准脆性材料抗压强度能量平衡尺寸效应模型

刘小宇; 杨政; 张慧梅

doi:10.6052/0459-1879-21-460

准脆性材料抗压强度能量平衡尺寸效应模型

doi: 10.6052/0459-1879-21-460

*.
西安科技大学理学院, 西安 710600
†.
西安交通大学人居环境与建筑工程学院, 西安 710049

基金项目: 国家自然科学基金(12172280, 51271138), 陕西省自然科学基金(2020Z-53)和陕西省教育厅专项科研计划(19JK0521)资助项目

详细信息

作者简介:
杨政, 教授, 主要研究方向: 结构失效破坏及其三维理论. E-mail: zyang@mail.xjtu.edu.cn

中图分类号: O346
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出版历程
- 收稿日期: 2021-09-09
- 录用日期: 2022-04-02
- 网络出版日期: 2022-04-02
- 刊出日期: 2022-06-18

ENERGY BALANCE SIZE EFFECT MODEL OF COMPRESSIVE STRENGTH FOR QUASI-BRITTLE MATERIALS

*.
College of Sciences, Xi'an University of Science and Technology, Xi’an 710600, China
†.
School of Human Settlement and Civil Engineering, Xi'an Jiaotong University, Xi’an 710049, China

摘要

摘要: 针对现有尺寸效应模型难以体现准脆性材料完整的抗压强度尺寸效应变化规律及其内在机理, 本文通过分析准脆性材料单轴压缩破坏过程中能量输入、储存、整体和局部能量耗散, 建立体现整体和局部损伤的力学模型及描述上述能量演化过程的双线性名义和真实应力应变曲线, 在此基础上确定了名义应力最大时输入能量、储存弹性能、整体和局部能量耗散的表达式, 最后基于能量平衡原理建立抗压强度尺寸效应模型. 抗压强度能量平衡尺寸效应模型能完整体现名义抗压强度尺寸效应, 即随试样尺寸增大, 名义抗压强度在试样尺寸小于等于局部损伤区尺寸时为真实强度, 然后逐渐减小, 最终当试样尺寸趋于无穷大时趋于弹性极限强度; 抗压强度能量平衡尺寸效应模型也能同时体现高径比和试样直径对名义强度的影响, 其包含的参数具有明确的物理意义, 可以反映真实强度、弹性极限强度、名义损伤模量非线性、局部损伤区大小和方向对准脆性材料名义抗压强度尺寸效应的影响; 通过把抗压强度能量平衡尺寸效应模型和现有尺寸效应模型应用于预测各种材料尺寸效应试验和数值模拟数据, 结果表明: 抗压强度能量平衡尺寸效应模型能很好描述试验和数值模拟尺寸效应的非线性变化规律及内在机理, 和现有尺寸效应模型相比, 其总体平均误差最小, 且小于5%.
- 准脆性材料 /
- 抗压强度 /
- 能量平衡 /
- 尺寸效应
Abstract: To address the problem of existing size effect models cannot reflect complete size effect of compressive strength and internal mechanism of quasi-brittle materials. In this paper, by analyzing energy input, storage, global-local energy dissipation during the failure process of quasi-brittle materials under uniaxial compression, mechanical model and bilinear nominal and true stress-strain curves are established to reflect global and local damage and describe the above energy evolution process respectively. On this basis, the expressions of input energy, stored elastic energy and global-local energy dissipation are determined when the nominal stress is the maximum. Finally, size effect model of compressive strength is established with energy balance principle. The energy balance size effect model of compressive strength can completely reflect the size effect of nominal compressive strength, namely with the increase of sample size, nominal compressive strength is the real strength when sample size is less than or equal to the size of local damage zone, and then gradually decreases, eventually tends to the elastic ultimate strength when the sample size approaches infinity. High to diameter ratio together with sample diameter can be taken into account in the energy balance size effect model of compressive strength. Its parameters, which can reflect the effect of real strength, elastic ultimate strength, nonlinear of nominal damage modulus, size and direction of local damage zone on nominal compressive strength size effect of quasi-brittle materials, have clear physical meaning. Experiment and numerical simulation data of various materials are utilized to validate and evaluate the energy balance size effect model of compressive strength and existing size effect models. The results indicate that the energy balance size effect model of compressive strength can well describe the nonlinear variation and internal mechanism of size effect of experiment and numerical simulation, and compared with the existing size effect models, its total average error is the smallest and less than 5%.
- quasi-brittle material /
- compressive strength /
- energy balance /
- size effect

