from microstructure properties to macroscopic I. Mechanical - PowerPoint PPT Presentation

Mechanical property from microstructure properties to macroscopic

I. Mechanical properties of crystals 2

Elastic moduli: isotropic form of Hooke’s law - Stress for isotropic elasticity - Strain for isotropic elasticity - Bulk Modulus - Shear Modulus - Young Modulus - Poison ratio - P-wave modulus 4

Requirements and relations of constants 5

Elastic stiffness tensor - 81 independent components - 36 independent components h.l. - 21 independent components 6

Voigt notation - Vectors of stress and strain - Hooke’s law, using the Voigt notation 7

Kelvin notation - Weight matrix Dellinger, J., Vasicek, D., & Sondergeld, C. (1998). Kelvin Notation for Stabilizing Elastic-Constant Inversion. 8 Revue de l’Institut Français Du Pétrole , 53(5), 709 – 719.

Third-order nonlinear elasticity Shear (transverse) plane wave Plane longitudinal (pressure) pulse wave - third-order nonlinear elasticity, the strain energy function E (for arbitrary anisotropy) Milholland, P., Manghnani, M.H., Schlanger, S.O., and Sutton, G.H., 1980. Geoacoustic modeling of deep-sea carbonate sediments. J. Acoust. Soc. Am., 68, 1351 – 1360 9

Phase velocities - phase velocity of wave propagation, for isotropic symmetry For transversely isotropic: - quasi-longitudinal mode - quasi-shear mode - pure shear mode Milholland, P., Manghnani, M.H., Schlanger, S.O., and Sutton, G.H., 1980. Geoacoustic modeling of deep- sea carbonate sediments. J. Acoust. Soc. Am., 68, 1351 – 1360 10

Elastic eigentensors - For an isotropic material, Hooke’s law - the strain energy U for an isotropic material 11

Modules defined for anisotropic elastic materials Transversely isotropic (TI) material with uniaxial stress: -Transversely isotropic Young’s modulus - TI Poisson ratios 12

Compliance matrix for anisotropic elastic materials 13

Thomsen’s notation for weak elastic anisotropy - Thomsen’s notation - Berryman extends the validity of Thomsen’s expressions for P - and quasi SV-wave velocities - For weak anisotropy, the constant e 14

Requirements to Thomsen`s notation - an additional anellipticity parameter 15

4-order nonlinear elasticity - The apparent fourth-rank stiffness 16

DFT methods of calculating mechanical properties • Symmetry-general least-squares extraction of elastic coefficients from ab initio total energy calculations • Universal linear-independent coupling strains 17

Symmetry-general least-squares extraction of elastic coefficients from ab initio total energy calculations Trends in the elastic response of binary early transition metal nitrides David Holec,1,* Martin Friak, ´ 2 Jorg Neugebauer, ¨ 2 and Paul H. Mayrhofer1 1Department of Physical Metallurgy and Materials Testing, Montanuniversitat Leoben, Franz-Josef-Strasse 18, AT-8700 Leoben, 18 Austria ¨ 2Max-Planck-Institut fur Eisenforschung GmbH, Max-Planck-Strasse 1, DE-40237 D ¨ usseldorf, Germany

Universal linear-independent coupling strains σ 𝑗 = 𝐷 𝑗𝑘 ε 𝑘 220 51,6 51,6 0 0 0 51,6 220 51,6 0 0 0 51,6 51,6 220 0 0 0 0 0 0 3,65 0 0 0 0 0 0 3,65 0 0 0 0 0 0 3,65 Siesta Kelvin Matrix First-Princiyles Calculation of Stress O. H. Nielsen and Richard M. Martin Xerox Palo Alto Research 19 Centers, Palo Alto, California 94304 (Received 20 December 1982)

II. Material models 20

Linear Elasticity models (LEM): - Orthotropic Linear Elasticity - Transversely Isotropic Linear Elasticity 21

Example of LEM Partial Strain energy 25 Strain Energy kJ 20 15 10 5 0 0 0,1 0,2 0,35 0,575 0,9125 1 Time, sec Mises stress 6000 Stress, Gpa 4000 2000 0 0 0,1 0,2 0,345 0,575 0,9125 1 Time, sec 22

Border cases of plastic models • Elastoplastic Material Model - Perfect plasticity - Isotropic strain hardening - Isotropic stress softening Buckley, C., Harding, J., Hou, J., Ruiz, C., Trojanowski, A.: Deformation of thermosetting resins at impact 23 rates of strain. Part I: experimental study. J. Mech. Phys. Solids 49(7), 1517 – 1538 (2001)

Constitutive Mathematics of Elasto- Plasticity • Onset of Yield • Yield Criterion 24

The hardening • The Isotropic Hardening • The Kinematic Hardening 25

Johnson-Cook Plasticity (JC) - The Johnson-Cook hardening formulation Johnson, G. R., & Cook, W. H. (1985). Fracture characteristics of three metals subjected to various 26 strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics, 21(1), 31 – 48.

