Effects of strain rate and elevated temperature on compressive flow stress and absorbed energy of polyimide foam

In this study, at first, the effect of strain rate on the strength and the absorbed energy of polyimide foam was experimentally examined by carrying out a series of compression tests at various strain rates, from 10−3 to 103 s−1. This polyimide foam has open cell structure with small cell size of 0.3 ∼ 0.6 mm. In the measurement of impact load, a special load cell with a small part for sensing load was adopted. For the measurement of the displacement, a high-speed camera was used. It was found that the flow stress of polyimide foam and the absorbed energy up to a strain of 0.4 increased with the increase of the strain rates. Secondly, the effect of ambient temperature on the strength and absorbed energy of polyimide foam was also investigated by using a sprit Hopkinson pressure bar apparatus and testing at elevated temperatures of 100 and 200 ◦C. With the increase of temperature, the strength and absorbed energy decreased and the effect is smaller in dynamic tests than static tests.


Introduction
As well known, polyethylene and polyurethane foams have generally been used as a shock absorber and an acoustic material in daily various situations.Therefore, there are many researches concerning the constitutive relations of polymer foams in compression [1][2][3], even in tension [4,5].These general foams are suitable and useful enough to absorb impact energy at room temperature.In the environment with an elevated temperature over 100 • C, however, they are not useful because of their lower glass transition temperature T G .The polymer foam with high T G , such as polyimide foam (T G ≥ 300 • C), has lately drawn considerable attention from the various fields, especially aerospace field.The mechanical properties of polyimide foams and polyimide foam filled honeycombs [6,7] were investigated and the recovery behaviour of rigid polymer foams, including polyimide foam, following long periods of compression was also examined at elevated temperatures [8].However, these researches were performed at quasi-static rate, not at dynamic rates.Therefore, it is quite meaningful to investigate the mechanical properties of polyimide foam under dynamic loading, because it is possibly used as a shock absorber at a higher temperature.
In this study, the effects of strain rate on the strength and the absorbed energy of polyimide foam were experimentally examined by carrying out a series of compression tests at various strain rates, from 10 −3 to 10 3 s −1 .The effect of ambient high temperature (100 and 200 • C) on the strength and the absorbed energy of polyimide foam was also investigated.

Materials and Specimen
Two kinds of foam made of a kind of polyimide provided by Industrial Summit Technology Co., Ltd.(I.S.T.) were used.They have two different foaming magnifica-tions of 10 and 5, denoted by Foam-A and Foam-B, respectively.The average densities of these foams were, respectively, ρ av = 128 and 254 kg/m 3 .The micro-structures of these forms are shown in figure 1.Typical open cells with fibrous structure are observed in the lighter foam (Foam-A), while filmy structure with many holes can be seen in the surface of Foam-B.
The specimens used are two types of rectangular parallelepiped specimen as shown in figure 2. Type-I specimen for quasi-static tests has a square cross section with a side of 14 mm and a height of 20 mm.This was also used for a series of dynamic tests at room temperature.Type-II has a square cross section with a side of 10 mm and 5.5 mm in height, which was only used for the tests at 20, 100 and 200 • C at dynamic rate.

Experimental methods
A series of quasi-static compression tests was carried out using Instron type universal testing machine at loading speed of 2 and 200 mm/min., which correspond to 1.7 × 10 −3 and 1.7 × 10 −1 s −1 , respectively.The load and the displacement of specimen were obtained from the output of the load cell and the displacement of crosshead, respectively.
The setup used for impact compression tests is shown in figure 3. A specimen located attached the flange was impacted by a striker which is accelerated by compressed air.The deformation of a specimen was measured by a high-speed video camera (Phantom V4.2) with 10,000 frames per second.The testing speed of about 20 m/s was attained, which was also obtained from the pictures of video camera.The applied load was measured using a special load cell [9], which has a small detective cylinder attached at one end of a stress-transmitted bar with large cross section.The striker rod has a cylindrical head of EPJ Web of Conferences    50 mm in dia., 300 mm in total length and about 0.5 kg.The output of the gauges glued on the detective cylinder surface was fed into a storage oscilloscope passing through a bridge box and amplifiers.
In order to measure the stress-strain relation of the foams at a medium loading speed like 1 m/s, a dropping weight (55 kg) testing apparatus was used as shown in  For the measurement of displacement, the highspeed video camera was also used.The dropping weight is hung by wire, which is cut for the tests.The initial height of weight, h = 60 mm, was adopted as the strain rate becomes around 50 s −1 .The load was measured by the load cell used in the impact tests.
For impact compression tests at elevated temperatures, 373 (100 • C) and 473 K (200 • C), the split Hopkinson bar apparatus as shown in figure 5 was used.The striker, input bar and output bar are made of SUS304 stainless steel rod with 16 mm in diameter.The specimen pre-heated up to the temperature over a desired temperature was quickly sandwiched between input and output bars.When the temperature of the specimen decreased to the desired temperature, impact compression test can be performed at a elevated temperature.To reduce the effect of heat from the specimen pre-heated, the location of strain gauges is as far as possible from the specimen.The ordinary relations to obtain the nominal stress, σ n , strain, ε n , and strain rate, , of the specimen in usual Hopkinson bar technique was used as follows: where, t is time and σ i , σ r and σ t are an incident pulse, a reflected pulse and a transmitted pulse propagating in the input and output bars.A, C and ρ are, respectively, the cross-sectional area, the velocity of elastic wave and the density of input and output bars.A S and L S are the cross-sectional area and the gauge length of specimen, respectively.
3 Experimental results and discussions

