Investigating the dynamic mechanical behaviors of polyurea through experimentation and modeling

Ho Wng ,Ximin Deng ,Hijun Wu ,*,Aiguo Pi ,Jinzhu Li ,Fenglei Hung

a State Key Laboratory of Explosion Science and Technology,Beijing Institute of Technology,Beijing,100081,China

b Wuhan Guide Infrared Co.,Ltd,Wuhan,430000,China

ABSTRACT Polyurea is widely employed as a protective coating in many fields because of its superior ability to improve the anti-blast and anti-impact capability of structures.In this study,the mechanical properties of polyurea XS-350 were investigated via systematic experimentation over a wide range of strain rates(0.001-7000 s-1)by using an MTS,Instron VHS,and split-Hopkinson bars.The stress-strain behavior of polyurea was obtained for various strain rates,and the effects of strain rate on the primary mechanical properties were analyzed.Additionally,a modified rate-dependent constitutive model is proposed based on the nine-parameter Mooney-Rivlin model.The results show that the stress-strain curves can be divided into three distinct regions:the linear-elastic stage,the highly elastic stage,and an approximate linear region terminating in fracture.The mechanical properties of the polyurea material were found to be highly dependent on the strain rate.Furthermore,a comparison between model predictions and the experimental stress-strain curves demonstrated that the proposed model can characterize the mechanical properties of polyurea over a wide range of strain rates.

Keywords:Polyurea Strain rate effect Dynamic mechanical properties Constitutive model

1. Introduction

Polyurea is a type of elastomeric polymer that is extensively used as a protective coating in many industrial applications,such as vehicles,ships,and buildings.Due to its superior physical and mechanical properties,it is regarded as an energy-absorbing material that can enhance structural resistance to blasts and ballistic loading.However,as the polyurea coating may experience a wide range of loading rates under impact,it is essential to characterize the mechanical properties of polyurea at different strain rates and establish a constitutive model to further improve the quality and efficiency of protective designs.

The absorbed-energy capacity of polyurea is closely related to its mechanical properties under high strain-rate loading.Scholars have studied the mechanical properties of polyurea over a limited range of strain rates.By using a universal testing machine and SHPB,Yi et al.[1]and Sarva et al.[2]observed an apparent nonlinear correlation and strain rate effect in the stress-strain curves for polyurea.They also observed a transition of polyurea from rubbery-regime behavior to leathery-regime behavior as the strain rate was increased.Similarly,Roland et al.[3]and Pathak et al.[4]performed stress-strain measurements of polyurea in uniaxial tension for low-to-moderate strain rates(10-3-102s-1).They found that the strength and stiffness gradually increased,whereas the damage strain gradually decreased as the strain rate was increased.Shim et al.[5]developed a modified SHPB system to perform compression tests on polyurea for strain rates ranging from 10 to 103s-1.

Numerical simulation has recently emerged as an important method to aid design,as an accurate material model of polyurea is essential.Hence,the establishment of a constitutive material model purposed for finite element analysis has been the focus of most related research.Shim et al.[6]proposed a rate-dependent finitestrain constitutive model to describe the continuous and multi-step compression behavior of polyurea for strain rates ranging from 10-3to 10 s-1.Amirkhizi et al.[7]developed a constitutive model of linear viscoelasticity that considered the effects of pressure;however,the applicability of their model is limited.Based on uniaxial tensile test results for polyurea,Raman et al.[8]proposed an empirical correlation to represent the effects of strain rate on the modulus of elasticity and yield stress.Mohotti[9]proposed a ratedependent Mooney-Rivlin model to predict the high strain-rate behavior of polyurea.Guo et al.[10,11]proposed different constitutive models focusing on strain rate and temperature.Zhang et al.[12]also proposed a bilinear constitutive model of polyurea to describe the temperature,strain rate,and pressure dependencies of the stress-strain behaviors.

