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Compressive behavior and constitutive model of polyurea at high strain rates and high temperatures
Materials Today Communications 22 (2020) 100834
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Compressive behavior and constitutive model of polyurea at high strain rates and high temperatures
T
Qiang Liua,b,*, Pengwan Chena,**, Yang Zhangc, Zhirong Lib a
State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, 100081, China Xi’an Modern Chemistry Research Institute, Xi’an, 710065, China c Northwestern Polytechnical University, Xi’an, 710072, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Polyurea Strain rate Temperature Constitutive model
To study the dynamic compressive mechanical properties of polyurea at strain rate range of 1400 s−1–5700 s−1 and high temperature range of 20 ℃–80 ℃, we conducted a dynamical mechanical experiment utilizing a split Hopkinson pressure bar (SHPB) system. The compressive stress-strain curves of polyurea under various strain rates and temperatures are analyzed and discussed. The experimental results indicated that the strain rate and high temperature can remarkably influence the mechanical behaviors of polyurea, compression behaviors increased with the increasing of strain rate and compression behaviors decreased with the increasing of temperature. In addition, a thermal-visco-hyperelastic constitutive model based on five-parameter Mooney-Rivlin model was proposed to describe the compression stress-strain behavior of polyurea over a wide range of strain rates and temperatures. The model parameters were obtained by fitting the experimental stress-strain curves. The model prediction results show good agreement with the experimental results.
1. Introduction Polyurea is an elastomeric polymer formed by chemical reaction between amine component and isocyanate component. It as a coating [1,2] or sandwich [3,4] has been extensively applied in composite structures for protection due to its high elongation, impact resistance and mechanical properties. Some researchers reported that polyurea coated outer facesheet of steel plate or a core layer could reduce deformation than uncoated structure during the process of blast and ballistic loading [5–7]. Several composite structures including polyurea materials are subjected to high strain rate during dynamic loading. In addition, the process of explosion can accompany with a significant thermal radiation [8]. Temperature have a significant influence on mechanical properties of polyurea. With the increase of temperature, the mechanical properties decrease. When temperature reach up to about 100 ℃, polyurea may appear slight mass loss. Once the mass loss of polyurea is to begin to change, the performance of the material will be affected. Therefore, it is essential to research the dynamic mechanical behavior of polyurea referring to the strain rate and temperature. Many literatures have also revealed that the mechanical properties of polyurea depend on strain rate and temperature [9–14]. The dynamic mechanical properties of polyurea were widely studied. Yi et al.
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[9] investigated the dynamic compression behavior of polyurea using split Hopkinson pressure bar (SHPB), the results revealed that polyurea was strong rate-dependence, and either a leathery or glassy-like behavior at high strain rate loading. Sarva et al. [10] reported that the uniaxial compression stress-strain behavior of an elastomeric- thermoset polyurea over a wide range of strain rates was strongly dependent on strain rate and seen to undergo transition from rubbery-regime behavior at low rates to leathery-regime behavior at the high rates. Shim and Mohr [11] utilized a modified split Hopkinson pressure bar (SHPB) system to study the strain rate sensitivity of polyurea. Wang et al. [12] investigated the effect of strain rate on polyurea compressive mechanical properties over the range of strain rates (0.001 s−1–7000 s−1) and found that the stress-strain curves can be divided into three distinct regions, and the mechanical properties of polyurea were found to be highly dependent on the strain rate. Considering the importance of temperature, it is essential to investigate the high stain rate mechanical properties of polyurea at different temperatures. Guo et al. [13] studied the compressive mechanical behavior of two formulations of polyurea over the temperature range from −40 ℃ to 20 ℃ and the strain rate range from 0.001 s−1 to 12,000 s−1, the experimental results shown the stress-strain curves of polyurea was sensitive to strain rate and temperature within and beyond linear
Corresponding author at: State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, 100081, China. Corresponding author. E-mail addresses: [email protected] (Q. Liu), [email protected] (P. Chen).
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https://doi.org/10.1016/j.mtcomm.2019.100834 Received 3 August 2019; Received in revised form 6 December 2019; Accepted 6 December 2019 Available online 06 December 2019 2352-4928/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. (a) Photograph of specimens, (b) physical specimen, and (c) geometrical dimensions.
the stress-strain behaviors. Wang et al. [12] modified a rate-dependent constitutive model based on the nine-parameter Moony-Rivlin model to characterize the mechanical properties of polyurea over a wide range of strain rates. Although the constitutive models mentioned above can describe the stress-strain behaviors of polyurea, some models seem to be too complicated for applications. However, the models to describe the stress-strain behavior of polyurea on the effect of strain rate and high temperature are rare. Therefore, it is necessary to develop new constitutive model for polyurea involving a wide range of strain rate and temperature. In this work, the dynamic compressive properties of polyurea were investigated by using a split Hopkinson pressure bar (SHPB) with the range of strain rates from 1400 s−1 to 5700 s−1 and temperatures from 20 ℃ to 80 ℃. This polyurea material can be used in building as a protective coating. Additionally, polyurea can also be applied to military vehicles or ships. The mechanical properties of polyurea are vulnerable to operating temperature and strain rate. On the basis of the experimental results, the effects of strain rate and temperature on the dynamic compressive behaviors were analyzed and discussed. Furthermore, a thermal-visco-hyperelastic constitutive model was proposed to describe the compression stress-strain behavior of polyurea in terms of the strain rate and temperature.
viscoelastic regime. Zhang et al. [14] carried out a systematic experimental program to study the stress-strain behaviors of two polyaspartic esters (PAE) polyureas over the temperature range of 233 K–293 K and strain rate range of 0.001 s−1–15000 s−1. The results indicated that the uniaxial compression stress-strain behaviors of polyureas were strongly dependent on temperature and strain rate, and the yield stress of the polyureas decreases linearly with increasing temperature. From the published studies, very few studies reported on the dynamic compressive properties of polyurea at high temperature. To describe the stress-strain behaviors of polyurea, some constitutive models were proposed. Amirkhizi et al. [15] proposed an experimentally-based linear viscoelastic constitutive model that incorporated the classical Williams-Landel-Ferry (WLF) time temperature transformation and pressure response. Li and Lua [16] proposed a hyper-visoelastic constitutive model for polyurea by including Ogden model and a nonlinear viscoelastic model. Shim and Mohr [17] developed a constitutive model based on a rheological model composed of two Maxwell elements. This model can be able to predict the experimentally-measured stress-strain curves for various loading and unloading histories. Gamonpilas and McCuiston [18] have established a non-linear viscoelastic constitutive model for polyurea in terms of a nine-parameter Mooney-Rivlin model. The model was found to be efficient for describing the compressive and tensile behavior of polyurea under a wide range of strain rates. Cho et al. [19] developed a microstructurally informed three-dimensional constitutive model to capture the behavior of polyurea over the range of strain rate (10−3 s-1–103 s-1). Nantasetphong et al. [20] combined the effect of pressure and temperature to propose a constitutive model based on Williams-LandelFerry (WLF) equation. The prediction of the model agreed well with pressure shift factor data about the confined compression tests of polyurea. Guo et al. [13] developed a visco-hyperealstic constitutive model based on the two order strain energy density function to describe the nonlinear mechanical behavior of polyurea over a wide range of strain rates and temperatures (−40 ℃ - +20 ℃). Meanwhile, Guo et al. [21] also developed a three-dimensional visco-hyperelastic constitutive model on the basis of the multiplicative decomposition of the deformation gradient tensor into hyperelastic and viscoelastic parts to describe the mechanical behavior of polyurea at various strain rates. Zhang et al. [14] proposed a bilinear constitutive model to describe the temperature (233 K–293 K), strain rate, and pressure dependences of
2. Experiments Polyurea consisting of 1.5 wt% nano silicon carbide was sprayed on the template to prepare the thin sheet with a thickness at around 4.7 mm. The thin sheet of specimen is presented in Fig. 1(a). The cylindrical specimens of 4.7 mm in length and 9.4 mm in diameter were obtained using mechanical processing method for dynamic compression experiments. The physical specimen and geometrical dimensions are presented in Fig. 1(b) and (c). All specimens were kept for at least three days to eliminate their residual stress. The dynamic compression experiment was conducted by using a split Hopkinson pressure bar (SHPB) apparatus, which consists of striker bar, incident bar, transmitter bar, absorber and data acquisition equipment. The schematic diagram of SHPB set-up was shown in Fig. 2. The diameter of all the bars were 15 mm. Both incident bar and transmitter bar were 1300 mm in length. The length of striker bar had two types of 400 mm (long striker bar) and 100 mm (short striker bar), 2
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Fig. 2. Schematic view of the SHPB set-up.
