THE VIRTUAL FIELDS METHOD VFM course, BSSM conference, Sept. 1st 2015 Professor Fabrice PIERRON Faculty of Engineering and the Environment Royal Society Wolfson Research Merit Award holder

Course overview  Introduction and basic idea  Principle of virtual work  The Virtual Fields Method (VFM): principle in elasticity  Complements on the VFM  MatchID demo (if time)  Conclusion Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Introduction and basic idea

Introduction  Huge progress in computational mechanics – Simulation of machining  Large strains elasto-plasticity  Large strain rates  Localization  Friction/thermal behaviour  Problem – Many material parameters required – How to obtain them? Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Introduction  Standard tests: Tensile test on rectangular specimen – Uniform stress state – Uniaxial stress strain curve – Very poor information – Very restrictive assumptions (constraints) Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015   ???? Develop the experimental identification procedures of the future ! 5/86

Alternative Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Extract material properties (Q 11, Q 22, Q 12, Q 66 ) from strain fields and load Composite specimen Grid method 6/86

Motivation  Extract more information from 1 test Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st ° tensile test 90° tensile test shear test (off-axis) 4 parameters 3 tests Local strain measurements Uniform stress fields (closed-form solution) Spatial stress distribution 7/86

Motivation  Novel strategy for orthotropic elastic moduli measurements Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Heterogeneous stress fields (no closed-form solution) T 1 test 8/86

Motivation  Complex test geometry Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Motivation  Heterogeneous materials – Identify spatial maps of parameters Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Longitudinal strain in a magnesium FSW weld (speckle interferometry) 10/86

Key issues: full-field measurements  Why? – Complex strain state – Need for more experimental information  Interferometry (moiré, speckle, etc…) – High sensitivity, high spatial resolution – Small displacements (rigid body), low strains and high strain gradients  White light techniques – Image correlation : large displacements, 3D – Grid method : intermediate Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Key issues: full-field measurements  Extraordinary potential – From a few measurements points to a few ’s!!  Qualitative / quantitative ? – Metrological properties not fully assessed yet – Often reduced to visual information – Very few quantitative use of the data!  International effort – DIC Challenge (Society for Experimental Mechanics) Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Key issues: inverse resolution  Extensive literature on inverse problems (mathematical)  Very few tools targeted at full-field processing  Very few groups with expertise in measurements and inverse problems at the same time  Inversion closely related to measurement performance!  Huge investigation area, critically underexplored Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Inverse resolution  Basic equations Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 I Equilibrium equations (static) + boundary conditions strong (local) or weak (global) II Constitutive equations (elasticity) III Kinematic equations (small strains/displacements) 14/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Geometry Boundary conditions Known Unknown Direct problem  Tools for solving this problem – Direct integration (closed-form solution) – Approximate solutions – Galerkin, Ritz – Finite elements, boundary elements… – etc… Inverse resolution 15/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Known Inverse problem  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… Geometry Some information on the boundary conditions (load cell) Unknown (measured) Inverse resolution 16/86

Inverse resolution  Tools for solving this problem – Model updating Idea: iterative use of tool for direct problem (analytical or approximate) Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 C0C0 Example g = F cost ( -) 0 17/86

 Model updating – Advantages  General method (full-field measurements not compulsory)  Tools already developed – Shortcomings  Sensitive to boundary conditions (generally badly known)  CPU intensive (for numerical approximations and non-linear equations…)  Not fully dedicated to full-field measurements Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Alternative tool: the Virtual Fields Method Inverse resolution 18/86

 Use global equations (and not local) Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st F/2 F x2x2 x1x1 L Integrate over x 2 Basic idea 19/86

Basic idea  Constitutive behaviour Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 In-plane linear elastic isotropy Material is homogeneous 20/86

Basic idea  Surface measurements only Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Constant strains through the thickness ? is the surface of each pixel If all pixels have the same size s (usually the case for CCD/CMOS based measurements) is the surface of the disc n is the number of strain data points 21/86

Basic idea  Finally Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st F/2 F x2x2 x1x1 L Integrate over x 1 22/86

Basic idea  What if the disc material is anisotropic? – Need for more equations  Tool to generate integral equilibrium equations: principle of virtual work Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

