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Creep deformation behavior of polymer materials with a 3D random pore structure: Experimental investigation and FEM modeling
Polymer Testing 80 (2019) 106097
Contents lists available at ScienceDirect
Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Material Behaviour
Creep deformation behavior of polymer materials with a 3D random pore structure: Experimental investigation and FEM modeling
T
Kanako Emori, Tatsuma Miura, Hiroshi Kishida, Akio Yonezu∗ Department of Precision Mechanics, Chuo University, 1-13-27 Kasuga, Bunkyo, Tokyo, 112-8551, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Porous polymer materials Pore structure Tension Creep Finite element method
This study investigates the creep deformation behavior of polyvinylidene difluoride (PVDF) with a 3D random pore structure. The test sample was a hollow fiber polymeric membrane with sub-micron-sized pores and an open cellular structure, which plays a critical role for water purification. Uniaxial tensile test was carried out for the polymeric membrane and it was found that the membranes underwent elasto-plastic deformation and creep deformation. In order to establish a numerical model, the finite element method (FEM) was employed. Using the Surface Evolver software, a 3D random pore structure was created in the representative volume element (RVE). The established computational model can predict both elastic and plastic deformation. Furthermore, the time–temperature–stress superposition principle (TTSSP) is employed for our FEM model to compute creep deformation. The present model enables the prediction of macroscopic and microscopic deformation behavior of porous materials that have a 3D random pore structure.
1. Introduction There are many porous polymer materials that are used for membrane separation, such as in filters, water purification, or ion exchange. Because these materials are mechanically deformed after a long period of use, it is important that the microscopic and macroscopic deformation behaviors be elucidated in order to improve structural integrity and develop high-performance products [1–5]. Although numerical simulation using the finite element method (FEM) is an effective way to predict deformation behavior, porous materials tend to possess an inhomogeneous pore shape (i.e. random pore structure) due to their manufacturing process (phase separation, drawing, foaming, etc.). The pore structure is three-dimensional and consists of a strut network of base material. Such a random structure can make it difficult to establish a computational model. There are several studies on computational modeling of random pore structures using FEM. To build up the microstructure of a porous material, the key geometric features are the network of struts and the strut geometry. In addition, an appropriate constitutive model is needed for the strut network of the base material, so that the FEM computation can reproduce all aspects of the mechanical response. An effective model is the 14-sided periodic Kelvin cell (Kelvin structure) [6–8]. It was reported that such models accurately capture the initial elastic modulus, yield strength, and the flat stress plateau. Since compressive
∗
loading onto porous materials is crucial for engineering, elements such as shock absorption, the buckling of struts, localized deformation instability, densification of pores, and crushing have been successfully modeled [9–12]. This kind of porous structure can be generated using the Surface Evolver software, in which the random foam models start as a soap froth microstructure, and the foaming factor can be changed randomly to make an actual random foam. Next, the ligaments (struts) are dressed with an appropriate distribution of solids to match that of the actual foam and its relative density. Gaitanaros et al. indicated that such random foam models are capable of capturing the mechanical responses of metallic foam subjected to quasi-static loading and crushing [11]. Furthermore, they investigated the simulation of dynamic crushing and its shock propagation [12]. Their modeling technique may be favored for various porous polymers, which is included in our target material. As mentioned, this study aims to investigate creep deformation behavior of porous polymer membranes subjected to uniaxial tension. It is expected that uniaxial tension causes the polymer struts to stretch, and then for the strut material to exhibit time-dependent creep deformation. Thus, our target sample may demonstrate a different result to metallic foam in the previous studies [6–12]. Indeed, the strut base material of polymer is significantly softer than metal, which presents a challenge in this study. This study created a FEM model that is capable of simulating the tensile creep deformation behavior of porous polymer materials. The
Corresponding author. E-mail address: [email protected] (A. Yonezu).
https://doi.org/10.1016/j.polymertesting.2019.106097 Received 15 July 2019; Received in revised form 29 August 2019; Accepted 3 September 2019 Available online 03 September 2019 0142-9418/ © 2019 Elsevier Ltd. All rights reserved.
Polymer Testing 80 (2019) 106097
K. Emori, et al.
test sample was a polyvinylidene difluoride (PVDF) hollow fiber membrane, which is commercially available for water purification. During purification and for long-term operation, the tensile creep deformation of the porous membrane is critical and deformation modeling is important [2,4,5,13]. In other words, when a water purification membrane is used for long period of time, creep deformation is inevitable, since the polymer strut easily deforms with time dependent. The test sample had an inhomogeneous, random 3D pore structure that the study attempted to replicate for FEM computation. This study first examined uniaxial tension followed by tensile creep deformation. For both deformations, macroscopic deformation behaviors were computed using FEM and compared with experimental data to verify the developed model. This model may be useful for understanding structural integrity of porous polymer membranes, especially investigation into the relationship between microscopic pore structure and macroscopic deformation behavior.
2. Materials and methods
Fig. 1. Microstructure of the hollow-fiber membrane: (a) macroscopic view and (b) microscopic view of cross section.
