Weak For Equation-Based Finite Eleent Modeling of Viscoelastic Asphalt Mixtures 1 Yuqing Zhang, Ph.D. A.M. ASCE Lecturer School of Engineering and Applied Science, Aston University Aston Triangle, Biringha, B4 7ET, U.K. Phone: +44(0)121 204 3391, Eail: y.zhang10@aston.ac.uk Bjorn Birgisson, Ph.D., P.E. Executive Dean, School of Engineering and Applied Science, Aston University Aston Triangle, Biringha, B4 7ET, UK Phone: +44(0)7825 125908, Eail: bjorn.birgisson@aston.ac.uk Robert L. Lytton, Ph.D., P.E., F. ASCE Professor, Fred J. Benson Chair Zachry Departent of Civil Engineering Texas A&M University 3136 TAMU, CE/TTI Bldg. 503A, College Station, Texas 77843 Phone: (979) 845-9964, Eail: r-lytton@civil.tau.edu 1 This is an Accepted Manuscript of an article published by ASCE in Journal of Materials in Civil Engineering. The final publication is available online via http://dx.doi.org/10.1061/(asce)mt.1943-5533.0001395
Zhang et al. 1 Abstract The objective of the study is to deonstrate using weak for partial differential equation (PDE) ethod for a finite eleent (FE) odeling of a new constitutive relation without the need of user subroutine prograing. The viscoelastic asphalt ixtures were odeled by weak for PDE based FE ethod as the exaples in the paper. A solid-like generalized Maxwell odel was used to represent the deforing echanis of a viscoelastic aterial, the constitutive relations of which were derived and ipleented in the weak for PDE odule of Cosol Multiphysics, a coercial FE progra. The weak for PDE odeling of viscoelasticity was verified by coparing Cosol and Abaqus siulations, which eployed the sae loading configurations and aterial property inputs in virtual laboratory test siulations. Both produced identical results in ters of axial and radial strain responses. The weak for PDE odeling of viscoelasticity was further validated by coparing the weak for PDE predictions with real laboratory test results of six types of asphalt ixtures with two air void contents and three aging periods. The viscoelastic aterial properties such as the coefficients of a Prony series odel for the relaxation odulus were obtained by converting fro the aster curves of dynaic odulus and phase angle. Strain responses of copressive creep tests at three teperatures and cyclic load tests were predicted using the weak for PDE odeling and found to be coparable with the easureents of the real laboratory tests. It was deonstrated that the weak for PDE based FE odeling can serve as an efficient ethod to ipleent new constitutive odels and free engineers fro user subroutine prograing. Keywords: Asphalt ixture, viscoelasticity, finite eleent odeling, weak for PDE, aster curve, generalized Maxwell odel
Zhang et al. 2 Introduction As a viscoelastic aterial, an asphalt ixture exhibits a tie and frequency dependent behavior. The current stress or strain responses of the asphalt ixtures are affected by the whole history of the strain or stress inputs prior to the current tie. Teperature has a significant influence on the aterial responses, which is coonly accounted for by odulus aster curves and tie-teperature shift functions. Extensive studies were perfored to deterine the appropriate shift functions and to develop the aster curve odels for dynaic odulus and phase angle of asphalt ixtures (Pellinen et al. 2002; Biligiri et al. 2010; Zhang et al. 2012a; Zhao et al. 2012). The dynaic odulus has becoe a ajor input for aterial properties in the Mechanical-Epirical Paveent Design Guide (MEPDG) (ARA 2004). Relaxation odulus and creep copliance are also eployed to characterize the viscoelastic properties of the asphalt ixtures, especially in a viscoelastic constitutive odeling (Gibson et al. 2003; Zhang et al. 2012b). Theoretically, all viscoelastic variables including coplex odulus (dynaic odulus as its agnitude), coplex copliance, relaxation odulus and creep copliance are interconvertible. Studies were also presented in the literature to introduce these conversions for use in practice (Park and Schapery 1999; Mun et al. 2007; Katicha et al. 2008; Hu and Zhou 2010). The viscoelasticity of asphalt ixtures has been well characterized by viscoelastic theories and laboratory tests as shown in the aforeentioned studies in the literature. However, applying viscoelastic odels and theories in aterial perforance predictions and paveent structural analysis are liited due to soe ipleentation difficulties. For instance, in the paveent research counity, Abaqus is one of the coonly-used finite eleent (FE) analysis progras that are utilized to odel asphalt ixtures perforance and siulate paveent structures (Huang et al. 2011; Darabi et al. 2012; Zhu and Sun 2013). Nevertheless, the following liitations restrict the FE odeling of asphalt ixtures in Abaqus. 1) The existing viscoelasticity odule cannot be used with any of the existing plasticity or daage odules in Abaqus (ABAQUS 2010). A user aterial subroutine ust be prograed in order
Zhang et al. 3 to siultaneously address different aterial echaniss such as viscoelasticity, plasticity, fracture, daage and their coupling effects. 2) The tie-teperature shift factor can only defined as the Willias Landel Ferry (WLF) and Arrhenius functions (supported as Input File ode only). A user defined tie-teperature shift factor ust be coded using user subroutine progras. 3) Prograing a user defined subroutine requires extensive experience with astering the coputer prograing language, which distracts the paveent researchers attention fro the aterial constitutive odeling and narrows the paveent structural siulations. 4) The users need to spend significant efforts to debug the user defined subroutines to avoid potential coputational errors and issues such as non-convergence, circular dependency, lowefficiency iteration, etc. For other nuerical progras developed by paveent researchers, siilar probles still exist with additional issues such as non-user-friendly interfaces, liited odeling abilities and restrictions on specific constitutive odels. A robust FE odeling ethod is needed to free paveent researchers fro the nuerical issues and siultaneously allow the to try their own aterial constitutive odels in an efficient way in paveent analysis. The weak for partial differential equation (PDE) based FE odeling is the tool that can achieve the objective. Therefore, the objective of the paper is to deonstrate how a constitutive relation is odeled by the weak for PDE ethod and ipleented in a finite eleent progra without a need of user subroutine prograing. The PDE based viscoelastic odeling of an asphalt ixture is deonstrated as an exaple in this paper to show how this can be done. Fro a physics perspective, elasticity, viscoelasticity, plasticity, and fracture are different aspects of echanical physics that ay occur siultaneously in a aterial. The control or constitutive equations of these physics can be derived by therodynaics using the virtual work principle and represented as ordinary differential equations (ODE) or partial differential equations (PDE) (ODE is treated as a special PDE). Thus different physics can be odeled siultaneously and evaluated for their interactions by solving the PDEs at one tie. The weak for of a PDE is regarded as a generalization of the virtual work
