- Email: [email protected]
Strain rate dependent compressive properties of glass microballoon epoxy syntactic foams
Materials Science and Engineering A 515 (2009) 19–25
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Strain rate dependent compressive properties of glass microballoon epoxy syntactic foams P. Li a,∗ , N. Petrinic a , C.R. Siviour a , R. Froud a , J.M. Reed b a b
Solid Mechanics and Materials Engineering Group, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK Rolls-Royce plc, Derby, UK
a r t i c l e
i n f o
Article history: Received 22 October 2008 Received in revised form 6 February 2009 Accepted 10 February 2009 Keywords: Syntactic foams Compressive properties Strain rate sensitivity Failure Compressibility
a b s t r a c t Lightweight glass microballoon epoxy syntactic foams have a high strength-to-weight ratio, making them attractive for transport applications. A better understanding of the compressive properties of such foams is required to improve predictive modelling tools and develop novel formulations. In this study, the response of a foam to compressive loading was experimentally investigated over strain rates from 0.001 to 4000 s−1 . The stress–strain response, deformation/damage history and volume change were examined quantitatively and/or qualitatively; all of these parameters exhibit strain rate sensitivity. Combined finite element stress analysis and microscopic observations reveal that heterogeneous (localised) damage arises in the foam due to the coexistence of two failure modes: (i) crushing of glass microballoons dominating in the central part and (ii) shear cracking of the epoxy matrix that forms and propagates from the corners. As the strength of the epoxy matrix increases with increasing strain rate, cracking of glass microballoons begins to dominate over the matrix/microballoon debonding, resulting in macroscopic strain rate dependency of compressive properties. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Glass microballoon epoxy syntactic foams have been increasingly used in automotive and aerospace applications due to their high strength-to-weight ratio, good impact resistance and excellent energy dissipation capacity. In order to simulate the service performance of syntactic foam components, a predictive tool requires accurate mechanical properties as input data. The mechanical behaviour of syntactic foams is intrinsically determined by basic material properties (e.g. chemical composition of matrix materials) and the unit cell structure (e.g. geometry of glass microballoons and their distribution) [1–6]. Extrinsic factors such as strain rate also significantly influence the mechanical response of cellular materials [2,3,5,7–9]. Therefore, a better understanding of compressive properties of glass microballoon epoxy syntactic foams at a wide range of loading rates is required to improve the predictive model and develop novel formulations in the future. Most studies of the compressive properties of metallic or polymeric syntactic foams have focused on quasi-static conditions (e.g. at a strain rate of 0.001 s−1 ) [1,6,10–12]. These reports characterise the stress–strain response, damage evolution, failure features and energy dissipation capacity, and extensively explore the effect of
∗ Corresponding author. Tel.: +44 1865 283496; fax: +44 1865 273906. E-mail addresses: [email protected], [email protected] (P. Li). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.02.015
intrinsic factors such as matrix material properties [1], geometry and volume fraction of microballoons, which in turn govern the relative density of the foam [6,12]. However, there is a paucity of studies in the literature focusing on the dynamic properties of glass microballoon epoxy syntactic foams, despite their applications at high loading rates [5,13,14]. These studies have demonstrated the strain rate dependency of the peak strength of syntactic foams [5,13], but none of them went to large enough strain to examine the energy dissipation capacity under deformations at different loading rates, an important characteristic of foams when used for packaging and protection. In addition, the strain rates available in these investigations are typically too sparse to develop a more complete constitutive equation necessary for structural analysis of components made of syntactic foams. To predict accurately the safety and service behaviour of a structural component also requires characterisation of volume (or density) history during compressive deformation [15–18]. The volume change in the plastic stage is an important parameter to quantify the constitutive response of cellular materials. However, this is not an area that has been addressed for the materials of interest here. The aim of the study reported in the current paper is to investigate the strain rate dependent compressive behaviour of glass microballoon epoxy syntactic foams, including stress–strain response, deformation history, failure mode and energy dissipation. The effect of strain rate on compressive properties was examined at
20
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
a number of strain rates ranging from 0.001 to 4000 s−1 . An empirical constitutive equation was developed to quantitatively describe the strain rate dependency of peak and plateau stresses, two key features to determine the fracture resistance and energy dissipation capacity of syntactic foams, respectively. The volume change history during compression was also quantified as a function of strain rate so as to provide information for more accurate structural impact simulation of syntactic foam components. The deformation/failure mechanisms involved during crushing of the foam were clarified using finite element stress analysis and microscopic observations, thus explaining the macroscopic stress–strain behaviour. 2. Experimental methods 2.1. Materials The syntactic foam was fabricated by mechanically mixing glass microballoons (thin-walled closed pore material) and epoxy resin pastes (matrix system) with very slow stir speed to minimise the damage of microballoons. Syntactic foam slabs were then cast, after which the resin was cured. The microstructure of the syntactic foam is almost homogeneous; a typical scanning electron micrograph is shown in Fig. 1. The entrapped air porosity was minimal. The glass microballoons, with an individual diameter of ∼100 m, a bulk density of ∼300 kg/m3 and a bulk crushing strength of ∼30 MPa, were evenly distributed in the foam with a volume fraction of approximately 70%. The relative density (*) of the foam can thus be estimated to be ∼30%. Good contact was observed between the epoxy matrix and the surface of the microballoons. 2.2. Mechanical testing Cylindrical specimens of diameter d = 5.0 mm and length l = 5.0 mm were cut from the syntactic foam slab. The linear dimension of the specimens, at ∼50 times the cell size (microballoons: ∼100 m in diameter), was large enough to ensure that the specimen behaviour is representative of bulk material. For cellular polymeric materials with high volume fraction of porosity (e.g. >60%), specimens with aspect ratio (l/d) of up to 1.0 should be used to avoid the formation of powdery fragments near cracks and thus achieve uniform compression [5,10]. However, a larger aspect ratio specimen can reduce friction and subsequent barrelling effect in compression tests. The friction coefficient between epoxy materials and steel platens could be as high as 0.1–0.25 [19,20]. Therefore, an aspect ratio of l/d = 1.0 was selected for the specimens in this investigation.
Fig. 2. Stress–strain curve at different loading rates. (Note: two representative test repeats are shown although more tests were performed for each loading rate; only three rates are shown in this figure, representing the full range of qualitative behaviour; stress levels at other rates are shown in Fig. 6.)
