Establishment of Mathematics Model and Study on Correlation between Complex Modulus and Temperature of Asphalt

doi:10.3969/j.issn.1006-396X.2022.01.002

Abstract

Abstract:

In order to understand the temperature characteristic of complex modulus,the correlation between the double logarithms of complex modulus and expressions at different temperatures was studied through ten original asphalts produced by two kinds of crude oil as well as these asphalts after RTFOT and PAV.For the first time,through correlation between the slope and intercept of linear fitting correlation,the optimized mathematics formulas between complex modulus and temperature were determined. Mathematics models for different aging stages were established. Mathematics models are simple,accurate,and practical and show universal applicability by comparing measured data at single temperature with predicted data of three asphalts produced by the third crude oil.The data predicted by mathematics model meet the requirements of reproducibility precision of the test method of complex modulus.

Key words: Asphalt, Rheological properties, Correlation, Temperature, Complex modulus

摘要：

为了充分理解复合模量的温度特性，采用两种原油生产的10个原样沥青以及其RTFOT后和PAV后沥青，研究其复合模量的双对数分别与不同温度表达方式之间的相关性，首次通过线性拟合关系斜率和截距的关联研究，明确最适合表征复合模量与温度之间的数学关系式，建立了适合不同老化阶段的数学模型。通过第三种原油生产的3个沥青实测数据与在单一温度下的预测数据的对比，证明数学模型具有简单、准确、实用且更好的普遍适用性。数学模型预测的数据满足复合模量实验方法再现性精密度的要求。

关键词: 沥青, 流变性质, 关联, 温度, 复合模量

CLC Number:

TE626.8

Xiaoying Zhang, Chuanjie Xu, Qiang Zhang. Establishment of Mathematics Model and Study on Correlation between Complex Modulus and Temperature of Asphalt[J]. Journal of Petrochemical Universities, 2022, 35(1): 10-17.

张小英, 徐传杰, 张强. 沥青复合模量与温度关联研究及数学模型的建立[J]. 石油化工高等学校学报, 2022, 35(1): 10-17.

Figures/Tables 7

Table 1 Properties of original asphalts

性质参数	数值
性质参数	KL⁃30	KL⁃50	KL⁃70	KL⁃90	KL⁃110	ZH⁃50	ZH⁃70	ZH⁃90	ZH⁃110	ZH⁃130
针入度（25 ℃）/ (0.1 mm)	29	51	68	87	113	46	67	87	115	124
针入度指数	-0.34	-0.77	-1.00	-0.63	-0.74	-1.64	-1.37	-1.67	-1.91	-1.80
软化点/℃	58.9	52.0	49.0	46.4	44.0	52.2	49.3	47.0	43.2	44.5
延度(10 ℃)/cm	6	23	141	>150	>150	0	103	>150	>150	>150
延度(15 ℃)/cm	11	>150	>150	>150	>150	>150	>150	>150	>150	>150
密度（25 ℃)/ （g $⋅$ cm^-3）	0.986 1	0.980 4	0.977 8	0.975 0	0.973 1	1.008 8	1.006 8	1.005 2	1.002 3	1.001 7
溶解度/%	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9
w(蜡)/%	1.7	1.7	1.5	1.6	1.6	2.5	2.6	2.5	2.6	2.5
黏度(60 ℃)/ （Pa $⋅$ s）	1 482.7	581.0	396.3	143.9	121.6	406.3	236.1	168.4	106.0	98.4
RTFOT后
质量变化/%	0.40×10^-3	-0.01	-0.02	-0.03	-0.05	0.07	-0.15	0.21	-0.22	-0.22
针入度比/%	90	82	81	80	78	74	72	68	68	64
延度(10 ℃)/cm	4.2	9	17.3	59.3	>150	0	9	32	>150	>150
延度(15 ℃)/cm	7.3	40	>150	>150	>150	37	>150	>150	>150	>150

