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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
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Publication Number: FHWA-HRT-08-073 Date: September 2009 |
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The constitutive modeling of asphalt concrete behavior is a topic that has gained national importance in the past few years. Such modeling efforts have the explicit goal of providing for better design and analysis of asphalt pavement structures to resist failure and/or better predict when failure will occur. These efforts should thus provide the tools necessary to better utilize available resources and/or to gain maximum results from limited resources. One such modeling effort that encompasses the two main forms of pavement distress, cracking and permanent deformation, is the multiaxial viscoelastoplastic continuum damage (MVEPCD) model and finite element package, finite element program (FEP++). The MVEPCD model combines elements of viscoelasticity, continuum damage mechanics, and viscoplasticity to model the material behavior, and FEP++ is used to model the interaction of material and structure.
The MVEPCD model has been characterized and verified using asphalt concrete mixtures tested at the Federal Highway Administration's Accelerated Load Facility in McLean, VA. A novel approach to modeling this process is suggested and verified in this work. In light of practical concerns related to constant rate tests using the Asphalt Mixture Performance Tester and due to the complexities of performing true time-dependent analysis of cyclic fatigue tests, a refined and simplified viscoelastoplastic continuum damage model is presented. A robust FEP++ has been developed to account for the effects of loading and boundary conditions. Analysis can be performed in either two-dimensional or three-dimensional configurations. The resulting predictions are deemed reasonable and, thus, a reliable simulation of pavement response.
Cheryl Allen Richter
Acting Director, Office of Infrastructure
Research and Development
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1. Report No. FHWA-HRT-08-073 |
2. Government Accession No. |
3. Recipient's Catalog No. |
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4. Title and Subtitle Development of a Multiaxial Viscoelastoplastic Continuum Damage Model for Asphalt Mixtures |
5. Report Date September 2009 |
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6. Performing Organization Code |
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7. Author(s) Y. Richard Kim, Ph.D. P.E., M.N. Guddati, Ph.D., B.S. Underwood, T.Y. Yun, V. Subramanian, S. Savadatti |
8. Performing Organization Report No. |
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9. Performing Organization Name and Address North Carolina State University |
10. Work Unit No. (TRAIS) |
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11. Contract or Grant No. DTFH61-05-H-00019 |
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12. Sponsoring Agency Name and Address Office of Research and Technology Services |
13. Type of Report and Period Covered |
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14. Sponsoring Agency Code FHWA |
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15. Supplementary Notes Project performed under FHWA DTFH61-05-H-00019; FHWA AOTR: Katherine Petros |
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16. Abstract This report highlights findings from the FHWA DTFH61-05-H-00019 project, which focused on the development of the multiaxial viscoelastoplastic continuum damage model for asphalt concrete in both compression and tension. Asphalt concrete pavement, one of the largest infrastructure components in the United States, is a complex system that involves multiple layers of different materials, various combinations of irregular traffic loading, and various environmental conditions. The performance of this structure is closely related to the performance of asphalt concrete. To predict the performance