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Multiaxial yield surface of transversely isotropic foams: Part II—Experimental
Author’s Accepted Manuscript Multiaxial Yield Surface of Transversely Isotropic Foams: Part II-Experimental Muhammad Shafiq, Ravi Sastri Mohammad Ehaab, Murat Vural
Ayyagari,
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PII: DOI: Reference:
S0022-5096(14)00207-5 http://dx.doi.org/10.1016/j.jmps.2014.10.009 MPS2550
To appear in: Journal of the Mechanics and Physics of Solids Received date: 1 February 2013 Revised date: 7 October 2014 Accepted date: 15 October 2014 Cite this article as: Muhammad Shafiq, Ravi Sastri Ayyagari, Mohammad Ehaab and Murat Vural, Multiaxial Yield Surface of Transversely Isotropic Foams: Part II-Experimental, Journal of the Mechanics and Physics of Solids, http://dx.doi.org/10.1016/j.jmps.2014.10.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Multiaxial Yield Surface of Transversely Isotropic Foams: Part II Experimental Muhammad Shafiq, Ravi Sastri Ayyagari, Mohammad Ehaab and Murat Vural1 Mechanical, Materials & Aerospace Engineering (MMAE) Department, Illinois Institute of Technology, Chicago, IL, 60616, USA Abstract. A robust understanding and modeling of the yield behavior in solid foams under complex stress states is essential to design and analysis of optimal structures using these lightweight materials. In pursuit of this objective a new custom–built Multi-Axial Testing Apparatus (MATA) is developed to probe the yield surface of transversely isotropic Divinycell H-100 PVC foam under a multitude of uniaxial, biaxial and triaxial strain paths. Experimental yield data produced constitutes the most comprehensive data set ever produced for any foam as it covers the entire spectrum of stress paths from hydrostatic compression to hydrostatic tension. Experimental results reveal that yielding in foams exhibits not only a quadratic pressure dependence, which is widely recognized in literature, but also a significant linear pressure dependence, which has been largely overlooked in previous studies. A new energy-based yield criterion developed for transversely isotropic foams is also validated using the experimental yield data.
Key words: Multiaxial loading, Yield surface, Solid foam, Experimental mechanics, Pressure dependence.
1
Corresponding author: [email protected]
1
INTRODUCTION Solid foams are finding a growing use as lightweight core materials in composite sandwich
constructions with primary load bearing functions mainly because of their high specific strength and stiffness along with their capability to mitigate impact energy. However, to exploit the full potential of these lightweight cellular solids it is necessary to analyze, understand, model and validate their constitutive response as accurately as possible under demanding multiaxial stress states. This necessitates the development of new testing methods and experimental data that properly address both the pressure dependence of these highly porous materials and the fabrication induced anisotropy commonly observed in commercial foams. A number of analytical and experimental work has been carried out to characterize the elastic and failure behavior of polymeric foams (Shaw and Sata, 1966; Triantafillou et al., 1989; Zhang et al., 1997; Deshpande and Fleck, 2001). Shaw and Sata (1966) conducted an array of experiments involving uniaxial, triaxial and hydrostatic compression on cylindrical polystyrene specimens, along with torsion and axial compression on tubular specimens, and proposed that the maximum principal stress could be an appropriate yield criterion. Fortes et al. (1989) performed hydrostatic compression experiments on polymeric foams (polyurethane and polystyrene) by immersing the specimens in water and plotting pressure vs. dilatation similar to a stress-strain plot. Based on their experiments they conclude that at low pressures the deformation is due to bending of cell walls, whereas the plateau region of the pressure-dilatation plot corresponds to cell wall buckling followed by densification at high pressures. They reported that the yield stress under hydrostatic compression is the same as that under uniaxial compression due to the value of plastic Poisson ratio being nearly zero in both cases, and suggested that the volume change depends only on maximum principal stress value. However both these works concentrate on the compressive states of stress alone and fail to address the difference in yield behavior under tension and compression. Experimental investigation of foams often requires accounting for the potential influence of specimen size on overall mechanical response. Andrews et al. (2001) investigated the specimen size effects on the overall mechanical properties of open and closed cell foams under uniaxial compression, shear loading and indentation tests and concluded that the specimen thickness and specimen geometry do influence the yield behavior of foams. It was observed that short specimens failed due to shear buckling, while longer specimens underwent non-uniform
compression at the holding grips. A key issue with testing of cellular solids is the premature failure of the specimens due to displacement constraints at the glued holding grips (Triantafillou et al., 1989). Chen and Fleck (2002) also reported the specimen size effect on biaxial yield envelope for foam plates. Wierzbicki and Doyoyo (2003) proposed to use tapered specimens to eliminate the influence of specimen size and localization bands on the stress-strain response of foams in uniaxial compression. This further led to butterfly shaped specimen geometry that has been investigated by Mohr and Doyoyo (2003) for testing Alporas aluminum foam under shear stress-normal stress loading by using so-called Arcan apparatus. Zhang et al. (1997) proposed a phenomenological hydrodynamic elasto-plastic constitutive law to explain the mechanical behavior of polymeric foams in uniaxial compression, hydrostatic compression and shear loading. However, foam behavior in tension-tension principal stress space has not been investigated and thus the overall applicability of their model remains to be tested for more general states of stress. Deshpande and Fleck (2000) conducted experiments to investigate the yield behavior of aluminum foams (Alporas and Duocel) under triaxial compressive states of stress. They proposed a phenomenological yield criterion that successfully captures the experimental data. However, their yield data in the mean stress - effective stress space were only limited to negative values of mean stress.
Later, Deshpande and Fleck (2001) conducted
multiaxial experiments on two different densities of PVC foams (Divinycell H100 and H200) and proposed the above mentioned yield surface capped by a maximum compressive principal stress criterion. Their experimental investigation, however, had very limited data on the tension side with positive mean stress component. It is also notable that, in contrast to the Zhang et al. (1997) model which used the non-associated flow rule, DF model (Deshpande and Fleck, 2000) considers associated flow rule to account for the low values of plastic Poisson’s ratio in polymeric foams. Our literature survey shows that there are about 12 works in literature that experimentally characterize the mechanical behavior of various polymeric foams (PVC, PE, PS, PP), as shown in Fig. 1. The stress states probed by each one of these studies are mapped in Fig. 1 in mean stress-effective stress (σ e − σ m ) space as it provides the most comprehensive frame for studying pressure dependent yield behavior of foams. However, none of these works conclusively establish the yield behavior of any given foam simply because of the limited stress paths investigated in each work. This becomes evident from the quantum of work undertaken
particularly in the triaxial loading regime. Although there exist some experimental data in triaxial compression regime for PVC foams (mostly due to Deshpande and Fleck, 2001), the lack of experimental work in triaxial tension regime is alarming. Unlike the yielding in metals and many other engineering materials, solid foams exhibit a pressure dependent yield behavior and, therefore, require dedicated multiaxial experiments to fully establish their yield surface under varying amount of positive and negative mean stress. Shear-compression (S/C) and shear-tension ! "#
1 Shaw and Sata (1969) 2 Triantafillou et al. (1989)
"#
4 Zhang et al. (1997, 1998) 5 Li et al. (2000) 6 Deshpande and Fleck (2001)
$% &% ' (
σe
3 Fortes et al. (1989)
7 Gilchrist and Mills (2001) 8 Gdoutos et al. (2002)
9 Gong et al. (2005) 10 Flores-Johnson et al. (2008) 11 Philippe viot (2009) 12 Kolupaev et al. (2010)
)*+ ,,-
)*+ ,,
σm
Fig. 1. Schematic of an elliptic yield surface in mean stress-effective stress space and mapping of the stress states explored in various studies for polymeric foams. Note that shear/compression (S/C), shear/tension (S/T) and biaxial experiments can only cover a limited portion of yield surface, and also no experimental data is available in triaxial tension regime.
(S/T) types of combined loading experiments, and even fully biaxial experiments, can probe only a limited portion of the yield surface, which is often not enough to reveal the true nature of pressure dependence mainly due to limited range of mean stress that can be introduced by these types of experiments. We must note that, although not included in Fig. 1, one can find a very similar picture for the scope of experimental work on aluminum foams where triaxial yield data is scarce for compressive states of stress and almost non-existent for tensile states of stress. It is obvious that fully triaxial experiments, in addition to biaxial ones, are of great importance not only to identify true constitutive behavior of solid foams but also to validate various yield models proposed for foams in a wider range of mean stress. But designing such experiments that can provide a representative mechanical response is a challenging work and
requires utmost attention to potential complexities such as specimen size effects and premature failure at the holding grips. It must be emphasized that the scarcity of experimental data coupled with the absence of an effective analytical framework capable of addressing complex stress states especially in 3D (Ruan et al., 2007) presents an obstacle to advance the current understanding and modeling of the mechanical response of solid foams. Present work attempts to address these issues (i) by reporting the development of a Multi-Axial Testing Apparatus (MATA) capable of biaxial and triaxial experiments along any arbitrary stress path, (ii) by presenting the most extensive experimental data set for the yield behavior of Divinycell H-100 PVC foam that cover the entire spectrum of stress paths from hydrostatic compression to hydrostatic tension, and (iii) by validating a pressure dependent yield criterion proposed in our companion paper (Ayyagari and Vural, 2013) for transversely isotropic solid foams. 2 EXPERIMENTAL 2.1 Material and Specimens For the present study, all specimens are machined from a 51mm thick sheet of Divinycell H100 foam, which is a closed cell PVC foam manufactured by DIAB. It will be referred to as H100 in the following discussions. The through-the-thickness stiffness and yield strength of H-100 is about 70% and 40% higher, respectively, than respective in-plane properties, making H-100 a transversely isotropic foam as will be shown in Section 2.3. Four types of specimen geometries shown in Fig. 2 are used to perform compression, tension and combined compression and tension tests. Cubic specimens measuring 51× 51× 51mm3 are used in biaxial and triaxial compression tests, while biaxial and triaxial tension and combined tension-compression tests are performed by using "Maltese-Cross Specimens" (MCS) and modified MCS. The MCS is specifically designed for triaxial loading conditions by cutting slots along each edge of the cube as shown in Fig. 2. Such a geometry ensures failure within the specimen gage section and prevents failure adjacent to the loading plates (Desphande and Fleck, 2001), providing required flexibility for the application of adhesive on the specimen surface during multiaxial tension and combined tension-
Fig. 2. Various specimen geometries used in experiments. (a) regular cubic specimen for uniaxial, biaxial and triaxial compression tests, (b) Maltese-cross specimen (MCS) for triaxial tension tests, (c) & (d) modified MCS for biaxial tension-tension and tension-compression loading.
