Practices for Unbound Aggregate Pavement Layers (2013)

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172 APPENDIX D Review of Current Resilient Modulus Models REvIEw of CuRRENt REsIlIENt MoDulus MoDEls Resilient models of granular materials increase with increas- ing stress states (stress-hardening), especially with confining pressure and/or bulk stress, and slightly with deviator stress (Lekarp et al. 2000a). Resilient behavior of unbound aggre- gate materials can be reasonably characterized by using stress dependent models which express the modulus as nonlinear functions of stress states. Such a characterization model must include in the formulation the two triaxial stress conditions, i.e., the confining pressure s3 and the deviator stress sd or, the applied mean pressure p and the deviator stress q, to account for the effects of both confinement and shear load- ing. The model parameters are traditionally obtained from the multiple regression analyses of the repeated load triaxial test data. In the following subsections, currently available models are discussed in detail. Confining Pressure Model Seed et al. (1967) introduced a simple model for the resil- ient modulus relating it to confining stresses. They con- ducted repeated load triaxial tests on sands and gravels, and expressed the results in the form: M KR K( )= σ (D-1)1 3 2 where s3 is confining pressure and K1 and K2 are regression analysis constants from experimental data. This model, how- ever, did not give high correlation coefficients. K- Model One of the most popular models was developed by Hicks and Monismith (1971). This model, known as the K-q model, has been the most widely used for modeling modulus as a func- tion of stress state applicable to granular materials. M KR n( )= θ (D-2) where q is bulk stress = (s1 + 2s3) or (sd + 3s3), sd is deviator stress = (s1 - s3) and K, n are regression analysis constants obtained from experimental data. Even though it is a popu- lar model, the K-q model has a shortcoming since it fails to adequately distinguish the effect of shear behavior. The impact of neglecting shear stress was illustrated by Uzan (1985) and the K-q model predicted an increasing resilient modulus as axial strains increased in contrast to the test data that showed a decrease in resilient modulus. According to Brown and Pappin (1981), the K-q model is not able to handle volumetric strains and therefore can only be applicable to a very limited stress range when confining pressure (s3) is less than deviator stress (sd). In addition, Nataatmadja (1989) reported that this model was not dimen- sionally satisfied as K had the same dimension with resilient modulus (MR). Despite of this weakness, the K-q model is still being used frequently for granular materials because of its simplicity. shackel’s Model After conducting repeated load triaxial tests on a silty-clayey soil, Shackel (1973) developed the following resilient modu- lus model in terms of octahedral shear stress and octahedral normal stress M KR K K ( ) ( )= τ σ     (D-3)1 oct oct 2 3 where Ki are material regression constants obtained from tri- axial test data. He proposed that his model was valid for both granular materials and cohesive soils. Since the model was defined in terms of stress invariants, it was considered to be one of the early advanced nonlinear models. I( )σ = σ + σ + σ =13 1 3 (D-4)oct 1 2 3 1 I I( ) ( ) ( ) ( )τ = σ − σ + σ − σ + σ − σ  = − 1 3 2 3 3 (D-5) oct 1 2 2 2 3 2 1 3 2 1 2 1 2 2 1 2 where I1 is the first stress invariant and I2 is the second invariant. Bulk-shear Modulus Model Boyce (1980) developed a nonlinear material model based on the secant bulk modulus (K) and the shear modulus (G). He found the influence of mean normal stress to resilient strain and the relationships were given as: K K pi n= ( )− (D-6)1 G G pi n= ( )− (D-7)1

173 where Ki is an initial value of bulk modulus, Gi is an initial value of shear modulus and n is a constant less than 1.0. Boyce (1980) also updated his model to satisfy Maxwell’s reciproc- ity theorem. Accordingly, the second order partial derivatives of a stress potential function are independent of the order of differentiation of volumetric and deviatoric stress compo- nents. Expressions of the moduli were given as follows: K K p q p i n = − β  ( )− 1 (D-8) 1 2 G G pi n= ( )− (D-9)1 where b is n KG i i ( )−1 6 , p is mean stress, q is deviator stress. In this model, the volumetric strains and deviatoric strains are related to mean normal stress (p) and deviatoric stress (q) as follows: K p q pv i n( )ε = − β  1 1 (D-10) 2 G p pqq i n( )ε =  13 (D-11) where ev and eq are the volumetric and shear strains, respec- tively. This model can successfully predict measured strains from the initial bulk and shear moduli and the applied stress states. uzan