The Virtual Fields Method Professor Fabrice Pierron University

The Virtual Fields Method Professor Fabrice Pierron University of Southampton

Summary § Full-field data processing – Global vs local equations § General inverse problem resolution – Equations – Different resolution strategies § The Virtual Fields Method – Back to the disc – Virtual field selection – Applications of the VFM Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 2/42

Global vs local equations (1/7) § Problem to solve Force + Geometry Extract E and n Analytical solution Make use of full-field data Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 3/42

Global vs local equations (2/7) § Main idea Plane stress linear elastic isotropy y d thickness: t F x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 4/42

Global vs local equations (3/7) § Main idea Integrate over y d Plane stress y Material is homogeneous d thickness: t F x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 5/42

Global vs local equations (4/7) § Main idea ? is the surface of each pixel n is the number of strain data points Note that these ‘pixels’ are strain pixels, ie, discrete areas where the strain is suppose constant Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 6/42

Global vs local equations (5/7) § Main idea If all pixels have the same size s (usually the case for CCD/CMOS based measurements) is the surface of the disc Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 7/42

Global vs local equations (6/7) § Main idea Integrate over x y d thickness: t F x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 8/42

Global vs local equations (7/7) § Finally Linear system, two equations, two unknowns Direct solution to a what is usually considered an inverse problem!! Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 9/42

Problem statement (1/3) § Basic equations I Equilibrium equations (static) + boundary conditions strong (local) or weak (global) II Constitutive equations (elasticity) III Kinematic equations (small strains/displacements) Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 10/42

Problem statement (2/3) § Direct problem Known Unknown Geometry Boundary conditions § Tools for solving this problem – Direct integration (closed-form solution) – Approximate solutions · Galerkin, Ritz · Finite elements, boundary elements, etc. Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 11/42

Problem statement (3/3) § Inverse problem Known Unknown Geometry Some information on the boundary conditions (load cell) § Tools for solving this problem – Statically determined tests: Closed form solution of Eq. I (uncoupled system) Force BC, simple geometry Ex. : tensile test, bending tests (on rect. beams) etc… Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 12/42

Resolution methods (1/4) § Tools for solving this problem – Model updating Idea: iterative use of tool for direct problem (analytical or approximate) Exemple C 0 g = Ferreur( - ) 0 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 13/42

Resolution methods (2/4) § Model updating – Advantages · General method (full-field measurements not compulsory) · Tools already developped – Shortcomings · Sensitive to boundary conditions (generally badly known) · CPU intensive (for numerical approximations and nonlinear equations…) · FE package needed to process test results · Not fully adapted to full-field measurements Alternative tool: the Virtual Fields Method Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 14/42

Resolution methods (3/4) § The Virtual Fields Method (VFM) – Basic idea Eq. I (weak form, static) Substitute stress from Eq. II Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 15/42

Resolution methods (4/4) § The Virtual Fields Method (VFM) – Homogeneous material 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 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 16/42

The Virtual Fields Method (1/14) § Simple example -F y Disc in diametrical compression Isotropic material Problem: extract E and n Eps y x F Eps x Eps s Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 17/42

The Virtual Fields Method (2/14) -F y x 1 st virtual field F Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 18/42

The Virtual Fields Method (3/14) Plane stress linear elastic isotropy -F y x F Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 19/42

The Virtual Fields Method (4/14) -F y Homogeneous material x F Plane stress Finally: Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 20/42

The Virtual Fields Method (5/14) -F y A F B Virtual work of external forces x Contribution of point A Coordinates of A: Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 21/42

The Virtual Fields Method (6/14) F A y Contribution of point B Coordinates of B: x -F B Finally Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 22/42

The Virtual Fields Method (7/14) F A y 1 st virtual field x -F B 2 nd virtual field Same system as with direct equilibrium Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 23/42

The Virtual Fields Method (8/14) y -F F Other virtual fields (an infinity!) x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 24/42

The Virtual Fields Method (9/14) y F -F Other virtual fields (an infinity!) x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 25/42

The Virtual Fields Method (10/14) y F -F x Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 26/42

The Virtual Fields Method (11/14) § Principal advantages – Independent from stress distribution – Independent from geometry – Direct identification (in elasticity) § Principal limitation – 2 D assumption (except if bulk measurements!) Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 27/42

