Experimental arrangement for measuring the high-strain-rate response of polymers under pressures

ABSTRACT

This study aims to investigate the high-strain-rate shear response of viscoelastic elastomeric coatings at large strains and under elevated levels of hydrostatic pressure. Results of this study shed light on the combined effects of deformation rate and pressure which might promote a transition from viscoelastic to glassy behavior. This work utilizes a Split Hopkinson Pressure Bar (SHPB) apparatus in conjunction with a customized version of the recently proposed Shear Compression Specimen (SCS) which consists of a polymer gage section with two metal ends that remain essentially rigid during deformation. Detailed finite element simulations were used to customize the adopted specimen, to determine its proper dimensions and promote its functionality. The customized specimen permits subjecting the tested specimen to a state of uniform pressure and shear stress, while allowing for measuring pressure, shear stress and shear strain directly. Results obtained using the customized specimen, which are included in this paper, illustrate its usefulness in measuring the effect of high-strain-rate, large strain and hydrostatic pressure on the shear stress-strain response of viscoelastic elastomers.

INTRODUCTION

Viscoelastic elastomers have been surprisingly effective in enhancing the resistance to penetration and fragmentation of armor when applied as armor coatings. During such processes they experience strains, strain-rates and complex states of stress under which their response might not be fully described by the commonly used constitutive models based on linear viscoelasticity or quasi-linear viscoelasticity, which were utilized in describing the constitutive behavior of polyurea in [1] and [2], respectively. The latter assumes the response to be a superposition between linear viscoelasticy and hyperelasticty. In this superposition, large strains are described by a hyperelastic potential while time-dependence follows linear viscoelastic principles. The deviation from linearviscoelasticy is anticipated because one observed that under high-strain-rates and when deformations achieve large metrics, elastomeric coatings provide excellent stiffness, highlighting the possibility of a deformation induced transition to the glassy state [3-5]. Theoretically and as presented in [6,7] such transition is generated by impeding relaxation mechanisms which can be achieved by negative dilatation derived from low temperature, high pressures or simply by deforming the material at a time scale shorter than the relaxation times of the material [5].

Although a detailed understanding of the mechanisms behind the effectiveness of elastomeric coatings is imperative to the development of such coatings, for design purposes and in order to employ coatings in engineering structures reliably, it might be sufficient to establish empirical constitutive models capable of predicting the response under any complex loading. Accordingly, the constitutive response should rely on experimental observations that cover the full range of the anticipated behavior and should represent all the principal parameters and their interactions. Hence, experimental data covering the full range of the strain, strain rates and pressures existing during impact or blast loadings are required for both developing constitutive models and understanding the different mechanisms.

So far experimental methodologies have been used effectively to provide significant amount of data. Split Hopkinson Pressure Bar (SHPB), which is a standard technique for extracting the high-strain-rate response of materials, was used to obtain the high-strain-rate response of polyurea and polyurethane at strain rates [8]. However, since viscoelastic elastomers are pressure dependent, with SHPB, the effect of strain-rate and mean stress are not easily separable and pressure-induced effects might be mistaken for strain-rate-sensitivity. Alternatively, one can directly measure the shear response independently of pressure by using a torsional Kolski apparatus or by using the specialized shear specimen proposed in [9], in which the specimen is placed at an angle of 45° with respect to the load. This specimen was used successfully in conjunction with SHPB to measure the response of polyurea at moderate to high strain rates. On the other hand, the effect of pressure at low strain rates was measured for polyurea by superposing a torsional strain field on confined specimens preloaded with an equitriaxial state of stress [10]. At the extreme end, the combined effect of pressure and high-strain-rates was measured for polyurea using the plate-impact experiment [11].

Although, the aforementioned experiments provided valuable data, they covered important but limited segments of the space defined by the potentially important strain-rates and pressures. Accordingly, an experimental arrangement is needed to measure the response of elastomeric coatings at moderate to high strain-rates and pressures. The present study aims to address this gap by merging two of the aforementioned techniques, namely the shear specimen [9] and plate impact experiment [11]. The remainder of this paper is organized as follows: We first present the experimental arrangement, then, a finite element verification is presented, followed by a presentation of the results and their discussion, with global conclusions presented last.