To achieve a wide range of strain rates, compression tests were conducted using (i) a commercial screw driven loading machine at quasi-static (low) loading rates (0.001–0.01 s−1 ), (ii) a hydraulic loading machine at medium loading rates (0.5–200 s−1 ), and (iii) Split-Hopkinson pressure configuration using Ø15 mm × 500 mm solid cylindrical steel bars at high loading rates (1000–4000 s−1 ). The compressive deformation of a specimen was captured using various imaging systems with framing rates ranging from 0.25 to 250,000 frames s−1 . Full details of the test procedure can be found in a previous paper [21]. To minimise interfacial friction all specimens were lubricated with CastrolTM LMX grease,1 which is widely used in high speed applications. At least three specimens were tested for each strain rate, allowing evaluation of the test reproducibility. The strain rate for each test was maintained approximately constant by the prescribed velocity boundary conditions. The nominal stress and strain were calculated from the measured force and displacement (see Fig. 2). Each stress–strain curve was plotted to the end of loading at an approximately constant strain rate, even though densification of the foam is expected to occur at a strain εd = ∼0.6 according to Gibson and Ashby [22]: εd = 1 − 1.4∗,
(1)
where the relative density, *, is ∼30%. Elastic properties are not discussed in this study because it is well established that (i) for quasi-static experiments, elastic properties would typically be measured by using a specimen with a larger aspect ratio, e.g. l/d > 2.0 and (ii) high strain rate experiments do not produce an accurate representation of elastic modulus [21]. 3. Results 3.1. Quasi-static compressive behaviour Representative nominal stress–strain curves of the syntactic foam at quasi-static rates of strain (0.001 s−1 ) are shown in Fig. 2. A good reproducibility of better than 5% in stress was achieved, indicative of consistent cell structure (glass microballoon distribution), specimen geometry and testing conditions between specimens. For clarity, Fig. 2 only shows a representative selection of stress–strain curves. A full set of data at all rates is given in Fig. 6. An initial, approximately linear, region corresponding to the elastic behaviour of the foam is observed in the stress–strain curves
Fig. 1. Typical microstructure of glass microballoon epoxy syntactic foams.
1
CastrolTM is a registered trademark of Castrol Ltd., Swindon, UK.
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
21
Fig. 3. Deformation sequence in compression tests at loading rates of (a) 0.001 s−1 and (b) 3500 s−1 . (Note: the interface between bars/platens and specimens is covered by the lubricants squeezed out; the solid line boxes A and B indicate the plastic and elastic deformation zones in the foam, respectively; the dashed line C represents the shear band inside the foam.)
until the peak stress (∼15 MPa) is reached (Fig. 2). A drop in stress then occurs, implying the onset of glass microballoon crushing. Due to the triaxiality of local stress states towards the corners of the specimen, discussed below, shear cracks form and then propagate in the epoxy matrix or along the interface between the matrix and microballoons. Therefore, the peak stress represents the fracture resistance of the syntactic foam. A nearly constant plateau region following the stress drop (Fig. 2) is due to progressive collapse of unit cells comprising a microballoon and its surrounding epoxy resin. The plateau region corresponds to the energy dissipation during compression. The plateau stress (∼13 MPa) defined by the average stress in the plateau region, indicates the capacity of energy dissipation of the foam. Shear cracking continues in the matrix; meanwhile crushing of glass microballoons exposes hollow space, which is consumed by the epoxy matrix material in compression. When a significant portion of microballoons is crushed, further loading causes densification of the foam. Fig. 3(a) illustrates the recorded deformation series of the syntactic foam in quasi-static compression. Visible barrelling occurred in the specimens. Although the hollow space on the end surfaces of the foam specimens can serve as a groove to hold lubricants in the initial stage of the experiment, lubricants were squeezed out during compression, causing significant interfacial restraining effect. The observation of barrelling during deformation suggests the nonuniformity of the internal deformation pattern due to friction at the contact surfaces, which then causes intense shear deformation arising along the ∼45◦ direction from the corners in a longitudinal section through the centre of the specimen. This shear stress will cause formation and growth of shear cracks, and thus non-uniform damage of the foam.
3.2. Dynamic compressive behaviour Fig. 2 also shows the nominal stress–strain curves of the syntactic foam at high strain rates (3500 s−1 ). Similar to quasi-static compression tests, Split-Hopkinson bar tests can be repeated with a deviation of <5%. The shape of the high rate stress–strain curve is similar to that at quasi-static rates. However, the peak stress increases with strain rate, e.g. ∼43 MPa versus ∼15 MPa at rates of 3500 and 0.001 s−1 respectively. This implies that the syntactic foam has a higher resistance to fracture under dynamic loading. The sharper stress drop observed at the high loading rate suggests that it is more difficult for cracks to initiate in the epoxy matrix at these rates than it was in the quasi-static experiments. However, relative to the initiation of shear cracks, their propagation is easier at high strain rates. Note that in absolute terms cracks do not propagate as easily as in quasi-static conditions, probably due to the different crack propagation modes at different loading rates as will be discussed below. Glass microballoons also start to crush after the peak stress is reached. Compared to low loading rates, a higher plateau stress (∼36 MPa) is observed at high strain rates, demonstrating enhanced energy dissipation capacity of the foam when subjected to dynamic loading. In high rate tests, two shear bands (labelled as C in Fig. 3(b)) evolve along the diagonal (40–60◦ ) towards the inside of the syntactic foam. Shear bands become more noticeable after the stress drops at a strain of ∼0.06. Compared to the extensive barrelling in low rate tests, catastrophic cracking occurring along the shear bands at high strain rates can contribute to a more considerable drop of stress. In all the high loading rate tests, the two shear bands can occur along the diagonal toward either the loading direction or
22
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
Fig. 4. (a) Hydrostatic and (b) maximum shear stress distribution in the longitudinal section through the centre of the specimen at a strain of 0.1; and (c) schematic of the longitudinal section illustrating the crushing of glass microballoons.
in the opposite direction (e.g. C in Fig. 3(b)). The deformation rates are different in the two sides of the shear bands. It seems that zone A deforms plastically whereas zone B elastically as shown in Fig. 3(b), while such separation is not observed in quasi-static compression tests (Fig. 3(a)). Due to the fracture of glass microballoons, compressing the syntactic foam consumes the space of the microballoons and thus increases the overall density of the foam. Less barrelling is observed at higher strain rate (compare Fig. 3(a) and (b)), which suggests that for a given longitudinal strain, the syntactic foam expands less and densifies more compared to quasi-static conditions. Therefore, the loading rate influences the volume change during plastic deformation.
By taking the strain of densification (εd ) as equal to ∼0.6, the energy dissipation capacity can therefore be estimated as ∼8 and ∼22 MJ/m3 for quasi-static (0.001 s−1 ) and dynamic (3500 s−1 ) strain rates, respectively. The syntactic foam does not ideally collapse during compression; it expands laterally. The volume change of the foam is observed to be dependent on the loading rate; in particular, during dynamic compression, both plastic and elastic zones exist in the foam. The effect of strain rate on peak and plateau stresses and volume change needs to be considered when the syntactic foam is used in applications that experience different loading rates.
