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Publication Number: FHWA-HRT-10-065
Date: December 2010

Modeling of Hot-Mix Asphalt Compaction: A Thermodynamics-Based Compressible Viscoelastic Model

508 Captions

Figure 1. Equation. Gradient and divergence operators with respect to reference configuration. Gradient of script a equals partial derivative of script a with respect to script X. In index notation, gradient of script a subscript ij equals the partial derivative of script a subscript i with respect to script X subscript j. Divergence of script T equals partial derivative of script T with respect to script X. In index notation, divergence of script T subscript i equals the partial derivative of script T subscript ij with respect to script X subscript j.

Figure 2. Equation. Gradient and divergence operators with respect to current configuration. Gradient of script a equals partial derivative of script a with respect to script x. In index notation, Gradient of script a subscript ij equals the partial derivative of script a subscript i with respect to script x subscript j. Divergence of script T equals partial derivative of script T with respect to script x. In index notation, divergence of script T subscript i equals the partial derivative of script T subscript ij with respect to script x subscript j

Figure 3. Illustration. Evolution of natural configurations associated with microstructural transformations resulting from the material response to an external stimulus. This graphic shows the stress-free state that occurs when tractions are removed. The form B subscript K subscript R can go through two processes: F subscript K subscript R or G. The process F subscript K subscript R produces the form B subscript K subscript (e times t). The process G produces the form B subscript K subscript (p times t), which goes through the process F subscript K subscript (p times t) and produces the form B subscript K subscript (e times t)

Figure 4. Equation. Energy-dissipation inequality. Sum of the following: T scalar product with L minus rho product with material time derivative of psi minus rho product (zeta product with material time derivative of theta) minus q scalar product with gradient of theta divided by theta, equals rho product (theta product Xi) equals xi, which is greater than or equal to zero

Figure 5. Equation. Reduced energy-dissipation inequality. Sum of the following: T scalar product with L minus rho product with material time derivative of psi equals symbol hat of xi, which is greater than or to equal zero

Figure 6. Equation. Helmholtz potential for mixture. psi equals mu (a function of roman numeral III subscript G) product (sum of: roman numeral I subscript (B subscript K subscript (p times t)) minus three minus natural logarithm of (roman numeral III subscript (B subscript K subscript (p times t))), the sum product divided by twice rho subscript K subscript (p times t)

Figure 7. Equation. Rate of dissipation function. Symbol hat of xi equals eta (a function of roman numeral III subscript G) times the scalar product of D subscript K subscript (p times t) with the product of B subscript K subscript (p times t) and D subscript K subscript (p times t)

Figure 8. Equation. Shear-modulus function. mu equals symbol hat of mu times (sum of: one plus lambda subscript one times (roman numeral III subscript G) exponent twice n subscript one) exponent q subscript one

Figure 9. Equation. Viscosity function. eta equals symbol hat of eta times (sum of: one plus lambda subscript two times (roman numeral III subscript G) exponent twice n subscript two) exponent q subscript two

Figure 10. Equation. Constitutive equation for stress. Stress T equals sum of the following: twice rho times roman numeral III subscript B subscript K subscript (p times t) times partial derivative of psi with respect to roman numeral III subscript B subscript K subscript (p times t) times the identity tensor I plus twice rho times partial derivative of psi with respect to roman numeral I subscript B subscript K subscript (p times t) times the tensor B subscript K subscript (p times t)

Figure 11. Equation. Evolution equation. Convected time derivative of tensor B subscript K subscript (p times t) equals minus two times sum of the following: inverse of tensor V subscript K subscript (p times t) times tensor T times tensor V subscript K subscript (p times t) minus rho times roman numeral III subscript G times partial derivative of psi with respect to roman numeral III subscript G times the identity tensor I, the total product divided by eta

Figure 12. Illustration. Schematic for the one-dimensional compression problem. This graphic shows a cylinder with downward forces T subscript zz. Other forces are shown on the sides and bottom. Three cylindrical coordinates are labeled z, r, and theta

Figure 13. Chart. Creep solution for the material model for HMA—creep load. This chart shows a plot obtained for the given material by calculating the solution to the semi-inverse creep problem using MATLAB®. The creep load chart shows T bar subscript zz as a straight line at 6.6 time parameter t bar for a range of stresses, T bar subscript zz. Stress, T bar subscript zz is on the y-axis and Time parameter t bar is on the x-axis.

Figure 14. Chart. Creep solution for the material model for HMA—Almansi strain. This chart shows a plot obtained for the given material by calculating the solution to the semi-inverse creep problem using MATLAB®. The Almansi strain chart shows the line e bar with e tilde times t bar on the y-axis and time parameter t bar on the x-axis

Figure 15. Chart. Total stretch calculated from the solution to the one-dimensional creep problem. This chart shows a plot of total stretch, one of the relevant field variables calculated. The material does return to a natural (stress-free) configuration upon unloading; therefore, the dissipation occurs because of a change in the microstructure of the material. The chart shows time parameter t bar on the x-axis and total stretch, lambda tilde on the y-axis.

Figure 16. Chart. Evolution of natural configuration calculated from the solution to the one-dimensional creep problem. This chart shows a plot of evolution of natural configuration, one of the relevant field variables calculated. The material does return to a natural (stress-free) configuration upon unloading; therefore, the dissipation occurs because of a change in the microstructure of the material. The chart shows time parameter t bar on the x-axis and evolution of natural configuration, b bar superscript 2 on the y-axis

Figure 17. Chart. Stored energy calculated from the solution to the one-dimensional creep problem. This chart shows a plot of stored energy, one of the relevant field variables calculated. The material does return to a natural (stress-free) configuration upon unloading; therefore, the dissipation occurs because of a change in the microstructure of the material. The chart shows time parameter t bar on the x-axis and stored energy, psi bar tilde on the y-axis.

