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

Mechanical property from microstructure properties to macroscopic

2

• I. Mechanical

properties of crystals

Elastic moduli: isotropic form of Hooke’s law

• Shear Modulus
• Bulk Modulus
• Young Modulus
• Poison ratio
• P-wave modulus
• Stress for isotropic elasticity
• Strain for isotropic elasticity

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. Revue de l’Institut Français Du Pétrole, 53(5), 709–719. 8

Third-order nonlinear elasticity

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

Plane longitudinal (pressure) pulse wave Shear (transverse) plane wave

• third-order nonlinear elasticity, the strain

energy function E (for arbitrary anisotropy)

Phase velocities

• phase velocity of wave propagation,

for isotropic symmetry

• pure shear mode

For transversely isotropic:

• quasi-shear mode
• quasi-longitudinal 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, Austria ¨ 2Max-Planck-Institut fur Eisenforschung GmbH, Max-Planck-Strasse 1, DE-40237 D ¨ usseldorf, Germany 18

Universal linear-independent coupling strains

First-Princiyles Calculation of Stress O. H. Nielsen and Richard M. Martin Xerox Palo Alto Research Centers, Palo Alto, California 94304 (Received 20 December 1982) 19

σ𝑗 = 𝐷𝑗𝑘ε𝑘 220 51,6 51,6 51,6 220 51,6 51,6 51,6 220 3,65 3,65 3,65 Siesta Kelvin Matrix

• II. Material models

20

21

Linear Elasticity models (LEM):

• Orthotropic Linear Elasticity
• Transversely Isotropic Linear Elasticity

Example of LEM

5 10 15 20 25 0,1 0,2 0,35 0,575 0,9125 1 Strain Energy kJ Time, sec

Partial Strain energy

2000 4000 6000 0,1 0,2 0,345 0,575 0,9125 1 Stress, Gpa Time, sec

Mises stress

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 rates of strain. Part I: experimental study. J. Mech. Phys. Solids 49(7), 1517– 1538 (2001) 23

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 strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics, 21(1), 31–48. 26

Johnson-Cook plasticity models parameters for some metals

27

Example of JC model

50 100 150 200 250 300 350 400 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 Energy, kJ Time, sec

Partial Strain energy

20 40 60 80 100 120 Stress, GPa Time, sec

Mises stress

28

Viscoelasticity

29

creep relaxation

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 University Press (1997) 32

The Standard Linear Solid Viscoelastic Model

33

The Generalized Maxwell Model (GM)

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

Example of GM model

35

• 0,001
• 0,0005

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 Energy, kJ Time, sec

Total energy

50000 100000 150000 0,00 0,10 0,20 0,30 Stress, Pa Time, sec

Mises

Temperature Dependence and Viscoelasticity

• Williams-Landel-Ferry (WLF) equation

36

• Arrhenius equation

Nonlinear Elasticity

37

Helmholtz free-energy function:

Classes Hyperelastic Material Models

Phenomenological models Mechanistic models

• Saint-Venant Kirchoff
• Polynomial
• Ogden
• Mooney-Rivlin
• Yeoh
• Arruda-Boyce
• Edwards-Vilgris
• Neo-Hookean

38

Neo-Hookean Hyperelastic Material Model (NHHM)

39

Example of NH model

• 200
• 150
• 100
• 50

50 100 150 0,00 0,05 0,07 0,07 0,07 0,08 0,08 0,08 0,09 0,09 0,09 0,09 0,09 0,10 0,11 0,11 0,11 0,11 Energy, kJ

40

1 2 3 0,000 0,059 0,073 0,075 0,078 0,092 0,093 0,095 0,101 0,109 0,110 0,110 Stress, GPa Time, sec

Mises stress

Arruda-Boyce Hyperelastic Model

Boyce, M.C., Arruda, E.M.: Constitutive models of rubber elasticity: a review. Rubber

• Chem. Technol. 73(3), 504–523 (2000)

41

Example of AB model

5 10 15 20 25 30 0,000 0,018 0,020 0,022 0,023 0,024 0,026 0,027 0,028 0,029 0,030 0,032 0,033 0,034 0,036 0,037 Energy, kJ Time, sec

Partial Strain energy

1 2 3 0,000 0,018 0,020 0,022 0,023 0,025 0,026 0,027 0,029 0,030 0,031 0,033 0,034 0,035 0,037 Stress, GPa Time, sec

Mises stress

42

Edwards-Vilgis Hyperelastic Model

Edwards, S., Vilgis, T.: The effect of entanglements in rubber elasticity. Polymer 27(4), 483– 492 (1986) 43

• strain energy due to cross-links
• the strain energy due to slip-links

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)

46

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

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)

Inductor current density distribution

60

matrx workpeace inductor

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

Electro-magnetic stamping (analytics)

10000 20000 30000 40000 50000 65 130 195 260 325 6,25 12,5 18,75 25 31,25 37,5 43,75 50 56,25 62,5

V, м/с t, мкс экспериментальная кривая расчетная кривая токовая кривая

• 40000
• 30000
• 20000
• 10000

10000 20000 30000 40000 50000 60000 50 52 54 56 58 60 62 64 66 68 70 6,25 12,5 18,75 25 31,25 37,5 43,75 50 56,25 62,5 68,75 75 81,25 87,5 93,75

D, мм t, мкс

62

Heat transfer analysis procedures and Electromagnetic application using FEM

64

Fluid dynamics application of FEM (ANSYS Fluent simulations)

Gas exchange Two phase flow Rocket engine Wind emulation

66

Discrete element modeling (Rocky DEM simulations)

68

CAE-systems

69

Literature:

• J. B. Ketterson - The Physics of solids. – Oxford 2016.
• G. Mavko, T. Mukerji, J. Dvorkin - The Rock physics, Second
• Edition. – Cambridge 2009
• An Introduction to Continuum Mechanics by Morton E. Gurtin
• Ellad B. Tadmor, Ronald E. Miller - Modeling Materials Continuum,

Atomistic and Multiscale Techniques. – Cambridge 2011

• Michael Okererke, Simeon Keates - Finite Element Applications, -

Springer 2018

• Jiyuan Tu Guan-Heng Yeoh, Chaoqun Liu - Computational Fluid
• Dynamics. – Butterworth-Heinemann 2018
• Tarek I. Zohdi - Modeling and Simulation of Functionalized

Materials for Additive Manufacturing and 3D Printing: Continuous and Discrete Media. – Springer 2018.

70