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Granular Micromechanics Model for Beams, Plates, and Shells CAUCHY HEXAGON VS. STRUCTURAL ELEMENT GRANULAR MICROMECHANICS FOR CONTINUUM ADVANTAGES cp3 National Science Foundation www.nsf.gov Center for Particulate Products and Processes engineering.purdue.edu/CP3 Payam Poorsolhjouy and Marcial Gonzalez www.marcialgonzalez.net • Macroscopic mechanical behavior of materials depends upon their microstructure and micromechanical properties. • Methods at various scales and with different levels of computational demand may be used for incorporating material microstructure and micromechanical properties. • In Granular Micromechanics approach, the material is envisioned as a collection of grains interacting with each other through pseudo-bonds that characterize material’s macroscopic behavior. INTRODUCTION • In this approach, material’s behavior is derived through micro-macro kinematic identification and an appropriate inter-granular constitutive relationship, followed by Principle of Virtual Work. Kinematic Identification u p i = u n i + @ u i @ x j (x p j - x n j ) δ i = u p i - u n i = ✏ ij l j ! " # $ # % Microscopic force-law f i = @ W ↵ @δ ↵ i = K ij δ ↵ j PVW W = σ ij ✏ ij = 1 V N c X ↵=1 W ↵ kl e ij s i f k d Tensorial Constitutive Equations Kinematic Assumption Principle of Virtual Work (PVW) Microscopic Constitutive Laws Principle of Virtual Work (PVW) Static Assumption Tensorial Continuum mechanics Granular Micromechanics Kinematic approach Granular Micromechanics Static approach Cauchy Stress σ ij = @ W @✏ ij = 1 V N c X ↵=1 @ W ↵ @✏ ij Granular system, Computational Demand 10 -2 10 -3 10 -5 10 -6 10 -9 10 -8 10 -7 10 -4 Length scale, meters Coarse grained – Molecular or bead-spring models. Atomic models Continuum models Meso-scale particle models Structures and their coarse graining Granular micromechanics – Granular Micromechanics Grains, boundaries, and contacts: ill-defined for complex materials Prohibitively large computations Ignores microstructure and micromechanical phenomena Stiffness tensor C ijkl = 1 V N c X ↵=1 h K ↵ ik l ↵ j l ↵ l i • Modeling materials with different levels of anisotropy using distribution functions • Capturing induced anisotropy automatically by using nonlinear inter- granular force laws. Transversely isotropic Orthotropic Isotropic 0.05 0.1 30 210 60 240 90 270 120 300 150 330 180 0 Force distribution Triaxial loading Stiffness tensor C ijkl = l 2 ⇢ ZZ h K ik l j l l i ⇠ (✓ , φ) sin ✓ d⌦ -60 -40 -20 0 -60 -40 -20 0 20 σ 22 (MPa) σ 11 (MPa) Initial Pre-loaded (0.2σy) Pre-loaded (0.4σy) Pre-loaded (0.6σy) Pre-loaded (0.8σy) Path-dependent failure • Minimal additional computational expense: o Only looking at different directions and not following every contact Δ Δ Δ && ’’ (( Δ ℎ && Element Size ×× ××ℎ Strain