Product: ABAQUS/Explicit
This example problem is used to verify the stiffness proportional material damping available via the *DAMPING option. 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 2.2.30–1. 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.
The materials are defined with either the *ELASTIC or the *HYPERELASTIC options. 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 hyperelastic material is a Mooney-Rivlin material, with the constants (for the polynomial strain energy function) 
551.6 kPa (80.0 lb/in2), 
137.9 kPa (20 lb/in2), and 
4.5322 × 10–3 kPa–1 (0.03125 psi–1). Its density is 1.069 × 107 kg/m3 (1.0 lbsec2 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 on the *DAMPING option for both materials is 0.01. 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 *BULK VISCOSITY option has been used to set the linear and quadratic bulk viscosities to zero. This isolates the effects of the stiffness proportional damping.
The time history of the energies for the C3D8R element model is shown in Figure 2.2.30–2. The value of ALLVD represents the amount of energy lost due to damping. When the stress pulse is between the ends of the bar, the kinetic and strain energies are equal. When a stress wave hits a free surface, the wave is reflected and its sign is reversed. Therefore, when the first half of the wave has hit the free end, the wave that it reflects exactly cancels the tail end of the original wave. At this point all the strain energy in the system has been converted to kinetic energy. Once the wave completely reflects off the end, half of the kinetic energy is transferred back to strain energy. As expected, the wave amplitude decreases. All other element types tested produce similar results.
This problem tests stiffness proportional material damping for all the available material models, but it does not provide independent verification.
Three-dimensional solid elements, elastic material definition.
Plane strain elements, elastic material definition.
Plane stress elements, elastic material definition.
Axisymmetric elements, elastic material definition.
Shell elements, elastic material definition.
Membrane elements, elastic material definition.
Two-dimensional beam elements, elastic material definition.
Three-dimensional beam elements, elastic material definition.
Two-dimensional truss elements, elastic material definition.
Three-dimensional truss elements, elastic material definition.
Three-dimensional solid elements, hyperelastic material definition.
Plane strain elements, hyperelastic material definition.
Plane stress elements, hyperelastic material definition.
Axisymmetric elements, hyperelastic material definition.
Shell elements, hyperelastic material definition.
Membrane elements, hyperelastic material definition.
