- Email: [email protected]
Behavior pattern and parametric characterization for low density crushable foams
Journal of Materials Processing Technology 191 (2007) 73–76
Behavior pattern and parametric characterization for low density crushable foams Qunli Liu, Brendan O’Toole ∗ Department of Mechanical Engineering, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, NV 89154-4027, USA
Abstract Crushable foams may be used for shock mitigation and impact absorption applications that take advantage of their complex compression behavior. It is highly desirable that their constitutive behavior be characterized by suitable models with meaningful parameters such as Young’s modulus for linear elasticity. This work describes constitutive behavior patterns and parametric characterization for low density crushable foams. Compression tests under large deformation were conducted on polyurethane foams of different densities. Various behavior patterns were observed that differ from the well-known three-stage behavior for porous materials. A set of multi-parameter models was proposed to parametrically characterize the behavior patterns. The constitutive model was intended to serve as a framework for crushability characterization. © 2007 Elsevier B.V. All rights reserved. Keywords: Crushable foam; Constitutive model; Behavior pattern; Parametric characterization
1. Introduction Polymeric foams have been widely used in many applications. Crushable foams are attracting increasing interest in application scenarios related to impact energy absorption and shock mitigation with their intrinsic properties not found in monolithic solid materials. These sorts of porous materials have a wide range of variations in their formula and properties. Therefore, it is convenient to tune the material to meet some specific requirements. Under compression, a large strain can be accommodated without a significant stress increase. Polymeric foams are of a non-linear nature, with high strain-rate dependence and temperature sensitivity. In order to select a foam for a specific application, the mechanical behavior of candidate materials must be determined experimentally. In developing a foam-enhanced structure for a specific application, it is a common practice to conduct a complete finite element analysis for optimal design before a prototype is made. Computer modeling of early candidates can reduce time and cost associated with the expensive prototype process. However, this computer modeling requires complete material property data to accurately characterize the mechanical behavior in the application conditions. It is highly desirable that the mechanical ∗
Corresponding author. E-mail addresses: [email protected] (Q. Liu), [email protected] (B. O’Toole). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.03.044
properties of foams be continually calculated from constitutive equations for various loading conditions. Also, it is of practical value to develop a comprehensive constitutive model for characterization of foam behavior under large deformation as Young’s modulus for linear elasticity. Research has been reported on micro-mechanical modeling [1,2], constitutive behavior [3,4] and strain rate effect [5–7] for polymeric foams. It is well-known that porous materials under compression have three phases, namely linear elasticity, hardening-like apparent plasticity (stress plateau), and densification phase. Recently, Liu and Subhash [8] proposed a phenomenological model to represent the three-stage behavior of foams. A few uniaxial-strain and uniaxial-stress tests were used to validate the model. Liu et al. [9] also employed the model to parametrically describe residual crushability (crushability of a partially deformed foam) of an open-cell epoxy-based foam. It was shown that the model has versatility to characterize behavior of polymeric foams with bulk density above 0.2 g/cm3 . The fundamental model has the following form: σ(ε) = A
eαε − 1 , 1 + eβε
(1)
where A, α, β are the model parameters that have physical meaning with respect to the stress–strain curve [8]. Parameters α and β determine whether the stress plateau phase is hardening-like (α > β) or softening-like (α < β) as shown in Fig. 1(a). To capture
74
Q. Liu, B. O’Toole / Journal of Materials Processing Technology 191 (2007) 73–76
Fig. 1. Typical patterns of foam constitutive behavior under large deformation: (a) three patterns that can be represented by Eq. (1) and (2) for medium density foam and (b) three patterns observed for low density foam.
the densification phase, another term in Eq. (2) is added in Eq. (1): σ(ε) = eC (eγε − 1).
