2. Characterization technology

Due to the action of high strain-rate loading [6–10], dynamic mechanical response of materials is different from that under static loading. Characterizing dynamic mechanical behaviour is full of interest for material applications in impact events. For polymers, because of their low wave impedance, a longer time is needed to reach the dynamic stress equilibrium during split Hopkinson bar (SHB) test [11, 12], which produces more difficulties in high-rate mechanical experiments.

In order to reduce the difference in wave impedance, the viscoelastic impact bar is used in SHB test [13]. However, the viscoelastic bar itself has a strain-rate sensitivity, which easily results in the dispersion and attenuation of stress wave during the travelling in bar [14]. Aluminium alloy bar seems to be effective for testing polymer materials, thanks to the match of wave impedance and its lower strain-rate sensitivity. For meeting the conditions of constant strain rate and dynamic stress equilibrium, it is necessary to accurately design the shape of incident pulse [15]. A pulse-shaping technique is applied to meet these requirements, which, in general, places a wafer at the end of incident bar. The wafer material can be copper. The advantage of this pulse-shaping technique is to reduce the oscillation of incident wave. By selecting the appropriate specimen shape and size, the constant strain-rate loading with uniform stress in specimen can be realized [8, 16, 17]. Besides, low wave impedance material in SHB experiment will make the transmitter signal weak, which results in a small difference between the incident signal and reflected signal. The semiconductor strain gauge is more sensitive [18], which can be used to detect the weak signal. So, it can be used in the SHB experiment to test the soft polymer materials.

Moreover, it is necessary to clarify the calculating methods of stress, strain and strain rate of the tested specimen from the signals recorded by strain gages in an SHB test. They are based on the assumption of one-dimensional stress wave theory. Two long slender elastic bars in SHB apparatus are shown in Figure 1, which sandwich a short cylindrical specimen in between them.

When the striker bar impacts the end of the incident bar, a compressive stress wave is produced that immediately begins to travel along the bar towards the specimen. Upon arriving at the specimen, the compressive stress wave partially reflects back towards the impact end of the incident bar. The remainder of the compressive stress wave transmits through the specimen

Figure 1. Schematic representation of the elastic waves travelling in a split Hopkinson bar (SHB) apparatus [8].

and enters into the transmitter bar, resulting in the deformation of the tested specimen [8]. Herein, the acoustic impedance of the specimen should be lower than the bar's impedance. The reflected pulse is a tensional wave, whereas the transmitted pulse remains in compression. The stress wave-induced strain histories of the incident bar and transmitter bar are recorded by strain gages A and B, respectively.

As long as the stress in the two bars remains under their elastic limits, specimen stress, strain and strain rate may be calculated from the recorded strain histories of the incident bar and transmitter bar. Herein, two important strain histories are needed to be identified, which are the reflected wave and the transmitted wave through the specimen. Kolsky [7, 8] applied onedimensional stress wave theory to develop the following relation for calculating the specimen's engineering stress, σSð Þt :

$$
\sigma\_S(t) = E \frac{A\_0}{A\_S} \varepsilon\_T(t) \tag{1}
$$

in which E is the transmitter bar's elastic modulus; A<sup>0</sup> is the transmitter bar's cross-sectional area; AS is the specimen's cross-sectional area before loading and εTð Þt is the transmitted strain history.

The specimen strain rate, ε 0 <sup>S</sup>ð Þt can be calculated as:

Polymer materials have become a widely concerned research focus in extreme conditions because of their outstanding performances, such as impact resistance, rate dependency, corrosion resistance, low density, easy moulding and so on. Even more, in some engineering applications, they show outstanding advantages and replace the traditional metals and non-

Therefore, understanding the dynamic mechanical behaviour of polymers is necessary. Especially, the investigations of polymers in explosion, shock, collision and other related mechanical behaviour under impact loading have both theoretical and practical significances. In this chapter, the current research program of dynamic mechanical behaviour of various represen-

Due to the action of high strain-rate loading [6–10], dynamic mechanical response of materials is different from that under static loading. Characterizing dynamic mechanical behaviour is full of interest for material applications in impact events. For polymers, because of their low wave impedance, a longer time is needed to reach the dynamic stress equilibrium during split Hopkinson bar (SHB) test [11, 12], which produces more difficulties in high-rate mechanical

In order to reduce the difference in wave impedance, the viscoelastic impact bar is used in SHB test [13]. However, the viscoelastic bar itself has a strain-rate sensitivity, which easily results in the dispersion and attenuation of stress wave during the travelling in bar [14]. Aluminium alloy bar seems to be effective for testing polymer materials, thanks to the match of wave impedance and its lower strain-rate sensitivity. For meeting the conditions of constant strain rate and dynamic stress equilibrium, it is necessary to accurately design the shape of incident pulse [15]. A pulse-shaping technique is applied to meet these requirements, which, in general, places a wafer at the end of incident bar. The wafer material can be copper. The advantage of this pulse-shaping technique is to reduce the oscillation of incident wave. By selecting the appropriate specimen shape and size, the constant strain-rate loading with uniform stress in specimen can be realized [8, 16, 17]. Besides, low wave impedance material in SHB experiment will make the transmitter signal weak, which results in a small difference between the incident signal and reflected signal. The semiconductor strain gauge is more sensitive [18], which can be used to detect the weak signal. So, it can be used in the SHB experiment to test the soft

