3. Various polymers for dynamic mechanical investigation

### 3.1. Rubber-like polymers

Rubber-like polymer has been used extensively as structural material of engineering components that are designed to resist impact, ranging from bus windows and eyeglasses to protective helmets and body armours. The choice of polymer materials for these applications has been made appealing by the relative low density and the transparency that are the characteristics of amorphous homopolymers. One of the studies on polymer materials focuses on the capability of absorbing dynamic strain energy and strain-rate dependency [1, 21–30]. Strain-rate dependency of the stress-strain behaviour of polymer materials has been well documented, where, in particular, yield stress is found to increase with the increasing strain rate. This feature of mechanical behaviour is highly relevant to engineering applications, when designing a polymer component required to resist an impact loading.

### 3.1.1. Polyurea

The thermoplastic elastomer polyurethane and the elastomeric thermoset polyurea are found to have new applications by increasing the survivability of structures under impact loading, including those encountered in blast and ballistic events. Yi et al. [31] studied the large deformation and rate-dependent stress-strain behaviour of polyurea and polyurethanes in dynamic compressive tests. A set of data was presented to quantify the rate-dependent behaviour of these materials from low strain rates (<1/s) to high strain rates (>1000/s), as shown in Figure 2.

The polyurea displayed a transition of deformation behaviour from rubber-like behaviour at low strain rates to leathery behaviour at high strain rates, whereas one of these three polyurethanes displayed a transition from rubber-like behaviour at low strain rates to glass-like behaviour at high strain rates. Figure 2 presents the rate-dependent behaviour by means of the stress vs logarithm strain rate, taking the stress evaluated at a strain level of 0.15 and 0.30 for

Figure 2. Flow stress vs strain rate relations of polyurea and three polyurethanes determined at the deformation strain of (a) 0.15 and (b) 0.30 [31].

Figure 2a and b, respectively. For both strain levels, the flow stress obviously demonstrated a close-to-linear dependency on the logarithm strain rate in both the high strain rate (≳10<sup>3</sup> /s) and low strain rate (≲100 /s) regions. So, mechanical behaviour, as illustrated by yield stress of the thermoplastic-elastomeric polyurethanes and elastomeric-thermoset polyureas, is strongly dependent on strain rate.

In addition, continuous investigation was conducted on the characterization of mechanical properties at very high strain rates under both dynamic compression and tension loadings [32]. The experimental results are shown in Figure 3. The uniaxial compression and tension data for polyurea are found to be consistent at the strain rates ranging from 0.001/s to 10,000/s, both of which increase with the increase in strain rate. Therefore, a strong dependency of flow stress on strain rate is clarified in the material of the thermoplastic-elastomeric polyurethane and elastomeric-thermoset polyurea, which is of particular interest when they play a role as a protective coating to enhance survivability of structures in high-rate loading events.

### 3.1.2. Polyurethane

3. Various polymers for dynamic mechanical investigation

Rubber-like polymer has been used extensively as structural material of engineering components that are designed to resist impact, ranging from bus windows and eyeglasses to protective helmets and body armours. The choice of polymer materials for these applications has been made appealing by the relative low density and the transparency that are the characteristics of amorphous homopolymers. One of the studies on polymer materials focuses on the capability of absorbing dynamic strain energy and strain-rate dependency [1, 21–30]. Strain-rate dependency of the stress-strain behaviour of polymer materials has been well documented, where, in particular, yield stress is found to increase with the increasing strain rate. This feature of mechanical behaviour is highly relevant to engineering applications, when designing a polymer component

The thermoplastic elastomer polyurethane and the elastomeric thermoset polyurea are found to have new applications by increasing the survivability of structures under impact loading, including those encountered in blast and ballistic events. Yi et al. [31] studied the large deformation and rate-dependent stress-strain behaviour of polyurea and polyurethanes in dynamic compressive tests. A set of data was presented to quantify the rate-dependent behaviour of these

The polyurea displayed a transition of deformation behaviour from rubber-like behaviour at low strain rates to leathery behaviour at high strain rates, whereas one of these three polyurethanes displayed a transition from rubber-like behaviour at low strain rates to glass-like behaviour at high strain rates. Figure 2 presents the rate-dependent behaviour by means of the stress vs logarithm strain rate, taking the stress evaluated at a strain level of 0.15 and 0.30 for

Figure 2. Flow stress vs strain rate relations of polyurea and three polyurethanes determined at the deformation strain of

materials from low strain rates (<1/s) to high strain rates (>1000/s), as shown in Figure 2.