3.3. Summary: rate dependent compressive properties
4.1. Failure mode
To facilitate structural modelling and optimise the design of both foam materials and components, compressive properties of glass microballoon epoxy syntactic foams have been characterised. The peak stress is one key property indicative of the fracture resistance of the foam. Damage, including crushing of microballoons and shear cracking, evolves after the peak stress is reached. The plateau stress ( pl ), another important feature of the foam, is calculated as the average stress in the plateau region. A linear relationship links pl and the energy dissipation capacity per unit initial volume, W:
The differences in macroscopic compressive properties between quasi-static and dynamic loading rates can be attributed to the different damage/fracture process in the glass microballoon epoxy syntactic foam (Figs. 2 and 3). During compression, the following two failure processes coexist in the foam: (i) crushing of glass microballoons and (ii) shear cracking of the epoxy matrix. However, the two failure modes arise and dominate at different locations in a specimen, causing heterogeneous (localised) damage of the foam. Such localised damage has been observed in aluminium [2] and epoxy [1,10] syntactic foams. Damage also occurs at the interface of the two phases (glass microballoons and epoxy matrix) in the foam. In low rate compression, a non-uniform stress distribution is expected according to the barrelling phenomena observed (Fig. 3(a)). An axisymmetric finite element model was established to
εd
W=
dε,
(2)
0
pl ≈
W . εd
(3)
4. Discussion
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
23
Fig. 5. A crack propagation near location L1 at (a) quasi-static and (b) dynamic (∼1500 s−1 ) strain rates. (After Woldesenbet et al. [5].)
predict the stress distribution. A crushable foam plasticity material formula [15] in ABAQUSTM2 was calibrated to data obtained from the experiments. The interfacial contact between the steel anvils and the epoxy foam specimen was defined by a frictional coefficient of 0.1 estimated from the literature [19,20]. Both the boundary and loading conditions were applied based on those observed in the real experiments. Fig. 4(a) and (b) illustrates the distributions of hydrostatic and deviatoric (shear) components of the stress field in the specimen at a strain of 0.1. The hydrostatic stress is responsible for compressive damage while the shear stress gives rise to shear cracking in the syntactic foam. The non-uniform hydrostatic/shear distribution can be associated with the localised damage in the syntactic foam as follows: 1. At the corners of the foam, e.g. L1 in Fig. 4(c), the maximum shear stress arises while the compressive stress is low. Shear type failure rather than compression damage therefore dominates in this region. The epoxy matrix is cracked under the shear stress; banded structure appearing in a stepwise fashion has been observed in similar epoxy syntactic foams by Gupta et al. [11]. Shear cracks propagate at a 40–60◦ angle to the loading axis. Most of the glass microballoons are uncrushed, causing only a small amount of debris. 2. In the central part of the foam (L3 in Fig. 4(c)), substantial compression damage prevails until the shear cracks originating in L1 grow to a considerable length and cause final failure of the specimen. Glass microballoons are completely crushed; and large amount of debris can be seen [1,11]. 3. A mixture of hydrostatic compression and shear damage occurs in the region between shear (L1 ) and compression (L3 ) zones (see L2 in Fig. 4(c)). A few uncrushed glass microballoons appear among the broken ones; shear forces drive the interfacial separation of microballoons from the epoxy.
syntactic foam at quasi-static and dynamic loading rates (see Fig. 5). In quasi-static compression, the crack tends to break the matrix and then pass around the glass microballoons or debond the interface between microballoons and the epoxy matrix. However, at high loading rates, the crack fractures microballoons and propagates through them, thus consuming a larger amount of energy compared to quasi-static compression. Equivalent variations in cracking mode have been observed in other two-phase materials, such as PBXs at different temperatures [23]. 4.2. Strain rate sensitivity In order to quantify the strain rate effect on compressive properties of the syntactic foam, additional tests were performed at different loading rates using the facilities described above. A wide range of strain rates with close intervals is essential to accurately quantify the rate sensitivity (Fig. 6). Both peak and plateau stresses increase with strain rate up to 4000 s−1 (see Figs. 2 and 6). An empirical constitutive model was developed to represent the relation between peak/plateau stresses () and strain ˙ rate (ε): ˙ , = 0 (1 + C log ε)
where 0 and C are material constants. A very good correlation (R2 > 94%) was achieved for both peak and plateau stresses (see Table 1). The selection of this empirical equation was justified because: (i) a constitutive model in the form of Eq. (4), widely
At high strain rates, the stress triaxiality in the syntactic foam also leads to formation of shear cracks along the direction of the maximum shear stress (40–60◦ to the loading axis). However, in this case, the increased strength of the rate sensitive epoxy, compared to the rate insensitive glass microballoons, means that the load transfer between the two components of the foam is different from that in the quasi-static experiments [2]. At high rates, cracking of glass microballoons begins to dominate over the matrix/microballoon debonding. Woldesenbet et al. [5] illustrated the difference between the crack propagation mechanisms in epoxy
2
ABAQUSTM is a registered trademark of ABAQUS Inc., RI, USA.
(4)
Fig. 6. Peak and plateau stresses as a function of strain rate.
24
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
Table 1 Estimated parameters with 95% confidence bounds for peak and plateau stresses. Strain rate range (s−1 ) Peak stress Plateau stress
0.001–4000 0.001–4000
0 (MPa)
C (s−1 )
R2
28.42 ± 1.19 23.27 ± 0.62
0.0702 ± 0.0093 0.0651 ± 0.0058
0.94 0.97
used in dynamic problems, has been proved a simple and effective way to introduce strain rate sensitivity in numerical and analytical calculations, particularly for polymeric materials [24]; (ii) for the syntactic foam studied, the stress–strain curve maintains a nearly similar shape over a wide range of strain rates (see Fig. 2); and (iii) experimentally, both peak and plateau stresses show an almost linear dependency on the logarithm of strain rate (Fig. 6). From a materials selection perspective, it is important to know the actual strain rate sensitivity for a specific application. The fitted parameters listed in Table 1 are applicable for strain rates ranging from 0.001 to 4000 s−1 (approximately 20 m s−1 , a typical automotive speed). For higher loading rate applications, e.g. 150–200 m s−1 in aerospace, further work is required to generate a larger set of stress–strain data. The strain rate dependency of syntactic foams can be attributed to a number of microscopic factors as follows: 1. The rate dependency of the foam is significantly determined by the rate sensitivity of the matrix material [2,25]. Viscoelasticity of the epoxy is likely to cause the strain rate dependency of the matrix and therefore of the foam. 2. As discussed previously, the shear crack propagates through glass microballoons at dynamic loading rates instead of bypassing or debonding them as happens in quasi-static deformation. Such a rate dependent damage mechanism will result in a stronger fracture resistance (peak stress) and better energy dissipation capacity (plateau stress) of the foam at high strain rates. 3. The inability of gas to escape from the inside after glass microballoons are crushed is likely to contribute to the increased energy dissipation of the epoxy (non-metallic) closed-cell syntactic foam at high strain rates. The gas pressure increases adiabatically in dynamic tests; more energy is required to compress the foam compared to a low loading rate test (a nearly isothermal process). 4. In low density (high porosity volume fraction) foams loaded at high rates of strain, there is commonly an inertia effect manifested in the initiation of higher order buckling modes of the characteristically slender meso-structure (open-cell framework and closed-cell membranes) before extensive deformation and fracture. However, these effects are not expected in this study because the relatively low porosity volume fraction and the hybrid nature of meso-structure of the given material were not susceptible to buckling. The relatively high proportion of polymeric resin and relatively high thickness-to-diameter ratio of the glass microballoons did not allow for the inertia effects to contribute to the rate dependent response of specimens loaded at relatively low boundary velocities (<25 m s−1 ). 4.3. Quantification of volume change The volume of the glass microballoon epoxy syntactic foam changes during the plastic stage of compression; the volume change at a specific strain is influenced by strain rate (Fig. 3). Quantification of the volume history is required to develop a constitutive dynamic model, which can be used to simulate the structural impact resistance of bulk components made from such foams [15,18]. Images of the deformation sequence at different loading rates were used to calculate the volume history (V) of a specimen during compression,
Fig. 7. The relation between the volumetric strain and nominal strain in quasistatic and dynamic compression tests. (Note: the linear trendlines are fitted on the measured data.)