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图 1 准脆性材料单轴压缩损伤力学模型

Figure 1. Damage mechanics model of quasi-brittle material under uniaxial compression

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图 2 双线性准脆性材料单轴压缩名义(OABT)和真实(OACK)应力应变曲线

Figure 2. Bilinear nominal (OABT) and true (OACK) stress-strain curves of quasi-brittle material under uniaxial compression

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图 3 不同阶段声发射位置(图4给出A-F阶段)

Figure 3. Locations of acoustic emission at different stages (A-F stages are shown in Fig. 4)

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图 4 花岗岩单轴压缩名义应力应变曲线

Figure 4. Nominal stress-strain curve of granite under uniaxial compression

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图 5 准脆性材料抗压强度能量平衡尺寸效应模型

Figure 5. Energy balance size effect model of compressive strength for quasi-brittle materials

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图 6 不同尺寸单晶钛合金单轴压缩变形破坏形貌^[57]

Figure 6. Deformation and failure morphology of single crystal titanium alloy with different sizes under uniaxial compression^[57]

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图 7 定比例框下不同单晶钛合金试样和局部剪切区尺寸

Figure 7. Size of different single crystal titanium alloy samples and local plastic zone under fix proportional frame

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图 8 参数α 对模型影响

Figure 8. Effect of the parameter α on model

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图 9 参数n对模型影响

Figure 9. Effect of the parameter n on model

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图 10 参数λ对模型影响

Figure 10. Effect of the parameter λ on model

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图 11 参数θ 对模型影响

Figure 11. Effect of the parameter θ on model

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图 13 参数a_t = b_t对模型影响

Figure 13. Effect of the parameter a_t = b_t on model

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图 12 参数a_t对模型影响

Figure 12. Effect of the parameter a_t on model

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图 14 不同尺寸效应模型预测编号1砂岩和试验数据对比

Figure 14. Comparison of experimental values and predicted results of different size effect models for No. 1 sandstone

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图 26 不同尺寸效应模型预测编号13裂隙岩石和模拟数据对比

Figure 26. Comparison of simulated values and predicted results of different size effect models for No. 13 crack-rock

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图 15 不同尺寸效应模型预测编号2白云岩和试验数据对比

Figure 15. Comparison of experimental values and predicted results of different size effect models for No. 2 dolomite

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图 16 不同尺寸效应模型预测编号3花岗岩和试验数据对比

Figure 16. Comparison of experimental values and predicted results of different size effect models for No. 3 granite

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图 17 不同尺寸效应模型预测编号4花岗岩和试验数据对比

Figure 17. Comparison of experimental values and predicted results of different size effect models for No. 4 granite

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图 18 不同尺寸效应模型预测编号5花岗岩和模拟数据对比

Figure 18. Comparison of simulated values and predicted results of different size effect models for No. 5 granite

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图 19 不同尺寸效应模型预测编号6煤岩和试验数据对比

Figure 19. Comparison of experimental values and predicted results of different size effect models for No. 6 coal

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图 20 不同尺寸效应模型预测编号7钛合金和试验数据对比

Figure 20. Comparison of experimental values and predicted results of different size effect models for No. 7 titanium alloy

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图 21 不同尺寸效应模型预测编号8完整岩石和模拟数据对比