This paper describes mechanical experiments on polyurea XS-350 under compression and tensile loading that were conducted for strain rates ranging from 0.001s-1to 7000s-1by using an MTS,Instron VHS,and split-Hopkinson bar experimental system.The experimental results were analyzed to determine the effects of strain rate on the compressive and tensile stress-strain behaviors.In addition,the nine-parameter Mooney-Rivlin(MR)model was modified by introducing a rate-dependent component to describe the visco-hyperelasticity of polyurea.The experimental data was used to calibrate the parameters for the new constitutive model.Finally,the accuracy of the model was validated with the experimental data obtained for polyurea.

2. Experimental procedures

2.1. Materials

XS-350 was employed as the polyurea specimens produced by company LINE-X in this study.This material is a two-component,high-performance elastomer spraying material,with the raw materials including an A component of isocyanate and a B component of RESIN.In the processing process,stir together the two compositions until the mixture is uniform,then spray on flat plate and let stand until cured.The basic performance parameters are presented in Table 1.

2.2. Mechanical tests

2.2.1. Quasi-static and low strain rate testing

Quasi-static and low strain rate tensile and compression tests were performed by using an MTS servo-hydraulic machine.Each test was repeated at least three times and conducted at room temperature(approximately 25°C).The force and displacement data were recorded by the data acquisition system and used to calculate the corresponding engineering stress-strain curves.Moreover, the true stress and true strain were calculated in consideration of the fact that polyurea is a bulk and(nearly)incompressible material;true stress is the load divided by the current cross-sectional area,and the true strain is the natural logarithm of the ratio of the current length to the initial length.

The test specimens were cut from a polyurea plate that was processed by using a spraying technique.The tensile samples were designed in accordance with the ISO 37 standard[13],as is shown in Fig.1.A polyurea sample implemented in tensile testing,and a photograph of a sample undergoing tensile testing,are depicted in Fig.2.The compression test samples were designed in accordance with the ISO 23592 standard[14];they were designed as a cylinder with a diameter of 10 mm and height of 4 mm.Fig.3(a)and(b)shows a polyurea compression test sample,and a sample undergoing compression testing,respectively.

Fig.1.Geometry of the quasi-static tensile test sample.(Measurements are in mm.)

Fig.2.(a)Polyurea tensile test sample,and(b)sample undergoing tensile testing.

Fig.3.(a)Polyurea compression test sample,and(b)sample undergoing compression testing.

2.2.2. Intermediate strain rate testing

The intermediate strain-rate experiment was performed on an Instron VHS dynamic experimental machine at the China Aircraft Strength Institute.The testing method and data processing method are similar to those implemented in the low strain-rate experiment.Because there is no applicable international or national standard for dynamic mechanical performance tests for polyurea elastomer,relevant literature was referenced to determine the sample size in this study[9].The tensile specimen thickness,which is illustrated in Fig.4,is 6 mm;furthermore,bolts were used to affix the clamp to the material(Fig.5).Fig.6 shows the Instron dynamic testing system and a tensile specimen undergoing the test.The compression sample was fabricated in accordance with the ISO 23592 standard[14],which suggests a cylinder with a diameter of 19 mm and height of 6 mm.

2.2.3. High strain rate testing

Split-Hopkinson compression bar(SHPB)and split-Hopkinson tensile bar (SHTB)were used to perform the high strain-rate compression and tensile tests on polyurea,respectively.Fig.7(a and b)presents a schematic of the split-Hopkinson compression and tensile bars.The strain gauges are located near the center of the incident bars,and near the specimen/transmission bar interface.Copper discs were employed as pulse shapers to maintain a dynamic stress equilibrium and facilitate constant strain rate deformation in the specimen by appropriately modifying the profile of the incident pulse[15].In addition,petroleum jelly was applied on the interfaces between the bars and the specimen to reduce friction.The material and geometric details of the bars are summarized in Table 2.

Table 1Basic performance parameters of XS-350.

Fig.4.Geometry of the intermediate strain-rate test sample.(Measurements are in mm.)

Fig.5.Schematic of the extender system:(a)plan view and(b)section view.(Measurements are in mm.)

