74 A. Alshaya et al. the FEM model. FEMU-U is attractive because displacements are first-order outputs of high-resolution full-field techniques of DIC and ESPI where strain is a second-order output and has greater noise associated with numerical differentiation. In 2-D models, the degree of freedom is (number of nodes) 2 – (number of constitutive parameters) – 1. For homogeneous, isotropic materials, the number of constitutive properties is two (E, v); for homogeneous, orthotropic materials, the number of constitutive parameters are four (E11, E22, G12, v12). For either case, the number of degrees of freedom is large and the problem is solved by minimizing least squares of the chosen cost function. The goal of this work is to evaluate the constitutive properties of a composite plate containing symmetrically-located sided-notches and vertically loaded in the strongest/stiffest material direction using IM and Airy stress function scheme. The authors are unaware of prior utilization of mapping and complex variables to experimentally determine the constitutive properties in notched composites from displacement data. 12.2 Relevant Equations For plane problems having rectilinear orthotropy and no body forces, the Airy stress function, F, can be expressed as a summation of two arbitrary analytical functions, F1 (z1) andF2 (z2), of the complex variables, z1 andz2, as [13] F D2ReŒF1 .z1/ CF2 .z2/ (12.1) such that zj DxC jy for j D1,2 and Re denotes the ‘real part’ of a complex number. The complex material properties 1 and 2 depend on the constitutive properties. The displacements in rectangular coordinates (x, y) of the physical z(DxCiy) plane can be expressed in terms of the stress functions. By introducing the new stress functions ˆ.z1/ D dF1 .z1/ dz1 ; and ‰.z2/ D dF2 .z2/ dz2 (12.2) one can write the displacements as u D2ReŒp1ˆ.z1/ Cp2‰.z2/ woy Cuo (12.3) v D2ReŒq1ˆ.z1/ Cq2‰.z2/ CwoyCvo (12.4) where wo , uo, and vo are constants of integration and characterize any rigid body translations (uo andvo) and rotation (wo). The other quantities, which depend on material properties, are p1 D 2 1 E11 12 E11 ; p2 D 2 2 E11 12 E11 ; q1 D 12 E11 1 C 1 E22 1 ; q2 D 12 E11 2 C 1 E22 2 (12.5) When the plate is loaded physically in a testing machine, the rigid body motions, uo , vo ,andwo are zero. Plane problems of elasticity classically involve determining the stress functions, ˆ(z1)and‰(z2), throughout a component and subject to the boundary conditions around its entire edge. For a region of a component adjacent to a traction free-edge, ˆ(z1) and ‰(z2) can be related to each other by the conformal mapping and analytic continuation techniques. The displacements can then be expressed in terms of the single stress function, ˆ(z1). Moreover, ˆ(z1) will be represented by a truncated power-series expansion whose unknown complex coefficients are determined experimentally. Once ˆ(z1) and ‰(z2) are fully evaluated, the individual displacements are known from Eqs. 12.3 through 12.4. For a significantly large region of interest in a finite structure, it may also be necessary to satisfy other boundary conditions at discrete locations. Conformal mapping is introduced to simplify the plane problem by mapping the region Rz of a complicated physical zDxCiy plane of a loaded component into a region R of a simpler shape in the D Ci plane, the latter being a unit circle if one represents the stress function as a Laurent series, Fig. 12.1 [13–21]. The new coordinate system (and resulting geometry) is usually chosen to aid in solving the equations and the obtained solution from this simplified domain can then be mapped back to the original physical geometry for a valid solution. Assume that a mapping function of the formzD¨( ) exists and which maps R of the simpler plane into Rz of themore complicated physical plane. For orthotropy, auxiliary planes and their induced mapping functions are defined in terms of j D C j , thereforezj D¨j( j), for j D1 , 2. The induced conformal mapping functions are one-to-one and invertible. The
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