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Characterization of the compressive deformation behavior with strain rate effect of low-density polymeric foams
Polymer Testing 50 (2016) 1e8
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Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Material behaviour
Characterization of the compressive deformation behavior with strain rate effect of low-density polymeric foams Akio Yonezu a, *, Keita Hirayama a, Hiroshi Kishida a, Xi Chen b, c, ** a
Department of Precision Mechanics, Chuo University, 1-13-27 Kasuga, Bunkyo, Tokyo 112-8551, Japan International Center for Applied Mechanics, SV Lab, School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, China c Department of Earth and Environmental Engineering, Columbia University 500 W 120th Street, New York, NY 10027, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 October 2015 Accepted 27 November 2015 Available online 2 December 2015
This study investigates the compressive deformation behavior of a low-density polymeric foam at different strain rates. The material tested has micron-sized pores with a closed cell structure. The porosity is about 94%. During a uni-axial compressive test, the macroscopic stressestrain curve indicates a plateau region during plastic deformation. Finite Element Method (FEM) simulation was carried out, in which the yield criterion considered both components of Mises stress and hydrostatic stress. By using the present FEM and experimental data, we established a computational model for the plastic deformation behavior of porous material. To verify our model, several indentation experiments with different indenters (spherical indentation and wedge indentation) were carried out to generate various tri-axial stress states. From the series of experiments and computations, we observed good agreement between the experimental data and that generated by the computational model. In addition, the strain rate effect is examined for a more reliable prediction of plastic deformation. Therefore, the present computational model can predict the plastic deformation behavior (including time-dependent properties) of porous material subjected to uni-axial compression and indentation loadings. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Porous material Compressive deformation behavior Strain rate Indentation Finite element method
1. Introduction Porous polymers with a low density are widely used for contact/ impact absorption and heat/acoustic insulation because of their good energy absorbing properties, good vibration attenuation and thermal/acoustic insulation [1e5]. In use, compressive loadings, including indentation and low velocity impacts by foreign objects, are often applied [6], and quasi-static indentation tests have been used to understand the low velocity impact response of composites [7] and protective coatings [8,9]. Consequently, analytical and numerical modeling for porous polymer materials is necessary in order to predict their deformation behavior, especially large plastic deformation. In general, the mechanical response is strongly dependent on inherent porous structure, polymer type and loading conditions. Indentation loading usually induces a complicated
* Corresponding author. ** Corresponding author. International Center for Applied Mechanics, SV Lab, School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, China. E-mail addresses: [email protected] (A. Yonezu), xichen@columbia. edu (X. Chen). http://dx.doi.org/10.1016/j.polymertesting.2015.11.021 0142-9418/© 2015 Elsevier Ltd. All rights reserved.
stress state compared with that of uni-axial loading. Such a complicated tri-axial stress may make modeling of macroscopic deformation more difficult when using an analytical/theoretical approach. Several analytical models have been developed to predict indentation resistance during quasi-static indentation of aluminum foams [10e12] and for polymeric foams [2]. Olurin et al. used an indentation test on aluminum foams to obtain material properties, i.e. plateau stress and tear energy [10]. Flores-Johnson et al. investigated the indentation responses of polymeric foam (polymethacrylimide and polyetherimide) with an analytical model based on experimental observation [2]. The deformation behavior was described based on total resistance force (which consisted of a crushing force, tearing force and friction force) developed during indentation loading. The analytical model was successfully established in order to calculate a resistance force during an indentation test. They verified the proposed model by conducting several indentation tests with different indenter shapes [2]. On the other hand, numerical simulations of quasi-static loading, including indentation into metallic polymeric foams, have been explored based on a continuum model of plasticity (constitutive model of plastic deformation). In fact, plastic theory
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A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
for porous media based on continuum models has been investigated. Miller has proposed a continuum plasticity framework for metallic foams [13]. The Drucker-Prager yield criterion was modified and introduced three adjustable parameters to match the yield surface from experimental data. These data are the uniaxial tensile and compressive yield stresses, and the ratio of radial to axial plastic strain rate, i.e. the “plastic Poisson's ratio”. An associated flow rule is also assumed to give a constitutive law of the plastic behavior. More simply, Deshpande and Fleck established a new yield criterion for porous media, which can be applied to both open cell and closed cell low density aluminum foam [14]. It is a very simple formula, which relies on only the plastic Poisson's ratio as the material parameter. These proposed criteria were verified based on the experimental data of yield surface for aluminum foam [15] and polymer foam [16]. Furthermore, the Deshpande and Fleck criterion was systematically investigated with regard to its applicability to plastic flow behavior with an assumption of associate flow [14,17,18]. Thus, this criterion may be readily employed to describe the plastic deformation of porous media with the aid of a numerical approach (e.g. FEM). As mentioned above, indentation loading (similar to low velocity impact) is critical for the use of porous materials, and it usually produces a strain rate effect in practice. Therefore, the modeling of plastic deformation, including indentation and rate-dependent behavior, is necessary for material design with a cellular structure. This study is motivated by the lack of a computational framework for the indentation mechanics with strain rate effect of lowdensity polymeric foams. In order to establish computational modeling for indentation responses subjected to various indenters with different loading rates, the yield criterion and plastic flow behavior proposed by Deshpande and Fleck [14] is used in the FEM computation. Note that strain rate effect is not included in the previous model [14]. Based on this, we will explore a computational framework, including the strain rate effect. In the experiment, a uniaxial compression test was first carried out to obtain the macroscopic stressestrain curve and the plastic Poisson's ratio. This leads to yield strength, work-hardening rate and the material parameter for the Deshpande-Fleck criterion. Subsequently, indentation tests were carried out on the porous polymer. Here, we used two indenters with different shapes (i.e. spherical type and wedge type) and different loading rates, to induce various indentation responses. Finally, we investigated the feasibility of our computational modeling, i.e., whether it can predict the indentation response obtained by the experiments. Our phenomenological approach may be more useful for explaining the indentation response (heterogeneous deformation) of low-density foam compared with the above analytical model based on the crush and fracture energy model. In Section 2, the material properties of the present polymer foam will be presented. Section 3 investigates the response to the uni-axial compression test. Section 4 describes the theory of yield criterion and plastic flow in order to establish the computational model for plastic deformation. Subsequently, the verification of our computational modeling through several indentation experiments is described. 2. Material and experimental procedure The material used in this study was commercial porous polymer material for heat/thermal insulating and impact absorption, more specifically a board of Porous Polypropylene (PP) with a closed cell structure (Zetlon®, Sekisui Chemical Co. Ltd) and thickness of 10 mm. Test specimens were taken from the board. The specimen size was different depending on the loading type (uni-axial compression and indentation), which will be described later. Fig. 1
Fig. 1. Pictures of the present specimen: (a) macroscopic picture; (b) X-ray CT image; and (c) surface layer observed through an optical microscope.
