A generalized Drucker–Prager viscoplastic yield surface model for asphalt concrete

Abstract

A generalized Drucker–Prager (GD–P) viscoplastic yield surface model was developed and validated for asphalt concrete. The GD–P model was formulated based on fabric tensor modified stresses to consider the material inherent anisotropy. A smooth and convex octahedral yield surface function was developed in the GD–P model to characterize the full range of the internal friction angles from 0° to 90°. In contrast, the existing Extended Drucker–Prager (ED–P) was demonstrated to be applicable only for a material that has an internal friction angle less than 22°. Laboratory tests were performed to evaluate the anisotropic effect and to validate the GD–P model. Results indicated that (1) the yield stresses of an isotropic yield surface model are greater in compression and less in extension than that of an anisotropic model, which can result in an under-prediction of the viscoplastic deformation; and (2) the yield stresses predicted by the GD–P model matched well with the experimental results of the octahedral shear strength tests at different normal and confining stresses. By contrast, the ED–P model over-predicted the octahedral yield stresses, which can lead to an under-prediction of the permanent deformation. In summary, the rutting depth of an asphalt pavement would be underestimated without considering anisotropy and convexity of the yield surface for asphalt concrete. The proposed GD–P model was demonstrated to be capable of overcoming these limitations of the existing yield surface models for the asphalt concrete.

Appendix: derivation of function \(\rho \left( {d,\theta } \right)\) in GD–P model

The resistance of an asphalt concrete to permanent deformation is provided by the material cohesion of asphalt mastic and the friction of aggregate skeleton. In general, material cohesion is an isotropic property and it only affects the size of a yield surface. The friction of the aggregate skeleton contributes the size and shape of the yield surface. Since $\rho \left( {d,\theta } \right)$ in GD–P model determines the shape of the yield surface on the octahedral plane, it should depend only on the aggregate skeleton of the mixture which behaves similarly to unbound aggregates or sands. This thinking urges the authors to take advantage of the yield surface model for cohesionless sand to model the yielding of aggregate skeleton of the asphalt concrete.

Matsuoka–Nakai [26, 27] model has been used to model the yield surface of the cohesionless sands and it is an inherently smooth and convex yield surface. Using the modified stress invariants, the Matsuoka–Nakai model is expressed as:

$$\frac{{\bar{I}_{1} \bar{I}_{2} }}{{\bar{I}_{3} }} = k$$

(25)

where $\bar{I}_{1}$ ($= \bar{\sigma }_{kk}$), $\bar{I}_{2}$ ($= \tfrac{1}{2}\left( {\bar{\sigma }_{ii} \bar{\sigma }_{jj} - \bar{\sigma }_{ij} \bar{\sigma }_{ji} } \right)$) and $\bar{I}_{3}$ ($= \det \left( {\bar{\sigma }_{ij} } \right)$) are first, second and third invariants of the modified stress tensor ($\bar{\sigma }_{ij}$). Parameter k can be expressed in terms of the material internal friction angle (or the extension ratio d) as follows [3]:

$$k = \frac{{9 - \sin^{2} \phi }}{{1 - \sin^{2} \phi }} = \frac{9d}{{\left( {2d - 1} \right)\left( {2 - d} \right)}}$$

(26)

To convert M–N model to an expression in terms of $\sqrt {\bar{J}_{2} }$ and $\bar{I}_{1}$ that are the first order of the stress, $\bar{I}_{2}$ and $\bar{I}_{3}$ are written as:

$$\bar{I}_{2} = \frac{1}{3}\bar{I}_{1}^{2} - \bar{J}_{2}$$

(27)

$$\bar{I}_{3} = \bar{J}_{3} - \frac{1}{3}\bar{I}_{1} \bar{J}_{2} + \frac{1}{27}\bar{I}_{1}^{3}$$

(28)

Substituting Eqs. 26, 27, and 27 into Eq. 25 obtains:

$$2\left( {d - 1} \right)^{2} \bar{I}_{1}^{3} - 6\left( {d^{2} - d + 1} \right)\bar{I}_{1} \bar{J}_{2} + 27d\bar{J}_{3} = 0$$