using the long striker bar as the strain rates at 1400 s−1 and 3000 s−1, and using the short striker bar as the strain rate at 5700 s−1 s in order to avoid the deformation of incident bar and transmitter bar. In the dynamic tests, the pressure of compressed air were set as 0.035 MPa, 0.085 MPa and 0.07 MPa, with the corresponding strain rates of about 1400 s−1, 3000 s−1 and 5700 s−1, respectively. In order to reduce the impedance difference between the specimen and the Hopkinson bars, all the bars were made of aluminum alloy with elastic modulus of 75 GPa and a density of 2700 kg/m3. Pulse shaper technology was used to achieve dynamic stress equilibrium and minimize the high frequency oscillations of stress waves [22–24]. A thin layer of grease was applied between the sides of specimen and the end faces of bars to reduce the friction [22]. Strain gauges were attached on the interface of incident bar and transmitter bar to capture the history of the strain signal. The captured voltage signals were converted into strain signals of specimen through a Wheatstone bridge and were stored in data acquisition equipment. A typical incident, reflected and transmitted signals from dynamic compression experiment on polyurea at a strain rate of 3000 s−1 are shown in Fig. 3. It can be found that the reflected signal has a nearly plateau region, indicating that the deformation of specimen is at a relatively constant strain rate. A temperature equipment was used to heat the specimen as
conducting the SHPB experiment. The high temperature equipment was shown in Fig. 2. The specimen was placed between the incident bar and the transmission bar, and then passed through the center of the heating furnace. In addition, K type thermocouple conductors contacted the lateral side of the specimen to monitor the temperature of heating. The dynamic experiments were performed with the temperature of room temperature, 50 ℃ and 80 ℃ respectively. The high temperatures were chosen at 50 ℃ and 80 ℃, which were regulated by adjusting the voltage regulator, the heating process took 3 min and 5 min, respectively. When the actual temperature reached the setting temperature, the specimens were controlled for at least 5 min within ± 2 ℃ of the fixed temperature. Then, the high temperature dynamic experiments could be carried out. Based on the one-dimensional stress wave assumption and uniformity assumption [25,26], the engineering stress, the engineering strain and strain rate can be determined by the following equations [22,27]
σE (t ) =
S0 EεT (t ) SS
(1)
ε˙E (t ) = −
2C0 εR (t ) Ls
(2)
εE (t ) = −
2C0 Ls
(3)
∫0
t
εR (t )
Where, σE (t ) is the engineering stress, εE (t ) is the engineering strain, ε˙E (t ) is the strain rate, εR (t ) and εT (t ) are the incident strain and transmitted strain, E is the Young’s modulus of aluminum alloy, C0 is the wave velocity in the bars, S0 is the cross sectional area of the bars, SS and Ls are the initial cross sectional area and the initial length of the specimen, respectively. The true stress-strain curves were obtained on the basis of one-dimensional stress wave theory [25]. The relationship between the compressive engineering stress-strain and the compressive true stress-strain can be determined as follows.
σtrue = σE (1 − εE )
(4)
εtrue = −ln(1 − εE )
(5)
Where εE is engineering strain, σE is engineering stress, σT is true stress, and εT is true strain. Fig. 4 illustrates the strain rates vs. time relations under various strain rates loading, including the strain rates of 1400 s−1, 3000 s−1
Fig. 3. Typical incident, reflected and transmitted signals of SHPB experiment. 3
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Fig. 4. Strain rates vs. time curves of compression testing. Fig. 6. Strain rates effect on strain energy (strain, ε = 0.13).
and 5700 s−1. Due to the difference of striker bars, as shown in Fig. 4, it can be observed that the pulse width of reflected signals is basically the same at the strain rates of 1400 s−1 and 3000 s−1, and the pulse width is narrow at the strain rate of 5700 s−1. Meanwhile, the loading time was so fast that oscillation of the curve occurred at the strain rate of 5700 s-1. This is because the specimen was subjected to high speed impact and shorter loading time at higher strain rate, leading to local bending of bars that making the oscillation of the curve occurred at the strain rate of 5700 s-1.
more obvious than the previous region. The stress begin to suddenly drop as the stress reach the maximum value, which is due to unloading happen rather than the failure of specimens. In practice, under dynamic impact loading, the specimens are not failure under the three strain rates. Since polyurea is a hyperplastic material, it will basically restore to their previous shapes after experiments. Only higher strain rate can cause specimen failure. To evaluate the energy absorption of polyurea at high strain rates, the relationship between absorbed energy and strain rates is presented in Fig. 6. The absorbed energy can be determined by integrating the stress-strain curve at a certain strain region [28,29]. Here, we calculated the absorbed energy of polyurea at strain from 0 to 0.13. For a given strain rate of 1400 s−1, the absorbed energy of polyurea is 3.14 MJ/m3. The absorbed energy increase to 4.17 MJ/m3 when the strain rate at 3000 s−1, and increase to 5.66 MJ/m3 in the case of 5700 s−1. With the strain rate of 5700 s−1, the strain is less than 0.15. As observed in Fig. 6. The absorbed energy shows a notable increase with the increasing strain rates. The changing regularity of the absorbed energy is similar to that of stress-strain.
3. Experimental results and discussion 3.1. Strain-rate effect on compression properties Fig. 5 presents the stress-strain curves over a wide range strain rates at room temperature. As can be seen from the Fig. 5, the flow stress corresponding to the same deformation of specimen increases with the increasing strain rate. Here, as the true strain is selected at 0.15, the flow stress increase from 30.30 MPa at strain rate of 1400 s−1 to 36.79 MPa at the strain rate of 3000 s−1. When the strain rate reaches up to 5700 s−1, the stress increase to 47.49 MPa. The reason of increment could be due to the strain rate sensitivity of polyurea. Compared with the strain rate of 1400 s−1 and 3000 s−1, it can be found that the deformation degree of specimen shows a notable increase with the increasing of strain rate, and the deformation value can be observed to be about 0.2 and 0.5, respectively. Although the flow stress is maximum, the deformation is not large at stain value of about 0.15 when the strain rate is 5700 s−1. In addition, at the strain rate of 3000 s−1, the increasing tendency of flow stress becomes fast when the deformation strain of specimens over the range of 0.45–0.55. The strain hardening is
3.2. Temperature effect on compression properties In addition to the rate dependent of polyurea, the stress-strain behavior is sensitive to temperature. Fig. 7 presents the stress-strain curves of polyurea at various temperatures under a certain strain rate. At a certain strain rate, it can be found that the flow stress corresponding to the same strain decreases with increasing temperature from 20 ℃ to 80 ℃. This is due to temperature rise lead to the softening of specimens. For instance, in the case of 1400 s−1, an increase at temperature from 20 ℃ to 80 ℃ lead to an around 42 % decrease in compressive strength. In addition, as the strain rate increases, flow stress increases at the same temperature. In the case of 50 ℃, the stress are 23.05 MPa, 31.03 MPa, and 33.70 MPa at strain of 0.15, with the corresponding strain rates of 1400 s−1, 3000 s−1 and 5700 s−1, respectively. At temperature of 80 ℃, the stress values at strain rates of 1400 s−1, 3000 s−1 and 5700 s-1 are 17.21 MPa, 18.88 MPa and 23.68 MPa, respectively. 4. Constitutive model According to the experimental results presented in previous section, it is well found that the compressive mechanical behavior of polyurea under dynamic loading is rate dependent and temperature dependent. For describing the dynamic mechanical behavior of polyurea under high strain rates and temperatures, a constitutive model including the effect of strain rate and temperature is presented. Regarding the significance of strain rate of rubber-like materials, a hyperelastic
Fig. 5. The stress-strain curves under various strain rates at room temperature. 4
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Fig. 7. The stress-strain curves of polyurea at various temperatures under a certain strain rate (a) 1400 s−1, (b) 3000 s−1, and (c) 5700 s−1.
