual samples, five samples were tested for each relative density. The average value of the stress-strain data of five samples with each relative density is taken as the effective value.Figure 2
Aluminum foam samples. (a)ρ=0.1. (b)ρ=0.15. (c)ρ=0.2.
(a)(b)(c)An MTS-810 hydraulic servo universal testing machine was used in the experiments. The whole experimental process is measured by the DIC system in real time, as shown in Figure3. DIC system has the advantages of no contact, a high measurement accuracy, full-field measurement, and strong anti-interference ability [31, 32].Figure 3
MTS-810 and DIC equipment.The loading head speed was set to 1.8 mm/min, corresponding to a quasi-static strain rate of 0.001s−1. When the axial loading reaches 24 mm, the nominal strain of the sample is 0.75, and the sample is essentially compacted. Figure 4 shows the compression process of three aluminum foams with different relative densities.Figure 4
Compression process of the aluminum foam samples. (a)ρ=0.1; (b)ρ=0.15; (c)ρ=0.2.
(a)(b)(c)The quasi-static stress-strain relationship obtained in the experiment was fitted using the Hanssen model through the CFtool module of MATLAB software. The discrete points in Figure5 are experimental effective values, and solid lines are fitting results. A good agreement is observed between the stress-strain responses obtained in the experiment and the fitting results, which proved the validity of the fitting. It should be noted that since the Hanssen model is an ideal rigid plastic hardening (R-PH) model, there is no elastic segment in the fitting curve.Figure 5
Uniaxial compression curve fitting of the Hanssen model.The parameters in (8) obtained by fitting are shown in Table 3. The material parameters involved in Table 3 are all related to the relative density ρ and can be expressed by (13) [30].(13)σp,γ,α2,1β=C0+C1ρfρf0n.Table 3
Parametersσp, γ, α2, and 1/β in equation (12).
ρσpMPaγα21/β0.101.5260.38112.360.13650.153.1821.56419.950.15030.207.9802.54034.810.2038The parameters in Table3 were substituted into (13), and the values of C0,C1,n were obtained by fitting with the relative density ρ, as shown in Table 4.Table 4
ParametersC0, C1, and n in equation (13).
σpMPaγα21/βC00000.1228C1750.289.07465.49.875n2.8292.1961.6213.0According to the data in Table4, we can obtain the uniaxial compressive stress-strain curves of aluminum foam samples with arbitrary relative density through (12) and (13).


## 3.2. Modified Equation of the Plastic Poisson’s Ratio

The plastic Poisson’s ratioνp, defined as the negative value of the ratio of the transverse and axial logarithmic strain rates, is difficult to measure. Some studies propose very different values. The uniaxial compression experiments in Section 3.1 were measured by the DIC system. Figure 6 shows the measurement process of the aluminum foam sample with a relative density of 0.15, and the serial numbers ①, ②, and ③ in Figure 6 are correspond to the numbers marked in Figure 7. The acquisition frequency of the system is 2 Hz, and the transverse true strain εTr and the axial true strain εT are calculated according to the acquisition results. To avoid errors caused by the nonuniformity of the samples, the average value of the calculation results of the five samples of each relative density aluminum foam is taken as the final effective value.Figure 6
Axial and transverse nominal strain contour maps obtained via DIC technology.Figure 7
Relationship between the transverse and axial true strain.Figure7 shows the corresponding relationship between the true transverse strain and the true axial strain of aluminum foam materials with different relative densities measured via DIC technology. By deriving the curve in Figure 7 to the true axial strain, the relationship between the instantaneous plastic Poisson’s ratio of the material and the true axial strain can be obtained, as shown in Figure 8. According to Figure 8, the plastic Poisson’s ratio of the aluminum foam with a higher relative density is larger. The plastic Poisson’s ratio of each relative density aluminum foam shows a trend of first decreasing and then increasing with an increase in the true axial strain, where the higher the relative density, the earlier the transition occurred.Figure 8
Variations in the plastic Poisson’s ratio with the axial true strain.As the compression progresses, the space occupied by the internal pores of the aluminum foam gradually decreases, until the compression reaches the limit. For an incompressible entity, the plastic Poisson’s ratio is always 0.5 [23]. Since the excellent energy absorption performance of the aluminum foam is mainly reflected in the platform stress section of plastic deformation when the axial nominal strain reaches 0.75, the aluminum foam is considered to be essentially compacted, and the experiment is stopped. Therefore, only a slight increase of the plastic Poisson’s ratio is observed at the end of the curve in Figure 8.Considering the variation characteristics of the plastic Poisson’s ratioνp with the true axial strain εT in Figure 8, it is assumed that the fitting equation is a quadratic polynomial function.(14)νp=b1εT2+b2εT+b3.Figure8 shows that the fitting effect is good. The fitting values of the parameters at each relative density in (14) are then shown in Table 5.Table 5
Fitting results of the parameters in equation (14).
ρb1b2b30.100.1301−0.30680.19210.150.1272−0.26710.23020.200.1250−0.22580.2714The parametersb1, b2, and b3 in (14) are regarded as functions of the relative density ρ and fitted by (15), which only contains two parameters k and n, and the fitting result is shown in Table 6.(15)b1,b2,b3=k+nlnρ.Table 6
Fitting results of the parameters in equation (15).
b1b2b3k0.1135−0.04260.4468n−0.00720.11570.1113According to equations (14) and (15) and the parameter data in Table 6, the relationship between the plastic Poisson’s ratio with respect to the relative density and the true axial strain can be established.


