DEVELOPMENT AND FINITE ELEMENT IMPLEMENTATION OF A

DEVELOPMENT AND FINITE ELEMENT IMPLEMENTATION OF A STRESS DEPENDENT ELASTO-VISCO-PLASTIC CONSTITUTIVE MODEL WITH DAMAGE FOR ASPHALT Andrew C. Collop+, A. (Tom) Scarpas*, Cor Kasbergen* and Arian de Bondt** +

Nottingham Centre for Pavement Engineering University of Nottingham, University Park Nottingham NG7 2RD, UK Tel: +44 (0)115 951 3935 Fax: +44 (0)115 951 3898 Email: [email protected] *

Section of Structural Mechanics Delft University of Technology, The Netherlands **

Ooms Avenhorn Holding The Netherlands

Paper submitted for the TRB 82nd Annual Meeting Washington DC, Jan 12-16, 2003 (Section A2D00)

Total Number of Words: 7,376 Last Revised: 05/11/2002

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Collop, Scarpas, Kasbergen & de Bondt

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ABSTRACT This paper describes the development and Finite Element (FE) implementation of a stress dependent elasto-visco-plastic constitutive model with damage. The model comprises elastic, delayed elastic and viscoplastic components. The strains (and strain rates) for each component are additive whereas they share the same stress (ie series model). This formulation was used so that a stress-based nonlinearity and sensitivity to confinement could be introduced into the viscoplastic component without affecting the behaviour of the elastic and delayed elastic components. A simple Continuum Damage Mechanics (CDM) formulation is introduced into the viscoplastic component to account for the effects of cumulative damage on the viscoplastic response of the material. The model is implemented in an incremental formulation into the CAPA-3D FE program developed at Delft University of Technology (The Netherlands). A local strain compatibility condition is utilised such that the incremental stresses are determined explicitly from the incremental strains at each integration point. The model is demonstrated by investigating the response of a semi-rigid industrial pavement structure subjected to container loading. Results show that the permanent vertical strains in the non-stress dependent case are significantly lower than the permanent vertical strains in the stress dependent case. Results also show that, in the stress dependent case, there is a more localised area of high permanent vertical compressive strain directly under the load at approximately half depth in the asphalt compared to the non stress-dependent case where the distribution is more even.




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INTRODUCTION The use of constitutive modelling is well established in many fields of engineering such as soil or rock mechanics and concrete technology. Asphaltic materials represent a difficult medium for the engineer to model due to their complex physical structure and corresponding complex behaviour. It is well documented that asphaltic materials are both loading rate and temperature dependent and exhibit elastic, viscous and plastic behaviour (1). Traditionally, the numerically intensive simulations required to model such complex material behaviour have been prohibitive in terms of computation processing time and storage space required. However the continuing increases in computing power and advances in numerical techniques now facilitate the implementation of complex constitutive models into incremental numerical techniques such as the Finite Element (FE) method (2,3). The development of an accurate constitutive model for asphaltic materials will provide a versatile tool for the analysis and study of asphaltic material response and performance. A particular sub-class of problems that is of interest to the pavement designer is related to situations where the loading is very slow moving or stationary (eg container loading, aircraft standing areas etc). Under these circumstances, the long loading times can lead to high levels of permanent strain (deformation) in the asphalt and many of the standard predictive techniques are not appropriate. Since the rate of accumulation of permanent deformation is time dependent, the most suitable forms of constitutive model are based on viscoelasticity and/or viscoplasticity (2,4). The objective of the research described in this paper is to develop a constitutive model that can be used to predict permanent deformation under conditions where the loading is very slow moving or stationary. Previous research has shown that the steady-state behaviour of a range of bitumens and asphalt mixtures is complex, ranging from linear viscous at low stress levels and power law creep at high stress levels with a creep exponent of between 2 and 2.5 (5,6,7). It has also been observed that the steady-state behaviour (in both the linear and non-linear regions) depends on the level of confining pressure (in a triaxial test) and temperature and during the latter stages of a creep test, the strain rate can increase dramatically as the specimen is progressively damaged (6,7). Consequently, to include all these factors requires a stress dependent elasto-visco-plastic constitutive model which is most easily incorporated into the FE method for modelling complex boundary value problems. This paper describes the development of a stress dependent elasto-visco-plastic constitutive model with damage and the procedure used to implement the model into the CAPA3D FE program developed at Delft University of Technology (8,9). To demonstrate the use of the model the problem of permanent deformation under container loading is simulated. ELASTO-VISCO-PLASTIC MODEL Typically, viscoelastic constitutive equations are developed from a number of spring and dashpot elements arranged in series and parallel. Two commonly used models (the generalised Maxwell model and the generalised Burger’s model) are shown in Figure 1. It can be shown that, for a linear material, spring and dashpot constants can be chosen such that the two models give identical behaviour (for the same number of elements in each model) (10). However, in the generalised Maxwell model the same strain is shared across all the elements and the stress is additive whereas in the generalised Burger’s model the strains are additive and the stress is the same for each element.