(2)
Parameter γ describes the intensity of densification. The densification phase becomes steeper as this parameter increases. These models meet the requirement for initial condition (zero strain at zero stress) and are continuously differentiable. Eq. (1) can satisfactorily represent softening-like or hardening-like stress–strain curves and the combination of Eqs. (1) and (2) can represent the response with significant densification shown in Fig. 1(a). A softening-like occurrence was observed in the stress–strain curves for low density foams prior to the stable stress-plateau phase, as shown in Fig. 1(b). Eq. (1) can be used to represent these patterns in small deformation (<12%) only. Another phenomenon was also observed in low density foams. A second softening-like occurrence showed up prior to densification. Immediately after this occurrence, the stress increased quickly for a small amount of strain and then increased a little slower, as shown in Fig. 1(b). The model presented in Eqs. (1) and (2) is not so versatile to capture this complicated behavior. The above model is modified in this investigation to capture the behavior patterns shown by low density foams under large compressive strain. 2. Constitutive models The model in Eq. (1) captures the primary compressive response of the foam. It is a good base on which to develop a more sophisticated model such that the complicated behavior shown in Fig. 1(b) can be satisfactorily characterized. A new extended model should attain the following requirements: (1) the model has a better capability in providing more accurate predictions of stress–strain relationship; (2) the model can be verified against experimental results on foams of different densities; (3) a unified expression should cover the current model such that it can describe all the behavior patterns shown in Fig. 1; (4) the new model should be kept as simple as possible and the parameters have explicit association with the behavior patterns. It can be seen from Fig. 1(a) that superposition of the softening-like response and the hardening-like response can con-
veniently produce a response shown in Fig. 1(b) in a strain range, i.e., less than 1. Inspired by this observation, we may assume another simple equation to represent the remaining portion of the response including the second softening-like occurrence. Therefore, the first step is to propose a model to cover the response up to the plateau stage: σ(ε) = A
eα1 ε − 1 eα2 ε − 1 +B , β ε 1+e 1 1 + eβ2 ε
(3)
where α1 > β1 and α2 < β2 . Recall the behavior patterns in Fig. 1(a); it is easy to perceive that the first term in Eq. (3) is primarily responsible for the linearly rising portion and the hardening-like portion, while the second term handles the softening-like portion. Another function must be found that deals with the second softening-like occurrence and the densification phase. The models will then be able to represent the entire response and its parameters will be associated with the response features. Eq. (3) is supposed to represent the response in the strain range prior to the second softening-like occurrence. The function representing the remaining (higher) strain range should therefore have negligible values in the lower strain range and be sensitive to the higher strain range only. This could be implemented on a right-shifted function that has the line at σ(ε) = 0 as its asymptote in negative strain range. The function should approach its asymptote quickly to mimic the negligible value situation. It was shown [8] that the first term approaches an asymptote located at σ(ε) = A in negative strain range. If another exponential function replaces the unity in the denominator to make it approach σ(ε) = 0 as strain goes to negative infinity, the above goal is attained. This function also should not alter the model in densification phase. The function is proposed as σ(ε) = C
eα3 (ε−δ) − 1 e−δα3 − 1 − C . e−β3 (ε−δ) + eβ3 (ε−δ) e−δβ3 + eδβ3
(4)
Parameter δ accounts for the right-shift and the second term is for zeroing the model under the initial condition (zero stress at zero strain). This function has a more complicated behavior than Eq. (1) or (3) and the roles of the parameters can be investigated separately. For conciseness, it is not presented here. Application of the models will be shown in the following section to paramet-
Q. Liu, B. O’Toole / Journal of Materials Processing Technology 191 (2007) 73–76
75
Fig. 2. Experimental results in comparison of model representation: (a) in uniaxial stress and (b) in uniaxial strain. Table 1 Model parameters for Foams X, Y and Z in uniaxial stress compression Foam
Model parameter A
α1
β1
B
α2
L G
0.454 0.449
38.2 38.0
37.9 37.7
5.22 5.14
L G
0.912 0.914
49.7 49.7
49.2 49.3
3.52 3.85
L G
2.16 2.18
57.5 57.3
57.1 56.8
2.28 2.64
β2
C
α3
β3
δ
3.01 3.10
28.4 28.2
0.082 0.082
16.2 12.9
12.3 9.05
0.814 0.826
7.65 7.01
37.0 36.9
0.081 0.109
11.4 8.89
6.99 4.68
0.613 0.669
44.4 43.4
0.356 0.555
5.94 6.09
2.28 2.69
0.538 0.650
X
Y
Z 20.9 18.0
Note: L and G represents local fitting and global fitting, respectively. Similar condition is applied in the subsequent table.
rically characterize various patterns of foam response obtained from experimental investigation.