(a) shows a macroscopic picture of the specimen. To observe the micro structure, the X-ray CT technique (micro-CT system, SkyScan 1172) was utilized. The specimens were scanned with the micro-CT system, whose scanning parameters were set as X-ray source voltage of 59 kV, image pixel size of 4.08 mm and rotation step of 0.2 . Fig. 1(b) shows the internal micro-CT cross-sectional images taken from the specimen. Many pores which seem to have elliptical geometry along the thickness direction were observed. A volume of 10 mm 10 mm 3 mm was used in order to measure the distribution of pore size. If the pore is assumed to be a spherical shape, its averaged diameter is 567 mm. Since the present specimen was of foam material, the structure was closed pore (closed-cell). The porosity was measured to be 94% based on the density measurement (the density of the PP material was 0.95 g/cm3). The foam had a surface layer of solid PP to protect the closed pore structure. To observe the cross section of a surface layer, the foam was potted in epoxy resin and the cross sectional surface was polished. Fig. 1 (c) shows the optical microscope image of the cross section. A surface layer of “solid” PP material was observed. Below the surface, a thin cell wall of PP ligament (to form the porous structure) was observed. This constitutes closed-cell PP foam. The thickness of the surface layer was measured in the range of about 5 mm length, resulting in an averaged thickness of 36 mm. The uniaxial compression test was performed on a universal testing machine with a ball screw type (LSC-1/30: Tokyo Testing Machine Inc.) using a 10 mm cube specimen. A compression jig with a flat face was used for uni-axial loading under displacement control. The rate of loading displacement was set to 1.7 102 mm/ s. For the measurement of displacement, two eddy current sensors (EX-305 and EX-201, Keyence) were used. These were mounted on the jigs so that they could directly measure the gap between the two compression jigs, which corresponds to the uniaxial deformation of the specimen. Subsequently, concentrated loading was applied to the specimen in order to investigate different mechanical conditions. This is typical indentation testing to produce a tri-axial stress state in the material. The stress state due to indentation is quite different to that of the uni-axial compression test. This study used two types of indenter whose geometries are spherical and wedge type. The
A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
3
Fig. 2. Nominal stress and strain curve of the present porous material during uni-axial compressive loading.
spherical indenter had a tip radius of 5 mm, which was made of bearing steel. For the wedge indenter, the tip angle was 90 and the width was 15 mm with a cemented carbide tip. The tip radius was observed to be 30 mm by using a Scanning Electron Microscope (SEM). The rate of loading displacement was set to 1.7 102 mm/s for both indenters. For the spherical indentation, the specimen size was 50 mm square and 10 mm thick, whereas that for the wedge indenter was 70 mm 12 mm 10 mm thickness. All tests were conducted at room temperature in air. 3. Uniaxial compressive test Fig. 2 shows the uni-axial stress and strain curves.1 In order to investigate the generality of the compressive test, we employed four specimens of identical size. Their data are plotted separately in this figure, where their results are identical, showing good reproducibility. During the initial stage, the stress linearly increases up to 0.6 MPa, and then it slightly decreases. After that, the stress appears almost constant, i.e. the plateau region. After a strain of 0.4, the stress increases gradually. Such a deformation trend is often observed in metallic and polymeric foams. It is expected that the yield stress would be about 0.6 MPa, and the plateau region corresponds to plastic, non-recoverable deformation. As shown in this figure, all tests show a similar trend. Note that the strain rate effect is present in many polymer materials and foam. Such a strain rate effect will be discussed in Section 4.4. To investigate elastic properties, a loading and unloading test was carried out.1 Below the yield stress of 0.6 MPa, repeated loading and unloading tests were carried out so that the specimen elastically deformed several times. Note that the maximum stress s was set to 0.25 MPa and it was unloaded down to 0.025 MPa (not complete unloading). At this minimum loading, the test was suspended for a long period of time (about 30 min) in order to terminate viscoelastic recovery of the foam. Thus, the interval time between each test is about 30 min. Fig. 3(a) shows the stress and strain curve during the repeated test. The test was conducted 11 times (the number of loading cycles was 11). At the first cycle, it appears that the loading jig partially contacted the specimen surface (it may not have been full contact), due to the roughness of the specimen and/or the misalignment of
1 The load was measured by the load cell and the displacement (of compressive deformation) was measured by the displacement sensor. The nominal stress was calculated from the measured load and initial cross-section area of specimen, and the nominal strain was obtained from the measured displacement and the initial thickness of specimen.
Fig. 3. Repeated loading-unloading test for measurement of Young's modulus: (a) nominal stressestrain curve during repeated testing; (b) the changes in Young's modulus as the test cycle was repeated.
the longitudinal axis. It is observed that the loading curve suddenly increases and has a hysteresis loop. Subsequent test cycles, however, show linear curves with little hysteresis loops. For the range between the nominal stresses of s ¼ 0.05 and 0.25 MPa, Young's modulus was measured. Fig. 3(b) shows the measured Young's modulus as a function of test cycle, and indicates that the value becomes constant when the test cycle increases. The last four data points were found to be stable, showing an average value of 39.2 MPa. In fact, the specimen was loaded beyond the yield stress of 0.6 MPa and the result of permanent deformation was observed. In Fig. 2 we observe a plateau region, indicating that the specimen underwent plastic deformation, possibly due to buckling of cell walls. Such a deformation mode is quite different from a homogenous solid material. For a macroscopic deformation mode, the changes in specimen volume are different. For a plastically deformed solid, the specimen volume does not change, whereas a porous material that includes foam undergoes apparent volume changes due to the buckling of the cell structure and collapse, leading to densification of its pore structure. Therefore, the plastic Poisson's ratio np is a key parameter when the deformation mechanics of an entirely foam material are investigated. As mentioned in Section 1, the plastic Poisson's ratio is important for the present yield criterion (which was proposed by Deshpande and Fleck and will be discussed later). Thus, both longitudinal and transverse strains were measured during the uni-axial compression test. Fig. 4 shows macroscopic pictures of the specimen during the compression test. Several marks were added to the specimen and we measured the distance between them to obtain transverse strain.
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Fig. 4. Changes in the plastic Poisson's ratio (C) and the parameter of a (B) with respect to the applied longitudinal compressive strain.
Therefore, the ratio of transverse strain and longitudinal strain yields the plastic Poisson's ratio np. Fig. 4 shows the plastic Poisson's ratio with respect to longitudinal strain (nominal strain), as indicated by the solid circle mark. Since this test was conducted five times, the averaged data with standard deviation are plotted in this figure. It is observed that the value of np appears almost constant, suggesting that it does not change in our deformation range.2 This value of np is employed for computational modeling in the next section.
The average value was a ¼ 1.98, which is employed in our computations in the next section. 4.2. Finite element method As mentioned above, the yield criterion of Eq. (1) was used for FEM computation in this study. Equivalent stress is given by Eq. (1). Similarly, when the associate rule is assumed equivalent plastic strain can be described in the following equation:
Z
4. Computational modeling
εp ≡ 4.1. Yield criterion Deshpande and Fleck phenomenologically developed a yield criterion that can cover the experimental data for the multi-axial yield behavior of open and closed-cell aluminum alloy foams [14]. They proposed a phenomenological yield surface and equivalent stress as given by:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1 s¼u 2 i s2e þ a2 s2m th 1þ a3
(1)
=
where se is the von Mises effective stress and sm is the mean stress. The yield criterion is s ¼ sY of the uniaxial tensile or compressive yield strength of the foam. It was also reported that the parameter a defines the shape of the yield surface. This is related to the plastic Poisson's ratio np, which can be calculated by: =
¼
2 a3
2 1þ a3
(2)
=
np ¼
ε_ p Tp ε_ L
12
=
where ε_ p is plastic strain rate and the subscripts indicate the loading direction, i.e. “L” is the longitudinal direction and “T” is the transverse direction. To determine the present yield surface, the parameter a (as described in Eq. (1)) is necessary. Thus, as shown in Fig. 4, this study measured the plastic Poisson's ratio np from the uniaxial compressive test, and converted it to the parameter a from Eq. (2). The parameter a was plotted in Fig.4 (with open marks), appearing almost constant within the present deformation range.
2 The constant plastic Poisson's ratio np suggested that the foam did not experience densification of its cellular structure owing to being a very low density foam.
b ε_ ≡
ffi Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 1 2 2 ε_ e þ 2 ε_ m 1þ 3 a
(3)
where b ε_ is the increment of strain (strain rate), ε_ e is the Mises effective plastic strain rate and ε_ m is the volumetric plastic strain rate. Equivalent plastic strain is the increment of the plastic strain rate. Here, we assumed that the hardening rule was governed by the following empirical equation:
s ¼ K$eðhg $εp Þ n
(4)
This is similar to the general hardening rule of the G'sell and Jonas law, which can describe non-linear responses of the stressestrain curve of a polymer material [19,20].3 When the plastic strain εp is 0 (at yield point), s becomes the yield stress (which corresponds to the value of K in Eq. (4)). As shown in Fig. 2, the apparent yield stress was 0.6 MPa, leading to K ¼ 0.6 MPa. The other unknown parameters (hg and n) were deduced from the comparison between the FEM computational data and the experimental data. This study created an axi-symmetric FEM model for uni-axial loading. As an example model, let the radius be 26 mm and thickness be 10 mm. Let the numbers of the element and node be 13,200 and 13,515, respectively. Then, let uni-axial loading under displacement control be applied to the model. Computations were carried out using the commercially available software, Marc (MSC, Marc 2012). Here, the elastic modulus was 39.2 MPa from Fig.3(b). Poisson's ratio (under elastic deformation) was set to 0.33,4 as a typical cellular material [4]. In addition, the yield criterion of Eq. (1) was employed and the associate rule was assumed. Such a plastic
3 Note that the strain rate effect is discussed in Section 4.4 where Eq. (4) is improved. 4 Elastic properties (E, G, n) of a typical cellular material with low-density is theoretically investigated based on bending of the cell frame [see Gibson- Asby, [4]].
