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. ButtlarProfessor 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

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Objectives

Develop efficient and accurate simulation scheme for viscoelastic functionally graded materials (VFGMs)

Correspondence Principle based formulation

Application: Asphalt concrete pavements (Part II)

E1 E2 E3 Eh

η1 η2 η3 ηh

Graded Finite Elements

Graded Elements: Account for material non-homogeneity within elementsunlike 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

Homogeneous Graded

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

x

y

z=E(x,y)

elementper nodes ofNumber

node, toingcorrespondfunction Shape

1

m

iN

ENE

i

m

i

ii

)),.(egPropertiesMaterial yΕ(x y

x

(0,0)

ConventionalHomogeneous

GIF

6

Viscoelasticity: Basics Constitutive Relationship for linear viscoelastic body:

o σij are stresses, εij are strains

o Superscript d represents deviatoric components

o Gijkl and Kijkl : shear and bulk moduli (space and time dependent)

o Assumptions: no body forces, small deformations

o Equilibrium:

o Strain-Displacement:

o ui : displacements

'

'

' ' ', 2 , ,

t t

d d

ij ijkl kl

t

x t G x t t x t dt

'

'

' ' ', 3 , ,

t t

kk kkll ll

t

x t K x t t x t dt

, ,

1

2ij i j j iu u

, 0ij j

7

Viscoelasticity: Correspondence Principle Correspondence Principle (Elastic-Viscoelastic

Analogy): “Equivalency between transformed (Laplace,

Fourier etc.) viscoelastic and elasticity equations”

K

G dd

3

2

:Law veConstituti

~~3~

~~2~

:Law veConstituti

K

G dd

Elastic Transformed Viscoelastic

Extensively utilized to solve variety of nonhomogeneousviscoelastic problems: Hilton and Piechocki (1962): Shear center of non-homogeneous

viscoelastic beams

Chang et al. (2007): Thermal stresses in graded viscoelastic films

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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)

E1 E2 E3 Eh

η1 η2 η3 ηh

i

ii

h

i

ii

E

tExpEtE

1

/)(

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Viscoelastic FGMs

Paulino and Jin (2001); Mukherjee and Paulino (2003)

– Material with “Separable Form”

, ( ) ( )t tE x E x f

E1(x)

η1 η2 η3 ηh

E2(x) E3(x) Eh(x)

10

General FE Implementation

Correspondence principle based implementation using Laplace transform (Yi and Hilton, 1998)

Variational Principle (Potential): (Taylor et al., 1970)

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FE Implementation: Basis Stationarity:

Ω : volume, S surface with traction Pi

Cijkl: constitutive properties

εij: mechanical strains, εij* : thermal strains, ui: displacements,

ξ: reduced time related to real time through time-temperature

superposition principle given by:

a is time-temperature shift factor, and T is temperature

' '' '

''

''

''

''

'' ' ' ' * ' ' ''

' ''

''

'' ''

''

,, , ,

,, 0.

δu

t t t t tkl

ijkl ijkl ijkl ij ij u

t t

t ti

i

t

x tC x t t t x t x t dt dt d

t t

u x tP x t t dt d

t

'

' '

0

( )

t

t a T t dt

12

FEM

Element stiffness matrix:

Force vectors:

Mechanical:

Thermal:

kij : element stiffness matrix,fi : element force (load) vectorui : displacement vectorεi : strains related to nodal degrees offreedom qj through isoparametric

shape functions Nij and theirderivatives Bij

, ,

u

T

ij ik kl lj uk x t B x C x t B x d

, ,i ij jf x t N x P x t d

* '

' '

'

,, , ( )

u

tlth

i ik kl u

x tf x t B x C x t t dt d

t

,i ij jx t B x q t

,i ij ju x t N x q t

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FEM: Assembly and Solution

Assembling provides global stiffness matrix, Kij and force vectors, Fi

Equilibrium:

Correspondence principle:

ã(s) is Laplace transform of a(t), s is transformation variable

'

' '

'

0

, 0 , , ,

tj th

ij j ij i i

U tK x t U K x t t dt F x t F x t

t

, , ,th

ij j i iK x s U s F x s F x s

0

( ) ( ) [ ]a s a t Exp st dt

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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 (x, s) and (s)

Solve linear system of equations to evaluate nodal displacement, (x,s)

Perform inverse Laplace transforms to get the solution, U(x,t)

Post-process to evaluate field quantities of interest

F~

Λ~

U~

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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)

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Graded Finite Element PerformanceBending example

2

100

)(

0

0

E

xExpExE

x

y

b 10

Analytical Solution (line)

Numerical Solution (markers)

17

Homogeneous Viscoelastic VerificationCreep example shown here

x

y

t

20

10

/)(

0

0

E

tExpEtE

Numerical inversion performed with

20-Collocation points

t

time

0.25

0.0

t

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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

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Reference Material Properties

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FEM Meshes

6-Layers / FGM / Homogeneous 3146 DOFs

6-node triangle elements

12-Layers 6878 DOFs

6-node triangle elements

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Numerical Results

(100-Collocation points)

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Concluding Remarks

Main Contribution:

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

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Thank you for your attention!!

x

y

z

E1 E2 EN

τ1 τ2 τN