3.1. Rubber-like polymers

198 Aspects of Polyurethanes

required to resist an impact loading.

3.1.1. Polyurea

(a) 0.15 and (b) 0.30 [31].

Zhang et al. [33] studied the dynamic mechanical behaviour of a polyurethane used as an interlayer in a laminated windshield construction at various strain rates (0.001/s to 7000/s) and

Figure 3. Stresses, taken at the true strain of 0.4 and 0.9, as a function of the true strain rate of the polyurea, where the value of the true strain rate used for each point is also taken at the true strain of 0.4 or 0.9 [32].

various temperatures (40C to 25C). The research results reveal that the mechanical behaviour of polyurethane interlayer is dependent on temperature and strain rate. Under dynamic loading, the transition from "rubbery" to "glassy" is exhibited in the stress-strain curves at 240C. In terms of the constitutive theory and experimental data, one-dimensional thermal-hyper-viscoelastic constitutive equation is recommended to characterize the compressive deformation response of polyurethane interlayer over a wide range of temperatures and strain rates.

Figure 4 shows the true stress-strain curves of polyurethane interlayer at different temperatures and at a certain strain rate. The temperature-dependent behaviour is clearly seen. When temperature decreases, the stress-strain curve goes up with the increase in yield stress, and the strainhardening behaviour becomes remarkable. At high strain rates and at low temperature, the flow stress date exhibits an obvious increase. Under the quasi-static (0.001/s) loading, the stress-strain results illustrate a common phenomenological mechanical behaviour of soft materials revealed in compression experiment at a low strain rate. As the temperature decreases from 20C to 40C, the significant changes occur in stress-strain curves. A rubbery behaviour transits into

Figure 4. True stress-strain curves of the polyurethane interlayer: (a) 0.001/s, (b) 2200/s, (c) 4800/s and (d) 6500/s [33].

glassy behaviour of the mechanical response, which is in line with the mechanical behaviour of the rubber-like materials below and above glass transition temperature. The inherent reason can be the glass transition temperature of the polyurethane interlayer of around 58C to 40C.

Figure 5 shows the true stress-strain curves and the corresponding true strain rate-strain curves at a temperature of 25C. The trend of strain rates seems to be relatively constant over the courses, which indicates that dynamic stress equilibrium is nearly stable. The uniaxial compressive stress-strain behaviour in the regime of high strain rate has a strong dependency of strain rate and temperature. At the same temperature, with the increase in strain rate, the flow stress increases and the strain hardening behaviour becomes more apparent.

### 3.1.3. Polyurethane elastomer

various temperatures (40C to 25C). The research results reveal that the mechanical behaviour of polyurethane interlayer is dependent on temperature and strain rate. Under dynamic loading, the transition from "rubbery" to "glassy" is exhibited in the stress-strain curves at 240C. In terms of the constitutive theory and experimental data, one-dimensional thermal-hyper-viscoelastic constitutive equation is recommended to characterize the compressive deformation

Figure 4 shows the true stress-strain curves of polyurethane interlayer at different temperatures and at a certain strain rate. The temperature-dependent behaviour is clearly seen. When temperature decreases, the stress-strain curve goes up with the increase in yield stress, and the strainhardening behaviour becomes remarkable. At high strain rates and at low temperature, the flow stress date exhibits an obvious increase. Under the quasi-static (0.001/s) loading, the stress-strain results illustrate a common phenomenological mechanical behaviour of soft materials revealed in compression experiment at a low strain rate. As the temperature decreases from 20C to 40C, the significant changes occur in stress-strain curves. A rubbery behaviour transits into

Figure 4. True stress-strain curves of the polyurethane interlayer: (a) 0.001/s, (b) 2200/s, (c) 4800/s and (d) 6500/s [33].

response of polyurethane interlayer over a wide range of temperatures and strain rates.

200 Aspects of Polyurethanes

Fan et al. [34] developed a soft polyurethane elastomeric material for impact-resistant applications. Stress-strain relations, characterized by using a split Hopkinson tension bar, are derived to reveal the mechanical properties at different strain rate levels at room temperature, as shown in Figure 6. The stress-strain curves from multiple tests at comparable strain rates are similar and partially overlap, illustrating the good reproducibility of the experimental data. The dynamic stress-strain curves show a different behaviour, compared to quasi-static stressstrain plot at a strain rate of 0.01/s. This difference has been also observed for other soft polymer materials [35, 36].