of asphalt concrete with reasonable accuracy, a better understanding of its deformation behavior under realistic conditions is urgently needed. Over the past decade, the authors have been successful in developing uniaxial material models that can accurately capture various critical phenomena such as microcrack-induced damage that is critical in fatigue modeling, strain-rate temperature interdependence, and viscoplastic flow, which is critical for high temperature modeling. The resulting model is termed the viscoelastoplastic continuum damage model. However, to consider the complicated nature of in-service stress states, a multidimensional model is needed. To predict the performance of the real pavement structures, it is also important to incorporate the material model in a pavement model that considers the vehicle and climatic loads as well as the boundary conditions; the in-house finite element package has been developed for this purpose. |
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17. Key Words Constitutive modeling, Asphalt concrete, Viscoelastic, Viscoplastic, Multiaxial, VECD, Damage mechanics |
18. Distribution Statement No restrictions. This document is available through the National Technical Information Service, Springfield, VA 22161. |
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19. Security Classif. (of this report) Unclassified |
20. Security Classif. (of this page) Unclassified |
21. No. of Pages 264 |
22. Price |
Form DOT F 1700.7 (8-72) Reproduction of completed page authorized
SI (Modern Metric) Conversion Factors
Figure 1. Graph. Schematic representation of dynamic modulus shifting process with unshifted data
Figure 2. Graph. Schematic representation of dynamic modulus shifting process with shifted data
Figure 4. Graph. Constant crosshead test results in stress-strain space
Figure 5. Graph. Constant crosshead test results in stress-pseudo strain space
Figure 6. Graph. Comparison of refined and approximate damage calculation techniques
Figure 7. Graph. Comparison of refined and approximate damage characteristic relationship
Figure 8. Illustration. Mechanical analog for the viscoplastic model
Figure 9. Illustration. Isotropic hardening diagram
Figure 10. Illustration. Kinematic hardening diagram
Figure 11. Illustration. Strain decomposition from creep and recovery testing
Figure 12. Illustration. Typical yield surface of HISS model
Figure 13. Graph. Mixture gradation chart
Figure 14. Graph. Comparison of test lane and laboratory gradations
Figure 15. Graph. Stress history of VL testing (unconfined and 140 kPa confinement VL)
Figure 16. Graph. Stress history of VL testing (500 kPa confinement VL)
Figure 17. Graph. Stress history of VLT testing (140 kPa confinement)
Figure 18. Graph. Stress history of VLT testing (500 kPa confinement)
Figure 19. Graph. Effect of 500 kPa confining pressure on the dynamic modulus in semi-log space
Figure 20. Graph. Effect of 500 kPa confining pressure on the dynamic modulus in log-log space
Figure 21. Graph. Effect of 500 kPa confining pressure on observed elasticity in the Control mixture
Figure 22. Graph. Effect of 500 kPa confining pressure on the log shift factor function
Figure 23. Graph. Effect of confining pressure on the relaxation spectrum
Figure 24. Graph. Use of the uniaxial relaxation spectrum for multiaxial test results
Figure 26. Graph. Multiaxial equilibrium characterization results
Figure 27. Graph. Multiaxial dynamic modulus model strength
Figure 28. Graph. C11 versus S for tension for Control-2006 mixture (5 °C reference)
Figure 29. Graph. Effect of time-dependent Poisson's ratio on C12 calculation
Figure 30. Graph. C12 characteristic curve for Control-2006 mixture
Figure 31. Graph. C12 characteristic curve for Control-2006 mixture in semi-logarithmic space
Figure 33. Graph. C12 versus S for tensile loading for Control-2006 mixture (5 °C reference)
Figure 46. Graph. C22 as a function of S calculated by different methodologies for 5-1-T
Figure 47. Graph. C22 as a function of S calculated by different methodologies for 5-3-T
