Research on nonlinear constitutive relationship of

Research on nonlinear constitutive relationship of permanent deformation in asphalt pavements

PENG Miaojuan1 & XU Zhihong2

1. Department of Civil Engineering, Shanghai University, Shanghai 200072, China;

2. School of Transportation Engineering, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Peng Miaojuan (email: mjpeng@staff.shu.edu.cn)

Received December 2, 2005; accepted May 30, 2006

Abstract To predict correctly the rut depths in asphalt pavements, a new nonlinear viscoelastic-elastoplastic constitutive model of permanent deformation in asphalt pave-ments is presented. The model combines a generalized Maxwell model with an elasto-plastic one. Then from the creep theory, the linear and nonlinear constitutive equations of the generalized Maxwell model are obtained. From the nonlinear finite element method for the rutting of the asphalt pavement, the rut depths of 4 asphalt-aggregate mixtures are obtained. And the results are compared with the ones from the finite element method by SHRP and the experiments by SWK/UN. The results in this paper are better than the ones by SHRP, and agree with the ones of the experiment by SWK/UN. This shows that the nonlinear viscoelastic-elastoplastic constitutive model, which is presented in this paper for the rutting of the asphalt pavement, is effective. The properties, such as nonlinear elastic-ity, plasticity, viscoelasticity and nonlinear viscoelasticity, which affect the rutting of an asphalt pavement, can be shown in the model. And the characteristics of the permanent deformation of the asphalt pavement can be presented entirely in the model.

Keywords: asphalt pavement, permanent deformation, nonlinear viscoelastic-elastoplastic constitutive rela-tionship, generalized Maxwell model, creep compliance.

1 Introduction

Rutting in asphalt pavements has become one of the destruction forms with the in-crease in traffic volume, tire pressure and axial load. And many researchers have done a lot of work on it[1]. At the present time, rutting in asphalt pavements is one of the hot spots and difficulties in road engineering. In recent years, viscoelasticity theory, espe-cially linear viscoelasticity theory, has been used to study the rutting in asphalt pave-ments. However, the development of rutting shows that asphalt material is not viscoelas-tic entirely. After the loads are removed, the asphalt mix does not recover to its unde-672 Science in China Series G: Physics, Mechanics & Astronomy formed state, then the permanent deformation is formed. Thus, a linear viscoelastic mate-rials model does not describe the rutting in asphalt pavements. On the other hand, the nonlinear viscoelastic theory and elastoplastic finite deformation theory will give a better description of the rutting because of the inclusions of the deformation of nonlinear elas-ticity and nonlinear viscoelasticity, and the non-recoverable deformation of plasticity. Asphalt mixture was regarded as the visco-elastoplastic material, and using Perzyna’s theory a visco-elastoplastic constitutive model was presented by Lu[2]. The nonlinear viscoelastic behaviour of conventional and modified asphalt concrete under creep was studied, and a mathematical model of nonlinear viscoelasticity of asphalt concrete was proposed by Judycki[3]. As a part of Strategic Highway Research Program (SHRP), a visco-elastoplastic constitutive model was developed to describe the permanent deforma-tion of asphalt concrete. SHRP model combined a generalized Maxwell model with an elastoplastic one in parallel connection. But the elastoplastic model in SHRP model did not agree with the results of the experiments and must be improved[4]. Erkens presented a constitutive model of asphalt concrete, and discussed 3-D finite element method for the asphalt concrete[5]. A 3-D viscoelastic finite element model of asphalt pavements was discussed by Blab[6]. A viscoelastic layer theory was used to discuss the permanent de-formation of asphalt concrete by Xu[7]. The linear viscoelasticity theory was applied and a “four components five parameters” model was presented to analyze permanent defor-mation of asphalt concrete by Xu[8]. The viscoelastic theory was applied to establishing a finite element model of asphalt mixture, then ABAQUS, a finite element analysis soft-ware, was used to analyze rutting by Feng[9]. A practical rheological model, which can reflect the consolidation effect during loading and the permanent deformation after unloading, for asphalt mixtures was proposed by Wang[10].

