Characterization of heterogeneity and nonlinearity in material properties of nuclear graphite using an inverse method Lianshan Lin a , Haiyan Li a , Alex S.L. Fok a, * , Mark Joyce b , James Marrow b a School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, P.O. Box 88, Sackville Street, Manchester M60 1QD, UK b School of Materials, The University of Manchester, Grosvenor Street, Manchester M1 7HS, UK article info PACS: 81.05.Uw 81.40.Jj 81.70.Bt 81.70.Fy abstract A finite-element-based inverse method has been developed to allow the characterization of the hetero- geneity and nonlinearity in material properties of isotropic nuclear graphite using a single experimental test. The method is implemented into the commercial finite element code ABAQUS using its User Material (UMAT) Subroutine for ease of application. The program has been verified using simulated examples with idealized distributions of elastic modulus and the results showed rapid convergence with good accuracy. Finally, the method was applied to actual mechanical testing of nuclear graphite for which the heteroge- neous distribution and nonlinearity of material properties were evaluated. Ó 2008 Published by Elsevier B.V. 1. Introduction The prediction of a component’s responses under loading using known material parameters forms the bulk of conventional engi- neering problems, which can also be called direct problems. In- verse problems, on the other hand, involve the determination of the unknown material parameters of a component using its mea- sured responses to loadings, with both the material parameters and loadings being nontrivial. The responses could be displace- ments, strains, frequencies of vibration, or temperature etc., depending on the material parameters under consideration. In- verse analysis is especially useful for determining material proper- ties that are highly nonlinear, heterogeneous and anisotropic. In general, methods for solving inverse problems require a solution of the responses in terms of the material parameters. Analytical solutions are only available for structures with relatively simple geometry and loading conditions. As a result, numerical methods, such as the finite element method, are usually employed to allow structures with arbitrary shapes and arbitrary loading conditions to be analyzed. For example, Grédiac et al. [1] employed a vir- tual-field method to determine the material parameters of thin anisotropic plates in bending; while Moussu and Niviot [2] used the method of superposition to determine the elastic constants of an orthotropic material by studying the free vibrations of a rectan- gular plate. Recent developments in speckle interferometry and digital image correlation technique [3], especially in Electronic Speckle Pattern Interferometry (ESPI), have made inverse methods even more powerful as a material characterization technique. In ESPI, the tested object is illuminated by laser light and the speckle pat- tern is recorded by a CCD camera. The interference of successive speckle patterns created by the laser and observed by the CCD camera includes information of the deformation in each visible point of the measured object with a potential resolution of 10 nm or 1 lm/m. With additional information from interference of reflected light, a full-field displacement map can therefore be obtained by this non-contact method, which provides a tremen- dous amount of information for performing inverse analysis, allow- ing complex material properties to be determined using a single experiment. For example, Grédiac et al. [4] calculated the in-plane elastic properties of orthotropic composite plates using the non- uniform strain fields induced in T-shape specimens; while Wang et al. [5] determined the elastic constants for a circular isotropic disc under diametral compression using the displacement map provided by Moire interferometry. A hybrid procedure was pre- sented by Genovese et al. [6] which combined speckle interferom- etry and numerical optimization to characterize homogeneous orthotropic materials (an eight-ply composite laminate). In this paper, an iterative inverse approach is presented for the characterization of the heterogeneity and nonlinearity exhibited in the material properties of isotropic nuclear graphite. A mixed numerical-experimental procedure, similar to that of Genovese et al. [6], is adopted, with the in-plane strain fields being obtained from optical methods such as ESPI. Changes in the local Young’s modulus were evaluated throughout the whole loading history up to the point of fracture, from which the local strain–stress 0022-3115/$ - see front matter Ó 2008 Published by Elsevier B.V. doi:10.1016/j.jnucmat.2008.07.042 * Corresponding author. Present address: Minnesota Dental Research Center for Biomaterials and Biomechanics, 16-212 Moos Tower, 515 Delaware Street, SE, Minneapolis, MN 55455, USA. Tel.: +44 161 275 4327; fax: +44 161 275 4328. E-mail address: alex.fok@manchester.ac.uk (A.S.L. Fok). Journal of Nuclear Materials 381 (2008) 158–164 Contents lists available at ScienceDirect Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat