measured in the deformed configuration (finding for example the intersection between the grid lines and the crack edge), but are not known in the undeformed configuration. As such, their initial positions are to be guessed, and then optimized according to some minimization criterion. The optimization criteria may vary from application to application; here two similar approaches have been adopted and compared: the former involves the minimization of the global strain energy, the latter consists in minimizing the strain difference (in a tensorial sense) between the conjugate boundaries of the two sub-elements. Obviously their results depends on the nodal displacements, however, preliminary virtual tests showed that they are likely to provide quite similar results, as shown in Fig. 4.3b, c. What is important is that both methods permit to compute the strain field up to the free edges of the through crack; this already represents an upgrading of the classical grid method where the elements cut by the crack are inevitably omitted from the strain computation. In order to assess the capability of the proposed method, two virtual experiments were conducted, the first regarding the measurement of strain in elements that are totally cut by the crack, the second dealing with the elements containing the crack tip. 4.3 First Virtual Experiment In the first virtual experiment, a FE model was used to simulate a sheet metal that is deformed by a spherical bulge. The sheet metal is a square of 100 mm size, and 1 mm thickness, whereas the rigid punch has a radius of 20 mm and is moved upward of 25 mm after the initial contact occurring at the bottom surface of the sheet metal. The sheet metal was assumed to have an elastic modulus of 200 GPa, a plastic tangent modulus of 1 GPa, and a yield stress of 200 MPa. The sheet metal has been modelled with shell 43 elements, 0.5 mm in size, by means of Ansys ® software, exploiting the double symmetry with respect tox andy axes (Fig. 4.4a). The nodes at the external sides were fixed, while the nodes on the internal side obeyed the symmetry conditions, exception made for the nodes from the centre to the x ¼24 mm coordinate, that was constrained in the y direction by means of uniaxial non-linear springs; the springs have an almost infinite stiffness up to 205 N, then their stiffness vanishes. In this way it was possible to reproduce a plausible fracture initiation and propagation, with accumulation of plastic strain at the crack edges, without the need for element kill (which would correspond to material loss). The nodal results of this simulation computed at the middle of the shell thickness, see Fig. 4.4b, were used as a virtual object to be acquired by means of optical grid method, both the classical one and the extended method here proposed. The first principal strain distribution, as computed by the fine meshed FE simulation, mapped onto the undeformed configuration, is reported in Fig. 4.5, where the symmetry has been expanded to show the entire map. Now, if the FEM shape is considered as real, the nodal displacement data can be “sub-sampled” thus to simulate what would happen in the optical grid method, where the markers are evenly spaced in the undeformed configuration; here a regular grid of 4 mm pitch is considered. X Y Z NODAL SOLUTION STEP=23 SUB =22 TIME=25 UZ TOP RSYS=0 DMX =25 SMN =-.318E-04 SMX =25 X Y Z ZV =1 *DIST=29.8803 *XF =24.014 *YF =23.1624 *ZF =-10 Z-BUFFER -.318E-04 2.77775 5.55553 8.33331 11.1111 13.8889 16.6667 19.4444 22.2222 25 fixed fixed fixed fixed a b Fig. 4.4 FEmodel (a) mesh and constraints, (b) resulting out-of-plane displacements 42 M. Sasso et al.
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