CML Lecture Series Primer on Homogenization

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1 C ecture Series Primer on Homogenization Bing C. Chen The University of ichigan, Department of echanical Engineering Computational echanics aboratory

2 utline odel problem: asymptotic analysis for a uni-directional problem Asymptotic analysis for elastic problem A more intuitive approach mplementation issues: symmetry & periodicity A Simply example: layered materials

3 Heterogeneous aterial Problem with highly-oscillating coefficients Need very refined meshes to find the local responses t esh refined to the microstructure level b X Γ t

4 ultiple-scale problem Periodic material properties: rapidly varying ε E E E2 E2 E E2 E E2 E E2 E E E E E E E E2 E2 E2 E2 E2 E2 E2 E(x) N a f d dx E x d ε dx u ε QP + f( x) ε 2ε 3ε 4ε nε (n+)ε ntroduce another scale y, to describe details of oscillation E(y) n (n+) x y y Ex ( + nε) Ex ( ) x / ε E( y + n) E( y)

5 Asymptotic Expansion u (x) 2.75 global coordinate slowly varying uε(x,y)u (x,y)+εu (x,y) x u (x,y) local coordinate rapid varying y x / ε y

6 Asymptotic Expansion: D Problem Equilibrium equation Two-scale problem: u i periodic a f a f a f d dx x + ε y d N dx E x d ε dx u ε QP + f( x) Replacing the original differential equation x + a f ε u x, y u x, y + εu x, y +, y x/ ε KJ N εa f KJ b a f a fg + + x ε y x ε y u P x, y εu x, y + f( x) Q

7 Expand the equation: rder analysis u ι (x,) u ι (x,) E E2 (ε -2 ) c 2 ( x) z N af a f a f QP a f u x, y y Ey u x, y y Ey y y y y c c x 2( x) Ey y u x y 2( ),, af a f Exy d y u x y d u x u x,,,, a f a f a f a f a f a f c ( x), u x, y u x only 2 z

8 (ε - ) rder analysis: (ε - ) + af u KJ P u N E y Q y x y af + u u E y KJ af c x x y z z afz af c x z u + x dy u y dy c x E y dy K J N u P af x E y dy Q icroscopic strain proportional to acroscopic strain u y H G N a f E y z af a fkj dy E y QP u x K J

9 rder analysis: (ε ) N a f y KJ Q u + P + u u E E y QP + f ( x) x x y y x z z z + N QP + x c a x f dy a f u E y dy f ( x) dy y x N x E y dy u x x z U K J V R S T N E H af af x u x K J P Q QP + f Homogenized coefficient: harmonic means E H a f x z N ( x ) E y dy N a f a f W + f ( x) P Q

10 z Energy consideration inimize the strain energy of the unit cell, subect to a given macroscopic strain <ε > a f E y min φ 2 z af + φ 2 E y d ε KJ H dy E ε dy 2 φ( ) φ( ) d φ dy d af E y ε + dy af E y c ε ε ε N z + + d φ dy E y dy a f c KJ d δφ dy dy d φ KJ KJ dy KJ P Q c N z z ε E( y) dy P Q + d φ dy ε KJ dy 2

11 Asymptotic Expansion: General Problem

12 Global-ocal analysis t ocal analysis ( PREAT ) b Unit Cell characteristic displacement: χ homogenized coefficient: E h X Γ t ocalization Process Unit Cell local stress distribution: σ(x,y) Global E analysis ( NASTRAN, ABAQUS ) t Homogeneous body b homogenized coefficient: E h acroscopic displacement u (x) macroscopic strain ε(u ) X Γ t

13 Asymptotic Expansion: general problem Equilibrium equation Two-scale problem Replacing the original differential equation x a f a f a f N ε ikl a f ε x f ( x) uk x ε < u x, y u > < x, y + εu > x, y +, y x/ ε + x x ε y l QP + i x + KJ N ikl + ε y x ε y l l KJ c a f k + a fh Q P < > < > u x, y εuk x, y + f( x)

14 (ε -2 ) Summary of the order analysis y N ikl a f xy, u < > k y a f x, y QP l a f x < > < > u u only (ε - ) y uk y N ikl uk y χ y QP y u p x < > < > k l uk x < > < > ikl l l q l KJ» Eq. of the characteristic displacement χ χ χ qp y N ikl χ y k l QP y i