Johnson-Cook plasticity models parameters for some metals 27

Example of JC model Partial Strain energy Energy, kJ 400 350 300 250 200 150 100 50 0 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 Time, sec Mises stress Stress, GPa 120 100 80 60 40 20 0 Time, sec 28

Viscoelasticity relaxation creep 29

Constitutive Models • Constitutive Models for Creep Response • Constitutive Models for Stress Relaxation Response 30

Dynamic Viscoelasticity 31

The Standard Linear Solid Viscoelastic Model McCrum, N., Buckley, C., Bucknall, C.: Principles of Polymer Engineering. Oxford Science Publications, Oxford 32 University Press (1997)

The Standard Linear Solid Viscoelastic Model 33

The Generalized Maxwell Model (GM) McCrum, N., Buckley, C., Bucknall, C.: Principles of Polymer Engineering. Oxford Science 34 Publications, Oxford University Press (1997)

Example of GM model Total energy Energy, kJ 0 0,00 1,30 2,60 3,90 5,20 6,50 7,80 9,10 10,40 11,70 13,00 14,30 15,60 16,90 18,20 19,50 -0,0005 -0,001 Time, sec Mises Stress, Pa 150000 100000 50000 0 0,00 0,10 0,20 0,30 Time, sec 35

Temperature Dependence and Viscoelasticity -Arrhenius equation - Williams-Landel-Ferry (WLF) equation 36

Nonlinear Elasticity Helmholtz free-energy function: 37

Classes Hyperelastic Material Models Phenomenological models Mechanistic models • • Saint-Venant Kirchoff Neo-Hookean • • Polynomial Arruda-Boyce • Ogden • Edwards-Vilgris • Mooney-Rivlin • Yeoh 38

Neo-Hookean Hyperelastic Material Model (NHHM) 39

Example of NH model Energy, kJ -200 -150 -100 100 150 -50 50 0 Stress, GPa 0,00 0 1 2 3 0,05 0,000 0,07 0,07 0,059 0,07 0,073 Mises stress 0,08 0,075 0,08 0,078 0,08 0,09 0,092 0,09 0,093 0,09 0,095 0,09 0,101 0,09 0,109 0,10 Time, sec 0,11 0,110 0,11 0,110 0,11 0,11 40

Arruda-Boyce Hyperelastic Model Boyce, M.C., Arruda, E.M.: Constitutive models of rubber elasticity: a review. Rubber 41 Chem. Technol. 73(3), 504 – 523 (2000)

Example of AB model Energy, kJ Stress, GPa 10 15 20 25 30 0 5 0 1 2 3 0,000 0,000 0,018 Partial Strain energy 0,018 0,020 0,020 0,022 0,022 Mises stress 0,023 0,023 0,024 0,025 0,026 0,026 0,027 0,027 0,028 0,029 0,029 0,030 0,030 0,031 0,032 0,033 0,033 0,034 Time, sec 0,034 Time, sec 0,035 0,036 0,037 0,037 42

Edwards-Vilgis Hyperelastic Model - strain energy due to cross-links -the strain energy due to slip-links Edwards, S., Vilgis, T.: The effect of entanglements in rubber elasticity. Polymer 27(4), 483 – 492 (1986) 43

Stress Formulation for Hyperelastic Material Models • Incompressible material with strain energy of form • Incompressible material with strain energy of form • Nominal stress for an incompressible material • Compressible material with strain energy of form • Nominal stress for an Compressible material: 44

III. Finite Elements Method 45

Computer-aided engineering ( CAE ) Computer-aided engineering ( CAE ) is the broad usage of computer software to aid in engineering analysis tasks. It includes finite element analysis (FEA), computational fluid dynamics (CFD), multibody dynamics (MBD), durability and optimization. It is included with computer- aided design (CAD) and computer-aided manufacturing (CAM) in the collective abbreviation "CAx 46

Finite Element Method 47

Numerical Methods for Computational Mechanics 48

The Need for the Finite Element Method • The PDE should have a solution that exists; • The solution must be unique; and finally, • The solution of the PDE should change continuously with the initial (boundary) conditions defined for the PDE Limitations: • The overly stiff/locking problem • The stress accuracy problem • Mesh distortion problem • Element shape problem • Discontinuity conundrum 49

Mesh Elements 51

Elements type 52

Mathematical theorems for mechanical calculation with FEM • Jacobian using Newton method • Cauchy-Green function • Boundary conditions: Dirichlet- Neumann • Von-Mises 54

Mechanical application of FEM (Abaqus simulations) 56

Dynamic analysis (Abaqus simulations) 58

Dynamic analysis example (electromagnetic stamping) matrx inductor workpeace Inductor current density distribution 60

Dynamic analysis example (LS-Dyna) Distribution of Lorentz forces in the workpiece and inductor in vector form The distribution of current density in the workpiece 61