The Virtual Fields Method (12/14) § Choice of virtual fields – Exact data: identified values are the same regardless of virtual fields – Noisy data: identified values are different for different sets of virtual fields – Choice of virtual fields? – Special optimized virtual fields · Expand on a set of functions (polynomials, piecewise etc…) · Write optimal robustness to noise (maximum likelyhood solution) Avril S. , Grédiac M. , Pierron F. , Sensitivity of the virtual fields method to noisy data, Computational Mechanics, vol. 34, n° 6, pp. 439 -452, 2004. http: //dx. doi. org/10. 1007/s 00466 -004 -0589 -6 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 28/42

The Virtual Fields Method (13/14) § Match. ID – Seamless integration of VFM with DIC https: //www. matchid. eu/en Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 29/42

The Virtual Fields Method (14/14) § Theory § Applications § Training Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 30/42

Applications of the VFM (1/12) § Homogeneous linear elasticity – Composites (anisotropy) Chalal H. , Avril S. , Pierron F. , Meraghni F. , Experimental identification of a nonlinear model for composites using the grid technique coupled to the virtual fields method, Composites Part A: Applied Science and Manufacturing , vol. 37, n° 2, pp. 315 -325, 2006. http: //dx. doi. org/10. 1016/j. compositesa. 2005. 04. 020 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. http: //dx. doi. org/10. 1016/j. compositesa. 2005. 050 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 31/42

Applications of the VFM (2/12) § Mechanical properties of large superconductiong coils (CEA Saclay / Siemens) Design and build the largest MRI facilities in the world (neurosciences) Iseult project 11. 7 T magnet Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 32/42

Applications of the VFM (3/12) § Magnetic field deforms coils – Deformed coils spoil magnetic field uniformity – Need to evaluate and correct for coil deformation VFM to measure orthotropic elastic constants of « pancakes » in cylindrical system Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 33/42

Applications of the VFM (4/12) § Full-scale testing Kim J. -H, Pierron F. , Nunio F. , Védrine P. , Characterizing elastic properties of superconducting windings by simulations and experiments, Superconductor Science and Technology, vol. 24, article 125001 (15 p. ), 2011. http: //dx. doi. org/10. 1088/09532048/24/12/125001 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 34/42

Applications of the VFM (5/12) § Homogeneous elasto-plasticity – Non-linear VFM Measured: Iterative but no direct problem solved Fast (minutes) Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 35/42

Applications of the VFM (6/12) § Isotropic hardening, monotonic loading Avril S. , Pierron F. , Pannier Y. , Rotinat R. , Stress reconstruction and constitutive parameter identification in elastoplasticity using measurements of deformation fields, Experimental Mechanics, vol. 48, n° 4, pp. 403 -420, 2008. Implemented in Camfit Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 36/42

Applications of the VFM (7/12) § Homogeneous elasto-plasticity – Kinematic/isotropic hardening, loading/unloading – Virtual fields selection Pierron F. , Avril S. , Tran T. V. , Extension of the virtual fields method to elasto-plastic material identification with cyclic loads and kinematic hardening, International Journal of Solids and Structures, vol. 47, pp. 2993– 3010, 2010. http: //dx. doi. org/10. 1016/j. ijsolstr. 2010. 06. 022 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 37/42

Applications of the VFM (8/12) § Composite damage – Damaged composite plate – Local bending stiffness reduction Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 38/42

Applications of the VFM (9/12) § Simulated impact – PTFE film between 1 & 2 and 2 & 3 surface plies BVID – Target: 1 -p = 0. 67 Kim J. -H. , Pierron F. , Wisnom M. , Syed. Muhamad K. , Composites Part A, 2007. Kim J. -H. , Pierron F. , Wisnom M. , Avril S. , Composites Part A, 2009. Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 39/42

Applications of the VFM (10/12) § Aluminium friction stir welds – Joint Ph. D with M. A Sutton and A. Reynolds at USC – Static and dynamic properties (Al 5456) 12. 7 mm Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 40/42

Applications of the VFM (11/12) § Aluminium friction stir welds – Yield stress along the weld Base material Axial strain (up to 5%) Le Louëdec G. Pierron F. , Sutton M. A. , Reynolds A. P. , Identification of the local elasto-plastic behavior of FSW welds using the Virtual Fields Method, Experimental Mechanics, available online, 2013. http: //dx. doi. org/10. 1007/s 11340 -012 -96790 41/42 Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method

Applications of the VFM (12/12) § High strain rate testing b F(t) x 1 QI carbon/epoxy ~2000 s-1 x www. photodyn. org Prof. F. Pierron – SESG 6045 and BSSM Exp. Mech. Course, University of Southampton, April 2019 – Virtual Fields Method 42/42