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The generalised Burger’s model will be adopted here because it shares the same framework as classical viscoplasticity models and allows non-linearities based on stress to be accommodated more easily. It can be seen from Figure 1(b) that the generalised Burger’s model comprises an elastic element in series with a number of viscoelastic (Voigt) elements and a viscoplastic element. As noted above, in this type of model the stress is transmitted through each element and the strains (and strain rates) are additive such that: ε (t ) = ε el (t ) + ε ve (t ) + ε vp (t ) (1) where ε , ε el , ε ve , ε vp are the total, elastic, viscoelastic and viscoplastic strain components and t is time. The elastic component can be readily calculated using: ε el (t ) = σ (t ) E 0 (2) where σ is the stress and E 0 is the modulus of elasticity of the elastic element. The viscoelastic and viscoplastic components can be calculated using the Hereditary Integral formulation (11): t dJ (t − t ′) ε ve (t ) = J ve (0 )σ (t ) + ∫ ve σ (t ′) dt ′ (3) ′ ( ) − d t t 0 t

ε vp (t ) = J vp (0 )σ (t ) + ∫ 0

dJ vp (t − t ′) d (t − t ′)

σ (t ′) dt ′

(4)

where J ve , J vp are the viscoelastic and viscoplastic creep compliances and t ′ is a dummy integration variable. Note that a Hereditary Integral formulation based on stress and rate of compliance has been chosen (rather than a formulation based on rate of stress and compliance) to avoid problems due to the sudden application of a stress where the rate of stress can be extremely high (eg in a creep test). For example, if a formulation based on the rate of stress had been used to simulate a creep test (ie step function loading), the rate of stress would be initially infinite and zero thereafter making numerical simulation difficult. It can be shown that the first derivatives of the viscoelastic and viscoplastic creep compliances and the initial creep compliances for the generalised Burger’s model shown in Figure 1(b) are given by: λ dJ ve (t − t ′) N 1 −(t −t ′ ) τ i , τi = i =∑ e d (t − t ) Ei i =1 λi dJ vp (t − t ′) 1 (5) = λ∞ d (t − t ) J ve (0 ) = J vp (0) = 0

where λi is the viscosity of the i’th Voigt viscoelastic element, E i is the modulus of elasticity of the i’th Voigt viscoelastic element and λ∞ is the viscosity of the viscoplastic element. The latest version of CAPA-3D is based on an incremental formulation where a change in stress is calculated from a change in displacement (full details of CAPA-3D are beyond the scope of this paper and can be found in (2,8,9)). Consequently, any new constitutive model needs to be expressed in incremental form. This is explained in the following section.




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INCREMENTAL FORMULATION The elastic strain increment ∆ε el is simply given by: ∆ε el =

ε el − t ε el = ∆σ E 0 (6) t + ∆t t ε el , ε el are the elastic strains at time t + ∆t and t respectively, and ∆σ is the where stress increment. The viscoelastic strain increment ∆ε ve is given by: ∆ε ve =

t + ∆t

ε ve − ε ve t

t + ∆t

 t + ∆t 1 −(t + ∆t −t ′ ) τ i = ∑ ∫ e σ (t ′) dt ′ − t ε vei  i =1  0 λ i  N

(7)

 N t + ∆t 1 −(t + ∆t −t ′ ) τ i  N  t 1 − (t + ∆t − t ′ ) τ i ′ ′ σ (t ) dt  + ∑  ∫ e σ (t ′) dt ′ − ∑ t ε vei = ∑ ∫ e i =1  0 λi  i =1  t λi  i =1 N

where N is the number of Voigt elements. The first term in Equation (7) can be simplified to give: N t  N  e − ∆t τ i t − ( t − t ′ ) τ i  N − ∆t τ i t i 1 − ( t + ∆t − t ′ ) τ i ′ ′ ′ ′ ( ) ( ) e σ t d t e σ t d t ( ε ve ) (8) =  ∑  = ∑e ∫ ∑ ∫ i =1  0 λi  i =1  λi 0  i =1 where t ε vei is the viscoelastic strain for the i’th Voigt element at time t. The second term in Equation (6) can be evaluated using the mid-point rule to give: N t + ∆t   N  1 −(t + ∆t −t ′ ) τ i 1  e σ (t ′) dt ′ ≅ ∑  e −(t + ∆t −t ′ ) τ i σ (t ′) dt ′ ∆t  ∫ ∑ λ i =1  t λi   i =1  t ′ = t + ∆t 2  i  (9) N  ∆t − ∆t 2τ i  t ∆σ   ≅ ∑ e  σ+  2   i =1  λi where t σ is the stress at time t. Substituting Equations (8) and (9) into Equation (7)

gives: N  ∆t ∆σ    (10) ∆ε ve ≅ ∑  t ε vei e −∆t τ i − 1 + e − ∆t 2τ i  t σ +  λi 2   i =1  A similar procedure can be followed for evaluation of the viscoplastic strain increment giving:

(

∆ε vp

)

∆ε vp ≅

∆t

λ∞

  

t

σ+

∆σ 2

  

(11)

VISCOPLASTIC STRESS DEPENDENCE As noted earlier, previous research has shown that at high stress levels, the steady-state (viscoplastic) strain-rate follows a power law relationship of the following form (12): dε = Kσ n = Kσ n −1σ dt (12) σ =

λ∞




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where K and n are material constants. Results from triaxial experiments have also shown that, in addition overall stress level, the steady-state strain rate also depends on the test temperature and the degree of confinement (6,7,12). Consequently, based on these results, a model of the following form can be formulated for determining the equivalent viscosity of the viscoplastic element as a function of the stress conditions, temperature and degree of confinement:

λ∞ = λuni (T )

σe  σ  0

1− n

   

10 B (η +1 3)

= λuni (T ) 10 B (η +1 3 )

(σ e > σ 0 )

(13)

(σ e ≤ σ 0 )

where σ e is the Von Mises equivalent stress (σ e = {3 2 s ij s ij } 1 2 ), s ij is the deviatoric stress tensor

( p = 1 3σ kk ) ,

(s

ij

= σ ij − 1 3 δ ij σ kk ) , δ ij is the Kronocker delta, p is the mean stress

η is the stress ratio (η = p σ e ) , λuni is the uniaxial viscosity measured from a uniaxial compression test (where η = − 1 3 ), T is temperature, and n, σ 0 , B , are material constants. Equation (13) is implemented numerically into CAPA-3D using the values of σ e and η calculated at the beginning of the time step (ie at time t). Figure 2 shows the fit from Equation (13) (solid lines) to experimental data for an idealised bituminous mixture (sand asphalt) tested by Collop and Khanzada (7). The fitted 20C C = 60 GPas , λ30 material parameters are: λuni uni = 5 GPas , σ 0 = 100 kPa , n = 2 and B = −3.2 . It can been from this figure that Equation (13) models the behaviour reasonably well, although the temperature and stress ratio ranges are somewhat limited. It should also be noted that, although Equation (13) is in a form where it can be applied to any stress state, the experimental data from which it was derived were for only negative stress ratios (ie compressive mean stress). This is currently being investigated as part of a joint research programme between the University of Nottingham (UK) and Delft University of Technology (The Netherlands). VISCOPLASTIC DAMAGE It can be seen from Equations (12) and (13) that a constant stress creep test will result in a constant value of effective viscosity and therefore a constant rate of permanent strain accumulation. However, as noted earlier, it is often observed that during the latter stages of a creep test the strain rate increases dramatically (tertiary creep). In terms of Equation (13) this implies that the effective viscosity is reducing as the material is becoming progressively damaged. An appropriate method to include this effect is to use Continuum Damage Mechanics (CDM). In its simplest form, CDM introduces a scalar quantity D which ranges from 0 (undamaged) to 1 (fully damaged). The crux of the CDM approach is an expression that defines the growth of D, often termed the evolution equation. Rabotonov proposed the following equations for creep and evolution of damage (13): C1σ n dε (14) = dt (1 − D )m C 2σ v dD = dt (1 − D )µ


(15)



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where C1 , C 2 , n, m, v, µ are material constants that, in general, depend on temperature. It can be seen that Equation (14) can be re-written in a similar form to Equation (13) giving: dε C1σ n −1σ 1 = = σ (16) dt (1 − D )m λ

λ=

σ 1−n (1 − D )m

(17) C1 It can be seen that Equation (17) is of the same general form as Equation (13) (for a constant stress ratio) with the addition of an extra term accounting for damage. Consequently, Equation (13) can be modified to account for cumulative damage giving:

λ∞ = λuni (T )

σe  σ  0

1− n

   

10 B (η +1 3 ) (1 − D )

(σ e > σ 0 )

m

(18)

= λuni (T ) 10 B (η +1 3) (1 − D ) (σ e ≤ σ 0 ) The evolution equation (Equation (15)) can be generalised in three-dimensions using the Von Mises equivalent stress to give: m

dD  σ e  1 = ~  dt  C  (1 − D )µ v

(19)

~ where C is a material constant. In incremental formulation, it can be shown that the damage increment is given by: 1

(

∆D = t + ∆t D − t D ≅ 1 − t D

 −  1− t D  

) (

)

µ +1



− (µ + 1)  

t

σ e + ∆σ e 2    

~ C

v

 ( µ +1) ∆t   

(20)

In order to reduce the number of constants in Equations (18) and (20), the use of the effective stress (σ~ = σ (1 − D )) in all physical relations has been proposed (14). Consequently, the number of exponents can be reduced to two (m = n, µ = v ) . It has also been suggested that v = (n + 1) 2 holds for a range of materials and temperatures in uniaxial creep further reducing the number of constants (14). Consequently, the number of independent constants to be ~ determined in Equations (18) and (20) reduces to just five λuni , σ 0 , B, n, C .