Moreover, it is necessary to clarify the calculating methods of stress, strain and strain rate of the tested specimen from the signals recorded by strain gages in an SHB test. They are based on the assumption of one-dimensional stress wave theory. Two long slender elastic bars in SHB apparatus are shown in Figure 1, which sandwich a short cylindrical specimen in between them.

When the striker bar impacts the end of the incident bar, a compressive stress wave is produced that immediately begins to travel along the bar towards the specimen. Upon arriving at the specimen, the compressive stress wave partially reflects back towards the impact end of the incident bar. The remainder of the compressive stress wave transmits through the specimen

metallic materials [1, 2].

194 Aspects of Polyurethanes

tative polymers is presented.

experiments.

polymer materials.

2. Characterization technology

$$d\varepsilon\_S(t) = \frac{d\varepsilon\_S(t)}{dt} = \frac{-2\mathcal{L}\_0}{L}\varepsilon\_R(t) \tag{2}$$

in which εRð Þt is the reflected strain history; L is the specimen length prior to impact and C<sup>0</sup> is the wave velocity in the incident bar. The reflected wave represents the strain-rate history in the specimen. So, the flat plateau in the reflected signal corresponds to a nearly constant rate of the specimen deformation during dynamic loading. The wave velocity in the incident bar, C0, can be calculated from the theory of elementary vibrations, as below:

$$\mathbf{C}\_0 = \sqrt{\mathbf{E}/\rho} \tag{3}$$

in which E and ρ are the incident bar's elastic modulus and density, respectively. Equation (2) can be integrated with time to attain the specimen's engineering strain history, εSð Þt , as below:

$$
\varepsilon\_S(t) = \frac{-2\mathcal{C}\_0}{L} \int\_0^t \varepsilon\_R(t) \tag{4}
$$

Thus, the data of specimen engineering stress, strain and strain rate can be derived from the recorded strain gage signals in an SHB experiment.

However, a uniform deformation of the specimen during SHB experiment should be maintained so that the experimental results can be clearly documented and interpreted for characterizing the dynamic stress-strain relation of a material.

Dynamic stress equilibrium is important for the validity of an SHB experiment. The stress in the specimen can be expressed in terms of the force exerted on each end surface of the specimen. When the specimen is sandwiched in between the incident bar and transmitter bar under dynamic compressive loading, the forces, F1ð Þt and F2ð Þt , are imposed on the specimen with a diameter of DS and a length of LS. The average force, Favg, applied on the specimen can be given as:

$$F\_{avg}(t) = \frac{F\_1(t) + F\_2(t)}{2} \tag{5}$$

Whereas, the average engineering stress on the tested cylindrical specimen is:

$$
\sigma\_{avg}(t) = \frac{F\_{avg}(t)}{\frac{1}{4}\pi D\_S^2} \tag{6}
$$

Herein, the forces, F1ð Þt and F2ð Þt applied on the specimen end surfaces are from the incident bar and transmitter bar, respectively. These two forces can be expressed in terms of elastic strains in the incident bar and transmitter bar, respectively, and can be calculated from the incident, reflected and transmitted signals, as below:

$$F\_1(t) = E[\varepsilon\_I(t) + \varepsilon\_R(t)]\frac{\pi D\_{bar}^2}{4} \tag{7}$$

$$F\_2(t) = E\varepsilon\_T(t)\frac{\pi D\_{bar}^2}{4} \tag{8}$$

in which Dbar is the diameter of the incident bar (and the transmitter bar) and εIð Þt , εRð Þt and εTð Þt are the strain histories of the incident, reflected and transmitted waves, respectively. For a specimen under high-speed loading, the dynamic stress equilibrium should be met for a constant strain rate, which means that F1ð Þt should be equal to F2ð Þt , as below:

$$F\_1(t) = F\_2(t) \tag{9}$$

However, in a real SHB experiment, the deviation exists between these two force histories. In order to analyse and quantify the deviation, a criterion for stress equilibrium is employed to compare the forces (or stresses) exerted on the specimen end surfaces. Ravichandran and Subhash [19] introduced a parameter, R tð Þ, to evaluate the proximity to stress equilibrium in the specimen, as below:

C<sup>0</sup> ¼

<sup>ε</sup>SðÞ¼ <sup>t</sup> �2C<sup>0</sup> L

recorded strain gage signals in an SHB experiment.

incident, reflected and transmitted signals, as below:

dynamic stress-strain relation of a material.

be given as:

196 Aspects of Polyurethanes

in which E and ρ are the incident bar's elastic modulus and density, respectively. Equation (2) can be integrated with time to attain the specimen's engineering strain history, εSð Þt , as below:

Thus, the data of specimen engineering stress, strain and strain rate can be derived from the

However, a uniform deformation of the specimen during SHB experiment should be maintained so that the experimental results can be clearly documented and interpreted for characterizing the

Dynamic stress equilibrium is important for the validity of an SHB experiment. The stress in the specimen can be expressed in terms of the force exerted on each end surface of the specimen. When the specimen is sandwiched in between the incident bar and transmitter bar under dynamic compressive loading, the forces, F1ð Þt and F2ð Þt , are imposed on the specimen with a diameter of DS and a length of LS. The average force, Favg, applied on the specimen can

F1ð Þþ t F2ð Þt

Favgð Þt 1

> πD<sup>2</sup> bar

πD<sup>2</sup> bar

FavgðÞ¼ t

σavgðÞ¼ t

F1ðÞ¼ t E½ � εIð Þþ t εRð Þt

F2ðÞ¼ t EεTð Þt

constant strain rate, which means that F1ð Þt should be equal to F2ð Þt , as below:

in which Dbar is the diameter of the incident bar (and the transmitter bar) and εIð Þt , εRð Þt and εTð Þt are the strain histories of the incident, reflected and transmitted waves, respectively. For a specimen under high-speed loading, the dynamic stress equilibrium should be met for a

Herein, the forces, F1ð Þt and F2ð Þt applied on the specimen end surfaces are from the incident bar and transmitter bar, respectively. These two forces can be expressed in terms of elastic strains in the incident bar and transmitter bar, respectively, and can be calculated from the

Whereas, the average engineering stress on the tested cylindrical specimen is:

ffiffiffiffiffiffiffiffi E=ρ q

ðt

εRð Þt ð4Þ

<sup>2</sup> <sup>ð</sup>5<sup>Þ</sup>

<sup>4</sup> <sup>π</sup>DS<sup>2</sup> <sup>ð</sup>6<sup>Þ</sup>

<sup>4</sup> <sup>ð</sup>7<sup>Þ</sup>

<sup>4</sup> <sup>ð</sup>8<sup>Þ</sup>

0

ð3Þ

$$R(t) = \left| \frac{\Delta \sigma(t)}{\sigma\_{\text{avg}}(t)} \right| = \left| \frac{\Delta F(t)}{F\_{\text{avg}}(t)} \right| = 2 \left| \frac{F\_1(t) - F\_2(t)}{F\_1(t) + F\_2(t)} \right| \tag{10}$$

in which, Δσð Þt and ΔF tð Þ are the differences of these two stresses and forces applied on the specimen end surfaces, respectively. σavgð Þt and Favgð Þt are the averages of these two stresses and forces, respectively (see Eqs. (5) and (6)). The specimen is assumed to be in dynamic stress equilibrium, when the value of R tð Þ is less than 0.05 [8]. This general criterion has been employed extensively to evaluate the dynamic stress equilibrium process in SHB experiments.

By substituting Eqs. (7) and (8) into Eq. (10), the parameter R tð Þ can be expressed by means of the relation between the incident, reflected and transmitted waves, as below:

$$R(t) = 2\left|\frac{F\_1(t) - F\_2(t)}{F\_1(t) + F\_2(t)}\right| = 2\left|\frac{\varepsilon\_I(t) + \varepsilon\_R(t) - \varepsilon\_T(t)}{\varepsilon\_I(t) + \varepsilon\_R(t) + \varepsilon\_T(t)}\right|\tag{11}$$

For a specimen under dynamic stress equilibrium, the forces applied on the specimen end surfaces can be expressed in terms of the two pressure bar strains, as expressed in Eqs. (7) and (8). Thus, according to Eq. (11), the following relation can be derived:

$$
\varepsilon\_I(t) + \varepsilon\_R(t) = \varepsilon\_T(t) \tag{12}
$$

Therefore, by applying an assumption of the positive travelling harmonic wave and the equation of motion, the expressions for the specimen's engineering stress, strain and strain rate can be derived in terms of the two pressure bars' strains, as shown in Eqs. (1), (4) and (2), respectively. In turn, under dynamic stress equilibrium, Eqs. (1), (4) and (2) are qualified to be used for calculating the specimen's engineering stress, strain and strain rate.

Therefore, the SHB technology for dynamic compression and tension experiments has been introduced. It is a mature method and a commonly used apparatus to characterize accurately the dynamic mechanical properties of various materials in the strain rates ranging from 1000/s to 10,000/s [20].

In this chapter, in order to characterize the high strain-rate mechanical response of polymer materials, the SHB technique is selected as the experimental equipment. On the one hand, the SHB equipment can accurately derive the dynamic mechanical properties of material under high-speed loading. On the other hand, it can be well controlled under laboratory working conditions [20].