compression tests. One additional advantage of MCS geometry is that the existence of slots provides necessary clearance to avoid any contact between the loading plates when at least one of the prescribed displacements is compressive. The modifications introduced into the basic MCS geometry, such as removing two of the “ears”, are only aimed at achieving a uniform deformation field in the gage section of H-100 foam in biaxial experiments. We must note that all specimens are machined from the same sheet of foam with a nominal density of ρ = 100kg / m3 and an average cell size of 450µ m . However, the density of each cubic specimen is measured before the experiments (which corresponds to about 40% of all specimens), and the average density of foam specimens is found to be ρ = 88 kg / m3 with a standard deviation of approximately 2% and maximum variation of 4.5%. 2.2 Apparatus Multi-Axial Testing Apparatus (MATA) shown in Fig. 3 is a custom built experimental setup developed for testing foam specimens in uniaxial, biaxial and triaxial stress states along arbitrary strain paths. MATA is capable of simultaneously applying compressive, tensile or any combination of compressive-tensile loads independently along three orthogonal directions at predefined or varying displacement rates in real time. A fully functional MATA used in a triaxial test utilizes three hydraulic actuators, six linear bearings, three load cells and three LVDTs. Two different sets of fixtures are designed for testing cubic and Maltese cross specimens. However, if required, the original configuration of MATA can also be altered to accommodate custom made specimens. MATA consists of six rigid blocks, aligned as three pairs (loading and restricting) along three orthogonal directions. Three blocks mounted on the actuator side are the loading blocks while
the other three serve as restricting mechanism in respective directions. Biaxial and triaxial stress states are achieved by the relative motion of these six rigid blocks. The blocks themselves are mounted on to six linear bearings aligned in different orientations, primarily to generate a relative triaxial motion and to avoid interference among them. Three linear bearings are installed on the actuators’ side and the other three are installed on the restricting side to enable relative motion of loading blocks without generating any resistance to each other. Apart from the linear motion of actuators, each block is allowed to move in only one direction normal to the loading axis, thereby restricting all other degrees of freedom including rotation. The linear bearings are
Fig. 3. Schematic of Multi-Axial Testing Apparatus (MATA) with three independent servo-hydraulic actuators and LVDTs.
designed for a relative motion of 25 mm on either side when they are centered. Due to this restriction on relative motion, the maximum nominal strain that can be achieved is limited to about 50% in any direction for the size of the specimens used in experiments.
Hydraulic actuators used in MATA have a maximum capacity of 3000 psi and are run by close loop servo controller device. Load is recorded using a load cell (with a maximum load rating of 2000 lbs) installed on each restricting side. Displacements are recorded using LVDTs installed on each actuator. The two LVDTs along horizontal axes have a maximum of ±2 inch span while the one along vertical axis has a maximum range of ±3 inch. Each axis can be operated under two modes, i.e., either in displacement controlled mode or load controlled mode. All experiments are conducted in the displacement controlled mode at predefined displacement rates.
Fig. 4. (a) Sectional view of MCS during triaxial loading, compression (specimen is behind the loading blocks).
(b) Cubic specimen loaded in triaxial
Hydraulic actuators are controlled by three independent micro-console controllers. The controllers are calibrated for displacement and load. The voltage values on controllers correspond to engineering units on LVDTs and load cells. A computer user interface is developed in LABVIEW to independently control three actuators mounted on three orthogonal axes. Three independent function generators are used in LABVIEW to generate desired ramp functions and subsequently provide command signals to actuators. Strain rates can also be provided and changed during operation on all the three actuators independently. Displacement and load data are recorded along three principal directions in a typical triaxial experiment as the raw data for subsequent processing. A cut-out view of MATA and MCS is given in Fig. 4a for a typical triaxial loading scenario while Fig. 4b is a pictorial view of the actual MATA configuration in triaxial compression.
2.3 Multiaxial Experiments Prior to multiaxial testing, fully instrumented uniaxial compression and tension experiments are performed along all the three principal material directions (i.e., through-the-thickness and transverse directions) in the quasi-static regime at a nominal strain rate of 5 ×10−4 s −1 . These experiments are carried out on an electro-mechanical Instron machine as shown in Fig. 5, and then repeated on MATA to establish the accuracy of its results. Longitudinal and transverse strains are measured using two extensometers and subsequently used for the determination of elastic moduli and Poisson's ratio under uniaxial compression and tension. Uniaxial compression
Fig. 5. Fully instrumented (a) uniaxial compression, and (b) uniaxial tension experiments employed to obtain elastic constants and yield strengths along three principal material axes.
tests are performed on cubic H-100 specimens while dog-bone specimens are used for uniaxial tension. Three specimens are tested along each direction to ensure repeatability. Based on uniaxial test results reported in Table 1, one can observe that the H-100 foam is transversely isotropic such that 1-2 plane (X-Y plane) is the plane of isotropy, and 3-direction (Z-direction) is the through-the-thickness direction along which elastic modulus and yield stress are considerably higher. Uniaxial tension and compression data plotted in Fig. 6 clearly show this transverse isotropy along with the amount of natural variation in stress-strain response for specimens of nominally the same density.
3.5
1.8
Uniaxial Tension
Uniaxial Compression 1.5
Stress (MPa)
Stress (MPa)
3 2.5 2 1.5 1 0.5
1-S1 1-S2 1-S3
a)
2-S1 2-S2 2-S3
3-S1 3-S2 3-S3
1.2 0.9 0.6 0.3
1-S1 1-S2 1-S3
b)
2-S1 2-S2 2-S3
3-S1 3-S2 3-S3
0
0 0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
Strain
Strain
Fig. 6. Stress-strain response of H-100 PVC foam in uniaxial tension and uniaxial compression. Note that 1-2 plane is the plane of isotropy in this transversely isotropic foam. Table 1 Material properties of transversely isotropic Divinycell H-100 PVC foam obtained from uniaxial compression and tension experiments in 1, 2 and 3 directions. Note that E, Y, e and ν are the Young's modulus (MPa), yield stress (MPa), yield strain and Poisson's ratios, respectively. Also note that 1-2 is the plane of isotropy.
Direction Specimen
1
Uniaxial Compression EC1 YC1 eC1 ν 12
2
54.642
0.995
0.020
S2
61.003
0.991
0.018
0.352
68.520
1.290
0.021
0.418
S3
59.030
1.060
0.019
0.359
60.449
1.379
0.025
0.382
Avg.
58.225
1.015
0.019
0.358
62.936
1.353
0.024
0.396
YC 2
eC 2
ν 21
YT 2
eT 2
ν 21
ET 2
1.390
0.025
0.389
S1
54.458
0.988
0.020
0.346
67.745
1.460
0.024
0.404
S2
54.321
0.983
0.020
0.332
67.246
1.634
0.026
0.410
S3
50.537
0.985
0.022
0.331
65.079
1.492
0.025
0.398
Avg.
53.105
66.690
..
3
59.840
ν 12
S1
EC 2
0.365
ET 1
Uniaxial Tension YT 1 eT 1
0.985
0.021
0.336
YC 3
eC 3
ν 31
ET 3
1.529
0.025
0.404
YT 3
eT 3
ν 31
S1
89.393
1.322
0.017
0.384
95.038
1.989
0.023
0.483
S2
93.091
1.419
0.017
0.383
101.260
2.131
0.023
0.544
S3
94.378
1.296
0.016
0.364
98.778
1.875
0.021
0.441
Avg.
92.287
1.346
0.017
0.377
98.359
1.998
0.022
0.489
Biaxial experiments are conducted at a predefined quasi-static strain rate in the plane of anisotropy (1-3 plane). In both biaxial and triaxial experiments, once the strain path is selected, the magnitudes of individual strain rates are adjusted such that the characteristic strain rate remains constant at 5 ×10−4 s −1 as in uniaxial experiments in order to avoid any potential rate effects one might expect from polymeric foams. The reader is referred to our companion paper
(Ayyagari and Vural, 2013) for a detailed account of characteristic strain. However, it must suffice to state here that it is a newly introduced scalar measure of strain for transversely isotropic compressible foams that is analogous to equivalent strain for isotropic bulk solids. Its definition that is provided in the next section must be enough for the discussion of current work. A total of 63 strain paths were selected for biaxial experiments in order to obtain sufficient number of data points to construct the yield surface in the plane of anisotropy. To conduct biaxial experiments, MATA has been modified to the biaxial configuration as shown in Fig. 7. It can be observed that the fixtures have additional plates with modified dimensions to match the
Fig. 7. Modified configuration of MATA for biaxial loading.
surface area of a modified MCS. Also, the third pair of actuating and restricting blocks is removed in this configuration to obtain a clean field of view for observation. In triaxial experiments, two types of specimen geometry are used as shown in Fig. 2a and 2b. Cubic specimens are used when all displacement components are compressive while MCS is used for triaxial tension and any tension-compression combinations. It must be noted that MCS are bonded to the loading platens using an epoxy adhesive not only in triaxial tension but in all
the experiments that include tensile loading. The application of adhesive is rather difficult on all the specimen surfaces and loading platens, and great care has been taken to ensure proper mounting and alignment of specimen within the fixture. At the expense of stating the obvious it must be noted that no glue is applied in triaxial compression tests where cubic specimens are used. Results of a typical triaxial compression experiment are presented in Fig. 8. In total, approximately 20 uniaxial, 50 biaxial and 60 triaxial experiments are carried out to probe the yield surface of H-100 foam for the entire spectrum of stress states from hydrostatic compression to hydrostatic tension.
Triaxial Compression
Stress (MPa)
1-direction 2-direction 3-direction
Strain Fig. 8. Individual stress components during a typical triaxial compression experiment of H-100.