(29)

Equations 3 and 8 in the text relate $\alpha$ to $d$ as follows:

$$\alpha = \frac{1 - d}{\sqrt 3 d}$$

(30)

Introducing Eq. 30 and the Lode angle defined by Eq. 7 into Eq. 29 gives:

$$d^{2} \left( {\alpha \bar{I}_{1} } \right)^{3} - \left( {d^{2} - d + 1} \right)\bar{J}_{2} \left( {\alpha \bar{I}_{1} } \right) + \left( {1 - d} \right)\left( {\bar{J}_{2} } \right)^{{{\raise0.5ex\hbox{$\scriptstyle 3$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} \cos \left( {3\theta } \right) = 0$$

(31)

Equation 31 is a transformed expression for the Matsuoka–Nakai model, which does not account for the temperature and strain rate dependent cohesion and strain hardening. To consider these material properties of asphalt concrete, the term $\kappa a_{T} a_{{\dot{\varepsilon }}}$ is added and Eq. 31 becomes:

$$d^{2} \left( {\alpha \bar{I}_{1} + \kappa a_{T} a_{{\dot{\varepsilon }}} } \right)^{3} - \left( {d^{2} - d + 1} \right)\bar{J}_{2} \left( {\alpha \bar{I}_{1} + \kappa a_{T} a_{{\dot{\varepsilon }}} } \right) + \left( {1 - d} \right)\left( {\bar{J}_{2} } \right)^{{{\raise0.5ex\hbox{$\scriptstyle 3$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} \cos \left( {3\theta } \right) = 0$$

(32)

To acquire an expression with the first order of stress, Eq. 32 is treated as a cubic equation that has a variable of $\alpha \bar{I}_{1} + \kappa a_{T} a_{{\dot{\varepsilon }}}$. Solving this cubic equation gives a new yield surface function as:

$$\sqrt {\bar{J}_{2} } \rho \left( {\theta ,\;d} \right) - \alpha \bar{I}_{1} - \kappa a_{T} a_{{\dot{\varepsilon }}} = 0$$

(33)

where

$$\rho \left( {\theta ,\;d} \right) = \mu \cos \left[ {\tfrac{1}{3}\arccos \left( {\gamma \cos 3\theta } \right)} \right]$$

(34)

and

$$\;\left\{ \begin{gathered} \mu = \frac{{2\sqrt {1 - d + d^{2} } }}{\sqrt 3 d} \hfill \\ \gamma = - \frac{3\sqrt 3 }{2}\frac{{\left( {1 - d} \right)d}}{{\left( {1 - d + d^{2} } \right)^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }} \hfill \\ \end{gathered} \right.\;\;\;\;\;\;\left( {d = \frac{3 - \sin \phi }{3 + \sin \phi }} \right)$$

(35)

Equation 33 is the GD–P yield surface model proposed in this study. Parameters $\mu$ and $\gamma$ are related to $d$ that is a function of internal friction angle ($\phi$). Equations 34 and 35 demonstrate that the shape of GD–P model on the octahedral plane only depends on the internal friction angle of the material that is determined by aggregate skeleton. $\theta$ is the Lode angle and $\theta$ = 0 indicates triaxial compression and $\theta$ = $\tfrac{\pi }{3}$ implies triaxial extension. Using Eqs. 34 and 35, one can get the following relations:

$$\left\{ \begin{gathered} \rho \left( {\theta = 0} \right) = \mu \cos \left[ {\frac{1}{3}\arccos \left( \gamma \right)} \right] = 1\;\;\;\;\;\;\;\;\;\;{\text{in}}\;{\text{compression}} \hfill \\ \rho \left( {\theta = \frac{\pi }{3}} \right) = \mu \cos \left[ {\frac{1}{3}\arccos \left( { - \gamma } \right)} \right] = \frac{1}{d}\;\;\;\;{\text{in}}\;{\text{extension}} \hfill \\ \end{gathered} \right.$$

(36)