## 3.3. Modified Equation of Ellipticity

The ellipticity can be obtained by fitting the von Mises effective stressσe and the mean stress σm of the aluminum foam under different loading conditions through the ellipse standard equation. On the plane of σe-σm, the ellipticity can also be determined by any two asymmetric points on the elliptic yield surface, as shown in (16)α2=σe22−σe12σm12−σm22.The two points corresponding to the uniaxial compression and the hydrostatic compression conditions are the farthest on theσe−σm plane. The experimental data of uniaxial compression and hydrostatic compression are thus sufficient to accurately determine the ellipticity [7].The hydrostatic compression experiment of aluminum foam was conducted through 3D Voronoi and finite element simulation technology. 3D Voronoi technology has been proven to have a high accuracy in the simulation of metal foam materials [6, 29, 33, 34]. The commercial software LS-DYNA was used for the finite element calculation. The model size is 30 mm × 30 mm × 30 mm, as shown in Figure 9(a), and the cell wall thickness of the Voronoi model is uniform. The relative density of the model can be adjusted by changing the thickness of the cell wall. Finally, the relative densities of the aluminum foam samples are 0.1, 0.15, and 0.2, which are the same as the experimental samples. All the models are meshed with shell elements. To ensure the accuracy of the model, elements that were too small size and had sharp angles at the cell wall junction were deleted, and finally, there were approximately 282,200 S4R shell elements in each sample. The matrix material of the model is pure aluminum, and the rate-independent bilinear strengthening model is adopted. The specific parameters are shown in Table 7 [11], and the loading method is shown in Figure 9(b). Six rigid plates are set on the six surfaces of the aluminum foam cube sample. The two parallel rigid plates form a group. In each group, one rigid plate is fixed, and the other moves in opposite directions at a constant quasi-static velocity, and the velocity of each rigid plate can be adjusted independently. In this section, three groups of rigid plates are set to move at the same velocity to achieve quasi-static hydrostatic compression. A surface-to-surface contact was employed to simulate the contact between the rigid plate and the Voronoi model, and the coefficient of friction was set to zero. A single surface contact was employed to simulate the contact among cell walls, and the coefficient of friction was 0.02.Figure 9
Finite element model (a) and loading method (b) .
(a)(b)Table 7
Matrix material parameters.
DensityYoung’s modulus (GPa)Poisson’s ratioYield stress (MPa)Tangent modulus (MPa)2.7 g/cm3700.338030In order to prove the validity of the above mesoscopic finite element model, uniaxial simulation experiments were conducted with only two loading plates in theZ direction. Comparisons between the experimental results in Section 3.1 and the simulation results show a good agreement, as shown in Figure 10, which proves the effectiveness of the finite element model in simulating foam metal.Figure 10
Uniaxial compression results of experiments and simulations.Then, the hydrostatic compression simulation was conducted through the simultaneous loading of the six loading plates in Figure9(b). Five samples were calculated for each relative density, and the average value of the five stress-strain curves was taken as the effective value. The relationship between the hydrostatic compression stress and the real volumetric strain of the samples with three relative densities is shown in Figure 11.Figure 11
Hydrostatic pressure and the true volumetric strain diagram.Based on the uniaxial compression experiment in Section3.1 and the hydrostatic compression simulation experiment in this section, the Mises effective stress σe and the mean stress σm of three kinds of relative density aluminum foam samples are fitted using (16). Figure 12 shows the fitting results when the true equivalent plastic strain is 0.3, 0.6, 0.9, and 1.2. The abscissa and ordinate axes in Figure 12 are the normalized mean stress and normalized equivalent stress. σ0 is the uniaxial yield stress of samples, which is equal to σp. As the true equivalent plastic strain increases, the stress level rises, and the elliptical yield surface gradually expands.Figure 12
A schematic diagram of ellipse fittings for three relative density samples with the equivalent plastic strains of 0, 1.2, 1.7, and 2.0.The curve of the square of ellipticityα2 with respect to the true equivalent plastic strain is obtained from the fitting result of the elliptical yield surface, as shown in Figure 13. The square α2 of the ellipticity is hardly affected by the relative density. The α2 values of the three types of relative density aluminum foam samples are similar, and the variation trend is the same. With an increase in the true equivalent plastic strain, the ellipticity decreases first and then increases. According to the variation trend of α2 with the true equivalent plastic strain, the fitting equation is assumed to be composed of a cubic polynomial.(17)α2=d1ε¯Tp3+d2ε¯Tp2+d3ε¯Tp+d4.Figure 13
The relationship between ellipticity and the true equivalent plastic strain.The fitting results ared1=−0.5746, d2=1.567, d3=−1.435, and d4=0.6665. The fitting curve of (17) is shown in Figure 13, and the fitting result is good.


