in materials/structural modellings and designs Liping Liu

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1 Closed-form solutions and their applications in materials/structural modellings and designs Liping Liu University it of Houston

Why composites and heterogeneous structures?

electric conductivity, density, elastic moduli, etc.

2 Why composites and heterogeneous structures? A material has many different properties in different context: electric conductivity, density, elastic moduli, etc. Applications may require a point that cannot be achieved by natural materials. New physics may emerge and new properties may be created Microstructure properties? 2 Picture from

3 Composites of 30% Terfenol-D and polymer Optic micrograph from Thesis of McKnight 2002, UCLA Copper+MoS 2 at/ Multi-Functional Composite and Honeycomb Panels, by Nasa /business/tg-img-honeycombpanels.html Boeing 787, 50% composites 3

4 Microstructure design Air bubles in self-assembled Top view of patterned media (Hard-drive polymer structure (Srinivasarao et al thin film). Graph from Rousseaux et al 2001, Science) (1995, J. Vac. Sci. Tech. B) 4

5 Overview Modeling: Predict the effective behaviors Homogenization, multiscale-multiphysics py analysis Eshelby s framework, Mori-Tanaka s method Optimization Hashin-Shtrikman bounds and their attainment A level-set based gradient method Designing (optimal) (micro)-structure with desired fields (Periodic) E-inclusions Magnets with precise multipole fields Flavor of Inverse problem: determine the unknown domains

6 Part I Modeling of heterogeneous media

7 Homogenization (1) Variational formulation: E-L equation: Periodic composites: Formal asymptotics: Energy expansion:

8 Unite cell problem (1) Periodic composites, structure volume fractions Effective tensor An optimal design problem (G-closure):

9 Dilute limit - Eshelby inclusion problem The governing elasticity equation Task: compute/minimize/maximize i i / i i the elastic energy 9

10 Eshelby s solution (1) HOMOGENEOUS Eshelby inclusion problem Ellipsoidal inclusion, Eshelby s Uniformity ypropertyp Elastic Energy 10

11 Eshelby s solution (2) INHOMOGENEOUS Eshelby inclusion problem Ellipsoidal inclusion, Eshelby s Uniformity ypropertyp Elastic Energy 11

12 Why Eshelby s solutions? Inhomogeneous Eshelby inclusion problem Transformation problem: model precipitates Crack, energy release rate or stress intensity factor Composites, effective properties, Mori-Tanaka See Mura (1987) for applications of Eshelby s solutions. Weng (1990) for relations between the two approaches of Mori-Tanaka vs Hashin-Shtrikman. 12

showed this fact Limitations Single inclusion, no

13 Uniformity and Optimality Least stress concentration Equilibrium / energy-minimizing shapes Roiturd (1986) showed this fact Limitations Single inclusion, no interaction can be accounted tdprecisely Copper+MoS 2 13

14 Eshelby s conjecture Eshelby (1961) conjectured: Among closed surfaces, the ellipsoid alone has this convenient (uniformity) property? Restricted to connected domains, the Eshelby conjecture was resolved by Asaro and Barnett 14 in 1975 in 2D, and in 3D, was recently resolved by Kang and Milton, and independently, by myself. This work is published in Proc. Roy. Soc. A (2008)

15 Generalizations of Ellipsoids Uniformity property: Polynomial field: Analyticity it is guaranteed:

16 E-inclusions Generalized Ellipsoids Eshelby-type solutions Extremal inclusion 16

17 E-inclusions TEM graph of Ni-based superalloy (from Cha, Yeo and Yoon 2006) 17

18 E-inclusions 18

19 Definition of Ellipsoids Ellipsoids have the property that a solution of satisfies the overdetermined condition Maxwell-Eshelby s solution 19

20 Definition of E-inclusions E-inclusions have the property that a solution of satisfies the overdetermined condition 20

21 Matrices and volume (fraction) 21

$(fractions) Existence of$ periodic simply-connected

22 Definition of periodic E-inclusions Periodic E-inclusions: satisfies the overdetermined condition inclusion Matrices and volume (fractions) Existence of periodic simply-connected E-inclusions in 2D was first shown by Vigdergauz (1986). 22

23 Definition of polynomial inclusions A polynomial inclusion corresponding to a polynomial of degree k is such that the solution of satisfies the overdetermined condition 23

24 Closed-form solution of the effective conductivity for two-phase composites Conductivity problems Bulk modulus of elastic composites Thermoelectric composites Thermo-elastic composites Hall-coefficients Magneto-electric composites 24

