4.2.3 The Virtual Fields Method In this study, the virtual fields method (VFM) was used for an inverse method to retrieve the constitutive parameters from the measured deformation fields. The VFM makes use of the principle of virtual work which describes the condition of global equilibrium. The equilibrium equation in the case of elasto-plasticity for dynamic loading, and in absence of body forces, can be written as follows: ð V ð t 0 _σi jdt 2 4 3 5 ε* i jdVþð Sf Tiu * i dS ¼ð V ρaiu * i dV ð4:3Þ where _σ is the stress rate which is a function of _ε (actual strain rate), σ (actual stress) and unknown constitutive parameters, V the measurement volume, T the distribution of applied forces acting on Sf, ε* the virtual strain field derived fromu* (the virtual displacement field), ρ the density and a the acceleration. Material parameters can be determined from the acceleration fields by choosing proper virtual fields which can get rid of the external virtual work (EVW) term including the loads. For an elasto-plasticity problem, the identification is carried out using an iterative procedure [6] to minimize the quadratic gap between the internal virtual work (IVW) and the acceleration term (the right hand side of (4.3)). Since the constitutive parameters are unknown, initial guesses are required to initiate the iteration. Then, the stress components are recalculated until the equilibrium equation is satisfied by updating the parameters. Nelder-Mead algorithm was used for the minimization. In this study, simple virtual fields were applied to find the material parameters as in (4.4). u* x ¼0, u * y ¼ y ymin ð Þ y ymax ð Þ ð4:4Þ where y is the vertical coordinate of the measurement points in the current (deformed) configuration. The chosen virtual fields cancels out the EVW term. The parameters were determined in less than 5 min. 4.2.4 Speed and Acceleration The speed fields can be obtained from the measured displacement fields using simpe finite difference. vi t þ Δt 2 ¼ ui t þΔt ð Þ ui tð Þ Δt ð 4:5Þ where i can be either x or y and t is time. The acceleration fields can be computed from the displacement fields by double temporal differentiation as in (4.6). Due to the nature of the quantities, the speed is defined at time t + Δt/2 and the acceleration is at time t. ai tð Þ¼ ui t þΔt ð Þþui t Δt ð Þ 2ui tð Þ Δt2 ð 4:6Þ 4.3 Results 4.3.1 FE Model A specimen geometry was chosen as in Fig. 4.1. The dimensions of the specimen were 80 mm (height) 30 mm (width) 1 mm (thickness). The width for the AOI was 10 mm. In ABAQUS/Explicit, input parameters for Swift hardening law were; K: 1300MPa, ε0: 0.0024, n: 0.16. Those parameters were obtained from a static uniaxial tensile test on a dual phase (DP) 780 steel specimen. Dynamic tensile tests were simulated by constraining the lower edge and by applying vertical load at the upper edge. During the deformation, the deformed coordinates of each measurement point were saved at evenly spaced 4 Characterization of the Dynamic Strain Hardening Behavior from Full-field Measurements 25
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