Product: Abaqus/Explicit
Large deformation kinematics, elastic-plastic material with strain hardening, user material, multi-point constraints.
The rotating cylinder problem was proposed by Longcope and Key (1977) as a means of exercising finite rotation algorithms. In this problem a cylinder with an initial angular velocity of 4000 rad/sec and a zero initial stress state is modeled. (This is physically impossible because the body forces would generate a stress field under this angular velocity. Nevertheless, these initial conditions are acceptable, since this is merely a numerical experiment.) The inside of the cylinder is subjected to an instantaneous application of a pressure of 67.3 MPa (9760 psi).
The elastic material properties are defined as Young's modulus of 71 GPa (1.03 × 107 psi), Poisson's ratio of 0.3333, and density of 2680 kg/m3 (2.508 × 10–4 lb sec2 in–4). An isotropic hardening plasticity model is used with an initial yield of 286 MPa (4.15 × 104 psi) and constant hardening modulus of 3.565 GPa (5.17 × 105 psi).
Only one-quarter of the ring is modeled using the *EQUATION and *MPC options to enforce the repeated symmetry boundary condition.
The *ORIENTATION option is used to define a local cylindrical coordinate system at each material point of the mesh.
The first case considered is a two-dimensional model using CPE4R elements. In this case two meshes are defined in the same problem, as shown in Figure 4.1.37–1. The lower mesh in Figure 4.1.37–1 uses the built-in Mises isotropic hardening plasticity model (*PLASTIC). The upper mesh in Figure 4.1.37–1 employs user subroutine VUMAT (*USER MATERIAL) with the kinematic hardening Mises model described in the Abaqus Analysis User's Manual. Figure 4.1.37–2 shows the time history of the maximum principal stress in the two-dimensional model for both cases. Figure 4.1.37–3 shows the time history of equivalent plastic strain in the two-dimensional model for both cases. Figure 4.1.37–4 shows the energy histories in the two-dimensional model. The energy history is particularly important in this analysis because it demonstrates that there is no energy lost in the enforcement of multi-point constraints.
The second case is a three-dimensional representation of the same problem using shells, membranes, and brick elements to model the ring with suitable boundary conditions to reproduce closely the original two-dimensional model. The built-in Mises isotropic hardening plasticity model is used. The meshes for the three-dimensional case are shown in Figure 4.1.37–5. Figure 4.1.37–6 shows the time history of the maximum principal stress in the three-dimensional model for both cases. Figure 4.1.37–7 shows the time history of the equivalent plastic strain in the three-dimensional model for both cases. Figure 4.1.37–8 shows the energy histories in the three-dimensional model. Note that each energy quantity is summed over the two cases.
The results compare well with those obtained by Longcope and Key (1977).
Input data for the two-dimensional case.
VUMAT subroutine for the two-dimensional case.
Input data for the three-dimensional case.
Longcope, D. B., and S. W. Key, “On the Verification of Large Deformation Inelastic Dynamic Calculations through Experimental Comparisons and Analytic Solutions,” PVP-PB-023, American Society of Mechanical Engineers, 1977.
