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Professor Fabrice PIERRON LMPF Research Group, ENSAM Châlons en Champagne, France THE VIRTUAL FIELDS METHOD Application to linear elasticity Paris Châlons en Champagne

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2/61 Basic equations or I Equilibrium equations (static) + boundary conditions strong (local) weak (global) II Constitutive equations (elasticity) III Kinematic equations (small strains/displacements)

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3/61 The Virtual Fields Method (VFM) Basic idea Eq. I (weak form, static) Substitute stress from Eq. II

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4/61 The Virtual Fields Method (VFM) valid for any kinematically admissible virtual fields For each choice of virtual field: 1 equation Choice of as many VF as unknowns: linear system Inversion: unknown stiffnesses Elasticity: direct solution to inverse problem !

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5/61 Simple example Fuuny shaped disc in diametric compression Isotropic material -F/2 F y x Eps y Eps x Eps s

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6/61 1 st virtual field: virtual compression field -F/2 F y x

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7/61 -F/2 F y x

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8/61 Homogeneous material Assumption: strain field uniform through the thickness Measurement: uniform strain over a « pixel » (N « pixels ») -F/2 F y x

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9/61 « Pixels » are of same area: Average strain Finally: -F/2 F y x

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10/61 Virtual work of external forces Contribution of point A Coordinates of A: -F/2 F y x A B C

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11/61 Contribution of point B Coordinates of B: Finally -F/2 F y x A B C L h

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12/61 1 st virtual field: uniform diametric compression 2 nd virtual field: transverse swelling -F/2 F y x A B C

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13/61 Finally -F/2 F y x Direct solution to inverse problem !!!

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14/61 Principal advantages Independent from stress distribution Independent from geometry Direct identification (no updating) Limitations Kinematic assumption through the thickness (plane stress, plane strain, bending...) y F -F x A B

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15/61 Anisotropic elasticity Example 2 Orthotropic material

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16/61 Choice of the virtual fields 1. Measurement on S 2 only (optical system) Over S 1 and S 3 : (rigid body) 2. A priori choice: over S 1 :

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17/61 Unknown force distribution over S 1 and S 3. Resultant P measured 3. Over S 3 (rigid body) : 2 possibilities tyityi txitxi

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18/61 tyityi txitxi No information on t x Distribution t y unknown Filtering capacity of the VF

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19/61 4. Continuity of the virtual fields Conditions over S 2 Virtual strain field discontinuous Choice of 4 virtual fields at least: example

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20/61 Over S 2 Over S 3 k = -L Uniform virtual shear y x

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21/61 Plane stress Plane orthotropic elasticity Homogeneous material 0dSTudV V * V * ij

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22/61 y x Field n°2: Bernoulli bending Sur S 2 Sur S 3 k = -L 3

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23/61 Field n°3: Global compression Over S 2 Sur S 3 k = 0 y x

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24/61 Field n°4: Local compression Over A 1 Over S 3 k = 0 y x Over A 2

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25/61 Field n°4: Local compression

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26/61 Final system AQ = B Q = A -1 B If VF independent !! Pierron F. et Grédiac M., Identification of the through-thickness moduli of thick composites from whole-field measurements using the Iosipescu fixture : theory and simulations, Composites Part A, vol. 31, pp , 2000.

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27/61 Experimental examples in linear elasticity

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28/61 Unnotched Iosipescu test Material: 0° glass-epoxy (2.1 mm thick)

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29/61 Polynomial fitting Noise filtering, extrapolation of missing data Displacements in the undeformed configuration Raw data Polynomial fitting Residual

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30/61 Strain fields Smooth fields local differentiation FE

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31/61 Identification: stiffness 6 specimens P = 600 N Reference (GPa) Coeff. var (%) Identified (GPa) Coeff. var (%) Predicted by VFM routine

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32/61 Through thickness stiffnesses of thick UD glass/epoxy composite tubes Optimized position of measurement area R. Moulart Master thesis Ref. 10

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33/61 Deformation maps

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34/61 Strain maps Polynomial fit, degree 3, transform to cylindrical and analytical differentiation

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35/61 Strain maps

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36/61 Strain maps

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37/61 Reference* (GPa) Identification results Identified (GPa) Coeff. var (%) – 5 tests Problem: not an in-plane test !!! * Typical values

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38/61 Problem with thick ring compression test

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39/61 Problem with thick ring compression test Solution: back to back cameras

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40/61 Set-up with two cameras

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41/61 Results Reference* (GPa) Identified (GPa) Coeff. var (%) – 9 tests Moulart R., Avril S., Pierron F., Identification of the through-thickness rigidities of a thick laminated composite tube, Composites Part A: Applied Science and Manufacturing, vol. 37, n° 2, pp , 2006.

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