Zhang et al. 4 principle and is an iportant atheatical analysis ethod to find the solutions to the PDE. More explanations of the weak for of the PDE is presented in the section Weak For PDE Modeling of Viscoelasticity of this paper, in which a structural analysis ethod is introduced to explain the concept of weak for PDE. As a general-purpose FE progra, Cosol Multiphysics provides an efficient coputational platfor to solve weak for PDEs and to address the coupling effects of different physics (COMSOL 2013a). The ajor advantage of using the weak for of PDE odeling in Cosol rather than traditional FE odeling is that no user subroutine is needed and the control/constitutive equations of different physics can be defined and solved by equation based odels such as weak for PDEs. Thus the different physics such as viscoelasticity, plasticity, fracture, heat transfer and oisture diffusion can be easily odeled using weak for PDEs. Their interactions and coupling effects can also be evaluated. This paper is focused on viscoelastic odeling using the weak for PDE in Cosol and asphalt ixtures are selected as the viscoelastic aterials to be tested for odel calibration and validation. The next section presents the asphalt ixture aterials and laboratory tests. Master curves of dynaic odulus and phase angle are constructed for six different asphalt ixtures. Then linear viscoelastic constitutive relations are introduced and relaxation odulus is deterined using dynaic odulus and phase angle aster curves. After that, weak for PDE odeling of viscoelasticity is presented based on a solid-like generalized Maxwell odel. The weak for PDE odeling of viscoelasticity is verified by coparing the predictions with Abaqus results and is further validated by coparing the predictions with laboratory test results. The last section presents the suary and conclusions. Laboratory Testing and Master Curves of Asphalt Mixtures The objectives of laboratory testing are 1) to deterine the viscoelastic properties of asphalt ixtures which are used as inputs to the weak for PDE based FE odeling; and 2) to validate the weak for PDE based odeling of viscoelasticity in the Cosol progra. Materials and Experients
Zhang et al. 5 Laboratory tests were perfored on lab-ixed-lab-copacted (LMLC) asphalt ixtures that were fabricated with one asphalt binder, at two air void contents and three aging periods. Three replicate speciens were fabricated for each cobination of the variables. The testing protocol (including test ethod, loading paraeters, and teperatures), aterials and easureents are suarized in Table 1. As for the aterials used in the tests, a coonly-used Texas Hanson liestone was selected in this study and the gradation of the aggregates was deterined based on a Type C (coarse surface) dense gradation specified by the Texas Departent of Transportation (TxDOT) (2004). The optiu asphalt content was calculated based on the TxDOT test procedure (TxDOT 2008) and was deterined as 4.4%. The asphalt concrete speciens were copacted using the Superpave gyratory copactor to a cylindrical saple with 150 in diaeter and 175 in height. Then the asphalt concrete saples were cored to 100 in diaeter and cut to 150 in height. The target air void contents had two levels including 4% and 7%. Soe of the asphalt concrete speciens were tested once they were fabricated, while soe of the speciens were stored in the aging roo and aged at a constant teperature of 60 C for 3 onths and 6 onths, respectively. Before testing, the speciens were put in an environental chaber at the testing teperature for at least 3 hours to reach the equilibriu teperature. Then they were tested using a Universal Testing Machine (UTM) based on the test plan shown in Table 1. It is noted that the test results of dynaic odulus tests including dynaic odulus and phase angle are used to construct the aster curves which are then eployed to deterine the viscoelastic properties such as relaxation oduli and used as the inputs to the FE odeling. The test results of creep tests and cyclic load tests are used to copare with the FE odeling predictions so as to validate the equation-based FE odeling of the viscoelasticity. Dynaic Modulus Tests and Master Curve Constructions Copressive dynaic odulus tests were perfored at six frequencies and four teperatures as indicated in Table 1. To ensure linear viscoelastic behavior of the aterial, different stress agnitudes were used in each level of teperature and frequency to liit the dynaic strains to less than 150 με which is recognized as the strain liit for the linear viscoelastic asphalt ixture under a copressive load
Zhang et al. 6 (Levenberg and Uzan 2004). Sae criterion was used in deterining the load levels in other laboratory tests. Trial tests were perfored on duy saples to deterine the stress levels and ensure the strain liit is aintained. It basically requires lower stresses being applied at the lower frequencies and higher teperatures or for the asphalt ixtures with higher air void content and shorter aging period, as shown in Table 1. The dynaic odulus and phase angle were easured using the built-in algorith in the UTM data acquisition and analysis progra. In linear viscoelastic theory, the coplex odulus is used to characterize the constitutive behavior of the viscoelastic aterials when subjected to a stress or strain oscillation. The agnitude of the coplex odulus is tered the dynaic odulus and the phase lag between stress and strain is the phase angle. The coplex odulus is expressed as cos sin E E ie E i (1) where E is the coplex odulus, E and E are the storage odulus and loss odulus which are the real part and iaginary part of the coplex odulus, E is the dynaic odulus, is the phase angle, i 1, is angular frequency in rad/sec, 2 f and f is the loading frequency in Hz. Typical results of the dynaic odulus and phase angle easureents are shown in Fig 1 and Fig 2, respectively. Based on the easureents at different teperatures and loading frequencies, the aster curves of dynaic odulus and phase angle can be constructed according to the tie teperature superposition principle. The aster curve odel for dynaic odulus eploys a sigoidal function as below (Pellinen et al. 2002; ARA 2004). log E logf r 1 e (2) where E is the dynaic odulus, is the iniu logarith of the dynaic odulus, is span of the logarith of the dynaic odulus, and are shape paraeters, f r is reduced frequency in Hz that has fr f at, where f is the loading frequency in Hz, a T is a tie-teperature shift factor