According to mechanics theories, asphalt concrete is the nonlinear visco-elastoplastic material. The rutting of an asphalt pavement is the permanent deformation under the wheel loads, so the constitutive model that describes the rutting of an asphalt pavement must consider both nonlinear viscoelasticity and plasticity. Now, there are fewer nonlin-ear constitutive models to describe the rutting of an asphalt pavement. And the general nonlinear viscoelastic constitutive models which have been presented in mechanics can-not describe the rutting of an asphalt pavement effectively.

In this paper, using the nonlinear viscoelasticity and elastoplasticity theories, a new nonlinear viscoelastic-elastoplastic constitutive model of permanent deformation in as-phalt pavements is presented. The model combines a generalized Maxwell model with an elastoplastic one. Then from the creep theory, the linear and nonlinear constitutive equa-tions of the generalized Maxwell model are obtained. From the nonlinear finite element method for the rutting of the asphalt pavement, the rut depths of 4 asphalt-aggregate mixtures are obtained. And the results are compared with the ones from the finite element method by SHRP and the experiments by SWK/UN. The results in this paper are better than the ones by SHRP, and agree with the ones of the experiment by SWK/UN. This shows that the nonlinear viscoelastic-elastoplastic constitutive model, which is presented in this paper for the rutting of the asphalt pavement, is effective. The properties, such as nonlinear elasticity, plasticity, viscoelasticity and nonlinear viscoelasticity, which affect

Nonlinear constitutive relationship of permanent deformation in asphalt pavements 673 the rutting of an asphalt pavement, can be shown in the model. And the characteristics of the permanent deformation of the asphalt pavement can be presented entirely in the model.

2 The constitutive relationship of the permanent deformation in asphalt pave-ments

According to the characteristics of the permanent deformation in asphalt pavements, a new nonlinear viscoelastic-elastoplastic permanent deformation constitutive model is proposed in this paper. The model combines a generalized Maxwell model with an elas-toplastic one (see Fig. 1). The constitutive relationship of each component is discussed as

follows.

Fig. 1. The permanent deformation model of asphalt pavements. 2.1 The constitutive relationship of nonlinear elasticity

According to the definition of the strain energy density function, the constitutive equa-tion of nonlinear elasticity is written as ,ij ij

W S E ∂=∂ (1) where 123(,,)W I I I is the strain energy density function.

The strain energy density function 123(,,)W I I I can be written as a Taylor series, i.e.

12312312321231

2331231

231(,,)(0,0,0)(0,0,0)1!1 ()(0,0,0)2!1 ()(0,0,0)3!1 (4!W I I I W I I I W I I I I I I W I I I I I I W I I I ⎛⎞∂∂∂=+

++⎜⎟∂∂∂⎝⎠∂∂∂+++∂∂∂∂∂∂+++∂∂∂∂+4123123)(0,0,0)I I I W I I I ∂∂+++∂∂∂" (2)

where 1I , 2I and 3I are the invariants of elastic strain tensor, and

1112233I E E E =++, (3)

2222112222333311122313()I E E E E E E E E E =++−++, (4)

674 Science in China Series G: Physics, Mechanics & Astronomy

22231122331223311123221333122()I E E E E E E E E E E E E =+−++. (5) If no initial stress is in the elastic body, i.e. 0ij E =, 0ij S =, then we have 0(0,0,0)()|0ij ij E W W E ===, (6) 0|0ij

E ij W E =∂=∂. (7) Substituting eqs. (6) and (7) into eq. (2) and omitting the terms whose order is greater than four, the strain energy function becomes e 2311223141253422617128139211()26111 ,2422

W C I C I C I C I I C I C I C I I C I I C I =++++++++E (8) where i C

(1,2,...,9i =) are the material constants and e E is the elastic strain tensor.

2.2 The constitutive relationship of plasticity [11]

In order to show the characteristics of asphalt concrete, the constitutive model which describes the rutting of an asphalt pavement must include plastic deformation. V on Mises yield rule and the kinematic hardening theory are used in this paper.

When the material is plastic, the strain increment can be divided into elastic strain in-crement and plastic strain increment, i.e.

e p d d d ij ij ij εεε=+, (9)

where the relation between the elastic strain increment e d ij ε and the stress increment d ij σ can be expressed as

e e d d ij ijkl kl D σε=. (10) And the plastic strain increment obeys the flow rule, i.e. p d ij ij

f ελσ∂=∂, (11) where λ is the proportion constant, and f is the yield surface.

The plastic increment constitutive relationship can be written as

ep d d ij kl ijkl D σε=, (12)

where ep ijkl D is called elastoplastic matrix, and

ep e e e ijkl ijkl pqrs ijkl pq rs f f D D D D A σσ∂∂=−∂∂, (13)

e p 2ijkl ij kl kl

f f f A D σσε⎛⎞∂∂∂=−⎜⎟⎜⎟∂∂∂⎝⎠

. (14)

Nonlinear constitutive relationship of permanent deformation in asphalt pavements 675

2.3 The nonlinear constitutive relationship of the generalized Maxwell model

(i) One-dimensional linear constitutive relationship of the generalized Maxwell model. One-dimensional linear constitutive relationship of Maxwell model is described as E σσεη

=+ . (15) One-dimensional linear creep constitutive relationship of Maxwell model is 01()1t t E εστ⎛⎞=+⎜⎟⎝⎠

, (16) and the creep function ()J t is defined as 1()1t J t E τ⎛⎞=+⎜⎟⎝⎠

. (17) The generalized Maxwell model consists of some separate Maxwell elements in paral-lel connection (see Fig. 2). Fig. 2 is also the viscoelastic part of the nonlinear viscoelas-tic-elastoplastic constitutive model (see Fig. 1) presented in this paper for rutting analysis

of an asphalt pavement.

Fig. 2. The generalized Maxwell model.

From the creep theory, the linear and nonlinear constitutive equations of the general-ized Maxwell model are obtained in this paper.

For the generalized Maxwell model, we have

e v εεε=+, (18)

()1N i i σσ==∑, (19)

where e ε and v ε are the elastic component and the plastic component of the strain, respectively; ()i σ (1,2,...,i N =) is the stress of the i -th Maxwell element; N is the number of Maxwell elements.

From eq. (16), the stress in each Maxwell element of the generalized Maxwell model can be obtained, and

()()i i i i

E t t τσετ=+, (20)

676 Science in China Series G: Physics, Mechanics & Astronomy where i E (1,2,...,i N =) is the elastic modules of the i -th Maxwell element, and i

i i E ητ=, (21)

where i η (1,2,...,i N =) is the viscosity constant of the i -th Maxwell element.

For the generalized Maxwell model, substituting eq. (20) into eq. (19), we have

01()()()N j j j j t E t J t t τεσετ===+∑, (22)

then

1001()()N j j j j t E J t t τεσστ−=⎛⎞==⎜⎟⎜⎟+⎝⎠

∑, (23) where 11()N j j j j J t E t ττ−=⎛⎞=⎜⎟⎜⎟+⎝⎠

∑ (24) is the creep function of the generalized Maxwell model.

The constitutive relationship of viscoelasticity can be derived on the basis of either differential operators or convolution integrals. The latter approach is followed in this pa-per.

We consider a creep experiment. A Maxwell model is loaded at time 0t t ≥ by a con-stant stress 0σ, then the strain is a function of time, and

000()()()t J t t H t t εσ=−−, (25) where 0()H t t − is the Heaviside unit step function.

According to the Boltzmann superposition principle, for the combined load history 1()()M

i i i t H t t σσ==−Δ∑, (26)

where M is the number of the load steps, the total strain is obtained directly by superpo-sition of the separate strain responses, and 11()()()()M M

i i i i i i i t t t J t t H t t εεσ===−=−−Δ∑∑. (27)

When the infinitesimal load steps i σΔ are applied, the total strain is determined by the integral representation ()()()d ()t

t J t H t εττστ−∞=−−∫ (28)

to sum up the deformation history. This hereditary integral is reduced to ()()()d t t J t t

στεττ−∞∂=−∂∫. (29) Eq. (29) is the integral constitutive relationship of the generalized Maxwell model, i.e. the viscoelasticity constitutive relationship in the nonlinear viscoelastic-elastoplastic

constitutive model presented in this paper for rutting analysis of an asphalt pavement. It can represent the basic experiment characteristic of asphalt material －memory character-istic.