Fig. 2(a) and (b) shows the stress-strain curves for Foams X, Y and Z in uniaxial stress and uniaxial strain, respectively. It can be seen that these foams have different behavior patterns under the two conditions considered. Under uniaxial stress, a noticeable softening-like occurrence exists following the elastic response and the densification phase is very smooth. While under uniaxial strain condition, a softening-like occurrence may or may not be observed close to the elastic response and the one prior to densification is considerable. Eq. (3) was first used to fit the uniaxial stress data prior to significant densification to ensure convergence as per the procedure presented in [8]. On the basis of the results from this first step (shown by dash dot lines in Fig. 2), Eq. (4) was used to represent the remaining portion in a specific strain range. The fitting range is set at a strain where the stress reaches 2.5 the maximum stress in the strain range of 0.1 (around 10%). Because Eqs. (3) and (4) were employed to fit different portions of the stress–strain curve locally (a local optimization procedure), the model parame-
3. Experimental validation Cylindrical samples were cut from polyurethane foam panels of thickness 25.4 mm with initial bulk densities of 0.065, 0.101 and 0.163 g/cm3 (called Foam X, Y and Z hereinafter), respectively. The samples had diameter of 26.72 mm, determined by the cutting tool. Compression tests were conducted on the foam samples under uniaxial stress and uniaxial strain conditions at low strain rate, namely around 10−3 s−1 . Uniaxial strain was achieved in a steel confinement cell that matched the samples. Experimental results are intended to verify that the constitutive models can reflect the effects of bulk density and confinement condition. Table 2 Model parameters for Foams X, Y and Z in uniaxial strain compression Foam
Model parameter A
α1
β1
L G
0.604 0.601
103.4 104.5
103.3 104.4
L G
1.20 1.20
77.1 75.6
77.0 75.4
L G
2.82 2.81
82.7 82.4
82.5 82.0
B
α2
β2
C
α3
β3
δ
0.206 0.207
25.9 22.9
21.9 19.0
1.03 1.03
X
Y 11.4 11.5
4.04 4.20
60.8 59.6
0.241 0.231
32.0 26.5
27.7 22.1
0.802 0.803
7.65 8.34
62.1 61.8
0.335 0.331
26.2 21.6
20.8 16.2
0.642 0.645
Z 9.981 9.391
76
Q. Liu, B. O’Toole / Journal of Materials Processing Technology 191 (2007) 73–76
ters may be improved in the entire fitting range. Therefore, combination of Eqs. (3) and (4) (dash lines in Fig. 2) was applied on the entire response in the fitting range with the parameter values obtained in the previous procedures as initial values (a global optimization procedure). The same procedure is also applied on uniaxial strain result for Foam X, with Eqs. (1) and (4) involved because of absence of softening-like occurrence. Experimental results from Foam Y and Z are intended to further validate the characterization procedure. They have similar behavior patterns in uniaxial stress or uniaxial strain, only minor differences were observed. Foam Y has sharp softening-like occurrence in uniaxial stress and Foam Z has that in uniaxial strain. For sharp softening-like occurrence, a scale parameter for the exponential function in the denominator of the second term in Eq. (3) can greatly improve the fitting situation. Details are not presented for conciseness. Table 1 presents the model parameters for Foams X, Y and Z in uniaxial stress compression, corresponding to Fig. 2(a). It can be seen from Table 1 that parameters A, α1 and β1 do not change much from local fitting to global fitting. The difference of α1 and β1 remains stable, dominating the hardeninglike response beyond the softening-like occurrence. Parameters B, α2 and β2 are basically responsible for the first softening-like occurrence, they do not change much either from local fitting to global fitting. Similar observation can be made for parameters C and δ. Although parameters α3 and β3 have considerable change from local fitting to global fitting, their difference remains stable. Table 2 presents the model parameters for test results in uniaxial strain, corresponding to Fig. 2(b). Because there is no noticeable softening-like occurrence prior to stress-plateau response for Foam X, Eqs. (1) and (4) are used to characterize the stress–strain curves. It can be seen that parameters in Table 2 have similar trend as those in Table 1.