Zhang et al. 7 shown as below which eploys a polynoial fitting function as the high teperature data are included (Francken and Verstraeten 1998; Biligiri et al. 2010). log 2 a at bt c (3) T where T is the teperature in Kelvin and a, b, and c are fitting paraeters. The aster curve odel for the phase angle is a -odel developed by the authors (Zhang et al. 2012a) which allows a non-syetric bell-shaped aster curve on the log-log plot of the phase angle versus the frequency. The -odel is presented as: Exp ax 1 1 f0 fr 1 1 fr f0 (4) where ax is the axiu phase angle, f 0 is the reference frequency at which ax occurs, is a fitting paraeter that deterines the curvature of the phase angle aster curve; f r is reduced frequency in Hz that has fr f at, where f is the loading frequency in Hz and a T is a tie-teperature shift factor that uses the sae forulation as in Eq. 3 (Biligiri et al. 2010). Eqs. 2 and 4 allow for the use of Excel spreadsheets and the Solver function to construct the aster curves of dynaic odulus and phase angle. The Solver function perfors nonlinear least square regression in Excel spreadsheets to deterine the odel coefficients in Eqs. 2, 3 and 4. Exaples are given for the aster curves of the dynaic odulus and phase angle in Fig. 1 and Fig. 2, respectively. Table 2 suarizes the aster curve odel coefficients for dynaic odulus and phase angle of the six types of asphalt ixtures with different air void contents and aging periods. It is noted that, to construct better aster curves (e.g., with higher R 2 values as indicated in Figs. 1 and 2), different a T values were deterined for dynaic odulus and phase angle, respectively. The curves in Figs. 1 and 2 that are arked as aster curve are the fitted dynaic odulus laboratory data using aster curve odels; the
Zhang et al. 8 curves labeled as predicted are those predicted fro the relaxation odulus odeled by Prony series as explained in the subsequent section. Constitutive Relations and Relaxation Modulus of Asphalt Mixtures The linear viscoelastic constitutive relations are firstly presented in ters of relaxation odulus for one-diensional condition and using relaxation bulk and shear oduli for ultiaxial condition. Then Prony series odel coefficients of the relaxation odulus are deterined based on dynaic odulus and phase angle aster curves. Linear Viscoelastic Constitutive Relations The constitutive relations for a linear viscoelastic aterial are generally expressed as voluetric and deviatoric coponents of stress and strain tensors. Under ulti-axial states of stress, the constitutive relations are presented as (Findley et al. 1989): 3 t kk kk K t d (5) 0 2 t eij sij Gt d (6) 0 is the first invariant of stress tensor which is the voluetric coponent of where kk 11 22 33 is the voluetric strain, stress, kk 11 22 33 K t is the relaxation bulk odulus, sij ij 13kkij is the deviatoric stress tensor and ij is the Kronecker delta, eij ij 13kkij is the deviatoric strain tensor, G t is the relaxation shear odulus, t is the current tie of interest, and is an integration variable that is a tie before current tie. Under a uniaxial state of stress, the constitutive relation is expressed as: t ij ij Et d 0 (7)
Zhang et al. 9 where ij and ij are the stress tensor and strain tensor, respectively. a solid-like generalized Maxwell odel is used, E t is the relaxation odulus. If E t can be characterized by a Prony series odel: M t E t E E exp 1 (8) where E is a long ter equilibriu odulus; E are coponents of the relaxation odulus; are coponents of relaxation tie; and M is the total nuber of the Maxwell eleents (one Maxwell eleent is coposed of one elastic spring and one viscous dashpot connected in series). Siilarly, the relaxation bulk odulus and shear odulus can also be expressed by a Prony series odel as below: M t K t K K exp K 1 (9) M t G t G G exp G 1 (10) where K and G are the long ter equilibriu bulk odulus and shear odulus, respectively; K and G are coponents of the relaxation bulk and shear odulus, respectively; and K G are relaxation K G ties for the bulk and shear responses, respectively, and it is norally assued that. The relationships between K t, using a tie dependent Poisson s ratio (e.g., Gt and E t can be established by a convolution integral t ) which is also a aterial property for viscoelastic aterials. The laboratory easuring and viscoelastic odeling of the tie dependent Poisson s ratio for asphalt concrete can be found in the literature (Zhang et al. 2012a; Zhang et al. 2014). To siplify the analysis, a constant Poisson s ratio was assued for the aterial, and one can have (Winean and Rajagopal 2001): K t E t 312 0 (11)
Zhang et al. 10 G t E t 2 1 0 (12) where 0 is the Poisson s ratio that is assued as a constant in this study. Based on Eqs. 11 and 12, K, G, K, G, can be calculated once E t and 0 are known. In this study, the values of Poisson s ratio for the six asphalt ixtures are obtained fro the authors previous study (Zhang et al. 2014) and the data are shown in Table 3. Deterination of Relaxation Modulus Based on Dynaic Modulus Test Results In the paveent counity, the dynaic odulus test has becoe a standard test, e.g., siple perforance test, to characterize the viscoelastic perforance of asphalt ixtures (Witczak et al. 2002) and the results of dynaic odulus have been used as a ajor input to the MEPDG (ARA 2004). However, the relaxation odulus is ore widely used in the perforance prediction of asphalt concrete, such as in the finite eleent (FE) siulations and other coputational progras. Thus it is convenient to deterine the relaxation odulus based on the dynaic odulus test results, which have been extensively studied in the literature as discussed in the Introduction. To deterine the coefficients of the relaxation odulus in Eq. 8 based on the dynaic odulus test results, the following two relationships are coonly used: E E E (13) E 2 2 i1 1 M 2 2 2 2 i1 1 M E (14) Then the dynaic odulus and phase angle are deterined as below: = 2 2 E E E (15) 1 E tan E (16)