The principle of virtual work

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 The Principle of Virtual Work Equilibrium equation Boundary conditions on Let be a vectorial function Integration over the whole domain 25/86

The Principle of Virtual Work  Integration by part and use of divergence theorem Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Virtual displacement field (cont. and diff) virtual strain tensor External volume forces Stress tensor Applied stress vector on solid boundary 26/86

Illustration of the PVW  Cantilever beam  Subsystem and FBD Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Section S F x1x1 x2x2 l L0L0 thickness: e F x1x1 x2x2 l L 0 -x 1 27/86

Illustration of the PVW  Resultant of internal forces Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 F x1x1 x2x2 l L 0 -x 1 x2x2 A x 28/86

Illustration of the PVW  Resultant of external force Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 F x1x1 x2x2 Section S L 0 -x 1 29/86 A x

Illustration of the PVW  Equilibrium Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Illustration of the PVW  Valid over any section S of the beam: – integration over x 1 Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Eq. 1 Eq. 2 Eq. 3 31/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Illustration of the PVW Principle of virtual work (static, no volume forces) Let us write a virtual field: x1x1 F x2x2 L0L0 l 32/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Illustration of the PVW Eq. 1 x1x1 F x2x2 L0L0 l 33/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Illustration of the PVW Let us write another virtual field: F x1x1 x2x2 L0L0 l 34/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Illustration of the PVW F x1x1 x2x2 L0L0 l Eq. 2 35/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Illustration of the PVW F x1x1 x2x2 L0L0 l Let us write a 3rd field: virtual bending 36/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Illustration of the PVW F x1x1 x2x2 L0L0 l Eq. 3 37/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Illustration of the PVW F x1x1 x2x2 L0L0 l Any virtual field can be selected 38/86

The Virtual Fields Method

 Back to the funny shaped disc  Principle of virtual work Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st F/2 F x2x2 x1x1 L No volume forces, static 40/86

 Let’s write two virtual fields – 1 st virtual field: virtual compression field Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 The Virtual Fields Method -F/2 F x2x2 x1x1 L 41/86

The Virtual Fields Method  2 nd virtual field: transverse shrinking  Other virtual fields (an infinity!) Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

The Virtual Fields Method  Principal advantages – Independent from stress distribution; etc. geometry – Direct identification in elasticity (no updating)  Limitations – Kinematic assumption through the thickness (plane stress, plane strain, bending...) Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VFM: more complex example  Orthotropic elasticity Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Orthotropic material x2x2 x1x1 44/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 VFM: more complex example 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 : x2x2 x1x1 45/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 x2x2 x1x1 VFM: more complex example Unknown force distribution over S 1 and S 3. Resultant P measured 3. Over S 3 (rigid body) : 2 possibilities T2iT2i T1iT1i 46/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 VFM: more complex example No information on t 1 Distribution T 2 unknown Filtering capacity of the VF x2x2 x1x1 T2iT2i T1iT1i 47/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 x2x2 x1x1 VFM: more complex example 4. Continuity of the virtual displacement field Conditions over S 2 Virtual strain field discontinuous Choice of 4 virtual fields at least: example 48/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 VFM: more complex example Over S 2 Over S 3 k = -L Uniform virtual shear x2x2 x1x1 x2x2 x1x1 49/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 VFM: more complex example Plane stress Plane stress orthotropic elasticity Homogeneous material 50/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 VFM: more complex example x2x2 x1x1 Field n°2: Bernoulli bending Over S 2 Over S 3 k = -L 3 51/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 VFM: more complex example Field n°3: Global compression x2x2 x1x1 Over S 2 Over S 3 k = 0 52/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 VFM: more complex example Field n°4: Local compression x2x2 x1x1 Over A 1 Over S 3 k = 0 Over A 2 53/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 VFM: more complex example Field n°4: Local compression 54/86

Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 VFM: more complex example Final system AQ = B Q = A -1 B If VF independent !! Pierron F. et Grédiac M., Composites Part A, vol. 31, pp , /86

Complements on the VFM

Summary  Virtual fields selection in elasticity  Heterogeneous materials  Non-linear constitutive models  Dynamics  Force identification  Equilibrium gap Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VF selection