A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
1.5 1.2 0.9
80
: Exp1 : Exp2 : Exp3 : Exp4 : FEM
, IndentaƟon force F, N
Nominal stress, MPa
1.8
0.6 0.3 0 0
0.2
5
0.4
0.6
0.8
60 50 40 30 20 10 0 0
Nominal strain
0.4
0.8
Indentation tests were carried out, as described in Section 2, in order to generate various stress fields for the verification of our computational model. Fig. 6 show the indentation curves of the spherical indenter and wedge indenter. Note that the wedge indentation responses were dependent on the width of the specimen. Thus, the indentation force was normalized by the specimen width, as shown in Fig.6 (b). For those experiments, several tests with different maximum loading were carried out, indicating that they are highly reproducible.5 To simulate the indentation responses, an FEM model was created, as shown in Fig.7. For the spherical indenter, an axisymmetric model with a thickness of 10 mm and radius of 40 mm was created. The numbers of elements and nodes were 27,528 and 27,994, respectively. The wedge indenter case was a two dimensional half model 10 mm thick, 47 mm long and 11 mm wide. Generalized plane strain condition was assumed. The numbers of the elements and nodes were 16,588 and 16,969, respectively. For both FEM models, the size of the specimen was very large (compared with the area of the indenter impression), and thus it could be considered as semi-infinite. It should be noted that, as shown in Fig.1(c), a solid PP layer (about 36 mm thick) existed on the top surface, since this layer may play a role in covering closed pores it may also affect indentation response. It is modeled in the present FEM computation (see Fig.7). As such, the present model consisted of two layers of materials; i.e. a top “solid” layer 36 mm thick and the bottom “foam” 10 mm thick. We assumed that the top layer was solid PP material, whose elastic properties were E ¼ 2350 MPa and Poisson's ratio n ¼ 0.5 [21]. Furthermore, this solid layer model employed the Mises yield criterion and the plastic property (stressestrain curve), which was obtained from a uni-axial
5 Mechanical response due to indentation (concentrated) loading seems to be dependent on indent place, since the present foam has many pores. Good reproducibility is still obtained, since the indenter size is significantly larger than the pore size.
1.6
2
2.4
(a) Normalized indentaƟon force F, N/mm
4.3. Model verifications through indentation loading
1.2
Displacement h, mm
Fig. 5. Nominal stress and strain curve of the experimental data of Fig. 2 and the computational data (indicated by an open circle mark) from FEM analysis.
constitutive behavior was embedded in the present FEM via a userdefined subroutine. In order to identify the material constants in Eq. (4), a parametric FEM study was carried out such that the FEM data matched the experimental data (in Fig.2). As shown in Fig.5, when hg ¼ 0.4 and n ¼ 1.7, the computational curve shows good agreement with the experimental curve. Thus, the material constants in Eq. (4) are K ¼ 0.6 MPa, hg ¼ 0.4 and n ¼ 1.7.
N N EXP N
: : : : FEM
70
4 3.5 3
: Exp (n=3) : FEM
2.5 2 1.5 1 0.5 0 0
0.4
0.8
1.2
1.6
Displacement h, mm
(b) Fig. 6. Indentation curves of (a) the spherical indenter and (b) the wedge indenter. The experimental data is indicated by a line and the computational data (from Fig. 7) is by an open circle mark.
Fig. 7. FEM model of indentation on the present porous material.
compressive test [21]. Here, the constitutive law of plastic deformation was employed in the following equation (which can effectively approximate the experimental stressestrain curve of [21]):
s½MPa ¼ 47 þ 40ε0:5 p
(5)
where εp is the plastic strain. The yield stress was 47 MPa. Subsequently, the foam (porous material) was assumed to obey the above yield criterion and plastic flow as described in Section 4.1 and 4.2 As shown in Fig.7, fine meshes were created around the contact region and a mesh converge test was carried out. Both indenters were assumed to be a rigid body. The friction coefficient was set to zero, which is reported to be a minor factor.
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A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
Fig. 8. Contour map of the equivalent stress field of (a) the spherical indenter and (b) the wedge indenter.
Fig. 9. Contour map of the equivalent strain field of (a) the spherical indenter and (b) the wedge indenter.
Fig. 10. Nominal stress and strain curve during uni-axial compressive test with different strain rates.
The computational indentation curves are plotted in Fig.6. The open mark of FEM shows good agreement with the experimental data for both indenters. Although the stress field under indentation loading is different to that under uni-axial loading, our FEM computation can predict the indentation deformation behavior. Finally, stress and strain distributions during indentation (with the spherical indenter and wedge indenter) were investigated. Fig.8 shows a contour map of equivalent stress seq, which is obtained from Eq. (1). Fig.9 shows a contour map of equivalent strain εeq, which is obtained from Eq. (3). It is found that both figures indicate a different distribution of seq and εeq. This indicates that a different indenter generates a different stress (and strain) distribution. Our computation, however, shows good agreement with the experimental data (see Fig.6). Thus, our FEM model is very useful for porous media subjected to various states of compressive loading. It is concluded that this study explores a computational model for predicting macroscopic plastic deformation of polymeric foam
A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
4.4. Strain rate effect
60 IndentaƟon force F, N
7
We conducted further investigation into the strain rate effect of plastic deformation using uni-axial compressive tests with different strain rates. Here, we selected three strain rates (8.3 102, 1.7 102, 8.3 104 1/s), spanning two orders of magnitude. Fig.10 shows the nominal stress and nominal strain curves for each strain rate. For each testing condition, three tests were carried out and the averaged data and standard deviation are plotted in this figure. It is found that the strain rate effect is prominent in the plastic deformation, which needs to be further justified into our computational model. As mentioned in Eq. (4), there are several constitutive equations for plastic deformation of polymer material. One simple equation is proposed as follows
25mm/min 1mm/min 0.05mm/min
45
n=3 30 15 0 0
0.5
1 1.5 Displacement h, mm
2
2.5
Fig. 11. Indentation curves of the spherical indenter with different loading rates. m s ¼ K$ehg εp $ε_ 2
under indentation. A uni-axial compression test was first carried out to obtain macroscopic stress and strain curves, yielding the apparent elastoplastic property and the plastic Poisson's ratio. These experimentally obtained parameters were used in our FEM model, which could simulate indentation responses to various indenters (i.e. spherical and wedge indenters). Note that the strain rate effect and time dependent property are often present in polymer materials, and thus addressed in the next section.
60
60
Exp. IndentaƟon force F, N
IndentaƟon force F, N
Exp. 45
FEM
30 15 0 0
(6)
In Eq. (6), εp is the plastic strain and ε_ is the strain rate. This is developed from the G'sell-Jonas law [22], and has been modified by Kermouche et al. [23], and Bucaille et al. [24]. In this equation, the material constant m represents the strain rate sensitivity, K is the consistency parameter and hg is the strain hardening modulus. Thus, this law is analogous to Eq. (4), except for the strain rate sensitivity m. Based on Eq. (6), parametric FEM studies were carried out to simulate the uni-axial compressive deformation, so that the
0.5
1
1.5
2
FEM
45 30 15 0 0
2.5
0.5
1
1.5
2
2.5
Displacement h, mm
Displacement h, mm
(a)
(b)
60
IndentaƟon force F, N
Exp. 45
FEM
30 15 0 0
0.5
1
1.5
2
2.5
Displacement h, mm
(c) Fig. 12. Comparison of experimental indentation curves (Fig. 11) with computational one for each loading rate ((a) 25 mm/min, (b) 1 mm/min, and (c) 0.05 mm/min).