4. Conclusion A few new behavior patterns were observed and reported for low density crushable foams. The foams tested have a softening-like occurrence following elastic regime and a smooth densification phase in uniaxial stress. In uniaxial strain, a softening-like occurrence was also observed prior to densification. Although the softening-like occurrence following elastic response can be attributed to buckling of cell walls, the cause for the softening-like occurrence prior to densification is not
ascertained yet. Nevertheless, new models were proposed for characterization of constitutive behavior of low density foams with the mentioned complicated patterns and were validated from experimental results. The models are applicable to various behavior patterns and model parameters are associated with local response features. The models have good maneuverability. This work, therefore, sets a foundation for characterization of foam crushability. Acknowledgements This work was supported in part by the U.S. Army cooperative agreement number DAAD19-03-2-0007, the U.S. Department of Energy (DOE) cooperative agreement DE-FC0898NV13410 and the General Plastics Manufacturing Company. References [1] C.M. Ford, L.J. Gibson, Uniaxial strength asymmetry in cellular materials: an analytical model, Int. J. Mech. Sci. 40 (1998) 521–531. [2] W.E. Warren, A.M. Kraynik, Nonlinear elastic behavior of open-cell foams, J. Appl. Mech. Trans. ASME 58 (1991) 376–381. [3] M.K. Neilsen, R.D. Krieg, H.L. Schreyer, Constitutive theory for rigid polyurethane foam, Polym. Eng. Sci. 35 (1995) 387–394. [4] J. Zhang, N. Kikuchi, V. Li, A. Yee, G. Nusholtz, Constitutive modeling of polymeric foam material subjected to dynamic crash loading, Int. J. Impact Eng. 21 (1998) 369–386. [5] H. Zhao, G. Gary, Behaviour characterization of polymeric foams over a large range of strain rates, Int. J. Vehicle Design 30 (2002) 135–145. [6] N.J. Mills, H.X. Zhu, High strain compression of closed-cell polymer foams, J. Mech. Phys. Solids 47 (1999) 669–695. [7] P. Viot, F. Beani, J.-L. Lataillade, Polymeric foam behavior under dynamic compressive loading, J. Mater. Sci. 40 (2005) 5829–5837. [8] Q. Liu, G. Subhash, A phenomenological constitutive model for polymer foams under large deformations, Polym. Eng. Sci. 44 (2004) 463–473. [9] Q. Liu, G. Subhash, X.-L. Gao, Parametric characterization for crushability of polymeric open-cell foams, J. Porous Mater. 12 (2005) 233–248.
Behavior pattern and parametric characterization for low density crushable foams Qunli Liu, Brendan O’Toole ∗ Department of Mechanical Engineering, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, NV 89154-4027, USA
Abstract Crushable foams may be used for shock mitigation and impact absorption applications that take advantage of their complex compression behavior. It is highly desirable that their constitutive behavior be characterized by suitable models with meaningful parameters such as Young’s modulus for linear elasticity. This work describes constitutive behavior patterns and parametric characterization for low density crushable foams. Compression tests under large deformation were conducted on polyurethane foams of different densities. Various behavior patterns were observed that differ from the well-known three-stage behavior for porous materials. A set of multi-parameter models was proposed to parametrically characterize the behavior patterns. The constitutive model was intended to serve as a framework for crushability characterization. © 2007 Elsevier B.V. All rights reserved. Keywords: Crushable foam; Constitutive model; Behavior pattern; Parametric characterization
1. Introduction Polymeric foams have been widely used in many applications. Crushable foams are attracting increasing interest in application scenarios related to impact energy absorption and shock mitigation with their intrinsic properties not found in monolithic solid materials. These sorts of porous materials have a wide range of variations in their formula and properties. Therefore, it is convenient to tune the material to meet some specific requirements. Under compression, a large strain can be accommodated without a significant stress increase. Polymeric foams are of a non-linear nature, with high strain-rate dependence and temperature sensitivity. In order to select a foam for a specific application, the mechanical behavior of candidate materials must be determined experimentally. In developing a foam-enhanced structure for a specific application, it is a common practice to conduct a complete finite element analysis for optimal design before a prototype is made. Computer modeling of early candidates can reduce time and cost associated with the expensive prototype process. However, this computer modeling requires complete material property data to accurately characterize the mechanical behavior in the application conditions. It is highly desirable that the mechanical ∗