15 Summary of the order analysis (contd) (ε ) x N ikl u + u k y KJ Q y u < > < > P < > k k + ikl l l y l y KJ + fi ( x) x N z ikl u + u k y < > < > k y l l KJ Q P + fi ( x) x x R S T N H i z u x δ δ ikl kp lq < > p q f + χ y ( x) KJ + i k l KJ Q P u p x < > q U V W + f x i ( )

16 Homogenized coefficients Homogenized elastic tensor: H i z ikl kp lq + χ δ δ y icroscopic equation for χ, in the variational form z χ k vi y y d v ikl δ kpδ lq + KJ i l Symmetric form of the homogenized elastic tensor H k mn im n ikl kp lq + zc h χ δ δ δ δ KJ dy y l z mn i k im n + ikl kp lq + χ χ δ δ δ δ dy y y KJ k l KJ dy l KJ aor/inor symmetries H H H mn mnqp nm H mn

17 ocalization process Given macroscopic strain field, find the microscopic stress field» microscopic strain field a up u χ u k k p εkl xy, f + δ pkδ ql + < > <> KJ < > xq yl yl xq» microscopic stress field a f a f a f χ u ε ε k p σ i ikl εkl ikl δ pkδ ql + xy, xy, xy, KJ yl x» Average stress (volume average) σ i z σ id H ikl u x < > p a f x KJ q < > q

18 ore ntuitive Approach

19 Two domains Engineering approach Domain: with heterogeneous material Ωε Domain: with homogeneous material X Ω + Unit Cell mpose a macroscopc strain: E Homogeneous stress: Σ acroscopic constitutive law: Σ h : E X Γ t ocal strain field ε(u) E + ε(u ) ocal strain ε(u) overall strain (E) + local corrector ε(u ) small st-order variation u contributes to zero order effect on derivative ε ui + x x ε y c a f a f KJ + < > < > ui ui x εui x, yh ε > x + ui y < > < >

20 Periodic strain field Average operator < f > Average of the gradient operator f (y) dy f f < f >< > d y i y V y V i i ( n f ) ds Γ f(γ ) f(γ 2 ) Γ2 n{-,} T n{,} T

21 Periodic strain field () Average of the strain corrector εcu h z <> <> <> <> <> u + u cnu + h i nu i ds 2 2 Average of the microscopic strain a f c h <> af c h ε u E+ ε u <> ε u E + ε u E

22 Periodic stress field ocal stress field σ satisfies» in equilibrium div σ a f σ.n are opposite on opposite side of unit cell Γ Γ2 f(γ ) f(γ 2 ) in σ.n σ.n verall stress (volume average) Σ σ

23 ocal problem Effective properties a f a fhi <>» constitutive law: σ y : E+ ε u y y a f» equilibrium div σ in, σ.n are opposite divd: c a f <> ε u yhi a f div : E z a f c h a f <> ε δu : : ε u d ε δu : : Ed linear equation with generalized loading E z d c Thermal-type oads z T T ε δu : :ε u d ε δu :αd z a f a f a f: β n i β n i

24 ocal problem: Solution by super-position Split the strain into fundamental eigen-strain z z E E, δ δ + δ δ / solve each sub-problem <> ε δ : : ε d ε δ : : d δ super-impose the results i z z a f c h a f a f c h a f c h ip q iq p 2 u u u E u T T ε δu: εχ d ε δu : d δu u <> E χ af c h E d c εχ hi ε u E+ u <> ε +

25 Three oad Cases in the homogenization ΙΙ (22) ΙΙ () ε (2) z ε δu : :εχ z a f ε δu : :ΙΙ a f c h () ΙΙ () d d z ε δu : :εχ z a f ε δu : :ΙΙ a f c h ( 22 ) ΙΙ (22) d d z ε δu : :εχ z a f ε δu : :ΙΙ a f c h ( 2 ) ΙΙ (2) d d

26 The thermal-load analogy for RHS R a f a f a f a fs T Thermal oads z z z T ε δu : :ε u d ε δu : :αd ε δu β d Thermal-type oads in the cell problem R a f c h a f a f S N z z z kl ε δu : :εχ ( ) d ε δu : :ΙΙ ( ) d ε δu λ λ + 2µ d T W T W T W µ () λ n i λ n i + 2µ n i β U V W λ + 2µ U V β n i R S λ U V β n i R U S V 4 QP 22 2 ()2 µ n i ()22 λ n i λ n i + 2µ n i 2µ n i µ n i