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A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
computational nominal stressestrain curve agrees with the experimental one (Fig.10). For each strain rate, we explored the appropriate materials constants (K, hg, and m) in Eq. (6). Minimization of the difference between the experiment and computation leads to K ¼ 0.6 MPa, hg ¼ 0.4, and m ¼ 0.03. To verify our computational framework, spherical indentation tests were carried out in a similar manner in Fig. 6(a). This study was conducted using three different loading rates (25, 1 and 0.05 mm/min), as shown in Fig.11. For each test, three tests were conducted and their averaged data is plotted with the standard deviation. The loading rate effect is readily recognizable, where a slow loading rate encourages plastic deformation. Subsequently, the computational model was used to simulate these indentation responses of Fig.11, and corresponding results are given in Figs.12(a)-(c). They showed a relatively agreement with each other. Note that the present study does not discuss the mechanism of strain rate effect. There are two known mechanisms for the strain rate effect in porous materials [4]. One is due to the mechanical property of base polymer materials, and the other is compressive gas in the closed cell. This was first investigated by Zhang and Ashby, who conducted compression tests for various closedcellular materials to investigate the effect of strain rate on the macroscopic stressestrain curve. When the effects of strain rate on plateau stress are compared in between cellular material and solid counterpart, it is found that the strain rate dependency comes mainly from the cellular wall of “solid” polymer material. The temperature increase of gas due to adiabatic compression, on the other hand, is not so large since the gas is surrounded by the cellular material having large thermal heat capacity. Therefore, the compression deformation proceeds almost isothermally (even if the strain rate changes significantly). Thus, we regard that our observed strain rate effect is mainly dependent of the cellular wall of polypropylene (PP) solid material. 5. Conclusions This study investigated the compressive deformation behavior of a low-density polymeric foam. The foam with 94% porosity had micron-sized pores with a closed cell structure. We first conducted a uni-axial compressive test, revealing that the macroscopic stressestrain curve indicated a plateau region when the material undergoes plastic deformation. Next, FEM was carried out to simulate the deformation behavior of the entire material. The FEM model employed the yield criterion, which is governed by both the stress component of Mises stress and hydrostatic mean stress (i.e. hydrostatic pressure sensitive yield criterion), in order to consider the porous structural effect, which is expressed by the plastic Poisson's ratio. By using our FEM and experimental data, we established a computational model for the plastic deformation behavior of porous material. To verify our FEM model, several indentation tests with different indenters (spherical indentation and wedge indentation) were carried out in order to generate various tri-axial stress states. For various stress distributions caused by different indenters, good agreement was observed between the experimental indentation curves and the computational curves. Furthermore, the strain rate effect of plastic deformation (viscoplastic property) was incorporated, by using the constitutive equation of the G'sell-Jonas law. The indentation responses with different loading rates were reasonably simulated. Therefore, the present phenomenological FEM model can predict the overall
plastic deformation behavior of low-density foam subjected to uniaxial compression and indentation loadings at various loading rates. Acknowledgments The work of A.Y. is supported by JSPS KAKENHI (Grant No. 26420025) from the Japan Society for the Promotion of Science (JSPS), and a Research Grant from the Takahashi Industrial and Economic Research Foundation (No.05-003-058) and The Sumitomo Foundation (Environmental Research Projects, No.143127). The work of X.C. is supported by the National Natural Science Foundation of China (11302163, 11172231 and 11372241), ARPA-E (DEeAR0000396) and AFOSR (FA9550-12-1-0159). References [1] Q. Liu, G. Subhash, A phenomenological constitutive model for foams under large deformations, Polym. Eng. Sci. 44 (2004) 463e473. [2] E.A. Flores-Johnson, Q.M. Li, Indentation into polymeric foams, Int. J. Solids Struct. 47 (2010) 1987e1995. [3] V.I. Rizov, Low velocity localized impact study of cellular foams, Mater. Des. 28 (2007) 2632e2640. [4] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, 1997. [5] V.P.W. Shim, Z.H. Tu, C.T. Lim, Two dimensional response of crushable polyurethane foam, Int. J. Impact Eng. 4 (2000) 703e731. [6] S. Abrate, Impact on Composite Structures, Cambridge University Press, New York, 1998. [7] A.T. Nettles, M.J. Douglas, A Comparison of Quasi-static Testing to Low Velocity Impact Testing, American Society for Testing and Materials International, West Conshohocken, PA, 2002. [8] X. Chen, J.W. Hutchinson, A.G. Evans, The mechanics of indentation induced lateral cracking, J. Am. Ceram. Soc. 88 (2005) 1233e1238. [9] X. Chen, R. Wang, N. Yao, et al., Foreign object damage in a thermal barrier system: mechanisms and simulations, Mater. Sci. Eng. A 352 (2003) 221e231. [10] O.B. Olurin, N.A. Fleck, M.F. Ashby, Indentation resistance of an aluminium foam, Scr. Mater. 43 (2000) 983e989. [11] S. Ramachandra, P. Sudheer Kumar, U. Ramamurty, Impact energy absorption in an Al foam at low velocities, Scr. Mater. 49 (2003) 741e745. [12] U. Ramamurty, M.C. Kumaran, Mechanical property extraction through conical indentation of a closed-cell aluminum foam, Acta Mater. 52 (2004) 181e189. [13] R.E. Miller, A continuum plasticity model for the constitutive and indentation behaviour of foamed metals, Int. J. Mech. Sci. 42 (2000) 729e754. [14] V.S. Deshpande, N.A. Fleck, Isotropic constitutive models for metallic foams, J. Mech. Phys. Solids 48 (2000) 1253e1283. [15] Z. Zhou, Z. Wang, L. Zhao, X. Shu, Loading rate effect on yield surface of aluminum alloy foams, Mater. Sci. Eng. A 543 (2012) 193e199. [16] V.S. Deshpande, N.A. Fleck, Multi-axial yield behaviour of polymer foams, Acta mater. 49 (2001) 1859e1866. [17] A. Reyes, O.S. Hopperstad, T. Berstad, et al., Constitutive modeling of aluminum foam including fracture and statistical variation of density, Eur. J. Mech. e A Solids 22 (2003) 815e835. [18] H. Bart-Smith, J.W. Hutchinson, A.G. Evans, Measurement and analysis of the structural performance of cellular metal sandwich construction, Int. J. Mech. Sci. 43 (2001) 1945e1963. [19] C.G. Sell, J.J. Jonas, Determination of the plastic behaviour of solid polymers at constant true strain rate, J. Mater. Sci. 14 (1979) 583e591. [20] J.L. Bucaille, E. Felder, Identification of the viscoplastic behavior of a polycarbonate based on experiments and numerical modeling of the nanoindentation test, J. Mater. Sci. 37 (2002) 3999e4011. [21] M. Jerabek, Z. Major, R.W. Lang, Uniaxial compression testing of polymeric materials, Polym. Test. 29 (2010) 302e309. [22] C. G'Sell, J.J. Jonas, Determination of the plastic behaviour of solid polymers at constant true strain rate, J. Mater. Sci. 14 (1979) 583e591. [23] G. Kermouche, J.L. Loubet, J.M. Bergheau, Extraction of stressestrain curves of elasticeviscoplastic solids using conical/pyramidal indentation testing with application to polymers, Mech. Mater. 40 (2008) 271e283. [24] J.L. Bucaille, E. Felder, G. Hochstetter, Identification of the viscoplastic behavior of a polycarbonate based on experiments and numerical modeling of the nano-indentation test, J. Mater. Sci. 37 (2002) 3999e4011.
