Progressive failure analysis of thin-wall aluminum extrusion under quasi-static and dynamic loads

Elastoplasticity

Hooputra et al. (2004) have shown that the extruded aluminum alloy EN AW-7108 T6 displays plastic orthotropy due to the nature of the extrusion processing and have used the Barlat symmetric yield locus (Barlat et al., 1991) to fit the experimental data. In this example we neglect the orthotropy and assume both the elastic and the plastic behavior to be isotropic with the yield surface described by the Mises yield function (see Inelastic Behavior). The assumption of isotropic plasticity may appear to be too restrictive for the accurate prediction of failure in extruded alloys. However, in crashworthiness simulations the assumption of isotropy usually yields acceptable results when compared with experimental observations, as shown in the results obtained in this example. Nevertheless, you should compare your simulation results with experimental data to check the validity of the isotropic plasticity assumption.

Damage initiation

Metal sheets and thin-walled extrusions made of aluminum alloys may fail due to one or a combination of the following failure mechanisms (Hooputra et al. 2004): ductile failure due to nucleation, growth, and coalescence of voids; shear failure due to fracture within shear bands; and failure due to necking instabilities. If the model consists of shell elements, a criterion for the last failure mechanism is necessary because the size of the localized neck is of the order of the sheet thickness and, hence, cannot be resolved with shell elements of dimensions one order of magnitude larger than the thickness.

Abaqus/Explicit offers a number of damage initiation criteria to model the onset of necking instabilities in sheet metals. These include the Forming Limit Diagram (FLD), Forming Limit Stress Diagram (FLSD), Müschenborn-Sonne Forming Limit Diagram (MSFLD), and Marciniak-Kuczynski (M-K) criteria. The first three criteria utilize the experimentally measured forming limit curves in the appropriate strain or stress spaces. The last criterion introduces virtual thickness imperfections in the sheet metal and analyzes the deformation in the imperfection zone to determine the onset of the instability (see Damage Initiation for Ductile Metals).

The strain-based FLD criterion is limited to applications where the strain path is linear. On the other hand, the stress-based FLSD criterion is relatively insensitive to changes in the strain path. However, this apparent independence of the stress-based limit curve due to the strain path may simply reflect the small sensitivity of the yield stress to changes in the plastic deformation. The M-K criterion can capture the effects of nonlinear strain paths accurately; however, it is computationally expensive, especially if large numbers of imperfection orientations are introduced. It has been verified that the results obtained using the MSFLD criterion are similar to those obtained using the M-K criterion but with a much reduced computational expense (see Progressive damage and failure of ductile metals). Therefore, in this example we choose the MSFLD damage initiation criterion for necking instability.

For specifying the MSFLD damage initiation criterion, the forming limit curve of the material is required. In Abaqus this criterion can be specified by converting the forming limit curve from the space of major versus minor strains to the space of equivalent plastic strain versus ratio of principal strain rates. Abaqus also allows direct specification of the forming limit curve for the MSFLD criterion (see Müschenborn-Sonne Forming Limit Diagram (MSFLD) Criterion). We use the forming limit curve based on the experimental work of Hooputra (2005). This curve is assumed to be valid at both the quasi-static and the dynamic strain rates. The parameter OMEGA used in conjunction with the MSFLD criterion to provide filtering of numerical noise in the evaluation of the ratio of principal strain rates is set to 0.001 in both cases (see Damage Initiation for Ductile Metals); this value is recommended for crashworthiness simulations.

Damage due to initiation, growth, and coalescence of voids leads to ductile failure in metals; the formation of cracks within shear bands leads to shear failure. Abaqus offers phenomenological damage initiation criteria for both of these mechanisms. The ductile criterion is specified by providing the equivalent plastic strain at the onset of ductile damage as a function of stress triaxiality and strain rate. Similarly, the shear criterion is specified by providing the equivalent plastic strain at the onset of shear damage as a function of shear stress ratio and strain rate (see Damage Initiation for Ductile Metals). The data required for both of these criteria may be difficult to obtain through direct experimentation since it would require experiments spanning a range of stress triaxiality and shear stress ratio that may be difficult to achieve. Hooputra et al. (2004) have given simplified analytical expressions for the ductile and the shear failure criteria that require only a limited number of experiments. In this example we adopt those expressions; however, we ignore the orthotropy of the ductile fracture to be consistent with the assumption of isotropic plasticity made earlier.

