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Professor Fabrice PIERRON LMPF Research Group, ENSAM Chlons en Champagne, France THE VIRTUAL FIELDS METHOD Application to linear elasticity Paris Chlons en Champagne Slide 2 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) Slide 3 3/61 The Virtual Fields Method (VFM) Basic idea Eq. I (weak form, static) Substitute stress from Eq. II Slide 4 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 ! Slide 5 5/61 Simple example Fuuny shaped disc in diametric compression Isotropic material -F/2 F y x Eps y Eps x Eps s Slide 6 6/61 1 st virtual field: virtual compression field -F/2 F y x Slide 7 7/61 -F/2 F y x Slide 8 8/61 Homogeneous material Assumption: strain field uniform through the thickness Measurement: uniform strain over a pixel (N pixels ) -F/2 F y x Slide 9 9/61 Pixels are of same area: Average strain Finally: -F/2 F y x Slide 10 10/61 Virtual work of external forces Contribution of point A Coordinates of A: -F/2 F y x A B C Slide 11 11/61 Contribution of point B Coordinates of B: Finally -F/2 F y x A B C L h Slide 12 12/61 1 st virtual field: uniform diametric compression 2 nd virtual field: transverse swelling -F/2 F y x A B C Slide 13 13/61 Finally -F/2 F y x Direct solution to inverse problem !!! Slide 14 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 Slide 15 15/61 Anisotropic elasticity Example 2 Orthotropic material Slide 16 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 : Slide 17 17/61 Unknown force distribution over S 1 and S 3. Resultant P measured 3. Over S 3 (rigid body) : 2 possibilities 3.1 3.2 tyityi txitxi Slide 18 18/61 tyityi txitxi No information on t x Distribution t y unknown Filtering capacity of the VF Slide 19 19/61 4. Continuity of the virtual fields Conditions over S 2 Virtual strain field discontinuous Choice of 4 virtual fields at least: example Slide 20 20/61 Over S 2 Over S 3 k = -L Uniform virtual shear y x Slide 21 21/61 Plane stress Plane orthotropic elasticity Homogeneous material 0dSTudV V * V * ij Slide 22 22/61 y x Field n2: Bernoulli bending Sur S 2 Sur S 3 k = -L 3 Slide 23 23/61 Field n3: Global compression Over S 2 Sur S 3 k = 0 y x Slide 24 24/61 Field n4: Local compression Over A 1 Over S 3 k = 0 y x Over A 2 Slide 25 25/61 Field n4: Local compression Slide 26 26/61 Final system AQ = B Q = A -1 B If VF independent !! Pierron F. et Grdiac 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. 309-318, 2000. Slide 27 27/61 Experimental examples in linear elasticity Slide 28 28/61 Unnotched Iosipescu test Material: 0 glass-epoxy (2.1 mm thick) Slide 29 29/61 Polynomial fitting Noise filtering, extrapolation of missing data Displacements in the undeformed configuration Raw data Polynomial fitting Residual Slide 30 30/61 Strain fields Smooth fields local differentiation FE Slide 31 31/61 Identification: stiffness 6 specimens P = 600 N Reference (GPa) 44.912.23.683.86 Coeff. var (%) 0.72.87.32.4 Identified (GPa) 39.7 6.6 10.4 23 3.65 2.4 3.03 13 Coeff. var (%) Predicted by VFM routine Slide 32 32/61 Through thickness stiffnesses of thick UD glass/epoxy composite tubes Optimized position of measurement area R. Moulart Master thesis Ref. 10 Slide 33 33/61 Deformation maps Slide 34 34/61 Strain maps Polynomial fit, degree 3, transform to cylindrical and analytical differentiation Slide 35 35/61 Strain maps Slide 36 36/61 Strain maps Slide 37 37/61 Reference* (GPa) 104043 Identification results Identified (GPa) 11.444.46.83.87 Coeff. var (%) 5 tests 87666959 Problem: not an in-plane test !!! * Typical values Slide 38 38/61 Problem with thick ring compression test Slide 39 39/61 Problem with thick ring compression test Solution: back to back cameras Slide 40 40/61 Set-up with two cameras Slide 41 41/61 Results Reference* (GPa) 104043 Identified (GPa) 11.445.46.782.62 Coeff. var (%) 9 tests 2910429 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. 326-336, 2006.