(

)

THREE-DIMENSIONAL FORMULATION For extension of the above equations to a general three-dimensional isotropic case, the viscoelastic equivalent of two independent material parameters are required. The simplest approach is to assume that the equivalent of Young’s modulus is elasto-visco-plastic and the equivalent of Poisson’s ratio (ν ) is a constant value. It should be noted that, strictly, Poisson’s ratio refers to the ratio of lateral strain to longitudinal strain in the region where the stress-strain curve for a material is linear. In this paper the terminology “equivalent value of Poisson’s ratio” has been used to refer to the ratio of lateral strain to longitudinal strain for elastic, viscoelastic and viscoplastic behaviour. Consequently, the three-dimensional expressions for the three strain increments are as follows: ∆ε el = C el ∆σ (21)




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∆ε ve = ∑  t ε ive (e−∆t τ N

i =1

 

∆ε vp

where C el =

1 E0

 1  el − ν − ν el   0  0   0

i

∆σ   − 1 + ∆t e −∆t 2τ i Cive  t σ +  2  

)



= ∆t C vp  t σ +

∆σ  2



(23)

 

− ν el

− ν el

0

0

0

1

− ν el

0

0

0

1

0

0

0

0

0

−ν

el

(

2 1 +ν

el

)

(22)

0

0

0

0

0

2 1 + ν el

0

0

0

0

(

)

(

0

2 1 + ν el

    ,    

C ive , C vp

are

)

respectively and E 0 replaced by λi and λ∞ similar with ν replaced by ν and ν respectively. In this type of formulation it is possible to assign different Poisson’s ratio equivalents to the different components. For example, it may be appropriate to set the equivalent of Poisson’s ratio equal to 0.3 for the recoverable components (elastic and viscoelastic) whereas it may be appropriate to set the equivalent of Poisson’s ratio equal to 0.5 for the irrecoverable component (viscoplastic) to simulate a constant volume permanent deformation process. el

ve i

vp

OVERALL METHODOLOGY For each loading step the following procedure is adopted: 1.

2.

Enter constitutive model subroutine with converged previous states of stress, viscoelastic strain and viscoplastic damage t σ, t ε ve , D and a trial total strain increment for that step (∆ε ) .

(

)

Use the constitutive model to determine an updated stress t + ∆t σ = t σ + ∆σ . To calculate ∆σ the fact that the strain increments calculated from the elastic, viscoelastic and viscoplastic components of the constitutive model should balance the strain increment passed into the subroutine is utilised. This results in the following local strain compatibility relationship: (∆ε el + ∆ε ve + ∆ε vp ) − ∆ε = 0 (24) Substituting Equations (21), (22) and (23) into Equation (24) and rearranging gives: N N  ∆t    − ∆t 2τ i  − ∆t τ i i t i −1   C vp + ∑ C ve e ∆σ ≅ ∆ε − ∑ ε ve e C el + 2  i =1 i =1     (25)

(

)



N





i =1



− ∆t  C vp + ∑ C ive e −∆t 2τ i  t σ

It can be seen that Equation (25) can be rearranged into a pseudo-elastic equation so that the stress increment can be explicitly determined: −1 ∆σ = C ∆ε (26) ∆t where C = C el + C1 , 2




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N

C1 = C vp + ∑ C ive e − ∆t 2τ i , i =1

∆ε = ε 1 − ∆t C 1 t σ and ε 1 = ∆ε −

3. 4.

N

∑ i =1

t

i

(

ε ve e

− ∆t τ i

)

−1

Calculate an updated viscoelastic strain

t + ∆t

ε ve = t ε ve + ∆ε ve

(using Equation (22)) and

updated damage t + ∆t D = t D + ∆D (using Equation (20)). If the element forces (calculated from the updated stresses) balance the external forces then t + ∆t σ, t +∆t ε ve , t + ∆t D become the converged stresses viscoelastic strains and damage for the next time step. If not, determine an updated trial total strain increment (∆ε ) and go back to step 1.