Table 1 The hyperelastic model parameters of polyuria. C10(MPa)
C01(MPa)
C20(MPa)
C02(MPa)
C11(MPa)
−146.296
−342.793
373.857
126.232
−120.574
Table 2 The viscoelastic parameters of polyurea at high strain rates. Strain rates(s−1)
E1 (MPa)
θ1 (μs)
1400 3000 5700
881.417 1156.481 6799.664
18.3 8.3 1.0
constitutive model is used to describe its mechanical behavior at low strain rates. Meanwhile, a viscoelastic constitutive model is appropriate for characterizing viscoelastic behavior at high strain rates. In the light of reporting of some literatures [30–32], the stress-strain mechanical behavior of polyurea can be described by combining hyperelastic model and viscoelastic model [13]. Therefore, a visco-hyperelastic constitutive model can be mentioned as
σ= σe + σv
Fig. 8. Comparison of model predictions with experimental data under various strain rates at room temperature.
isotropic material, considering more than two terms model of Rivlin series are accurate in the prediction of rubber-like materials, the Mooney-Rivlin strain energy function W with five material constants [33] can be expressed as follow
(6)
W = C10 (I1 − 3) + C01 (I2 − 3) + C20 (I1 − 3)2 + C02 (I2 − 3)2
Here σe is the Cauchy stress of hyperelastic part, σv is the Cauchy stress of viscoelastic part. Based on the assumption of polyurea is the incompressible and
+ C11 (I1 − 3)(I2 − 3)
(7)
Where C10 、 C01 、 C20 、 C02 and C11 are the material constants, I1 and 5
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Fig. 9. The relationship between stress and temperature of polyurea at various strain levels and strain rates: (a) strain rate at 1400 s−1, (b) strain rate at 3000 s−1, and (c) strain rate at 5700 s−1.
I2 are the first and second invariants which can be expressed as λ2 + 2/ λ and 2λ + 1/ λ2 , respectively, λ is the stretch ratio in the direction of uniaxial compression loading, and λ = 1 − ε , ε is the strain under the loading direction. Based on the equation of the Cauchy stress tensor in literature [12], the true stress of hyperelastic part of polyurea is derived as
σv =
t
E1 exp(−
t−τ ˙ ) λdτ θ1
(9)
Where E1 is the elastic constant of Maxwell element, θ1 is the relaxation time of Maxwell element, τ is the relaxation time. Thus, the visco-hyperelastic constitutive model of polyurea under high strain rates can be represented by adding up Eqs. (9) and (10) together as follow
σe = 2(λ−λ−2)[C10 + 2C20 (λ2 + 2λ−1 − 3) + C11 (2 λ+ λ−2 − 3) + λ−1 (C01 + 2C02 (2 λ+ λ−2 − 3) + C11 (λ2 + 2λ−1 − 3))]
∫0
σ= 2(λ−λ−2)[C10 + 2C20 (λ2 + 2λ−1 − 3) + C11 (2 λ+ λ−2 − 3) (8)
+ λ−1 (C01 + 2C02 (2 λ+ λ−2 − 3) + C11 (λ2 + 2λ−1 − 3)) t t−τ ˙ ) λdτ + E1 exp(− 0 θ1
∫
Here, λ = exp(−εT ) , εT is true strain. The ZWT constitutive model [30,34] have been widely used to describe the mechanical behavior of numerous polymer materials. In the ZWT constitutive model, the model is comprised of a nonlinear elastic equilibrium response and two Maxwell viscoelastic response. Regarding the viscoelastic of polymer, the first Maxwell element (low frequency) is used to describe the viscoelastic behavior at low strain rates, and the second Maxwell element (high frequency) is used to describe the viscoelastic behavior at high strain rates [35,36]. Under compression conditions of high strain rates, the impact experiment time is very short, and the low frequency Maxwell element has not enough time to relax until the test is over [30]. In order to reduce model parameters, the low frequency Maxwell element integral term can be neglected at high strain rates. Thus, the high frequency Maxwell element was used to describe the viscoelastic behavior at high strain rates. Therefore, the constitutive expression for the nonlinear viscoelastic response under dynamic compression can be given as
(10)
The material constants of Eq. (8) including C10 、 C01 、 C20 、 C02 and C11 hyperelastic parameters can be acquired by fitting the experimental results at a low strain rate of 0.1 s−1. The viscoelastic model parameters are fitted by substituting constants of Table 1 into Eq. (10), E1 and θ1 can then be obtain by fitting experimental results of polyurea at various high strain rates. The visco-hyperelastic model parameters of polyurea are shown in Tables 1 and 2. As observed in Table 2, the E1 coefficient has a prominent increase with the increase of strain rates. However, the θ1 relaxation time coefficient has an obvious reduction with the increasing strain rates. A comparison of predicted stress-strain results with experimental data is shown in Fig. 8. It can be found that the prediction results of visco-hyperelastic model is in a good agreement with experimental data under the strain rates of 0.1 s−1, 1400 s−1, 3000 s−1 and 5700 s−1. Hence, it can be considered that the visco-hyperelastic constitutive model is adapted to describe the 6
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Fig. 10. Comparison of the prediction results with experimental data for polyurea at different temperatures and strain rates: (a) strain rate at 1400 s−1, (b) strain rate at 3000 s−1, and (c) strain rate at 1400 s−1.
experimental data, the prediction results of proposed model shows in good agreement with the experimental data under different strain rates and temperatures, which indicates that the thermal-visco-hyperelastic constitutive model is applicable to describe the mechanical behavior of polyurea over a wide range of temperatures from 20 ℃ to 80 ℃.
mechanical behaviors of polyurea under a wide range of strain rates at room temperature. In addition to the effect of strain rates, the temperatures can also affect the mechanical properties of polyurea. To evaluate the testing temperature effect of polyurea at strain rates of 1400 s−1 and 3000 s−1, the relationship between stress and temperature at different strain conditions can be presented in Fig. 9, which indicates that stress decrease with increasing of temperature at different strain rates and strain levels. By fitting the relationship between stress and temperature, we can obtain the function as follow
σref σ
= 1 + a(
T T ) + b( )2 TR TR
5. Conclusions The dynamic compressive SHPB experiments of polyurea were conducted under different strain rates from 1400 s−1 to 5700 s−1 and temperatures from 20 ℃ to 80 ℃, and a thermal-visco-hyperelastic constitutive model was proposed to describe the mechanical behavior based on the experimental data, hyperelastic model, and viscoelastic model. The following conclusions can be drawn.