In order to study the rutting of an asphalt pavement under a dynamic load, the complex creep compliance of the generalized Maxwell model is discussed as follows.

To compute the response of a viscoelastic body under a dynamic load, the material is subjected to an oscillating load resulting in a state of stress

0exp(i )t σσω=, (30) which varies harmonically with frequency ω. And the stress amplitude is 0σ

, i =The accompanying strain response

0exp[i()]t εεωδ=− (31) oscillates at the same frequency ω but lags behind the stress by the phase angle δ. This phase shift δ, often called loss angle, is an important quantity for the characteriza-tion of viscoelastic dissipative properties.

Substituting eqs. (30) and (31) into eq. (15), we have

000i exp[i()]i exp(i )exp(i )E t t t τεωωδτσωωσω⋅−=⋅+, (32) i.e.

00i exp(i )(1i )E τεωσδωτ⋅=+. (33) Eq. (33) can be written as 00

11i exp(i )i E εωτδστω+−=. (34) Then the complex creep compliance for a Maxwell element is defined as *0

()11i exp(i )i t J E εωτδστω+=−=. (35) Splitting the complex creep compliance *J into a real and an imaginary part, we have *'''11i 1i J J J E ωτ⎛⎞=+=−⎜⎟⎝⎠

, (36) where the components 'J and ''J are called storage and loss compliance, respectively. The loss compliance does not recover because of translating into quantity of heat and losing.

For the generalized Maxwell model, from eq. (19) we have 220*222211()i ()()11N N

j j j j j j j j E E t t J t ωτωτεσεωτωτ==⎡⎤==+⎢⎥++⎢⎥⎣⎦

∑∑. (37)

Thus, the complex creep compliance for the generalized Maxwell model is computed as 1

22*22221i 11N j j j j j j j E E J ωτωτωτωτ−=⎛⎞⎡⎤⎜⎟=+⎢⎥⎜⎟++⎢⎥⎣

⎦⎝⎠∑. (38)

Let

222211N j j j j E A ωτωτ==+∑, (39)

2211N j j j j E B ωτωτ==+∑. (40)

Then we have *'''1i i (i )i A B A B J J J A B A B A B A B −−=+=+==−+++, (41) where 'J and ''J are the storage compliance and the loss compliance of the general-ized Maxwell model, respectively.

(ii) The 3-D linear viscoelastic constitutive relationship of the generalized Maxwell model. The 3-D linear viscoelastic constitutive relationship of the generalized Maxwell element is deduced in this section.

The stress and strain tensor can be decomposed into the spherical tensor and the de-viatoric tensor, then

111213111213212223212223313233313233000000σσσσσσσσσσσσσσσσσσσσσσσσ−⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥=+−⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦

, (42) 1112131112132122

23212223313233313233000000e e e e e e εεεεεεεεεεεεεεεεεε−⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥=+−⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦

. (43) In 3-D, there are two creep compliances 1()J t and 2()J t which describe the fea-tures of deviatoric and spherical parts, respectively.

In 3-D, the rheological property is obtained mainly by the shear deformation. Then we assume the volume deformation is elastic as follows.

For the Maxwell element, we can obtain the 3-D constitutive relationship ''2,3.

ij ij ij e G Ke ησσησ⎧+=⎪⎨⎪=⎩ (44) Similar to the 1-D Maxwell element, the 3-D integral constitutive relationship of the Maxwell element is obtained, i.e. '12d ()()()d ,d d ()()()d ,d t ij ij t e t J t e t J t σττττσττττ−∞−∞⎧=−⎪⎪⎨⎪=−⎪⎩

∫∫ (45) where

111()2t J t G η⎛⎞=+⎜⎟⎝⎠

, (46)

21()3J t K

=

. (47) Similar to the 1-D generalized Maxwell model, the 3-D integral constitutive relation-ship of the generalized Maxwel model is expressed as '12d ()()()d ,d d ()()()d ,d t ij ij t e t J t e t J t σττττσττττ−∞−∞⎧=−⎪⎪⎨⎪=−⎪⎩

∫∫ (48) where 1()J t and 2()J t are creep compliances of deviatoric and spherical parts, re-spectively, and

1

11()N j j j j j G J t G t ηη−=⎡⎤=⎢⎥+⎢⎥⎣⎦∑, (49) 21()3J t K

=. (50) 2.4 Nonlinear viscoelastic constitutive relationship of the generalized Maxwell model Nonlinear viscoelastic constitutive relationship of the generalized Maxwell model will be obtained as follows.

