Investigation of the dynamic stress–strain response of compressible polymeric foam using a non-parametric analysis
Introduction
The determination of a material's constitutive behavior at intermediate to high strain rate ranges has been a subject of interest for quite a long time [1], [2]. Historically, the importance of obtaining the equilibrium state has required special attention in the experimental setup, since configurations where equilibrium readily can be identified make it easier to study the dynamic stress–strain response of the materials. The issue is particularly significant in the case of engineering materials with low strength and low mechanical impedance [3], where conditions of stress/strain equilibrium oftentimes require a time duration that exceeds the time required for the material to undergo failure. In such cases, the determination of stress–strain response of the material based on the load and displacement measurement at specimen ends can introduce a significant degree of inaccuracy in the obtained constitutive response, with continued deterioration in accuracy at higher strain rates [4], [5], [6].
Polymeric foams, as a class of materials widely used in applications that require light weight structural design with superior energy dissipation characteristics, are among the materials possessing low wave propagation speed [7], [8], [9], [10]. Their use in areas such as automotive industries, ships and packaging requires precise knowledge of the deformation response of the material at different loading rates. Dynamic deformation and failure behavior of these materials have been studied both experimentally and numerically in recent years [11], [12], [13], [14], [15], [16]. In general, most polymeric foams possess remarkable strain rate sensitivity at strain rates above 500 s−1 [14]. However, owing to non-homogeneity in the stress and deformation states, particularly during the early stages of dynamic loading, quantifying the dynamic deformation of foams has been a major challenge. One of the most widely used methods in studying the dynamic deformation behavior of soft materials is the split Hopkinson pressure bar technique [3]. Different approaches have been practiced in recent years to increase the accuracy of the measurements in this technique, particularly in testing of low impedance materials. These approaches include pulse shaping techniques [3], use of polymeric bars [17] and long projectiles [18]. A recent study by Liu et al. [16] indicates that in addition to the solutions proposed above, full-field measurements must be incorporated in order to accurately measure the deformation response and reveal the active failure mechanisms in foam specimens subjected to high strain rate loading conditions.
The advent of full-field measurement techniques such as digital image correlation (DIC), in conjunction with different experimental techniques, has facilitated the study of deformation of materials over a wide range of time and length scales [16], [19], [20], [21], [22], [23]. More importantly, the recent work by Pierron and his group [20], [24], [25] using virtual fields and inverse methods has identified a unique way of analyzing the dynamic deformation of materials by using D'Alembert's principle and incorporating “inertia forces” into the analysis. Though the inverse methods have already been demonstrated to be effective in calculating the stress–stress response of materials at high strain rate loading conditions, the effect of compressibility has not been addressed in previous studies.
The present work focuses on the study of rigid closed-cell polymeric foams subjected to direct impact loading by accounting for material compressibility and the effect of inertia. A shock tube apparatus is used to apply dynamic loading on the foam specimens, while high speed stereovision imaging together with 3D digital image correlation is used to study the full-field deformation of the material under high strain rate loading conditions. A simple mathematical model has also been proposed to account for both material compressibility and the local variation of density during deformation of the specimens. Also, based on the full-field displacement distribution captured by DIC and considering the instantaneous change of material density, a non-parametric analysis is developed to incorporate “inertia effects” into the analysis, following references 5 and 6. Using the proposed methodology, the full-field stress–strain response of the material has been determined during high strain rate loading and the global constitutive behavior of the foam has been quantified. The original contribution here is the inclusion of the material compressibility into the analysis. In addition, attempts have been made to generalize the methodology to study the dynamic deformation of any low-impedance compressible solid. To the authors' knowledge, this is the first time a thorough analysis of the dynamic deformation of compressible polymeric foams has been performed taking into account the concurrent effects of material compressibility and inertia loading to capture the full-field stress–strain response of the material.
Section snippets
Material and specimen geometry
The material used in this study is a rigid closed-cell polyurethane foam of 560 kg/m3 (35 pcf) nominal density supplied by Sandia National Laboratories [26]. The initial density of the foam specimens is measured in-house and confirmed to be consistent with density values reported in the literature [26]; the measured value is 95% of literature data. A cylindrical foam specimen with a high contrast speckle pattern is shown in Fig. 1a. The specimen is 25.4 mm in diameter and 25.4 mm in height.
Modeling of material compressibility
The polymeric foam specimens examined in this work are compressible, and hence their density varies when subjected todeformation [30]. To take the material compressibility into account, a simple one-dimensional model is proposed in this work, which facilitates the calculation of material's instantaneous density as a function of its initial density and the applied strain components.
Let us consider a compressible cylindrical specimen of initial volume Ω0 and initial density ρ0 exposed to axial
Results and discussion
Fig. 8 shows the force measured at the load-cell side of the specimen for two independent experiments performed using one Mylar sheet. Consistent results shown here confirm the repeatability of the experimental results in this work.
The velocity of the projectile can be varied by changing the thickness and/or the number of diaphragms separating the driver and driven sections of the shock tube. Projectile velocity in this work is calculated by differentiating the projectile displacement with
Conclusions
The dynamic stress–strain response of closed-cell rigid polyurethane foam is investigated using high speed photography with 3D digital image correlation. Owing to the low wave propagation speed of the examined foam specimens, a highly non-homogeneous state of deformation is observed during impact loading, a condition that requires the inclusion of inertia stresses in the analysis. The change in density, inertia forces and stresses are first calculated using the full-field strain and
Acknowledgments
Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
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