(11)
Where σref is the stress at room temperature, TR is the room temperature at 20 ℃, a and b are the coefficient associated with temperature range from 20 ℃ to 80 ℃. The material coefficient a and coefficient b are calculated by Eq. (11) related to global optimal algorithm. At the strain rate of 1400 s−1, parameters a= -0.01336 and b = 0.05742 are obtained by fitting the test data of Fig. 9(a). Meanwhile, for a given strain rate of 3000 s−1, parameters a = −0.11971 and b = 0.07842 are obtained by fitting the test data of Fig. 9(b). The visco-hyperelastic constitutive model can be extended to various temperatures by substituting Eqs. (10) into (11), which is called thermal-visco-hyperelastic constitutive model. In order to verify fidelity of the thermal-visco-hyperelastic constitutive model, all the parameters are taken to the model, the comparison of model prediction with experimental data at different temperatures and strain rates were shown in Fig. 10. Comparison between the predictions curves and
(1) The dynamic compressive mechanical behavior of polyurea is strain rate dependent and temperature dependent. The stress corresponding to the same strain increases with the increasing of strain rate. The stress-strain curve of polyurea exhibited apparent hardening behavior in the late deformation stage under strain rate of 3000 s−1 and room temperature. However, this hardening phenomenon is not obvious due to material softening with increasing temperature. In addition, the stress demonstrates a distinct reduction with the increasing temperature. (2) Under dynamic loading, the absorbed energy of polyurea corresponding to the same deformation is significantly increased as the strain rate increases. (3) Based on the hyperelastic model and viscoelastic model of polymer, 7
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a thermal-visco-hyperelastic constitutive model considers the effect of strain rate and temperature is established to describe the mechanical properties of polyurea. The prediction results of proposed constitutive model show a good agreement with the experimental results at the strain rates of 1400 s−1–5700 s−1 and the temperature range of 20 ℃–80 ℃. (4) In view of the work, we can infer that studies involving the mechanical behavior of polyurea provide a good knowledge for industrial application under high temperature conditions. Moreover, the constitutive model has good potential in predicting the effect of strain rate and high temperature on the stress-strain behaviors of polyurea under impact loading such as civilian infrastructure and military protection systems.
2208–2213. [11] J. Shim, D. Mohr, Using split Hopkinson pressure bars to perform large strain compression tests on polyurea at low, intermediate and high strain rates, Int. J. Impact Eng. 36 (9) (2009) 1116–1127. [12] H. Wang, X. Deng, H. Wu, A. Pi, J. Li, F. Huang, Investigating the dynamic mechanical behaviors of polyurea through experimentation and modeling, Def. Technol. (2019). [13] H. Guo, W. Guo, A.V. Amirkhizi, R. Zou, K. Yuan, Experimental investigation and modeling of mechanical behaviors of polyurea over wide ranges of strain rates and temperatures, Polym. Test. 53 (2016) 234–244. [14] X. Zhang, J. Wang, W. Guo, R. Zou, A bilinear constitutive response for polyureas as a function of temperature, strain rate and pressure, J. Appl. Polym. Sci. 134 (35) (2017) 45256. [15] A.V. Amirkhizi, J. Isaacs, J. McGee, S. Nemat-Nasser, An experimentally-based viscoelastic constitutive model for polyurea, including pressure and temperature effects, Philos. Mag. 86 (36) (2006) 5847–5866. [16] C. Li, J. Lua, A hyper-viscoelastic constitutive model for polyurea, Mater. Lett. 63 (11) (2009) 877–880. [17] J. Shim, D. Mohr, Rate dependent finite strain constitutive model of polyurea, Int. J. Plast. 27 (6) (2011) 868–886. [18] C. Gamonpilas, R. McCuiston, A non-linear viscoelastic material constitutive model for polyurea, Polymer 53 (17) (2012) 3655–3658. [19] H. Cho, R.G. Rinaldi, M.C. Boyce, Constitutive modeling of the rate-dependent resilient and dissipative large deformation behavior of a segmented copolymer polyurea, Soft Matter 9 (27) (2013) 6319–6330. [20] W. Nantasetphong, A.V. Amirkhizi, S. Nemat-Nasser, Constitutive modeling and experimental calibration of pressure effect for polyurea based on free volume concept, Polymer 99 (2016) 771–781. [21] H. Guo, W. Guo, A.V. Amirkhizi, Constitutive modeling of the tensile and compressive deformation behavior of polyurea over a wide range of strain rates, Constr. Build. Mater. 150 (2017) 851–859. [22] J.T. Fan, J. Weerheijm, L.J. Sluys, Compressive response of multiple-particlespolymer systems at various strain rates, Polymer 91 (2016) 62–73. [23] J.T. Fan, J. Weerheijm, L.J. Sluys, Dynamic compressive mechanical response of a soft polymer material, Mater. Des. 79 (2015) 73–85. [24] F. Jitang, W. Cheng, Dynamic compressive response of a developed polymer composite at different strain rates, Compos. Part B Eng. 152 (2018) 96–101 S1359836818317773-. [25] Kolsky, An investigation of the mechanical properties of materials at very high rates of loading, Proc. Phys. Soc. Sect. B 62 (11) (1949) 676. [26] H. Kolsky, Stress waves in solids, Nature 1 (1) (1954) 88–110. [27] Y. Tao, H. Chen, K. Yao, H. Lei, Y. Pei, D. Fang, Experimental and theoretical studies on inter-fiber failure of unidirectional polymer-matrix composites under different strain rates, Int. J. Solids Struct. 113 (2016) S0020768316303390. [28] M.F. Omar, H.M. Akil, Z.A. Ahmad, Measurement and prediction of compressive properties of polymers at high strain rate loading, Mater. Des. 32 (8) (2011) 4207–4215. [29] W. Wang, X. Zhang, N. Chouw, Z.X. Li, Y. Shi, Strain rate effect on the dynamic tensile behaviour of flax fibre reinforced polymer, Compos. Struct. 200 (2018). [30] J. Jiang, J.-s. Xu, Z.-s. Zhang, X. Chen, Rate-dependent compressive behavior of EPDM insulation: experimental and constitutive analysis, Mech. Mater. 96 (2016) 30–38. [31] H. Yang, X.F. Yao, H. Yan, Y.N. Yuan, Y.F. Dong, Y.H. Liu, Anisotropic hyper-viscoelastic behaviors of fabric reinforced rubber composites, Compos. Struct. 187 (2017) S0263822317332154. [32] K. Narooei, M. Arman, Generalization of exponential based hyperelastic to hyperviscoelastic model for investigation of mechanical behavior of rate dependent materials, J. Mech. Behav. Biomed. Mater. 79 (2018) 104. [33] N.W. Tschoegl, Constitutive equations for elastomers, J. Polym. Sci. Part A-1: Polym. Chem. 9 (7) (1971) 1959–1970. [34] J. Zhu, S. Hu, L. Wang, An analysis of stress uniformity for concrete-like specimens during SHPB tests, Int. J. Impact Eng. 36 (1) (2019) 61–72. [35] A.D. Mulliken, M.C. Boyce, Mechanics of the rate-dependent elastic–plastic deformation of glassy polymers from low to high strain rates, Int. J. Solids Struct. 43 (5) (2006) 1331–1356. [36] L. Zhang, X. Yao, S. Zang, Q. Han, Temperature and strain rate dependent tensile behavior of a transparent polyurethane interlayer, Mater. Des. 65 (1) (2015) 1181–1188.