Zhang et al. 11 Based on the aster curves of the dynaic odulus and phase angle as constructed using Eqs. 2, and 4, one can deterine the Prony odel coefficients of the relaxation odulus, i.e., E, E and by iniizing the error between the aster curve values and the predictions using Eqs. 13 to 16. Note that, in this study, was controlled varying fro 10-6 to 10 4 s which covered the range of the relaxation tie obtained fro tie-teperature shifting, thus only E and E were used as the regression coefficients. The dashed lines in Figs 1 and 2 show the exaples for the dynaic oduli and phase angles predicted using the relaxation odulus Prony odel. Siilar figures to Figs 1 and 2 were also obtained for the other five asphalt ixtures, but not shown in the paper due to the length of the paper. The regressed relaxation odulus coefficients for the six asphalt ixtures are suarized in Table 3, which basically indicates a higher odulus (take instantaneous odulus, for exaple) as the asphalt ixture has a lower air void content or a longer aging period. It is noted that in this study, the dynaic odulus and phase angle were both utilized in the forulation of the regression iniization objective (as shown in Eq. 17) considering the fact that both dynaic odulus and phase angle are the two non-negligible coponents of the coplex odulus. Using only one coponent of coplex odulus ay not accurately represent the viscoelastic properties of the aterial. It is also noted that Eq. 17 noralizes the two regression variables, i.e., dynaic odulus and phase angle, to avoid the influences of the different unit and agnitude of the regression variables on the regression objectives (i.e., the error) so that the two different variables can be accounted for in a single regression forulation. The siilar regression techniques were also eployed in the literature (Levenberg and Shah 2008; Zhao et al. 2013). 1 error N 2 2 N E N i i Predicted Predicted 1 1 i1 E i i1 i MasterCurve MasterCurve (17) Weak For Equation-Based FE Modeling of Viscoelasticity
Zhang et al. 12 The stress and strain relations are derived for a solid-like generalized Maxwell odel to represent the viscoelastic behavior of the aterial. Then weak for PDE is introduced and ipleented step by step in Cosol to solve the viscous strains and predict the viscoelastic stress responses. Stress and Strain Analysis in Generalized Maxwell Model The echanical analogs of the Prony series odel (i.e., Eqs. 8, 9 and 10) can be represented by a solid-like generalized Maxwell odel which has one spring and M (nuber of) Maxwell eleent branches assebled in parallel. If characterizing the ulti-axial aterial properties using the bulk and shear oduli, the solid like generalized Maxwell odel is plotted in Fig. 3, in which ( 1, 2, M ) are the spring bulk and shear oduli of the -th Maxwell branch, and K and K G G are the bulk and shear coponents of the dashpot viscosity of the -th Maxwell branch. If the teperature of interest (T) differs fro the reference teperature (T R), the viscosity of the dashpot coplies with the K K G G tie-teperature superposition principle and one has T a T and T a T where a T is the tie-teperature shift factor. T R, T R The stress and strain are analyzed based on the echanical analogs of the solid-like generalized Maxwell odel in Fig 3. The total stress is the su of the stress in the single spring and the stresses in all Maxwell branches, while the strain reains the sae between the single spring and the Maxwell branches. Thus M ij ij ij 1 (18) (19) el vi ij ij ij ij ij where ij and ij are the stress and strain of the single spring, ij and ij are the stress and strain of the -th Maxwell branch. The constitutive relation for the single spring is odeled by an elastic Hooke s law as below:
Zhang et al. 13 1 ij kkij sij Kkk 2Geij (20) 3 For each Maxwell branch, the strain is decoposed into the elastic strain ( and the viscous strain ( vi ij el ij ) due to the spring ) due to the dashpot as in Eq. 19. It is noted that the elastic strain or viscous strain between different Maxwell branches can be different since the spring odulus and the dashpot viscosity vary between Maxwell branches. Furtherore, the elastic strain and the viscous strain are decoposed into voluetric and deviatoric coponents as below: 1 e 3 1 e 3 el el el ij kk ij ij vi vi vi ij kk ij ij (21) The stress within each Maxwell branch is expressed as the su of bulk stress and deviatoric stress (i.e., s ). Since the spring and the dashpot are connected in series in a Maxwell branch, the 1 ij 3 kk ij ij bulk stress or deviatoric stress of the spring is identical to the bulk stress or deviatoric stress of the dashpot for each Maxwell branch. Therefore, el K d kk kk 3Kkk 3 dt vi el G deij sij 2Geij 2 dt vi (22) el vi Expressing Eq. 19 in ters of bulk strain and deviatoric strain gives and kk kk kk K G e e e. Substituting these two equations and ak, ag into Eq. 22 gives: el vi ij ij ij T T vi dkk vi at kk kk 0 dt vi deij vi at eij eij 0 dt (23)
Zhang et al. 14 1 For a given strain history (e.g., t t e t ), Eq. 23 works as an ordinary differential ij 3 kk ij ij equation (ODE) which can be used to solve for the viscous strains (i.e., vi kk and vi e ij ) provided tie boundary conditions are known. Once the viscous strains are solved, the stress for each Maxwell branch is calculated by: 2 K G e e (24) vi vi ij kk kk ij ij ij The three-diensional stress-strain constitutive relation for the generalized Maxwell odel is obtained by cobining Eqs. 18, 20, and 24 and presented as: M vi vi K 2G e K 2G e e (25) ij kk ij kk kk ij ij ij 1 Then the solved viscous strains together with the given strain history and the aterial properties (bulk and shear oduli) are used in Eq. 25 to predict the stress responses due to the given strain history. Thus Eqs. 23 and 25 define the ulti-axial constitutive relations for a viscoelastic aterial and can be easily ebedded in a finite eleent odeling such as in a weak for PDE odeling. Weak For PDE Modeling of Viscoelasticity The weak for of a PDE is an iportant atheatical analysis ethod to find the solutions to the PDE. The original for PDE (e.g., Eq. 26) is a strong for which requires the PDE to be satisfied at every point and all ters ust be sufficiently continuous for derivatives. However, the PDEs doinating natural physics are not necessarily continuous and soe ters in the PDE are only defined over a sall region. A weak for PDE provides a better odel for these situations. Take a structural analysis as an exaple, Eq. 26 is the strong for PDE of the Poisson s equation. cu f (26) where u and u are gradient and divergence of a vector u such as a displaceent vector. f is a source, e.g., body force. c is a coefficient atrix such as a odulus atrix. Multiplying both sides of Eq. 26 by a test function v and integrating the both sides over a region Ω give:
