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# PhD ISG, Seminar by Dijmedo Kondo, 26 April 2017

Homogenization of ductile porous materials with pressure-sensitive matrix: theoretical formulation and numerical bounds.

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### PhD ISG, Seminar by Dijmedo Kondo, 26 April 2017

1. 1. HOMOGENIZATION OF DUCTILE POROUS MATERIALS WITH PRESSURE-SENSITIVE MATRIX: THEORETICAL FORMULATION AND NUMERICAL BOUNDS , . , . . Institut Jean Le Rond d’Alembert, UMR 7190 CNRS, UPMC, France La Sapienza, Roma - April 26, 2017 CHENG et al. (LML, France) Presentation EMMC14 - Goteborg EMMC14 - Goteborg, Aug. 27-29, 2014 1 / 18
2. 2. Outline Introduction and motivation 1 Limit analysis of porous material matrix 2 Conclusions CHENG et al. (LML, France) Presentation EMMC14 - Goteborg EMMC14 - Goteborg, Aug. 27-29, 2014 2 / 18
3. 3. Introduction Physical mechanisms of ductile fracture of metals Ductile fracture in metals Modiﬁcations of the Gurson’s model (Tvergaard, 1982; Tvergaard and Needleman, 1984; Needleman and Tvergaard, 1985). Weck and Wilkinson (2008). In-situ SEM images of the deformation sequence of an aluminum alloy 5052. Modeling ductile behavior of porous materials Limit analysis and non-linear homogenization approaches Gurson, (1977); Ponte Castaneda, (1991); Michel and Suquet, (1992); Guo et al., (2008); Anoukou et al., (2016). Gurson’s model (1977) F = ⌃2 eq 2 0 + 2p cosh ✓ tr⌃ 2 0 ◆ 1 p2 D = @F @⌃ ˙p = (1 p)trDwith ( = 0 if F(⌃ < 0) 0 if F(⌃ = 0) Strength criterion Normality rule Evolution law nucleation & growth of voids porosity (Puttick, 1960; Rogers, 1960, McClintock 1968, Rice and Tracey 1969)
4. 4. Carte min ral Carte de porosit <5 12,5 20 27,5 (-) 250 µm LML, UMR8107 CNRS, USTL Colloque SOIZE 2010 01-02 July 18 / 18 porosity plays a crucial role Geomaterials : example of argillite with a matrix (known to be pressure sensitive)
5. 5. Presentation of the model Experimental evidences of the behavior of chalk materials 30 35 40 Contrainteisotrope(MPa) 0 5 10 15 20 25 0 10 20 30 40 50 60 70 80 Déformation axiale (10-3) Contrainteisotrope(MPa) phase "élastique" phase plastique phase "élastique" "pore collapse" q l' espace des contraintes p' / q p' (contrainte effective moyenne) = (σ'1 + 2 σ'3) / 3 q (contrainte déviatorique) = σ'1 - σ'3 cisaillement "pore collapse" p' traction Complex phenomenological multi-surface-based plasticity models are usually considered, but they are not easy for numerical implementation Challenge : proposal of a constitutive model based on a micromechanically derived single yield surface
6. 6. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives ‚ Geomaterials and some ductile metals pressure-sensitive and Lode angle dependen (see for instance Xue 2007 ; Bai and Wierzbicki 2008 and 2010) ‚ Guo et al. 2008, have propose a strength criterion of a porous solid with pressure-sensitive dilatant matrix ‚ theoretical modelling of porous solids with a matrix obeying a pressure and Lode angle dependent plastic criterion : Mohr-Coulomb criterion F(‡) = ‡eq ` –(◊L )(‡m ´ H) § 0 –(◊L ) = sin(„) ’(◊L ) , H = C cot(„) ’(◊L ) = 1? 3 cos(◊L ) ´ 1 3 sin(„) sin(◊L ) ◊L = ´1 3 arcsin ´? 27J3 2‡3 eq ¯ ´30˝ § ◊L § 30˝ Kokou Anoukou EMMC-14 Sweden 3 / 16