Table 1 Properties of original asphalts

性质参数	数值
性质参数	KL⁃30	KL⁃50	KL⁃70	KL⁃90	KL⁃110	ZH⁃50	ZH⁃70	ZH⁃90	ZH⁃110	ZH⁃130
针入度（25 ℃）/ (0.1 mm)	29	51	68	87	113	46	67	87	115	124
针入度指数	-0.34	-0.77	-1.00	-0.63	-0.74	-1.64	-1.37	-1.67	-1.91	-1.80
软化点/℃	58.9	52.0	49.0	46.4	44.0	52.2	49.3	47.0	43.2	44.5
延度(10 ℃)/cm	6	23	141	>150	>150	0	103	>150	>150	>150
延度(15 ℃)/cm	11	>150	>150	>150	>150	>150	>150	>150	>150	>150
密度（25 ℃)/ （g $⋅$ cm^-3）	0.986 1	0.980 4	0.977 8	0.975 0	0.973 1	1.008 8	1.006 8	1.005 2	1.002 3	1.001 7
溶解度/%	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9	>99.9
w(蜡)/%	1.7	1.7	1.5	1.6	1.6	2.5	2.6	2.5	2.6	2.5
黏度(60 ℃)/ （Pa $⋅$ s）	1 482.7	581.0	396.3	143.9	121.6	406.3	236.1	168.4	106.0	98.4
RTFOT后
质量变化/%	0.40×10^-3	-0.01	-0.02	-0.03	-0.05	0.07	-0.15	0.21	-0.22	-0.22
针入度比/%	90	82	81	80	78	74	72	68	68	64
延度(10 ℃)/cm	4.2	9	17.3	59.3	>150	0	9	32	>150	>150
延度(15 ℃)/cm	7.3	40	>150	>150	>150	37	>150	>150	>150	>150

Table 2 Parameters of the linear relationship between lg（lgG*） and t

样品编号	B_lg（lg_G $*$ _）⁃_t			K_lg（lg_G $*$ _）⁃_t			r_lg（lg_G $*$ _)⁃_t
样品编号	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后
KL⁃30	1.009 70	1.005 60	0.934 91	-0.006 67	-0.006 16	-0.004 74	-0.999 79	-0.999 83	-0.999 80
KL⁃50	0.977 98	0.974 35	0.916 65	-0.006 91	-0.006 33	-0.004 75	-0.999 85	-0.999 81	-0.999 85
KL⁃70	0.968 05	0.963 33	0.908 12	-0.007 05	-0.006 47	-0.004 79	-0.999 76	-0.999 87	-0.999 78
KL⁃90	0.944 13	0.941 75	0.897 62	-0.007 09	-0.006 46	-0.005 05	-0.999 86	-0.999 87	-0.999 82
KL⁃110	0.937 23	0.936 85	0.890 31	-0.007 29	-0.006 70	-0.005 17	-0.999 70	-0.999 74	-0.999 75
ZH⁃50	1.057 98	1.022 33	0.941 14	-0.004 03	-0.007 34	-0.004 98	-0.999 77	-0.999 97	-0.999 81
ZH⁃70	1.053 30	1.010 37	0.937 79	-0.004 18	-0.007 37	-0.005 10	-0.999 79	-0.999 96	-0.999 97
ZH⁃90	1.057 60	0.970 81	0.926 09	-0.004 34	-0.007 19	-0.005 26	-0.999 85	-0.999 90	-0.999 81
ZH⁃110	1.022 53	0.980 23	0.911 26	-0.004 09	-0.007 63	-0.005 23	-0.999 76	-0.999 99	-0.999 75
ZH⁃130	1.023 68	0.968 55	0.912 38	-0.004 14	-0.007 52	-0.005 35	-0.999 86	-0.999 97	-0.999 86

