Microstructurally-Informed Random Field Description: Case Study on Chaotic Masonry

Transcription

1 Microstructurally-Informed Random Field Description: Case Study on Chaotic Masonry M. Lombardo 1 J. Zeman 2 M. Šejnoha 2,3 1 Civil and Building Engineering Loughborough University 2 Department of Mechanics 3 Centre for Integrated Design of Advanced Structures Faculty of Civil Engineering Czech Technical University in Prague 1 st International Symposium on Uncertainty Modelling in Engineering, Prague, M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 1 / 16

2 Motivation: Analysis of a historic masonry wall Key issues Heterogeneity and non-uniformity of the structure Size of constituents not negligible with respect to structural length Random morphology of the material Available modeling concepts 1 Homogenization approaches 2 Stochastic media theories M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 2 / 16

3 Motivation: Analysis of a historic masonry wall Key issues Heterogeneity and non-uniformity of the structure Size of constituents not negligible with respect to structural length Random morphology of the material Available modeling concepts 1 Homogenization approaches 2 Stochastic media theories M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 2 / 16

4 Motivation: Homogenization vs random fields Rapidly oscillating coefficients Assumes scale separation Periodic unit cell (PUC) Random materials SEPUC Explicit treatment of heterogeneity (ZEMAN & ŠEJNOHA, MSMSE, 27) Random coefficients No a-priory scale separation Random field construction Randomness is for free Link with microstructure? M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 3 / 16

5 Motivation: Homogenization vs random fields Rapidly oscillating coefficients Assumes scale separation Periodic unit cell (PUC) Random materials SEPUC Explicit treatment of heterogeneity (ZEMAN & ŠEJNOHA, MSMSE, 27) Random coefficients No a-priory scale separation Random field construction Randomness is for free Link with microstructure? M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 3 / 16

6 Outline 1 Quantification of random morphology 2 Statistics of structural response Improved perturbation technique Karhunen-Loève expansion Hashin-Shtrikman approach 3 Numerical example 4 Conclusions M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 4 / 16

7 Quantification of random morphology Elementary geometrical descriptors of random media One-point correlation function S s (1) γ s (volume fraction) Two-point correlation function S s (2) (m, n) STONES environment (FALSONE & LOMBARDO, 26) M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 5 / 16

8 Quantification of random morphology Implications for random field description FALSONE & LOMBARDO, 26 Consider a discrete binary random field χ s : Ω S {, 1} χ s (x, θ) = { 1 if x stone(θ) if x mortar(θ) E[χ s (x)] = γ s E[χ s (x)χ s (y)] = S (2) s (x y) For a general binary field f : Ω S R { f (s) if x stone(θ) f (x, θ) = f (m) if x mortar(θ) Basic statistical characterization E[f (x)] = γ s f (s) + (1 γ s )f (m) ( ) ( 2 R fg (x y) = (x y) γ s f (s) f (m)) ( g (s) g (m)) S (2) s Rigorous microstructure-based description M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 6 / 16

9 Statistics of structural response Improved perturbation technique ELISHAKOFF ET AL., 1995 Elasticity problem on Ω with stochastic material data C(x, θ) = χ s (x, θ)c (s) + (1 χ s (x, θ))c (m) Discretization of Ω by N e elements (2h correlation length) Midpoint discretization of random field: χ h s (θ) = [ χ h s,1 (θ), χh s,2 (θ)..., χh s,n e (θ) ] T {, 1} N e Discretized FEM equations: K(χ h s (θ))u h (θ) = F Elementary statistics of displacements A E[u h ] = F where N e N e A = K K i (K ) 1 K j E[χ s,i χ s,j ] i=1 j=1 M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 7 / 16

10 Statistics of structural response Karhunen-Loève expansion GHANEM AND SPANOS, 1991 Ω Karhunen-Loève expansion of a stationary random field g(x, θ) g(x, θ) = E[g] + M λi f i (x)ξ(θ), i=1 R g (x y)f i (y) dy = λ i f i (x) ξ i (θ) = 1 λi 3.5 x 14 Ω (g(x, θ) E[g]) f i (x) dx eigenvalues λ i eigenvalue index i M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 8 / 16

11 Statistics of structural response Karhunen-Loève expansion.25 f f i = y x i = y x f 1.2 f i = y x i = y x M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 9 / 16

12 Statistics of structural response Karhunen-Loève expansion Spatial dimensional x is accurately captured Stochastic dimension θ: Truncated Gaussian variables Evaluation of statistics: Monte-Carlo integration E[u] 1 N s with N s = 1, N s s=1 u h (θ i ) M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 1 / 16

13 Statistics of structural response eplacements Hashin-Shtrikman approach LUCIANO & WILLIS, 25,26 Ω C(x,θ) = + C C τ(x,θ) Reference problem Polarization problem Reference body Ω with stiffness C Stress equivalence condition σ(x, θ) = C(x, θ)ε(x, θ) = C ε(x, θ) + τ (x, θ) Approximate solution of reference problem using FEM ε h (x), Gh (x, y) = Nh u(x)(k h ) 1 B ht (y), Γ h (x, y) = Bh (x)(k h ) 1 B ht (y) M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 11 / 16

14 Statistics of structural response Hashin-Shtrikman approach: Polarization problem Polarization stress: τ h (x, θ) = N h (x) ( χ 1 (x, θ)d h 1 + χ 2(x, θ)d h ) 2 Discretized version of H-S variational principles K h i d h i + j K h ijd h j = R h i K h i = K h ij = R h i = Statistics of the solution: E[u h (x)] = u h (x) Ω Ω Ω γ i (x)n ht (x) [C i C ] 1 N h (x) dx N ht (x) S (2) ij (x y)γ h (x, y)nh (y) dx dy Ω γ i (x)n ht (x)ε h (x) dx Ω G h (x, y)nh (y) ( γ 1 d h 1 + γ 2d h 2) dy M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 12 / 16

15 Numerical example Phase E [MPa] ν [-] Stone Mortar Q1/(Q) elements Improved perturbation method (25 25 elements) x E[σ xx ] [MPa] E[σ yy ] [MPa] E[σ xy ] [MPa].15 5 M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 13 / 16

16 Numerical example H-S [C = γ s C (s) + γ m C (m) ] cements E[uy] [m] H-S [2C = C (s) + C (m) ] K-L -.18 H-S [C = min(c (s),c (m) )] Improved perturbation Relative position along the top edge M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 14 / 16

17 Numerical example -.1 E[uy] [m] H-S + K-L -.2 H-S Improved perturbation Relative position along the top edge M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 15 / 16

18 Conclusions Spatial correlation of a random field can be directly obtained from samples of microstructure Improved perturbation method seems to fully utilize second-order data, but it inconsistent with the H-S bounds Karhunen-Loeve expansion needs a large number of terms for realistic structures Results of Hashin-Shtrikman method strongly depend on the choice of reference media Need for a rational construction of non-gaussian fields with prescribed correlation structure Additional info available at M. Lombardo et al. (Loughborough & CTU) Microstructurally-Informed Random Fields ISUME 211, Prague 16 / 16