The tensile test specimen was formed into a dumbbell-shaped sheet with a thickness of 4 mm.The specimen was connected to the pull rod by a special fixture,and the sizes of the sample and fixture are shown in Fig.8.It should be noted that the thickness of the compression test specimen will affect the reliability of the data.Considering the low wave velocity of this material,the stress wave occurring within the specimen after multiple reflections allows the material to achieve a uniform state;thus,the sample must be designed to have a low ratio of length to diameter.Previous literature[5]has reported that the ratio of length to diameter of the elastomer specimen should range from 0.25 to 0.5 in order to significantly reduce the attenuation and frictional effects of the stress wave.Thus,in order to achieve the desired experimental strain rate, two compression samples of different sizes were required.The first and second samples were cylinders with diameters and heights of 6 mm and 2 mm,and 10 mm and 4 mm,respectively.

The basic theoretical assumptions employed in the split-Hopkinson bar experiment are the one-dimensional stress wave theory and dynamic stress equilibrium assumption.According to this stress uniformity assumption,the axial force at its frontF1,that at its back endF2follow the relationship

where εi(t),εr(t),and εt(t)are the incident,reflected,and transmitted strain pulses,respectively;E1,A1,E2,A2are the elastic modulus,the cross-sectional area of incident and transmitted bars,respectively.The dynamic average stress σs(t),strain εs(t),and strain rate ˙εs（t）of the specimen can be calculated via the following equations with consideringA1=A2in this paper:

whereC1andC2are the longitudinal wave velocity of the incident and transmitted bars,respectively;Asandlsare the cross-sectional area and length of the specimens,respectively.Fig.9 shows the typical oscilloscope records in SHPB.The force histories during this experiment are illustrated in Fig.10.TheFRcurve was almost coincided with that of the difference betweenFTandFI,demonstrating that the specimen deforms follows the hypothesis of the dynamic stress equilibrium.

Fig.6.Instron VHS dynamic experimental machine.

3. Results and discussion

The stress-strain curves for the samples under axial extension and compression are as shown in Figs.11 and 12,respectively.It is evident from these figures that the polyurea material exhibited remarkable strain-rate hardening and strain hardening effects,and high nonlinearity.The true stress-strain curves at low strain rates can be observed to have three distinct stages.The initial region represents an initial linear-elastic stage.In this regime,the stress is proportional to the strain,and the relationship between stress and strain satisfies Hooke's law.The second region represents a highly elastic stage in which the stress increases slowly with increased strain on the material.The third region is an approximate linear region that terminates in fracture; at this stage, the material exhibited a significant strain hardening effect.The material of the elastic modulus can clearly be observed to dramatically increase in response to an increasing strain rate.It increased about 30 times at strain rates of 4000/s in this paper.Additionally,the general trend of the stress-strain curves demonstrates that the yield strain decreased with increased strain rate.It is necessary to note that only the linear-elastic component of the relationship between strain and stress was recorded during the SHTB test because the tension specimen failure after several repetitions of stress wave loading.In addition,stress values obtained from the tensile and compression test data were found to be similar for a given strain rate and strain.

Fig.7.Configuration of the split-Hopkinson compression and tensile bars:(a)loading device and(b)bar component and measurement system.

Table 2Material and geometric details of the bars.

The polyurea did not display any distinct yield behavior.A 0.2%offset method was used to quantify the yield stress,which is shown in Fig.13 as a function of the logarithm of the strain rate.The yield stress was found to vary from 10 to 43 MPa over the wide range of strain rates implemented in this study,thereby demonstrating considerable sensitivity to the strain rate;in particular,at high strain rates,the yield stress sharply increased with increasing strain rate.Because strain rate increase can profoundly alter intermolecular interactions,polyurea undergoes a transition from rubberyregime behavior to leathery-regime behavior;this consequently contributed to the changing yield.Similar observations have also been reported by Yi[1]and Sarva[2].

Fig.8.(a)Geometry of high strain-rate tensile sample,and(b)schematic of the fixture used in tensile testing.(Measurements are in mm.)

Fig.9.Typical oscilloscope records in SHPB.