with the assumption that the specimen maintains a longitudinally axisymmetric shape. The volume change was averaged over the whole specimen because shear bands occurring inside a specimen make it impossible to calculate the real 3D volume from 2D images. The volumetric strain (εv ) was therefore calculated using the measured longitudinal area (A and A0 ) and length (L and L0 ) of the specimen in a 2D image. (Note: the subscript 0 symbolises the original dimension.) εv = 1 −
V A2 L0 . =1− V0 A0 2 L
(5)
The volumetric strain is plotted against the nominal strain up to 0.5 in Fig. 7. The syntactic foam becomes denser as the specimen is plastically compressed. The volumetric strain is almost directly proportional to the nominal strain (ε): εv = kε,
(6)
where k is a constant. The syntactic foam does not behave as an incompressible material such as metals (k = 0), nor as an ideal collapsed material with the specimen diameter maintaining constant (k = 1). The volume change increases with strain rate, e.g. k = 0.54 ± 0.01 versus 0.61 ± 0.03 (correlation R2 = 99%) for quasistatic and dynamic compression, respectively (Fig. 7). The two factors likely to contribute to the rate dependency of volume change by preventing lateral expansion in high rate compression are: (i) time-dependent properties of the epoxy matrix material (e.g. viscoelasticity and viscoplasticity) and (ii) changes in the collapse/damage mechanism at higher strain rates. 5. Conclusions The compressive properties of glass microballoon epoxy syntactic foams were experimentally characterised over a wide range of strain rates (0.001–4000 s−1 ). The underlying deformation and failure mechanisms were investigated using combined finite element stress analysis and microscopic examination. Strain rate effects on the strengths of the two components in the syntactic foam, in particular the epoxy, cause non-uniform deformation and heterogeneous (localised) damage. Two failure modes dominate in different regions of the foam: (i) crushing of glass microballoons in the central part due to substantial
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
hydrostatic stress and (ii) shear cracking of the epoxy matrix, which forms and propagates from the corners where the maximum shear stress occurs. As the epoxy matrix gets stronger at higher rates, cracking of glass microballoons begins to dominate over the matrix/microballoon debonding under dynamic loading, requiring more energy to be absorbed in the foam. Shear cracks propagate through glass microballoons at high strain rates instead of bypassing or debonding them as happens in quasi-static testing. The underlying deformation and failure mechanisms lead to the macroscopic rate dependent compressive response of the syntactic foam. Both peak and plateau stresses, indicative of fracture resistance and energy dissipation capacity of the foam, respectively, increase almost linearly with the logarithm of strain rate. Unlike quasi-static compression, separate plastic and elastic deformation zones, under dynamic loading, arise on the two sides of shear bands along which final catastrophic cracking occurs. The quantified volume history indicates that volumetric strain is almost directly proportional to nominal strain and increases with strain rate. Quantification of both rate dependent stress–strain relation and volume change will improve the accurate prediction of the service behaviour of bulk components made from such foams. Furthermore, from the viewpoint of materials selection, it is essential to know the actual strain rate dependency for a specific application. Acknowledgements The authors acknowledge the colleagues at University of Oxford for their help during the experiments. The Cordin 550 high speed camera used in this research was provided by the EPSRC instrument loan pool. The authors are particularly grateful to A. Walker for his advice and support while using this camera.
25
References [1] J. Adrien, E. Maire, N. Gimenez, V. Sauvant-Moynot, Acta Mater. 55 (2007) 1667–1679. [2] D.K. Balch, J.G. O’Dwyer, G.R. Davis, C.M. Cady, G.T. Gray, D.C. Dunand, Mater. Sci. Eng. A 391 (2005) 408–417. [3] I.W. Hall, M. Guden, C.J. Yu, Scripta Mater. 43 (2000) 515–521. [4] K.Y.G. McCullough, N.A. Fleck, M.F. Ashby, Acta Mater. 47 (1999) 2323– 2330. [5] E. Woldesenbet, N. Gupta, A. Jadhav, J. Mater. Sci. 40 (2005) 4009–4017. [6] G.H. Wu, Z.Y. Dou, D.L. Sun, L.T. Jiang, B.S. Ding, B.F. He, Scripta Mater. 56 (2007) 221–224. [7] K. Kitazono, Y. Takiguchi, Scripta Mater. 55 (2006) 501–504. [8] T. Mukai, H. Kanahashi, T. Miyoshi, M. Mabuchi, T.G. Nieh, K. Higashi, Scripta Mater. 40 (1999) 921–927. [9] B. Song, W. Chen, D.J. Frew, J. Compos. Mater. 38 (2004) 915–936. [10] N. Gupta, Kishore, E. Woldesenbet, S. Sankaran, J. Mater. Sci. 36 (2001) 4485–4491. [11] N. Gupta, E. Woldesenbet, Kishore, J. Mater. Sci. 37 (2002) 3199–3209. [12] M. Koopman, K.K. Chawla, K.B. Carlisle, G.M. Gladysz, J. Mater. Sci. 41 (2006) 4009–4014. [13] B. Song, W.W. Chen, W.Y. Lu, Int. J. Mech. Sci. 49 (2007) 1336–1343. [14] H. Zhao, G. Gary, Int. J. Veh. Des. 30 (2002) 135–145. [15] V.S. Deshpande, N.A. Fleck, J. Mech. Phys. Solids 48 (2000) 1253–1283. [16] C.L. Martin, D. Bouvard, Acta Mater. 51 (2003) 373–386. [17] C.D. Turner, M.F. Ashby, Acta Mater. 44 (1996) 4521–4530. [18] J. Zhang, N. Kikuchi, V. Li, A. Yee, G. Nusholtz, Int. J. Impact Eng. 21 (1998) 369–386. [19] R. Bassani, G. Levita, M. Meozzi, G. Palla, Wear 247 (2001) 125–132. [20] J. Wu, X.H. Cheng, Wear 261 (2006) 1293–1297. [21] P. Li, C.R. Siviour, N. Petrinic, Exp. Mech., in press, doi:10.1007/s11340-008-91291. [22] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, 1999. [23] D.M. Williamson, S.J. Palmer, W.G. Proud, R. Govier, Brazilian disc testing of a UK PBX above and below the glass transition temperature, in: M. Elert, M.D. Furnish, R. Chau, N.C. Holmes, J. Nguyen (Eds.), Proceedings of the Conference of the American Physical Society Topical Group on Shock Compression of Condensed Matter, AIP Conference Proceedings, vol. 955, Hawaii, 2007, pp. 803– 806. [24] N. Jones, Structural Impact, Cambridge University Press, Cambridge, 1998. [25] Z.Y. Dou, L.T. Jiang, G.H. Wu, Q. Zhang, Z.Y. Xiu, G.Q. Chen, Scripta Mater. 57 (2007) 945–948.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Strain rate dependent compressive properties of glass microballoon epoxy syntactic foams P. Li a,∗ , N. Petrinic a , C.R. Siviour a , R. Froud a , J.M. Reed b a b
Solid Mechanics and Materials Engineering Group, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK Rolls-Royce plc, Derby, UK
a r t i c l e
i n f o
Article history: Received 22 October 2008 Received in revised form 6 February 2009 Accepted 10 February 2009 Keywords: Syntactic foams Compressive properties Strain rate sensitivity Failure Compressibility