components Constant Varying in height Constitutive laws ./ = ./23 23 = ̅ = ̅ KIRCHHOFF PLATE ELEMENT ! " # u 1 (x, y, z )=¯ u(x, y ) - z @ ¯ w @ x u 2 (x, y, z )=¯ v (x, y ) - z @ ¯ w @ y u 3 (x, y, z )=¯ w(x, y ) Displacement Strain tensor ✏ ij = u (i,j ) = 2 4 ¯ ✏ xx + z ¯ xx ¯ ✏ xy + z ¯ xy 0 ¯ ✏ xy + z ¯ xy ¯ ✏ yy + z ¯ yy 0 0 0 0 3 5 Constitutive relationships 8 > > > > > > < > > > > > > : N 1 N 2 M 1 M 2 V 12 Q 12 9 > > > > > > = > > > > > > ; = Eh 1 - ⌫ 2 2 6 6 6 6 6 6 4 1 ⌫ 0 0 0 0 ⌫ 1 0 0 0 0 0 0 h 2 /12 h 2 ⌫ /12 0 0 0 0 h 2 ⌫ /12 h 2 /12 0 0 0 0 0 0 1-⌫ 2 0 0 0 0 0 0 h 2 12 1-⌫ 2 3 7 7 7 7 7 7 5 8 > > > > > > < > > > > > > : ¯ ✏ 11 ¯ ✏ 22 ¯ 11 ¯ 22 ¯ γ 12 2¯ 12 9 > > > > > > = > > > > > > ; PARTICLE-BINDER MATERIAL SYSTEM SHELLS FUNCTIONALLY GRADED MATERIALS (FGM) 0 0.002 0.004 0.006 0.008 0.01 0 2 4 6 8 1 2 4 6 N M Normal Force, N [KN ] Bending Moment, M [KN.m] Bending strain, [1/m] Homogeneous 0 0.002 0.004 0.006 0.008 0.0 N M Normal Force, N [KN ] Bending Moment, M [KN.m] Bending strain, [1/m] FGM Microscopic force law Compression 0 0.5 1 1.5 7 5 [1=m] -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 Normal force and Moment N M Bending 0 0.5 1 1.5 σ 11 ¯ CYCLIC LOADING Bending Axial Constant grain size FGM Strain components ✏ 1 = R 1 R 1 - z (¯ ✏ 1 + z ¯ 1 ) ✏ 2 = R 2 R 2 - z (¯ ✏ 2 + z ¯ 2 ) γ 12 = R 1 R 1 - z (¯ ✏ 12 + z ¯ 12 )+ R 2 R 2 - z (¯ ✏ 21 + z ¯ 21 ) γ 23 = γ 13 =0 In a spherical shell, constitutive laws will be identical to that of a Kirchhoff plate. ! " # $ % Coupled constitutive relationship 8 > > > > > > < > > > > > > : N 1 N 2 M 1 M 2 V 12 Q 12 9 > > > > > > = > > > > > > ; = C h 1 - ⌫ 2 2 6 6 6 6 6 6 6 4 ¯ k n ¯ k n ⌫ Δk n h h 2 12 Δk n h h 2 12 ⌫ 0 0 ¯ k n ⌫ ¯ k n Δk n h h 2 12 ⌫ Δk n h h 2 12 0 0 Δk n h h 2 12 Δk n h h 2 12 ⌫ ¯ k n h 2 /12 ¯ k n h 2 ⌫ /12 0 0 Δk n h h 2 12 ⌫ Δk n h h 2 12 ¯ k n h 2 ⌫ /12 ¯ k n h 2 /12 0 0 0 0 0 0 ¯ k n 1-⌫ 2 Δk n h h 2 12 1-⌫ 2 0 0 0 0 Δk n h h 2 12 1-⌫ 2 ¯ k n h 2 12 1-⌫ 2 3 7 7 7 7 7 7 7 5 8 > > > > > > < > > > > > > : ✏ k 11 ✏ k 22 k 11 k 22 γ k 12 2 k 12 9 > > > > > > = > > > > > > ; [1] [2] REFERENCES 1- Misra, Anil, and Poorsolhjouy, Payam, Acta Mechanica 227, no. 5 (2016): 1393. 2- Poorsolhjouy, Payam, and Misra, Anil, Intl. J. of Solids and Structures 108 (2017): 139-152. -20 -15 -10 -5 0 7 0 [%] -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 Normal force and Moment N M -1.5 -1 -0.5 0 0.5 1 1.5 7 5 [1=m] -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 Normal force and Moment N M -1.5 -1 -0.5 0 0.5 1 1.5 7 5 [1=m] -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 Normal force and Moment N M Loading progress -25 -20 -15 -10 -5 0 7 0 [%] -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 Normal force and Moment N M -25 -20 -15 -10 -5 0 7 0 [%] -1 -0.8 -0.6 -0.4 -0.2 0 Normal force and Moment N M