2.1. Material
3. Experimental results
This study employed a hollow fiber PVDF membrane, which is used for water purification. As shown in Fig. 1 (a), the cross section of the test sample is a tube shape with an outer diameter of 1.23 mm, an inner diameter of 0.67 mm, and a wall thickness of 0.28 mm. The sample was manufactured using the thermally induced phase separation (TIPS) method. When the surface and cross section of the test sample were observed using a field emission scanning electron microscope (FESEM), it was found that the microstructure was a heterogeneous open cellular structure (Fig. 1(b)). From the mass ratio (with a PVDF density of 1.78 g/cm3), the porosity of the sample was measured to be 65%. It is well known that PVDF is a crystalline polymer material, and there are several types of crystal structures, therefore, Fourier transform infrared spectrophotometer (FT-IR, AIM-8800, Shimadzu Corporation) was used to reveal that the sample had an α phase crystal structure.
3.1. Pore structure The microstructure of the entire cross section of the membrane was carefully observed. It was confirmed that the pore structure did not change significantly depending on position by observation of the test sample near the outer surface, the middle part, and near the inner surface in the cross section, which allows the test sample to be categorized as a symmetric membrane. Fig. 2 shows an enlarged view of the open cell structure of the pore which is composed of a PVDF matrix (ligament or strut). Fig. 2 also shows the shape of the PVDF ligament (strut), whose shape was not simply straight, but looks like a cylinder shape with a constricted diameter. The cross section of the PVDF ligament was not uniform in the length direction. Here, we assumed the cross section to be a perfect circle, and that each strut had a constricted shape with the largest diameter and smallest diameter (diameter ratio2 is defined in this study). Subsequently, in order to create the FEM model, the geometric information of the microstructure was extracted from the FESEM images. The pore size distribution (Fig. 3(a)) and the diameter ratio of the PVDF struts (Fig. 3(b)) were measured.3 Note that the pore is assumed to be a perfect circle in Fig. 3 (a), with a 160 nm average diameter. As shown in Fig. 3(b), the maximum diameter is located at the ends of the strut, and the minimum diameter is at the strut center. The diameter ratio was measured, and its distribution is plotted in Fig. 3(b). The average value was approximately 0.75. As described above, these distributions (Fig. 3(a) and (b)) appeared uniform in the thickness direction of the fiber membrane.
2.2. Experimental method This study conducted the uniaxial tension and creep tests using a universal testing machine (Auto Graph, AG-1, Shimadzu Corporation). The load was measured using a load cell, and the displacement was measured using the crosshead movement. The samples were attached to the testing machine using a capstan type clamp. The uniaxial tension test was performed under displacement control with four different strain rates (0.9 × 10 −4, 4.8 × 10 −4, 4.8 × 10 −3, 1.4 × 10 −2 s−1) at room temperature. The creep tests were performed under load control for all samples. First, the test sample was loaded at a strain rate of 0.9 × 10 −4 s−1 until a given load was reached, and then the load was sustained during the creep test, about 72 h. The test was conducted at room temperature, and seven different sustained loads (2.0, 2.5, 3.0, 3.5, 4.0, 4.5 N) were selected. These selected loads corresponded to nominal stresses of 2.4, 3.0, 3.6, 4.2, 4.8, and 5.3 MPa. More detail information is described in the footnote.1
2 Each strut had a constricted shape with a maximum diameter of ϕd_max and a minimum diameter of ϕd_min. The diameter ratio of each strut is the ratio of its minimum and maximum diameters. 3 We first measured area for each pore and assumed that a pore is true circle shape. Next, the dimeter of pore circle was investigated to make a distribution of pore diameter (in Fig. 3(a)). For Fig. 3(b), diameter ratio is the ratio of between minimum diameter and maximum diameter as described in the manuscript. We measured this diameter ratio from the pictures of SEM observation to investigate its distribution. For the simple modeling of strut, the average diameter of strut is determined from macroscopic porosity and overall strut volume in the RVE. Based on the average strut diameter and diameter ratio, each strut was modeled in the RVE. Indeed, to gradually change the diameter of strut (the changing law of strut), the diameter is changed in 5 steps to express neck shape of strut (a constricted diameter).
1 For each test (each loading condition), the test was conducted at least three times. The average data was employed in this study. The sustained load was controlled within about 5%. The creep test was conducted about 72 h. As discussed later (e.g. Fig. 5), longer time of creep test may result in linear relationship of creep deformation, which is easily predicted. Thus, we conducted creep tests up to 72 h.
2
Polymer Testing 80 (2019) 106097
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Fig. 5. Experimental creep strain curves for different loading values. Fig. 2. Magnified view of cross section of the hollow-fiber membrane.
Fig. 3. Distribution of (a) pore size and (b) diameter ratio of the strut of the hollow-fiber membrane.
3.2. Uni-axial tension
3.3. Creep test
Fig. 4 shows the nominal stress-strain curve of the uniaxial tension tests at different strain rates. At the initial stage, the stress increased linearly as the strain increased. When the stress reached about 4 MPa, the stress-strain curve bent, and then continued to increase linearly until fracture. Iio et al. conducted similar experiments, in which detailed examinations were conducted using FESEM to understand microscopic deformation behavior [5]. As a result, it was concluded that the initial linear process was in the elastic deformation regime, and the subsequent process was in the plastic deformation regime. This tendency was observed at all strain rates. Next, the influence of a change in strain rate on the deformation behavior was investigated. The yield stress increased slightly as the strain rate increased, but the strain rate dependency was not very significant in this study.