Corresponding author. E-mail addresses: [email protected] (Q. Liu), [email protected] (B. O’Toole). 0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.03.044
properties of foams be continually calculated from constitutive equations for various loading conditions. Also, it is of practical value to develop a comprehensive constitutive model for characterization of foam behavior under large deformation as Young’s modulus for linear elasticity. Research has been reported on micro-mechanical modeling [1,2], constitutive behavior [3,4] and strain rate effect [5–7] for polymeric foams. It is well-known that porous materials under compression have three phases, namely linear elasticity, hardening-like apparent plasticity (stress plateau), and densification phase. Recently, Liu and Subhash [8] proposed a phenomenological model to represent the three-stage behavior of foams. A few uniaxial-strain and uniaxial-stress tests were used to validate the model. Liu et al. [9] also employed the model to parametrically describe residual crushability (crushability of a partially deformed foam) of an open-cell epoxy-based foam. It was shown that the model has versatility to characterize behavior of polymeric foams with bulk density above 0.2 g/cm3 . The fundamental model has the following form: σ(ε) = A
eαε − 1 , 1 + eβε
(1)
where A, α, β are the model parameters that have physical meaning with respect to the stress–strain curve [8]. Parameters α and β determine whether the stress plateau phase is hardening-like (α > β) or softening-like (α < β) as shown in Fig. 1(a). To capture
74
Q. Liu, B. O’Toole / Journal of Materials Processing Technology 191 (2007) 73–76
Fig. 1. Typical patterns of foam constitutive behavior under large deformation: (a) three patterns that can be represented by Eq. (1) and (2) for medium density foam and (b) three patterns observed for low density foam.
the densification phase, another term in Eq. (2) is added in Eq. (1): σ(ε) = eC (eγε − 1).
(2)
Parameter γ describes the intensity of densification. The densification phase becomes steeper as this parameter increases. These models meet the requirement for initial condition (zero strain at zero stress) and are continuously differentiable. Eq. (1) can satisfactorily represent softening-like or hardening-like stress–strain curves and the combination of Eqs. (1) and (2) can represent the response with significant densification shown in Fig. 1(a). A softening-like occurrence was observed in the stress–strain curves for low density foams prior to the stable stress-plateau phase, as shown in Fig. 1(b). Eq. (1) can be used to represent these patterns in small deformation (<12%) only. Another phenomenon was also observed in low density foams. A second softening-like occurrence showed up prior to densification. Immediately after this occurrence, the stress increased quickly for a small amount of strain and then increased a little slower, as shown in Fig. 1(b). The model presented in Eqs. (1) and (2) is not so versatile to capture this complicated behavior. The above model is modified in this investigation to capture the behavior patterns shown by low density foams under large compressive strain. 2. Constitutive models The model in Eq. (1) captures the primary compressive response of the foam. It is a good base on which to develop a more sophisticated model such that the complicated behavior shown in Fig. 1(b) can be satisfactorily characterized. A new extended model should attain the following requirements: (1) the model has a better capability in providing more accurate predictions of stress–strain relationship; (2) the model can be verified against experimental results on foams of different densities; (3) a unified expression should cover the current model such that it can describe all the behavior patterns shown in Fig. 1; (4) the new model should be kept as simple as possible and the parameters have explicit association with the behavior patterns. It can be seen from Fig. 1(a) that superposition of the softening-like response and the hardening-like response can con-
veniently produce a response shown in Fig. 1(b) in a strain range, i.e., less than 1. Inspired by this observation, we may assume another simple equation to represent the remaining portion of the response including the second softening-like occurrence. Therefore, the first step is to propose a model to cover the response up to the plateau stage: σ(ε) = A
eα1 ε − 1 eα2 ε − 1 +B , β ε 1+e 1 1 + eβ2 ε
(3)
