Notched beam under cyclic loading

The geometry and mesh are shown in Figure 1. Figure 2 shows the discretization in the vicinity of the notch, which is the region of interest in this analysis. Only one-half of the beam is modeled since the geometry and loading are symmetric with respect to the x= 0 plane. All dimensions are given in millimeters. The beam is 1 mm thick and is modeled with plane strain, second-order, reduced-integration elements (type CPE8R). The mesh is chosen to be similar to the mesh used by Doghri (1993). No mesh convergence studies have been performed.

The material properties reported by Doghri (1993) for a low-carbon (AISI 1010), rolled steel are used in this example.

A Young's modulus of E= 210 GPa and a Poisson's ratio of $\nu$ = 0.3 define the elastic response of the material. The initial yield stress is ${\sigma |}_0$ = 200 MPa.

The nonlinear evolution of the center of the yield surface is defined by the equation

where $\mathit{\alpha }$ is the backstress, ${\sigma }^0$ is the size of the yield surface (size of the elastic range), ${\overline{\epsilon }}^{pl}$ is the equivalent plastic strain, and C = 25.5 GPa and $\gamma$ = 81 are the material parameters that define the initial hardening modulus and the rate at which the hardening modulus decreases with increasing plastic strain, respectively. The quantity $\sqrt{2/3}C/\gamma =$ 257 MPa defines the limiting value of the equivalent backstress $\overline{\alpha }=\sqrt{{\mathit{\alpha }}^{dev}:{\mathit{\alpha }}^{dev}}$; further hardening is possible only through the change in the size of the yield surface (isotropic hardening).

The isotropic hardening behavior of this material is modeled with the exponential law

where ${\sigma }^0$ is the size of the yield surface (size of the elastic range), $Q_{\mathrm{\infty }}$ = 2000 MPa is the maximum increase in the elastic range, and b = 0.26 defines the rate at which the maximum size is reached as plastic straining develops.

The material used for this simulation is cold rolled. This work hardened state is represented by specifying an initial equivalent plastic strain ${{\overline{\epsilon }}^{pl}|}_0$ = 0.43 (so that ${\sigma }^0$ = 411 MPa) and an initial backstress tensor

The beam is subjected to a 4-point bending load. Since only half of the beam is modeled, the model contains one concentrated load at a distance of 26 mm from the symmetry plane (see Figure 1). The pivot point is 42 mm from the symmetry plane. The simulation runs 3 1/2 cycles over 7 time units. In each cycle the load is ramped from zero to 675 N and back to zero. An amplitude curve is used to describe the loading and unloading. The increment size is restricted to a maximum of 0.125 to force Abaqus to follow the prescribed loading/unloading pattern closely.

This example demonstrates the Neuber and Glinka plasticity corrections for estimating the plastic solution near a notch region from the stress results of a linear elastic analysis. The plasticity correction capabilities require the definition of the plastic response of the material. This example uses an isotropic Mises plasticity definition that matches the uniaxial response of the original combined-hardening material. Figure 6 shows the isotropic hardening definition obtained this way.

The notched beam is subject to a load that increases monotonically from 0 N to 300 N. The results of the plasticity corrections are compared to those from a full nonlinear elastic-plastic analysis.

Figure 7 and Figure 8 show the distributions of stresses and equivalent plastic strains at the end of the analysis. In Figure 7 the stresses obtained by the nonlinear elastic-plastic analysis are compared with the Neuber and Glinka stresses evaluated by performing purely elastic analysis, and Figure 8 shows a similar comparison for equivalent plastic strains. The Neuber and Glinka stresses and plastic strains show good agreement with the values obtained by the elastic-plastic analysis as is expected for such localized plastic deformations. Neuber’s rule predictions of stresses and strains are higher (more conservative) than Glinka’s results as is usually the case.