27 Effective stiffness tensor H : E Σ σ a f d c hi <> : ε u : E+ ε u strain energy of a homogeneous body subect to macroscopic strain H : E E : + d Average strain energy of the unit cell c h εχ i : : + εχ c hi H Effective stiffness tensor H kl : + ε ikl d c kl χ h d i i d c hi d c hi ΙΙ + εχ : : ΙΙ + εχ H i i kl kl ikl

28 mplementation ssues: symmetry & periodicity

29 Periodicity conditions Symmetry conditions periodicity conditions enforced in the unit cell Elimination agrange multiplier: (PC)

30 Symmetry conditions unit cell has planes of symmetry» material properties» geometry local problem can be solved (/2 or /4 model) with standard B.C. on the plane of symmetry

31 Symmetry conditions () Symmetric loads (, 22) u σ2 ΙΙ () ΙΙ (22) u2 σ2 anti-symmetric load (2) u2 σ ΙΙ (2) u σ22

32 R S T R S v v v Periodicity condition: elimination Reduce D by eliminating the D on the opposite side of the unit cell 2 3 U V W T N K K K T K K K T K T K K R S QP T v U T V K K2 + K3 T T T T v W N 2 K2 + K3 K22 + K33 + K23 + K23 u u u 2 3 U R V W S T v v v 2 3 U V W T R S T f f f 2 3 U V W u (2) u (3) T u v f QP R U S R U R VW T S VW S u T 2 v T 2 f + f 2 3 U V W

33 z Special case: homogeneous material ε δu : :εχ d ε δu : :ΙΙ z ε R a f δu N ( ) z ( kl) a f c h a f U R S λ + 2µ V S λ λ λ + 2µ T W T U V W R U S V T W 4 µ QP 22 2 d d λ+2µ λ+2µ Zero RHS; χ : zero vector ocalization tensor: ( +ε( ε(χ ) ) GG + H JJ GG + K H JJ H ( i) i ( i) kl Eikl rs ε d rs χ afi Ers rs ε d χafi Eikl E λ λ K λ+2µ λ ikl

34 2D Unit Cell Problem, 6 load cases Elastic Problem z ( kl) χ p δu z i δui y y y d i ikl z φ p δu i y y λ q ψ y q Thermal-elastic Problem z i ( k) i i δt y Conductivity Problem z z β λ i ik Unit strain δui y d δt y d i δ u δ u i δ T i kl kl22 kl2 T Unit temperature rise i Unit heat flux i2

35 Homogenized thermoelastic and conductivity properties z z z Unit cell problems Homogenization of thermoelastic & conductivity z z z a f a f H + εχ ( ) : : + εχ ( ) d H a f β β+ β: ε( φ) d H a f a f λ + ψ λ + ψ d b g a f εδ ( u ) : : + εχ ( ) d δu b g a f εδ ( u ) : : εφ ( ) + β T d δu a f a f δt λ + ψ d δt Unit Cell periodicity conditions enforced in in the the unit unit cell cell

36 Simply Examples: ayered aterials

37 Homogenization of layered material E,α E2,α2 ayered along y, y N + χ y a f / y 2 KJ Q P k ikl δ kpδ lq l ayered material is orthotropic R S T 2 ρ a f ρ c h c h c h c h δ δ + χ + δ δ C ( ) p q, 22 2 p 2q δ δ + χ + δ δ C ( ) 22 p 2q 22 2, 2 p q 2

38 Homogenization Results H N Q P P P P P P P P 22 2 β H 22 H G K J N Q P P P P P β β β β 22

39 Homogenization Results.8 H 2222/,α 2,α2.6.4 H / ρ ρ a f ρ.8.6 H 22/ 22 2 α α , ρ

40 Red+ellow? range Adding the CTE is not like adding paints evin s example, 967 The effective CTE dose not form a convex set Unusual CTE occurs when ρ < α α , ρ α α 22,α 2,α2 a f ρ

41 Summary A problem with highly oscillating coefficients can be solved by global-local analysis ocal analysis: characteristic displacement of the unit cell, find homogenized coefficient Global analysis: the original domain is treated as if homogeneous to find the stress/strain distribution ocalization process: given the find the local stress distribution within the unit cell

Fig. 1. Circular fiber and interphase between the fiber and the matrix.

Fig. 1. Circular fiber and interphase between the fiber and the matrix. Finite element unit cell model based on ABAQUS for fiber reinforced composites Tian Tang Composites Manufacturing & Simulation Center, Purdue University West Lafayette, IN 47906 1. Problem Statement In