Table 2 Parameters of the linear relationship between lg（lgG*） and t

样品编号	B_lg（lg_G $*$ _）⁃_t			K_lg（lg_G $*$ _）⁃_t			r_lg（lg_G $*$ _)⁃_t
样品编号	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后
KL⁃30	1.009 70	1.005 60	0.934 91	-0.006 67	-0.006 16	-0.004 74	-0.999 79	-0.999 83	-0.999 80
KL⁃50	0.977 98	0.974 35	0.916 65	-0.006 91	-0.006 33	-0.004 75	-0.999 85	-0.999 81	-0.999 85
KL⁃70	0.968 05	0.963 33	0.908 12	-0.007 05	-0.006 47	-0.004 79	-0.999 76	-0.999 87	-0.999 78
KL⁃90	0.944 13	0.941 75	0.897 62	-0.007 09	-0.006 46	-0.005 05	-0.999 86	-0.999 87	-0.999 82
KL⁃110	0.937 23	0.936 85	0.890 31	-0.007 29	-0.006 70	-0.005 17	-0.999 70	-0.999 74	-0.999 75
ZH⁃50	1.057 98	1.022 33	0.941 14	-0.004 03	-0.007 34	-0.004 98	-0.999 77	-0.999 97	-0.999 81
ZH⁃70	1.053 30	1.010 37	0.937 79	-0.004 18	-0.007 37	-0.005 10	-0.999 79	-0.999 96	-0.999 97
ZH⁃90	1.057 60	0.970 81	0.926 09	-0.004 34	-0.007 19	-0.005 26	-0.999 85	-0.999 90	-0.999 81
ZH⁃110	1.022 53	0.980 23	0.911 26	-0.004 09	-0.007 63	-0.005 23	-0.999 76	-0.999 99	-0.999 75
ZH⁃130	1.023 68	0.968 55	0.912 38	-0.004 14	-0.007 52	-0.005 35	-0.999 86	-0.999 97	-0.999 86

Table 3 Parameters of the relationship between lg（lgG*） and T

样品编号	B_lg（lg_G $*$ _)⁃_T			K_lg（lg_G $*$ _)⁃_T			r_lg（lg_G $*$ _)⁃_T
样品编号	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后
KL⁃30	2.831 06	2.687 93	2.229 70	-0.006 67	-0.006 16	-0.004 74	-0.999 79	-0.999 83	-0.999 80
KL⁃50	2.864 17	2.703 39	2.214 31	-0.006 91	-0.006 33	-0.004 75	-0.999 85	-0.999 81	-0.999 85
KL⁃70	2.894 31	2.729 38	2.215 71	-0.007 05	-0.006 47	-0.004 79	-0.999 76	-0.999 87	-0.999 78
KL⁃90	2.881 44	2.705 71	2.277 58	-0.007 09	-0.006 46	-0.005 05	-0.999 86	-0.999 87	-0.999 82
KL⁃110	2.928 45	2.767 91	2.303 47	-0.007 29	-0.006 70	-0.005 17	-0.999 70	-0.999 74	-0.999 75
ZH⁃50	2.159 23	3.026 85	2.302 37	-0.004 03	-0.007 34	-0.004 98	-0.999 77	-0.999 97	-0.999 81
ZH⁃70	2.194 23	3.023 10	2.331 68	-0.004 18	-0.007 37	-0.005 10	-0.999 94	-0.999 96	-0.999 97
ZH⁃90	2.244 29	2.933 48	2.361 66	-0.004 34	-0.007 19	-0.005 26	-0.999 73	-0.999 90	-0.999 81
ZH⁃110	2.139 71	3.065 50	2.338 65	-0.004 09	-0.007 63	-0.005 23	-0.999 98	-0.999 99	-0.999 75
ZH⁃130	2.153 91	3.022 78	2.373 41	-0.004 14	-0.007 52	-0.005 35	-0.999 88	-0.999 97	-0.999 86