In statics, the initial stiffness of the soft polyurethane elastomeric polymer material is much negligible. While in dynamics, stiffness becomes significantly higher, and the length of the linear ascending branch increases with the increase in strain rate. Even though stress equilibrium is not

Figure 5. True stress-strain curves and the corresponding true strain rate-strain curves at a temperature of 25C [33].

Figure 6. Representative engineering stress-strain plots of the soft polyurethane elastomeric polymer material under dynamic tension loading with three curves per selected strain rate level [34].

attained in the specimen at the beginning of the initial dynamic loading, a linear link of yielding point and origin point (0, 0) in the stress-strain curve can be conducted to roughly evaluate the dynamic tensile modulus, considering that material yielding occurs after dynamic stress equilibrium. The line slope is the tangent modulus, which can indicate the material stiffness at the corresponding strain rate. The relation between tangent modulus and strain rate is shown in Figure 7. By linearly fitting the curve of tangent modulus versus log strain rate, a log strain rate value of 2.65 or about 450/s strain rate is attained. It indicates that the strain rate of 450/s is the critical transition point at which mechanical response of the soft polyurethane elastomeric polymer material changes from a rubber-like behaviour at low strain rates to a glass-like behaviour at high strain rates at room temperature [31, 37].

### 3.2. Glass-like polymers

The development of glass-like polymer materials that are more impact- and ballistic-resistant has many possible applications ranging from military vehicle windows to civilian products. Two common kinds of organic glasses that are used in these engineering areas include the polycarbonate (PC) and polymethyl methacrylate (PMMA). These materials are transparent and lightweight compared with their inorganic counterparts, which is important especially in vehicle and personal protection applications. Each of these materials has a unique inherent mechanism by which energy is absorbed during impact. Polycarbonate, like other ductile materials, has the capability to absorb a large amount of energy through yielding and plasticity. For PMMA, the majority of energy absorption is due to the creation of new surface area during fracture. There is an obvious difference between their macroscopic failure mechanisms.

Figure 7. The relation of tangent modulus versus strain rate curve for determining the critical strain rate of the soft polyurethane elastomeric polymer material for the transition from a rubber-like behaviour at low strain rates to a glasslike behaviour at high strain rates at room temperature [34].

Failure in PC is relatively localized while PMMA is effective at delocalizing failure in the form of radial cracking and Hertzian cone fracture (during impact). Moreover, the glass-like polymers also have an obvious rate dependency of mechanical properties [1, 38, 39], which directly affects the dynamic strength and determines the structure design, application and reliability for safety and security.

### 3.2.1. Polymethyl methacrylate (PMMA)

attained in the specimen at the beginning of the initial dynamic loading, a linear link of yielding point and origin point (0, 0) in the stress-strain curve can be conducted to roughly evaluate the dynamic tensile modulus, considering that material yielding occurs after dynamic stress equilibrium. The line slope is the tangent modulus, which can indicate the material stiffness at the corresponding strain rate. The relation between tangent modulus and strain rate is shown in Figure 7. By linearly fitting the curve of tangent modulus versus log strain rate, a log strain rate value of 2.65 or about 450/s strain rate is attained. It indicates that the strain rate of 450/s is the critical transition point at which mechanical response of the soft polyurethane elastomeric polymer material changes from a rubber-like behaviour at low strain rates to a glass-like behav-

Figure 6. Representative engineering stress-strain plots of the soft polyurethane elastomeric polymer material under

The development of glass-like polymer materials that are more impact- and ballistic-resistant has many possible applications ranging from military vehicle windows to civilian products. Two common kinds of organic glasses that are used in these engineering areas include the polycarbonate (PC) and polymethyl methacrylate (PMMA). These materials are transparent and lightweight compared with their inorganic counterparts, which is important especially in vehicle and personal protection applications. Each of these materials has a unique inherent mechanism by which energy is absorbed during impact. Polycarbonate, like other ductile materials, has the capability to absorb a large amount of energy through yielding and plasticity. For PMMA, the majority of energy absorption is due to the creation of new surface area during fracture. There is an obvious difference between their macroscopic failure mechanisms.

iour at high strain rates at room temperature [31, 37].

dynamic tension loading with three curves per selected strain rate level [34].