a b s t r a c t Lightweight glass microballoon epoxy syntactic foams have a high strength-to-weight ratio, making them attractive for transport applications. A better understanding of the compressive properties of such foams is required to improve predictive modelling tools and develop novel formulations. In this study, the response of a foam to compressive loading was experimentally investigated over strain rates from 0.001 to 4000 s−1 . The stress–strain response, deformation/damage history and volume change were examined quantitatively and/or qualitatively; all of these parameters exhibit strain rate sensitivity. Combined finite element stress analysis and microscopic observations reveal that heterogeneous (localised) damage arises in the foam due to the coexistence of two failure modes: (i) crushing of glass microballoons dominating in the central part and (ii) shear cracking of the epoxy matrix that forms and propagates from the corners. As the strength of the epoxy matrix increases with increasing strain rate, cracking of glass microballoons begins to dominate over the matrix/microballoon debonding, resulting in macroscopic strain rate dependency of compressive properties. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Glass microballoon epoxy syntactic foams have been increasingly used in automotive and aerospace applications due to their high strength-to-weight ratio, good impact resistance and excellent energy dissipation capacity. In order to simulate the service performance of syntactic foam components, a predictive tool requires accurate mechanical properties as input data. The mechanical behaviour of syntactic foams is intrinsically determined by basic material properties (e.g. chemical composition of matrix materials) and the unit cell structure (e.g. geometry of glass microballoons and their distribution) [1–6]. Extrinsic factors such as strain rate also significantly influence the mechanical response of cellular materials [2,3,5,7–9]. Therefore, a better understanding of compressive properties of glass microballoon epoxy syntactic foams at a wide range of loading rates is required to improve the predictive model and develop novel formulations in the future. Most studies of the compressive properties of metallic or polymeric syntactic foams have focused on quasi-static conditions (e.g. at a strain rate of 0.001 s−1 ) [1,6,10–12]. These reports characterise the stress–strain response, damage evolution, failure features and energy dissipation capacity, and extensively explore the effect of
∗ Corresponding author. Tel.: +44 1865 283496; fax: +44 1865 273906. E-mail addresses: [email protected], [email protected] (P. Li). 0921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2009.02.015
intrinsic factors such as matrix material properties [1], geometry and volume fraction of microballoons, which in turn govern the relative density of the foam [6,12]. However, there is a paucity of studies in the literature focusing on the dynamic properties of glass microballoon epoxy syntactic foams, despite their applications at high loading rates [5,13,14]. These studies have demonstrated the strain rate dependency of the peak strength of syntactic foams [5,13], but none of them went to large enough strain to examine the energy dissipation capacity under deformations at different loading rates, an important characteristic of foams when used for packaging and protection. In addition, the strain rates available in these investigations are typically too sparse to develop a more complete constitutive equation necessary for structural analysis of components made of syntactic foams. To predict accurately the safety and service behaviour of a structural component also requires characterisation of volume (or density) history during compressive deformation [15–18]. The volume change in the plastic stage is an important parameter to quantify the constitutive response of cellular materials. However, this is not an area that has been addressed for the materials of interest here. The aim of the study reported in the current paper is to investigate the strain rate dependent compressive behaviour of glass microballoon epoxy syntactic foams, including stress–strain response, deformation history, failure mode and energy dissipation. The effect of strain rate on compressive properties was examined at
20
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
a number of strain rates ranging from 0.001 to 4000 s−1 . An empirical constitutive equation was developed to quantitatively describe the strain rate dependency of peak and plateau stresses, two key features to determine the fracture resistance and energy dissipation capacity of syntactic foams, respectively. The volume change history during compression was also quantified as a function of strain rate so as to provide information for more accurate structural impact simulation of syntactic foam components. The deformation/failure mechanisms involved during crushing of the foam were clarified using finite element stress analysis and microscopic observations, thus explaining the macroscopic stress–strain behaviour. 2. Experimental methods 2.1. Materials The syntactic foam was fabricated by mechanically mixing glass microballoons (thin-walled closed pore material) and epoxy resin pastes (matrix system) with very slow stir speed to minimise the damage of microballoons. Syntactic foam slabs were then cast, after which the resin was cured. The microstructure of the syntactic foam is almost homogeneous; a typical scanning electron micrograph is shown in Fig. 1. The entrapped air porosity was minimal. The glass microballoons, with an individual diameter of ∼100 m, a bulk density of ∼300 kg/m3 and a bulk crushing strength of ∼30 MPa, were evenly distributed in the foam with a volume fraction of approximately 70%. The relative density (*) of the foam can thus be estimated to be ∼30%. Good contact was observed between the epoxy matrix and the surface of the microballoons. 2.2. Mechanical testing Cylindrical specimens of diameter d = 5.0 mm and length l = 5.0 mm were cut from the syntactic foam slab. The linear dimension of the specimens, at ∼50 times the cell size (microballoons: ∼100 m in diameter), was large enough to ensure that the specimen behaviour is representative of bulk material. For cellular polymeric materials with high volume fraction of porosity (e.g. >60%), specimens with aspect ratio (l/d) of up to 1.0 should be used to avoid the formation of powdery fragments near cracks and thus achieve uniform compression [5,10]. However, a larger aspect ratio specimen can reduce friction and subsequent barrelling effect in compression tests. The friction coefficient between epoxy materials and steel platens could be as high as 0.1–0.25 [19,20]. Therefore, an aspect ratio of l/d = 1.0 was selected for the specimens in this investigation.
Fig. 2. Stress–strain curve at different loading rates. (Note: two representative test repeats are shown although more tests were performed for each loading rate; only three rates are shown in this figure, representing the full range of qualitative behaviour; stress levels at other rates are shown in Fig. 6.)
