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Simulation of compression behavior and strain-rate effect for aluminum foam sandwich panels
Computational Materials Science 112 (2016) 205–209
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Simulation of compression behavior and strain-rate effect for aluminum foam sandwich panels Renjun Dou, Sawei Qiu, Yan Ju, Yuebo Hu ⇑ Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming 650500, Yunnan, China
a r t i c l e
i n f o
Article history: Received 23 August 2015 Received in revised form 23 October 2015 Accepted 25 October 2015 Available online 19 November 2015 Keywords: Micro-inertia Compression behavior Strain-rate effect Aluminum foam sandwich panel Matrix material
a b s t r a c t Based on the structure characteristics of aluminum foam sandwich (AFS) panels, two-dimensional random models with different relative densities are created by combing C++ and ANSYS/LS-DYNA software in this paper. Under different strain rates, compression behavior and strain-rate effect of AFS panels are investigated on the basis of the established finite element models. It is found that stress–strain curves of AFS panels are made up of three stages, namely elastic stage, plateau stage and densification stage. Moreover, the research results also confirm that the strain-rate effect of AFS panels is related to strainrate sensitivity of matrix materials and effect of micro-inertia for AFS panels. With the increase of the relative density, the strain-rate effect of AFS panels is more obvious. Through comparison and analysis, the obtained results in this work are in good agreement with other research results, which confirms the feasibility and rationality of the models. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction As new-type materials, aluminum foam sandwich (AFS) panels, which consist of a highly porous aluminum foam core and two face sheets bonded to the core on either side, have some excellent properties such as light weight, high strength-to-weight and stiffness-to-weight ratios, good buckling resistance, good energyabsorbing capacity and design versatility, and so on [1]. Therefore, AFS panels are widely used as structural materials and functional materials in aerospace, shipbuilding, electronics, automobile sectors, construction fields and other fields [2]. In recent years, the study on foam materials has been attracting considerable attention. Compressive properties of closed-cell aluminum foams were studied by Xia et al. [3] and the research results showed that ceramic microspheres were uniformly distributed in the cell walls and could improve the yield strength, energy absorption capacity of original foams, and so on Yu et al. [4] established finite element model of machine tool table with aluminum foam-filled structure and pointed out the machine tool table had the superiority in high speed machining. Computational model of a graded aluminum foam sandwich structure were created by Qi et al. [5] and the blast-resistant performances were analyzed and optimized.
⇑ Corresponding author. Tel.: +86 87165170917; fax: +86 87165194243. E-mail addresses: [email protected] (R. Dou), [email protected] (S. Qiu), [email protected] (Y. Ju), [email protected] (Y. Hu). http://dx.doi.org/10.1016/j.commatsci.2015.10.032 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
However, as far as we known, the numerical simulation study on compression behavior and strain-rate effect for AFS panels is few. In this paper, based on the structure characteristics of AFS panels, two-dimensional random models with different relative densities (20%, 30% and 40%) are created by combing C++ and ANSYS/LS-DYNA software. As shock-resistant and energy absorption materials, AFS panels have wide application prospects. In order to widely and reasonably use the foam materials, a major aim of the current study is to study the influence of strain-rate effect of matrix materials and effect of micro-inertia on mechanical properties under different strain rates. In addition, the numerical simulation results were analyzed and validated compared with previous study results.