3.2. Glass-like polymers

202 Aspects of Polyurethanes

As one of glass-like polymers, polymethyl methacrylate (PMMA) is reviewed for clarifying the dynamic mechanical behaviour [1]. A combined experimental and analytical method has been performed to investigate the mechanical behaviour of PMMA material at strain rates ranging from 10<sup>4</sup> /s to 10<sup>4</sup> /s. The relation of yield stress and strain rate is documented in Figure 8.

The yield stress was found to increase as a non-linear function with the logarithmic strain rate, displaying the strain-rate sensitivity. The mechanisms of the rate-dependent elastic-plastic deformation of PMMA material from low to high strain rates were also studied. A computational model was developed based on the concepts of both the Ree-Eyring yield theory and the viscoelastic theory, which indicates that intermolecular resistance to deformation may be decomposed into the contributions of different molecular processes, each with their own unique rate and temperature dependency. This model is probably suitable for two-component polymer materials.

Therefore, rate-dependent behaviour of polymer materials was revealed and mechanisms for impact resistance were also explored using the combined experimental and computational

Figure 8. True yield stress of PMMA material as a function of true strain rate (logarithmic scale)—low to high strain rates [1].

methods. A concerted effort has been made to investigate and develop new polymer materials with improved characteristics for impact resistance and damage tolerance. Concerns regarding rate dependency and impact resistance of polymer materials are the basis for extensive research.

### 3.2.2. Polycarbonate (PC)

Dar et al. [39] studied the mechanical behaviour of polycarbonate (PC) polymer under the effect of various temperatures and strain rates. Mechanical characterizations are carried out through uniaxial compression and split Hopkinson pressure bar (SHPB) for revealing low and high strain rate response, respectively. Meanwhile, the experiments are performed for strain rates varying from 10<sup>3</sup> /s to 103 /s and a temperature range of 213 K to 393 K. The experimental results reveal that the stress-strain behaviour of polycarbonates is much different at lower and higher strain rates. At higher strain rate, the polycarbonates yields at higher yield stress compared to that at low strain rate. At lower strain rate, yield stress increases with the increase in strain rate while it decreases significantly with the increase in temperature. Likewise, initial elastic modulus, yield and flow stress increase with the increase in strain rate, whereas decreases with the increase in temperature. Yield stress increases significantly for low temperature and higher strain rates.

SHPB tests were performed to determine the dynamic response of polycarbonate at strain rates varying from 1350/s to 9400/s. Dynamic tests were performed at five different strain rates, and the results in terms of true stress-strain curves are shown in Figure 9. The results show that yield stress increases with the increase in strain rate. The stress-strain curves show almost similar

Figure 9. High strain rate stress-strain response of polycarbonate [39].

mechanical response in which initial nonlinear elastic behaviour was observed followed by subsequent yielding, strain softening and hardening. Yield stress changes significantly with the increase in strain rate. An increase of 20.6% in yield stress was calculated with strain rate increase from 1350 to 9400/s. At all strain rates, ductile response of polycarbonate was observed and ductile-brittle transition was not found.

Dynamic tests at a strain rate of 1350/s were also performed at three different temperatures and the results are shown in Figure 10. The change in yield stress is more significant in case of temperature than strain rate. The 43.4% decrease in yield stress with the increase in temperature from 233 K to 333 K is revealed.

Yield stress summarized at different strain rates and temperatures were plotted and a liner relationship was found between them as shown in Figure 11. A 0.69 MPa/K decrease in yield stress was observed between temperature variations of 233–333 K. Dynamic stress sensitivity (ðσdynamic � σstaticÞ=σstatic of polycarbonate is computed to be 128% which is significantly less than PMMA [40] showing that polycarbonate is not a highly strain rate-sensitive polymer. σstatic in this case is considered as quasi-static or very low strain rate stress.

### 3.2.3. Polymethylene diisocyanate (PMDI)

methods. A concerted effort has been made to investigate and develop new polymer materials with improved characteristics for impact resistance and damage tolerance. Concerns regarding rate dependency and impact resistance of polymer materials are the basis for extensive research.