To achieve a wide range of strain rates, compression tests were conducted using (i) a commercial screw driven loading machine at quasi-static (low) loading rates (0.001–0.01 s−1 ), (ii) a hydraulic loading machine at medium loading rates (0.5–200 s−1 ), and (iii) Split-Hopkinson pressure configuration using Ø15 mm × 500 mm solid cylindrical steel bars at high loading rates (1000–4000 s−1 ). The compressive deformation of a specimen was captured using various imaging systems with framing rates ranging from 0.25 to 250,000 frames s−1 . Full details of the test procedure can be found in a previous paper [21]. To minimise interfacial friction all specimens were lubricated with CastrolTM LMX grease,1 which is widely used in high speed applications. At least three specimens were tested for each strain rate, allowing evaluation of the test reproducibility. The strain rate for each test was maintained approximately constant by the prescribed velocity boundary conditions. The nominal stress and strain were calculated from the measured force and displacement (see Fig. 2). Each stress–strain curve was plotted to the end of loading at an approximately constant strain rate, even though densification of the foam is expected to occur at a strain εd = ∼0.6 according to Gibson and Ashby [22]: εd = 1 − 1.4∗,
(1)
where the relative density, *, is ∼30%. Elastic properties are not discussed in this study because it is well established that (i) for quasi-static experiments, elastic properties would typically be measured by using a specimen with a larger aspect ratio, e.g. l/d > 2.0 and (ii) high strain rate experiments do not produce an accurate representation of elastic modulus [21]. 3. Results 3.1. Quasi-static compressive behaviour Representative nominal stress–strain curves of the syntactic foam at quasi-static rates of strain (0.001 s−1 ) are shown in Fig. 2. A good reproducibility of better than 5% in stress was achieved, indicative of consistent cell structure (glass microballoon distribution), specimen geometry and testing conditions between specimens. For clarity, Fig. 2 only shows a representative selection of stress–strain curves. A full set of data at all rates is given in Fig. 6. An initial, approximately linear, region corresponding to the elastic behaviour of the foam is observed in the stress–strain curves
Fig. 1. Typical microstructure of glass microballoon epoxy syntactic foams.
1
CastrolTM is a registered trademark of Castrol Ltd., Swindon, UK.
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
21
Fig. 3. Deformation sequence in compression tests at loading rates of (a) 0.001 s−1 and (b) 3500 s−1 . (Note: the interface between bars/platens and specimens is covered by the lubricants squeezed out; the solid line boxes A and B indicate the plastic and elastic deformation zones in the foam, respectively; the dashed line C represents the shear band inside the foam.)
until the peak stress (∼15 MPa) is reached (Fig. 2). A drop in stress then occurs, implying the onset of glass microballoon crushing. Due to the triaxiality of local stress states towards the corners of the specimen, discussed below, shear cracks form and then propagate in the epoxy matrix or along the interface between the matrix and microballoons. Therefore, the peak stress represents the fracture resistance of the syntactic foam. A nearly constant plateau region following the stress drop (Fig. 2) is due to progressive collapse of unit cells comprising a microballoon and its surrounding epoxy resin. The plateau region corresponds to the energy dissipation during compression. The plateau stress (∼13 MPa) defined by the average stress in the plateau region, indicates the capacity of energy dissipation of the foam. Shear cracking continues in the matrix; meanwhile crushing of glass microballoons exposes hollow space, which is consumed by the epoxy matrix material in compression. When a significant portion of microballoons is crushed, further loading causes densification of the foam. Fig. 3(a) illustrates the recorded deformation series of the syntactic foam in quasi-static compression. Visible barrelling occurred in the specimens. Although the hollow space on the end surfaces of the foam specimens can serve as a groove to hold lubricants in the initial stage of the experiment, lubricants were squeezed out during compression, causing significant interfacial restraining effect. The observation of barrelling during deformation suggests the nonuniformity of the internal deformation pattern due to friction at the contact surfaces, which then causes intense shear deformation arising along the ∼45◦ direction from the corners in a longitudinal section through the centre of the specimen. This shear stress will cause formation and growth of shear cracks, and thus non-uniform damage of the foam.
3.2. Dynamic compressive behaviour Fig. 2 also shows the nominal stress–strain curves of the syntactic foam at high strain rates (3500 s−1 ). Similar to quasi-static compression tests, Split-Hopkinson bar tests can be repeated with a deviation of <5%. The shape of the high rate stress–strain curve is similar to that at quasi-static rates. However, the peak stress increases with strain rate, e.g. ∼43 MPa versus ∼15 MPa at rates of 3500 and 0.001 s−1 respectively. This implies that the syntactic foam has a higher resistance to fracture under dynamic loading. The sharper stress drop observed at the high loading rate suggests that it is more difficult for cracks to initiate in the epoxy matrix at these rates than it was in the quasi-static experiments. However, relative to the initiation of shear cracks, their propagation is easier at high strain rates. Note that in absolute terms cracks do not propagate as easily as in quasi-static conditions, probably due to the different crack propagation modes at different loading rates as will be discussed below. Glass microballoons also start to crush after the peak stress is reached. Compared to low loading rates, a higher plateau stress (∼36 MPa) is observed at high strain rates, demonstrating enhanced energy dissipation capacity of the foam when subjected to dynamic loading. In high rate tests, two shear bands (labelled as C in Fig. 3(b)) evolve along the diagonal (40–60◦ ) towards the inside of the syntactic foam. Shear bands become more noticeable after the stress drops at a strain of ∼0.06. Compared to the extensive barrelling in low rate tests, catastrophic cracking occurring along the shear bands at high strain rates can contribute to a more considerable drop of stress. In all the high loading rate tests, the two shear bands can occur along the diagonal toward either the loading direction or
22
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
Fig. 4. (a) Hydrostatic and (b) maximum shear stress distribution in the longitudinal section through the centre of the specimen at a strain of 0.1; and (c) schematic of the longitudinal section illustrating the crushing of glass microballoons.
in the opposite direction (e.g. C in Fig. 3(b)). The deformation rates are different in the two sides of the shear bands. It seems that zone A deforms plastically whereas zone B elastically as shown in Fig. 3(b), while such separation is not observed in quasi-static compression tests (Fig. 3(a)). Due to the fracture of glass microballoons, compressing the syntactic foam consumes the space of the microballoons and thus increases the overall density of the foam. Less barrelling is observed at higher strain rate (compare Fig. 3(a) and (b)), which suggests that for a given longitudinal strain, the syntactic foam expands less and densifies more compared to quasi-static conditions. Therefore, the loading rate influences the volume change during plastic deformation.
By taking the strain of densification (εd ) as equal to ∼0.6, the energy dissipation capacity can therefore be estimated as ∼8 and ∼22 MJ/m3 for quasi-static (0.001 s−1 ) and dynamic (3500 s−1 ) strain rates, respectively. The syntactic foam does not ideally collapse during compression; it expands laterally. The volume change of the foam is observed to be dependent on the loading rate; in particular, during dynamic compression, both plastic and elastic zones exist in the foam. The effect of strain rate on peak and plateau stresses and volume change needs to be considered when the syntactic foam is used in applications that experience different loading rates.