2. Modeling and simulating During the finite element analysis, two-dimensional random model of AFS panels has the advantages of easy dividing of meshes and short computing time to analyze compression behavior and strain-rate effect, and it are confirmed that two-dimensional random model is widely adopted in Refs. [6,7]. In this work, based on above reasons, the model of AFS panels is simplified into twodimensional random model whose aluminum alloy core consists of plenty of circular holes. The modeling process is reported in our previous work [8]. In this simulation, according to Ref. [9], different relative densities (20%, 30% and 40%) and strain-rates (100 s1, 5000 s1 and 10,000 s1) are chosen to investigate the
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R. Dou et al. / Computational Materials Science 112 (2016) 205–209
influence of the strain-rate effect of the matrix material and the relative density. Meanwhile, in order to study the influence of effect of micro-inertia, the simulation analysis of the model with relative density 40% under strain-rates 100 s1, 2500 s1, 5000 s1, 7500 s1 and 10,000 s1 are carried out, respectively. The size of radius varies from 0.5 to 1.5 mm for AFS panels with different relative densities. As shown in Fig. 1, the model of AFS panels is composed of two thin aluminum sheets (60 mm length, 0.7 mm thickness) and a aluminum foam core (60 mm length, 12 mm thickness). In the work, the Cowper–Symonds model is used as the constitutive model of AFS panels because it has some applications in studying properties of aluminum alloys and is a simple but effective to incorporate the compression behavior and strain-rate effect of AFS panels. The Cowper–Symonds model is defined according to the formula (1) [10],
r ¼r d y
s y
1=P ! e_ 1þ C
Table 1 Properties of matrix material of AFS panels [9]. Density (kg/m3)
Young’s modulus (MPa)
Poisson ratio
Yield strength (MPa)
2700 Tangent modulus (MPa) 690
6.9 104 C (s1) 6500 or 6.5 106
0.3 p 4
76 b 1
ð1Þ
where rdy is the dynamic yield strength, rsy is the static yield strength of the material. C and P are corresponding parameters of strain-rates and e_ is the strain-rate. In the process of calculation, the strain-rate effect of matrix materials is studied by adjusting the parameter C. The strain-rate effect of matrix materials is considered when C is 6500. If C is 6.5 106, the strain-rate effect of matrix materials is not considered. The parameters of matrix material of AFS panels are shown in Table 1 [9]. In order to improve the efficiency and speed of analysis, adaptive meshing techniques are used for the grid divisions of the model of AFS panels. Auto-dimensional contact is chosen in model since two initially separated cell walls is potential to contact each other in the process of compression. In order to simulate real loading condition, one half of top of the model is compressed by a pressure head along the negative Y direction and the bottom of the model is fixed. Moreover, as shown above, the moving speed of the pressure head is applied by different strain-rates from 100 to 10,000 s1, respectively. The left and right sides of the model are free and the loading and boundary constraint is shown in Fig. 2. Finally, the simulation tests of AFS panels with different relative densities were carried out in the ANSYS/LS-DYNA software on the basis of above conditions.
Fig. 2. Loading and constraint diagram.
3. Results and discussions 3.1. Influences of strain-rate sensitivity of the matrix materials Figs. 3–5 show stress–strain curves of AFS panels with different relative densities when strain-rate effect of the matrix materials is
(a) Model of relative density 40%.
Fig. 3. Stress–strain curves of the model of relative density 40% with ratedependent matrix material.
(b) Model of relative density 30%.
(c) Model of relative density 20%. Fig. 1. Two-dimensional random models of AFS panels.
R. Dou et al. / Computational Materials Science 112 (2016) 205–209
Fig. 4. Stress–strain curves of the model of relative density 30% with ratedependent matrix material.
207
of these curves are not obvious due to the decrease of the relative density. But the curves have a certain level of strain-rate effect. The same conclusion is drawn that high strain-rate will lead to the plateau stress to increase and plateau width to shorten. In addition, seen from Figs. 3–5, the plateau stress decreases, but the width extends with the decreases of the relative density. On the one hand, the breaking of foam structure is mainly derived from narrowing of cell structure and plastic flow of cell walls for AFS panels with high relative density. Strain-rate effect of AFS panels continues to be influenced by strain-rate sensitivity of matrix material. On another hand, the destruction of the model originates mainly from buckling and instability of cell walls and mechanical properties of foam structure depend mainly on shapes and distributions of cells for AFS panels with low relative density. Therefore, properties of matrix materials are limited. Another conclusion can be drawn that AFS panels with lower relative density have lower strain-rate sensitivity. Figs. 6–8 shows stress–strain curves of AFS panels with different relative density (20%, 30% and 40%) when the strain-rate effect of the matrix material is not obvious. These curves have three similar stages, as shown above. As shown in Figs. 6–8, the differences caused by strain-rates are obviously decreased, which illustrates that strain-rate effect of AFS panels weakens if strain-rate effect of the matrix materials is not considered. Thus, the strain-rate effect of AFS panels depends much on strain-rate sensitivity of its matrix materials. The relationship between the dynamic plateau stress and relative density for AFS panels with rate-dependent matrix materials under different strain rates (100 s1, 5000 s1 and 10,000 s1) are shown in Fig. 9. The plateau stress is the stress of stress–strain curves, which corresponded to strain of 0.2. Gibson et al. [11] described the relationship between the plateau stress and relative density of foam materials by Eq. (2) [12],
rp ¼ aqn ry
ð2Þ
where rP is the plateau stress of foam material, ry is yield strength of matrix material, a and n are material parameters, q is the relative density of foam material. Because the face sheets are so thin, related performance of AFS panels is studied by using the equation. By fitting above results, the calculated parameters a and n under different strain-rates are summarized in Table 2. As seen from Table 2,
Fig. 5. Stress–strain curves of the model of relative density 20% with ratedependent matrix material.
