Finite element analysis is widely used in research and industry. In some cases, such as (Tarigopula, et al. 2008) and (Wierzbicki, et al. 2005), it is used to determine values such as stress triaxiality that could not be measured directly on a physical test. This approach will not be used here as it requires the assumption that the model is accurate. This is risky for modelling a material up to fracture as a large portion of the deformation will take place post-necking. The material input to a finite element simulation is typically extrapolated from pre-necking stress-strain curves. Higher strains are required for crash simulations and these can be obtained from layered sheet compression tests (Lanzerath, et al. 2007). However, it is not simple to obtain an accurate flow curve this way as the friction and shear from the die needs to be accounted for (Becker, Pöhlandt and Lange 1989). (Wierzbicki, et al. 2005) presented a flow curve extrapolated beyond necking that compared favourably with a value calculated from area reduction at fracture.
Page | 38 This section will discuss the elements, control, loading and material inputs to a finite element simulation suitable for designing coupon geometries. A sample file is shown in Appendix A2.
2.4.1 – Elements
There are two main types of elements – shell and solid.
Shell elements are substantially faster to process (Xiao, Hsiung and Zhao 2008). They are designed to be plane-stress and as such are intended to be significantly wider than they are thick. This requirement is to ensure accuracy in bending and folding. However, for tensile coupons, there is no out of plane loading and smaller elements may be used.
Solid elements give very accurate results as they correctly model stress in all three directions. However, they are not usually suitable for sheet material as they need to be approximately cubic to avoid numerical instabilities. This means that their dimensions in- plane need to be similar to the through-thickness dimension, which increases the number of elements and hence the processing time. To increase the accuracy in bending, multiple elements through thickness are required, which increases the computational time further. Within both shell and solid elements, there are several types available. These are essentially the number of integration points in the element and the way they are calculated. An element may be under-integrated or fully integrated. The former has one integration point at the centre of the element, whereas the latter has one per node. Under- integrated nodes are faster to process but can be too soft and suffer from hourglass modes, which is a mathematical instability that results in zero energy deformation. Fully integrated elements have no hourglassing and are more accurate but are more computationally expensive and more likely to have negative volumes at large deformations. Poor quality meshes with triangles or heavily distorted elements can also be overly stiff.
Page | 39 As shell elements are the standard for industrial modelling, they will be used in this thesis wherever possible to ensure that the techniques developed are compatible. Fully integrated elements will be used for the improved accuracy.
2.4.2 – Control
Implicit simulations are typically used for low speed deformation while explicit are used for high speed. This is because explicit calculations have a small time step for the element size, on the order of , which means that a lot of processor time is required to simulate low speed deformation. It is possible to run a model at a high velocity as an explicit simulation and obtain results similar to quasi-static testing, though there are two requirements. Firstly, the material model needs to be strain rate insensitive, otherwise it will predict the wrong strain hardening response. Secondly, the kinetic energy of the system needs to remain lower than of the internal energy (Prior 1994) and (Choi, et al. 2002).
An explicit simulation can be run at a higher time step by mass scaling. The time step is determined by √( ⁄ ) and thus adding mass to the system can increase the rate at which the simulation runs. Reducing the stiffness would have the same effect, though it would also invalidate the results.
Implicit simulations can suffer from non-convergence. As such, this work will use explicit simulations, run at a higher speed where necessary to reduce the computational time.Solid elements will be used if there are significant through thickness effects that are not correctly accounted for by shells.
2.4.3 – Loading
Specimen loading in finite element analysis can be done with either force control or displacement control. The testing machines are displacement controlled and so that
Page | 40 approach is used here too. The moving grip is assigned an initial velocity and a boundary prescribed motion.
This velocity is maintained until the strain in the expected fracture location reaches . The velocity can be a function of time, which would enable a simulation to accurately match the ramping of velocity in the low speed system and the drop-off associated with the grips clamping in the high speed system. However, this behaviour is unwanted and assumed to be negligible, mostly occurring at low strain. Therefore, the loading will be modelled as a constant velocity.
2.4.4 – Material
There are many different material models available in each finite element package. The main material being studied here, DP800 steel, is isotropic. This means that the LS-DYNA material MAT_PIECEWISE_LINEAR_PLASTICITY can be used (Du Bois, et al. 2006). This model uses Young's modulus with a limiting stress to model elasticity, while plasticity is modelled by a monotonically increasing curve with 100 samples. Strain rate dependency is modelled by using a table of plastic flow curves at different strain rates.
Failure can be simulated with this material model. Elements are deleted when they reach a threshold strain. This corresponds to the constant effective strain discussed in Section 1.1.3. This function will not be used as the fracture strain is not known in advance and it is useful to study the material behaviour beyond the expected limit.
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