2.4 Data Processing A total of 130 uniaxial, biaxial and triaxial experiments are conducted to probe the yield surface of Divinycell H-100 foam at predefined displacement rates. In each experiment, loaddisplacement data are recorded along the three principal directions. These data are used to determine the nominal stresses and strains in respective directions. Then, characteristic stresses and strains are computed by using individual stress and strain components. The reader is referred to our companion paper (Ayyagari and Vural, 2013) for a detailed account of the characteristic stress and characteristic strain, which are newly introduced scalar measures of stress and strain for transversely isotropic compressible foams that is analogous to equivalent stress and strain for isotropic bulk solids. The characteristic stress is defined as:
σˆ 2 = σˆ e2 + βˆ 2σˆ m2
(1)
where σˆ e is the effective stress component derived from the deviatoric part of strain energy density and σˆ m is the mean stress component based on hydrostatic part, which are defined by:
σˆ m =
σ 1 + σ 2 + r1σ 3 3
(2)
σˆ e = σ + σ + r σ − M σ 1σ 2 − Nr1 (σ 2σ 3 + σ 3σ 1 ) 2 1
2 2
2 1
2 3
The constant r1 = Y1 Y3 is a measure of the strength anisotropy defined by the ratio of yield strengths while M and N are functions of elastic constants as well as the strength anisotropy as defined in Table 2. It must be noted that Eq. (2) defines modified (hatted) versions of mean stress
Parameters
2E ª 2 2 1 −ν 13ν 31 ) + (ν 12 +ν 13ν 31 ) + ( 2 +ν 31Z )(1 +ν 12 )ν 31r1 º ( ¼ (1 +ν 12 ) ¬
2
R = − (1 − 2ν )
2
Q = (1 − 2ν )
2
P = (1 − 2ν )
3
σ 1 + σ 2 + r1σ 3
( 6 + 14ν 12 − 4ν 31Z ) , ( 6 + 2ν 12 + 2ν 31Z ) N=
( 6 − 4ν 12 + 5ν 31Z ) ( 6 + 2ν 12 + 2ν 31Z )
εˆe =
α P ( ε12 + ε 22 ) + Qr22ε 32 + Rε1ε 2 + Sr2 ( ε 2ε 3 + ε 3ε1 ) E α εˆv = ( ε1 + ε 2 )(1 +ν 31r1 ) + ε 3r2 (1 −ν 12 + 2 (ν 31 r2 ) ) 3K 9 ( 3 − 2ν 12 − 2ν 31Z ) 9 E1 , βˆ 2 = Eˆ = ( 6 + 2ν 12 + 2ν 31Z ) ( 6 + 2ν 12 + 2ν 31Z )
σˆ e = σ 12 + σ 22 + r12σ 32 − M σ 1σ 2 − Nr1 (σ 2σ 3 + σ 3σ 1 )
σˆ m =
M=
3
σ1 + σ 2 + σ 3
2 2 ε1 + ε 22 + ε 32 − ε1ε 2 − ε 2ε 3 − ε 3ε1 3
Eˆ =
9 (1 − 2ν ) 3E , βˆ 2 = 2 (1 +ν ) 2 (1 +ν )
εˆv = ε1 + ε 2 + ε 3
εˆe =
σˆ e = σ 12 + σ 22 + σ 32 − σ 1σ 2 − σ 2σ 3 − σ 3σ 1
σˆ m =
M = 1, N = 1
2
3 1 , K= 2 (1 + ν ) 3 (1 − 2ν )
1 (1 +ν )(1 − 2ν )
S = 2 (1 +ν 31r1 ) (1 −ν 12 + 2 (ν 31 r2 ) ) − E ª¬ 2 (1 +ν 12 )(ν 31 r2 ) + ( 2 +ν 31Z )(1 −ν 12 + 2ν 13ν 31 ) º¼ S = − (1 − 2ν )
R = 2 (1 +ν 31r1 ) −
2
2 2 Q = (1 −ν 12 + 2 (ν 31 r2 ) ) − 2 E ª(1 +ν 12 )(ν 31 r2 ) + (ν 31 r2 )(1 −ν 12 )( 2 + ν 31Z ) º ¬ ¼
P = (1 +ν 31r1 ) −
E ª 2 (1 −ν 13ν 31 )(ν 12 +ν 13ν 31 ) + ( 2 +ν 31Z )(1 +ν 12 )ν 31r1 º¼ (1 +ν 12 ) ¬
E=
9 1 , K= ( 6 + 2ν 12 + 2ν 31Z ) ( 3 − 2ν 12 − 2ν 31Z )
E=
2
α=
1 (1 −ν 12 − 2ν 13ν 31 )
r1 = 1, r2 = 1, Z = 2
Isotropic Material
α=
Table 2 Explicit definition of parameters that appear in Eq. (1) through (5) in the manuscript. Transversely Isotropic Material 1 r1 = Y1 Y3 , r2 = e1 e3 , Z = r1 + r2
and effective stress that proved to be quite convenient in the analysis of data when transverse isotropy is present. They reduce to conventional definitions in the absence of anisotropy where
r1 = M = N = 1 . In a similar way, the characteristic strain is defined by:
εˆ 2 = εˆe2 +
εˆv2 βˆ 2
(3)
where εˆv and εˆe are volumetric strain and effective strain that are work conjugates of mean stress and effective stress, respectively, as defined below:
εˆe =
α ( 6 + 2ν 12 + 2ν 31Z )
P ( ε12 + ε 22 ) + Qr22ε 32 + Rε1ε 2 + Sr2 ( ε 2ε 3 + ε 3ε1 )
9
(4)
α ( 3 − 2ν 12 − 2ν 31Z )
( ε1 + ε 2 )(1 +ν 31r1 ) + ε 3r2 (1 −ν 12 + 2 (ν 31 r2 ) ) 3 A sample of stress-strain curves is given in Fig. 9 for a typical biaxial compression εˆv =
experiment in the plane of anisotropy (i.e., 1-3 plane) along with the characteristic stress-strain plot. The individual stress-strain data in Figs. 9a and 9b are used to calculate characteristic stress-strain values by using Eqs. (1) through (4), and the result is plotted in Fig. 9c. This procedure to plot characteristic stress-strain curve is adopted for all multiaxial experiments. Some of these plots are presented in Fig. 10 for biaxial stress paths, where each experiment is continued until either failure (typically a tensile failure) or sufficiently large strain levels are achieved (typical in compression-compression paths and some tension-compression paths). One immediately notes from these plots that the elastic loading portions of characteristic stress-strain curves do not collapse onto a master line whose slope is uniquely defined by a characteristic elastic modulus, Eˆ . Whereas in FE simulations of a transversely isotropic Kelvin foam model (Ayyagari and Vural, 2013), the elastic portions of characteristic stress-strain plots nicely fall on
a)
1-direction
Stress (MPa)
Stress (MPa)
b)
3-direction
Strain
c)
Strain
Fig. 9. Results of a typical biaxial compression experiment in the plane of anisotropy (1-3 plane); (a) stress-strain response in 1-direction, (b) stress-strain response in 3-direction, (c) resulting characteristic stress-strain plot.
top of each other with a common slope of Eˆ . This discrepancy is attributed to the natural scatter in the elastic as well as strength properties of H-100 foam as can be observed from Table 1, which causes non-singular Eˆ values that falls in a finite band as evidenced from Fig. 10. Once the characteristic stress-strain curves are obtained the yield point is determined by using 0.2% strain offset technique on characteristic stress-strain curves, and the corresponding stress values are tagged as the yield point in principal stress space as well as mean stresseffective stress space.
3
3
a)
2 1.5 1
2 1.5 1
0.5
0.5
.
0 0.00
0.05
0.10
0.15
0 0.00
0.20
3
. 0.05
0.10
0.15
0.20
3
c)
d)
2.5
(MPa)
2.5
(MPa)
b)
2.5
(MPa)
(MPa)
2.5
2 1.5 1
2 1.5 1
0.5
0.5
.
0 0.00
0.05
0.10
0.15
0.20
0 0.00
. 0.05
0.10
0.15
0.20
Fig. 10. Characteristic stress-strain plots for biaxial loading in the plane of anisotropy (1-3 plane) within the four quadrants of principal stress space. Various strain paths in (a) Tension Tension, (b) Compression - Tension, (c) Compression - Compression, (d) Tension Compression quadrants.
3
RESULTS AND DISCUSSION Experimental yield points determined by the procedure outlined in preceding section are
plotted in Figs. 11 and 12 for Divinycell H-100 PVC foam in both principal stress space (in the plane of anisotropy) and mean stress-effective stress space, respectively. Both data sets, which describe the yield surface of H-100 in respective stress spaces, exhibit certain amount of scatter that is comparable to the scatter obtained from uniaxial experiments (see Table 1 as well as the solid markers in Figs. 11-12 for the variation in uniaxial data). Given that the density of foam specimens tested has a relatively small standard deviation of about 2%, we attribute the scatter in data to fabrication induced imperfections in foam sheets from which the specimens were machined, i.e., microstructural as well as density heterogeneity within the specimens, and potential human error in mounting procedure and specimen preparation. Experimental yield data presented in Figs. 11 and 12 constitute the most comprehensive data set in literature ever produced for any solid foam and reveal two important features of yielding behavior in foams which have been largely overlooked in previous studies. The first is the linear pressure dependence of the yield behavior which is clearly observed by the loss of symmetry about the origin in Fig. 11 and about the effective stress axis ( σˆ e ) in Fig. 12. The yield surface is shifted towards the first quadrant (tension-tension quadrant) in Fig. 11 and towards right along the mean stress axis in Fig. 12. Although the quadratic pressure dependence is often recognized by various yield criteria proposed in literature for foams, additional linear pressure dependence is largely overlooked which we believe is due partly to the scatter in experimental data but mostly due to lack of experimental data that cover the entire spectrum of stress states in a continuous manner from pure hydrostatic compression to hydrostatic tension. Considering that multiaxial experiments are difficult to run, require customized equipment and take significant amount of time, the yield data graphically presented in Figs. 11 and 12 are tabulated in Appendix A for the use of interested reader as well as the validation of new yield criteria that might be proposed in future studies. At the expense of repeating the obvious, it must be stressed that the elliptic shape of yield surface in Fig. 12 in mean stress-effective stress space is due to the effect of compressibility in yield behavior of foams and manifests itself as a quadratic mean stress term in many of the yield models proposed in literature. On the other hand, experimentally observed shift to the right in mean stress axis is a clear manifestation of an additional linear dependence that has not been properly addressed in previous studies.
Fig. 11. Yield surface of Divinycell H-100 PVC foam in the plane of anisotropy. Markers represent the yield points extracted from biaxial experiments while solid and dashed lines are the upper and lower bound predictions from the proposed yield criterion, i.e., Eq. (5).
Fig. 12. Yield surface of Divinycell H-100 PVC foam in mean stress-effective stress space. Markers represent the yield points extracted from uniaxial, biaxial and triaxial experiments while solid and dashed lines are the upper and lower bound predictions from the proposed yield criterion, i.e., Eq. (5).