Figure 18. Chart. Rate of dissipation calculated from the solution to the one-dimensional creep problem. This chart shows a plot of rate of dissipation, one of the relevant field variables calculated. The material does return to a natural (stress-free) configuration upon unloading; therefore, the dissipation occurs because of a change in the microstructure of the material. The chart shows show time parameter t bar on the x-axis and dissipation rate, xi bar tilde on the y axis

Figure 19. Illustration. Schematic for the one-dimensional stress-relaxation problem. This graphic shows a cylinder with downward forces e subscript zz times t equals C, a constant. Below the cylinder is the description r times t equals R times t, theta times t equals Theta times t, z times t equals lambda times t times Z times t

Figure 20. Chart. Applied strain calculated from the solution to the one-dimensional stress-relaxation problem. This chart shows the results of step-input strain calculations. This applied strain chart shows time parameter t bar on the x-axis and the line e tilde subscript zz with strain (times 10 superscript -3) on the y-axis.

Figure 21. Chart. Stress relaxation calculated from the solution to the one-dimensional stress-relaxation problem. This chart shows the results of step-input strain calculations. This stress relaxation chart shows time parameter t bar on the x-axis and the line T subscript zz with stress (times 10 superscript -4) on the y-axis.

Figure 22. Chart. Stored energy calculated from the solution to the one-dimensional stress-relaxation problem. This chart shows the results of step-input strain calculations. This stored energy chart shows time parameter t bar on the x-axis and the line psi bar tilde with stored energy (times 10 superscript -4) on the y-axis.

Figure 23. Chart. Rate of dissipation calculated from the solution to the one-dimensional stress-relaxation problem. This chart shows the results of step-input strain calculations. This rate of dissipation chart shows time parameter t bar on the x-axis and the line xi bar tilde with rate of dissipation (times 10 superscript -4) on the y-axis

Figure 24. Chart. FE solution to the model response to applied constant stress—creep load. This chart, with figure 25, shows the comparison for the constant applied stress. The chart shows load as a line, using two axes: stress, T bar subscript zz, and time parameter, t bar

Figure 25. Chart. FE solution to the model response to applied constant stress—total stretch. This chart, with figure 24, shows the comparison for the constant applied stress. The chart shows analytical as a line, using total stretch, lambda tilde, on the y-axis and time parameter, t bar, on the x-axis. Finite-element model (FEM) is a line that roughly parallels analytical. The calculated values (using MATLAB®) of the stretch and stress in response to applied stress and the corresponding finite-element (FE) solutions agree well

Figure 26. Chart. FE solution to the model response to applied constant strain (compressive 0.05)—applied strain. This chart, with figure 27, shows the comparison for the constant applied strain. The chart shows a line on two axes: strain (e tilde times t bar) and time parameter, t bar.

Figure 27. Chart. FE solution to the model response to applied constant strain (compressive 0.05)—normal stress response. This chart, with figure 26, shows the comparison for the constant applied strain. The chart shows analytical as a line, using two axes: stress, T bar subscript zz, on the y-axis and time parameter, t bar, on the x-axis. Finite-element model (FEM) is a line that roughly follows analytical. The calculated values (using MATLAB®) of the stretch and stress in response to applied strain and the corresponding finite-element (FE) solutions agree well

Figure 28. Illustration. Schematic of the constant shear loading applied to a unit cube. This graphic shows a unit cube labeled gamma with dimensions x, y, and z. The cube shears in only a lateral direction, x, indicating a shear deformation

Figure 29. Chart. Shear stress (T12) observed in response to constant shear loading. This chart represents the shear in the xy-plane. The material response corresponds with that of a nonlinear material due to the exhibition of normal stress differences. The chart shows the lines gamma equals 0.1 and gamma equals 0.5 using two axes: T subscript 12 (MPa) and time (seconds)

Figure 30. Chart. Comparison of the first normal stress (T11T22) response to constant shear loading. This chart represents the first normal stress. The material response corresponds with that of a nonlinear material due to the exhibition of normal stress differences. The chart shows the lines gamma equals 0.1 and gamma equals 0.5 using two axes: T subscript 11 through T subscript 22 (MPa) and time (seconds)

Figure 31. Chart. Comparison of the second normal stress (T22T33) response to constant shear loading. This chart represents the second normal stress. The material response corresponds with that of a nonlinear material due to the exhibition of normal stress differences. The chart shows the lines gamma equals 0.1 and gamma equals 0.5 using two axes: T subscript 22 through T subscript 33 (MPa) and time (seconds)

Figure 32. Illustration. Schematic of the constant shear rate applied to a unit cube. This graphic shows a unit cube labeled Kappa with dimensions x, y, and z. The cube shears in only a lateral direction, x, indicating a shear deformation

Figure 33. Chart. Shear stress (T12) observed in response to constant shear rate loading. This chart represents the shear in the xy-plane. The chart shows the lines kappa equals 0.5 and kappa equals 0.1 using two axes: T subscript 12 (MPa) and time (seconds). The material response corresponds with that of a nonlinear material due to the exhibition of normal stress differences

Figure 34. Chart. Shear stress (T12) as a function of the shear rate (using the model parameters in table 1). This chart represents the shear in the xy-plane. The chart shows a straight line that begins at 0 MPa (T subscript 12) and 0 seconds to the power of -1 (K) and ends at 1.5 MPa and 0.5 seconds to the power of -1

Figure 35. Chart. Comparison of the first normal stress (T11T22) response to constant shear rate loading. This chart represents the first normal stress. The chart shows the lines kappa equals 0.1 and kappa equals 0.5 using two axes: T subscript 11 through T subscript 22 (MPa) and time (seconds). The material response corresponds with that of a nonlinear material due to the exhibition of normal stress differences

Figure 36. Chart. Comparison of the second normal stress (T22T33) response to constant shear rate loading. This chart represents the second normal stress. The chart shows the lines kappa equals 0.1 and kappa equals 0.5 using two axes: T subscript 22 through T subscript 33 (MPa) and time (seconds). The material response corresponds with that of a nonlinear material due to the exhibition of normal stress differences

Figure 37. Photo. Superpave® gyratory compactor. This photo shows a piece of laboratory equipment, the Superpave® gyratory compactor. There is a metal drum housed in blue, metal casing that is attached to an electronic control system.