Figure 48. Graph. C22 as a function of S calculated by different methodologies for 5-4-T
Figure 49. Graph. C22 as a function of S calculated by different methodologies for 5-5-T
Figure 51. Graph. Effect of confining pressure on coefficient Y
Figure 59. Graph. t-TS with growing damage under confinement verification at a 0.0001ε level
Figure 60. Graph. t-TS with growing damage under confinement verification at a 0.0005ε level
Figure 61. Graph. t-TS with growing damage under confinement verification at a 0.001ε level
Figure 62. Graph. t-TS with growing damage under confinement verification at a 0.0022ε level
Figure 63. Graph. t-TS with growing damage under confinement verification at a 0.004ε level
Figure 64. Graph. t-TS with growing damage under confinement verification at a 0.007ε
Figure 65. Graph. Effect of 500 kPa confining pressure on strength mastercurves
Figure 66. Graph. Effect of 500 kPa confining pressure on ductility in constant crosshead rate tests
Figure 67. Illustration. A schematic representation of the concept of average dC/dt
Figure 69. Illustration. Mathematical equivalence of the formulation used by Lee, Daniel, and Kim
Figure 85. Graph. Stress-strain curves for unconfined constant strain-rate tests
Figure 86. Graph. Stress-strain curves for 500 kPa confinement constant strain-rate tests
Figure 87. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 5 °C
Figure 88. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 25 °C
Figure 89. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 40 °C
Figure 90. Graph. Comparison of 500 kPa confinement and unconfined constant rate tests for 55 °C
Figure 91. Graph. Viscoplastic strain versus cumulative loading time (unconfined VL)
Figure 92. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VL)
Figure 93. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VL)
Figure 94. Graph. Viscoplastic strain versus cumulative loading time (unconfined VT testing)
Figure 97. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement CLT)
Figure 98. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT)
Figure 115. Graph. Stress mastercurves for the Control mixture under uniaxial conditions
Figure 116. Graph. Stress mastercurves for the Control mixture under triaxial conditions (500 kPa)
Figure 117. Graph. Variation of strain rate during unloading
Figure 119. Graph. Confining stress effect on the relaxation modulus
Figure 122. Graph. Effect of test method on shift factor functions
Figure 125. Diagram. MVECD model characterization
Figure 126. Graph. C11 versus S for compression for Control mixture (5 °C reference)
Figure 127. Graph. C12 versus S for compression for Control mixture (5 °C reference)
Figure 128. Graph. C22 versus S for compression for Control mixture (5 °C reference)
Figure 129. Graph. Comparison of tension and compression of C11 damage function
Figure 130. Graph. Comparison of tension and compression of C12 damage function
Figure 131. Graph. Comparison of tension and compression of C22 damage function
Figure 133. Graph. Determined fitting results and coefficients of function a(tp)
Figure 134. Graph. Determined fitting results and coefficients of function D(tp σ)
Figure 135. Graph. VT predictions
Figure 136. Graph. VL predictions
Figure 137. Graph. CLT predictions (2.0 MPa deviatoric stress-0.1-s pulse time)
Figure 138. Graph. CLT predictions (2.0 MPa deviatoric stress-0.4-s pulse time)
Figure 139. Graph. CLT predictions (2.0 MPa deviatoric stress-1.6-s pulse time)
Figure 140. Graph. CLT predictions (2.0 MPa deviatoric stress-6.4-s pulse time)
Figure 141. Graph. CLT predictions (1.8 MPa deviatoric stress-1.6-s pulse time)
Figure 142. Graph. CLT predictions (2.2 MPa deviatoric stress-1.6-s pulse time)
Figure 143. Graph. Compressive and tensile peak stress in SQRT(J2) - I1 space
Figure 144. Graph. Determined γ0 parameter function
Figure 145. Graph. Determined R parameter function
Figure 146. Graph. Determined n parameter function
Figure 147. Graph. Determined α0 parameter function
Figure 148. Graph. Rate-dependent initial yield surface