considered. All the curves consist of three stages, namely elastic stage, plateau stage and densification stage. At the beginning, the curves present a approximatively linear stage, namely elastic stage. In the stage, foam structure can only produce elastic deformation, where subjects no severe stress concentration. After that, a slowly rising slope in curves represents the plateau stage. The curves show plastic plateau stage because of a large number of cell walls reach the yield limit, which leads to AFS panels to overwhelm and presents a large number of structure collapses. Finally, foam structure is compact and the stress–strain curves sharply rise, which behaves as behavior of the matrix materials. Besides, it can be found from Fig. 3 that the plateau stress increases with the increase of the strain-rate, but the plateau width shortens, which show that AFS panels have obvious strain-rate effect. In the calculations below, the effect of the strain-rate effect of the matrix material is studied. Figs. 4 and 5 show the stress– strain curves for AFS panels with relative density 20% and 30%, respectively. Compared with Fig. 3, the influences of strain-rate
Fig. 6. Stress–strain curves of the model of relative density 40% with rateindependent matrix material.
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R. Dou et al. / Computational Materials Science 112 (2016) 205–209
Fig. 9. Relationship of plateau stress of AFS panels versus relative density under different strain-rates. Fig. 7. Stress–strain curves of the model of relative density 30% with rateindependent matrix material.
Table 2 Fitting results of parameters.
e_ (s1)
a
n
100 5000 10,000
0.52 0.73 0.91
1.81 1.58 1.42
Fig. 8. Stress–strain curves of the model of relative density 20% with rateindependent matrix material.
with the increase of the strain-rate, the scaling factor a increases, implying that AFS panels have obvious strain-rate effects. In addition, the index n decreases gradually with the increase of the strain-rate, which illustrates strain-rate sensitivity of AFS panels is related to the relative density of foam material. Through comparative analysis, the study result is in accordance with the above simulation results, which can validate the correctness of the results. 3.2. Influences of the effect of micro-inertia Gibson et al. [11] found that the influence of gas pressure in cells on mechanical property of foam materials is negligible and
Fig. 10. Stress–strain curves AFS panels with relative density 40% under different strain- rates.