Table 3 Parameters of the relationship between lg（lgG*） and T

样品编号	B_lg（lg_G $*$ _)⁃_T			K_lg（lg_G $*$ _)⁃_T			r_lg（lg_G $*$ _)⁃_T
样品编号	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后
KL⁃30	2.831 06	2.687 93	2.229 70	-0.006 67	-0.006 16	-0.004 74	-0.999 79	-0.999 83	-0.999 80
KL⁃50	2.864 17	2.703 39	2.214 31	-0.006 91	-0.006 33	-0.004 75	-0.999 85	-0.999 81	-0.999 85
KL⁃70	2.894 31	2.729 38	2.215 71	-0.007 05	-0.006 47	-0.004 79	-0.999 76	-0.999 87	-0.999 78
KL⁃90	2.881 44	2.705 71	2.277 58	-0.007 09	-0.006 46	-0.005 05	-0.999 86	-0.999 87	-0.999 82
KL⁃110	2.928 45	2.767 91	2.303 47	-0.007 29	-0.006 70	-0.005 17	-0.999 70	-0.999 74	-0.999 75
ZH⁃50	2.159 23	3.026 85	2.302 37	-0.004 03	-0.007 34	-0.004 98	-0.999 77	-0.999 97	-0.999 81
ZH⁃70	2.194 23	3.023 10	2.331 68	-0.004 18	-0.007 37	-0.005 10	-0.999 94	-0.999 96	-0.999 97
ZH⁃90	2.244 29	2.933 48	2.361 66	-0.004 34	-0.007 19	-0.005 26	-0.999 73	-0.999 90	-0.999 81
ZH⁃110	2.139 71	3.065 50	2.338 65	-0.004 09	-0.007 63	-0.005 23	-0.999 98	-0.999 99	-0.999 75
ZH⁃130	2.153 91	3.022 78	2.373 41	-0.004 14	-0.007 52	-0.005 35	-0.999 88	-0.999 97	-0.999 86

Table 4 Equation and parameter of the relationship between lg（lgG*）and lgt

样品编号	B_lg(lg_G $*$ _)⁃lg_t			K_lg(lg_G $*$ _)⁃lg_t			r_lg(lg_G $*$ _)⁃lg_t
样品编号	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后
KL⁃30	2.498 69	2.381 14	1.283 56	-1.061 83	-0.980 89	-0.333 24	-0.997 44	-0.997 60	-0.997 02
KL⁃50	2.344 20	2.226 52	1.169 70	-1.003 25	-0.919 54	-0.267 21	-0.997 34	-0.997 16	-0.996 02
KL⁃70	2.362 79	2.242 62	1.116 65	-1.024 28	-0.939 40	-0.235 26	-0.996 96	-0.997 41	-0.994 68
KL⁃90	2.170 18	2.058 16	1.117 75	-0.931 04	-0.847 77	-0.248 32	-0.996 94	-0.996 99	-0.994 88
KL⁃110	2.196 57	2.095 03	1.067 37	-0.956 47	-0.879 63	-0.217 26	-0.996 28	-0.996 42	-0.993 03
ZH⁃50	1.813 58	2.396 40	1.214 30	-0.561 81	-1.021 90	-0.284 95	-0.999 51	-0.999 00	-0.998 53
ZH⁃70	1.835 87	2.389 40	1.217 67	-0.581 91	-1.025 72	-0.291 90	-0.999 43	-0.998 63	-0.999 08
ZH⁃90	1.870 38	2.315 19	1.164 09	-0.604 59	-1.000 00	-0.263 96	-0.998 13	-0.998 36	-0.998 27
ZH⁃110	1.788 05	2.409 52	1.100 15	-0.569 38	-1.062 99	-0.226 00	-0.998 73	-0.998 95	-0.997 67
ZH⁃130	1.799 08	2.377 03	1.105 80	-0.576 55	-1.047 43	-0.231 39	-0.999 54	-0.999 18	-0.998 06