Figure 149. Graph. Variation of α for 1,800 kPa CLT loading (500 kPa confinement)
Figure 150. Graph. Viscoplastic strain predictions for VT tests (500 kPa confinement)
Figure 151. Graph. Viscoplastic strain predictions for CLT tests (500 kPa confinement)
Figure 152. Graph. Viscoplastic strain predictions for RVT tests (500 kPa confinement)
Figure 153. Illustration. Variation of yield stress (Standard Linear Solid model)
Figure 154. Graph. Stress histories for rest period analysis
Figure 155. Graph. Yield stress versus cumulative loading time (rest period analysis)
Figure 156. Graph. Viscoplastic strain versus cumulative loading time (rest period analysis)
Figure 157. Graph. Stress history for loading time analysis
Figure 158. Graph. Viscoplastic strain versus cumulative loading time (loading time analysis)
Figure 159. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT)
Figure 160. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VL)
Figure 161. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VT)
Figure 162. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VL)
Figure 163. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT)
Figure 164. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VLT)
Figure 165. Graph. Viscoplastic strain versus cumulative loading time (140 kPa confinement VT)
Figure 172. Graph. Viscoplastic strain versus cumulative loading time (500 kPa confinement VLT)
Figure 174. Graph. Layout of the numerical experiment specimen
Figure 177. Graph. Evolution of damage parameter, S, for the monotonic test
Figure 178. Graph. Plot of function C(S) with time
Figure 179. Diagram. Domain module
Figure 180. Illustration. Infinite elastic layer on a rigid base
Figure 183. Diagram. Analysis module
Figure 184. Screen capture. Main window of the preprocessor
Figure 185. Screen capture. Control panel
Figure 186. Screen capture. Mesh discretization
Figure 187. Screen capture. Zoom operation on a mesh
Figure 188. Diagram. Stages involved in an FEP++ analysis
Figure 189. Screen capture. Sample data entry window
Figure 190. Screen capture. Error dialog for a semantic error
Figure 191. Screen capture. Analysis run-time window
Figure 192. Graph. Temperature variations used for simulations
Figure 193. Illustration. Vertical strains in winter
Figure 194. Illustration. Vertical strains in summer
Figure 195. Illustration. Longitudinal strains in winter
Figure 196. Illustration. Longitudinal strains in summer
Figure 197. Illustration. Transverse strains in winter
Figure 198. Illustration. Transverse strains in summer
Figure 199. Illustration. Vertical strains for Control mixture
Figure 200. Illustration. Vertical strains for SBS mixture
Figure 201. Illustration. Longitudinal strains for Control mixture
Figure 202. Illustration. Longitudinal strains for SBS mixture
Figure 203. Illustration. Transverse strains for Control mixture
Figure 204. Illustration. Transverse strains for SBS mixture
Figure 205. Illustration. Vertical strains for a wheel speed of 13.41 m/s
Figure 206. Illustration. Vertical strains for a wheel speed of 26.82 m/s
Figure 207. Illustration. Longitudinal strains for a wheel speed of 13.41 m/s
Figure 208. Illustration. Longitudinal strains for a wheel speed of 26.82 m/s
Figure 209. Illustration. Transverse strains for a wheel speed of 13.41 m/s
Figure 210. Illustration. Transverse strains for a wheel speed of 26.82 m/s
Figure 211. Screen capture. General Information dialog
Figure 212. Screen capture. Material Properties dialog
Figure 213. Screen capture. Elastic material properties dialog
Figure 214. Screen capture. Viscoelastic material properties dialog
Figure 215. Screen capture. Prony Coefficients dialog
Figure 216. Screen capture. Layer properties dialog
Figure 217. Screen capture. Mesh properties dialog
Figure 218. Screen capture. Load properties dialog
Figure 219. Screen capture. Analysis Parameters dialog
Figure 220. Screen capture. Summary dialog
Figure 221. Screen capture. FEP++ analysis in progress
Table 1. Relevant asphalt binder information
Table 2. Summary of constructed lanes' air void and asphalt content
