A Tensile Split Hopkinson Bar for Testing Particulate Polymer Composites Under Elevated Rates of Loading

Abstract

A tensile split Hopkinson bar apparatus is developed for testing high strain rate behavior of glass-filled epoxy. The apparatus uses a specimen gripping configuration which does not require fastening and/or gluing and can be readily used for castable materials. Details of the experimental setup, design of grips and specimen, specimen preparation method, benchmark experiments, and tensile responses are reported. Also, the effects of filler volume fraction (0–30%) and particle size (11–42 μm) are examined under high rates of loading and the results are compared with the ones obtained from quasi-static loading conditions. The results indicate that the increase in the loading rate contributes to a stiffer and brittle material response. In the dynamic case lower ultimate stresses are seen with higher volume fractions of filler whereas in the corresponding quasi-static cases an opposite trend exists. However, the absorbed specific energy values show a decreasing trend in both situations. The results are also evaluated relative to the existing micromechanical models. The tensile response for different filler sizes at a constant volume fraction (10%) is also reported. Larger size filler particles cause a reduction in specimen failure stress and specific energy absorbed under elevated rates of loading. In the quasi-static case, however, the ultimate stress is minimally affected by the filler size.

Appendix A

Grip and Specimen Shape Design

Specimen Grip Design: A dovetail shaped specimen gripping mechanism was chosen in an effort to minimize the attenuation that can occur in more complex attachment configurations such as specimens with threaded ends, clamps with fasteners, etc. Finite element analysis was used to arrive at a specific shape that would allow the bars to be used repeatedly without damage. The details of this process are outlined in the following.

A generic dovetail shape is shown in Fig. 18 along with the complementary dogbone specimen. There are several features that require attention in this particular arrangement. Some of these include (a) the dovetail width, (b) the dovetail length, (c) the angle between the dovetail and the axis of the bar, and (d) the filet radius.

Fig. 18

Generalized dovetail configuration showing features of interest

An initial geometry was selected, machined and tested in the actual setup. For this initial specimen–bar interface geometry the dovetail in the incident bar end failed. A closer look at this geometry with finite element analysis revealed that the stresses produced in the aluminum grips during the experiment due to the stress concentration were in excess of the failure stress for the 7075-T6 aluminum being used. Thus, this geometry was used as a benchmark for refining the specimen–bar interface shape such that the tensile testing would not result in failure of the grip region of the incident bar.

The finite element model used for this purpose consisted of two parts, the specimen and the incident bar ends. The geometry was modeled in Solid Edge® graphics tool and converted into an IGES (initial graphics exchange specification) format. This neutral file format was imported directly into ABAQUS® structural analysis environment. The two parts were then discretized independently and merged together as an assembly. Due to the complex geometry of the bar end, four-noded linear tetrahedral elements (C3D4 in ABAQUS®) were used for the mesh. The specimen itself was meshed using eight-noded linear brick elements (C3D8R in ABAQUS®). Quarter symmetry of the configuration was exploited to reduce the size of the model. This allowed the use of a much denser mesh for improving the accuracy of results. About 20,000 elements were used to discretize the incident bar end, and about 1,000 elements were used for the specimen. Figure 19(a) shows the loads and boundary conditions used in the model. The mesh is shown in Fig. 19(b).

Fig. 19

(a) Loads and boundary conditions for dovetail model, (b) finite element model of dovetail grip, (c) surface of contact enforcement

Contact elements were used along the interface between the bar and specimen on the hatched surface shown in Fig. 19(c). The contact was formulated for both the surface normal and tangential directions. The constitutive law for contact elements included a linear stress–strain behavior in the direction normal to the surface with a stiffness of approximately ten times the stiffness of the aluminum grip. The aluminum surface was chosen as the master surface. In the direction tangential to the surface, the friction between the aluminum grip (master surface) and specimen end (slave surface) was also accounted for using a stiffness (penalty) method. In this method, a certain amount of shear stress is carried across the interface between the master and slave surfaces. The shear stress is directly proportional to the normal load between the surfaces. This allowed for the estimation of the friction during the loading event. The end of the bar was constrained from translation in the horizontal direction, and a uniform pressure was applied on the end surface of the specimen as shown in Fig. 19(a).

The stresses through the thickness at the location of the least cross-sectional area of the grip (Fig. 20) were the primary output quantity of interest. Designing for a minimal stress ensures that the bar end will endure repetitive loading of the grips. Twelve iterations of the design were explored with the FE model. The particular parameters being studied included the angle of the dovetail, the maximum width of the dovetail, and the length the dovetail extends into the incident bar. The initial (a) and final (d) geometries that were analyzed are shown in Fig. 20(b). Also shown are two of the intermediate geometries (b and c).

Fig. 20

(a) Line of interest for stresses, (b) geometries of interest, (c) stress through the thickness for different geometries

The plots in Fig. 20(c) correspond to the normalized von-Mises stresses along the line shown in Fig. 20(a). The origin of the plot corresponds to the mid-plane of the cylindrical rod. As can be seen, iteration (a) exceeded the yield strength of the aluminum by 20% and iteration (d) of the dovetail design had significantly lower stresses.

Specimen Stress Distribution: For tensile specimens, it is necessary that the gage section has a uniform stress distribution, and that there are no obvious stress concentrations. A plane stress finite element analysis was completed to verify this. The finite element model consisted of the desired dogbone geometry registered against an analytically rigid surface. Since aluminum is essentially rigid in comparison to the polymer specimen, an analytically rigid surface could be used instead of a meshed deformable body, thus simplifying the model. Also, quarter symmetry was invoked. The mesh consisted of eight-noded biquadrilateral plane stress elements (CPS8R in ABAQUS®). The stress distribution contoured for von Mises stresses for an imposed stress of 40 MPa is shown in Fig. 21. The uniformity of von-Mises stresses in the gage section of the specimen is clearly evident.

Fig. 21

Typical specimen stress distribution