Zhang et al. 15 c u vd fvd (27) If the solution to Eq. 26 is V and Eq. 27 holds for any v V, one can say Eq.27 is equivalent to Eq. 26 in the region Ω. Integrating the left side of Eq. 27 by parts and using the Gauss theore to convert a body integration to a surface integration and applying a surface traction boundary condition (e.g., n cu P), one can get: c v fvd PvdS 0 u (28) where P is the surface traction and S is the area of the surface of the region Ω, i.e.,. Eq. 28 is the weak for for the strong for PDE in Eq. 26. In fact, the weak for PDE can be regarded as a generalization of the virtual work principle where the test function v is equivalent to the virtual displaceent. Thus, if let v u where u is a sall virtual displaceent disturbance, one can derive that the left side of Eq. 28 gives the virtual work increent due to the sall displaceent disturbance. The advantages of using the weak for PDE are 1) lower continuity requireent on the solution copared to the strong for PDE as the weak for reduces the axiu order of the spatial derivatives, e.g., Eq. 26 is a second-order derivative while Eq. 28 is a first-order derivative; 2) the boundary conditions are clearly specified in the weak for PDE, such as the surface integration in Eq. 28. Because of these, a weak for PDE is particularly suitable for discretization and nuerical solution using the finite eleent ethod. Cosol Multiphysics provides strong solvers and user-friendly interfaces to find the solutions to any fors of PDEs using weak for odeling. In Cosol PDE odeling, users only need to input Weak Expressions which are the integrands of the weak for PDE. Cosol will then perfor a nuerical integration as 0 Weak Expressions d. For exaple, the spatial integrand of Eq. 28 is cu v fv which can be rewritten in Weak Expressions as
Zhang et al. 16 c ux test ux uy test uy uz test uz f test u (29) where v testu is pre-defined in Cosol (COMSOL 2013a) and test u test u is used. ux, uy and uz are spatial derivatives of the vector u. The sybol * stands for ultiplied by. A negative sign is added to the weak expressions as the Cosol convention has the integral in the right-hand side of the equation (COMSOL 2013a). In this study, the viscoelastic odeling is conducted in Cosol using a Weak For PDE odule and a Linear Elastic Solid odule to solve the viscous and elastic responses of the generalized Maxwell odel, respectively. It is noted that the existing viscoelastic odule in Cosol assues that the viscous part of the deforation is incopressible and all viscous deforations are attributed to the shear stresses (COMSOL 2013b). Thus the voluetric deforation is purely elastic, which is not the case for ost of the geoaterials such as soils, sands and asphalt ixtures. The viscous bulk and shear strains in Eqs. 23 have as any coponents as the nuber of the strain coponents of a solid deforing proble. Thus the viscous bulk and shear strains are treated as additional degrees of freedo of the proble and can be solved using weak for PDE odeling. Descriptions of the key odeling steps in Cosol are shown as below: First, define the aterial properties in Cosol. The aterial properties including E, E, and 0 shown in Table 3 are input as Paraeters, and a T is input as a Variable that is a function of teperature, e.g., Eq. 3. The teperature can be defined as a Paraeter too or solved using a theral analysis of the structure. Then these aterial properties are converted to bulk and shear odulus coponents, i.e., K, G, K, and G using Eqs. 11 and 12. Second, define Dependent Variables in Weak For PDE odule. The viscous voluetric and shear strains (i.e., vi kk and vi e ij ) are treated as new Dependent Variables and input in the Weak For PDE odule. It is noted that 1,2, M and ij 11,12,13,22,23,33. Thus the total nuber of the
Zhang et al. 17 dependent variables for vi kk is M that is the nuber of Maxwell branches. The total nuber of the dependent variables for vi e ij is 6*M. In this study, u 1 and u2ij are assigned in the Weak For PDE odule as the dependent variables to represent vi kk and vi e ij, respectively. Third, input the Weak Expressions in the Weak For PDE odule. In this study, vi kk and vi e ij are Dependent Variables which do not include spatial derivatives as shown in Eq.23. Thus cu v 0. The source function f in Eq. 29 is deterined as the left side of Eq. 23. Thus the Weak Expressions for Eq. 23 are represented as the second ter of Eq. 29 (i.e., f test u ), which are at * * u1t u1 solid. eelvol* test u1 at * * u2ijt u2 ij solid. eeldevij* test u2ij (30) where 1, 2, M, ij 11,12,13,22,23,33 ; ut 1 and u2ijt are tie derivatives of the dependent variables u 1 and u2ij, which represent the coponents of the viscous voluetric and shear strains, respectively. solid. eelvol and solid. eeldevij are elastic voluetric strain and elastic deviatoric strain coponents in a long ter, respectively, which are deterined by the Linear Elastic Solid odule representing the strain responses of the single spring in the generalized Maxwell odel. Fourth and last, update stresses due to viscous responses. Once the viscous strains are deterined, the stress for each Maxwell branch can be calculated using Eq. 24 which is expressed in Cosol as:. 1 2. 2 Sij K solid eelvol u G solid eeldevij u ij (31) where Sij ( 1,2, M, and ij 11,12,13,22,23,33) are defined in Variable and expressed using Eq. 31. Then the total stress is updated based on Eq. 25 and expressed in Cosol as: M solid. Silij Sij (32) 1 where solid. Silij ( ij 11,12,13,22,23,33) is the initial stress coponents which are used to represent the stresses due to viscous strains and will be autoatically added to the total stresses in Cosol.