Table 4 Equation and parameter of the relationship between lg（lgG*）and lgt

样品编号	B_lg(lg_G $*$ _)⁃lg_t			K_lg(lg_G $*$ _)⁃lg_t			r_lg(lg_G $*$ _)⁃lg_t
样品编号	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后
KL⁃30	2.498 69	2.381 14	1.283 56	-1.061 83	-0.980 89	-0.333 24	-0.997 44	-0.997 60	-0.997 02
KL⁃50	2.344 20	2.226 52	1.169 70	-1.003 25	-0.919 54	-0.267 21	-0.997 34	-0.997 16	-0.996 02
KL⁃70	2.362 79	2.242 62	1.116 65	-1.024 28	-0.939 40	-0.235 26	-0.996 96	-0.997 41	-0.994 68
KL⁃90	2.170 18	2.058 16	1.117 75	-0.931 04	-0.847 77	-0.248 32	-0.996 94	-0.996 99	-0.994 88
KL⁃110	2.196 57	2.095 03	1.067 37	-0.956 47	-0.879 63	-0.217 26	-0.996 28	-0.996 42	-0.993 03
ZH⁃50	1.813 58	2.396 40	1.214 30	-0.561 81	-1.021 90	-0.284 95	-0.999 51	-0.999 00	-0.998 53
ZH⁃70	1.835 87	2.389 40	1.217 67	-0.581 91	-1.025 72	-0.291 90	-0.999 43	-0.998 63	-0.999 08
ZH⁃90	1.870 38	2.315 19	1.164 09	-0.604 59	-1.000 00	-0.263 96	-0.998 13	-0.998 36	-0.998 27
ZH⁃110	1.788 05	2.409 52	1.100 15	-0.569 38	-1.062 99	-0.226 00	-0.998 73	-0.998 95	-0.997 67
ZH⁃130	1.799 08	2.377 03	1.105 80	-0.576 55	-1.047 43	-0.231 39	-0.999 54	-0.999 18	-0.998 06

Table 5 Parameters of the relationship between lg（lgG*） and lgT

样品编号	B_lg(lg_G $*$ _)⁃lg_T			K_lg(lg_G $*$ _)⁃lg_T			r_lg(lg_G $*$ _)⁃lg_T
样品编号	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后
KL⁃30	13.891 80	12.904 67	9.028 92	-5.265 09	-4.863 31	-3.318 89	-0.999 52	-0.999 59	-0.999 68
KL⁃50	14.077 47	12.982 14	8.866 01	-5.357 25	-4.910 78	-3.260 65	-0.999 62	-0.999 55	-0.999 73
KL⁃70	14.345 05	13.228 60	8.836 36	-5.470 75	-5.016 08	-3.252 45	-0.999 47	-0.999 65	-0.999 64
KL⁃90	14.151 35	12.967 26	9.264 70	-5.404 45	-4.920 89	-3.432 48	-0.999 63	-0.999 65	-0.999 69
KL⁃110	14.510 72	13.418 83	9.370 35	-5.554 35	-5.107 69	-3.479 27	-0.999 38	-0.999 44	-0.99961
ZH⁃50	8.639 83	14.821 09	9.281 28	-3.101 52	-5.644 70	-3.420 89	-0.999 88	-0.999 94	-0.99975
ZH⁃70	8.907 96	14.864 79	9.478 27	-3.213 12	-5.667 49	-3.503 06	-0.999 99	-0.999 87	-0.99994
ZH⁃90	9.225 63	14.480 07	9.631 93	-3.341 34	-5.526 30	-3.571 40	-0.999 58	-0.999 76	-0.99975
ZH⁃110	8.712 65	15.334 72	9.478 30	-3.145 83	-5.872 04	-3.514 91	-0.999 90	-0.999 95	-0.99968
ZH⁃130	8.804 91	15.110 08	9.681 49	-3.183 08	-5.784 91	-3.597 81	-0.999 97	-0.999 97	-0.99981

