CAPITULO 2: GOBIERNO Y LAS JUNTAS PARROQUIALES EN EL MARCO DEL
2.1 Planificación gubernamental y políticas públicas
Figure 3.1: Experimental arrangements for tension tests, Cheung (1995)
ǫT ǫ
ǫr
ǫpl Time Time σ
=
b)
Time
σ
σmin
σmin
σmax
σmax
σm
R
c)
ǫT
Time Time σ
σp
∆p
∆p
∆g
∆g
Figure 3.2: (a) Schematic showing the stress and strain time histories in a creep recovery test. (b) Schematic of the applied stress as a function of time in the con-tinuous cyclic tests.(c) Schematic of the stress and strain time histories in the pulse train experiments.
0 0.1 0.2 0.3 0.4 0.5 0
0.5 1 1.5
σss
Nominal axial strain, ǫ
Nominalaxialstress,σ,(MPa)
Experiment Model
˙
ǫ= 0.02 s−1
˙
ǫ= 0.015 s−1
˙
ǫ= 0.007 s−1
˙
ǫ= 0.0005 s−1
Figure 3.3: Constant strain-rate tests at four selected values of the applied strain-rate on the 50 pen bitumen at 0◦C.
0 5 10 15 20 25 30 35 40
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
˙ ǫss
Nominalaxialstrain,ǫ
Time (s)
1.0
Experiment Model σ= 0.127 MPa
σ= 0.064 MPa
Figure 3.4: Constant stress creep tests at two selected creep stresses on the 50 pen bitumen at 10◦C.
10−5 10−4 10−3 10−2 10−1 100 101 10−5
10−4 10−3 10−2 10−1
10
✹
Steady-statestrainrate,˙ǫss,s−1
Steady-state stressσss, or mean stressσm (MPa) 20◦C 10◦C 0◦C −5◦C
Modified Cross Model Monotonic Strain Controlled Monotonic Stress Controlled Cyclic Stress Controlled
1.0
1.0 1.0
2.6
Figure 3.5: The monotonic and continuous cyclic steady-state behaviour of the 50 pen bitumen at the four temperatures investigated in the current study.
0 0.05 0.1 0.15 0.2 0.25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 x 10
−3
Nominal axial strain,ǫ
˙ǫoc,(s−1 )
50 pen 100 pen
Figure 3.6: Loading calibration curves ˙ǫoc(ǫ) for the 50 and 100 pen bitumens. Curves from a series of five constant strain-rate and creep tests at different temperatures are superimposed.
0 50 100 150 200 0
0.05 0.1 0.15 0.2 0.25
0 100 200 300 400 500 600 700
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
Experiment Experiment
Model Model a)
b)
NominalaxialstrainǫNominalaxialstrainǫ
Time (s) Time (s)
Figure 3.7: Creep recovery test results for 50 pen bitumen. (a) T = 10◦C, σ = 0.32 MPa. (b) T = 0◦C, σ = 0.2 MPa.
0 0.05 0.1 0.15 0.2 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14
✭
✹+
1.0
ǫr
ǫT =ǫr+ǫpl
ψ
σ= 0.41 MPa. 50 pen @10◦C σ= 0.32 MPa. 50 pen @10◦C σ= 0.16 MPa. 50 pen @10◦C
σ= 0.095 MPa. 100 pen @0◦C σ= 0.191 MPa. 50 pen @0◦C σ= 0.48 MPa. 50 pen @−5◦C
Figure 3.8: Summary of the creep recovery experimental results which show a linear relationship between ǫr and ǫT.
0 0.2 0.4 0.6 0.8 1
−5 0 5 10 15 x 10−4
50 pen 100 pen
ǫ ˆ
r˙ ǫ
uc, (s
−1)
Figure 3.9: Recovery calibration curves ˙ǫuc(ˆǫr) for the 50 and 100 pen bitumens.
Curves from a series of five creep recovery tests at different temperatures are super-imposed.
0 2 4 6 8 10 0
0.05 0.1 0.15 0.2 0.25
0 100 200 300 400 500
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
a)
b)
Experiment
Experiment Model
Model
N om in al ax ia l st ra in ǫ N om in al ax ia l st ra in ǫ
Time (s) Time (s)
σm= 0.064 MPa σm= 0.127 MPa
σm= 0.127 MPa σm= 0.255 MPa
Figure 3.10: Continuous cyclic stress controlled tests for 50 pen bitumen. (a) T = 10◦C, f = 2 Hz and R = 0.15. (b)T = 0◦C, f = 0.1 Hz and R = 0.15.
