Una Metodología Para Resolver Problemas Inversos en Mecánica Experimental de Sólidos

Transcription

1 Una Metodología Para Resolver Problemas Inversos en Mecánica Experimental de Sólidos J. F. Cárdenas-García, PhD, PE Becario Prometeo Escuela Politécnica Nacional Quito, ECUADOR 1 of 69

2 Outline Motivation Definitions Examples: Direct and Inverse Problems Inverse Problem Methodology Detailed Example: Atomic Force Microscope: MEMS Testing Summary of 69

3 Motivation Traditionally theory and experimentation are done independently Leading to an apocryphal Law of Research: Nobody believes the numerical results, except the person who generated them Everybody believes the experimental results, except the person who obtained them 3 of 69

4 Motivation Observations: Currently numerical techniques (e.g., ABAQUS) are of common usage Experimental techniques are more user friendly (?) and yield full field digital results (e.g., Digital Image Correlation) Goals: Satisfy the age old need to integrate theory with practice Do it seamlessly and with little human interaction Questions: What is gained by tackling this task? How can it be done? 4 of 69

5 Definitions Forward or Direct Problem Boundary Conditions Elastic Constants Input Body Geometry Output Displacements Strains, Stresses Reverse or Inverse Problem Boundary Conditions, or Elastic Constants Output Body Geometry Input Displacements Strains, Stresses, and Elastic Constants, or Boundary Conditions 5 of 69

6 Definitions: Forward Problem Simulation Experiment Define Input and Output Parameters Develop the Model Define the Forward Problem Develop the Experimental Model Define Input and Output Parameters Experimental 6 of 69

7 Definitions: Forward Problem Experiment Define the Forward Problem Develop the Experimental Model Define Input and Output Parameters Experimental 7 of 69

8 Definitions: Inverse Problem Methodology Simulation Experiment Define Input and Output Parameters Develop the Model Define the Forward Problem Develop the Experimental Model Define Input and Output Parameters Experimental 8 of 69

9 Definitions: Recursive Inverse Analysis Develop Recursive Inverse Analysis Recursive Inverse Analysis 9 of 69

10 Definitions: Recursive Inverse Analysis,, F P V F P V e i Data i Analytical i,,, Fi P V Fi P V Fi P V Analytical Data Relationship between data and least squares fitting Objective function relating data and analytical function F i F / /... k1 i F k i P P F k i V V H OT k F ap F1 F1 F P V 1 F F F P k V Fm k F m Fm P V k 1 T T where F ; a P V ; P k P c a F c a a Taylor s series expansion and its matrix form for m data points and k th iteration step Correction factor solution (convergence?) 10 of 69

11 Definitions: Inverse Problem Methodology Simulation Experiment Define Input and Output Parameters Develop the Model Define the Forward Problem Develop the Experimental Model Define Input and Output Parameters Experimental Develop Recursive Inverse Analysis Experimental Define the Inverse Problem Define Calculation Initiation Parameters Recursive Inverse Analysis Define Calculation Initiation Parameters Define the Inverse Problem Verify Convergence of Inverse Analysis Verify Verify Calculation Con vergence Error Verify Experimental 11 of 69

12 Definitions: Inverse Problem Methodology Simulation Experiment Define Input and Output Parameters Develop the Model Define the Forward Problem Develop the Experimental Model Define Input and Output Parameters Experimental Develop Recursive Inverse Analysis Experimental Define the Inverse Problem Define Calculation Initiation Parameters Recursive Inverse Analysis Define Calculation Initiation Parameters Define the Inverse Problem Verify Convergence of Inverse Analysis Verify Verify Calculation Con vergence Error Verify Experimental 1 of 69

13 Examples: Direct and Inverse Problems Interference fringes (Newton s rings) Interferometric moiré Photoelasticity 13 of 69

14 Definitions Forward or Direct Problem Boundary Conditions Elastic Constants Input Body Geometry Output Displacements Strains, Stresses Reverse or Inverse Problem Boundary Conditions, or Elastic Constants Output Body Geometry Input Displacements Strains, Stresses, and Elastic Constants, or Boundary Conditions 14 of 69

15 Anticlastic Behavior of a Beam in Pure Bending Newton s rings resulting from the deformation of the beam surface 1/ = tan α ; = 64.5 o => = 0.3 A. Cornu, Compt. Rend., Vol. 69, p. 333, S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd edition, McGraw-Hill Book Company, New York (1970), p of 69

