Evolution characteristics of elastic energy storage of marble under creep loading
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摘要: 为研究蠕变过程中大理岩对能量的吸收转化特征和应变能的释放与耗散机理，测试分析了大理岩在单轴循环加载和多级蠕变循环加载条件下的应力-应变关系，计算得到各自的弹性能密度曲线，采用幂函数拟合了大理岩弹性能密度与应力之间的关系。对比两种加载方式的测试数据，结果表明：单轴循环加载和多级蠕变循环加载试验的弹性能密度曲线基本重合，即加载模式不影响大理岩的弹性能密度变化特征；两种加载模式下大理岩材料都经历压密、稳定和劣化三个阶段。经历蠕变循环加载之后的大理岩试样，弹性储能能力会降低，约为单轴循环加载的43 %；强度也有明显下降，约为单轴循环加载的62 %。如果考虑长期承载问题，大理岩的最大应力值应控制在材料的压密阶段结束点对应的应力以下较为有利。Abstract: In order to study the energy absorption and transformation characteristics of marble, the release of strain energy and dissipation mechanism during creep process, this paper tested the stress-strain relationship of marble under uniaxial cyclic loading and multi-level creep cyclic loading respectively, and calculated their elastic energy density curves.The power function is used to fit the relationship between elastic energy density and stress of marble.By comparing the test data of the two loading modes, the results showed that the elastic energy density curves of uniaxial cyclic loading and multi-level creep cyclic loading were basically coincident, and they have the same elastic energy density curves, which means neither the creep loading mode nor the uniaxial cyclic loading mode will affect the elastic energy density curve of the marble.Comparing the ratio of elastic energy to external power in two loading modes, the results showed that the marble materials all went through three stages: compaction, stabilization and deterioration.After creep cyclic loading, the elastic energy storage capacity of marble would decrease to 43 % of uniaxial cyclic loading.Correspondingly, its strength also decreased obviously, which was 62 % of uniaxial cyclic loading.If the long-term bearing issue of this kind of marble is taken into account, the maximum stress value should be controlled below the stress that corresponds to the end point of the compaction stage of material.
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Key words:
- marble /
- creep loading /
- elastic energy density /
- elastic energy ratio
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图 3 大理岩单轴循环加卸载的应力-应变曲线
Figure 3. Stress-strain curve of marble sample under uniaxial cyclic loading and unloading
图 4 岩样单次加卸载弹性能密度计算示意图
Figure 4. Calculation diagram of elastic energy density under single loading and unloading for rock sample
图 5 单轴循环加卸载下大理岩试件破坏形态
Figure 5. Failure modes of marble samples under uniaxial cyclic loading and unloading
图 6 大理岩岩样多级蠕变循环加卸载模式示意图
Figure 6. Schematic diagram of marble sample under multi-level creep cyclic loading mode
图 7 大理岩多级蠕变循环加卸载试验应力-应变曲线