A creep test was conducted for each sustained load. The creep strain-time curves are shown in Fig. 5. Note that the start time of the creep test is at the origin in the figure. It was carefully confirmed that the stress-strain curves until the holding load are almost the same for all tests. As shown, the creep strain increased gradually with time and the slope tended to decrease with time. However, the creep strain behavior is different, depending on the sustained load. At the minimum force of F = 2 N, creep deformation is small. On the other hand, a higher sustained load (F = 3 or higher) induces large creep deformation. 4. Discussion 4.1. FEM model This study created a model of the heterogeneous microstructure with random pores for the FEM computation test sample. We first used Surface Evolver, which is a structural modeling software for random soap froth, including bubbles, foamed particles, and their cluster [14]. The calculation was setup to minimize the balance between surface energy and mechanical force by satisfying Plateau's laws [12,15]. As shown in Fig. 6 (a), in order to create the model, a cubic space was filled with a Kelvin structure as the initial condition. In this study, the number of Kelvin structures was 125 (5 × 5 x 5). This cube corresponds to a representative volume element (RVE) for FEM computation, which is described later. Next, to randomly change the volume of each Kelvin structure in the cube foaming analysis was performed using the Surface Evolver software. As shown in Fig. 6 (b), the created microstructure becomes random foam for a heterogeneous cellular structure. Since the present membrane has an open cellular structure (Fig. 1), a strut network in the foam is required. Using MATLAB, the strut data of each foam structure was extracted. The skeleton frame was then
Fig. 4. Experimental nominal stress-strain curves at different strain rates. 3
Polymer Testing 80 (2019) 106097
K. Emori, et al.
Fig. 6. Random bubble analysis process: (a) before foaming, (b) after foaming. Fig. 8. FEM model of the random pore structure.
imported to the FEM software (MSC, Marc). Fig. 7 shows an illustration of the FEM model. In this FEM model, beam elements were employed for the PVDF strut network. The cross section of the beam element was assumed to be a perfect circle. The diameter of the cross section was adjusted so that the porosity of the entire RVE was 65%. Note that, as shown in Fig. 7, the ratio of the minimum diameter to the maximum diameter of the PVDF strut was referred to as the actual diameter ratio (see Fig. 3(b)). To verify the microstructural model, the pore size and diameter ratio were measured and compared with the actual experiment as shown in Fig. 3. The created FEM model is indicated by the red “x”, shown overlapping on the actual microstructure indicated by an open circle. This demonstrates that the distribution of pore diameter and diameter ratio are well reproduced. In other words, the random pore structure is geometrically reproduced. Therefore, we successfully developed a FEM model with a heterogeneous microscopic structure for the polymeric membrane.
2
h ε m σ ‾ = Ke g ‾p⋅‾ε˙
(1)
Here, K is a work hardening coefficient, hg is a work hardening index, and m is a strain rate dependent coefficient. These material constants were obtained as K = 76.0 MPa, hg = 0.05, and m = 0.049 from previous studies [16]. The von Mises yield criterion is employed in this study. In addition, general polymer materials may show strain rate dependence even during elastic deformation. However, as shown in Fig. 5, the PVDF hollow fiber membrane showed very little dependency on strain rate in the linear curve. Therefore, the strain rate dependence of the elastic properties was not considered in this study. The computational result of FEM is shown in Fig. 9 together with the experimental result. It was found that the computational result generally agrees with the experimental one during elasto-plastic deformation. Thus, our model is capable of simulating elasto-plastic deformation behavior up to about 0.2 strain. 4.3. Creep deformation
4.2. Elasto-plastic deformation Polymer materials often exhibit time-dependent deformation behavior, such as creep and stress relaxation. Since the present sample is a porous polymer material, the base polymer and inherent pore structure encourages deformation as compared to a solid material. Such deformability in elasto-plastics and creep become important for water purification operations. Creep deformation of polymers is generally investigated using creep compliance as a parameter, which is defined as creep strain divided by applied strain. It has been reported that creep compliance is sometimes dependent on applied stress; this kind of phenomenon is called non-linear creep, which appears when testing times are longer, applied stress is larger, or when both conditions occur simultaneously [18–20]. Much research in modeling non-linear creep deformation behavior has already been performed. Jazouli et al. and Ikeshima et al. used the time–temperature–stress superposition
Fig. 8 shows the created FEM model with the beam element virtually displayed in three dimensions. Note that the diameter ratio is also indicated in this figure. It can be seen that the microstructure of the model is random pore, which is similar to the actual structure seen in Fig. 1. To conduct the elasto-plastic analysis a periodic boundary condition was employed, and uniaxial tensile loading was applied to the FEM model so that the microscopic and macroscopic deformation behavior of the polymeric membrane can be computed. It is noted that the elasto-plastic properties of the PVDF matrix for FEM computation were derived from previous studies [16], and elastic deformation was assumed to follow Hooke's law. In addition, the stress-strain (σ ‾ − εp ) curve during plastic deformation was influenced by strain rate (‾ε˙ m) , and the following work hardening rule [17] was employed.
Fig. 7. FEM model of random porous structure in the RVE (a) and the shape of the strut structure (b). 4
Polymer Testing 80 (2019) 106097
K. Emori, et al.
Fig. 9. Nominal stress-strain curves from the experimental results and FEM computation.
Fig. 11. Creep strain curves from the experimental results and FEM computation.
principle (TTSSP) [20,21] for polymer creep testing, in which the creep compliance curve at different temperatures or stress levels can be shifted along the time scale by using a master curve at a reference temperature or stress level. They successfully obtained a master curve for creep compliance, which can be acceptable for a wide range of applied stress levels. Therefore, non-linear creep deformation under different applied stresses can be predicted based on TTSSP. For this principal, creep compliance D(t) (creep strain normalized by the applied constant stress) is expressed by the following equation.