Figure 38. Photo. Static steel-wheel roller. This photo shows an unmanned static steel wheel roller on pavement. The vehicle has an operator's platform with a seat and controls and two large, steel rollers at the front and back

Figure 39. Illustration FE mesh used in modeling the SGC. This graphic shows an example of the finite-element (FE) mesh of the Superpave® gyratory compactor (SGC). A cylinder sits on top of a wider, shorter cylinder. The top cylinder shows S, mises, from +5.750e-01 to +6.900e 01, with one round spot in the lower part of the cylinder from +5.175e-01 to +5.750e 01. The bottom, wider but shorter cylinder shows S, mises, from +4.881e+06 to +6.900e-01. Below the cylinders is the following information: ODB: sh125sm1185387494.22096.odb, ABAQUS/Standard 6.4-5; Step: Step-2; Increment 13: Step Time equals 4.513; Primary Var: S, Mises; Deformed Var: U, Deformation Scale Factor: +1.000e+00

Figure 40. Chart. Analysis of the sensitivity of compaction to . This chart shows an example of the influence of mu, keeping all other parameters the same. More compaction is achieved as the estimated mu decreases. The charts shows three lines: estimated mu equals 1900, estimated mu equals 1700, and estimated mu equals 1500. The lines are on two axes: normalized height and time of compaction (seconds)

Figure 41. Chart. Analysis of the sensitivity of compaction to n1. This chart shows an analysis of the parameter n subscript 1. mu is significantly affected by n subscript 1, which controls the maximum compaction of a mixture. The chart shows three lines: n subscript 1 equals 5.0, n subscript 1 equals 4.0, and n subscript 1 equals 3.0. The lines are on two axes: normalized height and time of compaction (seconds)

Figure 42. Chart. Analysis of the sensitivity of compaction to . This chart shows that the parameter estimated eta controls the point at which the material behavior starts to change from a very low-viscosity fluidlike behavior to a high-viscosity fluidlike behavior. The chart shows three lines: estimated eta equals 1600, estimated eta equals 1400, and estimated eta equals 1300. The lines are on two axes: normalized height and time of compaction (seconds).

Figure 43. Chart. Analysis of the sensitivity of compaction to λ2. This chart shows that the parameter lambda subscript 2 is directly related to the initial slope of the compaction curve. The chart shows three lines: lambda subscript 2 equals 0.26, lambda subscript 2 equals 0.25, and lambda subscript 2 equals 0.22. The lines are on two axes: normalized height and time of compaction (seconds)

Figure 44. Chart. Analysis of the sensitivity of compaction to q2. This chart shows that the model parameter q subscript 2 contributes to the nonlinear change of viscosity from the start of the compaction process. The overall compaction of the mixture is higher at lower (more negative) values of q subscript 2. The chart shows three line: q subscript 2 equals -34, q subscript 2 equals -30, and q subscript 2 equals -26. The lines are on two axes: normalized height and time of compaction (seconds)

Figure 45. Equation. Modified shear-modulus function. mu equals symbol hat of mu times (sum of: one plus 0.25 times (roman numeral III subscript G) exponent twice n subscript one) exponent -25

Figure 46. Equation. Modified viscosity function. eta equals symbol hat of eta times (sum of: one plus lambda subscript two times (roman numeral III subscript G) exponent 5) exponent q subscript two

Figure 47. Chart. Illustration of the relationship of the model's parameters to the compaction process. This graphic shows a chart with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The chart contains a line and descriptions of points and planes on the line: the initial viscous dissipation limit, due lambda subscript 2; the onset of high viscosity, determined by the estimated eta; the degree of nonlinear transition, controlled through q subscript 2; the beginning of material consolidation, determined by the estimated mu; the rate of consolidation, governed by n subscript 1; and the asymptotic consolidation limit, due lambda subscript 1

Figure 48. Chart. Influence of angle of gyration on the compaction curve. This chart shows the finite-element (FE) simulations of the Superpave® gyratory compactor (SGC) at different angles of gyration. The chart shows normalized height on the y-axis and time of compaction (seconds) on the x-axis. It has lines for 0.5, 1.25, 2.0, and 3.0 degrees

Figure 49. Chart. Maximum shear stress at the top of the specimen for a gyration angle of 1.25 degrees. The finite-element (FE) model was used to determine the maximum shear stresses at the top of the specimens for a gyration angle of 1.25 degrees. This chart shows Tresca stress (MPa) on the y-axis and time of compaction (seconds) on the x-axis, and it depicts a series of lines. Initially, the shear stress decreases rapidly when the material behaves as a compressible fluid and then starts to increase gradually as the mixture starts to behave like a highly viscous fluid. The shear stresses are higher for higher angles of compaction. This is because an increase in the angle of gyration is associated with an increase in the applied shear stresses. Also, the point at which shear stress starts to increase occurs earlier at a 2.0-degree angle than at a 1.25-degree angle. This is because the mixture compacts and gains strength faster at a 2.0-degree angle of gyration