## 4. Results and Discussion

### 4.1. Modified D-F Model and Its Simulation Verification

After the above discussion, (14) of the plastic Poisson’s ratio with respect to the relative density and the true axial strain and (17) of ellipticity with respect to equivalent plastic strain are obtained. Based on the nonassociated flow rule, the D-F model is modified by (14) and (17). Then, the modified model is embedded in the ABAQUS solver through the subroutine VUMAT in ABAQUS/Explicit. In the subroutine realization process, the explicit integration algorithm of forward Euler iteration is adopted. The equivalent plastic strain is used as the independent variable to update the stress. In the implementation of subroutine, the explicit integration algorithm of forward Euler iteration is adopted. The equivalent plastic strain is used as the independent variable to update the stress. At each moment t during the numerical calculation process, the ABAQUS solver first calculates the strain increment at the material integral point according to the boundary conditions and then transfers the time increment Δt and strain increment into the VUMAT subroutine to calculate each variable at the moment t+Δt. Finally, the subroutine returns the updated variables to the ABAQUS solver.To prove the accuracy of the modified D-F model, numerical simulations of aluminum foam under bi-/triaxial compression were conducted with finite element code ABAQUS/Explicit. The sample is also a 30 × 30 × 30 mm3 cube with 8000 solid elements, see Figure 14. For triaxial compression, two loading conditions are considered: hydrostatic compression with the velocity ratio VX:VY:VZ=1:1:1 and proportional compression with the velocity ratio VX:VY:VZ=2:2:1. For biaxial compression, two loading conditions are also considered: biaxial loading with the velocity ratio VX:VY:VZ=1:1:− and proportional loading with the velocity ratio VX:VY:VZ=2:1:−, where “−” represents a free end (without constrain). There is no friction between the loading plates and the sample.Figure 14
Schematic diagram of the multiaxial loading of the D-F model and the modified D-F model: (a) the triaxial compression and (b) the biaxial compression.
(a)(b)The D-F model uses the constantνp=0.17 [23] and α2=2.54 according to Table 1. The modified D-F model uses variable plastic Poisson’s ratio in (14) and ellipticity in (17). As for the hardening law, the original model uses experimental discrete point data, and the modified model uses the Hanssen stress-strain relationship in (12).The prediction results of the two models for biaxial and triaxial loading conditions are shown in Figures15 and 16. It can be seen that the accuracy of the modified model is always better than that of the original model, especially under biaxial loading conditions. In addition, according to Figures 15 and 16, the higher the relative density, the more obvious the superiority of the modified model. This is because the effect of the plastic Poisson’s ratio will become more and more significant as the relative density of the foam material increases, which also confirms the importance of accurately measuring the Poisson’s ratio through experiments.Figure 15
Predictions of triaxial loading conditions by the D-F model and the modified D-F model when the velocity ratio of three principal stress directions is (a)1 : 1:1 and (b)2 : 2:1.
(a)(b)Figure 16
Predictions of biaxial loading conditions by the D-F model and the modified D-F model when the velocity ratio of three principal stress directions is (a) 1 : 1:- and (b) 2 : 1:-.
(a)(b)