7, random pore structures may be created for porous polymeric membranes, which are used for water purification. In addition, appropriate constitutive equations for elasto-plastic properties and creep were employed, and macroscopic deformation behavior can successfully be predicted. 5. Conclusion
(3)
This study employed FEM in order to establish a method of deformation modeling for a polymer membrane with an inhomogeneous porous structure. The test sample was a PVDF hollow fiber membrane used for water purification. Time-dependent tensile deformation behavior was examined. The obtained results are summarized below.
Here, b and c are material constants. Φσ is stress shift factor, which is expressed by constants C1 and C2 and the reference stress σ0. The material constants are derived from experimental results [16]. Fig. 10 shows the variation in creep compliance as a function of applied stress of PVDF bulk-solid material (note that it is not porous material) (Fig. 10(a)) [16]. Subsequently, the master curve of the creep compliance based on experimental data was drawn (Fig. 10(b)). When appropriate constants for C1 and C2 (reduced time) are selected, creep compliance curves can be unified by using the reduced time, as shown in this figure. This indicates that non-liner creep deformation can be modeled by this master curve. These constitutive equations (Eqs. (2) and (3)) were substituted into the FEM model as the base PVDF, and the macroscopic creep strain was computed (similar to Fig. 5). We conducted several tests with the different applied loadings of 2, 2.5, and 3 N. Fig. 11 shows the computed macroscopic creep strain curves for different loads together with the experimental results of Fig. 5, indicating very good agreement with each other. Thus, our FEM model is useful for deformation modeling of a random porous structure. By using the modeling process in Figs. 6 and
1. The test sample had an inhomogeneous three-dimensional open cellular structure, with submicron-sized pores and a porosity of 65%. In order to create a FEM model, the pore structure (the distribution of pore size and strut shape) was measured using FESEM images. 2. Stress-strain curves were obtained from a uniaxial tension test, which demonstrated linear-elastic deformation and plastic deformation. Creep tests were also conducted to investigate the timedependent deformation behavior of the test sample, showing that creep strain depends on applied load and creep time. 3. In order to create a sample with an inhomogeneous three-dimensional open cellular structure, we first created a representative volume element (RVE) where Kelvin structures were arranged, and the random foaming calculation was performed for each Kelvin structure. Finally, the strut of each structure was imported into FEM software. The created inhomogeneous structure reproduced well the actual porous structure. 4. The established FEM model successfully simulated the overall
D (t ) = b⋅(t⋅Φσ )c log Φσ =
C1 (σ − σ0) C2 + (σ − σ0)
(2)
Fig. 10. (a) Variation of creep compliances at a four different applied stress levels and (b) master curve for the creep compliances. 5
Polymer Testing 80 (2019) 106097
K. Emori, et al.
elastic-plastic deformation and creep deformation of the test sample. Moreover, the microscopic deformation of the random pore structure can be discussed. Thus, new strength design, such as optimization of microstructures, can be performed by utilizing the proposed deformation modeling of the 3D random porous structure.
[6] [7] [8]
Acknowledgements
[9] [10]
The authors are grateful to Mr. Takumi NAGAKURA at Chuo University for their important contributions. This work was supported by the “Nanotechnology Platform” of the Ministry of Education, Culture, Sports, Science and Technology, Japan. This work is supported by JSPS KAKENHI (Grantno.26420025) from the Japan Society for the Promotion of Science.
[11] [12] [13]
Appendix A. Supplementary data
[14] [15]
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.polymertesting.2019.106097.
[16]
References
[17]
[1] Amy E. Childress, Pierre Le-Clech, L. Joanne, Daugherty, et al., Mechanical analysis of hollow fiber membrane integrity in water reuse applications, Desalination 180 (2005) 5–14. [2] Jason A. Morehouse, Douglas R. Lloyd, Benny D. Freeman, et al., Modeling the stretching of microporous membranes, J. Membr. Sci. 283 (2006) 430–439. [3] P.L. Hanks, K.J. Kaczorowski, E.B. Becker, D.R. Lloyd, Modeling of uni-axial stretching of track-etch membranes, J. Membr. Sci. 305 (2007) 196–202. [4] A. Yonezu, S. Iio, T. Itonaga, et al., Tensile deformation of polytetrafluoroethylene (PTFE) hollow fiber membrane used for water purification, Water Sci. Technol. 70 (2014) 1244–1250. [5] S. Iio, A. Yonezu, H. Yamamura, X. Chen, Deformation modeling of
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polyvinylidenedifluoride (PVDF) symmetrical microfiltration hollow-fiber (HF) membrane, J. Membr. Sci. 497 (2016). L. Gong, S. Kyriakides, N. Triantafyllidis, On the stability of Kelvin cell foams under compressive loads, J. Mech. Phys. Solids 53 (2005) 771–794. L. Gong, S. Kyriakides, W.-Y. Jang, Compressive response of open-cell foams. Part I: morphology and elastic properties, Int. J. Solids Struct. 42 (2005) 1355–1379. W.Y. Jang, A.M. Kraynik, S. Kyriakides, On the microstructure of open-cell foams and its effect on elastic properties, Int. J. Solids Struct. 45 (2008) 1845–1875. W.Y. Jang, S. Kyriakides, On the crushing of aluminum open-cell foams: Part II analysis, Int. J. Solids Struct. 46 (2009) 635–650. W.Y. Jang, S. Kyriakides, A.M. Kraynik, On the compressive strength of open-cell metal foams with kelvin and random cell structures, Int. J. Solids Struct. 47 (2010) 2872–2883. S. Gaitanaros, S. Kyriakides, A.M. Kraynik, On the crushing response of random open-cell foams, Int. J. Solids Struct. 49 (2012) 2733–2743. S. Gaitanaros, S. Kyriakides, Dynamic crushing of aluminum foams: Part II – analysis, Int. J. Solids Struct. 