Table 5 Parameters of the relationship between lg（lgG*） and lgT

样品编号	B_lg(lg_G $*$ _)⁃lg_T			K_lg(lg_G $*$ _)⁃lg_T			r_lg(lg_G $*$ _)⁃lg_T
样品编号	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后	原样沥青	RTFOT后	PAV后
KL⁃30	13.891 80	12.904 67	9.028 92	-5.265 09	-4.863 31	-3.318 89	-0.999 52	-0.999 59	-0.999 68
KL⁃50	14.077 47	12.982 14	8.866 01	-5.357 25	-4.910 78	-3.260 65	-0.999 62	-0.999 55	-0.999 73
KL⁃70	14.345 05	13.228 60	8.836 36	-5.470 75	-5.016 08	-3.252 45	-0.999 47	-0.999 65	-0.999 64
KL⁃90	14.151 35	12.967 26	9.264 70	-5.404 45	-4.920 89	-3.432 48	-0.999 63	-0.999 65	-0.999 69
KL⁃110	14.510 72	13.418 83	9.370 35	-5.554 35	-5.107 69	-3.479 27	-0.999 38	-0.999 44	-0.99961
ZH⁃50	8.639 83	14.821 09	9.281 28	-3.101 52	-5.644 70	-3.420 89	-0.999 88	-0.999 94	-0.99975
ZH⁃70	8.907 96	14.864 79	9.478 27	-3.213 12	-5.667 49	-3.503 06	-0.999 99	-0.999 87	-0.99994
ZH⁃90	9.225 63	14.480 07	9.631 93	-3.341 34	-5.526 30	-3.571 40	-0.999 58	-0.999 76	-0.99975
ZH⁃110	8.712 65	15.334 72	9.478 30	-3.145 83	-5.872 04	-3.514 91	-0.999 90	-0.999 95	-0.99968
ZH⁃130	8.804 91	15.110 08	9.681 49	-3.183 08	-5.784 91	-3.597 81	-0.999 97	-0.999 97	-0.99981

Table 6 Equation and parameter of the relationship between slope and intercept in temperature expressions of different stages

拟合方程	老化阶段	拟合参数
拟合方程	老化阶段	b	k	r
$K l g (l g G ) ⁃ t = k l g (l g G ) ⁃ t B l g (l g G ) ⁃ t + b l g (l g G ) ⁃ t$	原样沥青	-0.035 29	0.029 56	0.898 93
	RTFOT后	-0.001 36	-0.005 68	-0.292 26
	PAV后	-0.007 64	0.002 83	0.218 48
$K l g (l g G ) ⁃ T = k l g (l g G ) ⁃ T B l g (l g G ) ⁃ T + b l g (l g G ) ⁃ T$	原样沥青	0.004 68	-0.004 06	-0.998 54
	RTFOT后	0.002 74	-0.003 37	-0.985 96
	PAV后	0.003 22	-0.003 60	-0.958 91
$K l g (l g G ) ⁃ l g t = k l g (l g G ) ⁃ l g t B l g (l g G ) ⁃ l g t + b l g (l g G ) ⁃ l g t$	原样沥青	0.854 56	-0.793 87	-0.983 40
	RTFOT后	0.267 83	-0.541 85	-0.958 12
	PAV后	0.355 89	-0.532 87	-0.991 46
$K l g l g (G ) ⁃ l g T = k l g l g (G ) ⁃ l g T B l g l g (G ) ⁃ l g T + b l g l g (G ) ⁃ l g T$	原样沥青	0.477 65	-0.414 80	-0.999 96
	RTFOT后	0.343 08	-0.405 00	-0.999 57
	PAV后	0.377 12	-0.410 29	-0.998 14

Table 6 Equation and parameter of the relationship between slope and intercept in temperature expressions of different stages