An alternative approach to assuming that the equivalent of Young’s modulus is elastovisco-plastic and the equivalent of Poisson’s ratio (ν ) is a constant value would be to divide the behaviour into deviatoric states of stress and strain and volumetric states of stress and strain. In this case, the elasto-visco-plastic model (Equations (21) to (23)) would be applied separately (with different parameters). Equations (24) and (25) would be formulated for deviatoric and volumetric states of strain and stress separately and the stress increment would be the sum of the deviatoric and volumetric stress increments calculated from the equivalents of Equation (26). ILLUSTRATIVE SIMULATION To demonstrate the capabilities of the model, the problem of container loading on a typical industrial pavement structure (of a semi-rigid nature) in The Netherlands has been simulated. The overall mesh is shown in Figure 3. 20-noded solid brick elements were used, each with 3 degrees of freedom at each node. The total number of elements was 847. Two planes of symmetry are utilised and semi-infinite elements are used to model the lateral and bottom boundaries. All materials (except the asphalt) are assumed to be linear elastic (see Table 1 for properties). The maximum likely vertical contact stress (for one container foot) was estimated to be 10.6 MPa (this corresponds to the worst case where three fully loaded containers are placed on top of each other). The loading is modelled as a uniform vertical contact stress (see hashed area in Figure 3) applied over 1 second, held at a constant level for 24 seconds, unloaded over 1 second and held at zero for 24 seconds (ie a total simulation time of 50 seconds). MATERIAL PROPERTIES Data for the idealised asphalt mixture (sand asphalt) were used to determine the asphalt material properties. The viscoplastic parameters (λuni , σ 0 , B, n,) were determined by fitting Equation (13) to the data shown in Figure 2 (the fit is shown by the solid lines). The elastic (E 0 , ) , viscoelastic (E1 , λ1 ) and additional damage parameter C~ were estimated by fitting the total predicted creep strain response (Equation (1)) to the measured creep strain response at an applied stress level of 700kPa and a temperature of 20oC taken from (12). The fitted and predicted responses are shown in Figure 4 and all the fitted parameters are shown in Table 2. It should be noted that, since this work is primarily investigating permanent deformation under quasi-static container

()




9

loading, the elastic and viscoelastic material properties are of less significance compared to the viscoplastic material properties. As noted above, the equivalent of Poisson’s ratio is required for an isotropic analysis. Previous research has shown that Poisson’s ration for bituminous materials is between 0.1 and 0.45. (15). Consequently, in this analysis it is assumed that the value of this parameter is 0.3 for all elements of the model. Further detailed triaxial testing is required to validate the assumption that the ratio of lateral to longitudinal strain is 0.3, particularly for the viscoplastic deformation. Figure 4 shows an experimental creep curve (dashed line) for the sand asphalt at an applied stress level of 700kPa and a temperature of 20oC taken from (12). Also shown in this figure (solid line) is the fit obtained (using one Voigt element) to the measured creep curve using the material parameters given in Table 2. It should be noted that this simulation is for illustration purposes only (sand asphalt is not a realistic material for this type of application). Testing is currently being undertaken by Ooms Avenhorn Holding to determine suitable material properties for a polymer modified asphalt that they would use under these circumstances. SIMULATION RESULTS It can be seen from Equation (18) that the equivalent viscosity primarily depends on the Von Mises stress, the stress ratio and the current level of viscoplastic damage. Consequently, it is necessary to examine the behaviour of each of these quantities to provide insight into the overall behaviour. In the following figures, results for a small section of the asphalt layer (300mm x 300mm) close to the applied load (see Figure 3) are shown superimposed on a magnification of the displaced mesh (the Displacement Magnification Factors (DMF) are given below). Von Mises Equivalent Stress It can be seen from Equation (18) that the Von Mises equivalent stress controls the non-linear part of the equivalent viscosity. Figure 5 shows contours of Von Mises equivalent stress (DMF=2) calculated just after the load has been fully applied (1 second). It can be seen from this figure that the Von Mises equivalent stress is greatest (approximately 6MPa) directly under the loaded area to a depth of approximately half of the asphalt layer thickness. It can also be seen from this figure that the Von Mises equivalent stress decays rapidly outside the loaded area. Consequently, according to Equation (18), the effective viscosity will be lowest (and hence the rate of viscoplastic deformation highest) in the region directly under the applied load to a depth of approximately half the asphalt layer thickness. Stress Ratio It can be seen from Equation (18) that the stress ratio accounts for the effect of confinement on the viscoplastic deformation process. It should be noted that a negative stress ratio indicates a compressive mean stress whereas a positive stress ratio indicates a tensile mean stress. For example, at a constant level of Von Mises equivalent stress, the lower the stress ratio, the greater the degree of confinement and hence the greater the equivalent viscosity. Figure 6 shows contours of stress ratio (DMF=2) calculated just after the load has been fully applied (1 second). It can be seen from this figure that in a large proportion of the asphalt the stress ratio is negative indicating compressive mean stresses. The minimum (negative) stress ratios occur directly under the loaded area. This is because, in this area, the compressive mean