For eq. (48), replacing the stress σ and the strain e with Kirchhoff stress S and Green strain E , respectively, we have ''12d ()()()d ,d d ()()()d ,d t ij ij t S E t J t S E t J t ττττττττ−∞−∞⎧=−⎪⎪⎨⎪=−⎪⎩

∫∫ (51) where '13

ij ij kk ij S S S δ=−, (52) '13

ij ij kk ij E E E δ=−, (53) 11223311()33

kk S S S S S =++=, (54) 11223311()33

kk E E E E E =++=. (55) In the total Lagrange description

()()t t t Δ=+Δ−S S S (56) and

()()t t t Δ=+Δ−E E E , (57) the nonlinear viscoelastic increment constitutive relationship of the generalized Maxwell model is

'111()[()()]d ()d t t t ij ij

ij t S S E t J t t J t J t t τττττττ+Δ−∞′′∂∂Δ=+Δ−−−++Δ−∂∂∫∫, (58) 222()[()()]d ()d t t t t S S E t J t t J t J t t τττττττ+Δ−∞∂∂Δ=+Δ−−−++Δ−∂∂∫∫. (59) 3 Numerical examples and analysis of the rutting in an asphalt pavement

A 3-D asphalt pavement is considered, and the cross section of the pavement is shown in Fig. 3. The length of the pavement is 300 cm, the width is 365.76 cm, and the depth is 140 cm. The tire pressure p =730 kPa. Distance between the tires is shown in Fig. 3.

Fig. 3. The cross section of a 3-D asphalt pavement.

In the FE analysis, four asphalt aggregate mixes, such as AAK-1-RD (V a = 3.7% and V a = 6.5%) and AAC-1-RD (V a = 3.8% and V a = 6.5%), are considered. Here AAK-1 and AAC-1 are the asphalt binders, RD (limestone, fully crushed quarry rock) is the aggre-gate, and V a is air-void content [4]. The nonlinear viscoelastic-elastoplastic constitutive relationship presented in this paper is used in asphalt layer and the material constants summarized in Tables 1―3. The subgrade is assumed as the linear elastic material, and Young’s modulus 138E = MPa, Poisson’s ratio 0.45μ=.

About the boundary conditions, the bottom surface of the subgrade is assumed to be fixed, which means that nodes on the bottom of the subgrade cannot move horizontally and vertically. Also, on the left and right surfaces of the pavement, all nodes are con-strained horizontally, but are free to move in the vertical direction.

A multiple loading step is used to simulate the moving wheel load. The duration of each wheel load is set to be 0.1 s. A single load repetition is that the wheel load moves from the end to another end of the pavement in the wheel moving direction.

The results of the rut depths obtained in this paper are shown in Table 4, and are com-pared with the ones from the finite element method by SHRP [4].

From Table 4, we can get the conclusions as follows:

(1) For AAK-1-RD (V a = 3.7%, V a = 6.5%) and AAC-1-RD (V a = 3.8%, V a = 6.5%), the results in this paper are approximately consistent with the ones in SHRP.

AAK-1-RD AAC-1-RD Material constants

V a = 3.7% V a = 6.5% V a = 3.8% V a = 6.5%

C1 5.06E+02 5.06E+02 4.36E+03

1.28E+03

C2−2.53E+02 −5.06E+02 −1.E+02 −3.22E+02

C3−5.06E+04 −5.06E+04 −2.73E+06 −2.96E+04

C4 5.06E+04 2.02E+05 1.09E+05

2.96E+05

C5 0.00E+00 0.00E+00 0.00E+00

0.00E+00

C6 1.26E+08 2.53E+08 5.46E+06

4.74E+08

C7−5.06E+07 −3.79E+07 −7.E+07 −2.37E+07

C8 5.06E+07 5.06E+07 1.09E+07

5.92E+07

C9 5.06E+05 5.06E+05 1.09E+06

8.88E+06

Table 2 Plastic material constants

AAK-1-RD AAC-1-RD Material

constants V

a

= 3.7% V a = 6.5% V a = 3.8% V a = 6.5% σy (MPa) 1.00E+00 1.00E+00 2.00E+00 1.00E-03 H′ 1.00E+03 1.00E+03 5.00E+02 5.00E+02