16 Four Point Loading of a Beam x 0 h c c x b P a a P y x 0 L Beam in Pure Bending M=Pa P Four-Point Loading of a Beam to Obtain Pure Bending 16 of 69

17 Anticlastic Behavior of a Beam in Pure Bending Newton s rings resulting from the deformation of the beam surface 1/ = tan α ; = 64.5 o => = 0.3 A. Cornu, Compt. Rend., Vol. 69, p. 333, S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd edition, McGraw-Hill Book Company, New York (1970), p of 69

18 Signature of a Beam in Pure bending 1/ = tan α ; = 61. o => = 0.30 Herbert Reissman and Peter S. Pawlik, Elasticity Theory and Applications, John Wiley & Sons, New York (1980) 18 of 69

19 Geometric Moiré: Superposition of two misaligned gratings 19 of 69

20 Moiré Interferometry: -beam setup 0 of 69

21 y Disc in Diametral Compression u P 1 1 x 1 q 1 q sin q1 sin Et u P 1 1 y ln r / r1 cosq1 cos Et x R q 1 y R q 1 P O q 1 q r 1 r A (x,y) x R q r x tan 1 R y x R 1 q tan 1 y x R 1 y r x R y P 1 of 69

22 of 69 x x x B E υ A E 1 u E and are coupled: Theoretically, E and can be obtained simultaneously using arbitrarily two points from either u x or u y field displacements q q q q R x sin 1 sin 1 t P A 1 1 x q q q q R x sin 1 sin 1 t P B 1 1 x y y y B E υ A E 1 u q q R y cos 1 cos 1 r / r ln t P A 1 1 y q q R y cos 1 cos 1 t P B 1 y Disc in Diametral Compression

23 Specimen in Loading Fixture 3 of 69

24 Experimental Setup High resolution CCD camera Moiré system Loading Fixture 4 of 69

25 Experimental Results u x -field at 1800 N Wrapped Phase map u x field pattern with an equal phase difference of / Unwrapped Phase map 5 of 69

26 Experimental results: E and Load E (GPa) 1800 N N Handbook [17] Area used in calculation (0.4 x 0.4 of diameter) 6 of 69

27 Advantages of Using Circular Disc for Determination of Material Constants Specimen is easy to fabricate and load Well-known theoretical displacement field Only one displacement field may be used to determine both Young s modulus E and Poisson s ratio Only two beam interferometry configuration is required 7 of 69

28 Finite Element Implementation: Moiré Hole Method 8 of 69

29 Two Inverse Problems pre-existing hole inverse problem => elastic properties y y y x x 1 x hole drilling inverse problem => residual stresses y y Surface Tractions y 1 1 x 1 x 1 x 9 of 69

30 100 MPa; 95 MPa; 59 MPa x y xy y x E 3.1 GPa; 0.35 yx y yx xy xy xy x x xy yx u x - general loading displacements y 30 of 69 yx

31 Two Inverse Problems pre-existing hole inverse problem => elastic properties y y y x x 1 x hole drilling inverse problem => residual stresses y y Surface Tractions y 1 1 x 1 x 1 x 31 of 69

32 100 MPa; 95 MPa; 59 MPa x y xy y x E 3.1 GPa; 0.35 yx y yx xy xy xy x x xy yx u x - general loading displacements y 3 of 69 yx

33 100 MPa; 95 MPa; 59 MPa x y xy y x E 3.1 GPa; 0.35 yx y yx xy xy xy x x xy yx u x - general loading displacements y 33 of 69 yx

34 yx yx y y xy xy x x x x xy xy y yx u x - residual stress displacements yx y y yx xy E 3.1 GPa; 0.35 x x x 100 MPa; 95 MPa; y xy xy 59 MPa y yx 34 of 69

35 Uniaxial Loading of a Finite Plate Y 3-node triangular isoparametric elements Number of elements: Number of nodes: Plate Size: W = 63-mm; L = 00-mm; t = 3-mm Hole Diameter = 13-mm 35 of 69

36 (a) (b) (c) Displacement isothetics for a tensile plate with a central hole, loaded in the vertical direction (adapted from Weissman and Post, 198 ) (a) vertical displacement field (100 lines per mm) (b) horizontal displacement field (100 lines per mm) (c) 45-degree displacement field (1700 lines per mm) 36 of 69