Table 3. Controlled crosshead testing matrix for Control-2006 in tension
Table 4. Controlled crosshead testing matrix for Control in compression
Table 5. Creep and recovery testing matrix for Control mixture in compression
Table 6. Test conditions for the VT and RVT tests
Table 7. Loading times for VLT test
Table 12. Cyclic tests performed
Table 13. Summary of cyclic correction factors
Table 14. Summary of αvalues for refined model
Table 15. Cyclic tests performed for 9.5-mm Fine mixture
Table 20. Delft material model coefficients functions
Table 21. Material coefficients used for the developed model analysis
Table 22. Compression viscoplastic material model coefficients
Table 23. Properties of pavement
Table 24. Properties of moving wheel load
| 2D |
Two-dimensional |
| 3D |
Three-dimensional |
| ALF |
Accelerated Load Facility |
| AMPT |
Asphalt Mixture Performance Tester |
| CLT |
Constant loading level and time test |
CR-TB |
Crumb Rubber Terminal Blend |
| CS |
Controlled stress |
| CX |
Controlled crosshead |
| EICM |
Enhanced Integrated Climatic Model |
| FEM |
Finite element method |
| FEP++ |
Finite element package (proper name) |
| FHWA |
Federal Highway Administration |
| GUI |
Graphical user interface |
| HISS |
Hierarchical single surface |
| HMA |
Hot mix asphalt |
| IDT |
Indirect tension test |
| LVDT |
Linear variable displacement transducers |
| LVE |
Linear viscoelastic |
| MVECD |
Multiaxial viscoelastic continuum damage |
| MVEPCD |
Multiaxial viscoelastoplastic continuum damage |
| NCHRP |
National Cooperative Highway Research Program |
| NCSU |
North Carolina State University |
| NMSA |
Nominal maximum size aggregate |
| RVT |
Reversed variable loading time test |
| SBS |
Styrene-butadiene-styrene |
| SHRP |
Strategic Highway Research Program |
| SPT |
Simple Performance Tester |
| t-TS |
Time-temperature superposition |
| TFHRC |
Turner-Fairbank Highway Research Center |
| TRS |
Thermorheologically simple |
| UVECD |
Uniaxial viscoelastic continuum damage |
| UVEPCD |
Uniaxial viscoelastoplastic continuum damage |
| VECD |
Viscoelastic continuum damage |
| VECD-FEP++ |
Viscoelastic continuum damage model included in finite element package |
| VEPCD |
Viscoelastoplastic continuum damage |
| VP |
Viscoplastic |
| VL |
Variable loading level test |
| VLT |
Variable loading level and time test |
| VT |
Variable loading time test |
| VTK |
Visualization Toolkit |
| εeif | elastic strain |
| εpif | plastic strain |
| εcif | creep strain |
| εvif | viscoplastic strain |
| εvpif | elastic plus linear viscoelastic strain due to damage |
| εve | elastic plus linear viscoelastic strain due to damage |
| ∫R | reduced physical frequency (Hz) |
| ωR | reduced angular frequency (=2π∫RΔt) |
| αt | time-temperature shift factor at specific temperature, T |
| π | 3.141593--- |
| σ | stress |
| ε | strain |
| εx | strain in direction x |
| με | microstrain |
| Ε(t) | relaxation modulus |
| D(t) | creep compliance |
| τ | dummy integration variable |
| Ρ subscript i | relaxation time (fitting coefficient) |
| τj | retardation time (fitting coefficient) |
| Ε∞ | relaxation modulus at infinite time |
| Εi | Prony coefficient for relaxation modulus |
| Dε | glassy compliance |
| Dj | Prony coefficient for creep compliance |
| Ε' | storage modulus |
| | Ε* | | dynamic modulus |
| Φ | phase angle |
| t | time |
| ξ | reduced time |
| ∂ | partial derivative |
| ΔX | delta, finite difference in X |
| ∂X | delta, finite difference in X |
| εR | pseudo strain |
| ΕR | reference modulus |
| C | normalized pseudo secant modulus |
| I | normalization parameter |
| S | damage |
| W | strain energy density |
| W dR | pseudo energy density |
| α | viscoelastic damage growth rate |
| Cikl | general stiffness matrix |
| ν | Poisson's ratio |
| Ε | elastic elongational modulus |
| G | elastic shear modulus |
| Zij | stiffness matrix for transversely isotropic material |
| Ε3 | modulus along axis of symmetry |
| Ε | modulus perpendicular to axis of symmetry |