The second important feature of current experimental data is the fact that Divinycell H-100 foam exhibits a transversely isotropic mechanical behavior, which suggests that many other solid foams (polymeric or metallic) produced by similar manufacturing techniques are likely to exhibit similar anisotropy. Therefore, a more general framework must be employed in the analysis of data to respect inherent anisotropy of solid foams. One such analytical framework is proposed in our companion paper (Ayyagari and Vural, 2013) whose results are extended to present experimental work through new scalar measures of mean stress, effective stress and characteristic stress along with respective work conjugate strains (see Eqs. (1) – (4)) that take the underlying transverse isotropy into account. This point becomes fairly important when analyzing yield data in the plane of anisotropy (or along triaxial stress paths) by using conventional isotropic definitions of mean stress and effective stress as it results in loosing crucial information and misinterpretation of data as exemplified in Fig. 6a of Ayyagari and Vural (2013). Finally, the experimental yield data presented in Figs. 11 and 12 are compared against a new yield criterion proposed in our companion paper (Ayyagari and Vural, 2013) as shown by solid and dashed lines in the figures. This new yield criterion is an extension of the yield model proposed for 2D foams (Alkhader and Vural, 2009) to 3D within the general framework of transverse isotropy, and it is given by:
§
βˆ 2 ·
§
©
9 ¹
©
σˆ e2 + βˆ 2σˆ m2 − 3 (Y1T + Y1C ) ¨¨1 +
¸¸ σˆ m = − ¨¨ 1 +
βˆ 2 ·
¸ Y1T Y1C 9 ¸¹
(5)
where Y1T and Y1C are uniaxial yield strengths in tension and compression, respectively, along 1direction. The βˆ factor is a function of elastic properties and strength anisotropy as defined in Table 2. Scalar measures of effective stress ( σˆ e ) and mean stress ( σˆ m ) that appear in Eq. (5) is slightly different from their conventional definitions to recognize the transverse isotropy of the foam as can be observed from Eq. (2). This yield criterion recognizes the linear pressure dependence observed in experimental data through the linear mean stress term which can be simply described by two uniaxial experiments ( Y1T and Y1C ). One more uniaxial experiment to find Y3T is required to fully define the yield surface for a transversely isotropic foam. One must note that this last experiment is only required for transversely isotropic foams to define βˆ which includes a strength anisotropy parameter hidden in Z term (see Table 2), otherwise Y1T and Y1C are enough to describe the yielding of isotropic foams. Other elastic constants that appear in βˆ term can be easily determined from the same uniaxial experiments. The yield criterion expressed by Eq. (5) needs minimal number of model constants obtainable from easy-to-perform uniaxial experiments because it is derived from the central hypothesis that total strain energy density drives the yielding behavior in solid foams, rather than a completely phenomenological approach. One important point that needs careful consideration, however, is the experimental scatter commonly observed in foam data. We propose to use the lowest and highest pairs of uniaxial strength values to plot a lower bound and upper bound to describe the yield surface. The terms “lower bound and upper bound” are loosely used here to denote the minimum and maximum expected boundaries of the yield surface based on the scatter in uniaxial data. Upper and lower bounds of yield surface plotted by this procedure, which are respectively denoted by solid and dashed elliptical curves in Figs. 11 and 12, satisfactorily capture the experimental data over a wide range stress states that effectively encapsulates the entire spectrum of stress paths one can possibly probe. The center of the ellipses has a slight offset in Figs. 11 and 12 which is a result of the linear pressure term introduced in the yield model. In this respect, the model prediction is completely in line with the experimental data and clearly shows the influence of linear pressure,
in addition to quadratic pressure, on the yield behavior of H-100 foam. However, underlying physical mechanisms for such a linear pressure dependence still remains to be investigated, which is the subject of a separate ongoing study.
4
CONCLUSIONS The yield surface of Divinycell H-100 PVC foam is experimentally investigated over a wide
range of complex stress states that cover the entire spectrum of stress paths from pure hydrostatic compression to hydrostatic tension. Uniaxial, biaxial and triaxial experiments are conducted by using a custom-built servo-hydraulic multiaxial testing apparatus (MATA) to probe the yield surface. Experimental results reveal the transversely isotropic nature of H-100 foam and also show a clear linear pressure dependence of its yield behavior, both of which have been largely overlooked in previous studies. Yield data obtained from extensive multiaxial experiments constitute the most comprehensive data set ever produced to date for any solid foam and, therefore, have been reported in Appendix A for the use of researchers as well as the validation of new yield criteria that might be proposed in future studies. Finally, the predictions of a new yield criterion, which is proposed in the companion paper (Ayyagari and Vural, 2013) for transversely isotropic foams, are compared against the experimental yield data with quite satisfactory results. The successful validation of this new yield criterion is attributed to (i) its general framework of transverse isotropy, (ii) its central hypothesis that the yield behavior in solid foams is driven by total strain energy density, and finally (iii) the recognition of linear pressure dependence which seems to lack in overwhelming majority of models proposed in the literature.
ACKNOWLEDGMENTS We are grateful for the support from the National Science Foundation (Grant No. 0728212 and 1030903).
REFERENCES Alkhader, M., Vural, M., 2009. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids. 57, 871–890. Andrews, E. W., Gioux, G., Onck, P., Gibson, L. J., 2001. Size effects in ductile cellular solids. Part II: experimental results. Int. J. Mech. Sci. 43, 701-713. Ayyagari, R. S., Vural, M., 2013. Multiaxial yield surface of transversely isotropic foams Part I: Modeling. (to be submitted). Chen, C., Fleck, N. A., 2002. Size effects in the constrained deformation of metallic foams. J. Mech. Phys. Solids. 50, 955-977. Desphande, V. S., Fleck, N. A., 2000. Isotropic constitutive model for metallic foams. J. Mech. Phys. Solids. 48, 1253-1283. Desphande, V. S., Fleck, N. A., 2001. Multiaxial yield behaviour of polymer foams. Acta Mater. 49, 1859-1866. Flores-Johnson, E. A., Li, Q. M., Mines, R. A. W., 2008. Degradation of elastic modulus of progressively crushable foams in uniaxial compression. J. Cell. Plast. 44, 415-434. Fortes, M. A., Fernandes, J. J., Serralheiro, I., Rosa, M. E., 1989. Experimental determinantion of hydrostatic compression versus volume change for cellular solids. J. Test. Eval. 17, 67-71. Gdoutos, E. E., Daniel, I. M., Wang, K. -A., 2002. Failure of cellular foams under multiaxial loading. Composites Part A. 33, 163-176. Gilchrist, A., Mills, N. J., 2001. Impact deformation of rigid polymeric foams experiments and FEA modelling. Int. J. Impact Eng. 25, 767-786. Gong, L., Kyriakides, S., Jang, W. -Y., 2005. Compressive response of open-cell foams Part I: Morphology and elastic properties. Int. J. Solids. Struct. 42, 1355-1379. Kolupaev, V. A., Bolchoun, A., Altenbach, H., 2010. Testing of Multi-Axial Strength Behavior of Hard Foams. ICEM 14, EPJ web of conferences, 6. Li, Q. M., Mines, R. A. W., Birch, R. S., 2000. The crush behaviour of Rohacell-51WF structural foam. Int. J. Solids. Struct. 37, 6321-6341. Mohr, D., Doyoyo, M., 2003. A new method for the biaxial testing of cellular solids. Exp. Mech. 43, 173-182. Philippe Viot, 2009. Hydrostatic compression on polypropylene foam. Int. J. Impact Eng. 36, 975-989.
Ruan, D., Lu, G., Ong, L. S., Wang, B., 2007. Triaxial compression of aluminum foams. Compos. Sci. Technol. 67, 1218-1234. Shaw, M. C., Sata, T., 1966. The plastic behaviour of cellular materials. Int. J. Mech. Sci. 8, 469478.Triantafillou, T. C., Zhang, J., Shercliff, T. L., Gibson, L. J., Ashby, M. F., 1989. Failure surfaces for cellular materials under multiaxial loads II. Comparison of models with experiment. Modeling. Int. J. Mech. Sci. 31, 665-678. Wierzbicki, T., Doyoyo, M., 2003. Determination of the local stress-strain response of foams. J. Appl. Mech. 70, 204-211. Zhang, J., Lin, Z., Wong, A., Kikuchi, N., Li, V. C., Yee, A. F., Nusholtz, G. S., 1997. Constitutive modeling and material characterization of polymeric foams. J. Eng. Mater. T. ASME. 119, 284-291.
APPENDIX A Tabulated yield values for Divinycell H-100 PVC foam obtained from multiaxial experiments. σ1 σ2 σ3 σˆ e σˆ m σ1 σ2 σ3 σˆ e σˆ m
Uniaxial
Biaxial
-1.000 -0.990 -1.060 0.000 0.000 0.000 1.390 1.290 1.380 0.000 0.000 0.000 -1.020 -1.042 -1.006 -1.039 0.000 0.000 0.000 0.000 0.000 1.566 -0.991 0.000 0.678 0.940 1.084 1.346 1.211 1.581 1.346 1.601 1.703 0.963 0.439 0.447 0.997 1.292 1.574 1.451 1.525 -0.804 -0.832 -0.784 -0.782 -0.525 -0.469 -0.339 -0.117 -0.009 -0.917 -0.741 -0.977 -1.025 -1.081 -1.018 -1.127
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 -1.320 -1.420 -1.300 0.000 0.000 0.000 1.990 2.130 1.880 0.000 0.000 0.000 0.000 -1.300 -1.617 -1.675 -1.632 1.738 0.000 0.000 -1.473 1.975 2.139 1.851 1.983 1.474 1.591 1.125 1.037 1.954 2.796 2.477 2.260 1.950 1.860 1.250 0.753 0.541 0.332 0.736 0.914 1.227 1.169 1.536 1.646 1.734 1.661 0.422 0.128 0.108 -0.358 -0.487 -0.522 -0.741
-0.333 -0.330 -0.353 -0.321 -0.346 -0.317 0.463 0.430 0.460 0.485 0.519 0.458 -0.340 -0.347 -0.335 -0.346 -0.317 -0.394 -0.408 -0.397 0.423 0.522 -0.330 -0.359 0.707 0.834 0.812 0.931 0.763 0.914 0.722 0.786 1.043 1.002 0.749 0.699 0.807 0.884 0.829 0.667 0.640 -0.187 -0.098 -0.039 0.038 0.110 0.217 0.288 0.383 0.401 -0.203 -0.216 -0.299 -0.429 -0.479 -0.466 -0.556
1.000 0.990 1.060 0.964 1.037 0.950 1.390 1.290 1.380 1.454 1.556 1.373 1.020 1.042 1.006 1.039 0.950 1.181 1.224 1.192 1.269 1.566 0.991 1.076 1.273 1.394 1.275 1.441 1.183 1.457 1.203 1.413 1.628 1.802 1.649 1.494 1.299 1.365 1.400 1.288 1.384 0.943 1.184 1.246 1.440 1.194 1.405 1.395 1.326 1.218 1.096 0.789 1.017 0.931 0.966 0.904 0.995
Biaxial
Triaxial