Zhang et al. 18 Verification and Validation of Equation Based FE Model The weak for PDE odeling of viscoelasticity in Cosol is verified through coparing the odeling results in Cosol and that in Abaqus using the sae aterial properties and loading configurations. It is further validated by coparing the weak for odeling results with laboratory test results of the asphalt ixtures. Verification by Coparing with Abaqus Viscoelastic Siulations The siulations in both Cosol and Abaqus platfors are perfored based on identical aterial properties and loading configurations. The responding strains are coputed by weak for PDE odeling of viscoelasticity in Cosol and by the viscoelastic odule in Abaqus. The siulated saple is an axisyetric cylinder with 100 diaeter and 200 height. The aterial properties are deterined based on one of the tested asphalt ixtures (i.e., the one with 4% air void content and 6 onth aging in Table 3). The aterial properties input in Cosol are E, E, and, which are directly obtained 0 fro Table 3. According to the user s anual (ABAQUS 2010), the aterial properties input in Abaqus include E, which are used as long-ter elastic odulus and Poisson s ratio, and 0 g, i k, and i for i the Prony viscoelastic odel, where M gi ki Ei E Ei i1 and i in which i, 1,2, M. The siulation tests are perfored at the reference teperature, thus 1 a for both platfors. The first siulation is a creep and recovery test with confineent. In both platfors, the sae loading configurations are applied on the saple: a hydrostatic confining pressure of 100 kpa is applied for 80 s; and a deviatoric stress is raped up to 200 kpa in 1 s and kept constant for the first 30 s and then unloaded to zero within 1 s and aintained at zero until 80 s. Fig. 4 copares the axial and radial strains deterined by the two platfors in a stress/strain vs. log (tie) coordinate. It is found that the strains predicted by the two platfors atched well with each other (relative error < 0.1% at each calculation point) in every stage of the test including loading rap, creep, unloading, and recovery. T
Zhang et al. 19 The second siulation is a cyclic load test with confineent. In both platfors, the sae loading configurations are applied on the saple: a hydrostatic confining pressure of 100 kpa is applied; and a deviatoric cyclic stress is loaded at a frequency of 0.05 Hz with an aplitude of 200 kpa. The test is terinated at 100 s (5 cycles). Fig. 5 shows the stresses and strains with loading tie which also atch well between the siulation results of the two platfors (relative error < 0.1% at each calculation point). Figs 4 and 5 verify the consistency and accuracy of the weak for PDE-based odeling of viscoelasticity in Cosol, which can produce identical results with other finite eleent odeling platfors providing all siulation conditions reain the sae. It is noted that, at the sae data acquisition rate, both Cosol weak for PDE-based viscoelastic odeling and Abaqus ebedded viscoelastic odeling can coplete the two siulations within reasonable coputational tie (i.e., approxiately, Cosol used 2 inutes and Abaqus used 1 inute). However, once further echaniss such as plasticity, fracture, and daage are coupled with viscoelasticity in Abaqus, the existing ebedded Abaqus viscoelastic odule will not be workable and user-defined prograing is needed for the coupled constitutive odeling. In contrast, the PDE-based odeling in Cosol will reain workable and copatible with PDE-based odeling of the other echaniss, which will be presented in future publications. It ust be ephasized that the tie and effort required for Cosol prograing based on user-interface inputs is super less than that required for Abaqus user-defined prograing based on coputer coding languages such as FORTRAN or C. Further evaluations of the coputational efficiency for PDE-based FE odeling on coupled echaniss are being perfored for a coplex aterial. Validation by Coparing with Laboratory Results More validations of the weak for PDE odeling of viscoelasticity are perfored by coparing the siulation results with the laboratory test results. Table 1 lists the lab tests that are used in validation including 1) creep tests at three teperatures and 2) cyclic load tests. The loading paraeters are also shown in Table 1. Axial strains in the two validation tests were recorded for the six types of asphalt ixtures which are copared with the viscoelastic predictions based on the weak for PDE-FE
Zhang et al. 20 siulations in Cosol. The aterial properties such as relaxation odulus and tie-teperature shift factor used as inputs to the FE siulations are deterined based on the dynaic odulus tests on the sae ixtures, which are shown in Table 3. Figs. 6-11 show the coparisons between the weak for PDE-FE predictions and the laboratory results of the creep tests at three teperatures for the six asphalt ixtures with two air void contents and three ageing periods, respectively. In the figures, the curves with the legend of Test Max and Test Min indicate that the strain curves are the axiu and iniu easureents of the three replicates at one teperature, respectively. It is found fro the coparisons for all of the six types of the asphalt ixtures that the predicted strains substantially atch with the easureents of the lab tests. It is noted that a lower static stress was applied at a higher teperature as shown in Table 1, thus the strain easured at a higher teperature can be less than that at a lower teperature (e.g., the strain at 40 C is less than that at 25 C in Figs. 8, 10, and 11). Figs. 12-17 show the coparison between the weak for PDE-FE predictions and the laboratory results of the cyclic load tests for the six asphalt ixtures with two air void contents and three ageing periods, respectively. In the figures, the easured total strains are decoposed into static strain and cyclic strain. The static strain is a ean strain based on which the total strain oscillates and the cyclic strain is the oscillation part of the total strain. Thus, the total strain is the su of the static strain and the cyclic strain. It is found, fro the coparisons for the six types of the asphalt ixtures, that the laboratory easured total strain is greater than the weak for PDE-FE predicted total strain. The possible reasons could be 1) a higher stress was used in the cyclic validation test than the dynaic odulus tests whose results were eployed to deterine the aterial properties as the FE inputs. Thus the nonlinear viscoelastic responses due to the higher stress were not be captured by the linear viscoelastic odel used in the FE odeling; 2) soe aount of plastic deforation ight be introduced into the saple which was not represented in the FE odeling. It ust be noted that the nonlinear viscoelasticity, plastic deforation and viscoplastic deforation can also be odeled by weak for PDE ethod, which have been achieved in the authors current work and will be presented in future publications.