Model Since the K-q model was not sufficient to describe the shear behavior of granular materials, Uzan (1985) made a modifi- cation to this model. An additional deviator stress component that includes the effect of shear behavior was shown to be in good agreement with test results. M KR K d K( ) ( )= θ σ (D-12)1 2 3 where q is bulk stress = (s1 + 2s3) or (sd + 3s3), sd is deviator stress = (s1 – s3), and K1, K2, and K3 are regression analysis constants obtained from experimental data. Considering in the formulation both bulk and deviator stresses, the Uzan model overcomes the deficiency of the K-q model that did not include shear effects and fits better with the test data than the K-q model. This was shown to be especially important when confining stress values applied on the specimen were larger than the applied deviator stresses during testing. lade and Nelson Model Lade and Nelson (1987) proposed an elastic material model based on energy conservation for closed-loop strain path. In this model, isotropic and nonlinear assumption was used in the elastic behavior of granular materials. With the assump- tion of energy conservation, the work during any arbitrary closed path stress cycle was written as:  W dW IK dI dJ G∫ ∫ ( )= = + =9 2 0 (D-13)cycle cycle 1 1 2 cycle where K is bulk modulus, G is shear modulus, I1 is the first stress invariant, and J2 is the second deviatoric invariant. The first order partial differential equation from Equation 5.24 is derived as follows: I K K J J G G I ∂ ∂ = ∂ ∂9 (D-14) 1 2 2 2 2 1 After substituting K E( )= − ν3 1 2 and G E ( )= + ν2 1 2 into Equation (D-14), the equation can be expressed in terms of E (Young’s modulus). J E J R I E I ∂ ∂ = ∂ ∂ 1 1 (D-15) 2 2 1 1 where R ( )( )= + ν − ν 6 1 1 2 . The final form of the stress-dependent modulus equation was proposed as follows: (D-16)1 2 2 =   +     λ E Mp I p R J pa a a where pa is atmospheric pressure and M and l are material constants. This Lade and Nelson model did not give good results due to the energy conservation principles adopted in this hyperelastic material model formulation since energy dissipates when granular materials are subjected to repeated loading. universal octahedral shear stress Model Witczak and Uzan (1988) proposed an improvement over the Uzan (1985) model by replacing the deviator stress term with octahedral shear stress. This model also used atmospheric pressure (pa) to normalize the bulk and shear stress terms to make the model parameters dimensionless. M K p I p pR a a K a K =        1 1 2 3 17τoct D-( ) where I1 is first stress invariant = (s1 + s2 + s3) or (s1 + 2s3), toct is octahedral shear stress = ¹⁄³{(s1 - s2)2 + (s1 - s3)2 + (s2 - s3)2}1/2 = 23 (s1 - s2), pa is atmospheric pressure, and K1, K2, and K3 are regression constants obtained from experimental data.

174 Itani Model An improved correlation between the resilient modulus and various stress state variables, such as deviator stress, mean stress, confining stress, and axial strain, was obtained from multiple regression analyses. Itani (1990) proposed the mate- rial model with a high correlation coefficient (R2 = 0.96) as follows: M K p pR a a K d K K( ) ( )= σ  σ σθ (D-18)1 3 2 3 4 where sq = (s1 + s2 + s3) = (s1 + 2s3), sd = s1-s3, s3 is con- fining stress, pa is atmospheric pressure, and K1, K2, K3 and K4 are multiple regression constants obtained from triaxial tests. With the goal of developing improved models to character- ize the resilient modulus, laboratory test data from different aggregate gradations were used in this study. Itani (1990) con- cluded that this model was useful to predict resilient modulus, although there was a slight multi-colinearity problem. This is due to the fact that two independent triaxial stress states are expressed in three stress terms in this equation. Crockford et al. Model Crockford et al. (1990) developed a resilient modulus model which was expressed as a function of volumetric water con- tent, suction stress, octahedral shear stress, unit weight of material normalized by the unit weight of water, and the bulk stress. The model was proposed as follows: M V VR W t W = +     ( )  β θ τ γγ β β β 0 1 3 2 4 Ψ oct D-( 19) where b0, b1, b2, and b3 are material constants, y is suction stress, V V W t is volumetric water content, toct is octahedral shear stress, and W γ γ is unit weight of material normalized by the unit weight of water. When eliminating moisture term and the normalized unit weight term, this equation simplifies to the octahedral shear stress model of Witczak and Uzan (1988). ut-Austin Model UT-Austin model was developed by Pezo (1993) with a good agreement of the resilient modulus data from the repeated load triaxial test. This model predicts the response vari- able, axial strain, instead of the resilient modulus