51 (2014) 1646–1661. Jingxuan Jia, Guodong Kang, Yiming Cao, Effect of stretching parameters on structure and properties of polytetrafluoroethylene hollow‐fiber membranes, Chem. Eng. Technol. (2016) 935–944. The Surface Evolver Version 2.70, August 25, 2013 http://facstaff.susqu.edu/ brakke/evolver/evolver.html. D.L. Weaire, S. Hutzler, The Physics of Foams, Oxford University press, Oxford, 1999. S. Castagnet, J. Gacougnolle, P. Dang, Correlation between macroscopical viscoelastic behavior and micromechanisms in strained a polyvinylidene fluoride (PVDF), Mater. Sci. Eng. A 276 (2000) 152–159. C.G. Sell, J.J. Jonas, Determination of the plastic behaviour of solid polymers at constant true strain rate, J. Mater. Sci. 14 (1979) 583–591. J. Lai, A. Bakker, Analysis of the non-linear creep of highdensity polyethylene, Polymer 36 (1995) 93–99. W. Luo, T. Yang, Q. An, Time-temperature-stress equivalence and its application to nonlinear viscoelastic materials, Acta Mech. Solida Sin. 14 (2001) 195–199. S. Jazouli, W. Luo, F. Bremand, T. Vu-Khanh, Application of time-stress equivalence to nonlinear creep of polycarbonate, Polym. Test. 24 (2005) 463–467. D. Ikeshima, A. Matsuzaki, T. Nagakura, et al., Non-linear creep deformation of polycarbonate at high stress level: experimental investigation and FEM modeling, J. Mater. Eng. Perform. 28 (2019) 1612–1617.
Contents lists available at ScienceDirect
Polymer Testing journal homepage: www.elsevier.com/locate/polytest
Material Behaviour
Creep deformation behavior of polymer materials with a 3D random pore structure: Experimental investigation and FEM modeling
T
Kanako Emori, Tatsuma Miura, Hiroshi Kishida, Akio Yonezu∗ Department of Precision Mechanics, Chuo University, 1-13-27 Kasuga, Bunkyo, Tokyo, 112-8551, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Porous polymer materials Pore structure Tension Creep Finite element method
This study investigates the creep deformation behavior of polyvinylidene difluoride (PVDF) with a 3D random pore structure. The test sample was a hollow fiber polymeric membrane with sub-micron-sized pores and an open cellular structure, which plays a critical role for water purification. Uniaxial tensile test was carried out for the polymeric membrane and it was found that the membranes underwent elasto-plastic deformation and creep deformation. In order to establish a numerical model, the finite element method (FEM) was employed. Using the Surface Evolver software, a 3D random pore structure was created in the representative volume element (RVE). The established computational model can predict both elastic and plastic deformation. Furthermore, the time–temperature–stress superposition principle (TTSSP) is employed for our FEM model to compute creep deformation. The present model enables the prediction of macroscopic and microscopic deformation behavior of porous materials that have a 3D random pore structure.
1. Introduction There are many porous polymer materials that are used for membrane separation, such as in filters, water purification, or ion exchange. Because these materials are mechanically deformed after a long period of use, it is important that the microscopic and macroscopic deformation behaviors be elucidated in order to improve structural integrity and develop high-performance products [1–5]. Although numerical simulation using the finite element method (FEM) is an effective way to predict deformation behavior, porous materials tend to possess an inhomogeneous pore shape (i.e. random pore structure) due to their manufacturing process (phase separation, drawing, foaming, etc.). The pore structure is three-dimensional and consists of a strut network of base material. Such a random structure can make it difficult to establish a computational model. There are several studies on computational modeling of random pore structures using FEM. To build up the microstructure of a porous material, the key geometric features are the network of struts and the strut geometry. In addition, an appropriate constitutive model is needed for the strut network of the base material, so that the FEM computation can reproduce all aspects of the mechanical response. An effective model is the 14-sided periodic Kelvin cell (Kelvin structure) [6–8]. It was reported that such models accurately capture the initial elastic modulus, yield strength, and the flat stress plateau. Since compressive
∗
loading onto porous materials is crucial for engineering, elements such as shock absorption, the buckling of struts, localized deformation instability, densification of pores, and crushing have been successfully modeled [9–12]. This kind of porous structure can be generated using the Surface Evolver software, in which the random foam models start as a soap froth microstructure, and the foaming factor can be changed randomly to make an actual random foam. Next, the ligaments (struts) are dressed with an appropriate distribution of solids to match that of the actual foam and its relative density. Gaitanaros et al. indicated that such random foam models are capable of capturing the mechanical responses of metallic foam subjected to quasi-static loading and crushing [11]. Furthermore, they investigated the simulation of dynamic crushing and its shock propagation [12]. Their modeling technique may be favored for various porous polymers, which is included in our target material. As mentioned, this study aims to investigate creep deformation behavior of porous polymer membranes subjected to uniaxial tension. It is expected that uniaxial tension causes the polymer struts to stretch, and then for the strut material to exhibit time-dependent creep deformation. Thus, our target sample may demonstrate a different result to metallic foam in the previous studies [6–12]. Indeed, the strut base material of polymer is significantly softer than metal, which presents a challenge in this study. This study created a FEM model that is capable of simulating the tensile creep deformation behavior of porous polymer materials. The
Corresponding author. E-mail address: [email protected] (A. Yonezu).
https://doi.org/10.1016/j.polymertesting.2019.106097 Received 15 July 2019; Received in revised form 29 August 2019; Accepted 3 September 2019 Available online 03 September 2019 0142-9418/ © 2019 Elsevier Ltd. All rights reserved.
