Professor Fabrice PIERRON LMPF Research Group, ENSAM Châlons en Champagne, France THE VIRTUAL FIELDS METHOD Introduction and Overview Paris Châlons en Champagne

A bit of history

3/61 A bit of history 1989 – First paper in Comptes rendus de lacadémie des sciences (principle) 1990 – PhD thesis of Michel GRÉDIAC (thin anisotropic plates, including experiments) 1994 – First collaboration between FP & MG (anisotropic in-plane properties, shear) 1996 – 98: First application in dynamics (vibration of thin plates, exp. & num.) 1998 – 2000: Series of work on in-plane elastic stiffnesses of composites (exp. & num.) 2001: first attempt at a non-linear law (anisotropic) 2002 – 04: Significant progress on virtual fields selection in elasticity (special virtual fields, minimization of noise effects)

4/61 A bit of history 2003: First application in vibration with damping (thin plates) 2005 – 06: Convincing experimental results (in-plane anisotropic composite stiffnesses) 2006 – First application to elasto-plasticity 2006: Theoretical framework in elasticity (relation between FEMU and VFM) 2006: Optimisation of test configuration (with Airbus UK) 2006: First application on heterogeneous materials – stiffness contrast in impacted composite plates (with Bristol Univ.)

5/61 A bit of history 2007: Application to elastography (MRI) 2007: First application to viscoplasticity (coll. M.A. Sutton) 2007: First application to heterogeneous plasticity (FSW joints) Ongoing Application to 3D bulk measurements (composites, biomechanics), project with Loughborough university (Prof. J.M. Huntley, Dr P.D. Ruiz) Optimisation of virtual fields in plasticity Friction Stir Welds (collaboration with ONERA, France) Development of a user-oriented software: CAMFIT

The principle of virtual work

or Equilibrium equations (static) + boundary conditions strong (local) weak (global) Valid for any KA virtual fields

Illustration of the PVW Section S F e1e1 e2e2 l L0L0

Over element 1 1F1F Local equilibrium:

21 Forces exerted by 2 over 1 F e1e1 e2e2 Section S L 0 -x 1

Resultant of internal forces 1 F1F1 21 F e1e1 e2e2 Section S L 0 -x 1

Equilibrium

Valid over any section S of the beam: integration over x 1 Eq. 1 Eq. 2 Eq. 3

Principle of virtual work (static, no volume forces) Let us write a virtual field: e1e1 F e2e2 L0L0 l

Eq. 1 e1e1 F e2e2 L0L0 l

Let us write another virtual field: F e1e1 e2e2 L0L0 l

Eq. 2 F e1e1 e2e2 L0L0 l

F e1e1 e2e2 L0L0 l Let us write a 3rd field: virtual bending

Eq. 3 F e1e1 e2e2 L0L0 l

The Virtual Fields Method

21/61 Basic equations or I Equilibrium equations (static) + boundary conditions strong (local) weak (global) II Constitutive equations (elasticity) III Kinematic equations (small strains/displacements)

22/61 The Virtual Fields Method (VFM) Basic idea Eq. I (weak form, static) Substitute stress from Eq. II

23/61 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 !

24/61 Simple example Fuuny shaped disc in diametric compression Isotropic material -F/2 F y x Eps y Eps x Eps s

25/61 1 st virtual field: virtual compression field -F/2 F y x

26/61 -F/2 F y x

27/61 Homogeneous material Assumption: strain field uniform through the thickness Measurement: uniform strain over a « pixel » (N « pixels ») -F/2 F y x

28/61 « Pixels » are of same area: Average strain Finally: -F/2 F y x

29/61 Virtual work of external forces Contribution of point A Coordinates of A: -F/2 F y x A B C

30/61 Contribution of point B Coordinates of B: Finally -F/2 F y x A B C L h

31/61 1 st virtual field: uniform diametric compression 2 nd virtual field: transverse swelling -F/2 F y x A B C

32/61 Finally -F/2 F y x Direct solution To inverse problem !!!

33/61 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

34/61 Anisotropic elasticity Example 2 Orthotropic material

35/61 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 :

36/61 Unknown force distribution over S 1 and S 3. Resultant P measured 3. Over S 3 (rigid body) : 2 possibilities tyityi txitxi

37/61 tyityi txitxi No information on t x Distribution t y unknown Filtering capacity of the VF

38/61 4. Continuity of the virtual fields Conditions over S 2 Virtual strain field discontinuous Choice of 4 virtual fields at least: example

39/61 Over S 2 Over S 3 k = -L Uniform virtual shear y x

40/61 Plane stress Plane orthotropic elasticity Homogeneous material 0dSTudV V * V * ij

41/61 y x Field n°2: Bernoulli bending Sur S 2 Sur S 3 k = -L 3

42/61 Field n°3: Global compression Over S 2 Sur S 3 k = 0 y x

43/61 Field n°4: Local compression Over A 1 Over S 3 k = 0 y x Over A 2

44/61 Field n°4: Local compression

45/61 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.

46/61 Experimental examples in linear elasticity

47/61 Unnotched Iosipescu test Material: 0° glass-epoxy (2.1 mm thick)

48/61 Polynomial fitting Noise filtering, extrapolation of missing data Displacements in the undeformed configuration Raw data Polynomial fitting Residual

49/61 Strain fields Smooth fields local differentiation FE

50/61 Identification: stiffness 6 specimens P = 600 N Reference (GPa) Coeff. var (%) Identified (GPa) Coeff. var (%) Predicted by VFM routine

51/61 Through thickness stiffnesses of thick UD glass/epoxy composite tubes Optimized position of measurement area R. Moulart Master thesis Ref. 10

52/61 Deformation maps

53/61 Strain maps Polynomial fit, degree 3, transform to cylindrical and analytical differentiation

54/61 Strain maps

55/61 Strain maps

56/61 Reference* (GPa) Identification results Identified (GPa) Coeff. var (%) – 5 tests Problem: not an in-plane test !!! * Typical values

57/61 Problem with thick ring compression test

58/61 Problem with thick ring compression test Solution: back to back cameras

59/61 Set-up with two cameras

60/61 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.

61/61 ACKNOWLEDGEMENTS Professor Michel GREDIAC Blaise Pascal University, France Colleagues and students from my research group: Dr Stéphane Avril, Dr Alain Giraudeau, Dr René Rotinat Dr Hocine Chalal, Mr Baoqiao Guo, Dr Yannick Pannier, Mr Raphaël Moulart French CNRS network (GDR): « full-field measurements and identification in solid mechanics »

62/61 ACKNOWLEDGEMENTS Funding French Ministry for Research French National Research Agency (ANR) Champagne Ardenne Regional Council Engineering and Physical Sciences Research Council (UK) Airbus UK