using the applied confining and deviator stresses. Since this model is independent of the response variables, it is very useful for any condition. M a a KR D r d d b c d b c d K K( ) ( ) ( )= σ ε = σ σ σ = σ σ = σ σ− − 1 (D-20) 3 1 3 1 3 2 3 where sd is deviator stress = (s1 - s3), s3 is confining stress and K1, K2 and K3 are regression analysis constants obtained from experimental data. lytton Model Lytton (1995) proposed that the principles of unsaturated soil mechanics could be applied to the universal octahedral shear stress model (Witczak and Uzan 1988) because unbound aggregate materials in pavements are normally unsaturated. To evaluate the effective resilient properties of unsaturated granular materials, Lytton added a suction term to the univer- sal octahedral shear stress model as follows: M K p I fh p pR a m a K a K = − θ  τ 3 (D-21)1 1 oct 2 3 where pa is atmospheric pressure, I1 is first stress invariant = (s1 + s2 + s3), is volumetric water content, f is function of the volumetric water content, hm is matric suction, toct is octa hedral shear stress = ¹⁄³{(s1 - s2)2 + (s1 - s3)2 + (s2 - s3)2}1/2, and K1, K2, and K3 are multiple regression constants obtained from triaxial tests. NCHRP 1-37A Mechanistic Empirical Pavement Design Guide (MEPDG) Model In the MEPDG (NCHRP 1-37A, 2004), a generalized con- stitutive model was adopted to characterize the resilient modulus of unbound aggregates. This equation combines both the stiffening effect of bulk stress and the softening effect of shear stress. Thus, the values of K2 should be posi- tive, since increasing the bulk stress produces a stiffening of the material. However, K3 should be negative to show a softening effect. To properly find the model constants, mul- tiple correlation coefficients determined from test results have to exceed 0.90. Note that this model is proposed for use with both unbound aggregates and fine-grained sub- grade soils. M K p p pR a a K a K = θ  τ + 1 (D-22)1 oct 2 3 where is the bulk stress = s1 + s2 + s3, toct is octahedral shear stress = ¹⁄³{(s1 - s2)2 + (s1 - s3)2 + (s2 - s3)2}1/2, pa is atmo- spheric pressure, and K1, K2, and K3 are constants obtained from experimental data. REfERENCEs Boyce, J.R., “A Nonlinear Model for the Elastic Behavior of Granular Materials Under Repeated Loading,” Inter- national Symposium on Soils Under Cyclic and Transient Loading, Swansea, 1980. Brown, S.F. and J.W. Pappin, “Analysis of Pavements with Granular Bases,” Transportation Research Record 810,

175 Transportation Research Board, National Research Coun- cil, Washington D.C., 1981, pp. 17–23. Crockford, W.W., L.J. Bendana, W.S. Yang, S.K. Rhee, and S.P. Senadheera, Modeling Stress and Strain States in Pavement Structures Incorporating Thick Granular Layers, Final Report, Contract F08635-87-C-0039, The Texas Transportation Institute, The Texas A&M Univer- sity, College Station, 1990. Hicks, R.G. and C.L. Monismith, “Factors Influencing the Resilient Properties of Granular Materials,” Transpor- tation Research Record 345, Transportation Research Board, National Research Council, Washington D.C., 1971, pp. 15–31. Itani, S.Y., Behavior of Base Materials Containing Large- Sized Particles, PhD Thesis, School of Civil and Envi- ronmental Engineering, Georgia Institute of Technology, Sep. 1990. Lade, P.V. and R.B. Nelson, “Modeling the Elastic Behavior of Granular Materials,” International Journal for Numer- ical and Analytical Methods in Geomechanics, Vol. 11, 1987, pp. 521–542. Lytton, R.L., “Foundations and Pavements on Unsaturated Soils,” 1st International Conference on Unsaturated Soils, Paris, France, 1995. Nataatmadja, A., Variability of Pavement Material Param- eters under Repeated Loading, PhD Dissertation, Monash University, Australia, 1989. NCHRP 1-37A, Mechanistic-Empirical Pavement Design Guide, Draft Report, Part 2 Design Inputs, 2004. Pezo, R.F., “A General Method of Reporting Resilient Modulus Tests of Soils—A Pavement Engineer’s Point of View,” Paper No: 93082, 72nd Annual Meeting of the Transportation Research Board, Washington, D.C., 1993. Seed, H.B., F.G. Mitry, C.L. Monismith, and C.K. Chan, NCHRP Report 35: Prediction of Flexible Pavement Deflections from Laboratory Repeated Load Tests, High- way Research Board, National Research Council, Wash- ington, D.C., 1967. Shackel, B., “Repeated Loading of Soils–A Review,” Austra- lian Road Research, Vol. 5, No. 3, 1973, pp. 22-49. Uzan, J., “Characterization of Granular Material,” Transpor- tation Research Record 1022, Transportation Research Board, National Research Council, Washington D.C., 1985, pp. 52–59. Witczak, M.W. and J. Uzan, The Universal Airport Pavement Design System, Report I of V: Granular Material Charac- terization, Department of Civil Engineering, University of Maryland, College Park, Md., 1988.