0 100 200 300 400 500 600 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14
0 100 200 300 400 500 600
0 0.02 0.04 0.06 0.08 0.1 0.12
Experiment
Experiment Model
Model R = 0.3
R = 0.6
R = 0.8
N om in al ax ia l st ra in , ǫ N om in al ax ia l st ra in , ǫ
Time (s) Time (s)
f = 0.1 Hz f = 1 Hz
f = 2 Hz
f = 10 Hz
b)
Figure 3.11: Continuous cyclic stress controlled tests at 0◦ C for 50 pen bitumen. (a) Results for three selected values of R with σm = 0.095 MPa and f = 0.1 Hz. (b) Results for four selected frequencies f with σm= 0.064 MPa and R = 0.15.
0 20 40 60 80 100 0
0.05 0.1 0.15 0.2 0.25
0 200 400 600 800 1000 1200
0 0.02 0.04 0.06 0.08 0.1 0.12
Experiment Experiment
Model
Model
a)
b)
N om in al ax ia l st ra in ǫ N om in al ax ia l st ra in ǫ
Time (s) Time (s)
∆g = 1 s
∆g = 2.5 s
∆g = 5 s
∆g = 30 s
∆g = 60 s
Figure 3.12: Pulse loading tests on the 50 pen bitumen. (a) T = 10◦C, σp = 0.32 MPa and ∆p = 0.2 s. (b) T = 0◦C, σp = 0.16 MPa, ∆p = 12 s.
0 200 400 600 800 1000 1200 0
0.02 0.04 0.06 0.08 0.1 0.12
Experiment Model
Nominalaxialstrainǫ
Time (s)
∆g = 30 s
∆g = 60 s
Figure 3.13: Pulse loading tests on 100 pen bitumen with T = 0◦C, σp = 0.095 MPa and ∆p = 12 s.
0.3 0.4 0.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0.2 0.1
0
Nominal axial strain, ǫ
Nominalaxialstress,σ(MPa)
Experiment Model
˙
ǫ= 0.01 s−1
˙
ǫ= 0.005 s−1
˙
ǫ= 0.001 s−1
σss
Figure 3.14: Monotonic constant strain-rate tests at three selected values of the applied strain-rate at 0◦C on Cariphalte TS.
0 50 100 150 200 250 300 350 400 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
˙ ǫss
1.0
Nominalaxialstrain,ǫ
Time, (s)
Experiment Model
σ= 0.06 MPa σ= 0.16 MPa
Figure 3.15: Monotonic constant stress creep tests on Cariphalte TS at two selected values of the applied stress at 0◦C.
10−2 10−1 100 101
10−4 10−3 10−2 10−1 100
Steady-statestrainrate,˙ǫss,s−1
Steady-state stressσss(MPa) 20◦C
10◦C
0◦C
◦
Modified Cross Model Monotonic Strain Controlled Monotonic Stress Controlled
Figure 3.16: Monotonic steady-state behaviour of the polymer-modified bitumen Ca-riphalte TS at the three temperatures investigated in the current study.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0
0.5 1 1.5 x 10
−3
▲
▲
▲
▲
▲
▲ ▲ ▲ ▲
▲ ▲
▲
▲
▲
▲
▲ ▲
▲
Nominal axial strain, ǫ
˙ ǫ
oc, (s
−1)
Cariphalte DM
Curves used in modelling Cariphalte TS
Figure 3.17: Loading calibration curves ˙ǫoc(ǫ) for Cariphalte TS and DM bitumens.
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
00 100 200 300 400 500
0 0.2 0.4 0.6 0.8 1
0 x
*
0.1 0.2 0.3 0.4 0.5 0.6 0.7
+ b)
a)
ǫ
rǫ
T= ǫ
r+ ǫ
pl1.0
ψ = 0.65 Experiment
Model
N om in al ax ia l st ra in , ǫ
Time (s)
σ= 0.18 MPa at 10◦C σ= 0.32 MPa at 10◦C σ= 0.56 MPa at 10◦C σ= 0.67 MPa at 0◦C σ= 0.77 MPa at 10◦C σ= 0.90 MPa at 10◦C σ= 1.60 MPa at 0◦C
Figure 3.18: Creep recovery tests.(a) Creep recovery test results for Cariphalte TS at σ = 0.64 MPa and T = 0◦C. (b) Summary of creep recovery experimental results which show a linear relationship between ǫr and ǫT.