| v3132 | Poisson's ratio between axis of symmetry and perpendicular plane |
| v1323 | Poisson's ratio between perpendicular plane and axis of symmetry |
| v12 | Poisson's ratio on perpendicular plane |
| Υ | Poisson's ratio term |
| Si | compliance matrix for transversely isotropic material |
| Aj | alterative stiffness matrix terms for transversely isotropic material |
| ev | volumetric strain (dilation) |
| e3 | major deviatoric strain |
| e2 | strain difference |
| Ρ | pressure |
| C11 | first material integrity term |
| C12 | second material integrity term |
| C22 | third material integrity term |
| υ | dilation |
| υR | pseudo dilation |
| Qi | generalized loads |
| qi | generalized displacements |
| υs | complementary strain energy |
| εvp | viscoplastic strain rate |
| λ | positive scalar factor |
| g | plastic potential function (general viscoplasticity) |
| Φ | overstress function |
| η | viscosity |
| Γ | fluidity |
| I1 | first stress invariant |
| J2 | second deviatoric stress invariant |
| J3 | third deviatoric stress invariant |
| sij | deviatoric stress |
| δIJ | Kroneker delta |
| Κ | isotropic hardening parameter |
| αIJ | viscoplastic kinematic hardening parameter |
| g | stress function (strain hardening model specific) |
| Pα | atmospheric pressure |
| R | tensile strength parameter |
| n | yield stress shape parameter |
| β | parameter determining shape of yield stress in deviatoric stress space |
| εin | inelastic strain rate |
| κ | isotropic hardening function |
| α | viscoplastic back stress parameter (kinematic hardening) |
| D | drag stress |
| R | isotropic hardening function |
| H | kinematic hardening function |
| G | back stress (viscoplastic model) |
| ζ | viscoplastic strain trajectory |
| θ | angular direction in axisymmetric coordinate system |
| α 1 | time-temperature shift factor function coefficient 1 |
| α2 | time-temperature shift factor function coefficient 2 |
| α3 | time-temperature shift factor function coefficient 3 |
| θ | bulk stress |
| εRpressure | pseudo strain due to pressure |
| υRpressure | pseudo dilation due to pressure |
| εRe | effective pseudo strain |
| εRs | permanent pseudo strain |
| εRm | total maximum pseudo strain |
| εRme | total effective pseudo strain at peak of loading |
| M | time change correction factor |
| tp | pulse time |
| ζp | reduced pulse time |
| N | number of points in a calculation |
| F | pseudo strain slope function |
| G | pseudo strain hysterisis function |
| H | healing function |
 |
analytical expression for pseudo strain as a function of time |
| Q | pseudo strain shape factor |
| Z | combined pseudo strain shape and pseudo stiffness time factor |
| εR0,ta | pseudo strain tension amplitude only |
| σ0,ta | stress tension amplitude only |
| σpp | peak-to-peak stress magnitude |
| β | factor quantifying time under tensile loading |
| σpeak | maximum value of stress in a cycle |
| σvalley | minimum value of stress in a cycle |
| ζi | reduced time within a cycle when tension loading begins |
| ζf | reduced time within a cycle when tension loading ends |
| Rt | form adjustment factor for characterization |
| κl | form adjustment factor for prediction |
| ε0,ta | strain tension amplitude only |
| α(tp) | phenomenological viscoplastic model slope pulse time function |
| D | phenomenological viscoplastic model intercept function |
| γ | viscoplastic softening parameter |
| Il,dilation | first stress invariant value at the beginning of dilation |
| J2,dilation | second deviatoric stress invariant value at the beginning of dilation |
| εreduced | reduced strain rate |
| D | viscosity parameter (final viscoplastic model) |
| G | orientation-dependent isotropic hardening function |
| Gn+1 | value of hardening function at next time step |
| Gn | value of hardening function at current time step |
| ζon | value of elastic state variable at current time step |
| ζon+l | value of elastic state variable at next time step |
| ζin | value of state variable for element i at current time step |
| ζin+l | value of state variable for element i at next time step |