10

stress is greater than the Von Mises stress. It can be seen by comparing Figures 5 and 6 that the low (negative) stress ratios directly under the applied load are concentrated at the surface whereas the high Von Mises stresses penetrate deeper into the asphalt layer. It can also be seen from Figure 6 that there is a region where the stress ratio is also low (negative) at the base of the asphalt layer under the load. The areas of highest (positive) stress ratios occur at the surface of the pavement in regions inside and outside the loaded area. In these regions the mean stress is tensile indicating that the asphalt is likely to dilate rather than compress in the areas where the stress ratio is negative. Equivalent Viscosity It can be seen from Equation (18) that the equivalent viscosity is a combination of the nonlinearity due to the Von Mises stress and the confinement due to the stress ratio. Figure 7 shows contours of equivalent viscosity (DMF=2) calculated just after the load has been fully applied (1 second). It can be seen from this figure that, in the region at the surface of the pavement directly under the load, the confining effect dominates and the equivalent viscosity is very high. However, it can also be seen from this figure that there is a region slightly deeper into the pavement (under the load) where the viscosity is significantly lower due to the higher (negative) stress ratios. Outside the loaded area on the pavement surface there is a region where the viscosity is lowest due to high (positive) stress ratios (the grey areas indicates that the viscosities are lower than 1x102MPas). Damage Figure 8 shows contours of damage (DMF=2) calculated just before the pavement is unloaded (25 seconds). It can be seen from this figure that the areas of highest damage are concentrated in regions directly under the applied load (extending to approximately half the thickness of the asphalt layer) where the Von Mises stress is greatest and the equivalent viscosity is relatively low. Figure 9 shows contours of equivalent viscosity (DMF=2) calculated just before the load is removed (25 seconds). It can be seen by comparing Figures 7 and 9 that the effect of the cumulative damage is to decrease the equivalent viscosity and hence result in an increased rate of viscoplastic deformation, particularly in an area directly beneath the loaded area, slightly beneath the pavement surface. Permanent Strain Figure 10 shows contours of permanent vertical strain (DMF=5) calculated after the pavement has been unloaded (50 seconds). It can be seen from this figure that the strains are negative (indicating a reduction in thickness) in the region directly under the load whereas they are positive (indicating an increase in thickness) in the regions outside and inside the loaded areas. This is due to material flowing from the area under the load to the adjacent areas. For comparison purposes, Figure 11 shows contours of permanent vertical strain (DMF=50) calculated after the pavement has been unloaded (50 seconds) assuming the equivalent viscosity of the asphalt to be constant (60GPas). It can be seen by comparing Figures 10 and 11 that the permanent vertical strains in the non-stress dependent case are significantly lower than the permanent vertical strains in the stress dependent case. This is because, in the stress dependent case, there are regions where the viscosity is significantly lower than 60GPas




11

(see Figures 7 and 9), partly due to the local stress conditions and partly due to the accumulation of viscoplastic damage which results in significantly more viscoplastic flow. It can also be seen that, in the stress dependent case (Figure 10), there is a more localised area of high permanent compressive vertical strain directly under the load at approximately half depth in the asphalt which corresponds to a localised area where the equivalent viscosity is low (Figure 9) and the stress is high (Figure 5). In the non-stress dependent case (Figure 11) the area of high permanent compressive vertical strain directly under the load is less localised and extends to the pavement surface. This is because the effective viscosity is constant and, consequently, the viscoplastic strain is driven by the stress conditions (see Figure 5). DISCUSSION The research described in this paper details the formulation and FE implementation of a constitutive model for predicting permanent deformation in asphaltic materials under quasi-static loading conditions. Since the focus was on permanent deformation, more attention has been given to the modelling of the viscoplastic component of material response. A stress dependence has been included into this component as well as sensitivity to confinement and damage accumulation. Linear elastic and viscoelastic components have been added in series with the viscoplastic component to make the model framework more general for future developments. Although material parameters have been determined for the elastic and viscoelastic components, permanent strain predictions depend only on the viscoplastic component. In the example presented, an idealised bituminous mixture (sand asphalt) was used because of the availability of material data. It should be noted that the simulation was for an idealised bituminous material (sand asphalt) that is not intended for the application of heavy containers. However, results show the effect of stress dependency and confinement on the permanent deformation is clearly important (compared to a linear viscoelastic simulation) particularly where the applied stress is high as in the case of container loading. Joint research between the University of Nottingham, Delft University of Technology and Ooms Avenhorn Holding is planned in the future to further develop/validate the model so that permanent deformation in the asphalt layers of a pavement structure under traffic loading can be predicted using CAPA-3D. In particular, the elastic and viscoelastic components will be generalised so that non-linear recoverable (transient) behaviour can be predicted. It is also recognised that higher levels of permanent deformation can occur under repeated loading compared to a single load application with the same overall loading time which has been attributed to plastic strain development in the aggregate skeleton. Since the existing model was developed using data from traditional creep testing it is important that the effects of repeated loading are included in future developments of the model to make its use more general. In the current model, Continuum Damage Mechanics (CDM) has been used to define a simple isotropic scalar damage variable and an associated evolution law based on the concept of an effective stress tensor. However, this type of model is independent of orientation and therefore cannot be used to model anisotropic damage growth. It is planned that the current CDM approach will be modified to account for anisotropic creep damage based on the use of a damage tensor. SUMMARY AND CONCLUSIONS • An elasto-visco-plastic constitutive model with viscoplastic damage has been developed.