Polymer Testing 80 (2019) 106097
K. Emori, et al.
test sample was a polyvinylidene difluoride (PVDF) hollow fiber membrane, which is commercially available for water purification. During purification and for long-term operation, the tensile creep deformation of the porous membrane is critical and deformation modeling is important [2,4,5,13]. In other words, when a water purification membrane is used for long period of time, creep deformation is inevitable, since the polymer strut easily deforms with time dependent. The test sample had an inhomogeneous, random 3D pore structure that the study attempted to replicate for FEM computation. This study first examined uniaxial tension followed by tensile creep deformation. For both deformations, macroscopic deformation behaviors were computed using FEM and compared with experimental data to verify the developed model. This model may be useful for understanding structural integrity of porous polymer membranes, especially investigation into the relationship between microscopic pore structure and macroscopic deformation behavior.
2. Materials and methods
Fig. 1. Microstructure of the hollow-fiber membrane: (a) macroscopic view and (b) microscopic view of cross section.
2.1. Material
3. Experimental results
This study employed a hollow fiber PVDF membrane, which is used for water purification. As shown in Fig. 1 (a), the cross section of the test sample is a tube shape with an outer diameter of 1.23 mm, an inner diameter of 0.67 mm, and a wall thickness of 0.28 mm. The sample was manufactured using the thermally induced phase separation (TIPS) method. When the surface and cross section of the test sample were observed using a field emission scanning electron microscope (FESEM), it was found that the microstructure was a heterogeneous open cellular structure (Fig. 1(b)). From the mass ratio (with a PVDF density of 1.78 g/cm3), the porosity of the sample was measured to be 65%. It is well known that PVDF is a crystalline polymer material, and there are several types of crystal structures, therefore, Fourier transform infrared spectrophotometer (FT-IR, AIM-8800, Shimadzu Corporation) was used to reveal that the sample had an α phase crystal structure.
3.1. Pore structure The microstructure of the entire cross section of the membrane was carefully observed. It was confirmed that the pore structure did not change significantly depending on position by observation of the test sample near the outer surface, the middle part, and near the inner surface in the cross section, which allows the test sample to be categorized as a symmetric membrane. Fig. 2 shows an enlarged view of the open cell structure of the pore which is composed of a PVDF matrix (ligament or strut). Fig. 2 also shows the shape of the PVDF ligament (strut), whose shape was not simply straight, but looks like a cylinder shape with a constricted diameter. The cross section of the PVDF ligament was not uniform in the length direction. Here, we assumed the cross section to be a perfect circle, and that each strut had a constricted shape with the largest diameter and smallest diameter (diameter ratio2 is defined in this study). Subsequently, in order to create the FEM model, the geometric information of the microstructure was extracted from the FESEM images. The pore size distribution (Fig. 3(a)) and the diameter ratio of the PVDF struts (Fig. 3(b)) were measured.3 Note that the pore is assumed to be a perfect circle in Fig. 3 (a), with a 160 nm average diameter. As shown in Fig. 3(b), the maximum diameter is located at the ends of the strut, and the minimum diameter is at the strut center. The diameter ratio was measured, and its distribution is plotted in Fig. 3(b). The average value was approximately 0.75. As described above, these distributions (Fig. 3(a) and (b)) appeared uniform in the thickness direction of the fiber membrane.
2.2. Experimental method This study conducted the uniaxial tension and creep tests using a universal testing machine (Auto Graph, AG-1, Shimadzu Corporation). The load was measured using a load cell, and the displacement was measured using the crosshead movement. The samples were attached to the testing machine using a capstan type clamp. The uniaxial tension test was performed under displacement control with four different strain rates (0.9 × 10 −4, 4.8 × 10 −4, 4.8 × 10 −3, 1.4 × 10 −2 s−1) at room temperature. The creep tests were performed under load control for all samples. First, the test sample was loaded at a strain rate of 0.9 × 10 −4 s−1 until a given load was reached, and then the load was sustained during the creep test, about 72 h. The test was conducted at room temperature, and seven different sustained loads (2.0, 2.5, 3.0, 3.5, 4.0, 4.5 N) were selected. These selected loads corresponded to nominal stresses of 2.4, 3.0, 3.6, 4.2, 4.8, and 5.3 MPa. More detail information is described in the footnote.1
2 Each strut had a constricted shape with a maximum diameter of ϕd_max and a minimum diameter of ϕd_min. The diameter ratio of each strut is the ratio of its minimum and maximum diameters. 3 We first measured area for each pore and assumed that a pore is true circle shape. Next, the dimeter of pore circle was investigated to make a distribution of pore diameter (in Fig. 3(a)). For Fig. 3(b), diameter ratio is the ratio of between minimum diameter and maximum diameter as described in the manuscript. We measured this diameter ratio from the pictures of SEM observation to investigate its distribution. For the simple modeling of strut, the average diameter of strut is determined from macroscopic porosity and overall strut volume in the RVE. Based on the average strut diameter and diameter ratio, each strut was modeled in the RVE. Indeed, to gradually change the diameter of strut (the changing law of strut), the diameter is changed in 5 steps to express neck shape of strut (a constricted diameter).