37 (a) (b) (c) Displacement isothetics for a tensile plate with a central hole, loaded in the vertical direction simulated using the Finite Element Method for yy = 1.67 MPa, E = 3.08 GPa, 0.38 (adapted from Weissman and Post, 198 ) (a) vertical displacement field (100 lines per mm) (b) horizontal displacement field (100 lines per mm) (c) 45-degree displacement field (1700 lines per mm) Reference Values for PMMA E = 3.10 GPa = of 69

38 Photoelasticity Plane of vibration x Plane of vibration A 1 F S A A 3 y A 7 A S 9 A A F 11 4 A 6 A 8 A o A 5 A 13 A 1 45 o 45 o + 45 o A 14 (Leaving) (Leaving) (Leaving) Light source Polarizer 1 st /4 - plate (S, F = principal axes) Doubly refracting plate (x, y = principal axes) nd /4 - plate (S, F = principal axes) Analyzer Schematic of a circular polariscope experimental setup (Adapted from Durelli and Riley, 1970) 38 of 69

39 Determination of external loads Tesar (193) 1 = 0 C ( 3 ) A (3 ) A Uniaxial case, at r = a: B B A r = ( 1 / c ) a r = a r = a B E 1 = ; 3 A 1 B of 69

40 Two Inverse Problems pre-existing hole inverse problem => elastic properties y y y x x 1 x hole drilling inverse problem => residual stresses y y Surface Tractions y 1 1 x 1 x 1 x 40 of 69

41 Objective: To determine external loads k = 1 / = 0.5 k = 1 / = 0.50 k = 1 / = 1.00 light-field isochromatics light-field isochromatics light-field isochromatics dark-field isochromatics dark-field isochromatics dark-field isochromatics Cárdenas-García (1999) 41 of 69

42 Objective: To determine residual stresses k = 1 / = 0.5 k = 1 / = 0.50 k = 1 / = 1.00 light-field isochromatics light-field isochromatics light-field isochromatics dark-field isochromatics dark-field isochromatics dark-field isochromatics Cárdenas-García (1999) 4 of 69

43 Inverse Problem Methodology Simulation Experiment Define Input and Output Parameters Develop the Model Define the Forward Problem Develop the Experimental Model Define Input and Output Parameters Experimental Develop Recursive Inverse Analysis Experimental Define the Inverse Problem Define Calculation Initiation Parameters Recursive Inverse Analysis Define Calculation Initiation Parameters Define the Inverse Problem Verify Convergence of Inverse Analysis Verify Verify Calculation Con vergence Error Verify Experimental 43 of 69

44 MEMS Example: Objectives Investigate the use of small scale specimens with nonuniform geometries (holes, notches, etc.) to compute the elastic properties of thin films Investigate the applicability of the AFM/DIC strain measurement method (Chasiotis & Knauss, EM 00) to reliably acquire small displacement field data from non-uniform strain fields Investigate the applicability of a least squares approach so as to take advantage of large amounts of collected displacement field data 44 of 69

45 Measurement of Nanoscale Deformations at the Notch Region 1 cm Characterization of notched geometries (Chasiotis & Knauss, JMPS 003) F Location of AFM imaging in a perforated tension specimen Polycrystalline silicon specimens MUMPs 35 microfabrication run 45 of 69

46 Specimen Handling Electrostatically assisted UV adhesive gripping (Chasiotis & Knauss, EM 00) No slip between grip and specimen during loading 46 of 69

47 Displacement Field for a Plate with a Hole homogeneous, isotropic infinite plate with a hole 47 of 69

48 The Pre-existing Hole Problem Relation between theory and experiment: F u f ( E,,, ) e k k k 1 k Resulting in the functionals: 1 f x A B u k x 1 x x E k 1 f A B u k E k y y 1 y y of 69

49 Pre-existing Hole Problem Taylor s series expansion results in: where, k k fk f i 1 k E i f f E E Ei Ei yielding in the limit for (f k ) i+1 = 0, 1 i1 i F f k f k i E F b g HG I K J HG f I k E K J 49 of 69

50 Pre-existing Hole Problem Or, expressed in matrix notation: f ap where, f f E f f f f E f f ; a E ; P m fm f m E 50 of 69

51 Pre-existing Hole Problem On solving for the parameter correction: E P c 1 a T f Parameter update: E E E where c a a i1 i i1 i T Iteration rule: E E E i1 i 1 i1 E i1 i1 i 1 51 of 69

