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CivilComp Proceedings
ISSN 17593433 CCP: 75
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and Z. Bittnar
Paper 20
Fitting Strains and Displacements by Minimizing Dislocation Energy C.A. Felippa and K.C. Park
Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado at Boulder, USA C.A. Felippa, K.C. Park, "Fitting Strains and Displacements by Minimizing Dislocation Energy", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 20, 2002. doi:10.4203/ccp.75.20
Keywords: finite element methods, dislocation energy minimization, strain fitting, displacement fitting, strainassumed elements, straingages, individual element test.
Summary
We present a procedure for matching a displacement field to a
given strain field, or viceversa, over an arbitrary domain,
which can be a finite element. The fitting criterion used
is minimization of a dislocation energy functional.
The strain field, whether given or fitted, need not be compatible.
The method has four immediate applications:
(i) finite element stiffness formulation based on fitting
assumednaturalstrain (ANS) fields to node
displacements; (ii) pointwise recovery of an internal displacement field
in ANS elements as required for consistent mass, body load
or geometric stiffness computations;
(iii) recovery of smoothed strains from node displacements for
stress postprocessing, and
(iv) system identification and damage detection from experimental data.
The article focuses on application (i) for the strain fitting (SF) problem
and (ii) for the displacement fitting (DF) problem.
The separation of mean and deviatoric strains is emphasized whenever
it is found convenient to simplify calculations.
We are given the strain field e(x) in the volume of a body or finite element, which contains free or specified parameters. This strain field is not necessarily compatible (derivable from a continuous displacement field). The source of e(x) could be experimental, from interpolation of strain gage readings. Or it may be one of the primary fields in strainassumed finite element formulations. Two related problems are studied: Strain fitting, or SF problem. Given a continuous displacement field u(x) and a strain field form e(x) that contains free parameters, find the parameters that best fit . The SF problem is trivial if e(x) is left completely free since if so e(x)Du(x) is obviously the solution. Displacement fitting, or DF problem. Given e(x), find an associated displacement field u(x) in so that the displacementderived strain field e(x)Du(x) matches over in the sense discussed below. Here D is the appropriate straindisplacement operator. The fitted displacement field is specified only within a rigid body motion. The symbol e follows the fielddependence notation developed for Parametrized Variational Principles [14]. Both problems DF and SF occur in finite element technology. Problem SF is important in the development of stiffness equations of assumedstrain elements, as well as in the recovery of smooth strain fields from node displacement information for postprocessing. Problem DF occurs when fitting a displacement field to an assumed strain element for constructing consistent masses, bodyloads node forces and geometric stiffnesses. Problem DF also occurs in system identification and damage detection [5,6], but this application is not addressed in the paper. The paper correlates variants of this method with the Free Formulation of Bergan and Nygård [7], application of the freefree flexibility to handle rigid body motions [8,9] use of Barlow points as optimal straingage locations [10] the Individual Element Test of Bergan and Hanssen [11], energy orthogonality [12], and the Assumed Natural Deviatoric Strain (ANDES) formulation [13,14]. It concludes with four examples of the DF problem in bars, plane beams [15,16], and the 4node bilinear quadrilateral. References
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