Figure 50. Chart. Maximum shear stress at the top of the specimen for a gyration angle of 2.0 degrees. The finite-element (FE) model was used to determine the maximum shear stresses at the top of the specimens for a gyration angle of 2 degrees. This chart shows Tresca stress (MPa) on the y-axis and time of compaction (seconds) on the x-axis, and it depicts a series of lines. Initially, the shear stress decreases rapidly when the material behaves as a compressible fluid and then starts to increase gradually as the mixture starts to behave like a highly viscous fluid. The shear stresses are higher for higher angles of compaction. This is because an increase in the angle of gyration is associated with an increase in the applied shear stresses. Also, the point at which shear stress starts to increase occurs earlier at a 2.0-degree angle than at a 1.25 degree angle. This is because the mixture compacts and gains strength faster at a 2.0-degree angle of gyration

Figure 51. Chart. Fitting of the compaction data at 1.25 degrees for project IH-35. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 52. Chart. Fitting of the compaction data at 1.25 degrees for project US-259. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 53. Chart. Fitting of the compaction data at 1.25 degrees for project SH-36. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 54. Chart. Fitting of the compaction data at 1.25 degrees for project SH-21. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 55. Chart. Fitting of the compaction data at 1.25 degrees for project US-87. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 56. Chart. Prediction of the compaction data at 2.0 degrees for project IH-35. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 57. Chart. Prediction of the compaction data at 2.0 degrees for project US-259. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 58. Chart. Prediction of the compaction data at 2.0 degrees for project SH-36. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 59. Chart. Prediction of the compaction data at 2.0 degrees for project SH-21. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 60. Chart. Prediction of the compaction data at 2.0 degrees for project US-87. This chart shows the final simulation results using the parameters in table 3. Simulation and experiment are plotted as lines with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The results show that the model has a reasonable representation of the compaction curves

Figure 61. Illustration. Pavement structure typically employed for studying field compaction. This graphic shows a base layer of 18 inches (457.2 mm), an old hot-mix asphalt (HMA) layer of 2 inches (50.8 mm), and a top HMA layer of 3 inches (76.2 mm). A very thin layer exists between the base and the old HMA layer and between the old HMA layer and the top HMA layer

Figure 62. Illustration. Sectional view of the FE mesh used for setting up field compaction simulations. This graphic shows a three-dimensional representation of asphalt layers. The newly laid loose mix is next to the compacted mix in the roller wake. Beneath this are the old asphalt layer and impedance layers around the top layers. Beneath these are the base layer and impedance layers around the base layers

Figure 63. Illustration. Schematic diagram illustrating the edges of the lane that correspond to fixed and free edges of the mesh in figure 62. This graphic shows the boundary conditions most frequently used. The layers are the compacted mix, new/newly compacted asphalt layer, and old asphalt layer. The fixed edge of the pavement corresponds to the inner edge of the pavement, while the free edge corresponds to the outer edge of the lane. The entire width of the pavement section is 17 ft (5.185 m). The width of the new asphalt layer is 12 ft (3.66 m). The roller is 7-ft (2.135-m) wide and covers the new asphalt layer in two overlapping passes

Figure 64. Chart. Typical displacement curve for a node under the cylindrical load for a cycle with forward and return passes followed by a forward pass with the load removed. This chart shows the typical response curve with the compaction experienced at a point in the roller path as the roller passes over it and returns for a second pass from the opposite direction. The chart shows the vertical displacement (mm) during the first pass, second pass, and third pass with no load. Vertical displacement from -5 mm–0 mm is labeled as instantaneous inelastic response; the material flows and has a short associated relaxation time

Figure 65. Chart. Vertical displacement observed at point P on the path of the roller. This plot represents the vertical deflection observed at a point P, measured on the y-axis in millimeters (mm). The chart notes the vertical deflection at point P at six different time instants during the motion. Four observations are made in the forward rolling in the first hundred time steps of motion, while two observations are made when the motion is reversed in the next hundred time steps. The observations are made at time steps 10, 60, 90, 100, 120 and 200. There is very little deflection at time steps 10 and 60, more than 5 mm at time step 90, about 3.5 mm at time step 100, about 6.5 mm at time step 120, and about 4 mm at time step 200. By the sixth observation, the amount of vertical displacement has leveled off

Figure 66. Chart. Roller location from observation point P. This chart represents the roller location from the observation point on the y-axis and the time steps needed for compaction on the x-axis. The roller location at six instants corresponding to the 10, 60, 90, 100 120 and 200 time steps are represented by vertical bars. The progression of the roller along the mat, relative to the point of observation, is represented by the progression of the vertical bars, which progress upward on the chart until point 4, and then return to the original position by point 6

Figure 67. Chart. Deflection at the node of interest when a load is applied for a short duration and then removed. This chart shows that as the roller passes over a point and moves further away, the material relaxes and is then subject to further deformation from this relaxed state during the immediate pass. The chart shows the creep relaxation as a line with normalized deflection on the y-axis and time steps on the x-axis. During constant loading (time steps 1–20), normalized height decreases. During the time when the load is removed (time steps 20–50), the normalized height rebounds and evens out.

Figure 68. Chart. Permanent deformation prediction of the model in multiple-pass loading. This chart shows how deformation builds up incrementally with each pass of the roller. However, the increments decrease in magnitude as the number of passes increases, due to material densification. The chart shows two, four, and six roller passes. The decreasing permanent deformation is given in three steps: delta x subscript 1 (time steps 0–200), delta x subscript 2 (time steps 200–400), and delta x subscript 3 (time steps 400–600)

Figure 69. Illustration. Roller contact geometry for static indentation and during motion. This graphic shows the distributed forces in the elements in contact during a static indentation, mimicking the cylindrical profile of the roller. The resolution of normal reaction forces due to a static indentation into a pliable material is represented by the total angle of contact given by theta. The angle of contact made by an arbitrary point on the profile of the roller with the material being compacted is represented by phi. The radius of the roller is represented by R. Employing the above quantities along with the weight of the roller and the mesh used for the material being compacted, the necessary loads to be applied during simulation are calculated