EXPERIMENTAL ARRANGEMENT

The shear-compression specimen which was originally proposed for polymers in [9] consists of two steel or aluminum blocks with the polymeric specimen (2mm by 2mm cross-section and 15.6 mm in length) placed between the blocks at an angle of 45 degrees as seen in Fig. 1. The polymeric specimen is glued to the aluminum blocks to apply the intended shear loading. The blocks, which remain essentially rigid during the specimen deformation, are then subjected to a compressive force, P(t) , which can be decomposed into compressive and shear stresses using

were w and A are known geometric parameters and 8 in [9] is set to 45°. In this work, 0 is varied so as to subject the sample to various levels of compressive stress.

Fig. 1 Shear compression specimen [9], used to measure the quasi-static and high strain rate shear response of polyurea.

During deformation, the polymeric specimen is subjected to a strain field that can be described analytically as [9]

such that u and v are the relative horizontal and vertical displacements of the blocks, respectively. Although in [9] the plane stress assumption was utilized since the out of plane thickness was small, which was also justified by way of finite element analysis, equations 1-3 are derived from force equilibrium and basic kinematics of deformation in the deformation (x-y) plane independently from plane stress assumption.

Although this shear-compression specimen was utilized successfully in [9], it inherently has two issues that conflict with the objectives of the current study. First, at small strains the polymer specimen is subjected to a plane state of stress but at lager strain a triaxial state of stress exists, however, only the normal stress component can be measured directly. More importantly, computing shear strains using Eqn. 3 requires knowing both components of displacements. But while the vertical component is easily measured, in [9] it was suggested to use optical techniques or a vibrometer to measure the horizontal component of relative displacement. In this work, we modified the geometry of the specimen by increasing its area normal to the applied load and decreasing its thickness, to generate a nearly equitriaxial state of stress in the polymeric specimen; hence the measured normal force component is enough to describe the state of stress in the specimen. The modified specimen, see Fig. 2, mimics specimens used in plate-impact experiments, except that loading is applied using SHPB for high strain rates and universal testing machines for quasi-static loading rates.

Fig. 2 Modified shear-compression specimen.

The incompressible nature of the tested specimens provides an easier alternative for determining the horizontal relative displacements of the blocks. Due to incompressibility and the geometrical properties of the modified specimen, the thickness (h) of the tested polymeric specimen can be assumed to remain unchanged during deformation. Accordingly, the relative deformation of the blocks has to satisfy

this equation is used in this work to compute the horizontal displacement instead of measuring it. Accordingly, by measuring the normal force and displacement either while using universal testing machines or SHPB, one can fully describe using Eqns. 1-4: the shear stress, pressure and shear strain in the tested polymeric specimen. Therefore, the new modified arrangement can be effective in quantifying the combined effect of pressure and strain rate on the shear stress of viscoelastic elastomers.

FINITE ELEMENT VERIFICATION

To ensure the accuracy of the measured response, finite element simulations were used to verify the assumptions behind this new configuration and establish the range of applicability of Eqns. 1-4. finite element verification was performed using two inclination angles 9 and 18, such that the smaller angle results in larger pressures. The metal blocks were made of steel and had a cross section of 20*10 mm2 as described in Fig. 2. The specimen thickness was 0.2 mm. Multiple constitutive models were employed in the finite element-based verification to ensure that the new specimen and Eqns. 1-4 were independent of the tested material. However, these constitutive models were incompressible, pressure independent and have a stress-strain behavior that lies within the bounds of the expected response of the tested materials. Although the different constitute models confirmed the validity of the assumptions, here we present the results obtained using a bilinear elastic-plastic constitutive model. This model had a stiffness of 10GPa, a yield stress of 2 MPa and a hardening modulus that allows the flow stress to attain the value of 30 MPa at a strain value of 1.0. Poisson ratio during the elastic phase was set to 0.33, while during the plastic phase it took the value of 0.5, following plasticity principles. Load was applied by means of displacement controlled boundary condition and friction was neglected at the top and bottom faces of the metallic blocks; since lubricant is present at these interfaces during experiments. The commercial FE software ABAQUS was used to perform all simulations.

For both angles, finite element results confirmed that the new modified geometry is effective in subjecting the specimen to a state of uniform hydrostatic pressure in conjunction with a uniform shear stress as can be seen in Figs. 3 and 4. Figure 3 illustrates the evolution of the normal components of stress with the applied prescribed displacement, while Figure 4 compares between the shear stress-strain relation reported by the finite element software and the values predicted by Eqns 14.

Fig. 3: Normal stress generated in the specimen. a) The inclination angle is 9°, while for b) angle is 18 °. As expected in both cases, the normal stresses are equal and are smaller for the larger angle.

Fig. 4: Shear stress-strain response. Showing a comparison between the stress-strain response reported by the finite element software and that computed by Eqns. 1-4. For a) the inclination angle is 9°, while for b) that angle angle is 18 °.

Figure 4 illustrates the ability of Eqns. 1-4 to determine the shear stress-strain relation accurately up to strains of at least 50%. Accordingly, in the following experimental work, the modified specimen is used to measure the shear stress-strain response of polyurea for strains that do not exceed 0.5.

EXPERIMENTS MEASUREMENTS

Steel blocks identical to those used in the simulations were machined. The face of each block at which specimens were to be glued was polished with 400 grit paper to increase the strength of the bond to the polyurea specimen. The process of creating the polyurea specimens commences by casting polyurea between two Teflon blocks to form 0.5 to 1.0 mm thick sheets. Specimens are then cut from the sheets and glued with superglue between the steel blocks. A 24 hours curing period was maintained consistently before testing to permit the glue at the interface to achieve its maximum strength. Specimens were then cleaned from any excess glue. Special attention was given to faces at which load was to be applied. These faces were cleaned thoroughly and lubricated with grease to minimize friction. Figure 5 presents one of the specimens.

Fig. 5 A specimen with a 9 inclination angle

RESULTS AND DISCUSSION

Quasi-static tests were performed using a 14000N capacity MTS system. Specimens with 9 ° inclination angles were tested at shear strain rates of 5 x 10-3 s-1 and 0.5 s-1, while specimens with 18° inclination angles were tested at shear strain rates of 0.03 s-1 and 0.3 s-1. Although these shear strain rates are small, their influence -when accompanied by large hydrostatic pressure- on the shear response of viscoelastic elastomers is of interest. Experimental results illustrating the measured shear stress-strain with respect to the accompanying pressure are presented in figure 6.

Fig. 6 Quasi-static response, showing: a) shear stress-strain response and b) the accompanying pressure.

Figure 6 shows that even at very slow rates (10e-3) the shear stress of polyurea monotonically increased and does not achieve a plateau flow stress at 1 to 2 MPa as was found in [12]. This suggests that the accumulated pressure during deformation promotes stiffening of the shear response of polyurea; this is in agreement with the observations presented in [10]. This figure also illustrates that the two orders of magnitude difference in strain rate has negligible effect within the range of quasi-static strain rates tested. One can argue that polyurea at room temperature behaves as an elastomer, the relaxation times of which are smaller than the time scale characterizing the tested quasi-static strain rates.

Fig. 7 Dynamic response: showing shear stress-strain response and the accompanying pressure for two dynamic tests performed at a shear strain rate of approximately 5000s-1. Two specimens (9° and 18°) were used the sample with smaller angle produced the larger pressure

Figure 7shows the dynamic response of polyurea at a shear strain rate of (5000s-1). During both tests, the glued interface was compromised at strains of about 0.15. Postmortem analysis of the specimens revealed that the specimens sustained damage during the test. This damage was repeatedly localized at the center of tested specimens. Figure 7 not only illustrates the strain-rate sensitivity of polyurea but also highlights the important role of the accompanying pressure; higher pressures stiffen the response, which agrees with the quasi-static tests.

CONCLUSIONS

In this work the shear compression specimen proposed in [9] was modified to investigate the shear response of polymers under a wide range of strain-rates and pressures. Principles behind the proposed modifications were outlined and the effectiveness and accuracy of the modified specimen was verified using FE simulations. Experiments were performed on polyurea under a wide range of strain rates (5×10-3 to 5×103 s-1) and pressures (reaching 90 MPa). Results illustrated the strain-rate sensitivity of polyurea, but more importantly it highlights the effect of pressure on the shear stress-strain response.