7 Use of Bulge Test Geometry for Material Property Identification 45 where t is the membrane thickness, Qis the in-plane stiffness from DQ", and is Poisson’s ratio. The simplification in Eq. 7.4 is unique for this geometry because all shear components are 0. Similarly, for the virtual work for bending V bending DZ V x" x C y" y C s" s dV DDZ S kxk x Ckyk y C 1 2 ksk s dSC DZ S kxk y Ckyk x 1 2 ksk s dS DDZ S kxk x Ckyk y dSC DZ S kxk y Ckyk x dS D 1 12 Qt3 Z S kxk x Ckyk y dSC 1 12 Qt3 Z S kxk y Ckyk x dS (7.5) where ki are the component curvatures andDis the bending stiffness, that is DDQt 3=12. Final implementation of PVW requires strain-displacement relations which are "x D @u @x (7.6) "y D @v @y (7.7) kx D @2w @x2 (7.8) ky D @2w @y2 (7.9) where u, v are the displacements in the x, y directions, respectively. Substituting virtual displacements in Eqs. 7.6, 7.7, 7.8, and 7.9 gives the relations for the virtual strains and curvatures. 7.4 Discussion and Conclusion Use of the formulation in the previous section presents some experimental challenges. The curvatures, ki, are assumed to be uniform through the material thickness. The in-plane strains, "i, are those at the thickness mid-point. Surface strain measurement is a combination of bending and membrane strains, which need to be separated for use in PVW. An obvious, but not trivial, experimental solution is to measure the strains on both the front and back surfaces of the membrane. A pointby-point average of front and back surface strain will be the membrane strain. Recognizing that k6 D0, the curvatures are related to the surface strains by ki D 1 t " front i " back i (7.10) Full field displacements are required for this analysis, including w, the out-of-plane displacement. Most full field displacement techniques require a reference image, including DIC, however capturing a reference image is problematic. If the specimen is exhibiting membrane-like behavior then a flat reference image is difficult to produce. Use of a reference image with very low magnitude q must be used with care because the problem is nonlinear, e.g. increasing the pressure magnitude to 2qdoes not increase the central deflection to 2w. The presentation will include a FEA simulation and experimental results demonstrating the use of VFM with the bulge geometry to identify material behavior.
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