• • • • • •

•

12

The model comprises linear elastic, linear viscoelastic and non-linear viscoplastic components. The strains (and strain rates) for each component are additive whereas they share the same stress (ie series model). A stress-based non-linearity and sensitivity to confinement has been introduced into the viscoplastic component. A simple Continuum Damage Mechanics (CDM) formulation has been introduced into the viscoplastic component to account the effects of cumulative damage on the viscoplastic response of the material. The model is implemented in an incremental formulation into the CAPA-3D Finite Element (FE) program developed at Delft University of Technology (The Netherlands). A local strain compatibility condition is utilised such that the incremental stresses are determined explicitly from the incremental strains at each integration point. The model has been demonstrated by investigating the response of a semi-rigid industrial pavement structure subjected to container loading It should be noted that this simulation is for illustration purposes only (sand asphalt is not a realistic material for this type of application). Results show that the permanent vertical strains in the non-stress dependent case are significantly lower than the permanent vertical strains in the stress dependent case. Results also show that, in the stress dependent case, there is a more localised area of high permanent compressive vertical strain directly under the load at approximately half depth in the asphalt compared to the non-stress dependent case where the distribution is more even.

REFERENCES (1) Desai, C. Mechanistic Pavement Analysis and Design using Unified Material and Computer Models., Proc. 3rd Int. Symp. on 3D Finite Elements for Pavement Analysis, Design and Research, Amsterdam, The Netherlands, April 2002 pp 1-64. (2) Scarpas, A., Al-Khoury, R., Gurp, C.A.P.M. and Erkens, S.M.J.G. Finite Element Simulation of Damage Development in Asphalt Concrete Pavements. Proc. 8th International Conference on Asphalt Pavements, Vol. 1, University of Washington, Seattle, Washington, 1997 pp 673-692. (3) Abbas, A., Papagiannakis, A.T. and Masad, E. A Formulation for the Non-Linear Viscoelastic Behaviour of Asphalt Binders. International Journal of Geomechanics (Submitted), July 2001. (4) Long, F. and Monismith, C.L. Use of a Nonlinear Viscoelastic Constitutive Model for Permanent Deformation in Asphalt Concrete. Proc. 3rd Int. Symp. on 3D Finite Elements for Pavement Analysis, Design and Research, Amsterdam, The Netherlands, April 2002 pp 91-110. (5) Cheung, C.Y. and Cebon, D. Deformation Mechanisms of Pure Bitumen. Journal of Materials in Civil Engineering, Vol. 9(3), 1997, pp 117-129. (6) Deshpande, V. and Cebon, D. Steady-State Constitutive Relationship for Idealized Asphalt Mixes. Mechanics of Materials Vol. 31(4), 1999, pp 271-297. (7) Collop, A.C. and Khanzada, S. Permanent Deformation Behaviour of Idealised Bituminous Mixtures. Proc. 3rd European Symposium on Performance and Durability of Bituminous Materials and Hydraulic Stabilised Composites, Leeds (UK), 1999.




(8)

(9)

(10) (11) (12)

(13) (14) (15)