Figure 8. True yield stress of PMMA material as a function of true strain rate (logarithmic scale)—low to high strain

Dar et al. [39] studied the mechanical behaviour of polycarbonate (PC) polymer under the effect of various temperatures and strain rates. Mechanical characterizations are carried out through uniaxial compression and split Hopkinson pressure bar (SHPB) for revealing low and high strain rate response, respectively. Meanwhile, the experiments are performed for strain

results reveal that the stress-strain behaviour of polycarbonates is much different at lower and higher strain rates. At higher strain rate, the polycarbonates yields at higher yield stress compared to that at low strain rate. At lower strain rate, yield stress increases with the increase in strain rate while it decreases significantly with the increase in temperature. Likewise, initial elastic modulus, yield and flow stress increase with the increase in strain rate, whereas decreases with the increase in temperature. Yield stress increases significantly for low temper-

SHPB tests were performed to determine the dynamic response of polycarbonate at strain rates varying from 1350/s to 9400/s. Dynamic tests were performed at five different strain rates, and the results in terms of true stress-strain curves are shown in Figure 9. The results show that yield stress increases with the increase in strain rate. The stress-strain curves show almost similar

/s and a temperature range of 213 K to 393 K. The experimental

3.2.2. Polycarbonate (PC)

rates [1].

204 Aspects of Polyurethanes

rates varying from 10<sup>3</sup>

ature and higher strain rates.

/s to 103

Song et al. [41] reported the dynamic mechanical response of three polymer foam materials made by rigid polymethylene diisocyanate (PMDI), varied in density (310 kg/m3 , 410 kg/m<sup>3</sup> and 550 kg/m<sup>3</sup> ), at strain rates as high as 3000/s and at temperatures ranging from 219 K to 347 K. The effects of material density, strain rate and temperature on the compressive response of the polymer foam materials were determined. Compressive stress-strain curves of the three foam materials (with the densities of 310 kg/m3 , 410 kg/m3 and 550 kg/m3 ) at various strain rates are shown in Figure 12a–c, respectively.

Figure 10. Effect of temperature on high strain rate stress-strain response [39].

Figure 11. Effect of strain rate and temperature on yield stress [39].

Figure 10. Effect of temperature on high strain rate stress-strain response [39].

206 Aspects of Polyurethanes

Figure 11. Effect of strain rate and temperature on yield stress [39].

Figure 12. Compressive stress-strain curves at various strain rates for our three foam-material densities: (a) 310 kg/m<sup>3</sup> , (b) 410 kg/m<sup>3</sup> and (c) 550 kg/m<sup>3</sup> [41].

They are obtained at room temperature (295 K) and have a common characteristic: an initial linearly elastic deformation followed by a collapse process of cell structures. When all the cell structures were collapsed, the condensation initiates, as revealed by the increasing stress amplitude in the stress-strain response [42]. For each polymer foam material, this characteristic is varied slightly with strain rates, and the cell-structure collapse process has an apparent variation. Under quasi-static loading, the stress-strain curves exhibit a long stress plateau and/or slow strain-hardening behaviour after yielding. This stress plateau indicates the plastic buckling of cell structures. However, under dynamic loading, stress drops from the peak date after yielding, causing the formation of N-shaped stress-strain curves. The stress drop is caused by the sudden collapse of cell structures under high strain-rate loading. So, the deformation and damage mechanisms are influenced by the strain rates in the polymer foam material.

Yield stress is also found to be dependent on strain rate. Figure 13 shows the details of the increase in yield stress with strain rate for these three polymer foam materials. The yield strength of the three foam materials linearly increases with the logarithm of strain rate, as

Figure 13. Strain-rate sensitivities of the three polymer foam materials with different densities [41].

shown by equation: σ<sup>y</sup> ¼ A þ Blog ε<sup>0</sup> =ε<sup>0</sup> ð Þ<sup>0</sup> , wherein A and B are constant data, and ε<sup>0</sup> <sup>0</sup> is the reference strain rate. The constant B is the slope of the solid line, which represents the strainrate sensitivity of yield stress. The two parallel lines at the bottom of Figure 13 imply that the strain-rate sensitivities of the 310 kg/m<sup>3</sup> and 410 kg/m<sup>3</sup> polymer foam materials are nearly equal. Both are lower than that of the 550 kg/m3 polymer foam material.

### 4. Conclusion

In this chapter, the characterization method of mechanical response of polymer materials under high strain-rate loading is firstly introduced. Then, two kinds of polymer materials, rubber-like and glass-like, are reviewed to illustrate their dynamic mechanical response. Herein, three polymers are presented as the representatives of each kind of polymer materials. The rate-dependent mechanical data are given and the influence of temperature is also clarified. These knowledge outputs not only guide the research of developing new impact-resistant polymer materials, but also support the protection engineering of applying polymer materials in dynamic events.