Average stress and density of specimen
In order to revise the σ − ε curves due to the difference of the density, we considered an average stress between strains of 0.2 and 0.3 as a representative stress value, σ av ,.
Here, we took two ratios: one is the ratio of the specimen density, ρ * (= ρ 1 /ρ 2 ), and the other is the ratio of average stress, σ * (= σ av1 /σ av2 ), where the subscripts of 1 and 2 indicate arbitrary two specimens.can say that the σ * is proportional to the n-th power of ρ * and independent on the strain rate.The relation between ρ * and σ * can be expressed by By using eq.( 4), all σ-ε curves of foam-A and foam-B were revised to have the density shown in catalogue, ρ cat = 120 kg/m 3 and ρ cat = 240 kg/m 3 , respectively.

Stress-strain curves of foam-A and -B
The stress-strain curves of the Foam-A & -B compressed at various strain rates from 1.7 × 10 −3 s −1 to 9.5 × 10 2 s −1 are shown in figures 7(a) and 7(b), respectively.The solid line indicates the nominal stress-strain curve at lowest strain rate of 1.7 × 10 −3 s −1 .The curve shows an initial linear elastic region, a relatively gentle increase of plastic flow stress and a region in which a little steeper increase of stress appears, although these three regions are not so clearly distinguished.Since the flow stress at a strain increases with the increase of stain rate, polyimide foam appears to have a positive strain-rate dependency.This may be caused by the strain-rate dependency of polyimide itself and the air failing to get out specimen due to high speed deformation.The level of the flow stress observed in the results of Foam-B is about sixth times greater than that of Foam-A, although the density of Foam-B is only twice as mentioned in the section 2.1.This means that the change of strength by foaming process is much greater than the change of density.

Effect of strain rate on strength and absorbed energy
In order to examine the strain-rate dependency of polyimide foams in detail, we plotted the flow stress at strains of 0.1, 0.2, 0.3 and 0.4 against the strain rate, as shown in Fig. 8.For the lighter foam, Foam-A, the clear strain-rate dependency only appears at the strain rate of over 10 s −1 , at least up to the strain of 0.3.For Foam-B, however, relatively large strain-rate dependency are observed in all strain rate range from 10 −3 ∼ 10 3 s −1 .The lines were drawn by applying the least-square method to these data, shown by an equation of where a and b are constants.The slope b representing the intensity of strain-rate dependency increases with the increase of plastic deformation.Thus, Foam-B with larger density shows greater strain-rate dependency than Foam-A, at least in small deformation.
Since the polymer foams are usually used for a shock absorber, it is important to examine their absorbed energy.The energy absorbed up to the strain of 0.4, U, was examined.The change of U with strain rate is shown in figure 9.The line of Foam-B was drawn by using the leastsquare method as used in figure 8.It is also found that the absorbed energy of Foam-B depends on the strain rate in the whole range of strain rate of 10 −3 ∼ 10 3 s −1 , although the strain rate dependency of Foam-A only appears  1.The flow stress during deformation and the absorbed energy up to the strain of 0.4 of Foam-B depend on the strain rate, i.e. they increase withe the increase of strain rate.While, the strain-rate dependency of flow stress and the absorbed energy of Foam-A only appear in the range of strain rate of over 10 s −1 .2. The flow stress and the absorbed energy up to the strain of 0.6 of the both foams clearly decrease with the increase of temperature.Although this tendency appears not only at static rate but also dynamic rate, the latter dependency of temperature is smaller than the former one.

Fig. 2 .
Fig. 2. Compressive specimens : (a) Type-I for quasi-static tests and (b) Type-II for dynamic tests at high temperatures.

Fig. 4 .
Fig. 4. Dropping weight machine for compression tests at medium loading rate.

Fig. 5 .
Fig. 5. Schematic of split Hopkinson pressure bar apparatus for impact compression test.

figure 4 .
figure4.For the measurement of displacement, the highspeed video camera was also used.The dropping weight is hung by wire, which is cut for the tests.The initial height of weight, h = 60 mm, was adopted as the strain rate becomes around 50 s −1 .The load was measured by the load cell used in the impact tests.For impact compression tests at elevated temperatures, 373 (100 • C) and 473 K (200 • C), the split Hopkinson bar apparatus as shown in figure5was used.The striker, input bar and output bar are made of SUS304 stainless steel rod with 16 mm in diameter.The specimen pre-heated up to the temperature over a desired temperature was quickly sandwiched between input and output bars.When the temperature of the specimen decreased to the desired temperature, impact compression test can be performed at a elevated temperature.To reduce the effect of heat from the specimen pre-heated, the location of strain gauges is as far as possible from the specimen.The ordinary relations to obtain the nominal stress, σ n , strain, ε n , and strain rate, , of the specimen in usual Hopkinson bar technique was used as follows:

2 DYMAT 2012 Fig. 6 .
Fig. 6.Relation between average stress ratio and density ratio in Foams-A and -B.

Fig. 7 .
Fig. 7. Stress-strain curves of (a) Foam-A and (b) Foam-B obtained at various strain rates.

Fig. 13 .
Fig. 13.Temperature dependency of absorbed energy up to = 0.6 of (a) Foam-A and (b) Foam-B.