3.3. Summary: rate dependent compressive properties
4.1. Failure mode
To facilitate structural modelling and optimise the design of both foam materials and components, compressive properties of glass microballoon epoxy syntactic foams have been characterised. The peak stress is one key property indicative of the fracture resistance of the foam. Damage, including crushing of microballoons and shear cracking, evolves after the peak stress is reached. The plateau stress ( pl ), another important feature of the foam, is calculated as the average stress in the plateau region. A linear relationship links pl and the energy dissipation capacity per unit initial volume, W:
The differences in macroscopic compressive properties between quasi-static and dynamic loading rates can be attributed to the different damage/fracture process in the glass microballoon epoxy syntactic foam (Figs. 2 and 3). During compression, the following two failure processes coexist in the foam: (i) crushing of glass microballoons and (ii) shear cracking of the epoxy matrix. However, the two failure modes arise and dominate at different locations in a specimen, causing heterogeneous (localised) damage of the foam. Such localised damage has been observed in aluminium [2] and epoxy [1,10] syntactic foams. Damage also occurs at the interface of the two phases (glass microballoons and epoxy matrix) in the foam. In low rate compression, a non-uniform stress distribution is expected according to the barrelling phenomena observed (Fig. 3(a)). An axisymmetric finite element model was established to
εd
W=
dε,
(2)
0
pl ≈
W . εd
(3)
4. Discussion
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
23
Fig. 5. A crack propagation near location L1 at (a) quasi-static and (b) dynamic (∼1500 s−1 ) strain rates. (After Woldesenbet et al. [5].)
predict the stress distribution. A crushable foam plasticity material formula [15] in ABAQUSTM2 was calibrated to data obtained from the experiments. The interfacial contact between the steel anvils and the epoxy foam specimen was defined by a frictional coefficient of 0.1 estimated from the literature [19,20]. Both the boundary and loading conditions were applied based on those observed in the real experiments. Fig. 4(a) and (b) illustrates the distributions of hydrostatic and deviatoric (shear) components of the stress field in the specimen at a strain of 0.1. The hydrostatic stress is responsible for compressive damage while the shear stress gives rise to shear cracking in the syntactic foam. The non-uniform hydrostatic/shear distribution can be associated with the localised damage in the syntactic foam as follows: 1. At the corners of the foam, e.g. L1 in Fig. 4(c), the maximum shear stress arises while the compressive stress is low. Shear type failure rather than compression damage therefore dominates in this region. The epoxy matrix is cracked under the shear stress; banded structure appearing in a stepwise fashion has been observed in similar epoxy syntactic foams by Gupta et al. [11]. Shear cracks propagate at a 40–60◦ angle to the loading axis. Most of the glass microballoons are uncrushed, causing only a small amount of debris. 2. In the central part of the foam (L3 in Fig. 4(c)), substantial compression damage prevails until the shear cracks originating in L1 grow to a considerable length and cause final failure of the specimen. Glass microballoons are completely crushed; and large amount of debris can be seen [1,11]. 3. A mixture of hydrostatic compression and shear damage occurs in the region between shear (L1 ) and compression (L3 ) zones (see L2 in Fig. 4(c)). A few uncrushed glass microballoons appear among the broken ones; shear forces drive the interfacial separation of microballoons from the epoxy.
syntactic foam at quasi-static and dynamic loading rates (see Fig. 5). In quasi-static compression, the crack tends to break the matrix and then pass around the glass microballoons or debond the interface between microballoons and the epoxy matrix. However, at high loading rates, the crack fractures microballoons and propagates through them, thus consuming a larger amount of energy compared to quasi-static compression. Equivalent variations in cracking mode have been observed in other two-phase materials, such as PBXs at different temperatures [23]. 4.2. Strain rate sensitivity In order to quantify the strain rate effect on compressive properties of the syntactic foam, additional tests were performed at different loading rates using the facilities described above. A wide range of strain rates with close intervals is essential to accurately quantify the rate sensitivity (Fig. 6). Both peak and plateau stresses increase with strain rate up to 4000 s−1 (see Figs. 2 and 6). An empirical constitutive model was developed to represent the relation between peak/plateau stresses () and strain ˙ rate (ε): ˙ , = 0 (1 + C log ε)
where 0 and C are material constants. A very good correlation (R2 > 94%) was achieved for both peak and plateau stresses (see Table 1). The selection of this empirical equation was justified because: (i) a constitutive model in the form of Eq. (4), widely
At high strain rates, the stress triaxiality in the syntactic foam also leads to formation of shear cracks along the direction of the maximum shear stress (40–60◦ to the loading axis). However, in this case, the increased strength of the rate sensitive epoxy, compared to the rate insensitive glass microballoons, means that the load transfer between the two components of the foam is different from that in the quasi-static experiments [2]. At high rates, cracking of glass microballoons begins to dominate over the matrix/microballoon debonding. Woldesenbet et al. [5] illustrated the difference between the crack propagation mechanisms in epoxy
2
ABAQUSTM is a registered trademark of ABAQUS Inc., RI, USA.
(4)
Fig. 6. Peak and plateau stresses as a function of strain rate.
24
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
Table 1 Estimated parameters with 95% confidence bounds for peak and plateau stresses. Strain rate range (s−1 ) Peak stress Plateau stress
0.001–4000 0.001–4000
0 (MPa)
C (s−1 )
R2
28.42 ± 1.19 23.27 ± 0.62
0.0702 ± 0.0093 0.0651 ± 0.0058
0.94 0.97
used in dynamic problems, has been proved a simple and effective way to introduce strain rate sensitivity in numerical and analytical calculations, particularly for polymeric materials [24]; (ii) for the syntactic foam studied, the stress–strain curve maintains a nearly similar shape over a wide range of strain rates (see Fig. 2); and (iii) experimentally, both peak and plateau stresses show an almost linear dependency on the logarithm of strain rate (Fig. 6). From a materials selection perspective, it is important to know the actual strain rate sensitivity for a specific application. The fitted parameters listed in Table 1 are applicable for strain rates ranging from 0.001 to 4000 s−1 (approximately 20 m s−1 , a typical automotive speed). For higher loading rate applications, e.g. 150–200 m s−1 in aerospace, further work is required to generate a larger set of stress–strain data. The strain rate dependency of syntactic foams can be attributed to a number of microscopic factors as follows: 1. The rate dependency of the foam is significantly determined by the rate sensitivity of the matrix material [2,25]. Viscoelasticity of the epoxy is likely to cause the strain rate dependency of the matrix and therefore of the foam. 2. As discussed previously, the shear crack propagates through glass microballoons at dynamic loading rates instead of bypassing or debonding them as happens in quasi-static deformation. Such a rate dependent damage mechanism will result in a stronger fracture resistance (peak stress) and better energy dissipation capacity (plateau stress) of the foam at high strain rates. 3. The inability of gas to escape from the inside after glass microballoons are crushed is likely to contribute to the increased energy dissipation of the epoxy (non-metallic) closed-cell syntactic foam at high strain rates. The gas pressure increases adiabatically in dynamic tests; more energy is required to compress the foam compared to a low loading rate test (a nearly isothermal process). 4. In low density (high porosity volume fraction) foams loaded at high rates of strain, there is commonly an inertia effect manifested in the initiation of higher order buckling modes of the characteristically slender meso-structure (open-cell framework and closed-cell membranes) before extensive deformation and fracture. However, these effects are not expected in this study because the relatively low porosity volume fraction and the hybrid nature of meso-structure of the given material were not susceptible to buckling. The relatively high proportion of polymeric resin and relatively high thickness-to-diameter ratio of the glass microballoons did not allow for the inertia effects to contribute to the rate dependent response of specimens loaded at relatively low boundary velocities (<25 m s−1 ). 4.3. Quantification of volume change The volume of the glass microballoon epoxy syntactic foam changes during the plastic stage of compression; the volume change at a specific strain is influenced by strain rate (Fig. 3). Quantification of the volume history is required to develop a constitutive dynamic model, which can be used to simulate the structural impact resistance of bulk components made from such foams [15,18]. Images of the deformation sequence at different loading rates were used to calculate the volume history (V) of a specimen during compression,
Fig. 7. The relation between the volumetric strain and nominal strain in quasistatic and dynamic compression tests. (Note: the linear trendlines are fitted on the measured data.)