Zhang et al. 21 Nevertheless, the cyclic strains reain alost identical within each load cycle between the easured and the predicted ones for all of the six types of the asphalt ixtures, as illustrated in Figs. 12-17. The dynaic odulus and phase angle were calculated based on the cyclic strains predicted by the weak for PDE-FE odeling and easured by the laboratory tests, respectively. As illustrated in Fig. 18, the dynaic odulus and phase angle are found to be coparable and atched well between the laboratory easureents and the PDE odeling predictions. This finding validates the accuracy of the viscoelastic odeling using the weak for PDE odeling in the FE progra. In su, Figs. 6-18 deonstrate that the weak for PDE-based viscoelastic odeling can reliably predict the responses of the viscoelastic aterial such as the asphalt ixtures. It can also be concluded fro the coparisons that the viscoelastic aterial properties deterined fro dynaic odulus tests and aster curves basically provide accurate odel inputs for the viscoelastic siulations in the FE odeling. Suary and Conclusions The constitutive behavior of viscoelastic aterials (asphalt ixtures were taken as exaples) was odeled by a weak for partial differential equation (PDE) based finite eleent (FE) odeling. A solidlike generalized Maxwell odel was used to represent the deforation echanis of the viscoelastic aterials. The viscoelastic stress and strain relations were derived and ipleented in weak for PDE odeling in the Cosol Multiphysics progra, which was verified by coparing virtual laboratory test siulations with Abaqus viscoelastic siulations. Creep and recovery tests and cyclic loading tests, both with confineents, were siulated in Cosol and Abaqus using the sae loading configurations and aterial property inputs. Both produced the identical results in ters of axial and radial strain responses. The weak for PDE odeling of viscoelasticity was further validated by coparing real laboratory test results with Cosol FE predictions. The aterial viscoelastic properties such as the Prony series odel coefficients for relaxation odulus were deterined by converting fro dynaic odulus and phase angle aster curves that were constructed using dynaic odulus test results. Strains in creep tests at
Zhang et al. 22 three teperatures and cyclic load tests were predicted and found to be coparable with the easureents of the real laboratory tests. It was deonstrated that the weak for PDE based FE odeling can serve as an efficient ethod to ipleent new constitutive odels and free engineers fro user subroutine prograing. Specifically, 1) the viscoelastic FE odeling using the weak PDE forulation is applicable for all linear viscoelastic aterials, and asphalt ixtures were taken as an exaple in this study; 2) no user subroutine prograing is needed in order to ipleent a new constitutive odel using the weak for PDE ethod. The users only need to define their variables and input the weak for integrand expressions in the Weak For PDE odule interface; 3) the weak for PDE odeling allows the users to focus on aterial constitutive characterizations, but not to be bothered by the nuerical issues such as convergence, iteration technologies, etc. Recoended future work includes extending the weak for PDE odeling to odeling coplicated physics/echaniss such as nonlinearity, plasticity, fracture, daage, theral and oisture diffusion and evaluate their coupling and interaction effects. These odels are being under perfored in the authors current work and will be presented in future publications. References ABAQUS (2010). Abaqus Analysis User's Manuals, Hibbit, Karlsson & Sorensen Inc., Pawtucket, Rhode Island. ARA (2004). "Guide for Mechanistic-Epirical Design of New and Rehabilitated Paveent Structures." National Cooperative Highway Research Progra (NCHRP), Report 1-37A, by Applied Research Associates, Transportation Research Board, National Research Council, Washington, D.C. Biligiri, K. P., Kaloush, K., and Uzan, J. (2010). "Evaluation of Asphalt Mixtures' Viscoelastic Properties Using Phase Angle Relationships." International Journal of Paveent Engineering, 11(2), 143-152.
Zhang et al. 23 COMSOL (2013a). Cosol Multiphysics Reference Manual, Version 4.3b, Available at: www.cosol.co. COMSOL (2013b). Structural Mechanics Module User's Guide, Version 4.3b, Available at: www.cosol.co. Darabi, M. K., Abu Al-Rub, R. K., Masad, E. A., Huang, C.-W., and Little, D. N. (2012). "A Modified Viscoplastic Model to Predict the Peranent Deforation of Asphaltic Materials under Cyclic- Copression Loading at High Teperatures." International Journal of Plasticity, 35, 100-134. Findley, W. N., Lai, J. S., and Onaran, K. (1989). Creep and Relaxation of Nonlinear Viscoelastic Materials with an Introduction to Linear Viscoelasticity, Dover Publication, Inc., Mineola, New York. Francken, L., and Verstraeten, J. (1998). "Interlaboratory Test Progra on Coplex Modulus and Fatigue." RILEM Report 17, Bituinous Binders and Mixes, London, UK, 182-215. Gibson, N. H., Schwartz, C. W., Schapery, R. A., and Witczak, M. W. (2003). "Viscoelastic, Viscoplastic, and Daage Modeling of Asphalt Concrete in Unconfined Copression." Transportation Research Record: Journal of the Transportation Research Board, No. 1860, Transportation Research Board of the National Acadeies, Washington, DC, 3-15. Hu, S., and Zhou, F. (2010). "Developent of a New Interconversion Tool for Hot Mix Asphalt (Ha) Linear Viscoelastic Functions." Canadian Journal of Civil Engineering, 37(8), 1071-1081. Huang, C. W., Abu Al-Rub, R. K., Masad, E. A., and Little, D. N. (2011). "Three-Diensional Siulations of Asphalt Paveent Peranent Deforation Using a Nonlinear Viscoelastic and Viscoplastic Model." Journal of Materials in Civil Engineering, 23(1), 56-68. Katicha, S., Flintsch, G., Loulizi, A., and Wang, L. (2008). "Conversion of Testing Frequency to Loading Tie Applied to the Mechanistic-Epirical Paveent Design Guide." Transportation Research Record: Journal of the Transportation Research Board, No. 2087, Transportation Research Board of the National Acadeies, Washington, DC, 99-108.
