Eshan V. Dave, Secretary of M&FGM2006 (Hawaii) Research Assistant and Ph.D. Candidate. Glaucio H. Paulino, Chairman of M&FGM2006 (Hawaii)

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Asphalt Pavement Aging and Temperature Dependent Properties through a Functionally Graded Viscoelastic Model I: Development, Implementation and Verification Eshan V.

1 Asphalt Pavement Aging and Temperature Dependent Properties through a Functionally Graded Viscoelastic Model I: Development, Implementation and Verification Eshan V. Dave, Secretary of M&FGM2006 (Hawaii) Research Assistant and Ph.D. Candidate Glaucio H. Paulino, Chairman of M&FGM2006 (Hawaii) Donald Biggar Willett Professor of Engineering William G. Buttlar Professor and Narbey Khachaturian Faculty Scholar Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign

2 Outline Part I Graded Finite Elements Viscoelasticity and FGMs Finite Element Formulations Verification Concluding Remarks Part II (Companion presentation) Asphalt Pavements Effect of Aging Simulations Concluding Remarks 2

3 Objectives Develop efficient and accurate simulation scheme for viscoelastic functionally graded materials (VFGMs) Correspondence Principle based formulation Application: Asphalt concrete pavements (Part II) E 1 E 2 E 3 E h η 1 η 2 η 3 η h 3

Graded Finite Elements Homogeneous Graded Graded Elements: Account for material non-homogeneity within elements unlike conventional (homogeneous) elements Lee and Erdogan (1995) and Santare and

4 Graded Finite Elements Homogeneous Graded Graded Elements: Account for material non-homogeneity within elements unlike conventional (homogeneous) elements Lee and Erdogan (1995) and Santare and Lambros (2000) Direct Gaussian integration (properties sampled at integration points) Kim and Paulino (2002) Generalized isoparametric formulation (GIF) Paulino and Kim (2007) and Paulino et al. (2007) further explored GIF graded elements Proposed patch tests GIF elements should be preferred for multiphysics applications Buttlar et al. (2006) demonstrated need of graded FE for asphalt pavements (elastic analysis) 4

Generalized Isoparametric Formulation (GIF) Material properties are

material properties at integration points Natural extension of the

Ε(x, y)) y z=e(x,y) x y (0,0) x E N N i m i1 i E i Shape function

5 Generalized Isoparametric Formulation (GIF) Material properties are sampled at the element nodes Iso-parametric mapping provides material properties at integration points Natural extension of the conventional isoparametric formulation Material Properties (eg. Ε(x, y)) y z=e(x,y) x y (0,0) x E N N i m i1 i E i Shape function corresponding m Number of nodes per element to node, i Conventional Homogeneous GIF 5

6 Viscoelasticity: Basics Constitutive Relationship for linear viscoelastic body: o σ ij are stresses, ε ij are strains o Superscript d represents deviatoric components o G ijkl and K ijkl : shear and bulk moduli (space and time dependent) o Assumptions: no body forces, small deformations o Equilibrium: o Strain-Displacement: o u i : displacements t t d ij x, t 2 Gijkl x, t t kl x, t dt d t t t kk x, t 3 Kkkll x, t t ll x, t dt ij, j 0 t 1 2 u u ij i, j j, i 6

Viscoelasticity: Correspondence Principle Correspondence Principle (Elastic-Viscoelastic Analogy): Equivalency between transformed (Laplace, Fourier etc.

7 Viscoelasticity: Correspondence Principle Correspondence Principle (Elastic-Viscoelastic Analogy): Equivalency between transformed (Laplace, Fourier etc.) viscoelastic and elasticity equations Elastic Constitutive Law : d 3 2G d K Transformed Viscoelastic Constitutive Law : ~ ~~ d 2G d ~ ~ 3K ~ Extensively utilized to solve variety of nonhomogeneous viscoelastic problems: Hilton and Piechocki (1962): Shear center of non-homogeneous viscoelastic beams Chang et al. (2007): Thermal stresses in graded viscoelastic films 7

8 Viscoelastic Model Prony series form: Generalized Maxwell Model Equivalency between compliance and relaxation forms Flexibility in fitting experimental data Transformations are well established Readily applicable to asphaltic and other viscoelastic materials (polymers, etc) E( t) i i E i h E i1 i Exp t / i E 1 E 2 E 3 E h η 1 η 2 η 3 η h 8