Figure 21. Comparison of simulated values and predicted results of different size effect models for No. 8 intact rock

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图 22 不同尺寸效应模型预测编号9 C20混凝土和试验数据对比

Figure 22. Comparison of experimental values and predicted results of different size effect models for No. 9 C20 concrete

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图 23 不同尺寸效应模型预测编号10 C40混凝土和试验数据对比

Figure 23. Comparison of experimental values and predicted results of different size effect models for No. 10 C40 concrete

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图 24 不同尺寸效应模型预测编号11 C60混凝土和试验数据对比

Figure 24. Comparison of experimental values and predicted results of different size effect models for No. 11 C60 concrete

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图 25 不同尺寸效应模型预测编号12裂隙岩石和模拟数据对比

Figure 25. Comparison of simulated values and predicted results of different size effect models for No. 12 crack-rock

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表 1 不同高径比H/D和直径D试样准脆性材料抗压强度试验和数值模拟数据

Table 1. Experiment and numerical simulation data of compressive strength for quasi-brittle materials with different height to diameter ratios H/D and diameters D

Type of test	No.	Rock	Sample sizes	Data sources
	1	sandstone	0.5~4	Ref. [63]
	2	dolomite	1.25~4	Ref. [64]
H/D	3	granite	1.25~4	Ref. [64]
	4	granite	0.5~6	Ref. [29]
	5	simulated granite	0.5~4	Ref. [58]
D	6	coal	0.02~2.14 m	Ref. [31]
	7	titanium alloy	0.24~8 μm	Ref. [57]
	8	simulated intact rock	0.2~2 m	Ref. [65]
	9	concrete-C20	0.1~1.2 m	Ref. [66]
	10	concrete-C40	0.1~1.2 m	Ref. [66]
	11	concrete-C60	0.1~1.2 m	Ref. [66]
	12	simulated crack-rock	0.6~6 m	Ref. [65]
	13	simulated crack-rock	0.05~12 m	Ref. [4]

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表 2 本文尺寸效应模型-式(16)或式(17)参数值

Table 2. Parameter values of size effect model proposed in this paper for Eq. (16) or (17)

Type of test	No.	λ_c/D_c	f_ce	f_c	n
H/D	1	0.413 1	0.768 9	144.0	0.803 3
	2	1.110 0	200.2	235.8	0.834 6
	3	0.222 7	236.4	988.5	1.500
	4	0.410 0	52.15	168.8	6.792
	5	0.540 0	183.5	257.0	0.350 5
D	6	0.042 63	5.206	33.10	0.028 4
	7	6.916 × 10⁻⁷	2045	4410	0.590 1
	8	0.050	45.45	96.18	3.452 × 10⁻⁴
	9	3.696 × 10⁻³	16.12	20.35	5.795 × 10⁻⁷
	10	3.672 × 10⁻³	26.13	38.88	6.117 × 10⁻⁷
	11	3.824 × 10⁻³	29.71	50.79	5.926 × 10⁻⁷
	12	0.550 1	1.778	15.00	0.814 2
	13	3.266 × 10⁻³	11.58	150.3	3.296 × 10⁻⁷

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表 3 Bazant尺寸效应模型-式(1)参数值

Table 3. Parameter values of Bazant size effect model for Eq. (1)

Type of test	No.	A₁₁	A₁₂	A₁₃	r
H/D	1	70.2	0.023 4	0.095 0	0.058 6
	2	191.6	0.083 5	0.128 7	0.302 8
	3	224.9	4.566 × 10⁻⁴	3.971 × 10⁻⁶	2.508 × 10⁻³
	4	43.56	0.065 4	0.505 9	0.049 0
	5	164.6	0.075 9	1.266 0	0.092 2
D	6	1.673	0.034 3	0.394 9	0.025 6
	7	1.388	0.130 9	1.323 0	0.067 8
	8	34.95	0.053 5	0.242 9	0.115 1
	9	15.24	0.024 3	0.094 0	0.429 6
	10	23.87	0.011 1	0.073 9	0.131 9
	11	25.53	0.013 2	0.119 3	0.086 5
	12	0.853 2	7.741 × 10⁻⁴	2.349 × 10⁻³	4.719 × 10⁻⁴
	13	7.078	0.048 4	0.661 3	0.021 0