For the ductile damage initiation criterion the equivalent plastic strain is given by the following function of the stress triaxiality, $\eta$ (Hooputra et al, 2004):

where ${\epsilon }_T^{+}$ and ${\epsilon }_T^{-}$ correspond to the equivalent plastic strain at ductile damage initiation for equibiaxial tensile and equibiaxial compressive deformation, respectively. For isotropic materials the stress triaxiality in equibiaxial tensile deformation state, ${\eta }^{+}$ , is $2/3$ , and in equibiaxial compressive deformation state, ${\eta }^{-}$ , is $-2/3$ . The definition of $\eta$ in Abaqus, as a ratio of the equivalent mean stress to the Mises equivalent stress, differs from that used by Hooputra et al. (2004) by a factor of $1/3$ . Consequently, the value of $k_0$ used in the above expression is three times the value used in Hooputra et al. (2004). The above expression has three parameters that must be obtained experimentally: ${\epsilon }_T^{+}$ , ${\epsilon }_T^{-}$ , and $k_0$ . These parameters depend on the material, strain rate, and possibly the temperature. For each strain rate of interest, three experiments are needed at different values of stress triaxiality to obtain the three material parameters. ${\epsilon }_T^{+}$ can be obtained directly from the Erichsen test ( $\eta ={\eta }^{+}$ ). Three-point bending of sheet coupons (with width/thickness > 4) under plane strain tension ( $\eta =1/\sqrt{3}$ ) and fracture at the notch root of waisted tensile coupons in uniaxial tension ( $\eta =1/3$ ) may provide two additional experiments to determine ${\epsilon }_T^{-}$ and $k_0$ . In the Erichsen and three-point bending experiments the local fracture strain can be derived by placing a grid on the specimen's surface; in the waisted tensile experiment the fracture strain can be obtained from the sheet thickness in the fracture plane (Hooputra et al., 2004). For the aluminum alloy used in this example the experimentally obtained ductile failure parameters at quasi-static and dynamic strain rates (250 s⁻¹) are listed in Table 1.

For the shear damage initiation criterion the equivalent plastic strain at the onset of damage is given by the following function of the shear stress ratio, ${\theta }_s$ (Hooputra et al, 2004):

where ${\theta }_s=\left(1-k_s\eta \right)/\varphi$ with $\varphi ={\tau }_{max}/{\sigma }_{eq}$ , and ${\epsilon }_S^{+}$ and ${\epsilon }_S^{-}$ correspond to the equivalent plastic strain at shear damage initiation for equibiaxial tensile and equibiaxial compressive deformation, respectively. The parameters ${\theta }_s^{+}$ and ${\theta }_s^{-}$ correspond to the values of ${\theta }_s$ at $\eta ={\eta }^{+}$ and $\eta ={\eta }^{-}$ , respectively. This expression has four parameters that must be determined experimentally: $k_s$ , ${\epsilon }_S^{+}$ , ${\epsilon }_S^{-}$ , and $f$ . These parameters depend on the material and strain rate. Hooputra et al. (2004) have used tensile specimens with a groove (rectangular cross-section and groove depth=half the sheet thickness) at ${45}^{\circ }$ to the loading direction ( ${\theta }_s=1.469$ ), specially designed tensile specimens with a groove parallel to the loading direction (pure shear, ${\theta }_s=1.732$ ), and Erichsen tests ( ${\theta }_s=1.6$ ) in conjunction with the above expression to determine ${\epsilon }_S^{+}$ , ${\epsilon }_S^{-}$ , and $f$ . The value of the material parameter $k_s$ is taken as 0.3. For the aluminum alloy used in this example the experimentally obtained shear failure parameters at quasi-static and dynamic strain rates (250 s⁻¹) are listed in Table 2.

Using the aforementioned expressions and the material parameters listed in Table 1 and Table 2, tabular data for ductile and shear damage initiation criteria can be generated as a function of stress triaxiality and shear stress ratio, respectively. This tabular data is provided in the Abaqus input files. The above expressions may give very high values of the equivalent plastic strain at damage initiation when the stress triaxiality or the shear stress ratio is very small. A cutoff value of the equivalent plastic strain can be provided in such cases.

Damage evolution and element removal

Damage evolution occurs once the damage initiation criteria are satisfied. Plastic displacement-based linear damage evolution law is used for each of the three damage initiation criterion. The value of the plastic displacement at which the damage variable reaches 1 is taken as 0.1. The default maximum degradation rule is used, and the elements are removed from the mesh once the maximum degradation has occurred (see Maximum Degradation and Choice of Element Removal).