Figure 7. Stress-strain curve of marble sample under multi-level creep cyclic loading and unloading
图 8 大理岩试样单次蠕变加卸载应力- 应变关系示意图
Figure 8. Relationship diagram between stress and strain of marble sample under single creep loading and unloading
图 9 多级蠕变循环加卸载下大理岩试件破坏形态
Figure 9. Failure modes of marble samples under multi-level creep cyclic loading and unloading
图 11 A3试样的拟合曲线与理论计算曲线
Figure 11. Fitting curve and theoretical calculation curve of A3
图 12 大理岩在2种加载模式下弹性能比例曲线
Figure 12. elastic energy ratio curves of marble under two loading modes
图 13 大理岩在2种加载模式下的耗散能密度曲线
Figure 13. Dissipative energy density curves of marble under two loading modes
表 1 大理岩岩样X-射线衍射分析
Table 1. X-ray diffraction analysis of marble samples
岩样 颜色 矿物种类和含量/% 方解石 白云石 1 白色 100.0 — 2 灰色 93.0 7.0 
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表 2 单轴循环加卸载试验结果
Table 2. Test results of uniaxial loading and unloading cycles
试样编号 应力/ MPa 总外力功密度U/ (kJ·m-3) 弹性能密度Ue/ (kJ·m-3) 耗散能密度Ud/ (kJ·m-3) 弹性能比例K/% 试样编号 应力/ MPa 总外力功密度U/ (kJ·m-3) 弹性能密度Ue/ (kJ·m-3) 耗散能密度Ud/ (kJ·m-3) 弹性能比例K/% A1 2.53 0.34 0.17 0.17 50.26 A3 2.53 0.25 0.14 0.11 55.07 5.08 0.72 0.52 0.20 72.24 5.07 0.57 0.46 0.11 79.98 7.62 1.31 1.04 0.27 79.07 7.62 1.09 0.94 0.15 85.63 10.17 2.03 1.68 0.35 83.04 10.17 1.78 1.57 0.21 87.96 12.71 2.88 2.45 0.43 85.30 12.71 2.58 2.29 0.29 88.90 15.26 3.83 3.32 0.51 86.73 15.26 3.52 3.14 0.38 89.42 17.80 4.85 4.26 0.59 87.76 17.81 4.49 4.05 0.44 90.08 20.36 6.04 5.32 0.72 88.16 20.36 5.58 5.05 0.53 90.55 22.90 7.29 6.44 0.85 88.38 22.90 6.75 6.12 0.63 90.75 25.45 8.67 7.68 0.99 88.63 25.44 8.07 7.32 0.75 90.73 28.00 10.13 9.01 1.12 88.88 28.00 9.50 8.59 0.91 90.37 30.53 11.66 10.39 1.27 89.07 30.54 11.03 9.92 1.11 89.97 33.09 13.33 11.91 1.42 89.33 32.98 12.65 11.28 1.37 89.21 35.63 15.06 13.48 1.58 89.53 35.38 14.41 12.79 1.62 88.81 37.91 15.99 14.41 1.58 90.11 48.37 25.28 22.75 2.53 90.01 40.42 18.00 16.01 1.99 88.92 50.91 27.77 24.98 2.79 89.96 43.26 21.09 17.94 3.15 85.06 A2 2.52 0.22 0.16 0.06 72.23 A4 2.52 0.36 0.18 0.18 49.28 5.08 0.64 0.49 0.15 76.50 5.08 0.80 0.52 0.28 65.33 7.60 1.20 0.98 0.22 81.22 10.17 1.92 1.59 0.33 83.07 7.62 1.42 1.02 0.38 71.98 12.72 2.76 2.32 0.44 83.95 15.25 3.71 3.14 0.57 84.83 10.17 2.11 1.65 0.46 78.32 17.81 4.74 4.08 0.66 86.08 20.35 5.83 5.08 0.75 87.15 12.71 2.89 2.41 0.48 83.22 22.90 7.07 6.23 0.84 88.05 25.45 8.37 7.41 0.96 88.59 15.26 3.82 3.28 0.54 85.97 30.53 9.81 8.73 1.08 89.51 33.09 11.34 10.15 1.19 89.79 17.81 4.85 4.23 0.62 87.30 35.64 12.99 11.66 1.33 89.87 38.18 14.79 13.29 1.50 90.11 20.36 5.99 5.29 0.70 88.17 40.71 16.58 14.94 1.64 90.25 43.27 18.58 16.77 1.81 90.37 22.90 7.24 6.41 0.83 88.53 45.82 20.63 18.64 1.99 90.41 25.45 8.62 7.64 0.98 88.67 48.36 22.76 20.58 2.18 90.45 28.00 10.09 8.95 1.14 88.63 50.91 25.09 22.69 2.40 90.46 32.94 13.98 11.34 2.64 81.08 53.46 27.45 24.83 2.62 90.14 35.33 16.58 12.28 4.30 74.07