拟合方程	老化阶段	拟合参数
拟合方程	老化阶段	b	k	r
$K l g (l g G ) ⁃ t = k l g (l g G ) ⁃ t B l g (l g G ) ⁃ t + b l g (l g G ) ⁃ t$	原样沥青	-0.035 29	0.029 56	0.898 93
	RTFOT后	-0.001 36	-0.005 68	-0.292 26
	PAV后	-0.007 64	0.002 83	0.218 48
$K l g (l g G ) ⁃ T = k l g (l g G ) ⁃ T B l g (l g G ) ⁃ T + b l g (l g G ) ⁃ T$	原样沥青	0.004 68	-0.004 06	-0.998 54
	RTFOT后	0.002 74	-0.003 37	-0.985 96
	PAV后	0.003 22	-0.003 60	-0.958 91
$K l g (l g G ) ⁃ l g t = k l g (l g G ) ⁃ l g t B l g (l g G ) ⁃ l g t + b l g (l g G ) ⁃ l g t$	原样沥青	0.854 56	-0.793 87	-0.983 40
	RTFOT后	0.267 83	-0.541 85	-0.958 12
	PAV后	0.355 89	-0.532 87	-0.991 46
$K l g l g (G ) ⁃ l g T = k l g l g (G ) ⁃ l g T B l g l g (G ) ⁃ l g T + b l g l g (G ) ⁃ l g T$	原样沥青	0.477 65	-0.414 80	-0.999 96
	RTFOT后	0.343 08	-0.405 00	-0.999 57
	PAV后	0.377 12	-0.410 29	-0.998 14

Table 7 Comparison between testing data and predicted data

老化阶段	样品名称	温度/℃	$G *$ /（Pa $⋅$ s）		B_lg（lg_G $*$ _）⁃lg_T	相对误差/%
老化阶段	样品名称	温度/℃	实测	预测	B_lg（lg_G $*$ _）⁃lg_T	相对误差/%
原样沥青	HL⁃70	52	1.43×10⁴	1.22×10⁴	14.011 35	-14.79
		58	5.58×10³	5.08×10³		-8.89
		64	2.33×10³	2.33×10³		0
		70	1.02×10³	1.16×10³		13.74
		76	4.81×10²	6.23×10²		29.48
	HL⁃90	46	2.52×10⁴	2.33×10⁴	14.375 76	-7.46
		52	9.27×10³	8.79×10³		-5.15
		58	3.70×10³	3.70×10³		0
		64	1.57×10³	1.71×10³		9.05
		70	7.14×10²	8.61×10²		20.60
	HL⁃110	46	1.80×10⁴	1.70×10⁴	14.737 40	-5.78
		52	6.80×10³	6.43×10³		-5.36
		58	2.73×10³	2.73×10³		0
		64	1.17×10³	1.27×10³		8.91
		70	5.36×10²	6.47×10²		20.69
PTFOT后	HL⁃70	46	1.43×10⁴	2.62×10⁴	15.147 59	3.75
		52	5.58×10³	9.25×10³		-0.19
		58	2.33×10³	2.33×10³		0
		64	1.02×10³	1.64×10³		4.68
		70	4.81×10²	8.00×10²		12.08
	HL⁃90	46	2.52×10⁴	2.02×10⁴	15.942 57	12.06
		52	9.27×10³	6.95×10³		2.22
		58	3.70×10³	3.70×10³		0
		64	1.57×10³	1.20×10³		2.45
		70	7.14×10²	5.80×10²		8.20
	HL⁃110	46	1.80×10⁴	2.02×10⁴	15.942 57	12.06
		52	6.80×10³	6.95×10³		2.22
		58	2.73×10³	2.73×10³		0
		64	1.17×10³	1.20×10³		2.45
		70	5.36×10²	5.80×10²		8.20
PAV后	HL⁃70	28	2.15×10⁶	2.85×10⁶	7.339 60	32.72
		25	3.74×10⁶	4.24×10⁶		13.28
		22	6.40×10⁶	6.40×10⁶		0
		19	1.08×10⁷	9.81×10⁶		-9.38
		16	1.78×10⁷	1.53×10⁷		-14.17
	HL⁃90	25	2.22×10⁶	2.74×10⁶	8.164 53	23.50
		22	3.94×10⁶	4.31×10⁶		9.50
		19	6.91×10⁶	6.91×10⁶		0
		16	1.18×10⁷	1.13×10⁷		-3.98
		13	1.95×10⁷	1.88×10⁷		-3.23
	HL⁃110	25	1.18×10⁶	1.31×10⁶	9.624 10	10.81
		22	2.13×10⁶	2.20×10⁶		3.29
		19	3.78×10⁶	3.78×10⁶		0
		16	6.59×10⁶	6.68×10⁶		1.32
		13	1.12×10⁷	1.21×10⁷		8.41