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Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Material behaviour
Characterization of the compressive deformation behavior with strain rate effect of low-density polymeric foams Akio Yonezu a, *, Keita Hirayama a, Hiroshi Kishida a, Xi Chen b, c, ** a
Department of Precision Mechanics, Chuo University, 1-13-27 Kasuga, Bunkyo, Tokyo 112-8551, Japan International Center for Applied Mechanics, SV Lab, School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, China c Department of Earth and Environmental Engineering, Columbia University 500 W 120th Street, New York, NY 10027, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 October 2015 Accepted 27 November 2015 Available online 2 December 2015
This study investigates the compressive deformation behavior of a low-density polymeric foam at different strain rates. The material tested has micron-sized pores with a closed cell structure. The porosity is about 94%. During a uni-axial compressive test, the macroscopic stressestrain curve indicates a plateau region during plastic deformation. Finite Element Method (FEM) simulation was carried out, in which the yield criterion considered both components of Mises stress and hydrostatic stress. By using the present FEM and experimental data, we established a computational model for the plastic deformation behavior of porous material. To verify our model, several indentation experiments with different indenters (spherical indentation and wedge indentation) were carried out to generate various tri-axial stress states. From the series of experiments and computations, we observed good agreement between the experimental data and that generated by the computational model. In addition, the strain rate effect is examined for a more reliable prediction of plastic deformation. Therefore, the present computational model can predict the plastic deformation behavior (including time-dependent properties) of porous material subjected to uni-axial compression and indentation loadings. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Porous material Compressive deformation behavior Strain rate Indentation Finite element method
1. Introduction Porous polymers with a low density are widely used for contact/ impact absorption and heat/acoustic insulation because of their good energy absorbing properties, good vibration attenuation and thermal/acoustic insulation [1e5]. In use, compressive loadings, including indentation and low velocity impacts by foreign objects, are often applied [6], and quasi-static indentation tests have been used to understand the low velocity impact response of composites [7] and protective coatings [8,9]. Consequently, analytical and numerical modeling for porous polymer materials is necessary in order to predict their deformation behavior, especially large plastic deformation. In general, the mechanical response is strongly dependent on inherent porous structure, polymer type and loading conditions. Indentation loading usually induces a complicated
* Corresponding author. ** Corresponding author. International Center for Applied Mechanics, SV Lab, School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, China. E-mail addresses: [email protected] (A. Yonezu), xichen@columbia. edu (X. Chen). http://dx.doi.org/10.1016/j.polymertesting.2015.11.021 0142-9418/© 2015 Elsevier Ltd. All rights reserved.
stress state compared with that of uni-axial loading. Such a complicated tri-axial stress may make modeling of macroscopic deformation more difficult when using an analytical/theoretical approach. Several analytical models have been developed to predict indentation resistance during quasi-static indentation of aluminum foams [10e12] and for polymeric foams [2]. Olurin et al. used an indentation test on aluminum foams to obtain material properties, i.e. plateau stress and tear energy [10]. Flores-Johnson et al. investigated the indentation responses of polymeric foam (polymethacrylimide and polyetherimide) with an analytical model based on experimental observation [2]. The deformation behavior was described based on total resistance force (which consisted of a crushing force, tearing force and friction force) developed during indentation loading. The analytical model was successfully established in order to calculate a resistance force during an indentation test. They verified the proposed model by conducting several indentation tests with different indenter shapes [2]. On the other hand, numerical simulations of quasi-static loading, including indentation into metallic polymeric foams, have been explored based on a continuum model of plasticity (constitutive model of plastic deformation). In fact, plastic theory
2
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for porous media based on continuum models has been investigated. Miller has proposed a continuum plasticity framework for metallic foams [13]. The Drucker-Prager yield criterion was modified and introduced three adjustable parameters to match the yield surface from experimental data. These data are the uniaxial tensile and compressive yield stresses, and the ratio of radial to axial plastic strain rate, i.e. the “plastic Poisson's ratio”. An associated flow rule is also assumed to give a constitutive law of the plastic behavior. More simply, Deshpande and Fleck established a new yield criterion for porous media, which can be applied to both open cell and closed cell low density aluminum foam [14]. It is a very simple formula, which relies on only the plastic Poisson's ratio as the material parameter. These proposed criteria were verified based on the experimental data of yield surface for aluminum foam [15] and polymer foam [16]. Furthermore, the Deshpande and Fleck criterion was systematically investigated with regard to its applicability to plastic flow behavior with an assumption of associate flow [14,17,18]. Thus, this criterion may be readily employed to describe the plastic deformation of porous media with the aid of a numerical approach (e.g. FEM). As mentioned above, indentation loading (similar to low velocity impact) is critical for the use of porous materials, and it usually produces a strain rate effect in practice. Therefore, the modeling of plastic deformation, including indentation and rate-dependent behavior, is necessary for material design with a cellular structure. This study is motivated by the lack of a computational framework for the indentation mechanics with strain rate effect of lowdensity polymeric foams. In order to establish computational modeling for indentation responses subjected to various indenters with different loading rates, the yield criterion and plastic flow behavior proposed by Deshpande and Fleck [14] is used in the FEM computation. Note that strain rate effect is not included in the previous model [14]. Based on this, we will explore a computational framework, including the strain rate effect. In the experiment, a uniaxial compression test was first carried out to obtain the macroscopic stressestrain curve and the plastic Poisson's ratio. This leads to yield strength, work-hardening rate and the material parameter for the Deshpande-Fleck criterion. Subsequently, indentation tests were carried out on the porous polymer. Here, we used two indenters with different shapes (i.e. spherical type and wedge type) and different loading rates, to induce various indentation responses. Finally, we investigated the feasibility of our computational modeling, i.e., whether it can predict the indentation response obtained by the experiments. Our phenomenological approach may be more useful for explaining the indentation response (heterogeneous deformation) of low-density foam compared with the above analytical model based on the crush and fracture energy model. In Section 2, the material properties of the present polymer foam will be presented. Section 3 investigates the response to the uni-axial compression test. Section 4 describes the theory of yield criterion and plastic flow in order to establish the computational model for plastic deformation. Subsequently, the verification of our computational modeling through several indentation experiments is described. 2. Material and experimental procedure The material used in this study was commercial porous polymer material for heat/thermal insulating and impact absorption, more specifically a board of Porous Polypropylene (PP) with a closed cell structure (Zetlon®, Sekisui Chemical Co. Ltd) and thickness of 10 mm. Test specimens were taken from the board. The specimen size was different depending on the loading type (uni-axial compression and indentation), which will be described later. Fig. 1
Fig. 1. Pictures of the present specimen: (a) macroscopic picture; (b) X-ray CT image; and (c) surface layer observed through an optical microscope.
(a) shows a macroscopic picture of the specimen. To observe the micro structure, the X-ray CT technique (micro-CT system, SkyScan 1172) was utilized. The specimens were scanned with the micro-CT system, whose scanning parameters were set as X-ray source voltage of 59 kV, image pixel size of 4.08 mm and rotation step of 0.2 . Fig. 1(b) shows the internal micro-CT cross-sectional images taken from the specimen. Many pores which seem to have elliptical geometry along the thickness direction were observed. A volume of 10 mm 10 mm 3 mm was used in order to measure the distribution of pore size. If the pore is assumed to be a spherical shape, its averaged diameter is 567 mm. Since the present specimen was of foam material, the structure was closed pore (closed-cell). The porosity was measured to be 94% based on the density measurement (the density of the PP material was 0.95 g/cm3). The foam had a surface layer of solid PP to protect the closed pore structure. To observe the cross section of a surface layer, the foam was potted in epoxy resin and the cross sectional surface was polished. Fig. 1 (c) shows the optical microscope image of the cross section. A surface layer of “solid” PP material was observed. Below the surface, a thin cell wall of PP ligament (to form the porous structure) was observed. This constitutes closed-cell PP foam. The thickness of the surface layer was measured in the range of about 5 mm length, resulting in an averaged thickness of 36 mm. The uniaxial compression test was performed on a universal testing machine with a ball screw type (LSC-1/30: Tokyo Testing Machine Inc.) using a 10 mm cube specimen. A compression jig with a flat face was used for uni-axial loading under displacement control. The rate of loading displacement was set to 1.7 102 mm/ s. For the measurement of displacement, two eddy current sensors (EX-305 and EX-201, Keyence) were used. These were mounted on the jigs so that they could directly measure the gap between the two compression jigs, which corresponds to the uniaxial deformation of the specimen. Subsequently, concentrated loading was applied to the specimen in order to investigate different mechanical conditions. This is typical indentation testing to produce a tri-axial stress state in the material. The stress state due to indentation is quite different to that of the uni-axial compression test. This study used two types of indenter whose geometries are spherical and wedge type. The
A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
3
Fig. 2. Nominal stress and strain curve of the present porous material during uni-axial compressive loading.