9 Viscoelastic FGMs Paulino and Jin (2001); Mukherjee and Paulino (2003) Material with Separable Form E x, t E( x) f( t) E 1 (x) E 2 (x) E 3 (x) E h (x) η 1 η 2 η 3 η h 9

10 General FE Implementation Correspondence principle based implementation using Laplace transform (Yi and Hilton, 1998) Variational Principle (Potential): (Taylor et al., 1970) 10

11 δ Stationarity: FE Implementation: Basis t t t tt xt, C x t t t x t x t dt dt d t t u t t * kl ijkl, ijkl ijkl ij, ij, u t t u x, t Pi x t t dt d t t, i 0. Ω : volume, S surface with traction Pi C ijkl : constitutive properties ε ij : mechanical strains, ε ij * : thermal strains, u i : displacements, ξ: reduced time related to real time through time-temperature superposition principle given by: t () t a T t dt 0 a is time-temperature shift factor, and T is temperature 11

12 Element stiffness matrix: Force vectors: Mechanical: Thermal: FEM T,, k x t B x C x t B x d ij ik kl lj u u,, f x t N x P x t d i ij j xt, f x t B x C x t t dt d t * th l i, ik kl, ( ) u t u k ij : element stiffness matrix, shape functions N ij and their f i : element force (load) vector derivatives B ij u i : displacement vector ε i : strains related to nodal degrees of ui x, t Nij xq j t freedom q j through isoparametric x, t B xq t i ij j 12

13 FEM: Assembly and Solution Assembling provides global stiffness matrix, K ij and force vectors, F i Equilibrium: Correspondence principle: t U K x t U K x t t dt F x t F x t t j ij, j 0 ij,,, i i t 0,, th, K x s U s F x s F x s ij j i i ã(s) is Laplace transform of a(t), s is transformation variable a( s) a( t) Exp[ st] dt 0 th 13

14 FEM: Implementation Define problem in time-domain (evaluate load vector, F(x, t) and stiffness matrix components K(x) and Λ(t)) Perform Laplace transform to evaluate F ~ (x, s) and Λ ~ (s) Solve linear system of equations to evaluate nodal displacement, U ~ (x,s) Perform inverse Laplace transforms to get the solution, U(x,t) Post-process to evaluate field quantities of interest 14

15 FEM: Verification MATLAB code using GIF and correspondence principle GIF Compare analytical and numerical solutions for graded boundary value problems Viscoelasticity Compare analytical and numerical solutions for viscoelastic bar imposed with creep loading Comparison with Commercial Code ABAQUS (Layered Approach) 15

16 Graded Finite Element Performance Bending example Analytical Solution (line) y b 10 Numerical Solution (markers) x E( x) E E Exp x 16

17 Homogeneous Viscoelastic Verification Creep example shown here y t Numerical inversion performed with 20-Collocation points t time x E( t) E E Exp t / t 17

18 FGM Verification with ABAQUS Simply supported beam in 3-point bending 100-second creep loading Graded viscoelastic material properties FE simulation: Homogeneous: Averaged properties Layered (ABAQUS): 6-Layers 12-Layers Graded: Same mesh structure as 6-Layers 18

19 Reference Material Properties 19

20 FEM Meshes 6-Layers / FGM / Homogeneous 3146 DOFs 6-node triangle elements 12-Layers 6878 DOFs 6-node triangle elements 20

21 Numerical Results (100-Collocation points) 21

22 Main Contribution: Concluding Remarks development of graded viscoelastic elements Extension of the Generalized Isoparametric Formulation (Elastic) to rate-dependent materials (viscoelastic) Correspondence Principle based formulation: separable material properties Companion presentation (paper) demonstrates application of this work to field of asphalt pavements Extension: Graded Viscoelastic formulation in time domain 22

23 Thank you for your attention!! z y x E 1 E 2 E N τ 1 τ 2 τ N 23

William G. Buttlar Glaucio H. Paulino Harry H. Hilton Phillip B. Blankenship Hervé DiBenedetto. All colleagues and friends

William G. Buttlar Glaucio H. Paulino Harry H. Hilton Phillip B. Blankenship Hervé DiBenedetto. All colleagues and friends Ph.D. Final Exam Asphalt Pavement Aging and Temperature Dependent Properties using a Functionally Graded Viscoelastic Model Eshan V. Dave June 5th 29 Department of Civil and Environmental Engineering University