1 For each test (each loading condition), the test was conducted at least three times. The average data was employed in this study. The sustained load was controlled within about 5%. The creep test was conducted about 72 h. As discussed later (e.g. Fig. 5), longer time of creep test may result in linear relationship of creep deformation, which is easily predicted. Thus, we conducted creep tests up to 72 h.
2
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Fig. 5. Experimental creep strain curves for different loading values. Fig. 2. Magnified view of cross section of the hollow-fiber membrane.
Fig. 3. Distribution of (a) pore size and (b) diameter ratio of the strut of the hollow-fiber membrane.
3.2. Uni-axial tension
3.3. Creep test
Fig. 4 shows the nominal stress-strain curve of the uniaxial tension tests at different strain rates. At the initial stage, the stress increased linearly as the strain increased. When the stress reached about 4 MPa, the stress-strain curve bent, and then continued to increase linearly until fracture. Iio et al. conducted similar experiments, in which detailed examinations were conducted using FESEM to understand microscopic deformation behavior [5]. As a result, it was concluded that the initial linear process was in the elastic deformation regime, and the subsequent process was in the plastic deformation regime. This tendency was observed at all strain rates. Next, the influence of a change in strain rate on the deformation behavior was investigated. The yield stress increased slightly as the strain rate increased, but the strain rate dependency was not very significant in this study.
A creep test was conducted for each sustained load. The creep strain-time curves are shown in Fig. 5. Note that the start time of the creep test is at the origin in the figure. It was carefully confirmed that the stress-strain curves until the holding load are almost the same for all tests. As shown, the creep strain increased gradually with time and the slope tended to decrease with time. However, the creep strain behavior is different, depending on the sustained load. At the minimum force of F = 2 N, creep deformation is small. On the other hand, a higher sustained load (F = 3 or higher) induces large creep deformation. 4. Discussion 4.1. FEM model This study created a model of the heterogeneous microstructure with random pores for the FEM computation test sample. We first used Surface Evolver, which is a structural modeling software for random soap froth, including bubbles, foamed particles, and their cluster [14]. The calculation was setup to minimize the balance between surface energy and mechanical force by satisfying Plateau's laws [12,15]. As shown in Fig. 6 (a), in order to create the model, a cubic space was filled with a Kelvin structure as the initial condition. In this study, the number of Kelvin structures was 125 (5 × 5 x 5). This cube corresponds to a representative volume element (RVE) for FEM computation, which is described later. Next, to randomly change the volume of each Kelvin structure in the cube foaming analysis was performed using the Surface Evolver software. As shown in Fig. 6 (b), the created microstructure becomes random foam for a heterogeneous cellular structure. Since the present membrane has an open cellular structure (Fig. 1), a strut network in the foam is required. Using MATLAB, the strut data of each foam structure was extracted. The skeleton frame was then
Fig. 4. Experimental nominal stress-strain curves at different strain rates. 3
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Fig. 6. Random bubble analysis process: (a) before foaming, (b) after foaming. Fig. 8. FEM model of the random pore structure.
imported to the FEM software (MSC, Marc). Fig. 7 shows an illustration of the FEM model. In this FEM model, beam elements were employed for the PVDF strut network. The cross section of the beam element was assumed to be a perfect circle. The diameter of the cross section was adjusted so that the porosity of the entire RVE was 65%. Note that, as shown in Fig. 7, the ratio of the minimum diameter to the maximum diameter of the PVDF strut was referred to as the actual diameter ratio (see Fig. 3(b)). To verify the microstructural model, the pore size and diameter ratio were measured and compared with the actual experiment as shown in Fig. 3. The created FEM model is indicated by the red “x”, shown overlapping on the actual microstructure indicated by an open circle. This demonstrates that the distribution of pore diameter and diameter ratio are well reproduced. In other words, the random pore structure is geometrically reproduced. Therefore, we successfully developed a FEM model with a heterogeneous microscopic structure for the polymeric membrane.
