Material damping in Abaqus/Explicit

This example problem is used to verify stiffness proportional material damping. A one-dimensional wave is propagated through a single row of elements and allowed to attenuate over time. Both continuum and structural elements are used. The C3D8R element model is shown in Figure 1. The row of elements is restrained on one side in the y-direction for the two-dimensional element models and restrained in the y- and z-directions for the three-dimensional element models. All the models are free at both ends in the x-direction. For the structural elements the loading is in-plane and all the rotational degrees of freedom are fixed. The damping will cause the amplitude and the frequency of the initial pulse to decrease until the internal energy of the system becomes zero and the bar has a constant longitudinal velocity.

Linear elastic, equation of state, and hyperelastic materials are tested. The elastic material has Young's modulus of 4.4122 × 10⁸ N/m² (6.4 × 10⁴ lb/in²), Poisson's ratio of 0.33, and density of 1.069 × 10¹⁰ kg/m³ (1.0 × 10³ lb sec² in⁻⁴). The behavior of the equation of state material is equivalent to that of the linear elastic material. The elastic shear modulus is 1.6589 × 10⁸ N/m² (2.4060 × 10⁴ lb/in²). For the tabular equation of state material model, the functions are defined as $f_1\left({\epsilon }_{vol}\right)=-K{\epsilon }_{vol}$ and $f_2\left({\epsilon }_{vol}\right)=$ 0.0, where $K=$ 4.3261 × 10⁸ N/m² (6.2745 × 10⁴ lb/in²). The linear $U_s-U_p$ type of equation of state material has $c_0=$ 2.0120 × 10^-1 m/sec (7.9212 in/sec). The hyperelastic material is a Mooney-Rivlin material, with the constants (for the polynomial strain energy function) $C_{10}=$ 551.6 kPa (80.0 lb/in²), $C_{01}=$ 137.9 kPa (20 lb/in²), and $D_1=$ 4.5322 × 10⁻³ kPa⁻¹ (0.03125 psi⁻¹). Its density is 1.069 × 10⁷ kg/m³ (1.0 lb sec² in⁻⁴). In both cases the densities have been increased to slow the wave speed down so that the wavelength of the stress pulse is just shorter than the length of the bar.

The stiffness proportional damping coefficient for both materials is 0.01. A variable stiffness proportional damping can also be defined by specifying the damping coefficient as a tabular function of temperature and/or field variables in Abaqus/Explicit. A large damping coefficient is chosen to illustrate clearly the effects of material damping. In general, this material property is meant to model low level damping of the system, in which case the value of the damping coefficient will be much smaller. In all cases the linear and quadratic bulk viscosities are set equal to zero. This isolates the effects of the stiffness proportional damping.

The models are also used to test band-limited damping. In all cases, the stiffness proportional damping is replaced by band-limited damping. The damping parameters are chosen as the damping ratio ${\eta }_d=$ =0.102525, the high-frequency cutoff $f_h=$ 10 (Hz), and the low-frequency cutoff $f_l=$ 4 (Hz).