CRediT authorship contribution statement Qiang Liu: Writing - original draft, Formal analysis, Writing - review & editing. Pengwan Chen: Investigation, Methodology. Yang Zhang: Resources, Validation. Zhirong Li: Funding acquisition. Declaration of Competing Interest None. Acknowledgements This work is supported by the State Administration of Science, Technology and Industry for National Defense (NO. WDZCKYXM20190503). The work also benefits from the Xi'an Modern Chemistry Institute. The authors also thank Prof. Weiguo Guo for experimental support in Northwestern Polytechnical University. References [1] E. Gauch, J. LeBlanc, A. Shukla, Near field underwater explosion response of polyurea coated composite cylinders, Compos. Struct. 202 (2018) 836–852. [2] N. Iqbal, P.K. Sharma, D. Kumar, P.K. Roy, Protective polyurea coatings for enhanced blast survivability of concrete, Constr. Build. Mater. 175 (2018) 682–690. [3] N. Gardner, E. Wang, P. Kumar, A. Shukla, Blast mitigation in a sandwich composite using graded core and polyurea interlayer, Exp. Mech. 52 (2) (2012) 119–133. [4] J. Leblanc, A. Shukla, Response of E-glass/vinyl ester composite panels to underwater explosive loading: effects of laminate modifications, Int. J. Impact Eng. 38 (10) (2011) 796–803. [5] K. Ackland, C. Anderson, T.D. Ngo, Deformation of polyurea-coated steel plates under localised blast loading, Int. J. Impact Eng. 51 (1) (2013) 13–22. [6] A. Remennikov, T. Ngo, D. Mohotti, B. Uy, M. Netherton, Experimental investigation and simplified modeling of response of steel plates subjected to close-in blast loading from spherical liquid explosive charges, Int. J. Impact Eng. 101 (2017) S0734743X16309666. [7] L. Xue, W.M. Jr, T. Belytschko, Penetration of DH-36 steel plates with and without polyurea coating, Mech. Mater. 42 (11) (2010) 981–1003. [8] G. Toader, E. Rusen, M. Teodorescu, A. Diacon, P.O. Stanescu, C. Damian, T. Rotariu, A. Rotariu, New polyurea MWCNTs nanocomposite films with enhanced mechanical properties, J. Appl. Polym. Sci. 134 (28) (2017) 45061. [9] J. Yi, M.C. Boyce, G.F. Lee, E. Balizer, Large deformation rate-dependent stress–strain behavior of polyurea and polyurethanes, Polymer 47 (1) (2006) 319–329. [10] S.S. Sarva, S. Deschanel, M.C. Boyce, W. Chen, Stress–strain behavior of a polyurea and a polyurethane from low to high strain rates, Polymer 48 (8) (2007)
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Compressive behavior and constitutive model of polyurea at high strain rates and high temperatures
T
Qiang Liua,b,*, Pengwan Chena,**, Yang Zhangc, Zhirong Lib a
State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, 100081, China Xi’an Modern Chemistry Research Institute, Xi’an, 710065, China c Northwestern Polytechnical University, Xi’an, 710072, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Polyurea Strain rate Temperature Constitutive model
To study the dynamic compressive mechanical properties of polyurea at strain rate range of 1400 s−1–5700 s−1 and high temperature range of 20 ℃–80 ℃, we conducted a dynamical mechanical experiment utilizing a split Hopkinson pressure bar (SHPB) system. The compressive stress-strain curves of polyurea under various strain rates and temperatures are analyzed and discussed. The experimental results indicated that the strain rate and high temperature can remarkably influence the mechanical behaviors of polyurea, compression behaviors increased with the increasing of strain rate and compression behaviors decreased with the increasing of temperature. In addition, a thermal-visco-hyperelastic constitutive model based on five-parameter Mooney-Rivlin model was proposed to describe the compression stress-strain behavior of polyurea over a wide range of strain rates and temperatures. The model parameters were obtained by fitting the experimental stress-strain curves. The model prediction results show good agreement with the experimental results.
1. Introduction Polyurea is an elastomeric polymer formed by chemical reaction between amine component and isocyanate component. It as a coating [1,2] or sandwich [3,4] has been extensively applied in composite structures for protection due to its high elongation, impact resistance and mechanical properties. Some researchers reported that polyurea coated outer facesheet of steel plate or a core layer could reduce deformation than uncoated structure during the process of blast and ballistic loading [5–7]. Several composite structures including polyurea materials are subjected to high strain rate during dynamic loading. In addition, the process of explosion can accompany with a significant thermal radiation [8]. Temperature have a significant influence on mechanical properties of polyurea. With the increase of temperature, the mechanical properties decrease. When temperature reach up to about 100 ℃, polyurea may appear slight mass loss. Once the mass loss of polyurea is to begin to change, the performance of the material will be affected. Therefore, it is essential to research the dynamic mechanical behavior of polyurea referring to the strain rate and temperature. Many literatures have also revealed that the mechanical properties of polyurea depend on strain rate and temperature [9–14]. The dynamic mechanical properties of polyurea were widely studied. Yi et al.
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[9] investigated the dynamic compression behavior of polyurea using split Hopkinson pressure bar (SHPB), the results revealed that polyurea was strong rate-dependence, and either a leathery or glassy-like behavior at high strain rate loading. Sarva et al. [10] reported that the uniaxial compression stress-strain behavior of an elastomeric- thermoset polyurea over a wide range of strain rates was strongly dependent on strain rate and seen to undergo transition from rubbery-regime behavior at low rates to leathery-regime behavior at the high rates. Shim and Mohr [11] utilized a modified split Hopkinson pressure bar (SHPB) system to study the strain rate sensitivity of polyurea. Wang et al. [12] investigated the effect of strain rate on polyurea compressive mechanical properties over the range of strain rates (0.001 s−1–7000 s−1) and found that the stress-strain curves can be divided into three distinct regions, and the mechanical properties of polyurea were found to be highly dependent on the strain rate. Considering the importance of temperature, it is essential to investigate the high stain rate mechanical properties of polyurea at different temperatures. Guo et al. [13] studied the compressive mechanical behavior of two formulations of polyurea over the temperature range from −40 ℃ to 20 ℃ and the strain rate range from 0.001 s−1 to 12,000 s−1, the experimental results shown the stress-strain curves of polyurea was sensitive to strain rate and temperature within and beyond linear
Corresponding author at: State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, 100081, China. Corresponding author. E-mail addresses: [email protected] (Q. Liu), [email protected] (P. Chen).
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https://doi.org/10.1016/j.mtcomm.2019.100834 Received 3 August 2019; Received in revised form 6 December 2019; Accepted 6 December 2019 Available online 06 December 2019 2352-4928/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. (a) Photograph of specimens, (b) physical specimen, and (c) geometrical dimensions.
the stress-strain behaviors. Wang et al. [12] modified a rate-dependent constitutive model based on the nine-parameter Moony-Rivlin model to characterize the mechanical properties of polyurea over a wide range of strain rates. Although the constitutive models mentioned above can describe the stress-strain behaviors of polyurea, some models seem to be too complicated for applications. However, the models to describe the stress-strain behavior of polyurea on the effect of strain rate and high temperature are rare. Therefore, it is necessary to develop new constitutive model for polyurea involving a wide range of strain rate and temperature. In this work, the dynamic compressive properties of polyurea were investigated by using a split Hopkinson pressure bar (SHPB) with the range of strain rates from 1400 s−1 to 5700 s−1 and temperatures from 20 ℃ to 80 ℃. This polyurea material can be used in building as a protective coating. Additionally, polyurea can also be applied to military vehicles or ships. The mechanical properties of polyurea are vulnerable to operating temperature and strain rate. On the basis of the experimental results, the effects of strain rate and temperature on the dynamic compressive behaviors were analyzed and discussed. Furthermore, a thermal-visco-hyperelastic constitutive model was proposed to describe the compression stress-strain behavior of polyurea in terms of the strain rate and temperature.