with the assumption that the specimen maintains a longitudinally axisymmetric shape. The volume change was averaged over the whole specimen because shear bands occurring inside a specimen make it impossible to calculate the real 3D volume from 2D images. The volumetric strain (εv ) was therefore calculated using the measured longitudinal area (A and A0 ) and length (L and L0 ) of the specimen in a 2D image. (Note: the subscript 0 symbolises the original dimension.) εv = 1 −
V A2 L0 . =1− V0 A0 2 L
(5)
The volumetric strain is plotted against the nominal strain up to 0.5 in Fig. 7. The syntactic foam becomes denser as the specimen is plastically compressed. The volumetric strain is almost directly proportional to the nominal strain (ε): εv = kε,
(6)
where k is a constant. The syntactic foam does not behave as an incompressible material such as metals (k = 0), nor as an ideal collapsed material with the specimen diameter maintaining constant (k = 1). The volume change increases with strain rate, e.g. k = 0.54 ± 0.01 versus 0.61 ± 0.03 (correlation R2 = 99%) for quasistatic and dynamic compression, respectively (Fig. 7). The two factors likely to contribute to the rate dependency of volume change by preventing lateral expansion in high rate compression are: (i) time-dependent properties of the epoxy matrix material (e.g. viscoelasticity and viscoplasticity) and (ii) changes in the collapse/damage mechanism at higher strain rates. 5. Conclusions The compressive properties of glass microballoon epoxy syntactic foams were experimentally characterised over a wide range of strain rates (0.001–4000 s−1 ). The underlying deformation and failure mechanisms were investigated using combined finite element stress analysis and microscopic examination. Strain rate effects on the strengths of the two components in the syntactic foam, in particular the epoxy, cause non-uniform deformation and heterogeneous (localised) damage. Two failure modes dominate in different regions of the foam: (i) crushing of glass microballoons in the central part due to substantial
P. Li et al. / Materials Science and Engineering A 515 (2009) 19–25
hydrostatic stress and (ii) shear cracking of the epoxy matrix, which forms and propagates from the corners where the maximum shear stress occurs. As the epoxy matrix gets stronger at higher rates, cracking of glass microballoons begins to dominate over the matrix/microballoon debonding under dynamic loading, requiring more energy to be absorbed in the foam. Shear cracks propagate through glass microballoons at high strain rates instead of bypassing or debonding them as happens in quasi-static testing. The underlying deformation and failure mechanisms lead to the macroscopic rate dependent compressive response of the syntactic foam. Both peak and plateau stresses, indicative of fracture resistance and energy dissipation capacity of the foam, respectively, increase almost linearly with the logarithm of strain rate. Unlike quasi-static compression, separate plastic and elastic deformation zones, under dynamic loading, arise on the two sides of shear bands along which final catastrophic cracking occurs. The quantified volume history indicates that volumetric strain is almost directly proportional to nominal strain and increases with strain rate. Quantification of both rate dependent stress–strain relation and volume change will improve the accurate prediction of the service behaviour of bulk components made from such foams. Furthermore, from the viewpoint of materials selection, it is essential to know the actual strain rate dependency for a specific application. Acknowledgements The authors acknowledge the colleagues at University of Oxford for their help during the experiments. The Cordin 550 high speed camera used in this research was provided by the EPSRC instrument loan pool. The authors are particularly grateful to A. Walker for his advice and support while using this camera.
25
References [1] J. Adrien, E. Maire, N. Gimenez, V. Sauvant-Moynot, Acta Mater. 55 (2007) 1667–1679. [2] D.K. Balch, J.G. O’Dwyer, G.R. Davis, C.M. Cady, G.T. Gray, D.C. Dunand, Mater. Sci. Eng. A 391 (2005) 408–417. [3] I.W. Hall, M. Guden, C.J. Yu, Scripta Mater. 43 (2000) 515–521. [4] K.Y.G. McCullough, N.A. Fleck, M.F. Ashby, Acta Mater. 47 (1999) 2323– 2330. [5] E. Woldesenbet, N. Gupta, A. Jadhav, J. Mater. Sci. 40 (2005) 4009–4017. [6] G.H. Wu, Z.Y. Dou, D.L. Sun, L.T. Jiang, B.S. Ding, B.F. He, Scripta Mater. 56 (2007) 221–224. [7] K. Kitazono, Y. Takiguchi, Scripta Mater. 55 (2006) 501–504. [8] T. Mukai, H. Kanahashi, T. Miyoshi, M. Mabuchi, T.G. Nieh, K. Higashi, Scripta Mater. 40 (1999) 921–927. [9] B. Song, W. Chen, D.J. Frew, J. Compos. Mater. 38 (2004) 915–936. [10] N. Gupta, Kishore, E. Woldesenbet, S. Sankaran, J. Mater. Sci. 36 (2001) 4485–4491. [11] N. Gupta, E. Woldesenbet, Kishore, J. Mater. Sci. 37 (2002) 3199–3209. [12] M. Koopman, K.K. Chawla, K.B. Carlisle, G.M. Gladysz, J. Mater. Sci. 41 (2006) 4009–4014. [13] B. Song, W.W. Chen, W.Y. Lu, Int. J. Mech. Sci. 49 (2007) 1336–1343. [14] H. Zhao, G. Gary, Int. J. Veh. Des. 30 (2002) 135–145. [15] V.S. Deshpande, N.A. Fleck, J. Mech. Phys. Solids 48 (2000) 1253–1283. [16] C.L. Martin, D. Bouvard, Acta Mater. 51 (2003) 373–386. [17] C.D. Turner, M.F. Ashby, Acta Mater. 44 (1996) 4521–4530. [18] J. Zhang, N. Kikuchi, V. Li, A. Yee, G. Nusholtz, Int. J. Impact Eng. 21 (1998) 369–386. [19] R. Bassani, G. Levita, M. Meozzi, G. Palla, Wear 247 (2001) 125–132. [20] J. Wu, X.H. Cheng, Wear 261 (2006) 1293–1297. [21] P. Li, C.R. Siviour, N. Petrinic, Exp. Mech., in press, doi:10.1007/s11340-008-91291. [22] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, 1999. [23] D.M. Williamson, S.J. Palmer, W.G. Proud, R. Govier, Brazilian disc testing of a UK PBX above and below the glass transition temperature, in: M. Elert, M.D. Furnish, R. Chau, N.C. Holmes, J. Nguyen (Eds.), Proceedings of the Conference of the American Physical Society Topical Group on Shock Compression of Condensed Matter, AIP Conference Proceedings, vol. 955, Hawaii, 2007, pp. 803– 806. [24] N. Jones, Structural Impact, Cambridge University Press, Cambridge, 1998. [25] Z.Y. Dou, L.T. Jiang, G.H. Wu, Q. Zhang, Z.Y. Xiu, G.Q. Chen, Scripta Mater. 57 (2007) 945–948.