Table 3 Viscoelastic material constants (MPa)

AAK-1-RD AAC-1-RD V a= 3.7% V a = 6.5% V a = 3.8% V a = 6.5%

Element i

E iηi E iηi E iηi E iηi

1 3.67E-04 8.25E-01 3.67E-04 8.25E-01 3.E-048.E-01 1.81E-03 8.69E-01

2 6.24E-04 1.42E-01 6.34E-04 1.42E-01 6.28E-04 1.17E-01 2.37E-0

3 1.13E-01

3 1.10E-03 2.47E-02 1.10E-03 2.47E-027.20E-0

4 1.60E-02 3.11E-03 1.49E-02

4 2.61E-03 5.86E-03 2.61E-03 5.86E-039.70E-04 2.15E-03 4.02E-03 1.93E-03

5 4.23E-03 9.51E-04 4.23E-03 9.51E-04 1.40E-03 3.10E-04 4.98E-03 2.39E-04

6 1.92E+02 4.31E-04 1.92E-02

4.31E-04

5.53E-03 1.23E-04 4.16E-02

1.99E-04

7 1.71E-01

3.86E-04 1.72E+03 3.86E-048.92E-02 1.98E-049.36E+03

4.49E-04

8 8.00E-01 1.80E-048.00E-01 1.80E-049.00E-01 2.00E-040.00E+00 0.00E+04

Table 4 Rut depths of four asphalt aggregate mixes (mm)

Asphalt aggregate mixes Asphalt Aggregate Air void (%) Results in this paper at 300

load repetitions (mm)

Results by SHRP at 300 load

repetitions (mm)

AAK-1 RD 3.7 0.8562 0.71 AAK-1 RD 6.5 0.9102 0.61 AAC-1 RD 3.8 0.8365 0. AAC-1 RD 6.5 0.8632 0.81

(2) The results in this paper are bigger than the ones in SHRP, and the reason is that the viscoelastic finite deformation and nonlinear elastic deformation are considered. The results in this paper are closer to actual conditions than the ones in SHRP. In order to show the validity of the method and the reliability of the results in this paper, the rut depth of an asphalt aggregate mixes (AAG-1-RH- V a = 4.7%) at 5000 load repetitions is computed, and the result is 1.3526 mm. The result in SWK/UN experiment is 1.44 mm[4]. Then the relative error is 6.069%. It is obvious that the result in this paper agrees with the one of the experiment by SWK/UN.

(3) The constitutive relationship in this paper considers both the upheaval at the edgeof each tire and the upheaval between tires. Then the model considered in this paper is identical to the actual pavement. The properties, such as nonlinear elasticity, nonlinear viscoelasticity and plasticity, of asphalt pavements are considered in the constitutive rela-tionship in this paper, then the characteristics of the permanent deformation of the asphalt pavement can be shown entirely.

4 Conclusions

From the creep theory, the linear and nonlinear constitutive equations of the general-ized Maxwell model are obtained in this paper.

A new nonlinear viscoelastic-elastoplastic constitutive model of permanent deforma-tion in asphalt pavements is presented in this paper. The properties, such as nonlinear elasticity, nonlinear viscoelasticity and plasticity, of asphalt pavement are considered in the constitutive relationship, then the characteristics of the permanent deformation of the asphalt pavement can be shown entirely.

The constitutive relationship in this paper considers both the upheaval at the edge of each tire and the upheaval between tires. Then the model considered in this paper is iden-tical to the actual pavement.

The results in this paper are better than the ones in SHRP and agree with the ones in SWK/UN experiment. This shows that the nonlinear viscoelastic-elastoplastic constitu-tive model, which is presented in this paper for the rutting of the asphalt pavement, is effective.

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