is not major reason which causes strain-rate effect of the foam materials. However, from Figs. 6–8, AFS panels have a certain degree of strain-rate effect. So, the following analysis will complete this work and the study on the effect of micro-inertia for AFS panels is conducted. Fig. 10 shows dynamic compressive stress–strain curves of AFS panels with rate-independent matrix material under
R. Dou et al. / Computational Materials Science 112 (2016) 205–209
different strain-rates (100 s1, 2500 s1, 5000 s1, 7500 s1 and 10,000 s1). In the ranges from 100 to 5000 s1, with the increase of the strain-rate, plateau stress and plateau width change less, which show that AFS panels have less strain-rate effect. In the ranges from 5000 to 10,000 s1, compared with above ranges, the plateau stress increase faster, which indicates AFS panels have strain-rate effect to a certain extent. A conclusion can be drawn that the effect of micro-inertia is a important factor and the influence of effect of micro-inertia for AFS panels is more obvious under high strain-rate during the compression process. The simulation results obtained are in good agreement with other theoretical results and experimental results of foam materials. Feng et al. [13] analyzed influence of strain-rate on mechanical property and energy absorption capacity of aluminum foam and pointed out the strain-rate effect mainly depended on strain-rate sensitivity of matrix materials, effect of micro-inertia and gas pressure in cells. Hu et al. [14] selected three kinds of foam materials and found that both rate-dependent foam material and rateindependent foam material had obvious strain-rate effect, and strain-rate effect of foam materials was related to effect of micro-inertia of matrix materials. Compared with the above study results, the simulation results is slightly different because the properties of the AFS panel models may be affected by matrix material, the structure of foams and the neglect of gas pressure, which can confirm that the finite element models and the research results are reliable. 4. Conclusion In the paper, two-dimensional random models of AFS panels are established and the simulation analysis of compression behavior
209
and strain-rate effect of AFS panels with different relative densities is conducted by C++ and ANSYS/LS-DYNA software. It was found that stress–strain curves of AFS panels were composed of three stages. Moreover, it was confirmed that strain-rate sensitivity of matrix materials and effect of micro-inertia decided strain-rate effect of AFS panels, and AFS panels with high porosity had poor strain-rate effect. At the same time, the simulation study provide theoretical basis for the related research, and it is of great significance to widely and reasonably use the foam materials in related industrial field. Acknowledgements We would like to thank Mr. Qingxian Hao, Professor Xing Wu and Professor Xiaocong He for providing valuable advices and recommendations with regards to the preparation of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
J. Banhart, H.W. Seeliger, Adv. Eng. Mater. 9 (2008) 793–802. Z.H. Wang, L. Jing, J.G. Ning, et al., Compos. Struct. 93 (2011) 1300–1308. X.C. Xia, X.W. Chen, Z. Zhang, et al., Mater. Des. 56 (2014) 353–358. Y.H. Yu, C.G. Li, D.L. Ruan, et al., Ordnance Mater. Sci. Eng. 37 (2014) 16–18. C. Qi, L.J. Yang, S. Yang, J. Vibr. Shock 32 (2013) 70–75. H.B. Su, X.Q. Huang, L.Q. Yang, et al., Sci. Technol. Eng. 9 (2009) 6991–6996. V. Hardenacke, J. Hohe, Int. J. Solids Struct. 46 (2009) 989–1006. S.W. Qiu, X.N. Zhang, Q.X. Hao, et al., Mater. Trans. 56 (2015) 687–690. Y. Liu, W.W. Gong, X. Zhang, Comput. Mater. Sci. 91 (2014) 223–230. J. Zhang, G.P. Zhao, T.J. Lu, J. Xi’an Jiaotong Univ. 44 (2010) 97–101. L.J. Gibon, M.F. Ashby, Cellular Solids: Structure and Properties, second ed., Cambridge University Press, Cambridge, 1997. [12] Z.G. Fan, C.Q. Chen, Q. Wan, Expl. Shock Waves 34 (2014) 742–747. [13] Y. Feng, Z.G. Zhu, Y. Pan, et al., Trans. Mater. Heat Treat. 2 (2004) 268–271. [14] S.S. Hu, W. Wang, Y. Pan, et al., Expl. Shock Waves 1 (2003) 13–18.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Simulation of compression behavior and strain-rate effect for aluminum foam sandwich panels Renjun Dou, Sawei Qiu, Yan Ju, Yuebo Hu ⇑ Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming 650500, Yunnan, China
a r t i c l e
i n f o
Article history: Received 23 August 2015 Received in revised form 23 October 2015 Accepted 25 October 2015 Available online 19 November 2015 Keywords: Micro-inertia Compression behavior Strain-rate effect Aluminum foam sandwich panel Matrix material
a b s t r a c t Based on the structure characteristics of aluminum foam sandwich (AFS) panels, two-dimensional random models with different relative densities are created by combing C++ and ANSYS/LS-DYNA software in this paper. Under different strain rates, compression behavior and strain-rate effect of AFS panels are investigated on the basis of the established finite element models. It is found that stress–strain curves of AFS panels are made up of three stages, namely elastic stage, plateau stage and densification stage. Moreover, the research results also confirm that the strain-rate effect of AFS panels is related to strainrate sensitivity of matrix materials and effect of micro-inertia for AFS panels. With the increase of the relative density, the strain-rate effect of AFS panels is more obvious. Through comparison and analysis, the obtained results in this work are in good agreement with other research results, which confirms the feasibility and rationality of the models. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction As new-type materials, aluminum foam sandwich (AFS) panels, which consist of a highly porous aluminum foam core and two face sheets bonded to the core on either side, have some excellent properties such as light weight, high strength-to-weight and stiffness-to-weight ratios, good buckling resistance, good energyabsorbing capacity and design versatility, and so on [1]. Therefore, AFS panels are widely used as structural materials and functional materials in aerospace, shipbuilding, electronics, automobile sectors, construction fields and other fields [2]. In recent years, the study on foam materials has been attracting considerable attention. Compressive properties of closed-cell aluminum foams were studied by Xia et al. [3] and the research results showed that ceramic microspheres were uniformly distributed in the cell walls and could improve the yield strength, energy absorption capacity of original foams, and so on Yu et al. [4] established finite element model of machine tool table with aluminum foam-filled structure and pointed out the machine tool table had the superiority in high speed machining. Computational model of a graded aluminum foam sandwich structure were created by Qi et al. [5] and the blast-resistant performances were analyzed and optimized.