7. 7. Kinematical Limit Analysis as an appropriate tool to predict strength spherical void eR iR .v xD Perfectly plastic matrix Strength of the solid phase deﬁned by a the convex set Gs of admissible stress states Gs = {σ, fs (σ) ≤ 0} Dual deﬁnition by means of the support function πs(d) of Gs πs (d) = sup(σ : d, σ ∈ Gs ) πs(d) represents the maximum "plastic" dissipation Séminaire LMA - Marseille (USTL) Micromécanique et Milieux poreux 02/02/2010 4 / 25
8. 8. Methodology Kinematical Limit Analysis again Macroscopic counterpart of πs(d) in the absence of interface effect Πhom (D) = (1 − f) inf v∈V(D) πs(d) s with d = 1 2 (grad v +t grad v) Support function of the domain Ghom of macroscopic admissible stresses Πhom (D) = sup(Σ : D, Σ ∈ Ghom ) Ghom : domain of macroscopic admissible stresses limit stress states at the macroscopic scale : Σ = ∂Πhom/∂D Séminaire LMA - Marseille (USTL) Micromécanique et Milieux poreux 02/02/2010 5 / 25
9. 9. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives Mohr-Coulomb criterion and the associated plastic dissipation function – Perfect plasticitywith Mohr-Coulomb strength criterion: In terms of principal stresses the M-C criterion reads F(‡) = sup i,j P t1,2,3u t| ‡i ´ ‡j | `(‡i ` ‡j ) sin(„)u ´ 2C cos(„) = 0 Kokou Anoukou EMMC-14 Sweden 4 / 16 By means of the stress invariants F(‡) = ‡ s eq ` –(◊L )(‡m ´ H) § 0 –(◊L ) = 1 3 ’ in ( ( ◊ „ ) ) , H = C cot(„) ’(◊L ) = ?1 3 cos L (◊L ) ´ ´? sin 27 ( J „) ¯ sin(◊L ) 1 3 ◊L = ´ arcsin 3 2‡3 eq ´30˝ § ◊L § 30˝ ﬁ(d) = `8, if tr(d) † (|d1 | ` |d2 | ` |d3 |) sin(„) ﬁ(d) = H.tr(d), if tr(d) • (|d1 | ` |d2 | ` |d3 |) sin(„)
10. 10. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives Atrialvelocityﬁeldforthekinematicallimitanalysis A first step of the analysis: ice of a relevant trial velocity field that is one which must be kinematically admissible (K.A) and comply with the plastic admissibility (P.A.) condition. A -parameter velocity field basedon , CR Mecanique is considered: v(x) = A0 ´b r ¯3— x ` A¨x with — = 3 ´ ‘ sin(„) 3(1 ` ‘ sin(„)) , and A = A1 ` eﬂ beﬂ ` eÏ beÏ ˘ ` A2ez bez (9) ‚ Kinematic admissibility (K.A.) condition : v(x = ber ) = D.x # ﬂ(A0 ` A1) = ﬂDﬂ z(A0 ` A2) = zDz ñ \$ & % A1 = Dm ´ Deq 2 sign(J3) ´ A0 A2 = Dm ` Deqsign(J3) ´ A0 ‚ Let us recall the plastic admissibility (P.A.) condition tr(d) • (|d1 | ` |d2 | ` |d3 |) sin(„) Kokou Anoukou EMMC-14 Sweden 6 / 16
11. 11. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives Minimization problem and resolution (see Anoukou et al. JMPS Part I, 91, 145-171 – Minimization problem P := \$ ’’’’’& ’’’’’% ⇧(D) = min v xﬁ(d)y⌦ s.c. A1 = Dm ´ Deq 2 sign(J3) ´ A0 (a) A2 = Dm ` Deqsign(J3) ´ A0 (b) dm • Gi (c) which by considering the limit load relation ⌃:D =⇧(D), translates in the following auxiliary minimization problem P˚ : P˚ := \$ ’’’’’& ’’’’’% min v ` xﬁ(d)y⌦ ´ 3⌃mDm ` ⌃eqDeq ˘ s.c. A0 ` 1 3 (2A1 ` A2) = Dm (a) 2 3 (A1 ´ A2) = Deqsign(J3) (b) xdmy⌦´Ê • xGiy⌦´Ê (c) with xﬁ(d)y⌦ = 3(1 ´ f )H xdmy⌦´Ê = 3H ” Dm(1 ´ f ) ´ A0 ` f 1´— ´ f ˘ı Kokou Anoukou EMMC-14 Sweden 8 / 16