### 4.2. Verification of Experiments

Two kinds of experiments, that is, uniaxial compression and hydrostatic compression, were conducted by Sridhar and Fleck [17]. The closed-cell Alulight aluminum foam was used, and the relative densities of the samples were 0.12, 0.22, 0.25, and 0.3. The uniaxial compression is completed by the Instron testing machine, and the hydrostatic compression is loaded by a 20 MPa manual hydraulic oil pump. Quasi-static strain rates of 10−3s−1 were adopted for both loading conditions. The uniaxial compression and hydrostatic compression stress-strain curves of samples with relative densities of 0.22 and 0.3 are given in the literature, as shown in Figure 17. The square of ellipticity, α2, calculated from the experimental data is plotted in Figure 18, which has similar variation characteristics with Figure 13. So, we also take equation (17) to fit α2, and the fitting equation is(18)α2=−2.201ε¯Tp3+4.096ε¯Tp2−2.267ε¯Tp+0.861.Figure 17
Uniaxial and hydrostatic stress-strain curves for closed-cell aluminum foams with different relative densities.Figure 18
The square of ellipticity for two kinds of aluminum foams and their fitting results.We predict the hydrostatic pressure of samples with relative densities of 0.12 and 0.25. It should be noted that the modified model adopts the varying plastic Poisson’s ratio in (14) and ellipticity in (17), while the original D-F model adopts the constant νp=0.17 [23] and α2=2.54 according to Table 1. In addition, only two uniaxial compressive stress-strain relations with relative densities of 0.22 and 0.3 are given in Ref. [17], which results the infeasibility of fitting the Hanssen model. Thus, the two kinds of models both adopt the scattered data in the uniaxial compression tests as the hardening law. The comparison between the predicted results and experimental results in Ref. [17] is shown in Figure 19. The results show that the prediction of the modified model with variable Poisson’s ratio and variable ellipticity is in better agreement with the experimental data, especially when the strain exceeds 0.2.Figure 19
Predictions of pressure under hydrostatic compression according to the D-F model and modified D-F model: (a)ρ=0.12 and (b)ρ=0.25.
(a)(b)