viscoelastic regime. Zhang et al. [14] carried out a systematic experimental program to study the stress-strain behaviors of two polyaspartic esters (PAE) polyureas over the temperature range of 233 K–293 K and strain rate range of 0.001 s−1–15000 s−1. The results indicated that the uniaxial compression stress-strain behaviors of polyureas were strongly dependent on temperature and strain rate, and the yield stress of the polyureas decreases linearly with increasing temperature. From the published studies, very few studies reported on the dynamic compressive properties of polyurea at high temperature. To describe the stress-strain behaviors of polyurea, some constitutive models were proposed. Amirkhizi et al. [15] proposed an experimentally-based linear viscoelastic constitutive model that incorporated the classical Williams-Landel-Ferry (WLF) time temperature transformation and pressure response. Li and Lua [16] proposed a hyper-visoelastic constitutive model for polyurea by including Ogden model and a nonlinear viscoelastic model. Shim and Mohr [17] developed a constitutive model based on a rheological model composed of two Maxwell elements. This model can be able to predict the experimentally-measured stress-strain curves for various loading and unloading histories. Gamonpilas and McCuiston [18] have established a non-linear viscoelastic constitutive model for polyurea in terms of a nine-parameter Mooney-Rivlin model. The model was found to be efficient for describing the compressive and tensile behavior of polyurea under a wide range of strain rates. Cho et al. [19] developed a microstructurally informed three-dimensional constitutive model to capture the behavior of polyurea over the range of strain rate (10−3 s-1–103 s-1). Nantasetphong et al. [20] combined the effect of pressure and temperature to propose a constitutive model based on Williams-LandelFerry (WLF) equation. The prediction of the model agreed well with pressure shift factor data about the confined compression tests of polyurea. Guo et al. [13] developed a visco-hyperealstic constitutive model based on the two order strain energy density function to describe the nonlinear mechanical behavior of polyurea over a wide range of strain rates and temperatures (−40 ℃ - +20 ℃). Meanwhile, Guo et al. [21] also developed a three-dimensional visco-hyperelastic constitutive model on the basis of the multiplicative decomposition of the deformation gradient tensor into hyperelastic and viscoelastic parts to describe the mechanical behavior of polyurea at various strain rates. Zhang et al. [14] proposed a bilinear constitutive model to describe the temperature (233 K–293 K), strain rate, and pressure dependences of
2. Experiments Polyurea consisting of 1.5 wt% nano silicon carbide was sprayed on the template to prepare the thin sheet with a thickness at around 4.7 mm. The thin sheet of specimen is presented in Fig. 1(a). The cylindrical specimens of 4.7 mm in length and 9.4 mm in diameter were obtained using mechanical processing method for dynamic compression experiments. The physical specimen and geometrical dimensions are presented in Fig. 1(b) and (c). All specimens were kept for at least three days to eliminate their residual stress. The dynamic compression experiment was conducted by using a split Hopkinson pressure bar (SHPB) apparatus, which consists of striker bar, incident bar, transmitter bar, absorber and data acquisition equipment. The schematic diagram of SHPB set-up was shown in Fig. 2. The diameter of all the bars were 15 mm. Both incident bar and transmitter bar were 1300 mm in length. The length of striker bar had two types of 400 mm (long striker bar) and 100 mm (short striker bar), 2
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Fig. 2. Schematic view of the SHPB set-up.
using the long striker bar as the strain rates at 1400 s−1 and 3000 s−1, and using the short striker bar as the strain rate at 5700 s−1 s in order to avoid the deformation of incident bar and transmitter bar. In the dynamic tests, the pressure of compressed air were set as 0.035 MPa, 0.085 MPa and 0.07 MPa, with the corresponding strain rates of about 1400 s−1, 3000 s−1 and 5700 s−1, respectively. In order to reduce the impedance difference between the specimen and the Hopkinson bars, all the bars were made of aluminum alloy with elastic modulus of 75 GPa and a density of 2700 kg/m3. Pulse shaper technology was used to achieve dynamic stress equilibrium and minimize the high frequency oscillations of stress waves [22–24]. A thin layer of grease was applied between the sides of specimen and the end faces of bars to reduce the friction [22]. Strain gauges were attached on the interface of incident bar and transmitter bar to capture the history of the strain signal. The captured voltage signals were converted into strain signals of specimen through a Wheatstone bridge and were stored in data acquisition equipment. A typical incident, reflected and transmitted signals from dynamic compression experiment on polyurea at a strain rate of 3000 s−1 are shown in Fig. 3. It can be found that the reflected signal has a nearly plateau region, indicating that the deformation of specimen is at a relatively constant strain rate. A temperature equipment was used to heat the specimen as
conducting the SHPB experiment. The high temperature equipment was shown in Fig. 2. The specimen was placed between the incident bar and the transmission bar, and then passed through the center of the heating furnace. In addition, K type thermocouple conductors contacted the lateral side of the specimen to monitor the temperature of heating. The dynamic experiments were performed with the temperature of room temperature, 50 ℃ and 80 ℃ respectively. The high temperatures were chosen at 50 ℃ and 80 ℃, which were regulated by adjusting the voltage regulator, the heating process took 3 min and 5 min, respectively. When the actual temperature reached the setting temperature, the specimens were controlled for at least 5 min within ± 2 ℃ of the fixed temperature. Then, the high temperature dynamic experiments could be carried out. Based on the one-dimensional stress wave assumption and uniformity assumption [25,26], the engineering stress, the engineering strain and strain rate can be determined by the following equations [22,27]
σE (t ) =
S0 EεT (t ) SS
(1)
ε˙E (t ) = −
2C0 εR (t ) Ls
(2)
εE (t ) = −
2C0 Ls
(3)
∫0
t
εR (t )
Where, σE (t ) is the engineering stress, εE (t ) is the engineering strain, ε˙E (t ) is the strain rate, εR (t ) and εT (t ) are the incident strain and transmitted strain, E is the Young’s modulus of aluminum alloy, C0 is the wave velocity in the bars, S0 is the cross sectional area of the bars, SS and Ls are the initial cross sectional area and the initial length of the specimen, respectively. The true stress-strain curves were obtained on the basis of one-dimensional stress wave theory [25]. The relationship between the compressive engineering stress-strain and the compressive true stress-strain can be determined as follows.