0 0.2 0.4 0.6 0.8 1
−2 0 2 4 6 8 10 12 14 16 18 x 10−4
ǫˆr
˙ǫuc,(s−1 ) Cariphalte DM Cariphalte TS 50 pen bitumen 100 pen bitumen
Figure 3.19: Recovery calibration curves ˙ǫuc( ˆǫr) for 50 and 100 pen bitumens and polymer-modified bitumens Cariphalte TS and DM.
80 60
40 20
0 100
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Nominalaxialstrain,ǫ
Time (s)
Experiment Model
σm= 0.8 MPa
σm= 0.32 MPa
Figure 3.20: Continuous cyclic stress controlled tests for Cariphalte TS at 0◦C, R=0.15 and f=2 Hz.
00 50 100 150 200 0.02
0.04 0.06 0.2 0.18 0.16 0.14 0.12 0.1 0.08
00 50 100 150 200
0.02 0.04 0.06 0.2 0.18 0.16 0.14 0.12 0.1 0.08
b)
R= 0.3
R = 0.6 R= 0.9
N om in al ax ia l st ra in , ǫ N om in al ax ia l st ra in , ǫ
Time (s)
Time (s)
f = 0.2 Hz
f = 1 Hz f = 2 Hz
f = 10 Hz
R =
σσminmax
Figure 3.21: Continuous cyclic stress controlled tests at 0◦C for Cariphalte TS. (a) Results for three selected values ofR withσm = 0.36 MPa and f = 2 Hz. (b) Results for four selected frequencies f with σm= 0.36 MPa and R = 0.3.
0 200 400 600 800 1000 1200 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Nominalaxialstrain,ǫ
Time (s)
Experiment Model
∆g= 60s
∆g= 30s
Figure 3.22: Pulse loading tests on Cariphalte TS with T = 0◦C, σp = 0.4 MPa and
∆p = 13s.
200
0 400 600 800
0.5
0.4
0.3
0.2
0.1
0 1000
Nominalaxialstrain,ǫ
Time (s)
Experiment Model
∆g= 90s
∆g= 30s
Figure 3.23: Pulse loading tests on Cariphalte DM with T = 0◦C, σp = 0.4 MPa and ∆p = 13s. Model calibration performed using a set of four simple tension and recovery tests.
Indentation behaviour of bitumen
4.1 Introduction
Indentation tests provide a cheap and easy method to measure the mechanical proper-ties of materials and also serve to validate multi-axial constitutive models of materials.
The focus of this chapter is to investigate the monotonic and cyclic spherical inden-tation response of bitumen with the aim of (i) validating the multi-axial constitutive model for bitumen developed in chapter 3 and (ii) investigating the repeated inden-tation response of bitumen which serves as a unit problem for road surfaces under vehicle loads.
The standard indentation test on creeping solids involves either applying a con-stant load and measuring the indentation creep with time or by pressing the indenter into the material at a prescribed rate and measuring the load as a function of time. To interpret these results many researchers have developed models to relate the inden-tation pressure to the constitutive response of the materials. Notably, Tabor (1951) proposed empirical relations to correlate the indentation pressure for rate indepen-dent strain hardening solids to the uniaxial tensile response of the material, while
65
Mulhearn and Tabor (1960) extended these empirical relations to power-law creeping materials. Using the similarity transformations for the indentation of metals devel-oped by Hill et al. (1989), Bower et al. (1993) provided a rigorous theoretical basis for the empirical relations developed by Mulhearn and Tabor (1960) for rate dependent solids. A source of error in the interpretation of creep properties from indentation tests is the neglect of the primary creep response (or the strain hardening behaviour) of rate dependent materials in the above analyses. Ogbonna et al. (1995) extended the scaling procedure of Hill et al. (1989) and Bower et al. (1993) to a class of creep constitutive laws that account for strain hardening. Such analyses provides the basis for the investigation of the indentation response of bitumen reported in this chapter.
In this chapter, the indentation model for power-law creeping solids of Bower et al. (1993) is summarised and then extended to the constitutive model for bitumen described in chapter 3. Then, an extensive experimental study of the monotonic, recovery and cyclic spherical indentation behaviour of bitumen is reported for a range of temperatures. Finally, the predictions of the model are compared with experimental measurements.