spherical indenter had a tip radius of 5 mm, which was made of bearing steel. For the wedge indenter, the tip angle was 90 and the width was 15 mm with a cemented carbide tip. The tip radius was observed to be 30 mm by using a Scanning Electron Microscope (SEM). The rate of loading displacement was set to 1.7 102 mm/s for both indenters. For the spherical indentation, the specimen size was 50 mm square and 10 mm thick, whereas that for the wedge indenter was 70 mm 12 mm 10 mm thickness. All tests were conducted at room temperature in air. 3. Uniaxial compressive test Fig. 2 shows the uni-axial stress and strain curves.1 In order to investigate the generality of the compressive test, we employed four specimens of identical size. Their data are plotted separately in this figure, where their results are identical, showing good reproducibility. During the initial stage, the stress linearly increases up to 0.6 MPa, and then it slightly decreases. After that, the stress appears almost constant, i.e. the plateau region. After a strain of 0.4, the stress increases gradually. Such a deformation trend is often observed in metallic and polymeric foams. It is expected that the yield stress would be about 0.6 MPa, and the plateau region corresponds to plastic, non-recoverable deformation. As shown in this figure, all tests show a similar trend. Note that the strain rate effect is present in many polymer materials and foam. Such a strain rate effect will be discussed in Section 4.4. To investigate elastic properties, a loading and unloading test was carried out.1 Below the yield stress of 0.6 MPa, repeated loading and unloading tests were carried out so that the specimen elastically deformed several times. Note that the maximum stress s was set to 0.25 MPa and it was unloaded down to 0.025 MPa (not complete unloading). At this minimum loading, the test was suspended for a long period of time (about 30 min) in order to terminate viscoelastic recovery of the foam. Thus, the interval time between each test is about 30 min. Fig. 3(a) shows the stress and strain curve during the repeated test. The test was conducted 11 times (the number of loading cycles was 11). At the first cycle, it appears that the loading jig partially contacted the specimen surface (it may not have been full contact), due to the roughness of the specimen and/or the misalignment of
1 The load was measured by the load cell and the displacement (of compressive deformation) was measured by the displacement sensor. The nominal stress was calculated from the measured load and initial cross-section area of specimen, and the nominal strain was obtained from the measured displacement and the initial thickness of specimen.
Fig. 3. Repeated loading-unloading test for measurement of Young's modulus: (a) nominal stressestrain curve during repeated testing; (b) the changes in Young's modulus as the test cycle was repeated.
the longitudinal axis. It is observed that the loading curve suddenly increases and has a hysteresis loop. Subsequent test cycles, however, show linear curves with little hysteresis loops. For the range between the nominal stresses of s ¼ 0.05 and 0.25 MPa, Young's modulus was measured. Fig. 3(b) shows the measured Young's modulus as a function of test cycle, and indicates that the value becomes constant when the test cycle increases. The last four data points were found to be stable, showing an average value of 39.2 MPa. In fact, the specimen was loaded beyond the yield stress of 0.6 MPa and the result of permanent deformation was observed. In Fig. 2 we observe a plateau region, indicating that the specimen underwent plastic deformation, possibly due to buckling of cell walls. Such a deformation mode is quite different from a homogenous solid material. For a macroscopic deformation mode, the changes in specimen volume are different. For a plastically deformed solid, the specimen volume does not change, whereas a porous material that includes foam undergoes apparent volume changes due to the buckling of the cell structure and collapse, leading to densification of its pore structure. Therefore, the plastic Poisson's ratio np is a key parameter when the deformation mechanics of an entirely foam material are investigated. As mentioned in Section 1, the plastic Poisson's ratio is important for the present yield criterion (which was proposed by Deshpande and Fleck and will be discussed later). Thus, both longitudinal and transverse strains were measured during the uni-axial compression test. Fig. 4 shows macroscopic pictures of the specimen during the compression test. Several marks were added to the specimen and we measured the distance between them to obtain transverse strain.
4
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Fig. 4. Changes in the plastic Poisson's ratio (C) and the parameter of a (B) with respect to the applied longitudinal compressive strain.
Therefore, the ratio of transverse strain and longitudinal strain yields the plastic Poisson's ratio np. Fig. 4 shows the plastic Poisson's ratio with respect to longitudinal strain (nominal strain), as indicated by the solid circle mark. Since this test was conducted five times, the averaged data with standard deviation are plotted in this figure. It is observed that the value of np appears almost constant, suggesting that it does not change in our deformation range.2 This value of np is employed for computational modeling in the next section.
The average value was a ¼ 1.98, which is employed in our computations in the next section. 4.2. Finite element method As mentioned above, the yield criterion of Eq. (1) was used for FEM computation in this study. Equivalent stress is given by Eq. (1). Similarly, when the associate rule is assumed equivalent plastic strain can be described in the following equation:
Z
4. Computational modeling
εp ≡ 4.1. Yield criterion Deshpande and Fleck phenomenologically developed a yield criterion that can cover the experimental data for the multi-axial yield behavior of open and closed-cell aluminum alloy foams [14]. They proposed a phenomenological yield surface and equivalent stress as given by:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1 s¼u 2 i s2e þ a2 s2m th 1þ a3
(1)
=
where se is the von Mises effective stress and sm is the mean stress. The yield criterion is s ¼ sY of the uniaxial tensile or compressive yield strength of the foam. It was also reported that the parameter a defines the shape of the yield surface. This is related to the plastic Poisson's ratio np, which can be calculated by: =
¼
2 a3
2 1þ a3
(2)
=
np ¼
ε_ p Tp ε_ L
12
=
where ε_ p is plastic strain rate and the subscripts indicate the loading direction, i.e. “L” is the longitudinal direction and “T” is the transverse direction. To determine the present yield surface, the parameter a (as described in Eq. (1)) is necessary. Thus, as shown in Fig. 4, this study measured the plastic Poisson's ratio np from the uniaxial compressive test, and converted it to the parameter a from Eq. (2). The parameter a was plotted in Fig.4 (with open marks), appearing almost constant within the present deformation range.
2 The constant plastic Poisson's ratio np suggested that the foam did not experience densification of its cellular structure owing to being a very low density foam.
b ε_ ≡
ffi Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 1 2 2 ε_ e þ 2 ε_ m 1þ 3 a
(3)
where b ε_ is the increment of strain (strain rate), ε_ e is the Mises effective plastic strain rate and ε_ m is the volumetric plastic strain rate. Equivalent plastic strain is the increment of the plastic strain rate. Here, we assumed that the hardening rule was governed by the following empirical equation:
s ¼ K$eðhg $εp Þ n
(4)
This is similar to the general hardening rule of the G'sell and Jonas law, which can describe non-linear responses of the stressestrain curve of a polymer material [19,20].3 When the plastic strain εp is 0 (at yield point), s becomes the yield stress (which corresponds to the value of K in Eq. (4)). As shown in Fig. 2, the apparent yield stress was 0.6 MPa, leading to K ¼ 0.6 MPa. The other unknown parameters (hg and n) were deduced from the comparison between the FEM computational data and the experimental data. This study created an axi-symmetric FEM model for uni-axial loading. As an example model, let the radius be 26 mm and thickness be 10 mm. Let the numbers of the element and node be 13,200 and 13,515, respectively. Then, let uni-axial loading under displacement control be applied to the model. Computations were carried out using the commercially available software, Marc (MSC, Marc 2012). Here, the elastic modulus was 39.2 MPa from Fig.3(b). Poisson's ratio (under elastic deformation) was set to 0.33,4 as a typical cellular material [4]. In addition, the yield criterion of Eq. (1) was employed and the associate rule was assumed. Such a plastic
3 Note that the strain rate effect is discussed in Section 4.4 where Eq. (4) is improved. 4 Elastic properties (E, G, n) of a typical cellular material with low-density is theoretically investigated based on bending of the cell frame [see Gibson- Asby, [4]].