Figure 70. Chart. Indentation of a cylindrical roller into pavement. This chart depicts vertical deflection (mm) on the y-axis for six full contact elements, four full and two half contact elements, four full contact elements, two full and one half contact elements, and two full contact elements. Nodal x-coordinate (mm) is listed on the x-axis of the chart. The y-axis is disproportionate in comparison to the x-axis

Figure 71. Chart. Change in nodal reaction forces as the load is applied over a smaller area. This chart shows the nodal reaction forces in the elements in response to the applied dead load of the roller as the contact area (represented by the number of elements in contact) is changed. A1 and A2 represent the edge and middle nodal forces of the same orientation in an element when six elements are in contact between the roller and the material. B1 and B2 are nodal forces at the edge and middle nodes of an element when four elements are in contact, and C1 and C2 are nodal forces when two elements are in contact

Figure 72. Chart. Comparison of vertical displacement for the two different patterns of loading. This chart shows the differences observed in the vertical displacement response in an element in the direct path of the traversing load over 200 time steps. Vertical displacement (mm) is on the y-axis, and the time steps are on the x-axis. The vertical displacement of the uniform pressure is relatively steady, but the vertical displacement of the nonuniform pressure varies

Figure 73. Chart. Comparison of the normal stress distribution at the pavement's top surface due to different loading patterns. This chart shows the differences observed in the stress response in an element in the direct path of the traversing load over 200 time steps. Normal stress (MPa) is on the y-axis, and the time steps are on the x-axis. The normal stress of the uniform pressure is relatively steady, but the normal stress of the nonuniform pressure varies

Figure 74. Chart. Comparison of the shear-stress (XY) distribution of the pavement's top surface due to different loading patterns (on the plane of symmetry). This chart shows the differences observed in the stress response in an element in the direct path of the traversing load over 200 time steps. XY-shear stress (MPa) is on the y-axis, and the time steps are on the x-axis. The shear stress of the uniform pressure is relatively steady, but the shear stress of the nonuniform pressure varies

Figure 75. Equation. Mean contact pressure. Script p subscript m equals script P divided by the product of pi with square of script a

Figure 76. Equation. Normal stress in elastic medium. Sigma subscript z divided by script p subscript m equals -4 divided by pi times (sum of: 1 minus square of script x divided by square of a) to the power one-half

Figure 77. Chart. Comparison of the normal stress predicted by the model to the normal stress predicted for an elastic medium (for the same mean contact pressure, Pm). This chart shows the normal stress predicted for hot mix asphalt (HMA) material and an elastic medium. Normal stress (MPa) is on the y-axis, and X coordinate (mm) is on the x-axis. The comparatively higher stresses in the elastic medium at a given mean contact pressure are due to the lack of any dissipative mechanism in the elastic material

Figure 78. Chart. Comparison of pavement vertical response as the normal stiffness of the interface elements is lowered by an order of magnitude from 1,451 ksi (10,000 MPa) to 145 ksi (1,000 MPa). This chart shows the results of the finite-element (FE) analysis using the interface layer. Vertical displacement (mm) is on the y-axis, and time steps are on the x-axis. The normal stiffness is varied (stiff interface and soft interface) for the first and second passes. The use of interface elements does not adversely affect the vertical displacement response of the model

Figure 79. Chart. Comparison of pavement response in the rolling direction as the normal stiffness of the interface elements is lowered by an order of magnitude from 1,450 ksi (10,000 MPa) to 145 ksi (1,000 MPa). This chart shows the results of the finite-element (FE) analysis using the interface layer. X displacement (mm) is on the y-axis, and time steps are on the x-axis. The normal stiffness is varied (stiff interface and soft interface) for the first and second passes. The use of interface elements does not adversely affect the x-displacement response of the model

Figure 80. Chart. Comparison of pavement response to increasing the shear stiffness in the x direction. This chart shows the results of the finite-element (FE) analysis using the interface layer. Vertical displacement (mm) is on the y-axis, and time steps are on the x-axis. The shear stiffness in the x-direction is varied (stiff shear x-direction and soft shear x-direction) for the first and second passes. The use of interface elements does not adversely affect the vertical displacement response of the model

Figure 81. Chart. Comparison of pavement response to increasing the shear stiffness in the z direction. This chart shows the results of the finite-element (FE) analysis using the interface layer. Vertical displacement (mm) is on the y-axis, and time steps are on the x-axis. The shear stiffness in the z-direction is varied (stiff shear y-direction and soft shear y direction) for the first and second passes. The use of interface elements does not adversely affect the vertical displacement response of the model

Figure 82. Chart. Comparison of the dampening effect provided by impedance layers surrounding the structure laterally in the x-direction. This chart shows response plots that have impedance layers (soft impedance and stiff impedance), which are useful in reducing horizontal reflections significantly. Horizontal displacement (mm) is on the y-axis, and time steps are on the x-axis

Figure 83. Chart. Comparison of the dampening effect provided by impedance layers surrounding the structure laterally in the y-direction. This chart shows response plots that have impedance layers (soft impedance and stiff impedance), which are useful in reducing vertical reflections significantly. Vertical displacement (mm) is on the y-axis, and time steps are on the x-axis

Figure 84. Chart. Comparison of the horizontal dampening effect provided by impedance layers surrounding the top layer (pavement and old asphalt) laterally in the x-direction. This chart shows response plots that have impedance layers surrounding the top layer (soft impedance and stiff impedance), which are useful in reducing horizontal reflections significantly—to the point that they are completely eliminated. Horizontal displacement (mm) is on the y-axis, and time steps are on the x-axis

Figure 85. Chart. Comparison of the vertical dampening effect provided by impedance layers surrounding the top layer laterally in the x-direction. This chart shows response plots that have impedance layers surrounding the top layer (soft impedance and stiff impedance), which are useful in reducing vertical reflections significantly—to the point that they are completely eliminated. Vertical displacement (mm) is on the y-axis, and time steps are on the x-axis