where α1 > β1 and α2 < β2 . Recall the behavior patterns in Fig. 1(a); it is easy to perceive that the first term in Eq. (3) is primarily responsible for the linearly rising portion and the hardening-like portion, while the second term handles the softening-like portion. Another function must be found that deals with the second softening-like occurrence and the densification phase. The models will then be able to represent the entire response and its parameters will be associated with the response features. Eq. (3) is supposed to represent the response in the strain range prior to the second softening-like occurrence. The function representing the remaining (higher) strain range should therefore have negligible values in the lower strain range and be sensitive to the higher strain range only. This could be implemented on a right-shifted function that has the line at σ(ε) = 0 as its asymptote in negative strain range. The function should approach its asymptote quickly to mimic the negligible value situation. It was shown [8] that the first term approaches an asymptote located at σ(ε) = A in negative strain range. If another exponential function replaces the unity in the denominator to make it approach σ(ε) = 0 as strain goes to negative infinity, the above goal is attained. This function also should not alter the model in densification phase. The function is proposed as σ(ε) = C
eα3 (ε−δ) − 1 e−δα3 − 1 − C . e−β3 (ε−δ) + eβ3 (ε−δ) e−δβ3 + eδβ3
(4)
Parameter δ accounts for the right-shift and the second term is for zeroing the model under the initial condition (zero stress at zero strain). This function has a more complicated behavior than Eq. (1) or (3) and the roles of the parameters can be investigated separately. For conciseness, it is not presented here. Application of the models will be shown in the following section to paramet-
Q. Liu, B. O’Toole / Journal of Materials Processing Technology 191 (2007) 73–76
75
Fig. 2. Experimental results in comparison of model representation: (a) in uniaxial stress and (b) in uniaxial strain. Table 1 Model parameters for Foams X, Y and Z in uniaxial stress compression Foam
Model parameter A
α1
β1
B
α2
L G
0.454 0.449
38.2 38.0
37.9 37.7
5.22 5.14
L G
0.912 0.914
49.7 49.7
49.2 49.3
3.52 3.85
L G
2.16 2.18
57.5 57.3
57.1 56.8
2.28 2.64
β2
C
α3
β3
δ
3.01 3.10
28.4 28.2
0.082 0.082
16.2 12.9
12.3 9.05
0.814 0.826
7.65 7.01
37.0 36.9
0.081 0.109
11.4 8.89
6.99 4.68
0.613 0.669
44.4 43.4
0.356 0.555
5.94 6.09
2.28 2.69
0.538 0.650
X
Y
Z 20.9 18.0
Note: L and G represents local fitting and global fitting, respectively. Similar condition is applied in the subsequent table.
rically characterize various patterns of foam response obtained from experimental investigation.
Fig. 2(a) and (b) shows the stress-strain curves for Foams X, Y and Z in uniaxial stress and uniaxial strain, respectively. It can be seen that these foams have different behavior patterns under the two conditions considered. Under uniaxial stress, a noticeable softening-like occurrence exists following the elastic response and the densification phase is very smooth. While under uniaxial strain condition, a softening-like occurrence may or may not be observed close to the elastic response and the one prior to densification is considerable. Eq. (3) was first used to fit the uniaxial stress data prior to significant densification to ensure convergence as per the procedure presented in [8]. On the basis of the results from this first step (shown by dash dot lines in Fig. 2), Eq. (4) was used to represent the remaining portion in a specific strain range. The fitting range is set at a strain where the stress reaches 2.5 the maximum stress in the strain range of 0.1 (around 10%). Because Eqs. (3) and (4) were employed to fit different portions of the stress–strain curve locally (a local optimization procedure), the model parame-
3. Experimental validation Cylindrical samples were cut from polyurethane foam panels of thickness 25.4 mm with initial bulk densities of 0.065, 0.101 and 0.163 g/cm3 (called Foam X, Y and Z hereinafter), respectively. The samples had diameter of 26.72 mm, determined by the cutting tool. Compression tests were conducted on the foam samples under uniaxial stress and uniaxial strain conditions at low strain rate, namely around 10−3 s−1 . Uniaxial strain was achieved in a steel confinement cell that matched the samples. Experimental results are intended to verify that the constitutive models can reflect the effects of bulk density and confinement condition. Table 2 Model parameters for Foams X, Y and Z in uniaxial strain compression Foam
Model parameter A
α1
β1
L G
0.604 0.601
103.4 104.5
103.3 104.4
L G
1.20 1.20
77.1 75.6
77.0 75.4
L G
2.82 2.81
82.7 82.4
82.5 82.0
B
α2
β2
C
α3
β3
δ
0.206 0.207
25.9 22.9