-1.029 -1.122 -1.023 -1.039 -0.833 -0.957 -0.868 -0.963 -0.894 -0.866 -0.685 -0.578 -0.517 -0.111 1.193 1.112 0.138 0.067 0.432 0.690 1.542 0.471 0.588 0.936 1.618 1.361 1.663 1.574 1.391 1.347 1.618 1.533 1.709 1.666 1.636 1.665 1.100 1.446 1.563 1.460 1.465 -1.010 -1.111 -1.095 -0.728 -1.137 -1.066 -0.998 -1.078 -1.176 -1.059 -1.103 -1.185 -1.033 -0.904 -1.162
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.351 1.607 1.553 1.415 1.792 1.761 1.863 1.795 1.739 1.894 1.105 1.646 1.800 1.638 1.502 -0.906 -1.014 -0.969 -0.684 -0.915 -0.936 -1.013 -0.941 -0.858 -0.802 -0.981 -0.962 -0.934 -0.600 -0.805
-0.988 -0.992 -1.058 -0.976 -0.973 -1.150 -1.151 -1.278 -1.250 -1.403 -1.375 -1.607 -1.464 -1.231 -1.181 -0.847 -1.227 -1.098 -1.096 -0.993 -0.431 -1.061 -0.887 -0.929 -0.158 -0.661 1.269 1.510 1.981 2.422 0.851 0.824 1.247 0.685 1.249 2.180 2.171 1.681 0.994 0.085 0.143 -0.441 -0.710 -1.248 -1.255 -0.284 -0.742 -0.801 -0.581 -0.290 -0.763 -1.089 -0.486 -0.276 -1.296 -0.432
-0.583 -0.615 -0.599 -0.584 -0.515 -0.599 -0.569 -0.632 -0.602 -0.630 -0.563 -0.584 -0.529 -0.337 0.110 0.164 -0.253 -0.245 -0.123 -0.012 0.409 -0.101 -0.020 0.086 0.501 0.293 0.555 0.383 0.130 0.467 0.989 0.963 0.774 1.128 0.673 0.190 0.545 0.272 0.875 1.401 1.288 0.584 0.484 0.110 0.252 0.780 0.411 0.356 0.535 0.793 0.386 0.204 0.681 0.726 0.357 0.684
0.939 1.009 0.949 0.944 0.801 0.930 0.879 0.976 0.930 0.983 0.912 1.036 0.944 0.853 1.770 1.505 0.969 0.836 1.074 1.214 1.713 1.080 1.060 1.391 1.675 1.644 1.314 1.428 1.464 1.511 1.344 1.299 1.494 1.321 1.429 1.717 1.264 1.440 1.363 1.054 1.024 -0.746 -0.881 -0.992 -0.776 -0.754 -0.848 -0.865 -0.815 -0.749 -0.806 -0.960 -0.834 -0.723 -0.817 -0.761
Ayyagari,
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PII: DOI: Reference:
S0022-5096(14)00207-5 http://dx.doi.org/10.1016/j.jmps.2014.10.009 MPS2550
To appear in: Journal of the Mechanics and Physics of Solids Received date: 1 February 2013 Revised date: 7 October 2014 Accepted date: 15 October 2014 Cite this article as: Muhammad Shafiq, Ravi Sastri Ayyagari, Mohammad Ehaab and Murat Vural, Multiaxial Yield Surface of Transversely Isotropic Foams: Part II-Experimental, Journal of the Mechanics and Physics of Solids, http://dx.doi.org/10.1016/j.jmps.2014.10.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Multiaxial Yield Surface of Transversely Isotropic Foams: Part II Experimental Muhammad Shafiq, Ravi Sastri Ayyagari, Mohammad Ehaab and Murat Vural1 Mechanical, Materials & Aerospace Engineering (MMAE) Department, Illinois Institute of Technology, Chicago, IL, 60616, USA Abstract. A robust understanding and modeling of the yield behavior in solid foams under complex stress states is essential to design and analysis of optimal structures using these lightweight materials. In pursuit of this objective a new custom–built Multi-Axial Testing Apparatus (MATA) is developed to probe the yield surface of transversely isotropic Divinycell H-100 PVC foam under a multitude of uniaxial, biaxial and triaxial strain paths. Experimental yield data produced constitutes the most comprehensive data set ever produced for any foam as it covers the entire spectrum of stress paths from hydrostatic compression to hydrostatic tension. Experimental results reveal that yielding in foams exhibits not only a quadratic pressure dependence, which is widely recognized in literature, but also a significant linear pressure dependence, which has been largely overlooked in previous studies. A new energy-based yield criterion developed for transversely isotropic foams is also validated using the experimental yield data.
Key words: Multiaxial loading, Yield surface, Solid foam, Experimental mechanics, Pressure dependence.
1
Corresponding author: [email protected]
1
INTRODUCTION Solid foams are finding a growing use as lightweight core materials in composite sandwich
constructions with primary load bearing functions mainly because of their high specific strength and stiffness along with their capability to mitigate impact energy. However, to exploit the full potential of these lightweight cellular solids it is necessary to analyze, understand, model and validate their constitutive response as accurately as possible under demanding multiaxial stress states. This necessitates the development of new testing methods and experimental data that properly address both the pressure dependence of these highly porous materials and the fabrication induced anisotropy commonly observed in commercial foams. A number of analytical and experimental work has been carried out to characterize the elastic and failure behavior of polymeric foams (Shaw and Sata, 1966; Triantafillou et al., 1989; Zhang et al., 1997; Deshpande and Fleck, 2001). Shaw and Sata (1966) conducted an array of experiments involving uniaxial, triaxial and hydrostatic compression on cylindrical polystyrene specimens, along with torsion and axial compression on tubular specimens, and proposed that the maximum principal stress could be an appropriate yield criterion. Fortes et al. (1989) performed hydrostatic compression experiments on polymeric foams (polyurethane and polystyrene) by immersing the specimens in water and plotting pressure vs. dilatation similar to a stress-strain plot. Based on their experiments they conclude that at low pressures the deformation is due to bending of cell walls, whereas the plateau region of the pressure-dilatation plot corresponds to cell wall buckling followed by densification at high pressures. They reported that the yield stress under hydrostatic compression is the same as that under uniaxial compression due to the value of plastic Poisson ratio being nearly zero in both cases, and suggested that the volume change depends only on maximum principal stress value. However both these works concentrate on the compressive states of stress alone and fail to address the difference in yield behavior under tension and compression. Experimental investigation of foams often requires accounting for the potential influence of specimen size on overall mechanical response. Andrews et al. (2001) investigated the specimen size effects on the overall mechanical properties of open and closed cell foams under uniaxial compression, shear loading and indentation tests and concluded that the specimen thickness and specimen geometry do influence the yield behavior of foams. It was observed that short specimens failed due to shear buckling, while longer specimens underwent non-uniform
compression at the holding grips. A key issue with testing of cellular solids is the premature failure of the specimens due to displacement constraints at the glued holding grips (Triantafillou et al., 1989). Chen and Fleck (2002) also reported the specimen size effect on biaxial yield envelope for foam plates. Wierzbicki and Doyoyo (2003) proposed to use tapered specimens to eliminate the influence of specimen size and localization bands on the stress-strain response of foams in uniaxial compression. This further led to butterfly shaped specimen geometry that has been investigated by Mohr and Doyoyo (2003) for testing Alporas aluminum foam under shear stress-normal stress loading by using so-called Arcan apparatus. Zhang et al. (1997) proposed a phenomenological hydrodynamic elasto-plastic constitutive law to explain the mechanical behavior of polymeric foams in uniaxial compression, hydrostatic compression and shear loading. However, foam behavior in tension-tension principal stress space has not been investigated and thus the overall applicability of their model remains to be tested for more general states of stress. Deshpande and Fleck (2000) conducted experiments to investigate the yield behavior of aluminum foams (Alporas and Duocel) under triaxial compressive states of stress. They proposed a phenomenological yield criterion that successfully captures the experimental data. However, their yield data in the mean stress - effective stress space were only limited to negative values of mean stress.
Later, Deshpande and Fleck (2001) conducted
multiaxial experiments on two different densities of PVC foams (Divinycell H100 and H200) and proposed the above mentioned yield surface capped by a maximum compressive principal stress criterion. Their experimental investigation, however, had very limited data on the tension side with positive mean stress component. It is also notable that, in contrast to the Zhang et al. (1997) model which used the non-associated flow rule, DF model (Deshpande and Fleck, 2000) considers associated flow rule to account for the low values of plastic Poisson’s ratio in polymeric foams. Our literature survey shows that there are about 12 works in literature that experimentally characterize the mechanical behavior of various polymeric foams (PVC, PE, PS, PP), as shown in Fig. 1. The stress states probed by each one of these studies are mapped in Fig. 1 in mean stress-effective stress (σ e − σ m ) space as it provides the most comprehensive frame for studying pressure dependent yield behavior of foams. However, none of these works conclusively establish the yield behavior of any given foam simply because of the limited stress paths investigated in each work. This becomes evident from the quantum of work undertaken
particularly in the triaxial loading regime. Although there exist some experimental data in triaxial compression regime for PVC foams (mostly due to Deshpande and Fleck, 2001), the lack of experimental work in triaxial tension regime is alarming. Unlike the yielding in metals and many other engineering materials, solid foams exhibit a pressure dependent yield behavior and, therefore, require dedicated multiaxial experiments to fully establish their yield surface under varying amount of positive and negative mean stress. Shear-compression (S/C) and shear-tension ! "#
1 Shaw and Sata (1969) 2 Triantafillou et al. (1989)
"#
4 Zhang et al. (1997, 1998) 5 Li et al. (2000) 6 Deshpande and Fleck (2001)
$% &% ' (
σe
3 Fortes et al. (1989)
7 Gilchrist and Mills (2001) 8 Gdoutos et al. (2002)
9 Gong et al. (2005) 10 Flores-Johnson et al. (2008) 11 Philippe viot (2009) 12 Kolupaev et al. (2010)
)*+ ,,-
)*+ ,,
σm
Fig. 1. Schematic of an elliptic yield surface in mean stress-effective stress space and mapping of the stress states explored in various studies for polymeric foams. Note that shear/compression (S/C), shear/tension (S/T) and biaxial experiments can only cover a limited portion of yield surface, and also no experimental data is available in triaxial tension regime.