 Exact data – All sets of VF produce the SAME results  Noisy data – Different sets provide different results – Optimal selection? Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VF selection  Strategy for virtual fields selection in elasticity – Projection of VF on a certain basis of functions – Polynomial – Choice of VF: choice of a set of m, n, a ij,b ij Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VF selection  Strategy to find a ij,b ij for given m and n – Special virtual fields – For instance, special virtual field to identify Q 11 – 4 linear equations linking the a ij and b ij coefficients – Virtual BCs: a few more linear equations – Still a very large number of VF in general! =1=0 Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VF selection  Strategy to find a ij,b ij for given m and n – Minimization of the sensitivity to noise – Constrained minimization problem where a ij and b ij coefficients are the unknown – Provides the maximum likelihood solution, with: is the standard deviation of strain noise are the sensitivity to noise coefficients Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VF selection  Validation – 30 copies of noise: distribution of stiffness – Increasing levels of noise  dev (Q ij ) GPa Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VF selection  Actual and virtual optimized virtual fields for Q 11 (m=4,n=5) Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VF selection  Piecewise virtual fields – Finite element mesh – Virtual Dofs unknown of minimization problem Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VF selection  Piecewise virtual fields – Advantage: more flexible – Easy for virtual BCs All dofs To zero All u 1 * =0 All u 2 * =cst Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

VF selection  Optimized piecewise virtual fields for Q 11 – Virtual strains discontinuous Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Heterogeneous materials

 Discrete parameterization – Multimaterials – Sharp transitions C2C2 C3C3 C1C1 Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Heterogeneous materials  Discrete parameterization – 6 unknowns in isotropic linear elasticity – Contrast of stiffness: strain derivative!! – Subject to noise Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Heterogeneous materials  Continuous parameterization – Functionally graded material – Smooth transitions – Polymer/composite with moisture ingress from one face – Ceramic with varying pore density – Etc… Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 C=f(x 2 ) x2x2 71/86

Heterogeneous materials  Continuous parameterization x2x2 Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Heterogeneous materials  Mixed parameterization – Plate with local damage or manufacturing defect Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 C1C1 C 2 =f(r) 73/86

Non-linear constitutive model

 Non-linear model with linear VFM formulation – Composite with shear strain softening Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 66 66 75/86

… in elasto-plasticity  Virtual fields method for non-linear constitutive equations Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Iterative but no direct problem solved Fast (minutes) Measured: 76/86

Dynamics

VFM in dynamics  Principle of virtual work in dynamics Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 x2x2 F(t) x1x1 L If the load is unknown In statics: with other virtual fields Only stiffness ratios can be identified (Poisson’s ratio) 78/86

VFM in dynamics Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Measurements Single spatial differentiation Double temporal differentiation x2x2 F(t) x1x1 L 79/86

VFM in dynamics Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 x2x2 F(t) x1x1 L Volume distributed load cell Embedded in the full-field data!! More on Wednesday! 80/86

Force identification

 Standard VFM  Force VFM Force identification Measured Unknown Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Berry, A., Robin, O., & Pierron, F. (2014). Journal of Sound and Vibration, 333(26), /86

Equilibrium gap

 Procedure – Equilibrium  Elasticity measured knownselected 84/42 Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015

Equilibrium gap  Choice of the virtual field – Piecewise functions  What is detected? – Bad strain data – Inaccurate material parameters – Inappropriate material model – Damage –…–… n data points k data points Spatial map of 85/42 Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015

Conclusion

 VFM: operational tool, computationally simple and efficient  Future challenges – Design and optimize test configurations, predict error bars (new operational standards) – Apply to 3D (X-ray CT + DVC) – High rate dynamics (see talk on Tuesday) – Optimization of VFs for non-linear VFM – Integrate with measurements: MatchID platform Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

MatchID  Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

More on the VFM  Theory  Applications  Self-training Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st /86

Supplementary material

The Principle of Virtual Work  A few notations – Vector of index i: T i – Second order tensor of index i and j:  ij – Convention of summation on repeated indices Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Scalar product between two vectors Scalar product between two second order tensors 91/86

The Principle of Virtual Work  A few notations – Scalar product between two second order tensors – Comma to represent differentiation Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 Matrix constructed through term by term multiplication + x 92/86

The Principle of Virtual Work  Voigt notations Prof. F. Pierron - VFM course, BSSM conference, Sept. 1st 2015 engineering shear strains 93/86