$volume fractions,$

25 Linear multiphase composites Periodic composites, microstructure volume fractions, effective tensor If, then 25

26 Nonlinear composites At two-phase periodic composite with a nonlinear inclusion i Effective energy density Application of the Eshelby s solution to nonlinear composites is first realized by Hill (1965) 26

27 Part II Optimality of E-inclusions

28 Why bounds? Effective medium theories: Maxwell (1862): effective dielectric constants Einstein i (1906): effective viscosity i of two-phase fluids Voigt-Reussg bounds, Hashin-Shtrikman bounds (1965) 28

29 HS bounds --- Derivation (1) Difficulty: differential constraints or kinematics Classic bounds: A different way of derivation. Find microstructure-independent bounds for Similar argument has been used by Silvestre (2007) for cross-property bounds

30 HS bounds --- Derivation (2) For the upper bound, assume that

31 HS bounds --- Derivation (3) Add and subtract: Substituting it back yields HS bounds:

32 HS bounds Lower HS bounds. Upper HS bounds.

33 Attainability conditions (1)

34 Attainability conditions(2) Checking the conditions for equality, we find the fll following necessary and sufficient i condition i where

35 Examples of optimal two-phase composites: conductivity Multi-rank laminates (laminates within ihi laminates): Coated spheres (confocal ellipses) Periodic structures: periodic E- inclusions

36 Two-phase well-ordered conductive composite Existence of periodic E-inclusion for any HS bounds describe the G-closure More than two phases or non-wellordered, or elasticity problem? Tartar (1986) and Grabovsky (1993) first solved the G-closure of two-phase conductive composites

37 New optimal microstructures Three-phase coated spheres Optimal multiphase composites

38 Attainable HS bounds (1) Attainability of HS bounds for m=0 has been shown by Milton (1981)

39 Attainable HS bounds (2)

40 Restrictions (1): the null lagrangian Null lagrangian: Ericksen (1962) showed that all null lagrangian can be written as a linear combinations of sub-determinants. See Kinderlehrer and Pedregal (1991) for theory of gradient Young measure

41 Restrictions (2): the Maximum principle Many other choices. But it is hard to find the optimal choices. Apply the maximum principle to Find the such that t

42 Restrictions (3) In particular, if For two dimensional space and three phases, this was shown by Albin, Cherkaev and Nesi (2007).

43 Attainable & unattainable bounds

44 Microstructure evolution Equilibrium shapes TEM graph of Ni3Ge precipitates (Courtesy of Kim & Ardell) 44

45 Minimizing field concentration Minimum stress concentration Phase splitting TEM graph of Ni-based superalloy (from Cha, Yeo and Yoon 2006) 45

46 Part III Designing (optimal) (micro-) structures with prescribed fields

47 Designing for prescribed fields Field expelling: for precision measurement and advanced research lab Field confinement: MRI, tokomak devices Syncrotrons: multipole fields DC motors: multipole fields mumetal.co.uk

48 Magnetic field of a p-inclusion Maxwell equation: In terms of Newtonian potential, Newtonian potential problem: Therefore, for a p-inclusion, the interior field is given by

49 Uniform field

50 Self-shielding uniform field

51 Quadrupole field

52 Sextupole field

53 Part IV The method: a simple variational inequality

54 Existence of E-inclusions Variational inequality Coincident (contact) set Piecewise quadratic obstacle Coincident set is an E-inclusion Other obstacles: other special fields Coincident set 54

55 Uniqueness of p-inclusions of degree Eshelby s s conjecture Uniqueness of variational inequality then implies the uniqueness of the p-inclusions of degree 2. Uniqueness of p-inclusions of higher degrees?

56 Numerical scheme Numerical schemes Finite element discretization Quadratic programming 56

57 Periodic E-inclusions 57

58 Examples of E-inclusions (1)

59 Uniform fields 59

60 Open questions Improve Hashin-Strickman bounds Statement: Uniqueness of p-inclusions of degree higher than three Statement: Analytic parametrization of p-inclusions of degree higher than three

61 Thank you!

Hashin-Shtrikman bounds for multiphase composites and their attainability

ARMA manuscript No. (will be inserted by the editor) Hashin-Shtrikman bounds for multiphase composites and their attainability Liping Liu October 16, 2007 Manuscript submitted to Archive for Rational Mechanics