σtrue = σE (1 − εE )
(4)
εtrue = −ln(1 − εE )
(5)
Where εE is engineering strain, σE is engineering stress, σT is true stress, and εT is true strain. Fig. 4 illustrates the strain rates vs. time relations under various strain rates loading, including the strain rates of 1400 s−1, 3000 s−1
Fig. 3. Typical incident, reflected and transmitted signals of SHPB experiment. 3
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Fig. 4. Strain rates vs. time curves of compression testing. Fig. 6. Strain rates effect on strain energy (strain, ε = 0.13).
and 5700 s−1. Due to the difference of striker bars, as shown in Fig. 4, it can be observed that the pulse width of reflected signals is basically the same at the strain rates of 1400 s−1 and 3000 s−1, and the pulse width is narrow at the strain rate of 5700 s−1. Meanwhile, the loading time was so fast that oscillation of the curve occurred at the strain rate of 5700 s-1. This is because the specimen was subjected to high speed impact and shorter loading time at higher strain rate, leading to local bending of bars that making the oscillation of the curve occurred at the strain rate of 5700 s-1.
more obvious than the previous region. The stress begin to suddenly drop as the stress reach the maximum value, which is due to unloading happen rather than the failure of specimens. In practice, under dynamic impact loading, the specimens are not failure under the three strain rates. Since polyurea is a hyperplastic material, it will basically restore to their previous shapes after experiments. Only higher strain rate can cause specimen failure. To evaluate the energy absorption of polyurea at high strain rates, the relationship between absorbed energy and strain rates is presented in Fig. 6. The absorbed energy can be determined by integrating the stress-strain curve at a certain strain region [28,29]. Here, we calculated the absorbed energy of polyurea at strain from 0 to 0.13. For a given strain rate of 1400 s−1, the absorbed energy of polyurea is 3.14 MJ/m3. The absorbed energy increase to 4.17 MJ/m3 when the strain rate at 3000 s−1, and increase to 5.66 MJ/m3 in the case of 5700 s−1. With the strain rate of 5700 s−1, the strain is less than 0.15. As observed in Fig. 6. The absorbed energy shows a notable increase with the increasing strain rates. The changing regularity of the absorbed energy is similar to that of stress-strain.
3. Experimental results and discussion 3.1. Strain-rate effect on compression properties Fig. 5 presents the stress-strain curves over a wide range strain rates at room temperature. As can be seen from the Fig. 5, the flow stress corresponding to the same deformation of specimen increases with the increasing strain rate. Here, as the true strain is selected at 0.15, the flow stress increase from 30.30 MPa at strain rate of 1400 s−1 to 36.79 MPa at the strain rate of 3000 s−1. When the strain rate reaches up to 5700 s−1, the stress increase to 47.49 MPa. The reason of increment could be due to the strain rate sensitivity of polyurea. Compared with the strain rate of 1400 s−1 and 3000 s−1, it can be found that the deformation degree of specimen shows a notable increase with the increasing of strain rate, and the deformation value can be observed to be about 0.2 and 0.5, respectively. Although the flow stress is maximum, the deformation is not large at stain value of about 0.15 when the strain rate is 5700 s−1. In addition, at the strain rate of 3000 s−1, the increasing tendency of flow stress becomes fast when the deformation strain of specimens over the range of 0.45–0.55. The strain hardening is
3.2. Temperature effect on compression properties In addition to the rate dependent of polyurea, the stress-strain behavior is sensitive to temperature. Fig. 7 presents the stress-strain curves of polyurea at various temperatures under a certain strain rate. At a certain strain rate, it can be found that the flow stress corresponding to the same strain decreases with increasing temperature from 20 ℃ to 80 ℃. This is due to temperature rise lead to the softening of specimens. For instance, in the case of 1400 s−1, an increase at temperature from 20 ℃ to 80 ℃ lead to an around 42 % decrease in compressive strength. In addition, as the strain rate increases, flow stress increases at the same temperature. In the case of 50 ℃, the stress are 23.05 MPa, 31.03 MPa, and 33.70 MPa at strain of 0.15, with the corresponding strain rates of 1400 s−1, 3000 s−1 and 5700 s−1, respectively. At temperature of 80 ℃, the stress values at strain rates of 1400 s−1, 3000 s−1 and 5700 s-1 are 17.21 MPa, 18.88 MPa and 23.68 MPa, respectively. 4. Constitutive model According to the experimental results presented in previous section, it is well found that the compressive mechanical behavior of polyurea under dynamic loading is rate dependent and temperature dependent. For describing the dynamic mechanical behavior of polyurea under high strain rates and temperatures, a constitutive model including the effect of strain rate and temperature is presented. Regarding the significance of strain rate of rubber-like materials, a hyperelastic
Fig. 5. The stress-strain curves under various strain rates at room temperature. 4
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Fig. 7. The stress-strain curves of polyurea at various temperatures under a certain strain rate (a) 1400 s−1, (b) 3000 s−1, and (c) 5700 s−1.
Table 1 The hyperelastic model parameters of polyuria. C10(MPa)
C01(MPa)
C20(MPa)
C02(MPa)
C11(MPa)
−146.296
−342.793
373.857
126.232
−120.574
Table 2 The viscoelastic parameters of polyurea at high strain rates. Strain rates(s−1)
E1 (MPa)
θ1 (μs)
1400 3000 5700
881.417 1156.481 6799.664
18.3 8.3 1.0
constitutive model is used to describe its mechanical behavior at low strain rates. Meanwhile, a viscoelastic constitutive model is appropriate for characterizing viscoelastic behavior at high strain rates. In the light of reporting of some literatures [30–32], the stress-strain mechanical behavior of polyurea can be described by combining hyperelastic model and viscoelastic model [13]. Therefore, a visco-hyperelastic constitutive model can be mentioned as
σ= σe + σv
Fig. 8. Comparison of model predictions with experimental data under various strain rates at room temperature.
isotropic material, considering more than two terms model of Rivlin series are accurate in the prediction of rubber-like materials, the Mooney-Rivlin strain energy function W with five material constants [33] can be expressed as follow
(6)
W = C10 (I1 − 3) + C01 (I2 − 3) + C20 (I1 − 3)2 + C02 (I2 − 3)2
Here σe is the Cauchy stress of hyperelastic part, σv is the Cauchy stress of viscoelastic part. Based on the assumption of polyurea is the incompressible and
+ C11 (I1 − 3)(I2 − 3)
(7)
Where C10 、 C01 、 C20 、 C02 and C11 are the material constants, I1 and 5
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Fig. 9. The relationship between stress and temperature of polyurea at various strain levels and strain rates: (a) strain rate at 1400 s−1, (b) strain rate at 3000 s−1, and (c) strain rate at 5700 s−1.
I2 are the first and second invariants which can be expressed as λ2 + 2/ λ and 2λ + 1/ λ2 , respectively, λ is the stretch ratio in the direction of uniaxial compression loading, and λ = 1 − ε , ε is the strain under the loading direction. Based on the equation of the Cauchy stress tensor in literature [12], the true stress of hyperelastic part of polyurea is derived as
σv =
t
E1 exp(−
t−τ ˙ ) λdτ θ1
(9)
Where E1 is the elastic constant of Maxwell element, θ1 is the relaxation time of Maxwell element, τ is the relaxation time. Thus, the visco-hyperelastic constitutive model of polyurea under high strain rates can be represented by adding up Eqs. (9) and (10) together as follow
σe = 2(λ−λ−2)[C10 + 2C20 (λ2 + 2λ−1 − 3) + C11 (2 λ+ λ−2 − 3) + λ−1 (C01 + 2C02 (2 λ+ λ−2 − 3) + C11 (λ2 + 2λ−1 − 3))]
∫0
σ= 2(λ−λ−2)[C10 + 2C20 (λ2 + 2λ−1 − 3) + C11 (2 λ+ λ−2 − 3) (8)
+ λ−1 (C01 + 2C02 (2 λ+ λ−2 − 3) + C11 (λ2 + 2λ−1 − 3)) t t−τ ˙ ) λdτ + E1 exp(− 0 θ1
∫
Here, λ = exp(−εT ) , εT is true strain. The ZWT constitutive model [30,34] have been widely used to describe the mechanical behavior of numerous polymer materials. In the ZWT constitutive model, the model is comprised of a nonlinear elastic equilibrium response and two Maxwell viscoelastic response. Regarding the viscoelastic of polymer, the first Maxwell element (low frequency) is used to describe the viscoelastic behavior at low strain rates, and the second Maxwell element (high frequency) is used to describe the viscoelastic behavior at high strain rates [35,36]. Under compression conditions of high strain rates, the impact experiment time is very short, and the low frequency Maxwell element has not enough time to relax until the test is over [30]. In order to reduce model parameters, the low frequency Maxwell element integral term can be neglected at high strain rates. Thus, the high frequency Maxwell element was used to describe the viscoelastic behavior at high strain rates. Therefore, the constitutive expression for the nonlinear viscoelastic response under dynamic compression can be given as
(10)
The material constants of Eq. (8) including C10 、 C01 、 C20 、 C02 and C11 hyperelastic parameters can be acquired by fitting the experimental results at a low strain rate of 0.1 s−1. The viscoelastic model parameters are fitted by substituting constants of Table 1 into Eq. (10), E1 and θ1 can then be obtain by fitting experimental results of polyurea at various high strain rates. The visco-hyperelastic model parameters of polyurea are shown in Tables 1 and 2. As observed in Table 2, the E1 coefficient has a prominent increase with the increase of strain rates. However, the θ1 relaxation time coefficient has an obvious reduction with the increasing strain rates. A comparison of predicted stress-strain results with experimental data is shown in Fig. 8. It can be found that the prediction results of visco-hyperelastic model is in a good agreement with experimental data under the strain rates of 0.1 s−1, 1400 s−1, 3000 s−1 and 5700 s−1. Hence, it can be considered that the visco-hyperelastic constitutive model is adapted to describe the 6
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Fig. 10. Comparison of the prediction results with experimental data for polyurea at different temperatures and strain rates: (a) strain rate at 1400 s−1, (b) strain rate at 3000 s−1, and (c) strain rate at 1400 s−1.