Figure 86. Chart. Comparison of the volumetric component of the viscous-evolution gradient for soft (72.5 ksi (500 MPa)) versus stiff (290 ksi (2,000 MPa)) bases. This chart shows plots of the volumetric response for a soft base and stiff base. The plots indicate that the material response is not very sensitive to changes in stiffness of the base when the response is viewed in an averaged sense. The chart shows det(G)-1 on the y-axis and time steps on the x-axis

Figure 87. Chart. Comparison of the volumetric component of the viscous-evolution gradient at two base-stiffness moduli of interest. This chart shows plots of the volumetric response for E equals 482.6 MPa and E equals 965.3 MPa. The plots indicate that the material response is not very sensitive to changes in stiffness of the base when the response is viewed in an averaged sense. The chart shows det(G)-1 on the y-axis and time steps on the x-axis

Figure 88. Chart. Comparison of the effect on the x-displacement of a node in the roller path as the base stiffness is varied from 72.5 ksi (500 MPa) (soft base) to 290 ksi (2,000 MPa) (stiff base). This chart shows that the finite-element (FE) model is responsive to a drastic increase in base modulus due to increasing rigidity in the base structure, leading to interference waves reflected internally. Horizontal displacement (mm) is on the y-axis, and time steps are on the x-axis. The horizontal displacement is given for soft base x-displacement and for stiff base x-displacement

Figure 89. Chart. Comparison of the effect on the y-displacement of a node in the roller path as the base stiffness is varied from 72.5 ksi (500 MPa) (soft base) to 290 ksi (2,000 MPa) (stiff base). This chart shows that the finite-element (FE) model is responsive to a drastic increase in base modulus due to increasing rigidity in the base structure, leading to interference waves reflected internally. Vertical displacement (mm) is on the y-axis, and time steps are on the x-axis. The vertical displacement is given for soft base y-displacement and for stiff base y-displacement

Figure 90. Chart. Comparison of the deflection for two base-stiffness moduli of interest. This chart shows that the finite-element (FE) model is responsive to a drastic increase in base modulus due to increasing rigidity in the base structure, leading to interference waves reflected internally. Vertical displacement (mm) is on the y-axis, and time steps are on the x-axis. The vertical displacement is given for E equals 482.6 MPa and for E equals 965.3 MPa

Figure 91. Chart. Compaction over a sequence of passes as the amplitude of vibration increases. This chart shows the material response according to the mechanics of the loading algorithm implemented. Compaction (percent) is on the y-axis, and pass number is on the x-axis. An increase in the amplitude of the vibratory load (3 mm, 5 mm, and 7 mm) leads to an increase in the amount of compaction achieved, as observed in field compaction

Figure 92. Chart. Compaction over a sequence of passes at different frequencies. This chart shows the material response according to the mechanics of the loading algorithm implemented. Compaction (percent) is on the y-axis, and pass number is on the x-axis. An increase in the frequency of the vibration (60 Hz and 3,600 vpm, 50 Hz and 3,000 vpm, and 40 Hz and 2,400 vpm) leads to an increase in the amount of compaction achieved, as observed in field compaction

Figure 93. Chart. Material response to change in dead load carried by each roller. This chart shows the material response according to the mechanics of the loading algorithm implemented. Compaction (percent) is on the y-axis, and pass number is on the x-axis. An increase in the load (30,000 lb, 27,000 lb, and 24,000 lb) leads to an increase in the amount of compaction achieved

Figure 94. Chart. Field compaction response at a constant frequency over multiple passes on a point. This chart shows the material response according to the mechanics of the loading algorithm implemented. Compaction (per unit) is on the y-axis, and time (seconds) is on the x axis. A decrease in the material's viscous nature (lambda equals 0.1 and lambda equals 4.0) leads to an increase in the amount of compaction achieved, as observed in field compaction

Figure 95. Chart. Evolution of the volumetric viscous gradient with a change in values of individual parameters , , λ1, and λ2. This chart shows the viscous evolution of the model as the different parameters are varied. The chart shows det(G)-1 on the y-axis and time steps on the x-axis. The parameters are estimated mu equals 810, estimated eta equals 1,700, lambda subscript 1 equals 0.2, and lambda subscript 2 equals 0.2. At the end of the time steps, lambda subscript 2 has the highest det(G)-1. Estimated eta has the next highest, and the other parameters are roughly the same

Figure 96. Chart. Evolution of the volumetric viscous gradient with a change in values of individual parameters n1, n2, q1, and q2. This chart shows the viscous evolution of the model as the different parameters are varied. The chart shows det(G)-1 on the y-axis and time steps on the x-axis. The parameters are estimated n subscript 1 equals 5, n subscript 2 equals 1.5, q subscript 1 equals -10, and q subscript 2 equals -15. At the end of the time steps, n subscript 1 has the highest det(G)-1. Next are q subscript 2 and q subscript 1; n subscript 2 has the lowest det(G)-1

Figure 97. Graphic. Regions of influence of model parameters in gyratory compaction.(51) This graphic shows a chart with normalized height on the y-axis and time of compaction (seconds) on the x-axis. The chart has a line and descriptions of points and planes on the line: the initial viscous dissipation limit, due lambda subscript 2; the onset of high viscosity, determined by estimated eta; the degree of nonlinear transition, controlled through q subscript 2; the beginning of material consolidation, determined by the estimated mu; the rate of consolidation, governed by n subscript 1; and the asymptotic consolidation limit, due lambda subscript 1