(S/T) types of combined loading experiments, and even fully biaxial experiments, can probe only a limited portion of the yield surface, which is often not enough to reveal the true nature of pressure dependence mainly due to limited range of mean stress that can be introduced by these types of experiments. We must note that, although not included in Fig. 1, one can find a very similar picture for the scope of experimental work on aluminum foams where triaxial yield data is scarce for compressive states of stress and almost non-existent for tensile states of stress. It is obvious that fully triaxial experiments, in addition to biaxial ones, are of great importance not only to identify true constitutive behavior of solid foams but also to validate various yield models proposed for foams in a wider range of mean stress. But designing such experiments that can provide a representative mechanical response is a challenging work and
requires utmost attention to potential complexities such as specimen size effects and premature failure at the holding grips. It must be emphasized that the scarcity of experimental data coupled with the absence of an effective analytical framework capable of addressing complex stress states especially in 3D (Ruan et al., 2007) presents an obstacle to advance the current understanding and modeling of the mechanical response of solid foams. Present work attempts to address these issues (i) by reporting the development of a Multi-Axial Testing Apparatus (MATA) capable of biaxial and triaxial experiments along any arbitrary stress path, (ii) by presenting the most extensive experimental data set for the yield behavior of Divinycell H-100 PVC foam that cover the entire spectrum of stress paths from hydrostatic compression to hydrostatic tension, and (iii) by validating a pressure dependent yield criterion proposed in our companion paper (Ayyagari and Vural, 2013) for transversely isotropic solid foams. 2 EXPERIMENTAL 2.1 Material and Specimens For the present study, all specimens are machined from a 51mm thick sheet of Divinycell H100 foam, which is a closed cell PVC foam manufactured by DIAB. It will be referred to as H100 in the following discussions. The through-the-thickness stiffness and yield strength of H-100 is about 70% and 40% higher, respectively, than respective in-plane properties, making H-100 a transversely isotropic foam as will be shown in Section 2.3. Four types of specimen geometries shown in Fig. 2 are used to perform compression, tension and combined compression and tension tests. Cubic specimens measuring 51× 51× 51mm3 are used in biaxial and triaxial compression tests, while biaxial and triaxial tension and combined tension-compression tests are performed by using "Maltese-Cross Specimens" (MCS) and modified MCS. The MCS is specifically designed for triaxial loading conditions by cutting slots along each edge of the cube as shown in Fig. 2. Such a geometry ensures failure within the specimen gage section and prevents failure adjacent to the loading plates (Desphande and Fleck, 2001), providing required flexibility for the application of adhesive on the specimen surface during multiaxial tension and combined tension-
Fig. 2. Various specimen geometries used in experiments. (a) regular cubic specimen for uniaxial, biaxial and triaxial compression tests, (b) Maltese-cross specimen (MCS) for triaxial tension tests, (c) & (d) modified MCS for biaxial tension-tension and tension-compression loading.
compression tests. One additional advantage of MCS geometry is that the existence of slots provides necessary clearance to avoid any contact between the loading plates when at least one of the prescribed displacements is compressive. The modifications introduced into the basic MCS geometry, such as removing two of the “ears”, are only aimed at achieving a uniform deformation field in the gage section of H-100 foam in biaxial experiments. We must note that all specimens are machined from the same sheet of foam with a nominal density of ρ = 100kg / m3 and an average cell size of 450µ m . However, the density of each cubic specimen is measured before the experiments (which corresponds to about 40% of all specimens), and the average density of foam specimens is found to be ρ = 88 kg / m3 with a standard deviation of approximately 2% and maximum variation of 4.5%. 2.2 Apparatus Multi-Axial Testing Apparatus (MATA) shown in Fig. 3 is a custom built experimental setup developed for testing foam specimens in uniaxial, biaxial and triaxial stress states along arbitrary strain paths. MATA is capable of simultaneously applying compressive, tensile or any combination of compressive-tensile loads independently along three orthogonal directions at predefined or varying displacement rates in real time. A fully functional MATA used in a triaxial test utilizes three hydraulic actuators, six linear bearings, three load cells and three LVDTs. Two different sets of fixtures are designed for testing cubic and Maltese cross specimens. However, if required, the original configuration of MATA can also be altered to accommodate custom made specimens. MATA consists of six rigid blocks, aligned as three pairs (loading and restricting) along three orthogonal directions. Three blocks mounted on the actuator side are the loading blocks while
the other three serve as restricting mechanism in respective directions. Biaxial and triaxial stress states are achieved by the relative motion of these six rigid blocks. The blocks themselves are mounted on to six linear bearings aligned in different orientations, primarily to generate a relative triaxial motion and to avoid interference among them. Three linear bearings are installed on the actuators’ side and the other three are installed on the restricting side to enable relative motion of loading blocks without generating any resistance to each other. Apart from the linear motion of actuators, each block is allowed to move in only one direction normal to the loading axis, thereby restricting all other degrees of freedom including rotation. The linear bearings are
Fig. 3. Schematic of Multi-Axial Testing Apparatus (MATA) with three independent servo-hydraulic actuators and LVDTs.
designed for a relative motion of 25 mm on either side when they are centered. Due to this restriction on relative motion, the maximum nominal strain that can be achieved is limited to about 50% in any direction for the size of the specimens used in experiments.
Hydraulic actuators used in MATA have a maximum capacity of 3000 psi and are run by close loop servo controller device. Load is recorded using a load cell (with a maximum load rating of 2000 lbs) installed on each restricting side. Displacements are recorded using LVDTs installed on each actuator. The two LVDTs along horizontal axes have a maximum of ±2 inch span while the one along vertical axis has a maximum range of ±3 inch. Each axis can be operated under two modes, i.e., either in displacement controlled mode or load controlled mode. All experiments are conducted in the displacement controlled mode at predefined displacement rates.
Fig. 4. (a) Sectional view of MCS during triaxial loading, compression (specimen is behind the loading blocks).
(b) Cubic specimen loaded in triaxial
Hydraulic actuators are controlled by three independent micro-console controllers. The controllers are calibrated for displacement and load. The voltage values on controllers correspond to engineering units on LVDTs and load cells. A computer user interface is developed in LABVIEW to independently control three actuators mounted on three orthogonal axes. Three independent function generators are used in LABVIEW to generate desired ramp functions and subsequently provide command signals to actuators. Strain rates can also be provided and changed during operation on all the three actuators independently. Displacement and load data are recorded along three principal directions in a typical triaxial experiment as the raw data for subsequent processing. A cut-out view of MATA and MCS is given in Fig. 4a for a typical triaxial loading scenario while Fig. 4b is a pictorial view of the actual MATA configuration in triaxial compression.
2.3 Multiaxial Experiments Prior to multiaxial testing, fully instrumented uniaxial compression and tension experiments are performed along all the three principal material directions (i.e., through-the-thickness and transverse directions) in the quasi-static regime at a nominal strain rate of 5 ×10−4 s −1 . These experiments are carried out on an electro-mechanical Instron machine as shown in Fig. 5, and then repeated on MATA to establish the accuracy of its results. Longitudinal and transverse strains are measured using two extensometers and subsequently used for the determination of elastic moduli and Poisson's ratio under uniaxial compression and tension. Uniaxial compression
Fig. 5. Fully instrumented (a) uniaxial compression, and (b) uniaxial tension experiments employed to obtain elastic constants and yield strengths along three principal material axes.
tests are performed on cubic H-100 specimens while dog-bone specimens are used for uniaxial tension. Three specimens are tested along each direction to ensure repeatability. Based on uniaxial test results reported in Table 1, one can observe that the H-100 foam is transversely isotropic such that 1-2 plane (X-Y plane) is the plane of isotropy, and 3-direction (Z-direction) is the through-the-thickness direction along which elastic modulus and yield stress are considerably higher. Uniaxial tension and compression data plotted in Fig. 6 clearly show this transverse isotropy along with the amount of natural variation in stress-strain response for specimens of nominally the same density.
3.5
1.8
Uniaxial Tension
Uniaxial Compression 1.5
Stress (MPa)
Stress (MPa)
3 2.5 2 1.5 1 0.5
1-S1 1-S2 1-S3
a)
2-S1 2-S2 2-S3
3-S1 3-S2 3-S3
1.2 0.9 0.6 0.3
1-S1 1-S2 1-S3
b)
2-S1 2-S2 2-S3
3-S1 3-S2 3-S3
0
0 0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
Strain
Strain
Fig. 6. Stress-strain response of H-100 PVC foam in uniaxial tension and uniaxial compression. Note that 1-2 plane is the plane of isotropy in this transversely isotropic foam. Table 1 Material properties of transversely isotropic Divinycell H-100 PVC foam obtained from uniaxial compression and tension experiments in 1, 2 and 3 directions. Note that E, Y, e and ν are the Young's modulus (MPa), yield stress (MPa), yield strain and Poisson's ratios, respectively. Also note that 1-2 is the plane of isotropy.
Direction Specimen
1
Uniaxial Compression EC1 YC1 eC1 ν 12
2
54.642
0.995
0.020
S2
61.003
0.991
0.018
0.352
68.520
1.290
0.021
0.418
S3
59.030
1.060
0.019
0.359
60.449
1.379
0.025
0.382
Avg.
58.225
1.015
0.019
0.358
62.936
1.353
0.024
0.396
YC 2
eC 2
ν 21
YT 2
eT 2
ν 21
ET 2
1.390
0.025
0.389
S1
54.458
0.988
0.020
0.346
67.745
1.460
0.024
0.404
S2
54.321
0.983
0.020
0.332
67.246
1.634
0.026
0.410
S3
50.537
0.985
0.022
0.331
65.079
1.492
0.025
0.398
Avg.
53.105
66.690
..
3
59.840
ν 12
S1
EC 2
0.365
ET 1
Uniaxial Tension YT 1 eT 1
0.985
0.021
0.336
YC 3
eC 3
ν 31
ET 3
1.529
0.025
0.404
YT 3
eT 3
ν 31
S1
89.393
1.322
0.017
0.384
95.038
1.989
0.023
0.483
S2
93.091
1.419
0.017
0.383
101.260
2.131
0.023
0.544
S3
94.378
1.296
0.016
0.364
98.778
1.875
0.021
0.441
Avg.
92.287
1.346
0.017
0.377
98.359
1.998
0.022
0.489
Biaxial experiments are conducted at a predefined quasi-static strain rate in the plane of anisotropy (1-3 plane). In both biaxial and triaxial experiments, once the strain path is selected, the magnitudes of individual strain rates are adjusted such that the characteristic strain rate remains constant at 5 ×10−4 s −1 as in uniaxial experiments in order to avoid any potential rate effects one might expect from polymeric foams. The reader is referred to our companion paper
(Ayyagari and Vural, 2013) for a detailed account of characteristic strain. However, it must suffice to state here that it is a newly introduced scalar measure of strain for transversely isotropic compressible foams that is analogous to equivalent strain for isotropic bulk solids. Its definition that is provided in the next section must be enough for the discussion of current work. A total of 63 strain paths were selected for biaxial experiments in order to obtain sufficient number of data points to construct the yield surface in the plane of anisotropy. To conduct biaxial experiments, MATA has been modified to the biaxial configuration as shown in Fig. 7. It can be observed that the fixtures have additional plates with modified dimensions to match the
Fig. 7. Modified configuration of MATA for biaxial loading.
surface area of a modified MCS. Also, the third pair of actuating and restricting blocks is removed in this configuration to obtain a clean field of view for observation. In triaxial experiments, two types of specimen geometry are used as shown in Fig. 2a and 2b. Cubic specimens are used when all displacement components are compressive while MCS is used for triaxial tension and any tension-compression combinations. It must be noted that MCS are bonded to the loading platens using an epoxy adhesive not only in triaxial tension but in all
the experiments that include tensile loading. The application of adhesive is rather difficult on all the specimen surfaces and loading platens, and great care has been taken to ensure proper mounting and alignment of specimen within the fixture. At the expense of stating the obvious it must be noted that no glue is applied in triaxial compression tests where cubic specimens are used. Results of a typical triaxial compression experiment are presented in Fig. 8. In total, approximately 20 uniaxial, 50 biaxial and 60 triaxial experiments are carried out to probe the yield surface of H-100 foam for the entire spectrum of stress states from hydrostatic compression to hydrostatic tension.