12. 12. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives Minimization problem : Lagrangian method and Karush-Kuhn-Tucker (KKT) conditions – Lagrangian method and Karush-Kuhn-Tucker (KKT) conditions ` xdmy⌦´Ê´xGiy⌦´Ê ˘ L(v, ⁄) = ´3(1´f )H xdmy⌦´Ê`3⌃mDm`⌃eqDeq`⁄(1-f) where ⁄ is the Lagrange-KKT multiplier. or explicitly using (1 ´ f ) xdmy⌦´Ê = “ Dm(1 ´ f ) ´ A0 ` f 1´— ´ f ˘‰ , we have : L(A0, Dm, Deq, ⁄) = ´(3H ´ ⁄) ´ (1 ´ f )Dm ´ (f 1´— ´ f )A0 ¯ ` 3⌃mDm` ⌃eqDeq ´ ⁄Ji(A0, Deq) with Ji(A0, De) = (1 ´ f ) xGiy⌦´Ê The KKT conditions are given in system (S) as follows : (S) := \$ ’’’’’’& ’’’’’’% @L @A0 = 0, @L @Dm = 0, @L @Deq = 0 @L @⁄ • 0 ô xdmy⌦´Ê • xGiy⌦´Ê ⁄ • 0 ⁄(xdmy⌦´Ê ´ xGiy⌦´Ê) = 0 ô ⁄ = 0 or xdmy⌦´Ê = xGiy⌦´Ê Kokou Anoukou EMMC-14 Sweden 9 / 16
13. 13. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives Parametric form of the strength criterion and void growth evaluation – Parametric form of the strength criterion F(Σm, Σeq) := \$ ’’’’’’& ’’’’’’% Σm C = ` 1 ´ f ˘Ψm 3γ ` f 1´β ´ f ˘ ` Ψm Σeq C = ` f 1´β ´ f ˘Ψeq γ ` f 1´β ´ f ˘ ` Ψm with Ψm = ω 1 β ´1 2 ˜ 1 β ´ I(ω) ` m ¯ ` ωI1 (ω) ¸ Ψeq = ω 1 β 4β ˜ β ´ 1 β ´ I(ω) ` eq ¯ ´ ωI1 (ω) ¸ – Void growth : evolution law 9f = 3 Deqω 2β ´ f 1´β ´ f ¯ or 9f = 3Dm(1 ´ f ) ´ 3Deqω 1 β 4β I(ω) Kokou Anoukou EMMC-14 Sweden 11 / 16
14. 14. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives Some features of the obtained criterion f =25% f =10% f =5% C = 1, f = 20 ° -15 -10 -5 0 -4 -2 0 2 4 6 Sm Sr-Sz f=0.01° f=1° f=10° C = 1, f = 25 % -2 -1 0 1 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Sm Sr-SzC = 1, f = 10 %, f = 10 ° J3 < 0 HqL = 30 °L J3 > 0 HqL = -30 °L -5 -4 -3 -2 -1 0 1 2 0.0 0.5 1.0 1.5 2.0 Sm Seq Kokou Anoukou EMMC-14 Sweden 12 / 16 Few comments on cap models
15. 15. Numerical assessment
16. 16. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives Numerical bounds (Pastor et al. 2010 6 JMPS, Part II) – Finite element discretization combined with a non-linear optimization procedure using MOSEK software – Numerical results The lower and upper bounds of the macroscopic criterion are Kokou Anoukou EMMC-14 Sweden 13 / 16
17. 17. 10 8 6 4 2 0 2 4 2 1 0 1 2 3 = 5 f = 1% f = 5% f = 10% ⌃m/c ⌃gps/c Lower bound Upper bound New criterion (a) 16 14 12 10 8 6 4 2 0 2 4 4 2 0 2 4 = 10 f = 1% f = 25% ⌃m/c ⌃gps/c Lower bound Upper bound New criterion (b) Comparison of the derived strength criterion predictions with the numerical bounds for two ﬁxed values of friction angle and several values of porosity: (a) = 5 , f = 1%, 5% and 10%, (b) = 10 and f = 1% and 25%.
18. 18. 16 14 12 10 8 6 4 2 0 2 4 6 4 2 0 2 4 f = 1% = 10 = 5 ⌃m/c ⌃gps/c Lower bound Upper bound New criterion (a) 12 10 8 6 4 2 0 2 4 2 0 2 4 f = 10% = 20 = 5 ⌃m/c ⌃gps/c Lower bound Upper bound New criterion (b) Comparison of the derived strength criterion predictions with the numerical bounds, for ﬁxed porosity f = 10%and two friction angles =5 and 20 .