Figure 98. Chart. Evolution of the volumetric viscous gradient with a change in λ1. This chart shows the results of simulations performed at different stiffnesses where lambda subscript 1 is varied and its effect on volumetric viscous gradient is studied. The chart shows det(G)-1 on the y-axis and time steps on the x-axis. Three lines represent lambda subscript 1 equals 0.2, lambda subscript 1 equals 0.25, and lambda subscript 1 equals 0.3. The profiles of the deflection and volumetric part of viscous evolution indicate that the model response is not sensitive to lambda subscript 1. The parameter can be a model constant and can be the same constant as in the Superpave® gyratory compactor (SGC) simulations

Figure 99. Chart. Evolution of the volumetric viscous gradient with a change in q1. This chart shows the results of simulations performed at different stiffnesses where q subscript 1 is varied and its effect on volumetric viscous gradient is studied. The chart shows det(G)-1 on the y-axis and time steps on the x-axis. Three lines represent q subscript 1 equals -20, q subscript 1 equals -15, and q subscript 1 equals -10. The model is sensitive to q subscript 1. The volumetric part of the viscous evolution exhibits no sensitivity to q subscript 1. The same constant that was used in the Superpave® gyratory compactor (SGC) simulations can be used for this parameter

Figure 100. Chart. Evolution of the volumetric viscous gradient with a change in n2. This chart shows the results of simulations performed at different stiffnesses where n subscript 2 is varied and its effect on volumetric viscous gradient is studied. The chart shows det(G)-1 on the y-axis and time steps on the x-axis. Three lines represent n subscript 2 equals 1.5, n subscript 2 equals 2.5, and n subscript 2 equals 3.5. The model exhibits considerable sensitivity to changes in n subscript 2. This response from the model shows deviation from the response exhibited by the model during the Superpave® gyratory compactor (SGC) simulations

Figure 101. Chart. Evolution of the volumetric viscous gradient with a change in λ2. This chart shows the results of simulations performed at different stiffnesses where lambda subscript 2 is varied and its effect on volumetric viscous gradient is studied. The chart shows det(G)-1 on the y-axis and time steps on the x-axis. Three lines represent lambda subscript 2 equals 0.30, lambda subscript 2 equals 0.22, and lambda subscript 2 equals 0.15. The model indicates that lambda subscript 2 in the Superpave® gyratory compactor (SGC) simulation is also significant for field compaction

Figure 102. Chart. Evolution of the volumetric viscous gradient with a change in q2. This chart shows the results of simulations performed at different stiffnesses where q subscript 2 is varied and its effect on volumetric viscous gradient is studied. The chart shows det(G)-1 on the y-axis and time steps on the x-axis. Three lines represent q subscript 2 equals -25, q subscript 2 equals -20, and q subscript 2 equals -15. The model indicates that q subscript 2 in the Superpave® gyratory compactor (SGC) simulation is also significant for field compaction

Figure 103. Chart. Evolution of the volumetric viscous gradient with a change in n1. This chart shows the results of simulations performed at different stiffnesses where n subscript 1 is varied and its effect on volumetric viscous gradient is studied. The chart shows det(G)-1 on the y-axis and time steps on the x-axis. Three lines represent n subscript 1 equals 3.0, n subscript 1 equals 4.0, and n subscript 1 equals 5.0. The model indicates that n subscript 1 in the Superpave® gyratory compactor (SGC) is also significant for field compaction. However, while n subscript 1 is less than or equal to 4.0, the model does not exhibit any sensitivity to this parameter, whereas at a higher value of 5.0, the model exhibits significantly higher stiffness. Hence, a correlation exists in parametric behavior for both compaction processes

Figure 104. Chart. Plot representing the final compacted state of the material along the width of the pavement. This chart shows the overlap zone between two roller passes. The zone contains the longitudinal joint where material shoving occurs to accommodate the compaction of the material. The chart shows the compaction predicted with a refined mesh for roller position 1 (passes 1 and 2) and roller position 2 (passes 3 and 4). Material shoves laterally in the first and second passes, and the free edge compacts less than at the middle of the lane. This agrees with observations in the field

Figure 105. Illustration. Schematic of a roller on a material with three locations for density measurements.(49) This graphic shows how the finite-element (FE) model is used to simulate field rolling compaction on material 12 ft wide with the roller centered on the material. The graphic gives pavement quality indicator measurement locations A, B, and C in the material. C is depicted at a location marked 4.5 ft from the left side of the graphic, B is depicted at a location marked 7.5 ft from the left side of the graphic, and A is depicted at a location marked 9 ft from the left side of the graphic

Figure 106. Chart. Measurements of the %AV in the asphalt mix.(49) This chart shows the measured percent air voids in the material for locations A, B, and C. Percent air voids (percent) are on the y-axis, and pass number is on the x-axis. A comparison of the typical simulated responses in using the finite-element (FE) model shows that the model developed does predict a trend of compaction over multiple passes similar to that measured in the field.

Figure 107. Chart. Measurements of change in %AV in the asphalt mix.(49) This chart shows the measured change in the percent air voids in the material for locations A, B, and C. Change in air voids (percent) is on the y-axis, and pass number is on the x-axis. A comparison of the typical simulated responses in using the finite-element (FE) model shows that the model developed does predict a trend of compaction over multiple passes similar to that measured in the field

Figure 108. Chart. Measurement and modeling results of %AV at point A of the pavement locations shown in figure 105. The chart is a comparison of the percent air voids measured at a point A of the pavement location shown in figure 105 against the simulation results obtained from using the model. The percent air voids are represented on the y-axis and the roller pass numbers are represented on the x-axis. The two lines are fairly close together, with the measurement line slightly higher than the model line

Figure 109. Chart. Measurement and modeling results of %AV at point B of the pavement locations shown in figure 105. The chart is a comparison of the percent air voids measured at a point B of the pavement location shown in figure 105 against the simulation results obtained from using the model. The percent air voids are represented on the y-axis and the roller pass numbers are represented on the x-axis. The two lines are close together, with the model line starting higher than the measurement line and the two intersecting at pass number 2

