Chapter12 Determination of Constitutive Properties in Inverse Problem Using Airy Stress Function A. Alshaya, John M. Considine, and R. Rowlands Abstract A new inverse problem formulation is developed using the Airy stress function. Inverse methods are used to determine the constitutive properties of a graphite/epoxy laminated composite loaded vertically by processing measured values of v-displacement component with an Airy stress function in complex variables. Displacements are recorded using digital image correlation. The traction-free conditions on the symmetrically located sided notches are satisfied analytically using conformal mappings and analytic continuation. The traction-free on the vertical free edge and a symmetrical condition on horizontal line of symmetry are imposed discretely. The primary advantage of this new formulation is the direct use of displacement data, eliminating the need for numerical differentiation when strain data is required. The inverse method algorithm determined the constitutive properties with errors range from 2% to 10%. Selection of Airy coefficients, test geometry configuration and comparison with other inverse methods will be addressed. Keywords Composites • Airy stress function • Inverse problems • Digital image correlation • Complex variables 12.1 Introduction The Airy stress function in complex variables was used extensively in determining stresses from measured displacements [1–4]. The Airy stress function can be processed with other measured data using thermoelasticity [5–7], photoelasticity [8], digital image correlation [2], moiré [9] or strain gages [10]. These hybrid methods do not necessitate knowing the applied loads, smooths the measured data and determines individual stresses throughout, including on the edge of the hole. All of the prior applications of the mapping technique evaluated the stresses by using the constitutive properties found experimentally from standard tensile tests whereas the present approach only evaluated these properties using the measured displacement from Digital Image Correlation. One method of evaluating constitutive properties of orthotropic materials is the use of inverse methods (IM). Avril and Pierron [11] reviewed several IM approaches and showed their general equivalency. IM can be generally described as the iterative adjustments of parameters (constitutive properties) in a numerical model (Airy stress function scheme) to minimize the difference between an experimentally measured quantity (displacement) and the numerically calculated quantity. By comparing FEM calculated out-of-plane displacement with those measured by shadow moiré, Le Magorou et al. [12] determined bending/torsion rigidities in composite wood panels by the resolution of IM. Molimard et al. [13] evaluated constitutive properties of a composite material by minimizing the difference between moiré-measured displacements and those predicted by FEM in a perforated tensile plate. Similarly, Genovese et al. [14] used IM procedures to evaluate a truss system and a composite plate. Considine [15] determined material properties in heterogeneous materials from full-field simulated displacement data using IM. Each of these references incorporated a specific type of IM entitled FEMU-U (finite element method updating – displacement). The root mean square of displacement differences, also called a cost function, between the measured values and those predicted by FEM are minimized by iteratively changing constitutive properties in A. Alshaya ( ) Kuwait University, Kuwait, Kuwait e-mail: alshaya@wisc.edu J.M. Considine Materials Research Engineer, U.S. Forest Service, Forest Products Laboratory, One Gifford Pinchot Drive, Madison, WI 53726, USA e-mail: jconsidine@wisc.edu R. Rowlands University of Wisconsin-Madison, Madison, WI, USA © The Society for Experimental Mechanics, Inc. 2018 A. Baldi et al. (eds.), Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-62899-8_12 73
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