Analysis and Simulations of Quasi-static Torsion Tests on Nearly Incompressible Soft Materials

Abstract

A new torsional Kolsky bar test has been developed at Purdue University in an effort to eliminate the effects of radial inertia that complicate the analysis of compression Kolsky bar tests on soft materials such as brain tissue (Nie X, Prabhu R, Chen W, Caruthers JM, Weerasooriya T (Exp Mech 51(9):1527–1534, 2011); Sanborn B, Nie X, Chen W, Weerasooriya T (J Biomech 45:434–439, 2012); Nie X, Sanborn B, Weerasooriya T, Chen W (Int J Impact Eng 53:56–61, 2013). In an effort to quantify the complete stress and strain states in this torsion test as well as to determine the influence of any inertial effects on the test data, we have undertaken numerical simulations of both dynamic and quasi-static torsion tests on very soft and nearly incompressible materials. Here we present the results of our quasi-static torsion simulations on nonlinear elastic specimens. Results are presented for thin solid (disc-shaped) specimens and thin annular (washer-shaped) specimens, although only the latter are used in the Purdue Kolsky bar tests. On the other hand, a pure torsional deformation is possible only for incompressible solid specimens. The differences between the stress states in solid and annular specimens provides insight into axially nonuniform stresses that develop in annular specimens. In addition, explicit closed-form solutions for the full stress state for pure torsion of an incompressible solid specimen have been used to verify the numerical simulations.

Appendix: Details of the Numerical Simulations and Methods

The numerical modeling of was divided into two parts: meshing and the solver followed by post processing and analysis. Meshes were generated in Cubit (V12.1; Sandia National Laboratory) and in TrueGrid (V2.3.4; XYZ Scientific Applications, Inc.). Simulations were performed using Sierra SolidMechanics (Adagio 4.24; Sandia National Laboratory). Adagio is an implicit, nonlinear preconditioned conjugant gradient solver. Reduced integration on HEX8 element meshes was performed to avoid volumetric locking. Due to the highly constrained nature of the problem, extreme care had to be taken to ensure the mesh was structured and symmetric to prevent mesh artifacts. Postprocessing of simulation results was carried out in ParaView (V3.14.0; Kitware) and MATLAB (The MathWorks, Natick, MA).

Utilizing the symmetry of the problem, the state of the material was reduced to five, 1-D radial slices in the reference configuration in ParaView for further analysis in Matlab. Averages taken over the five circular faces assumed rotational symmetry and were implemented using a composite Simpson’s integration. Similarly, volumetric averages were calculated from the five circular slices utilizing 5-point Simpson integration. Continuous plots over the radius of the specimens have been linearly interpolated between elements.

In practice, enforcing the glued boundary conditions in the simulations for the solid specimens resulted in erroneously large pressures. These pressures were concentrated in a small region containing the axis of the specimen (r = 0). These pressures were alleviated by relaxing the glued boundary conditions on the specimen faces to allow for a radial expansion. This resulted in extremely small radial displacements ( ∼ microns), keeping the deformation close to pure torsion (Figs. 11.1 and 11.2). We suspect the large pressures resulted from highly constrained hex elements at the center. These elements undergo what can be approximated as a shear and a rotation. Outer elements, however, are closer to a pure shear strain. Due to the compressible Mooney-Rivlin model implemented in the code, the elements underwent volume changes producing pressures larger than in the incompressible case. Similar simulations were conducted on the annular specimen that relaxed the glued boundary condition. These, in contrast, resulted in deformations consistent with Eq. (11.13) where radial strains were on the order of 10 % and neither the deformation nor the stress state resembled that of pure torsion.