13

Scarpas, A. and Blaauwendraad, J. Experimental Calibration of a Constitutive Model for Asphaltic Concrete. Proceedings of the Euro-C Conference on the Computational Modelling of Concrete Structures, Badgastein, Oostenrijk, April 1998, pp 193-202. Erkens, S.M.J.G., Liu, X., Scarpas, A. and Kasbergen, C. Issues in the Constitutive Modelling of Asphalt Concrete. Proc. 3rd Int. Symp. on 3D Finite Elements for Pavement Analysis, Design and Research, Amsterdam, The Netherlands, April 2002 pp 475-494. Roscoe, R. Mechanical Models for the Representation of Visco-Elastic Properties. British Journal of Applied Physics, Vol. 1, 1950, pp 171-173. Flugge, W., Viscoelasticity., Blaisdell Publishing Company, Massachusetts, US, 1967. Collop, A.C. and Khanzada, S. Permanent deformation in idealised "Sand Asphalt" bituminous mixtures. International Journal of Road Materials and Pavement Design, Vol. 2, No. 1, 2001, pp 7-28. Rabotnov, Yu.N. Creep Problems in Structural Members. North-Holland, Amsterdam, 1969. Zyczkowski, M. Creep Damage Evolution Equations Expressed in Terms of Dissipated Power. International Journal of Mechanical Sciences, Vol. 42, 2000, pp 755-769. Read, J.M. Fatigue cracking of bituminous paving mixtures, PhD Thesis, School of Civil Engineering, University of Nottingham, 1996.




14

LIST OF TABLES AND FIGURES

TABLE 1 General Material Properties TABLE 2 Asphalt Material Properties

FIGURE 1 (a) Generalised Maxwell viscoelastic model, (b) Generalised Burger’s viscoelastic model. FIGURE 2 Experimental viscosity data. FIGURE 3 Finite Element (FE) Mesh. FIGURE 4 Fitted Creep Curve. FIGURE 5 Contours of Von Mises Equivalent Stress. FIGURE 6 Contours of Stress Ratio. FIGURE 7 Contours of Equivalent Viscosity (Initial). FIGURE 8 Contours of Accumulated Damage. FIGURE 9 Contours of Equivalent Viscosity (Final). FIGURE 10 Permanent Vertical Strain Distribution (Stress Dependent Case). FIGURE 11 Permanent Vertical Strain Distribution (Non-Stress Dependent Case).




15

TABLE 1 General Material Properties Material Clay/Peat Sand Cement Treated Base


Young’s Modulus (MPa) 50 100 5,000

Poisson’s Ratio 0.4 0.35 0.2

Thickness (mm) 1,600 2,000 400



16

TABLE 2 Asphalt Material Properties Property C λ20 uni (MPas) σ 0 (MPa) n B

ν e ,ν ve ,ν vp

Value 60,000 0.1 2 -3.2 0.3


Property ~ C (MPa) E 0 (MPa) E1 (MPa) λ1 (MPas)

Value 110 70 120 600



17

FIGURES

σ(t)

E1

λ1

En

E2

λ2

λn

σ(t) (a) Voigt element E1 E0

En

λ∞ σ(t)

σ(t) λ1 (b)

λn

FIGURE 1 (a) Generalised Maxwell viscoelastic model, (b) Generalised Burger’s viscoelastic model.




η=-0.8, 20oC

12

10

Effective Viscosity / Pa.s

18

η=-0.6, 20oC 11

10

η=-0.33, 20oC 10

10

η=-0.33, 30oC 9

10

8

10

4

5

10

10

6

7

10

10

Equivalent Stress / Pa

FIGURE 2 Experimental viscosity data.




19

Loading Detail

y

120mm Asphalt 400mm Cement Treated Base

2,000mm Sand

1,600mm Clay/Peat x z Clay/Peat SemiInfinite Boundary

2,000mm 2,000mm

FIGURE 3 Finite Element (FE) Mesh.




20

0.06

0.05

Strain / m/m

0.04

0.03

0.02

0.01

0

0

50

100

150 200 250 Time / Seconds

300

350

400

FIGURE 4 Fitted Creep Curve.




21

Von Mises Equivalent Stress (MPa) 0

2.0

x z

4.0

6.0

FIGURE 5 Contours of Von Mises Equivalent Stress.




22

Stress Ratio -1.5 -1.0 -0.5 0

x z

0.5 1.0 1.5

FIGURE 6 Contours of Stress Ratio.




23

Viscosity (MPas) 1x102 1x103 1x104

x

1x105

z

1x106 1x107 1x108

FIGURE 7 Contours of Equivalent Viscosity (Initial).




24

Damage 0

0.1

0.2

x 0.3

z 0.4

0.5

FIGURE 8 Contours of Accumulated Damage.




25

Viscosity (MPas) 1x102 1x103 1x104

x

1x105

z

1x106 1x107 1x108

FIGURE 9 Contours of Equivalent Viscosity (Final).




26

Strain (%) -2.0

0

x z 2.0

4.0

FIGURE 10 Permanent Vertical Strain Distribution (Stress Dependent Case).




27

Strain (%) -0.3

-0.15

x

0

z 0.025

0.05

FIGURE 11 Permanent Vertical Strain Distribution (Non-Stress Dependent Case).



DEVELOPMENT AND FINITE ELEMENT IMPLEMENTATION OF A

DEVELOPMENT AND FINITE ELEMENT IMPLEMENTATION OF A

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