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Publication Number: FHWAHRT10065 Date: December 2010 
The modeling framework and the proposed model used for the compaction study were introduced in chapter 3. The model was further developed and improved to account for the behavior of asphalt concrete subjected to compaction in the field. To this end, the model was implemented in the FE software CAPA3D, developed at Delft University of Technology.^{(53,54)} CAPA3D was developed as an FEbased platform to serve the computational needs of the research group Mechanics of Structural Systems at Delft University of Technology and international teams that cooperate with the group. Over the years, the software has evolved into a fullfledged FE system for static or dynamic analysis of largescale threedimensional pavement and soil engineering models. The software consists of a sophisticated user interface; a powerful, highly parallelized bandoptimizing mesh generator; a highquality, usercontrolled graphical output; several material and element types; and a variety of specialized algorithms for more efficient analysis of pavement construction. These algorithms include a moving loadsimulation algorithm and a contact algorithm, which are essential for modeling the compaction process.
Researchers solved the constrained compression problems using the FE implementation of the model in CAPA3D. In finite elements, these problems were solved by employing a 20noded threedimensional solid element. The representative material element was constrained to move only in a normal direction (zdirection). A comparison was drawn between the solutions obtained from the analysis (with aid from calculations in MATLAB^{®}) and the solutions obtained using the method of finite elements.^{(54)} The compressive strain applied was 0.05, or 5 percent. The material properties used are presented in table 1. An explanation of the parameters listed in table 1 is presented in chapter 5 of this report. These parameters were also used for FE calculations.
Table 1. Parameters employed for the study of the material response when subject to onedimensional constrained compression.
(MPa) 
n_{1} 
λ_{1} 
q_{1} 
(MPa·s) 
n_{2} 
λ_{2} 
q_{2} 

810 
4.0 
0.25 
−15 
1,400 
2.5 
0.25 
−25 
Inputs to the model are given by linearly increasing the input to a constant value and then keeping the input at this value. Removing the input is then performed by linearly decreasing from the maximum value to no input. The comparison for constant applied stress is presented in figure 24 and figure 25, and the comparison for constant applied strain is presented in figure 26 and figure 27. The calculated values (using MATLAB^{®}) of the stretch and stress in response to applied stress and strain, respectively, and the corresponding FE solutions agree well, as can be observed from figure 24 through figure 27. This agreement serves as validation for the implementation of the model in CAPA3D using the method of finite elements.
Figure 24. Chart. FE solution to the model response to applied constant stress—creep load.
Figure 25. Chart. FE solution to the model response to applied constant stress—total stretch.
Figure 26. Chart. FE solution to the model response to applied constant strain (compressive 0.05)—applied strain.
Figure 27. Chart. FE solution to the model response to applied constant strain (compressive 0.05)—normal stress response.
In the developed FE implementation, a constantshear problem is solved. The schematic representation for such a model is given in figure 28. The unit cube is subject to a ratetype shear deformation, and the boundary conditions to be applied to the unit cube are such that the cube is constrained to move only in a lateral direction, indicating a shear deformation.
Figure 28. Illustration. Schematic of the constant shear loading applied to a unit cube.
In figure 29 through figure 31, T_{12} represents the shear in the xyplane, T_{11}–T_{22} represents the first normal stress, and T_{22}–T_{33} represents the second normal stress. As shown in the figures, the material response corresponds with that of a nonlinear material due to the exhibition of normal stress differences.
Figure 29. Chart. Shear stress (T_{12}) observed in response to constant shear loading.
Figure 30. Chart. Comparison of the first normal stress (T_{11}–T_{22}) response to constant shear loading.
Figure 31. Chart. Comparison of the second normal stress (T_{22}–T_{33}) response to constant shear loading.
Using the developed FE implementation, a constantshearrate problem is solved. The schematic representation for such a model is given in figure 32. The unit cube is subject to a ratetype shear deformation, and the boundary conditions applied to the unit cube are such that the cube is constrained to move only in a lateral direction, indicating a shear deformation.
Figure 32. Illustration. Schematic of the constant shear rate applied to a unit cube.
In figure 33 through figure 36, T_{12} represents the shear in the xyplane, T_{11}–T_{22} represents the first normal stress, and T_{22}–T_{33} represents the second normal stress.
Figure 33. Chart. Shear stress (T_{12}) observed in response to constant shear rate loading.
Figure 34. Chart. Shear stress (T_{12}) as a function of the shear rate (using the model parameters in table 1).
Figure 35. Chart. Comparison of the first normal stress (T_{11}–T_{22}) response to constant shear rate loading.
Figure 36. Chart. Comparison of the second normal stress (T_{22}–T_{33}) response to constant shear rate loading.
As shown in figure 33, figure 35, and figure 36, the material response corresponds with that of a nonlinear material due to the exhibition of normal stress differences (though small in comparison to the shear stress T_{12}). For the chosen set of parameters, the model seems to exhibit nearlinear shearstress (T_{12}) versus shearrate characteristics. Such characteristics typically represent the response of a fluid with a rate of dissipation similar to Newtonian fluid. However, the model does exhibit normal stress differences, a typical characteristic of nonNewtonian fluids, as shown in figure 35 and figure 36. Hence, the model behaves like a nonNewtonian fluid, and the choice of the parameters enhances the shearthickening nature of the model (unlike in figure 34). This feature of the model is very useful for the purposes of modeling compaction in the laboratory or in the field as the material in those situations undergoes shearthickening behavior. The choice of model functions for shear modulus and viscosity ensure such a transition in the material's property.