19. 19. Conclusions micromechanical procedure - . The approach The criterion illustrated and validated by comparison with . Extension to saturated geomaterials by means of an effective stress concept Implemen t and structural computations with the complete model Extension to nanoporous materials (joint work with Prof. G. Vairo (Roma 2) : co-supervised PhD Thesis of Stella Brach (MOM, 2016; IJP, 2017) CHENG et al. (LML, France) Presentation EMMC14 - Goteborg EMMC14 - Goteborg, Aug. 27-29, 2014 18 / 18
20. 20. Introduction Can porous strength criteria be eﬀectively used at sub-micron lengthscales? Hutchinson (2000), International Journal of Solids and Structures, 37, pp. 225-238. At nanoscales, mechanical features are dramatically diﬀerent from those of the same material at a larger lengthscale Biener et al., (2005), (2006); Cheng et al., (2013); Hakamada et al., (2007); Hodge et al., (2007)
21. 21. SEM of a nanoporous alumina (Kustandi et al., 2010) Organic or inorganic solid matrix a  100 nmUniform void size High speciﬁc surface area [Hodge et al. (2007)] [Volkerta et al. (2006)] Nano-indentation test  Fan and Fang, (2009) Ligament size [nm] Yieldstrength[GPa] Void-size eﬀects Ligament size a  100 nm Some experimental observations on anoporous materials Zhang et al., (2007, 2008, 2010); Goudarzi et al., (2010); Moshtaghin et al., (2012); Dormieux and Kondo, (2013). Limit analysis Dormieux and Kondo, (2010); Monchiet and Kondo, (2013). Non-linear homogenization Strength Properties of Nanoporous materials : A molecular Dynamics based approach by S. Brach, L. Dormieux, D. Kondo & G. Vairo (work with Univ. Tor Vergata; MOM 2016)
22. 22. Stella Brach PhD dissertation, 29 November 2016 Introduction Non-linear homogenization Molecular Dynamics Limit Analysis Conclusions Objective: strength properties nanoporous samples at the nanoscale and void-size eﬀects. State-of-the-art: Marian et al. (2004, 2005), Traiviratana et al. (2008), Zhao et al. (2009), Bringa et al. (2010), Tang et al. (2010), Mi et al. (2011). Material strength under uniaxial/hydrostatic tests; Few attention has been paid to void-size eﬀects. X Y Z x y z a0 a0/2 < 100 > [100] [010] [001] b L 2R L L a0/2 Single-crystal/single-void FCC Aluminium Fixed values of porosity and void sizes in 0.271 nm - 3.247 nm Periodic boundary conditions Diﬀerent deformation paths and triaxiality levels Simulations carried out in LAMMPS Brach, S., Dormieux, L., Kondo, D., & Vairo, G. (2016b). Mechanics of Materials, 101, 102-117. 16 / 39 Molecular Dynamics
23. 23. Stella Brach PhD dissertation, 29 November 2016 Conclusions Void-size eﬀects: overall expansion of strength domain as void size reduces A signiﬁcant dependency on the three stress invariants I1, J2, ✓⌃ −6 −4 −2 0 2 4 6 8 10 12 14 16 18 0 1 2 3 4 5 6 7 8 9 10 [GPa] r[GPa] Bulk L/B=30 L/B=40 L/B=70 L/B=100 L/B=110 Bulk TXE ( =0) SHR ( = /6) TXC ( = /3) R = 0.812 nm R = 1.082 nm R = 1.894 nm R = 2.706 nm R = 2.977 nm Void-size eﬀects: shape transition in deviatoric proﬁles r = 2 GPa r = 4 GPa r = 6 GPa r = 8 GPa L/B=20 L/B=40 L/B=60 L/B=70 SHR⌃ SHR⌃ SHR⌃ SHR⌃ SHR⌃ SHR⌃ TXC⌃ TXC⌃ TXC⌃ TXE⌃ TXE⌃ TXE⌃ ✓D = ⇡ 6 ✓D = ⇡ 3 ✓D = 0 ✓⌃ = ⇡/3 ✓⌃ = ⇡/6 ✓⌃ = 0 r=2 GPa r=4 GPa r=6 GPa r=8 GPa 210 60 90 120 150 180 L/B=20 L/B=40 L/B=60 L/B=70 r=2 GPa r=4 GPa r=6 GPa r=8 GPa 210 60 90 120 150 180 L/B=20 L/B=40 L/B=60 L/B=70 r=2 GPa r=4 GPa r=6 GPa r=8 GPa 60 90 120 150 180 L/B=20 L/B=40 L/B=60 L/B=70 r=2 GPa r=4 GPa r=6 GPa r=8 GPa 210 60 90 120 150 180 L/B=20 L/B=40 L/B=60 L/B=70 R = 1.082 nm R = 1.624 nm R = 2.165 nm R = 0.541 nm= 1.353 nm, p = 1% ⇣ = I⌃ 1 p 3 , r = q 2J⌃ 2 , cos 3✓⌃ = 3 p 3J⌃ 3 2J⌃ 2 3/2 Brach, S., Dormieux, L., Kondo, D., & Vairo, G. (2016b). Mechanics of Materials, 101, 102-117. Strength domain of a nanoporous nanoscale sample