2
h ε m σ ‾ = Ke g ‾p⋅‾ε˙
(1)
Here, K is a work hardening coefficient, hg is a work hardening index, and m is a strain rate dependent coefficient. These material constants were obtained as K = 76.0 MPa, hg = 0.05, and m = 0.049 from previous studies [16]. The von Mises yield criterion is employed in this study. In addition, general polymer materials may show strain rate dependence even during elastic deformation. However, as shown in Fig. 5, the PVDF hollow fiber membrane showed very little dependency on strain rate in the linear curve. Therefore, the strain rate dependence of the elastic properties was not considered in this study. The computational result of FEM is shown in Fig. 9 together with the experimental result. It was found that the computational result generally agrees with the experimental one during elasto-plastic deformation. Thus, our model is capable of simulating elasto-plastic deformation behavior up to about 0.2 strain. 4.3. Creep deformation
4.2. Elasto-plastic deformation Polymer materials often exhibit time-dependent deformation behavior, such as creep and stress relaxation. Since the present sample is a porous polymer material, the base polymer and inherent pore structure encourages deformation as compared to a solid material. Such deformability in elasto-plastics and creep become important for water purification operations. Creep deformation of polymers is generally investigated using creep compliance as a parameter, which is defined as creep strain divided by applied strain. It has been reported that creep compliance is sometimes dependent on applied stress; this kind of phenomenon is called non-linear creep, which appears when testing times are longer, applied stress is larger, or when both conditions occur simultaneously [18–20]. Much research in modeling non-linear creep deformation behavior has already been performed. Jazouli et al. and Ikeshima et al. used the time–temperature–stress superposition
Fig. 8 shows the created FEM model with the beam element virtually displayed in three dimensions. Note that the diameter ratio is also indicated in this figure. It can be seen that the microstructure of the model is random pore, which is similar to the actual structure seen in Fig. 1. To conduct the elasto-plastic analysis a periodic boundary condition was employed, and uniaxial tensile loading was applied to the FEM model so that the microscopic and macroscopic deformation behavior of the polymeric membrane can be computed. It is noted that the elasto-plastic properties of the PVDF matrix for FEM computation were derived from previous studies [16], and elastic deformation was assumed to follow Hooke's law. In addition, the stress-strain (σ ‾ − εp ) curve during plastic deformation was influenced by strain rate (‾ε˙ m) , and the following work hardening rule [17] was employed.
Fig. 7. FEM model of random porous structure in the RVE (a) and the shape of the strut structure (b). 4
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Fig. 9. Nominal stress-strain curves from the experimental results and FEM computation.
Fig. 11. Creep strain curves from the experimental results and FEM computation.
principle (TTSSP) [20,21] for polymer creep testing, in which the creep compliance curve at different temperatures or stress levels can be shifted along the time scale by using a master curve at a reference temperature or stress level. They successfully obtained a master curve for creep compliance, which can be acceptable for a wide range of applied stress levels. Therefore, non-linear creep deformation under different applied stresses can be predicted based on TTSSP. For this principal, creep compliance D(t) (creep strain normalized by the applied constant stress) is expressed by the following equation.
7, random pore structures may be created for porous polymeric membranes, which are used for water purification. In addition, appropriate constitutive equations for elasto-plastic properties and creep were employed, and macroscopic deformation behavior can successfully be predicted. 5. Conclusion
(3)
This study employed FEM in order to establish a method of deformation modeling for a polymer membrane with an inhomogeneous porous structure. The test sample was a PVDF hollow fiber membrane used for water purification. Time-dependent tensile deformation behavior was examined. The obtained results are summarized below.
Here, b and c are material constants. Φσ is stress shift factor, which is expressed by constants C1 and C2 and the reference stress σ0. The material constants are derived from experimental results [16]. Fig. 10 shows the variation in creep compliance as a function of applied stress of PVDF bulk-solid material (note that it is not porous material) (Fig. 10(a)) [16]. Subsequently, the master curve of the creep compliance based on experimental data was drawn (Fig. 10(b)). When appropriate constants for C1 and C2 (reduced time) are selected, creep compliance curves can be unified by using the reduced time, as shown in this figure. This indicates that non-liner creep deformation can be modeled by this master curve. These constitutive equations (Eqs. (2) and (3)) were substituted into the FEM model as the base PVDF, and the macroscopic creep strain was computed (similar to Fig. 5). We conducted several tests with the different applied loadings of 2, 2.5, and 3 N. Fig. 11 shows the computed macroscopic creep strain curves for different loads together with the experimental results of Fig. 5, indicating very good agreement with each other. Thus, our FEM model is useful for deformation modeling of a random porous structure. By using the modeling process in Figs. 6 and
1. The test sample had an inhomogeneous three-dimensional open cellular structure, with submicron-sized pores and a porosity of 65%. In order to create a FEM model, the pore structure (the distribution of pore size and strut shape) was measured using FESEM images. 2. Stress-strain curves were obtained from a uniaxial tension test, which demonstrated linear-elastic deformation and plastic deformation. Creep tests were also conducted to investigate the timedependent deformation behavior of the test sample, showing that creep strain depends on applied load and creep time. 3. In order to create a sample with an inhomogeneous three-dimensional open cellular structure, we first created a representative volume element (RVE) where Kelvin structures were arranged, and the random foaming calculation was performed for each Kelvin structure. Finally, the strut of each structure was imported into FEM software. The created inhomogeneous structure reproduced well the actual porous structure. 4. The established FEM model successfully simulated the overall
D (t ) = b⋅(t⋅Φσ )c log Φσ =
C1 (σ − σ0) C2 + (σ − σ0)
(2)
Fig. 10. (a) Variation of creep compliances at a four different applied stress levels and (b) master curve for the creep compliances. 5
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elastic-plastic deformation and creep deformation of the test sample. Moreover, the microscopic deformation of the random pore structure can be discussed. Thus, new strength design, such as optimization of microstructures, can be performed by utilizing the proposed deformation modeling of the 3D random porous structure.
[6] [7] [8]
Acknowledgements
[9] [10]
The authors are grateful to Mr. Takumi NAGAKURA at Chuo University for their important contributions. This work was supported by the “Nanotechnology Platform” of the Ministry of Education, Culture, Sports, Science and Technology, Japan. This work is supported by JSPS KAKENHI (Grantno.26420025) from the Japan Society for the Promotion of Science.
[11] [12] [13]
Appendix A. Supplementary data
[14] [15]
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.polymertesting.2019.106097.
[16]
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