21.9 19.0
1.03 1.03
X
Y 11.4 11.5
4.04 4.20
60.8 59.6
0.241 0.231
32.0 26.5
27.7 22.1
0.802 0.803
7.65 8.34
62.1 61.8
0.335 0.331
26.2 21.6
20.8 16.2
0.642 0.645
Z 9.981 9.391
76
Q. Liu, B. O’Toole / Journal of Materials Processing Technology 191 (2007) 73–76
ters may be improved in the entire fitting range. Therefore, combination of Eqs. (3) and (4) (dash lines in Fig. 2) was applied on the entire response in the fitting range with the parameter values obtained in the previous procedures as initial values (a global optimization procedure). The same procedure is also applied on uniaxial strain result for Foam X, with Eqs. (1) and (4) involved because of absence of softening-like occurrence. Experimental results from Foam Y and Z are intended to further validate the characterization procedure. They have similar behavior patterns in uniaxial stress or uniaxial strain, only minor differences were observed. Foam Y has sharp softening-like occurrence in uniaxial stress and Foam Z has that in uniaxial strain. For sharp softening-like occurrence, a scale parameter for the exponential function in the denominator of the second term in Eq. (3) can greatly improve the fitting situation. Details are not presented for conciseness. Table 1 presents the model parameters for Foams X, Y and Z in uniaxial stress compression, corresponding to Fig. 2(a). It can be seen from Table 1 that parameters A, α1 and β1 do not change much from local fitting to global fitting. The difference of α1 and β1 remains stable, dominating the hardeninglike response beyond the softening-like occurrence. Parameters B, α2 and β2 are basically responsible for the first softening-like occurrence, they do not change much either from local fitting to global fitting. Similar observation can be made for parameters C and δ. Although parameters α3 and β3 have considerable change from local fitting to global fitting, their difference remains stable. Table 2 presents the model parameters for test results in uniaxial strain, corresponding to Fig. 2(b). Because there is no noticeable softening-like occurrence prior to stress-plateau response for Foam X, Eqs. (1) and (4) are used to characterize the stress–strain curves. It can be seen that parameters in Table 2 have similar trend as those in Table 1.
4. Conclusion A few new behavior patterns were observed and reported for low density crushable foams. The foams tested have a softening-like occurrence following elastic regime and a smooth densification phase in uniaxial stress. In uniaxial strain, a softening-like occurrence was also observed prior to densification. Although the softening-like occurrence following elastic response can be attributed to buckling of cell walls, the cause for the softening-like occurrence prior to densification is not
ascertained yet. Nevertheless, new models were proposed for characterization of constitutive behavior of low density foams with the mentioned complicated patterns and were validated from experimental results. The models are applicable to various behavior patterns and model parameters are associated with local response features. The models have good maneuverability. This work, therefore, sets a foundation for characterization of foam crushability. Acknowledgements This work was supported in part by the U.S. Army cooperative agreement number DAAD19-03-2-0007, the U.S. Department of Energy (DOE) cooperative agreement DE-FC0898NV13410 and the General Plastics Manufacturing Company. References [1] C.M. Ford, L.J. Gibson, Uniaxial strength asymmetry in cellular materials: an analytical model, Int. J. Mech. Sci. 40 (1998) 521–531. [2] W.E. Warren, A.M. Kraynik, Nonlinear elastic behavior of open-cell foams, J. Appl. Mech. Trans. ASME 58 (1991) 376–381. [3] M.K. Neilsen, R.D. Krieg, H.L. Schreyer, Constitutive theory for rigid polyurethane foam, Polym. Eng. Sci. 35 (1995) 387–394. [4] J. Zhang, N. Kikuchi, V. Li, A. Yee, G. Nusholtz, Constitutive modeling of polymeric foam material subjected to dynamic crash loading, Int. J. Impact Eng. 21 (1998) 369–386. [5] H. Zhao, G. Gary, Behaviour characterization of polymeric foams over a large range of strain rates, Int. J. Vehicle Design 30 (2002) 135–145. [6] N.J. Mills, H.X. Zhu, High strain compression of closed-cell polymer foams, J. Mech. Phys. Solids 47 (1999) 669–695. [7] P. Viot, F. Beani, J.-L. Lataillade, Polymeric foam behavior under dynamic compressive loading, J. Mater. Sci. 40 (2005) 5829–5837. [8] Q. Liu, G. Subhash, A phenomenological constitutive model for polymer foams under large deformations, Polym. Eng. Sci. 44 (2004) 463–473. [9] Q. Liu, G. Subhash, X.-L. Gao, Parametric characterization for crushability of polymeric open-cell foams, J. Porous Mater. 12 (2005) 233–248.