A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
1.5 1.2 0.9
80
: Exp1 : Exp2 : Exp3 : Exp4 : FEM
, IndentaƟon force F, N
Nominal stress, MPa
1.8
0.6 0.3 0 0
0.2
5
0.4
0.6
0.8
60 50 40 30 20 10 0 0
Nominal strain
0.4
0.8
Indentation tests were carried out, as described in Section 2, in order to generate various stress fields for the verification of our computational model. Fig. 6 show the indentation curves of the spherical indenter and wedge indenter. Note that the wedge indentation responses were dependent on the width of the specimen. Thus, the indentation force was normalized by the specimen width, as shown in Fig.6 (b). For those experiments, several tests with different maximum loading were carried out, indicating that they are highly reproducible.5 To simulate the indentation responses, an FEM model was created, as shown in Fig.7. For the spherical indenter, an axisymmetric model with a thickness of 10 mm and radius of 40 mm was created. The numbers of elements and nodes were 27,528 and 27,994, respectively. The wedge indenter case was a two dimensional half model 10 mm thick, 47 mm long and 11 mm wide. Generalized plane strain condition was assumed. The numbers of the elements and nodes were 16,588 and 16,969, respectively. For both FEM models, the size of the specimen was very large (compared with the area of the indenter impression), and thus it could be considered as semi-infinite. It should be noted that, as shown in Fig.1(c), a solid PP layer (about 36 mm thick) existed on the top surface, since this layer may play a role in covering closed pores it may also affect indentation response. It is modeled in the present FEM computation (see Fig.7). As such, the present model consisted of two layers of materials; i.e. a top “solid” layer 36 mm thick and the bottom “foam” 10 mm thick. We assumed that the top layer was solid PP material, whose elastic properties were E ¼ 2350 MPa and Poisson's ratio n ¼ 0.5 [21]. Furthermore, this solid layer model employed the Mises yield criterion and the plastic property (stressestrain curve), which was obtained from a uni-axial
5 Mechanical response due to indentation (concentrated) loading seems to be dependent on indent place, since the present foam has many pores. Good reproducibility is still obtained, since the indenter size is significantly larger than the pore size.
1.6
2
2.4
(a) Normalized indentaƟon force F, N/mm
4.3. Model verifications through indentation loading
1.2
Displacement h, mm
Fig. 5. Nominal stress and strain curve of the experimental data of Fig. 2 and the computational data (indicated by an open circle mark) from FEM analysis.
constitutive behavior was embedded in the present FEM via a userdefined subroutine. In order to identify the material constants in Eq. (4), a parametric FEM study was carried out such that the FEM data matched the experimental data (in Fig.2). As shown in Fig.5, when hg ¼ 0.4 and n ¼ 1.7, the computational curve shows good agreement with the experimental curve. Thus, the material constants in Eq. (4) are K ¼ 0.6 MPa, hg ¼ 0.4 and n ¼ 1.7.
N N EXP N
: : : : FEM
70
4 3.5 3
: Exp (n=3) : FEM
2.5 2 1.5 1 0.5 0 0
0.4
0.8
1.2
1.6
Displacement h, mm
(b) Fig. 6. Indentation curves of (a) the spherical indenter and (b) the wedge indenter. The experimental data is indicated by a line and the computational data (from Fig. 7) is by an open circle mark.
Fig. 7. FEM model of indentation on the present porous material.
compressive test [21]. Here, the constitutive law of plastic deformation was employed in the following equation (which can effectively approximate the experimental stressestrain curve of [21]):
s½MPa ¼ 47 þ 40ε0:5 p
(5)
where εp is the plastic strain. The yield stress was 47 MPa. Subsequently, the foam (porous material) was assumed to obey the above yield criterion and plastic flow as described in Section 4.1 and 4.2 As shown in Fig.7, fine meshes were created around the contact region and a mesh converge test was carried out. Both indenters were assumed to be a rigid body. The friction coefficient was set to zero, which is reported to be a minor factor.
6
A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
Fig. 8. Contour map of the equivalent stress field of (a) the spherical indenter and (b) the wedge indenter.
Fig. 9. Contour map of the equivalent strain field of (a) the spherical indenter and (b) the wedge indenter.
Fig. 10. Nominal stress and strain curve during uni-axial compressive test with different strain rates.
The computational indentation curves are plotted in Fig.6. The open mark of FEM shows good agreement with the experimental data for both indenters. Although the stress field under indentation loading is different to that under uni-axial loading, our FEM computation can predict the indentation deformation behavior. Finally, stress and strain distributions during indentation (with the spherical indenter and wedge indenter) were investigated. Fig.8 shows a contour map of equivalent stress seq, which is obtained from Eq. (1). Fig.9 shows a contour map of equivalent strain εeq, which is obtained from Eq. (3). It is found that both figures indicate a different distribution of seq and εeq. This indicates that a different indenter generates a different stress (and strain) distribution. Our computation, however, shows good agreement with the experimental data (see Fig.6). Thus, our FEM model is very useful for porous media subjected to various states of compressive loading. It is concluded that this study explores a computational model for predicting macroscopic plastic deformation of polymeric foam
A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
4.4. Strain rate effect
60 IndentaƟon force F, N
7
We conducted further investigation into the strain rate effect of plastic deformation using uni-axial compressive tests with different strain rates. Here, we selected three strain rates (8.3 102, 1.7 102, 8.3 104 1/s), spanning two orders of magnitude. Fig.10 shows the nominal stress and nominal strain curves for each strain rate. For each testing condition, three tests were carried out and the averaged data and standard deviation are plotted in this figure. It is found that the strain rate effect is prominent in the plastic deformation, which needs to be further justified into our computational model. As mentioned in Eq. (4), there are several constitutive equations for plastic deformation of polymer material. One simple equation is proposed as follows
25mm/min 1mm/min 0.05mm/min
45
n=3 30 15 0 0
0.5
1 1.5 Displacement h, mm
2
2.5
Fig. 11. Indentation curves of the spherical indenter with different loading rates. m s ¼ K$ehg εp $ε_ 2
under indentation. A uni-axial compression test was first carried out to obtain macroscopic stress and strain curves, yielding the apparent elastoplastic property and the plastic Poisson's ratio. These experimentally obtained parameters were used in our FEM model, which could simulate indentation responses to various indenters (i.e. spherical and wedge indenters). Note that the strain rate effect and time dependent property are often present in polymer materials, and thus addressed in the next section.
60
60
Exp. IndentaƟon force F, N
IndentaƟon force F, N
Exp. 45
FEM
30 15 0 0
(6)
In Eq. (6), εp is the plastic strain and ε_ is the strain rate. This is developed from the G'sell-Jonas law [22], and has been modified by Kermouche et al. [23], and Bucaille et al. [24]. In this equation, the material constant m represents the strain rate sensitivity, K is the consistency parameter and hg is the strain hardening modulus. Thus, this law is analogous to Eq. (4), except for the strain rate sensitivity m. Based on Eq. (6), parametric FEM studies were carried out to simulate the uni-axial compressive deformation, so that the
0.5
1
1.5
2
FEM
45 30 15 0 0
2.5
0.5
1
1.5
2
2.5
Displacement h, mm
Displacement h, mm
(a)
(b)
60
IndentaƟon force F, N
Exp. 45
FEM
30 15 0 0
0.5
1
1.5
2
2.5
Displacement h, mm
(c) Fig. 12. Comparison of experimental indentation curves (Fig. 11) with computational one for each loading rate ((a) 25 mm/min, (b) 1 mm/min, and (c) 0.05 mm/min).