Fig.10.Typical force histories in SHPB.

Fig.11.Tensile stress-strain curves for various strain rates.

Fig.12.Compression stress-strain curves for various strain rates.

Fig.13.Yield stress as a function of the logarithm of the strain rate.

Fig.14.Modulus of elasticity of the polyurea for various strain rates.

Fig.14 illustrates how the modulus of elasticity of the polyurea varies according to the strain rate.In low and intermediate strain rate regions ranging from 100 to 200 MPa, the modulus of elasticity was observed to linearly increase as the logarithm of the strain rate increased,exhibiting dramatic increases(up to 2 GPa)at high strain rates.This fniding suggests that the initial stiffness of the material was enhanced at higher strain rates.Alternatively,the tangent modulus derived from the highly elastic stage of the compression or tensile stress-strain curves was generally consistent for low and intermediate strain rates (20-27 MPa in compression and 26-35 MPa in tension); however, its value increased to 120 MPa,which is nearly a six-fold increase,when the strain rate was high(Fig.15).In particular,because the failure strain in uniaxial tension is larger than that tested in uniaxial compression, the tensile strain-hardening effect is more pronounced,and the tangent modulus in tension is slightly higher than that in compression.

4. Constitutive model

4.1. Modified RDMR model

The MR model is a classical hyperelastic material model that is available to model the nonlinear behavior of elastomeric materials that is associated with large strain. As the simple constitutive equation and its parameters can be relatively easily determined via experimentation,the MR model has been widely used in many studies and encoded in some commercial finite element software,e.g.,ANSYS.However,the original MR model is independent of strain rate,and as the materials would undergo rapid plastic deformation during an explosion or impact loading,the influence of strain rate on the mechanical properties of materials need to be considered in this constitutive model.

Fig.15.Tangent modulus of the polyurea for various strain rates.

In continuum mechanics,the deformation of a rubber-like material is assumed to be a uniform deformation of an isotropic super elastomer.The strain energy of material can be expressed as a function of an invariant of the left Cauchy-Green deformation tensor.For an incompressible MR material,the most common strain energy function,which was proposed by Rivlin[16]in 1951,is as follows:

There are many forms of expansions for the above formula,with two, five, and nine expansions being most commonly implemented.In this study,a strain energy function with nine material constants was adopted.

whereDandcijare the material constants,I1andI2are the two invariants of the left Cauchy-Green deformation tensor,which can be expressed as

where λi=1,2,3=1+εEis the stretch ratio in the primary direction,εEis the engineering strain in the loading direction.In consideration of the incompressible material assumption,J=1.When the material is subjected to uniaxial loading,and λ1=λ.

The Cauchy(true)stress can be expressed by using the first and second deformation tensor invariants,as follows:

In order to consider the strain rate effect in the constitutive model,Mohotti[9]proposed a rate-dependent MR(RDMR)model by adding a rate-dependent component to the strain energy function.Consequently,the RDMR model predictions were found to be in agreement with the corresponding experimental results for varying strain rates within the range of 20-400 s-1.Analysis and fitting of the stress-strain curves derived as based on the results of the experiments carried out in this study indicated that the RDMR model is only applicable in the low and intermediate strain rate regions,as it results in large error when it incorporates the high strain-rate experimental data.To resolve this problem,the strain rate correlation factor was defined as a function of the standardized strain rate for a wide range of strain rates;it can be expressed as

where μ is the strain rate correlation factor,andis the standardized strain rate,which is defined as

The strain energy function(Eq.(6))and stress(Eq.(9))were respectively derived as follows:

The above equations represent the modified RDMR model proposed in this paper.The strain rate correlation factor function was determined as based on the experimental tension and compression test results for various strain rates.The derivation and parameter calibration methods are presented in the next section.

4.2. Parameter fitting

To validate the nine-parameter MR model(Eq.(6))for different strain rates,the stretch ratio-strain energy density curves and experimental data were compared,as is shown in Fig.16;the corresponding fitting parameters are given in Table 3.As can be seen from the figure, the fitting curves agree quite well with the experiment data,indicating that the nine-parameter MR strain energy function can accurately describe the nonlinear mechanical response of polyurea at individual strain rate.Note that these parameters are only reliable under the condition that the strain rate is unchanged throughout the deformation process.