Triaxial Compression
Stress (MPa)
1-direction 2-direction 3-direction
Strain Fig. 8. Individual stress components during a typical triaxial compression experiment of H-100.
2.4 Data Processing A total of 130 uniaxial, biaxial and triaxial experiments are conducted to probe the yield surface of Divinycell H-100 foam at predefined displacement rates. In each experiment, loaddisplacement data are recorded along the three principal directions. These data are used to determine the nominal stresses and strains in respective directions. Then, characteristic stresses and strains are computed by using individual stress and strain components. The reader is referred to our companion paper (Ayyagari and Vural, 2013) for a detailed account of the characteristic stress and characteristic strain, which are newly introduced scalar measures of stress and strain for transversely isotropic compressible foams that is analogous to equivalent stress and strain for isotropic bulk solids. The characteristic stress is defined as:
σˆ 2 = σˆ e2 + βˆ 2σˆ m2
(1)
where σˆ e is the effective stress component derived from the deviatoric part of strain energy density and σˆ m is the mean stress component based on hydrostatic part, which are defined by:
σˆ m =
σ 1 + σ 2 + r1σ 3 3
(2)
σˆ e = σ + σ + r σ − M σ 1σ 2 − Nr1 (σ 2σ 3 + σ 3σ 1 ) 2 1
2 2
2 1
2 3
The constant r1 = Y1 Y3 is a measure of the strength anisotropy defined by the ratio of yield strengths while M and N are functions of elastic constants as well as the strength anisotropy as defined in Table 2. It must be noted that Eq. (2) defines modified (hatted) versions of mean stress
Parameters
2E ª 2 2 1 −ν 13ν 31 ) + (ν 12 +ν 13ν 31 ) + ( 2 +ν 31Z )(1 +ν 12 )ν 31r1 º ( ¼ (1 +ν 12 ) ¬
2
R = − (1 − 2ν )
2
Q = (1 − 2ν )
2
P = (1 − 2ν )
3
σ 1 + σ 2 + r1σ 3
( 6 + 14ν 12 − 4ν 31Z ) , ( 6 + 2ν 12 + 2ν 31Z ) N=
( 6 − 4ν 12 + 5ν 31Z ) ( 6 + 2ν 12 + 2ν 31Z )
εˆe =
α P ( ε12 + ε 22 ) + Qr22ε 32 + Rε1ε 2 + Sr2 ( ε 2ε 3 + ε 3ε1 ) E α εˆv = ( ε1 + ε 2 )(1 +ν 31r1 ) + ε 3r2 (1 −ν 12 + 2 (ν 31 r2 ) ) 3K 9 ( 3 − 2ν 12 − 2ν 31Z ) 9 E1 , βˆ 2 = Eˆ = ( 6 + 2ν 12 + 2ν 31Z ) ( 6 + 2ν 12 + 2ν 31Z )
σˆ e = σ 12 + σ 22 + r12σ 32 − M σ 1σ 2 − Nr1 (σ 2σ 3 + σ 3σ 1 )
σˆ m =
M=
3
σ1 + σ 2 + σ 3
2 2 ε1 + ε 22 + ε 32 − ε1ε 2 − ε 2ε 3 − ε 3ε1 3
Eˆ =
9 (1 − 2ν ) 3E , βˆ 2 = 2 (1 +ν ) 2 (1 +ν )
εˆv = ε1 + ε 2 + ε 3
εˆe =
σˆ e = σ 12 + σ 22 + σ 32 − σ 1σ 2 − σ 2σ 3 − σ 3σ 1
σˆ m =
M = 1, N = 1
2
3 1 , K= 2 (1 + ν ) 3 (1 − 2ν )
1 (1 +ν )(1 − 2ν )
S = 2 (1 +ν 31r1 ) (1 −ν 12 + 2 (ν 31 r2 ) ) − E ª¬ 2 (1 +ν 12 )(ν 31 r2 ) + ( 2 +ν 31Z )(1 −ν 12 + 2ν 13ν 31 ) º¼ S = − (1 − 2ν )
R = 2 (1 +ν 31r1 ) −
2
2 2 Q = (1 −ν 12 + 2 (ν 31 r2 ) ) − 2 E ª(1 +ν 12 )(ν 31 r2 ) + (ν 31 r2 )(1 −ν 12 )( 2 + ν 31Z ) º ¬ ¼
P = (1 +ν 31r1 ) −
E ª 2 (1 −ν 13ν 31 )(ν 12 +ν 13ν 31 ) + ( 2 +ν 31Z )(1 +ν 12 )ν 31r1 º¼ (1 +ν 12 ) ¬
E=
9 1 , K= ( 6 + 2ν 12 + 2ν 31Z ) ( 3 − 2ν 12 − 2ν 31Z )
E=
2
α=
1 (1 −ν 12 − 2ν 13ν 31 )
r1 = 1, r2 = 1, Z = 2
Isotropic Material
α=
Table 2 Explicit definition of parameters that appear in Eq. (1) through (5) in the manuscript. Transversely Isotropic Material 1 r1 = Y1 Y3 , r2 = e1 e3 , Z = r1 + r2
and effective stress that proved to be quite convenient in the analysis of data when transverse isotropy is present. They reduce to conventional definitions in the absence of anisotropy where
r1 = M = N = 1 . In a similar way, the characteristic strain is defined by:
εˆ 2 = εˆe2 +
εˆv2 βˆ 2
(3)
where εˆv and εˆe are volumetric strain and effective strain that are work conjugates of mean stress and effective stress, respectively, as defined below:
εˆe =
α ( 6 + 2ν 12 + 2ν 31Z )
P ( ε12 + ε 22 ) + Qr22ε 32 + Rε1ε 2 + Sr2 ( ε 2ε 3 + ε 3ε1 )
9
(4)
α ( 3 − 2ν 12 − 2ν 31Z )
( ε1 + ε 2 )(1 +ν 31r1 ) + ε 3r2 (1 −ν 12 + 2 (ν 31 r2 ) ) 3 A sample of stress-strain curves is given in Fig. 9 for a typical biaxial compression εˆv =
experiment in the plane of anisotropy (i.e., 1-3 plane) along with the characteristic stress-strain plot. The individual stress-strain data in Figs. 9a and 9b are used to calculate characteristic stress-strain values by using Eqs. (1) through (4), and the result is plotted in Fig. 9c. This procedure to plot characteristic stress-strain curve is adopted for all multiaxial experiments. Some of these plots are presented in Fig. 10 for biaxial stress paths, where each experiment is continued until either failure (typically a tensile failure) or sufficiently large strain levels are achieved (typical in compression-compression paths and some tension-compression paths). One immediately notes from these plots that the elastic loading portions of characteristic stress-strain curves do not collapse onto a master line whose slope is uniquely defined by a characteristic elastic modulus, Eˆ . Whereas in FE simulations of a transversely isotropic Kelvin foam model (Ayyagari and Vural, 2013), the elastic portions of characteristic stress-strain plots nicely fall on
a)
1-direction
Stress (MPa)
Stress (MPa)
b)
3-direction
Strain
c)
Strain
Fig. 9. Results of a typical biaxial compression experiment in the plane of anisotropy (1-3 plane); (a) stress-strain response in 1-direction, (b) stress-strain response in 3-direction, (c) resulting characteristic stress-strain plot.
top of each other with a common slope of Eˆ . This discrepancy is attributed to the natural scatter in the elastic as well as strength properties of H-100 foam as can be observed from Table 1, which causes non-singular Eˆ values that falls in a finite band as evidenced from Fig. 10. Once the characteristic stress-strain curves are obtained the yield point is determined by using 0.2% strain offset technique on characteristic stress-strain curves, and the corresponding stress values are tagged as the yield point in principal stress space as well as mean stresseffective stress space.
3
3
a)
2 1.5 1
2 1.5 1
0.5
0.5
.
0 0.00
0.05
0.10
0.15
0 0.00
0.20
3
. 0.05
0.10
0.15
0.20
3
c)
d)
2.5
(MPa)
2.5
(MPa)
b)
2.5
(MPa)
(MPa)
2.5
2 1.5 1
2 1.5 1
0.5
0.5
.
0 0.00
0.05
0.10
0.15
0.20
0 0.00
. 0.05
0.10
0.15
0.20
Fig. 10. Characteristic stress-strain plots for biaxial loading in the plane of anisotropy (1-3 plane) within the four quadrants of principal stress space. Various strain paths in (a) Tension Tension, (b) Compression - Tension, (c) Compression - Compression, (d) Tension Compression quadrants.
3
RESULTS AND DISCUSSION Experimental yield points determined by the procedure outlined in preceding section are
plotted in Figs. 11 and 12 for Divinycell H-100 PVC foam in both principal stress space (in the plane of anisotropy) and mean stress-effective stress space, respectively. Both data sets, which describe the yield surface of H-100 in respective stress spaces, exhibit certain amount of scatter that is comparable to the scatter obtained from uniaxial experiments (see Table 1 as well as the solid markers in Figs. 11-12 for the variation in uniaxial data). Given that the density of foam specimens tested has a relatively small standard deviation of about 2%, we attribute the scatter in data to fabrication induced imperfections in foam sheets from which the specimens were machined, i.e., microstructural as well as density heterogeneity within the specimens, and potential human error in mounting procedure and specimen preparation. Experimental yield data presented in Figs. 11 and 12 constitute the most comprehensive data set in literature ever produced for any solid foam and reveal two important features of yielding behavior in foams which have been largely overlooked in previous studies. The first is the linear pressure dependence of the yield behavior which is clearly observed by the loss of symmetry about the origin in Fig. 11 and about the effective stress axis ( σˆ e ) in Fig. 12. The yield surface is shifted towards the first quadrant (tension-tension quadrant) in Fig. 11 and towards right along the mean stress axis in Fig. 12. Although the quadratic pressure dependence is often recognized by various yield criteria proposed in literature for foams, additional linear pressure dependence is largely overlooked which we believe is due partly to the scatter in experimental data but mostly due to lack of experimental data that cover the entire spectrum of stress states in a continuous manner from pure hydrostatic compression to hydrostatic tension. Considering that multiaxial experiments are difficult to run, require customized equipment and take significant amount of time, the yield data graphically presented in Figs. 11 and 12 are tabulated in Appendix A for the use of interested reader as well as the validation of new yield criteria that might be proposed in future studies. At the expense of repeating the obvious, it must be stressed that the elliptic shape of yield surface in Fig. 12 in mean stress-effective stress space is due to the effect of compressibility in yield behavior of foams and manifests itself as a quadratic mean stress term in many of the yield models proposed in literature. On the other hand, experimentally observed shift to the right in mean stress axis is a clear manifestation of an additional linear dependence that has not been properly addressed in previous studies.
Fig. 11. Yield surface of Divinycell H-100 PVC foam in the plane of anisotropy. Markers represent the yield points extracted from biaxial experiments while solid and dashed lines are the upper and lower bound predictions from the proposed yield criterion, i.e., Eq. (5).