### 4.3. The Influence of Ellipticity and the Plastic Poisson’s Ratio on the Modified D-F Model

The shape of the yield surface of the modified D-F model is controlled by the ellipticity, and the shape of the plastic flow potential surface is controlled by the plastic Poisson’s ratio. The accuracy of the modified D-F model is proven in Section4.1 and 4.2. However, the influence of the ellipticity and the plastic Poisson’s ratio on the prediction accuracy of the modified D-F model is not shown separately. The modified D-F model is based on the nonassociated flow rule, and the ellipticity and plastic Poisson’s ratio are calculated independently. This section analyzes and compares the influence of the ellipticity and the plastic Poisson’s ratio on the prediction accuracy by a control study.In this section, the relative density of the aluminum foam sample is 0.2, and the hydrostatic compression condition is applied. The model size is the same as that in Figure14 in Section 4.1. Two types of D-F models are used. In the first type, the ellipticity α is taken as the variable value, which is obtained by (17), and the plastic Poisson’s ratio νp is taken as the fixed values of 0 [7], 0.17 [23], and 0.3 [11] given in the literature. In the second type, the plastic Poisson’s ratio νp is taken as the variable value, obtained by (14), and the square of the ellipticity α2 is taken as the fixed values 4.5, 1.6, and 0.6, which are obtained from Table 1 using the plastic Poisson’s ratio νp=0, Poisson’s ratio ν=0.27, and the experimental values of σ0 and p0, respectively. The prediction results of the above two types of D-F models and the modified D-F model proposed in Section 4.1 are compared with the experimental data, as shown in Figures 20 and 21.Figure 20
Predictions of the pressure under hydrostatic compression according to the first-type D-F model and the modified D-F model.Figure 21
Predictions of the pressure under hydrostatic compression according to the second-type D-F model and the modified D-F model.The pressure under hydrostatic compression predicted by the first-type D-F model and the modified D-F model is compared with FE results in Figure20. Figure 20 shows that different fixed-value plastic Poisson’s ratios will not affect the initial yield of the sample. After the sample enters the plastic deformation section, the predictions of the three fixed plastic Poisson’s ratios begin to show errors, and with an increasing strain, the predictions of the three plastic Poisson’s ratios become increasingly dispersed. However, the prediction of the modified D-F model is always satisfactory. This indicates that the effect of the plastic Poisson’s ratio on the D-F model appears in the plastic compression stage. The aluminum foam sample possesses both volumetric plastic flow and shear flow when the plasticity is initiated [35], so the influence of the plastic Poisson’s ratio begins to appear.The pressure under hydrostatic compression predicted by the second-type D–F model and the modified D-F model is compared with the FE results in Figure21. According to Figure 21, the prediction accuracy of the modified D-F model is better than that of the fixed ellipticity model. The value of the fixed ellipticity will directly affect the initial yield position and the subsequent yield in the experiment. The ellipticity of 0.6 calculated by Table 1 can accurately predict the initial yield point, but the error gradually becomes larger after the true equivalent plastic strain reaches 0.5. However, the predictions of the other two fixed ellipticities have a large error with the simulation experiment results, whether it is the initial yielding or the subsequent yielding stage.Ellipticity and the plastic Poisson’s ratio have different influences on the accuracy of the D-F model under the nonassociated flow rule. The modified D-F model can obtain a higher accuracy only by considering the two parameters as the variable values.


## 4.1. Modified D-F Model and Its Simulation Verification

After the above discussion, (14) of the plastic Poisson’s ratio with respect to the relative density and the true axial strain and (17) of ellipticity with respect to equivalent plastic strain are obtained. Based on the nonassociated flow rule, the D-F model is modified by (14) and (17). Then, the modified model is embedded in the ABAQUS solver through the subroutine VUMAT in ABAQUS/Explicit. In the subroutine realization process, the explicit integration algorithm of forward Euler iteration is adopted. The equivalent plastic strain is used as the independent variable to update the stress. In the implementation of subroutine, the explicit integration algorithm of forward Euler iteration is adopted. The equivalent plastic strain is used as the independent variable to update the stress. At each moment t during the numerical calculation process, the ABAQUS solver first calculates the strain increment at the material integral point according to the boundary conditions and then transfers the time increment Δt and strain increment into the VUMAT subroutine to calculate each variable at the moment t+Δt. Finally, the subroutine returns the updated variables to the ABAQUS solver.To prove the accuracy of the modified D-F model, numerical simulations of aluminum foam under bi-/triaxial compression were conducted with finite element code ABAQUS/Explicit. The sample is also a 30 × 30 × 30 mm3 cube with 8000 solid elements, see Figure 14. For triaxial compression, two loading conditions are considered: hydrostatic compression with the velocity ratio VX:VY:VZ=1:1:1 and proportional compression with the velocity ratio VX:VY:VZ=2:2:1. For biaxial compression, two loading conditions are also considered: biaxial loading with the velocity ratio VX:VY:VZ=1:1:− and proportional loading with the velocity ratio VX:VY:VZ=2:1:−, where “−” represents a free end (without constrain). There is no friction between the loading plates and the sample.Figure 14
Schematic diagram of the multiaxial loading of the D-F model and the modified D-F model: (a) the triaxial compression and (b) the biaxial compression.
(a)(b)The D-F model uses the constantνp=0.17 [23] and α2=2.54 according to Table 1. The modified D-F model uses variable plastic Poisson’s ratio in (14) and ellipticity in (17). As for the hardening law, the original model uses experimental discrete point data, and the modified model uses the Hanssen stress-strain relationship in (12).The prediction results of the two models for biaxial and triaxial loading conditions are shown in Figures15 and 16. It can be seen that the accuracy of the modified model is always better than that of the original model, especially under biaxial loading conditions. In addition, according to Figures 15 and 16, the higher the relative density, the more obvious the superiority of the modified model. This is because the effect of the plastic Poisson’s ratio will become more and more significant as the relative density of the foam material increases, which also confirms the importance of accurately measuring the Poisson’s ratio through experiments.Figure 15
Predictions of triaxial loading conditions by the D-F model and the modified D-F model when the velocity ratio of three principal stress directions is (a)1 : 1:1 and (b)2 : 2:1.
(a)(b)Figure 16
Predictions of biaxial loading conditions by the D-F model and the modified D-F model when the velocity ratio of three principal stress directions is (a) 1 : 1:- and (b) 2 : 1:-.
(a)(b)