experimental data, the prediction results of proposed model shows in good agreement with the experimental data under different strain rates and temperatures, which indicates that the thermal-visco-hyperelastic constitutive model is applicable to describe the mechanical behavior of polyurea over a wide range of temperatures from 20 ℃ to 80 ℃.
mechanical behaviors of polyurea under a wide range of strain rates at room temperature. In addition to the effect of strain rates, the temperatures can also affect the mechanical properties of polyurea. To evaluate the testing temperature effect of polyurea at strain rates of 1400 s−1 and 3000 s−1, the relationship between stress and temperature at different strain conditions can be presented in Fig. 9, which indicates that stress decrease with increasing of temperature at different strain rates and strain levels. By fitting the relationship between stress and temperature, we can obtain the function as follow
σref σ
= 1 + a(
T T ) + b( )2 TR TR
5. Conclusions The dynamic compressive SHPB experiments of polyurea were conducted under different strain rates from 1400 s−1 to 5700 s−1 and temperatures from 20 ℃ to 80 ℃, and a thermal-visco-hyperelastic constitutive model was proposed to describe the mechanical behavior based on the experimental data, hyperelastic model, and viscoelastic model. The following conclusions can be drawn.
(11)
Where σref is the stress at room temperature, TR is the room temperature at 20 ℃, a and b are the coefficient associated with temperature range from 20 ℃ to 80 ℃. The material coefficient a and coefficient b are calculated by Eq. (11) related to global optimal algorithm. At the strain rate of 1400 s−1, parameters a= -0.01336 and b = 0.05742 are obtained by fitting the test data of Fig. 9(a). Meanwhile, for a given strain rate of 3000 s−1, parameters a = −0.11971 and b = 0.07842 are obtained by fitting the test data of Fig. 9(b). The visco-hyperelastic constitutive model can be extended to various temperatures by substituting Eqs. (10) into (11), which is called thermal-visco-hyperelastic constitutive model. In order to verify fidelity of the thermal-visco-hyperelastic constitutive model, all the parameters are taken to the model, the comparison of model prediction with experimental data at different temperatures and strain rates were shown in Fig. 10. Comparison between the predictions curves and
(1) The dynamic compressive mechanical behavior of polyurea is strain rate dependent and temperature dependent. The stress corresponding to the same strain increases with the increasing of strain rate. The stress-strain curve of polyurea exhibited apparent hardening behavior in the late deformation stage under strain rate of 3000 s−1 and room temperature. However, this hardening phenomenon is not obvious due to material softening with increasing temperature. In addition, the stress demonstrates a distinct reduction with the increasing temperature. (2) Under dynamic loading, the absorbed energy of polyurea corresponding to the same deformation is significantly increased as the strain rate increases. (3) Based on the hyperelastic model and viscoelastic model of polymer, 7
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a thermal-visco-hyperelastic constitutive model considers the effect of strain rate and temperature is established to describe the mechanical properties of polyurea. The prediction results of proposed constitutive model show a good agreement with the experimental results at the strain rates of 1400 s−1–5700 s−1 and the temperature range of 20 ℃–80 ℃. (4) In view of the work, we can infer that studies involving the mechanical behavior of polyurea provide a good knowledge for industrial application under high temperature conditions. Moreover, the constitutive model has good potential in predicting the effect of strain rate and high temperature on the stress-strain behaviors of polyurea under impact loading such as civilian infrastructure and military protection systems.
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CRediT authorship contribution statement Qiang Liu: Writing - original draft, Formal analysis, Writing - review & editing. Pengwan Chen: Investigation, Methodology. Yang Zhang: Resources, Validation. Zhirong Li: Funding acquisition. Declaration of Competing Interest None. Acknowledgements This work is supported by the State Administration of Science, Technology and Industry for National Defense (NO. WDZCKYXM20190503). The work also benefits from the Xi'an Modern Chemistry Institute. The authors also thank Prof. Weiguo Guo for experimental support in Northwestern Polytechnical University. References [1] E. Gauch, J. LeBlanc, A. Shukla, Near field underwater explosion response of polyurea coated composite cylinders, Compos. Struct. 202 (2018) 836–852. [2] N. Iqbal, P.K. Sharma, D. Kumar, P.K. Roy, Protective polyurea coatings for enhanced blast survivability of concrete, Constr. Build. Mater. 175 (2018) 682–690. [3] N. Gardner, E. Wang, P. Kumar, A. Shukla, Blast mitigation in a sandwich composite using graded core and polyurea interlayer, Exp. Mech. 52 (2) (2012) 119–133. [4] J. Leblanc, A. Shukla, Response of E-glass/vinyl ester composite panels to underwater explosive loading: effects of laminate modifications, Int. J. Impact Eng. 38 (10) (2011) 796–803. [5] K. Ackland, C. Anderson, T.D. Ngo, Deformation of polyurea-coated steel plates under localised blast loading, Int. J. Impact Eng. 51 (1) (2013) 13–22. [6] A. Remennikov, T. Ngo, D. Mohotti, B. Uy, M. Netherton, Experimental investigation and simplified modeling of response of steel plates subjected to close-in blast loading from spherical liquid explosive charges, Int. J. Impact Eng. 101 (2017) S0734743X16309666. [7] L. Xue, W.M. Jr, T. Belytschko, Penetration of DH-36 steel plates with and without polyurea coating, Mech. Mater. 42 (11) (2010) 981–1003. [8] G. Toader, E. Rusen, M. Teodorescu, A. Diacon, P.O. Stanescu, C. Damian, T. Rotariu, A. Rotariu, New polyurea MWCNTs nanocomposite films with enhanced mechanical properties, J. Appl. Polym. Sci. 134 (28) (2017) 45061. [9] J. Yi, M.C. Boyce, G.F. Lee, E. Balizer, Large deformation rate-dependent stress–strain behavior of polyurea and polyurethanes, Polymer 47 (1) (2006) 319–329. [10] S.S. Sarva, S. Deschanel, M.C. Boyce, W. Chen, Stress–strain behavior of a polyurea and a polyurethane from low to high strain rates, Polymer 48 (8) (2007)
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