Because strain energy is not a uniform function of strain,the modified RDMR model parameters determined via the compression and tensile stress-strain curves are different.The strain rate correlation factor for compression tests was determined as follows.The parameter values for the nine-parameter MR model were obtained for the reference strain rate of 0.01 s-1,and are listed in Table 4.The value of the base stretch ratio was set as λ0=0.6.The strain rate correlation factor values are shown in Fig.17 for various strain rates;this figure indicates that polyurea has good energy absorption capacity at high strain rates.Moreover,it is clear from Fig.17 that the relationship between the strain rate correlation factor and standardized strain rate is not linear over the range of strain rates employed in this study.Based on the variation trend presented and the analysis performed,the following bilinear form was used to fit the relationship:

whereaandcare coefficients.The calculated values,which are provided in Table 5,were obtained from the compression test.

Fig.16.Comparison of the nine-parameter MR model and experimental data.

Table 3Nine-parameter MR model parameter values derived from tension test results.

Table 4Nine-parameter MR model parameter values derived from compression test results.

Fig.17.Strain rate correlation factor as a function of the standardized strain rate.

Table 5Coefficients a and c derived from compression test results.

The true stress-strain curves predicted by the modified RDMR model were compared to the experimental results,as is illustrated in Fig.18.It can be seen that the prediction of the true stress agrees well with the experimental results for most of the true strain rates;this indicates that the modified RDMR model has the potential describe the mechanical properties of polyurea under compression for a relatively wide range of strain rates.However,at high strain rates,the predicted values are less than the experimental values at the linear-elastic stage,as is illustrated in Fig.18(c).This discrepancy is due to the fact that,when the strain rate is relatively high,increases of the stress range value in the linear-elastic stage are significantly larger than those observed after the material has entered the yield stage.

For tensile testing validation,a reference strain rate of 0.01 s-1and a base stretch ratio of 1.6 were implemented.As previously mentioned,Table 3 lists the parameters of the nine-parameter MR model that were derived from the tensile stress-strain curves.The coefficientsaandc,which were derived as based on the tension test results,are listed in Table 6.The comparison between model predictions and experimental date is shown in Fig.19.The model predictions agree well with the experimental results.

Fig.18.Comparison between the modified RDMR model predictions and results of compression testing:(a)Low strain rate,(b)Intermediate strain rate,and(c)High strain rate.

Table 6Coefficients a and c derived from tension test results.

Fig.19.Comparison between modified RDMR model predictions and experimental data obtained from tensile testing.

5. Conclusion

Uniaxial compression and tension testing was used to investigate the dynamic mechanical properties of polyurea XS-350 for strain rates ranging from 0.001 to 7000 s-1.In this study,we quantified the stress-strain relationship of polyurea,and analyzed the effects of strain rate on the yield strength,modulus of elasticity,and tangent modulus.A modified RDMR model was proposed to describe the mechanical properties of polyurea for different strain rates.The main conclusions are as follows:

(1)The stress-strain curve of polyurea XS-350 exhibits distinct nonlinear characteristics, and can be divided into the following three regions: an initial linear-elastic stage, a highly elastic stage,and an approximate linear region that terminates in fracture.

(2)When polyurea XS-350 is subjected to dynamic loading,the influence of the strain rate is significant;specifically,with the increase of the strain rate,the yield stress and modulus of elasticity increase,while the yield strain decreases.Also during this time,polyurea transitions from rubbery-regime behavior to leathery-regime behavior.

(3)A modified rate-dependent constitutive model was developed as based on the nine-parameter MR model;the proposed model can better predict the mechanical properties of polyurea under low,intermediate,and high strain rates.

Acknowledgments

The authors would like to acknowledge the Provincial Basic Research Program of China(NO.2016209A003,NO⋅2016602B003).