52 Inverse Problem Methodology Simulation Define Input and Output Parameters Develop the Model Define the Forward Problem Develop Recursive Inverse Analysis Define the Inverse Problem Define Calculation Initiation Parameters Recursive Inverse Analysis Verify Convergence of Inverse Analysis Verify Verify Calculation Con vergence Error 5 of 69

53 Simulations - Horizontal Isothetics Loading Direction = 0.1 = 0.9 Uniaxial loading pitch = 4.17 nm/fringe 53 of 69

54 Simulations Vertical Isothetics Loading Direction = 0.1 = 0.9 Uniaxial loading pitch = 4.17 nm/fringe 54 of 69

55 Simulations Horizontal Isothetics Loading Direction = 0.1 = 0. = 0.9 Equibiaxial loading pitch = 4.17 nm/fringe 55 of 69

56 Specimen Geometry Polycrystalline silicon 1 cm specimen dimensions: Width: 160 m Thickness: m Length: 700 m Hole radius: 8 m F Characterization of notched geometries (Chasiotis & Knauss, JMPS 003) 56 of 69

57 Optical Microscope Assembly Atomic Force Microscope (AFM) Pico-motor actuator Load cell Air bearing Air supply 57 of 69

58 XYZ Manual Positioner Cruciform Specimen Holder AFM Head 58 of 69

59 Inverse Problem Methodology Experiment Define the Forward Problem Develop the Experimental Model Define Input and Output Parameters Experimental Develop Recursive Inverse Analysis Experimental Recursive Inverse Analysis Define Calculation Initiation Parameters Define the Inverse Problem Verify Convergence of Inverse Analysis Verify Calculation Con vergence Error Verify Experimental 59 of 69

60 AFM Imaging Experimental Results F Area of Image Correlation F F 1 cm Left: Schematic of the gage section of a microscale tension specimen. AFM measurements were conducted in the area next to the hole marked with a dashed line. Center: AFM image of the polycrystalline silicon specimen including portion of the 8-m radius hole. (Note Loading Direction) Right: Horizontal displacement field in a 14x1 m area that was obtained using AFM/DIC. The black arrow shows the direction of the applied far-field force, F. 60 of 69

61 Simulations - Horizontal Isothetics Loading Direction = 0.1 = 0.9 Uniaxial loading pitch = 4.17 nm/fringe 61 of 69

62 Computed elastic modulus and Poisson s ratio using horizontal displacements Test Load, P, in gm (mn) Elastic Modulus, E, GPa Poisson s ration, (76.5) (77.5) (76.5) (77.5) Average of 69

63 AFM Imaging Experimental Results F Area of Image Correlation F F Left: Schematic of the geometry of a polysilicon specimen used to measure vertical (transverse) displacements. AFM measurements were conducted in the area next to the hole marked with a dashed line. Center: AFM image of the polycrystalline silicon specimen including portion of the 6.3 m radius hole. (Note Loading Direction) Right: Transverse displacement field in a 14x1 m area that was obtained using AFM/DIC. The black arrow shows the direction of the applied far-field force F. 63 of 69

64 Simulations Vertical Isothetics Loading Direction = 0.1 = 0.9 Uniaxial loading pitch = 4.17 nm/fringe 64 of 69

65 Computed elastic modulus and Poisson s ratio using vertical displacements Test Hole radius = 6.3 E (GPa) Test Test Test Test Test Test Test Average Reference Values E = 164 ± 7 GPa = 0.11 ± of 69

66 Summary An inverse problem methodology is useful to explore many problems of scientific and practical interest, as it allows us: To systematize the way to approach and to solve these types of problems To create a symbiosis between theoretical and experimental elements in solid mechanics To promote the best utilization of resources oriented to solve practical problems in solid mechanics 66 of 69

67 Thank You! W.F. Riley and J. W. Dally (1969) 67 of 69

68 Inverse Problem Methodology Simulation Experiment Define Input and Output Parameters Develop the Model Define the Forward Problem Develop the Experimental Model Define Input and Output Parameters Experimental Develop Recursive Inverse Analysis Experimental Define the Inverse Problem Define Calculation Initiation Parameters Recursive Inverse Analysis Define Calculation Initiation Parameters Define the Inverse Problem Verify Convergence of Inverse Analysis Verify Verify Calculation Con vergence Error Verify Experimental 68 of 69

69 An Inverse Problem Methodology for Experimental Solid Mechanics Questions? and / or Discussion 69 of 69