Table 7 Comparison between testing data and predicted data

老化阶段	样品名称	温度/℃	$G *$ /（Pa $⋅$ s）		B_lg（lg_G $*$ _）⁃lg_T	相对误差/%
老化阶段	样品名称	温度/℃	实测	预测	B_lg（lg_G $*$ _）⁃lg_T	相对误差/%
原样沥青	HL⁃70	52	1.43×10⁴	1.22×10⁴	14.011 35	-14.79
		58	5.58×10³	5.08×10³		-8.89
		64	2.33×10³	2.33×10³		0
		70	1.02×10³	1.16×10³		13.74
		76	4.81×10²	6.23×10²		29.48
	HL⁃90	46	2.52×10⁴	2.33×10⁴	14.375 76	-7.46
		52	9.27×10³	8.79×10³		-5.15
		58	3.70×10³	3.70×10³		0
		64	1.57×10³	1.71×10³		9.05
		70	7.14×10²	8.61×10²		20.60
	HL⁃110	46	1.80×10⁴	1.70×10⁴	14.737 40	-5.78
		52	6.80×10³	6.43×10³		-5.36
		58	2.73×10³	2.73×10³		0
		64	1.17×10³	1.27×10³		8.91
		70	5.36×10²	6.47×10²		20.69
PTFOT后	HL⁃70	46	1.43×10⁴	2.62×10⁴	15.147 59	3.75
		52	5.58×10³	9.25×10³		-0.19
		58	2.33×10³	2.33×10³		0
		64	1.02×10³	1.64×10³		4.68
		70	4.81×10²	8.00×10²		12.08
	HL⁃90	46	2.52×10⁴	2.02×10⁴	15.942 57	12.06
		52	9.27×10³	6.95×10³		2.22
		58	3.70×10³	3.70×10³		0
		64	1.57×10³	1.20×10³		2.45
		70	7.14×10²	5.80×10²		8.20
	HL⁃110	46	1.80×10⁴	2.02×10⁴	15.942 57	12.06
		52	6.80×10³	6.95×10³		2.22
		58	2.73×10³	2.73×10³		0
		64	1.17×10³	1.20×10³		2.45
		70	5.36×10²	5.80×10²		8.20
PAV后	HL⁃70	28	2.15×10⁶	2.85×10⁶	7.339 60	32.72
		25	3.74×10⁶	4.24×10⁶		13.28
		22	6.40×10⁶	6.40×10⁶		0
		19	1.08×10⁷	9.81×10⁶		-9.38
		16	1.78×10⁷	1.53×10⁷		-14.17
	HL⁃90	25	2.22×10⁶	2.74×10⁶	8.164 53	23.50
		22	3.94×10⁶	4.31×10⁶		9.50
		19	6.91×10⁶	6.91×10⁶		0
		16	1.18×10⁷	1.13×10⁷		-3.98
		13	1.95×10⁷	1.88×10⁷		-3.23
	HL⁃110	25	1.18×10⁶	1.31×10⁶	9.624 10	10.81
		22	2.13×10⁶	2.20×10⁶		3.29
		19	3.78×10⁶	3.78×10⁶		0
		16	6.59×10⁶	6.68×10⁶		1.32
		13	1.12×10⁷	1.21×10⁷		8.41

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