⇑ Corresponding author. Tel.: +86 87165170917; fax: +86 87165194243. E-mail addresses: [email protected] (R. Dou), [email protected] (S. Qiu), [email protected] (Y. Ju), [email protected] (Y. Hu). http://dx.doi.org/10.1016/j.commatsci.2015.10.032 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
However, as far as we known, the numerical simulation study on compression behavior and strain-rate effect for AFS panels is few. In this paper, based on the structure characteristics of AFS panels, two-dimensional random models with different relative densities (20%, 30% and 40%) are created by combing C++ and ANSYS/LS-DYNA software. As shock-resistant and energy absorption materials, AFS panels have wide application prospects. In order to widely and reasonably use the foam materials, a major aim of the current study is to study the influence of strain-rate effect of matrix materials and effect of micro-inertia on mechanical properties under different strain rates. In addition, the numerical simulation results were analyzed and validated compared with previous study results.
2. Modeling and simulating During the finite element analysis, two-dimensional random model of AFS panels has the advantages of easy dividing of meshes and short computing time to analyze compression behavior and strain-rate effect, and it are confirmed that two-dimensional random model is widely adopted in Refs. [6,7]. In this work, based on above reasons, the model of AFS panels is simplified into twodimensional random model whose aluminum alloy core consists of plenty of circular holes. The modeling process is reported in our previous work [8]. In this simulation, according to Ref. [9], different relative densities (20%, 30% and 40%) and strain-rates (100 s1, 5000 s1 and 10,000 s1) are chosen to investigate the
206
R. Dou et al. / Computational Materials Science 112 (2016) 205–209
influence of the strain-rate effect of the matrix material and the relative density. Meanwhile, in order to study the influence of effect of micro-inertia, the simulation analysis of the model with relative density 40% under strain-rates 100 s1, 2500 s1, 5000 s1, 7500 s1 and 10,000 s1 are carried out, respectively. The size of radius varies from 0.5 to 1.5 mm for AFS panels with different relative densities. As shown in Fig. 1, the model of AFS panels is composed of two thin aluminum sheets (60 mm length, 0.7 mm thickness) and a aluminum foam core (60 mm length, 12 mm thickness). In the work, the Cowper–Symonds model is used as the constitutive model of AFS panels because it has some applications in studying properties of aluminum alloys and is a simple but effective to incorporate the compression behavior and strain-rate effect of AFS panels. The Cowper–Symonds model is defined according to the formula (1) [10],
r ¼r d y
s y
1=P ! e_ 1þ C
Table 1 Properties of matrix material of AFS panels [9]. Density (kg/m3)
Young’s modulus (MPa)
Poisson ratio
Yield strength (MPa)
2700 Tangent modulus (MPa) 690
6.9 104 C (s1) 6500 or 6.5 106
0.3 p 4
76 b 1
ð1Þ
where rdy is the dynamic yield strength, rsy is the static yield strength of the material. C and P are corresponding parameters of strain-rates and e_ is the strain-rate. In the process of calculation, the strain-rate effect of matrix materials is studied by adjusting the parameter C. The strain-rate effect of matrix materials is considered when C is 6500. If C is 6.5 106, the strain-rate effect of matrix materials is not considered. The parameters of matrix material of AFS panels are shown in Table 1 [9]. In order to improve the efficiency and speed of analysis, adaptive meshing techniques are used for the grid divisions of the model of AFS panels. Auto-dimensional contact is chosen in model since two initially separated cell walls is potential to contact each other in the process of compression. In order to simulate real loading condition, one half of top of the model is compressed by a pressure head along the negative Y direction and the bottom of the model is fixed. Moreover, as shown above, the moving speed of the pressure head is applied by different strain-rates from 100 to 10,000 s1, respectively. The left and right sides of the model are free and the loading and boundary constraint is shown in Fig. 2. Finally, the simulation tests of AFS panels with different relative densities were carried out in the ANSYS/LS-DYNA software on the basis of above conditions.