Fig. 12. Yield surface of Divinycell H-100 PVC foam in mean stress-effective stress space. Markers represent the yield points extracted from uniaxial, biaxial and triaxial experiments while solid and dashed lines are the upper and lower bound predictions from the proposed yield criterion, i.e., Eq. (5).
The second important feature of current experimental data is the fact that Divinycell H-100 foam exhibits a transversely isotropic mechanical behavior, which suggests that many other solid foams (polymeric or metallic) produced by similar manufacturing techniques are likely to exhibit similar anisotropy. Therefore, a more general framework must be employed in the analysis of data to respect inherent anisotropy of solid foams. One such analytical framework is proposed in our companion paper (Ayyagari and Vural, 2013) whose results are extended to present experimental work through new scalar measures of mean stress, effective stress and characteristic stress along with respective work conjugate strains (see Eqs. (1) – (4)) that take the underlying transverse isotropy into account. This point becomes fairly important when analyzing yield data in the plane of anisotropy (or along triaxial stress paths) by using conventional isotropic definitions of mean stress and effective stress as it results in loosing crucial information and misinterpretation of data as exemplified in Fig. 6a of Ayyagari and Vural (2013). Finally, the experimental yield data presented in Figs. 11 and 12 are compared against a new yield criterion proposed in our companion paper (Ayyagari and Vural, 2013) as shown by solid and dashed lines in the figures. This new yield criterion is an extension of the yield model proposed for 2D foams (Alkhader and Vural, 2009) to 3D within the general framework of transverse isotropy, and it is given by:
§
βˆ 2 ·
§
©
9 ¹
©
σˆ e2 + βˆ 2σˆ m2 − 3 (Y1T + Y1C ) ¨¨1 +
¸¸ σˆ m = − ¨¨ 1 +
βˆ 2 ·
¸ Y1T Y1C 9 ¸¹
(5)
where Y1T and Y1C are uniaxial yield strengths in tension and compression, respectively, along 1direction. The βˆ factor is a function of elastic properties and strength anisotropy as defined in Table 2. Scalar measures of effective stress ( σˆ e ) and mean stress ( σˆ m ) that appear in Eq. (5) is slightly different from their conventional definitions to recognize the transverse isotropy of the foam as can be observed from Eq. (2). This yield criterion recognizes the linear pressure dependence observed in experimental data through the linear mean stress term which can be simply described by two uniaxial experiments ( Y1T and Y1C ). One more uniaxial experiment to find Y3T is required to fully define the yield surface for a transversely isotropic foam. One must note that this last experiment is only required for transversely isotropic foams to define βˆ which includes a strength anisotropy parameter hidden in Z term (see Table 2), otherwise Y1T and Y1C are enough to describe the yielding of isotropic foams. Other elastic constants that appear in βˆ term can be easily determined from the same uniaxial experiments. The yield criterion expressed by Eq. (5) needs minimal number of model constants obtainable from easy-to-perform uniaxial experiments because it is derived from the central hypothesis that total strain energy density drives the yielding behavior in solid foams, rather than a completely phenomenological approach. One important point that needs careful consideration, however, is the experimental scatter commonly observed in foam data. We propose to use the lowest and highest pairs of uniaxial strength values to plot a lower bound and upper bound to describe the yield surface. The terms “lower bound and upper bound” are loosely used here to denote the minimum and maximum expected boundaries of the yield surface based on the scatter in uniaxial data. Upper and lower bounds of yield surface plotted by this procedure, which are respectively denoted by solid and dashed elliptical curves in Figs. 11 and 12, satisfactorily capture the experimental data over a wide range stress states that effectively encapsulates the entire spectrum of stress paths one can possibly probe. The center of the ellipses has a slight offset in Figs. 11 and 12 which is a result of the linear pressure term introduced in the yield model. In this respect, the model prediction is completely in line with the experimental data and clearly shows the influence of linear pressure,
in addition to quadratic pressure, on the yield behavior of H-100 foam. However, underlying physical mechanisms for such a linear pressure dependence still remains to be investigated, which is the subject of a separate ongoing study.
4
CONCLUSIONS The yield surface of Divinycell H-100 PVC foam is experimentally investigated over a wide
range of complex stress states that cover the entire spectrum of stress paths from pure hydrostatic compression to hydrostatic tension. Uniaxial, biaxial and triaxial experiments are conducted by using a custom-built servo-hydraulic multiaxial testing apparatus (MATA) to probe the yield surface. Experimental results reveal the transversely isotropic nature of H-100 foam and also show a clear linear pressure dependence of its yield behavior, both of which have been largely overlooked in previous studies. Yield data obtained from extensive multiaxial experiments constitute the most comprehensive data set ever produced to date for any solid foam and, therefore, have been reported in Appendix A for the use of researchers as well as the validation of new yield criteria that might be proposed in future studies. Finally, the predictions of a new yield criterion, which is proposed in the companion paper (Ayyagari and Vural, 2013) for transversely isotropic foams, are compared against the experimental yield data with quite satisfactory results. The successful validation of this new yield criterion is attributed to (i) its general framework of transverse isotropy, (ii) its central hypothesis that the yield behavior in solid foams is driven by total strain energy density, and finally (iii) the recognition of linear pressure dependence which seems to lack in overwhelming majority of models proposed in the literature.
ACKNOWLEDGMENTS We are grateful for the support from the National Science Foundation (Grant No. 0728212 and 1030903).
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APPENDIX A Tabulated yield values for Divinycell H-100 PVC foam obtained from multiaxial experiments. σ1 σ2 σ3 σˆ e σˆ m σ1 σ2 σ3 σˆ e σˆ m
Uniaxial
Biaxial
-1.000 -0.990 -1.060 0.000 0.000 0.000 1.390 1.290 1.380 0.000 0.000 0.000 -1.020 -1.042 -1.006 -1.039 0.000 0.000 0.000 0.000 0.000 1.566 -0.991 0.000 0.678 0.940 1.084 1.346 1.211 1.581 1.346 1.601 1.703 0.963 0.439 0.447 0.997 1.292 1.574 1.451 1.525 -0.804 -0.832 -0.784 -0.782 -0.525 -0.469 -0.339 -0.117 -0.009 -0.917 -0.741 -0.977 -1.025 -1.081 -1.018 -1.127
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 -1.320 -1.420 -1.300 0.000 0.000 0.000 1.990 2.130 1.880 0.000 0.000 0.000 0.000 -1.300 -1.617 -1.675 -1.632 1.738 0.000 0.000 -1.473 1.975 2.139 1.851 1.983 1.474 1.591 1.125 1.037 1.954 2.796 2.477 2.260 1.950 1.860 1.250 0.753 0.541 0.332 0.736 0.914 1.227 1.169 1.536 1.646 1.734 1.661 0.422 0.128 0.108 -0.358 -0.487 -0.522 -0.741
-0.333 -0.330 -0.353 -0.321 -0.346 -0.317 0.463 0.430 0.460 0.485 0.519 0.458 -0.340 -0.347 -0.335 -0.346 -0.317 -0.394 -0.408 -0.397 0.423 0.522 -0.330 -0.359 0.707 0.834 0.812 0.931 0.763 0.914 0.722 0.786 1.043 1.002 0.749 0.699 0.807 0.884 0.829 0.667 0.640 -0.187 -0.098 -0.039 0.038 0.110 0.217 0.288 0.383 0.401 -0.203 -0.216 -0.299 -0.429 -0.479 -0.466 -0.556
1.000 0.990 1.060 0.964 1.037 0.950 1.390 1.290 1.380 1.454 1.556 1.373 1.020 1.042 1.006 1.039 0.950 1.181 1.224 1.192 1.269 1.566 0.991 1.076 1.273 1.394 1.275 1.441 1.183 1.457 1.203 1.413 1.628 1.802 1.649 1.494 1.299 1.365 1.400 1.288 1.384 0.943 1.184 1.246 1.440 1.194 1.405 1.395 1.326 1.218 1.096 0.789 1.017 0.931 0.966 0.904 0.995
Biaxial
Triaxial
-1.029 -1.122 -1.023 -1.039 -0.833 -0.957 -0.868 -0.963 -0.894 -0.866 -0.685 -0.578 -0.517 -0.111 1.193 1.112 0.138 0.067 0.432 0.690 1.542 0.471 0.588 0.936 1.618 1.361 1.663 1.574 1.391 1.347 1.618 1.533 1.709 1.666 1.636 1.665 1.100 1.446 1.563 1.460 1.465 -1.010 -1.111 -1.095 -0.728 -1.137 -1.066 -0.998 -1.078 -1.176 -1.059 -1.103 -1.185 -1.033 -0.904 -1.162
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.351 1.607 1.553 1.415 1.792 1.761 1.863 1.795 1.739 1.894 1.105 1.646 1.800 1.638 1.502 -0.906 -1.014 -0.969 -0.684 -0.915 -0.936 -1.013 -0.941 -0.858 -0.802 -0.981 -0.962 -0.934 -0.600 -0.805
-0.988 -0.992 -1.058 -0.976 -0.973 -1.150 -1.151 -1.278 -1.250 -1.403 -1.375 -1.607 -1.464 -1.231 -1.181 -0.847 -1.227 -1.098 -1.096 -0.993 -0.431 -1.061 -0.887 -0.929 -0.158 -0.661 1.269 1.510 1.981 2.422 0.851 0.824 1.247 0.685 1.249 2.180 2.171 1.681 0.994 0.085 0.143 -0.441 -0.710 -1.248 -1.255 -0.284 -0.742 -0.801 -0.581 -0.290 -0.763 -1.089 -0.486 -0.276 -1.296 -0.432
-0.583 -0.615 -0.599 -0.584 -0.515 -0.599 -0.569 -0.632 -0.602 -0.630 -0.563 -0.584 -0.529 -0.337 0.110 0.164 -0.253 -0.245 -0.123 -0.012 0.409 -0.101 -0.020 0.086 0.501 0.293 0.555 0.383 0.130 0.467 0.989 0.963 0.774 1.128 0.673 0.190 0.545 0.272 0.875 1.401 1.288 0.584 0.484 0.110 0.252 0.780 0.411 0.356 0.535 0.793 0.386 0.204 0.681 0.726 0.357 0.684
0.939 1.009 0.949 0.944 0.801 0.930 0.879 0.976 0.930 0.983 0.912 1.036 0.944 0.853 1.770 1.505 0.969 0.836 1.074 1.214 1.713 1.080 1.060 1.391 1.675 1.644 1.314 1.428 1.464 1.511 1.344 1.299 1.494 1.321 1.429 1.717 1.264 1.440 1.363 1.054 1.024 -0.746 -0.881 -0.992 -0.776 -0.754 -0.848 -0.865 -0.815 -0.749 -0.806 -0.960 -0.834 -0.723 -0.817 -0.761