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表 3 多级蠕变循环加卸载试验结果
Table 3. Test results of multi-level creep cyclic loading and unloading
试样编号 单次循环第一阶段 单次循环第二阶段 应力/ MPa 总外力功密度U/(kJ·m-3) 弹性能密度Ue1 /(kJ·m-3) 耗散能密度Ud/(kJ·m-3) 弹性能比例K/% 弹性能密度Ue2/(kJ·m-3) B1 3.39 0.45 0.23 0.22 51.33 0.21 6.67 0.85 0.72 0.13 84.56 0.74 10.09 1.66 1.49 0.17 90.19 1.50 13.49 2.67 2.43 0.24 91.07 2.47 17.30 4.07 3.75 0.32 91.98 3.77 20.20 5.26 4.86 0.40 92.39 4.93 23.57 6.78 6.25 0.53 92.09 6.40 26.93 8.81 8.00 0.81 90.81 8.17 B2 3.39 0.41 0.26 0.15 64.69 0.28 6.69 1.01 0.84 0.17 83.24 0.86 10.08 1.88 1.66 0.22 88.49 1.70 13.49 2.96 2.65 0.31 89.54 2.71 16.97 4.25 3.85 0.39 90.72 4.04 20.19 5.56 5.10 0.46 91.72 5.12 23.57 7.15 6.59 0.56 92.17 6.72 26.93 8.86 8.13 0.73 91.72 8.39 30.30 10.97 10.05 0.92 91.59 10.23 B3 3.39 0.51 0.26 0.25 50.48 0.25 6.69 1.01 0.81 0.20 80.40 0.82 10.08 1.83 1.58 0.25 86.55 1.60 13.49 2.88 2.56 0.32 88.79 2.59 16.79 4.04 3.64 0.40 90.00 3.62 19.79 5.22 4.73 0.49 90.61 4.76 23.18 6.73 6.14 0.59 91.20 6.17 26.99 8.71 7.88 0.83 90.53 7.99 29.99 10.44 9.48 0.96 90.72 9.60 B4 3.39 0.43 0.24 0.19 56.05 0.25 6.69 0.93 0.77 0.16 83.48 0.81 10.08 1.81 1.58 0.23 87.46 1.62 13.49 2.91 2.59 0.32 89.02 2.65 16.79 4.14 3.72 0.42 89.81 3.77 19.78 5.36 4.84 0.52 90.25 4.93 23.18 7.01 6.24 0.77 89.06 6.36 26.99 9.11 8.08 1.03 88.69 8.12
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表 4 单轴循环加载拟合的参数a与b
Table 4. Fitting parameters a and b by uniaxial cyclic loading
试件编号 a b A1 0.032 1.69 A2 0.035 1.66 A3 0.027 1.74 A4 0.046 1.57 均值 0.035 1.70 标准差 0.007 0.03
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表 5 蠕变循环加载拟合的参数a与b
Table 5. Fitting parameters a and b by creep cyclic loading
试件编号 a b B1 0.028 1.72 B2 0.036 1.65 B3 0.035 1.65 B4 0.035 1.65 均值 0.034 1.67 标准差 0.003 0.03
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表 6 弹性能储能极限值
Table 6. Limit value of stored elastic energy
加载方式 编号 极限荷载/ kN 极限应力/ MPa 总外力功密度U/(kJ·m-3) 极限弹性能密度Ue/(kJ·m-3) 弹性能比例K/% 单轴循环加载 A1 99.96 50.91 27.77 24.98 89.96 A2 104.97 53.46 30.03 27.06 90.14 A3 84.94 43.26 21.09 17.94 85.06 A4 69.37 35.33 16.58 12.28 74.07 平均值 89.81 45.74 23.87 20.57 84.81 标准差 13.91 7.09 5.34 5.86 6.53 多级蠕变循环加载 B1 52.88 26.93 8.81 8.00 90.81 B2 59.49 30.30 10.97 10.05 91.59 B3 58.89 29.99 10.44 9.48 90.72 B4 52.99 26.99 9.11 8.08 88.69 均值 56.06 28.55 9.83 8.90 90.45 标准差 3.13 1.60 0.90 0.89 1.07
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