Fig. 2. Loading and constraint diagram.
3. Results and discussions 3.1. Influences of strain-rate sensitivity of the matrix materials Figs. 3–5 show stress–strain curves of AFS panels with different relative densities when strain-rate effect of the matrix materials is
(a) Model of relative density 40%.
Fig. 3. Stress–strain curves of the model of relative density 40% with ratedependent matrix material.
(b) Model of relative density 30%.
(c) Model of relative density 20%. Fig. 1. Two-dimensional random models of AFS panels.
R. Dou et al. / Computational Materials Science 112 (2016) 205–209
Fig. 4. Stress–strain curves of the model of relative density 30% with ratedependent matrix material.
207
of these curves are not obvious due to the decrease of the relative density. But the curves have a certain level of strain-rate effect. The same conclusion is drawn that high strain-rate will lead to the plateau stress to increase and plateau width to shorten. In addition, seen from Figs. 3–5, the plateau stress decreases, but the width extends with the decreases of the relative density. On the one hand, the breaking of foam structure is mainly derived from narrowing of cell structure and plastic flow of cell walls for AFS panels with high relative density. Strain-rate effect of AFS panels continues to be influenced by strain-rate sensitivity of matrix material. On another hand, the destruction of the model originates mainly from buckling and instability of cell walls and mechanical properties of foam structure depend mainly on shapes and distributions of cells for AFS panels with low relative density. Therefore, properties of matrix materials are limited. Another conclusion can be drawn that AFS panels with lower relative density have lower strain-rate sensitivity. Figs. 6–8 shows stress–strain curves of AFS panels with different relative density (20%, 30% and 40%) when the strain-rate effect of the matrix material is not obvious. These curves have three similar stages, as shown above. As shown in Figs. 6–8, the differences caused by strain-rates are obviously decreased, which illustrates that strain-rate effect of AFS panels weakens if strain-rate effect of the matrix materials is not considered. Thus, the strain-rate effect of AFS panels depends much on strain-rate sensitivity of its matrix materials. The relationship between the dynamic plateau stress and relative density for AFS panels with rate-dependent matrix materials under different strain rates (100 s1, 5000 s1 and 10,000 s1) are shown in Fig. 9. The plateau stress is the stress of stress–strain curves, which corresponded to strain of 0.2. Gibson et al. [11] described the relationship between the plateau stress and relative density of foam materials by Eq. (2) [12],
rp ¼ aqn ry
ð2Þ
where rP is the plateau stress of foam material, ry is yield strength of matrix material, a and n are material parameters, q is the relative density of foam material. Because the face sheets are so thin, related performance of AFS panels is studied by using the equation. By fitting above results, the calculated parameters a and n under different strain-rates are summarized in Table 2. As seen from Table 2,
Fig. 5. Stress–strain curves of the model of relative density 20% with ratedependent matrix material.
considered. All the curves consist of three stages, namely elastic stage, plateau stage and densification stage. At the beginning, the curves present a approximatively linear stage, namely elastic stage. In the stage, foam structure can only produce elastic deformation, where subjects no severe stress concentration. After that, a slowly rising slope in curves represents the plateau stage. The curves show plastic plateau stage because of a large number of cell walls reach the yield limit, which leads to AFS panels to overwhelm and presents a large number of structure collapses. Finally, foam structure is compact and the stress–strain curves sharply rise, which behaves as behavior of the matrix materials. Besides, it can be found from Fig. 3 that the plateau stress increases with the increase of the strain-rate, but the plateau width shortens, which show that AFS panels have obvious strain-rate effect. In the calculations below, the effect of the strain-rate effect of the matrix material is studied. Figs. 4 and 5 show the stress– strain curves for AFS panels with relative density 20% and 30%, respectively. Compared with Fig. 3, the influences of strain-rate
Fig. 6. Stress–strain curves of the model of relative density 40% with rateindependent matrix material.