8
A. Yonezu et al. / Polymer Testing 50 (2016) 1e8
computational nominal stressestrain curve agrees with the experimental one (Fig.10). For each strain rate, we explored the appropriate materials constants (K, hg, and m) in Eq. (6). Minimization of the difference between the experiment and computation leads to K ¼ 0.6 MPa, hg ¼ 0.4, and m ¼ 0.03. To verify our computational framework, spherical indentation tests were carried out in a similar manner in Fig. 6(a). This study was conducted using three different loading rates (25, 1 and 0.05 mm/min), as shown in Fig.11. For each test, three tests were conducted and their averaged data is plotted with the standard deviation. The loading rate effect is readily recognizable, where a slow loading rate encourages plastic deformation. Subsequently, the computational model was used to simulate these indentation responses of Fig.11, and corresponding results are given in Figs.12(a)-(c). They showed a relatively agreement with each other. Note that the present study does not discuss the mechanism of strain rate effect. There are two known mechanisms for the strain rate effect in porous materials [4]. One is due to the mechanical property of base polymer materials, and the other is compressive gas in the closed cell. This was first investigated by Zhang and Ashby, who conducted compression tests for various closedcellular materials to investigate the effect of strain rate on the macroscopic stressestrain curve. When the effects of strain rate on plateau stress are compared in between cellular material and solid counterpart, it is found that the strain rate dependency comes mainly from the cellular wall of “solid” polymer material. The temperature increase of gas due to adiabatic compression, on the other hand, is not so large since the gas is surrounded by the cellular material having large thermal heat capacity. Therefore, the compression deformation proceeds almost isothermally (even if the strain rate changes significantly). Thus, we regard that our observed strain rate effect is mainly dependent of the cellular wall of polypropylene (PP) solid material. 5. Conclusions This study investigated the compressive deformation behavior of a low-density polymeric foam. The foam with 94% porosity had micron-sized pores with a closed cell structure. We first conducted a uni-axial compressive test, revealing that the macroscopic stressestrain curve indicated a plateau region when the material undergoes plastic deformation. Next, FEM was carried out to simulate the deformation behavior of the entire material. The FEM model employed the yield criterion, which is governed by both the stress component of Mises stress and hydrostatic mean stress (i.e. hydrostatic pressure sensitive yield criterion), in order to consider the porous structural effect, which is expressed by the plastic Poisson's ratio. By using our FEM and experimental data, we established a computational model for the plastic deformation behavior of porous material. To verify our FEM model, several indentation tests with different indenters (spherical indentation and wedge indentation) were carried out in order to generate various tri-axial stress states. For various stress distributions caused by different indenters, good agreement was observed between the experimental indentation curves and the computational curves. Furthermore, the strain rate effect of plastic deformation (viscoplastic property) was incorporated, by using the constitutive equation of the G'sell-Jonas law. The indentation responses with different loading rates were reasonably simulated. Therefore, the present phenomenological FEM model can predict the overall
plastic deformation behavior of low-density foam subjected to uniaxial compression and indentation loadings at various loading rates. Acknowledgments The work of A.Y. is supported by JSPS KAKENHI (Grant No. 26420025) from the Japan Society for the Promotion of Science (JSPS), and a Research Grant from the Takahashi Industrial and Economic Research Foundation (No.05-003-058) and The Sumitomo Foundation (Environmental Research Projects, No.143127). The work of X.C. is supported by the National Natural Science Foundation of China (11302163, 11172231 and 11372241), ARPA-E (DEeAR0000396) and AFOSR (FA9550-12-1-0159). References [1] Q. Liu, G. Subhash, A phenomenological constitutive model for foams under large deformations, Polym. Eng. Sci. 44 (2004) 463e473. [2] E.A. Flores-Johnson, Q.M. Li, Indentation into polymeric foams, Int. J. Solids Struct. 47 (2010) 1987e1995. [3] V.I. Rizov, Low velocity localized impact study of cellular foams, Mater. Des. 28 (2007) 2632e2640. [4] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, 1997. [5] V.P.W. Shim, Z.H. Tu, C.T. Lim, Two dimensional response of crushable polyurethane foam, Int. J. Impact Eng. 4 (2000) 703e731. [6] S. Abrate, Impact on Composite Structures, Cambridge University Press, New York, 1998. [7] A.T. Nettles, M.J. Douglas, A Comparison of Quasi-static Testing to Low Velocity Impact Testing, American Society for Testing and Materials International, West Conshohocken, PA, 2002. [8] X. Chen, J.W. Hutchinson, A.G. Evans, The mechanics of indentation induced lateral cracking, J. Am. Ceram. Soc. 88 (2005) 1233e1238. [9] X. Chen, R. Wang, N. Yao, et al., Foreign object damage in a thermal barrier system: mechanisms and simulations, Mater. Sci. Eng. A 352 (2003) 221e231. [10] O.B. Olurin, N.A. Fleck, M.F. Ashby, Indentation resistance of an aluminium foam, Scr. Mater. 43 (2000) 983e989. [11] S. Ramachandra, P. Sudheer Kumar, U. Ramamurty, Impact energy absorption in an Al foam at low velocities, Scr. Mater. 49 (2003) 741e745. [12] U. Ramamurty, M.C. Kumaran, Mechanical property extraction through conical indentation of a closed-cell aluminum foam, Acta Mater. 52 (2004) 181e189. [13] R.E. Miller, A continuum plasticity model for the constitutive and indentation behaviour of foamed metals, Int. J. Mech. Sci. 42 (2000) 729e754. [14] V.S. Deshpande, N.A. Fleck, Isotropic constitutive models for metallic foams, J. Mech. Phys. Solids 48 (2000) 1253e1283. [15] Z. Zhou, Z. Wang, L. Zhao, X. Shu, Loading rate effect on yield surface of aluminum alloy foams, Mater. Sci. Eng. A 543 (2012) 193e199. [16] V.S. Deshpande, N.A. Fleck, Multi-axial yield behaviour of polymer foams, Acta mater. 49 (2001) 1859e1866. [17] A. Reyes, O.S. Hopperstad, T. Berstad, et al., Constitutive modeling of aluminum foam including fracture and statistical variation of density, Eur. J. Mech. e A Solids 22 (2003) 815e835. [18] H. Bart-Smith, J.W. Hutchinson, A.G. Evans, Measurement and analysis of the structural performance of cellular metal sandwich construction, Int. J. Mech. Sci. 43 (2001) 1945e1963. [19] C.G. Sell, J.J. Jonas, Determination of the plastic behaviour of solid polymers at constant true strain rate, J. Mater. Sci. 14 (1979) 583e591. [20] J.L. Bucaille, E. Felder, Identification of the viscoplastic behavior of a polycarbonate based on experiments and numerical modeling of the nanoindentation test, J. Mater. Sci. 37 (2002) 3999e4011. [21] M. Jerabek, Z. Major, R.W. Lang, Uniaxial compression testing of polymeric materials, Polym. Test. 29 (2010) 302e309. [22] C. G'Sell, J.J. Jonas, Determination of the plastic behaviour of solid polymers at constant true strain rate, J. Mater. Sci. 14 (1979) 583e591. [23] G. Kermouche, J.L. Loubet, J.M. Bergheau, Extraction of stressestrain curves of elasticeviscoplastic solids using conical/pyramidal indentation testing with application to polymers, Mech. Mater. 40 (2008) 271e283. [24] J.L. Bucaille, E. Felder, G. Hochstetter, Identification of the viscoplastic behavior of a polycarbonate based on experiments and numerical modeling of the nano-indentation test, J. Mater. Sci. 37 (2002) 3999e4011.