## 4.2. Verification of Experiments

Two kinds of experiments, that is, uniaxial compression and hydrostatic compression, were conducted by Sridhar and Fleck [17]. The closed-cell Alulight aluminum foam was used, and the relative densities of the samples were 0.12, 0.22, 0.25, and 0.3. The uniaxial compression is completed by the Instron testing machine, and the hydrostatic compression is loaded by a 20 MPa manual hydraulic oil pump. Quasi-static strain rates of 10−3s−1 were adopted for both loading conditions. The uniaxial compression and hydrostatic compression stress-strain curves of samples with relative densities of 0.22 and 0.3 are given in the literature, as shown in Figure 17. The square of ellipticity, α2, calculated from the experimental data is plotted in Figure 18, which has similar variation characteristics with Figure 13. So, we also take equation (17) to fit α2, and the fitting equation is(18)α2=−2.201ε¯Tp3+4.096ε¯Tp2−2.267ε¯Tp+0.861.Figure 17
Uniaxial and hydrostatic stress-strain curves for closed-cell aluminum foams with different relative densities.Figure 18
The square of ellipticity for two kinds of aluminum foams and their fitting results.We predict the hydrostatic pressure of samples with relative densities of 0.12 and 0.25. It should be noted that the modified model adopts the varying plastic Poisson’s ratio in (14) and ellipticity in (17), while the original D-F model adopts the constant νp=0.17 [23] and α2=2.54 according to Table 1. In addition, only two uniaxial compressive stress-strain relations with relative densities of 0.22 and 0.3 are given in Ref. [17], which results the infeasibility of fitting the Hanssen model. Thus, the two kinds of models both adopt the scattered data in the uniaxial compression tests as the hardening law. The comparison between the predicted results and experimental results in Ref. [17] is shown in Figure 19. The results show that the prediction of the modified model with variable Poisson’s ratio and variable ellipticity is in better agreement with the experimental data, especially when the strain exceeds 0.2.Figure 19
Predictions of pressure under hydrostatic compression according to the D-F model and modified D-F model: (a)ρ=0.12 and (b)ρ=0.25.
(a)(b)


