Another indirect tensile test method utilizes theta-shaped specimen geometries, such as that shown in Fig. 28.1. The application of compressive forces normal to the central web section results in the development of a uniform uniaxial tensile stress state in the central web. This test method was initially developed by [3] in the late 1960s, but never saw wide adoption due to the complex specimen geometry. More recently [4, 5] made modifications to the initial specimen geometry to perform strength measurements of Si wafer material on small-scale specimens. The stress state developed in the modified specimen geometry was analyzed using finite element analysis, and was shown to maintain a state uniform uniaxial stress in the gage section [5]. Similar the Brazilian disk test, the stress in the web is related to the applied load Pand takes the form σ ¼ KP Dt , where K is a constant that depends on the geometry of the sample. Durelli et al. [3] found K¼13:8 via photoelastic analysis, and Gaither et al. [5] found K¼14:239 using 3-D finite element analysis. In this paper we aim to investigate the use of theta-shaped specimens in conjunction with a compressive Kolsky bar to measure the dynamic tensile strength of materials. A slightly modified version of the “arch” specimen geometry [4, 5] is analyzed via dynamic finite element simulations. The simulation results are used to evaluate the uniformity and the magnitude of the stress state in the central web for different loading pulses. Specific attention is paid to the excitation of transverse deflection of the web due to acceleration of the specimen during loading. 28.2 Methodology The specimen geometry of the theta-shaped specimen studied here is shown in Fig. 28.1. Units of measure are scaled to a diameter of unity for ease of adaptation to various sizes or systems of measure. The internal geometry is identical to that used by [4, 5] but the external utilizes flats instead of the “top hat” geometry in the load bearing area. These flats aid with specimen alignment, and alleviate the potentially high contact stresses associated with a disk in contact with a flat plate. For the study presented here, the outside diameter was chosen to be 10 mm and the thickness 5 mm. The material of the specimen was poly-methyl methacrylate with Young’s modulus Es ¼3:2 GPa, Poisson’s ratio νs ¼0:35, and mass density ρs ¼1190 kg=m3. The constant K in the load–stress relation has not yet been determined for the geometry studied here, but we expect it to be close to that for the geometry of Gaither et al. [5] because the internal geometries are so similar. A Kolsky bar system with incident bar length of 1.0 m and transmitted bar length of 0.5 m was modeled to apply the loading to the specimen. All bars had diameter of 12.7 mm and were made of aluminum with Young’s modulus Eb ¼69GPa, Poisson’s ratio νb ¼0:3, and mass density ρb ¼2730kg=m3. All components were modeled in the finite element software Comsol Multiphysics version 5.0 as two-dimensional plane stress entities (note: the bars were modeled as having square cross section 12.7 mm 10.0 mm resulting in cross sectional area equivalent to a 12.7 mm-diameter circular section bar). All components were meshed with linear elastic, 4-node quadrilateral elements, as shown in Fig. 28.2. The majority of the bar areas were meshed with a structured mesh with seven elements across and aspect ratio of 1. Near the specimen this mesh transitions to a free mesh to allow for better mapping of nodes between specimen and bar. The specimen mesh resulted in eight elements across the thickness of the central web and a free mesh outside of the web region. A convergence study was performed and further refinement of the mesh size in the specimen did not significantly affect the results. Fig. 28.1 Specimen geometry, scaled to an overall diameter of unity. The specimen studied here has a diameter of 10 mm. Specimen thickness is chosen to be half the diameter, but this is not a strict requirement 204 J. Kimberley and A. Garcia
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