208
R. Dou et al. / Computational Materials Science 112 (2016) 205–209
Fig. 9. Relationship of plateau stress of AFS panels versus relative density under different strain-rates. Fig. 7. Stress–strain curves of the model of relative density 30% with rateindependent matrix material.
Table 2 Fitting results of parameters.
e_ (s1)
a
n
100 5000 10,000
0.52 0.73 0.91
1.81 1.58 1.42
Fig. 8. Stress–strain curves of the model of relative density 20% with rateindependent matrix material.
with the increase of the strain-rate, the scaling factor a increases, implying that AFS panels have obvious strain-rate effects. In addition, the index n decreases gradually with the increase of the strain-rate, which illustrates strain-rate sensitivity of AFS panels is related to the relative density of foam material. Through comparative analysis, the study result is in accordance with the above simulation results, which can validate the correctness of the results. 3.2. Influences of the effect of micro-inertia Gibson et al. [11] found that the influence of gas pressure in cells on mechanical property of foam materials is negligible and
Fig. 10. Stress–strain curves AFS panels with relative density 40% under different strain- rates.
is not major reason which causes strain-rate effect of the foam materials. However, from Figs. 6–8, AFS panels have a certain degree of strain-rate effect. So, the following analysis will complete this work and the study on the effect of micro-inertia for AFS panels is conducted. Fig. 10 shows dynamic compressive stress–strain curves of AFS panels with rate-independent matrix material under
R. Dou et al. / Computational Materials Science 112 (2016) 205–209
different strain-rates (100 s1, 2500 s1, 5000 s1, 7500 s1 and 10,000 s1). In the ranges from 100 to 5000 s1, with the increase of the strain-rate, plateau stress and plateau width change less, which show that AFS panels have less strain-rate effect. In the ranges from 5000 to 10,000 s1, compared with above ranges, the plateau stress increase faster, which indicates AFS panels have strain-rate effect to a certain extent. A conclusion can be drawn that the effect of micro-inertia is a important factor and the influence of effect of micro-inertia for AFS panels is more obvious under high strain-rate during the compression process. The simulation results obtained are in good agreement with other theoretical results and experimental results of foam materials. Feng et al. [13] analyzed influence of strain-rate on mechanical property and energy absorption capacity of aluminum foam and pointed out the strain-rate effect mainly depended on strain-rate sensitivity of matrix materials, effect of micro-inertia and gas pressure in cells. Hu et al. [14] selected three kinds of foam materials and found that both rate-dependent foam material and rateindependent foam material had obvious strain-rate effect, and strain-rate effect of foam materials was related to effect of micro-inertia of matrix materials. Compared with the above study results, the simulation results is slightly different because the properties of the AFS panel models may be affected by matrix material, the structure of foams and the neglect of gas pressure, which can confirm that the finite element models and the research results are reliable. 4. Conclusion In the paper, two-dimensional random models of AFS panels are established and the simulation analysis of compression behavior
209
and strain-rate effect of AFS panels with different relative densities is conducted by C++ and ANSYS/LS-DYNA software. It was found that stress–strain curves of AFS panels were composed of three stages. Moreover, it was confirmed that strain-rate sensitivity of matrix materials and effect of micro-inertia decided strain-rate effect of AFS panels, and AFS panels with high porosity had poor strain-rate effect. At the same time, the simulation study provide theoretical basis for the related research, and it is of great significance to widely and reasonably use the foam materials in related industrial field. Acknowledgements We would like to thank Mr. Qingxian Hao, Professor Xing Wu and Professor Xiaocong He for providing valuable advices and recommendations with regards to the preparation of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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