## 4.3. The Influence of Ellipticity and the Plastic Poisson’s Ratio on the Modified D-F Model

The shape of the yield surface of the modified D-F model is controlled by the ellipticity, and the shape of the plastic flow potential surface is controlled by the plastic Poisson’s ratio. The accuracy of the modified D-F model is proven in Section4.1 and 4.2. However, the influence of the ellipticity and the plastic Poisson’s ratio on the prediction accuracy of the modified D-F model is not shown separately. The modified D-F model is based on the nonassociated flow rule, and the ellipticity and plastic Poisson’s ratio are calculated independently. This section analyzes and compares the influence of the ellipticity and the plastic Poisson’s ratio on the prediction accuracy by a control study.In this section, the relative density of the aluminum foam sample is 0.2, and the hydrostatic compression condition is applied. The model size is the same as that in Figure14 in Section 4.1. Two types of D-F models are used. In the first type, the ellipticity α is taken as the variable value, which is obtained by (17), and the plastic Poisson’s ratio νp is taken as the fixed values of 0 [7], 0.17 [23], and 0.3 [11] given in the literature. In the second type, the plastic Poisson’s ratio νp is taken as the variable value, obtained by (14), and the square of the ellipticity α2 is taken as the fixed values 4.5, 1.6, and 0.6, which are obtained from Table 1 using the plastic Poisson’s ratio νp=0, Poisson’s ratio ν=0.27, and the experimental values of σ0 and p0, respectively. The prediction results of the above two types of D-F models and the modified D-F model proposed in Section 4.1 are compared with the experimental data, as shown in Figures 20 and 21.Figure 20
Predictions of the pressure under hydrostatic compression according to the first-type D-F model and the modified D-F model.Figure 21
Predictions of the pressure under hydrostatic compression according to the second-type D-F model and the modified D-F model.The pressure under hydrostatic compression predicted by the first-type D-F model and the modified D-F model is compared with FE results in Figure20. Figure 20 shows that different fixed-value plastic Poisson’s ratios will not affect the initial yield of the sample. After the sample enters the plastic deformation section, the predictions of the three fixed plastic Poisson’s ratios begin to show errors, and with an increasing strain, the predictions of the three plastic Poisson’s ratios become increasingly dispersed. However, the prediction of the modified D-F model is always satisfactory. This indicates that the effect of the plastic Poisson’s ratio on the D-F model appears in the plastic compression stage. The aluminum foam sample possesses both volumetric plastic flow and shear flow when the plasticity is initiated [35], so the influence of the plastic Poisson’s ratio begins to appear.The pressure under hydrostatic compression predicted by the second-type D–F model and the modified D-F model is compared with the FE results in Figure21. According to Figure 21, the prediction accuracy of the modified D-F model is better than that of the fixed ellipticity model. The value of the fixed ellipticity will directly affect the initial yield position and the subsequent yield in the experiment. The ellipticity of 0.6 calculated by Table 1 can accurately predict the initial yield point, but the error gradually becomes larger after the true equivalent plastic strain reaches 0.5. However, the predictions of the other two fixed ellipticities have a large error with the simulation experiment results, whether it is the initial yielding or the subsequent yielding stage.Ellipticity and the plastic Poisson’s ratio have different influences on the accuracy of the D-F model under the nonassociated flow rule. The modified D-F model can obtain a higher accuracy only by considering the two parameters as the variable values.


## 5. Conclusion

The self-similar isotropic hardening foam constitutive proposed by Deshpande and Fleck is widely used to describe the constitutive behavior of compressible foam materials due to its concise and effective characteristics. In this paper, combined with DIC technology, a uniaxial compression experiment of aluminum foam was conducted. The simulation experiments of multiaxial compressions were conducted based on the Voronoi model. Based on the experiments, a modified D-F model for multiaxial compression is proposed. The specific research content is summarized as follows.The uniaxial compression experiments of three kinds of relative density aluminum foam samples were conducted by an MST-810 universal testing machine. Combined with DIC technology, the plastic Poisson’s ratioνp of the aluminum foam was measured. It was determined that the greater the relative density, the greater the plastic Poisson’s ratio. The plastic Poisson’s ratio decreases with an increasing true axial strain and then increases slightly before densification. The fitting equation of the plastic Poisson’s ratio with respect to the relative density and the true axial strain was proposed. The Hanssen model fitted with uniaxial compression data was modified by considering the effect of plastic Poisson’s ratio.Voronoi technology was used to conduct hydrostatic compression simulation experiments of aluminum foams. The ellipticityα was determined by the experimental data of uniaxial compression and hydrostatic compression. It was found that the ellipticity is independent of the relative density and is not a fixed value but varies with the equivalent plastic strain. A fitting equation of the ellipticity with respect to the equivalent plastic strain is proposed.Based on the nonassociated flow rule, a modified D-F model considering a variable plastic Poisson’s ratio and a variable ellipticity is proposed by the subroutine VUMAT in ABAQUS/Explicit. Through the bi-/triaxial compression simulation experiments, good agreement is observed between the experimental data and the predictions of the modified D-F model. The influence mechanism of plastic Poisson’s ratio and ellipticity on the predictions of the modified D-F model were analyzed. The predictions are not accurate when either parameter is invariable. The accuracy of the prediction of the modified D-F model can only be guaranteed when both the ellipticity and the plastic Poisson’s ratio are taken as variable values.



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*Source: 9235106-2022-03-16.xml*