Zhang et al. 24 Levenberg, E., and Shah, A. (2008). "Interpretation of Coplex Modulus Test Results for Asphalt- Aggregate Mixes." J. Test. Eval., 36(4), 326-334. Levenberg, E., and Uzan, J. (2004). "Triaxial Sall-Strain Viscoelastic-Viscoplastic Modeling of Asphalt Aggregate Mixes." Mechanics of Tie-Dependent Materials, 8(4), 365-384. Mun, S., Chehab, G. R., and Ki, Y. R. (2007). "Deterination of Tie-Doain Viscoelastic Functions Using Optiized Interconversion Techniques." Road Mater. Paveent Des., 8(2), 351-365. Park, S. W., and Schapery, R. A. (1999). "Methods of Interconversion between Linear Viscoelastic Material Functions. Part I a Nuerical Method Based on Prony Series." International Journal of Solids and Structures, 36(11), 1653-1675. Pellinen, T. K., Bonaquist, R. F., and Witczak, M. W. (2002). "Asphalt Mix Master Curve Construction Using Sigoidal Fitting Function with Non-Linear Least Squares Optiization." 15th Engineering Mechanics Division Conference, Colubia University, New York, United States. TxDOT (2004). "Standard Specifications for Construction and Maintenance of Highways, Streets, and Bridges." Texas Departent of Transportation, Austin, TX. TxDOT (2008). "Test Procedure for Design of Bituinous Mixtures." TxDOT Designation: Tex-204-F, Texas Departent of Transportation, Austin, TX. Winean, A. S., and Rajagopal, K. R. (2001). Mechanical Response of Polyers, an Introduction, Cabridge University Press, New York, NY. Witczak, M. W., Kaloush, K., Pellinen, T., and El-Basyouny, M. (2002). "Siple Perforance Test for Superpave Mix Design." National Cooperative Highway Research Progra (NCHRP) Report 465, Transportation Research Board, National Research Council, Washington, D.C. Zhang, Y., Luo, R., and Lytton, R. L. (2014). "Anisotropic Characterization of Crack Growth in Tertiary Flow of Asphalt Mixtures in Copression." Journal of Engineering Mechanics, 140(6), in press. Zhang, Y., Luo, R., and Lytton, R. L. (2012a). "Anisotropic Viscoelastic Properties of Undaaged Asphalt Mixtures." Journal of Transportation Engineering, 138(1), 75-89.
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Zhang et al. 26 List of Tables Table 1 Testing aterials, protocol and paraeters for asphalt concrete. Asphalt Mixtures Binder: NuStar (PG67-22) Air Void: 4% and 7% Aging Periods at 60 C for: 0, 3, 6 onths (Note: in Loading Paraeters colun, 0-4% stands for the ixtures with 0- onth aging and 4% air void content, and siilar notation for the rests) Test Methods Dynaic odulus tests Copressive creep tests Cyclic load tests Teperatures 10 C 25 C 40 C 55 C 10 C 25 C 40 C 40 C Loading Paraeters 25 Hz 10 Hz 5 Hz 1 Hz 0.5 Hz 0.1 Hz 300 kpa (0-4%) 200 kpa (0-7%) 500 kpa (3-4%) 400 kpa (3-7%) 500 kpa (6-4%) 500 kpa (6-7%) 60 kpa (0-4%) 40 kpa (0-7%) 120 kpa (3-4%) 80 kpa (3-7%) 200 kpa (6-4%) 120 kpa (6-7%) 20 kpa (0-4%) 20 kpa (0-7%) 30 kpa for rests 55 kpa (0-4%) 50 kpa (0-7%) 65 kpa (3-4%) 60 kpa (3-7%) 80 kpa (6-4%) 65 kpa (6-7%) All at 1 Hz Results Obtained Dynaic odulus; Phase angle Axial strains vs. loading tie Axial strains vs. loading cycles Purpose of Tests Deterination of viscoelastic odel paraeters Validation of equation based FE odeling of viscoelasticity
Zhang et al. 27 Table 2 Master curve odel coefficients of dynaic oduli and phase angles and corresponding tieteperature shift factor odel coefficients for the six different asphalt ixtures Binder NuStar Binder (PG67-22) Aging Periods 0 onth 3 onths 6 onths Air Void Contents 4% 7% 4% 7% 4% 7% Master Curve Coefficients for E* (MPa) a T for E* Master Curve Coefficients for ( ) a T for δ 1.254 0.756 1.562 1.195 1.583 1.533 α 3.381 3.723 3.028 3.259 3.082 2.995 η -0.345-0.203-0.361-0.483-0.560-0.381 γ 0.486 0.424 0.508 0.463 0.432 0.469 a 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 b -0.193-0.160-0.170-0.171-0.194-0.184 c 50.62 40.17 43.32 43.85 51.09 47.77 ax 31.9 32.4 32.2 33.5 32.3 31.7 f 0 0.2016 0.6557 0.1146 0.0563 0.0371 0.1212 β 0.0187 0.0277 0.0255 0.0225 0.0220 0.0241 a 0.0010 0.0011 0.0010 0.0011 0.0010 0.0010 b -0.712-0.746-0.683-0.739-0.706-0.714 c 125.0 125.7 115.8 123.4 123.1 125.6