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表 4 Weiss尺寸效应模型-式(2)参数值

Table 4. Parameter values of Weiss size effect model for Eq. (2)

Type of test	No.	A₂₁	A₂₂	m
H/D	1	28.49	0.858 4	7.204
	2	50.40	0.974 7	192.7
	3	45.97	0.404 0	236.3
	4	47.10	0.850 3	49.72
	5	175.10	4.978 0	59.27
D	6	1.313	0.955 1	3.649
	7	1.049	1.435	1.621
	8	19.95	1.386	30.69
	9	1.544	1.789	14.49
	10	2.888	1.377	22.92
	11	6.461	1.585	22.78
	12	2.284	0.317 1	1.694
	13	35.78	1.898	1.01 × 10⁻⁶

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表 5 刘宝琛尺寸效应模型-式(3)参数值

Table 5. Parameter values of Lius size effect model for Eq. (3)

Type of test	No.	A₃₁	A₃₂	A₃₃
H/D	1	140.0	1.864	79.64
	2	97.43	0.956 1	203.1
	3	219.1	1.755	238.0
	4	183.3	1.235	57.25
	5	106.2	0.845 8	190.3
D	6	42.31	7.538	4.82
	7	3.423	1.020	1.872
	8	88.61	2.923	44.32
	9	6.427	4.912	16.02
	10	21.77	5.932	25.91
	11	34.92	5.332	29.27
	12	159.3	4.403	1.781
	13	176.4	3.421	11.42

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表 6 杨圣奇尺寸效应模型-式(4)参数值

Table 6. Parameter values of Yangs size effect model for Eq. (4)

Type of test	No.	A₄₁	A₄₂	A₄₃
H/D	1	2.188	3.499	0.318 0
	2	1.392	4.934	0.232 0
	3	2.930	4.342	0.178 9
	4	1.546	3.546	0.549 1
	5	5.632	3.528	0.165 8
D	6	1.411	2.030	0.025 61
	7	0.691 1	1.211	0.170 7
	8	2.316	2.925	0.160 3
	9	2.252	1.936	0.029 55
	10	1.454	2.853	0.043 72
	11	1.888	2.725	0.059 11
	12	0.315 8	1.081	1.577
	13	0.518 1	4.038	0.090 04

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表 7 不同尺寸效应模型预测平均误差AE

Table 7. Prediction average errors (AEs) of different size effect models

No.	Eq. (1)	Eq. (2)	Eq. (3)	Eq. (4)	Eq. (16) or (17)
1	1.39	1.38	0.82^min	1.81	1.07
2	0.74	0.74	0.65^min	0.73	0.65^min
3	0.68	0.33	0.30^min	0.67	0.33
4	2.89	2.47^min	2.96	4.54	2.55
5	1.77	1.99	1.45	3.22	1.38^min
6	18.20	16.18	8.06	79.83	7.57^min
7	5.96	5.69	6.85	13.85	3.04^min
8	8.61	6.20	6.14^min	8.29	6.83
9	0.34^min	0.47	0.42	1.08	0.98
10	0.92^min	0.98	0.94	2.14	1.61
11	1.27	1.48	0.85^min	4.07	2.14
12	21.13	12.41^min	13.46	18.15	13.08
13	13.49	21.06	12.01	117.11	10.89^min
TAE	5.95	5.49	4.22	34.68	4.01^min
Note: The superscript of min denotes the minimum average error (AE) or total average error (TAE) of different size effect models in above table.

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