Figure 110. Chart. Measurement and modeling results of %AV at point C of the pavement locations shown in figure 105. The chart is a comparison of the percent air voids measured at a point C of the pavement location shown in figure 105 against the simulation results obtained from using the model. The percent air voids are represented on the y-axis and the roller pass numbers are represented on the x-axis. The two lines are close together, with the model line starting higher than the measurement line and the two intersecting at pass number 2

Figure 111. Illustration. Pavement structure for the US-87 project. This graphic shows the structure of the pavement for the US-87 pavement project. The bottom layer, depicted in magenta, is 6 inches (152.4 mm) of lime-treated subgrade. The next layer, depicted in red, is 6 inches (152.4 mm) of flexible base. Next, depicted in yellow, is 3.5 inches (88.9 mm) of Type B hot-mix asphalt (HMA). The surface layer, depicted in blue, is 2 inches (50.8 mm) of Type C HMA

Figure 112. Chart. Schematic for the rolling patterns for the US-87 project. This graphic shows the simulation of the sequence and location of the roller for the US-87 project, along with the boundary conditions representative of the restrained and unrestrained edges of the pavement. Line segments represent rollers with arrows indicating their rolling directions. An upward arrow indicates forward rolling, and a downward arrow indicates the reverse. The scale is 1 division equals 1 ft (0.304.8 m). The roller is 7 ft (2.135 m) wide

Figure 113. Illustration. Pavement structure for the US-259 project. This graphic shows the structure of the pavement for the US-259 pavement project. The bottom layer, depicted in red, is 10 inches (254 mm) of flexible base. The next layer, depicted in yellow, is 9 inches (228.6 mm) of Type B hot-mix asphalt (HMA) and asphalt concrete pavement. The surface layer, depicted in blue, is 2 inches (50.8 mm) of Type C HMA

Figure 114. Chart. Schematic for the rolling patterns for the US-259 project. This graphic shows the simulation of the sequence and location of the roller for the US-259 project, along with the boundary conditions representative of the restrained and unrestrained edges of the pavement. Line segments represent rollers with arrows indicating their rolling directions. An upward arrow indicates forward rolling, and a downward arrow indicates the reverse. The scale is 1 division equals 1 ft (0.305 m). The roller is 7 ft (2.135 m) wide

Figure 115. Chart. Comparison of the total percent compaction from simulations with the general trend of the %AV measured at the end of the field compaction process for US-87. This chart shows that the compaction for US-87 predicted by the Superpave® gyratory compactor (SGC) parameters is outside the range of the change in percent air void (%AV) measured in the field. Therefore, the parameters (shear modulus) are adjusted so that the compaction obtained in simulations is contained within the range of measured %AV values. Within the compaction zone, the behavior of the mix in the simulations correlates well with the trends observed in the field. In the chart, compaction (%) is shown on the y-axis and cores (group number) are shown on the x axis. The comparison is made per core group, which represents a different location across the material relative to the edge

Figure 116. Chart. Total percent compaction from simulations compared to the general trend of the %AV measured at the end of the field compaction process for US-259. This chart shows that the compaction for US-259 predicted by the Superpave® gyratory compactor (SGC) parameters is outside the range of the change in percent air voids (%AV) measured in the field. Therefore, the parameters (shear modulus) are adjusted so that the compaction obtained in simulations is contained within the range of measured %AV values. Within the compaction zone, the behavior of the mix in the simulations correlates well with the trends observed in the field. In the chart, compaction (%) is shown on the y-axis and cores (group number) are shown on the x axis. The comparison is made per core group, which represents a different location across the material relative to the edge

Figure 117. Chart. Comparison of prediction of percent compaction per roller pass for US 87 and US-259. This chart shows that the simulations predict that the US-87 material will undergo more compaction by the end of the whole process than the US-259 material. Compaction is calculated at a common location on the material for both projects. Percent compaction is on the y-axis, and number of passes over entire mat is on the x-axis. The calculations were taken at a distance of 1 ft (0.305 m) from the edge

Figure 118. Chart. Prediction of percent compaction per roller pass across the material for US-87 (cores taken at four locations). This chart shows the material behavior of US-87 pavement undergoing compaction. Percent compaction is on the y-axis, and number of passes over entire mat is on the x-axis. Core group 4 undergoes the most compaction. Core groups 2 and 3 are next in percentage of compaction, respectively. Core group 1 undergoes the least compaction

Figure 119. Chart. Prediction of percent compaction per roller pass across the material for US-259 (cores taken at four locations). This chart shows the material behavior of US-259 pavement undergoing compaction. Percent compaction is on the y-axis, and number of passes over entire mat is on the x-axis. Core group 2 undergoes the most compaction. Core groups 1, 5, and 3 are next in percentage of compaction, respectively. Core group 4 undergoes the least compaction

Figure 120. Illustration. Micromechanical response of an asphalt mix. This figure is a schematic illustration of the combination of a continuum level multiplicative split of the total deformation, script F, into the inelastic deformation gradient, script G, and the elastic deformation gradient, script F subscript (script e), that takes into account the micromechanical response of asphalt mix

Figure 121. Chart. Representation of tasks involved in modeling asphalt compaction. This figure is a schematic representation of the modeling steps involved in relating the laboratory characterization of asphalt mixes through the conduct of gyratory experiments, simulating the gyratory experiments, and their field compaction behavior. The relationship between the continuum model parameters and the mixture compositional properties is first understood through the laboratory experiment and its simulation. The understanding gained is used to simulate and study laboratory and field compaction is more general cases to characterize asphalt mix compaction. Finally, an understanding is gained of the influence of the compositional properties on the field compaction process.

 


The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
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United States Department of Transportation - Federal Highway Administration