A NEW ANISOTROPIC PLASTIC-DAMAGE MODEL AND ITS NUMERICAL IMPLEMENTATION FOR PLAIN CONCRETE
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摘要: 该文的目的是建立一种新的、相对简单的混凝土各向异性塑性损伤本构模型，以方便的模拟混凝土结构的破坏行为。为了更好地描述混凝土在拉、压荷载作用下的不同损伤机制，建立了拉、压不同的两种损伤演化方程，用于确定各向异性的拉、压损伤变量。另外，根据应变等效假设，假定有效构型和损伤构型的应变相等，该方法不仅大大简化了模型的推导过程，而且可方便的通过解耦算法进行有效应力和损伤及名义应力的计算，也即塑性部分计算可通过现有的隐式算法实现，损伤部分及名义应力的计算则可通过较为简便的显式算法实现，从而可大大提高计算效率。模型结果与试验结果的对比分析表明：该模型能较好地描述混凝土在三维应力状态下的非线性行为；对双边开口四点弯曲梁试件的模拟也表明：该模型能反应混凝土损伤各向异性的特点，计算结果相比ABAQUS软件自带的混凝土损伤塑性本构模型(CDP模型)更符合实际情况，计算效率也更高。Abstract: A new and relatively simple anisotropic plastic-damage constitutive model is developed to facilitate the numerical simulation of the failure of concrete structures. For better describe the different damage mechanisms of concrete under tensile and compressive loadings, two different damage evolution equations, being suitable for determining the anisotropic tensile and compressive damage variables, are established. Moreover, the strain equivalence hypothesis is used to assume that the strains in both the effective and nominal configurations are equal. Through this method, the deducing process of the model can be simplified, and the effective stress calculation and the damage evolution are able to be implemented by a decoupled algorithm. Thusly, the plastic part of the model can be implemented implicitly, and the damage part and nominal stress calculation can be implemented explicitly, which is convenient for numerical implementation. The model response is compared to a wide range of test results. The results show that the model is able to describe the nonlinear behavior of concrete in three-dimensional stress state well. The simulation of double-edge-notched (DEN) specimen shows that: the model can reflect the characteristics of concrete damage anisotropy. The results are more consistent with the actual situation, and the computational efficiency is also higher than that of the concrete damage plasticity (CDP) model provided by ABAQUS software.
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Key words:
- concrete /
- plastic-damage model /
- anisotropic damage /
- tensile damage /
- compressive damage
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图 1 单轴拉伸荷载作用下混凝土柱有效构型与名义构型对比
Figure 1. The comparison of the nominal and effective configurations of cylindrical bar under uniaxial tension
图 11 第1主应变方向对应的损伤区分布
Figure 11. Distribution of damage zone corresponding to the first principal strain direction
图 12 第2主应变方向对应的损伤区分布
Figure 12. Distribution of damage zone corresponding to the second principal strain direction
图 13 第3主应变方向对应的损伤区分布
Figure 13. Distribution of damage zone corresponding to the third principal strain direction
图 15 由CDP模型计算的拉损伤云图
Figure 15. Distribution of damage in tension calculated by CDP model
图 16 由CDP模型计算的压损伤云图
Figure 16. Distribution of damage in compression calculated by CDP model
材料参数 数值 材料参数 数值 初始弹性
模量${E_0}/{\rm{GPa}} $31.00 峰值应力$f_{\rm u}^ -$对应应变$\varepsilon _{\rm u}^ - /( \times {10^{ - 3} })$ −1.19 初始未损伤弹性模量${ {\bar E}_0}/{\rm{GPa}} $ 33.08 极限应变$\varepsilon _{\rm f}^ - /( \times {10^{ - 3} })$ −8.53 泊松比ν 0.20 压缩硬化参数Q−/MPa 57.50 初始损伤d0 0.063 压缩硬化参数b− 1,082 单轴抗压屈服
强度$f_0^ - /{\rm{MPa}} $−17.08 控制压缩应力-应变
曲线形状参数c13.21 单轴受压峰值
应力$f_{\rm u}^ - /{\rm{MPa} }$−28.13 控制压缩应力-应变
曲线形状参数c2−0.12 注：Q−和b−分别代表单轴压缩饱和应力及饱和率，用来控制单轴压缩时硬化曲线的形状。 
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表 2 根据文献[40]试验结果得到的材料参数表
Table 2. Material constants identified from the experimental results of literature[40]
材料参数 数值 材料参数 数值 初始弹性
模量${E_0}/{\rm{GPa} } $31.00 抗拉强度$f_0^ + $对应
应变$\varepsilon _0^ + /( \times {10^{ - 4}}) $1.12 初始未损伤弹性
模量${ {\bar E}_0}/{\rm{GPa}} $33.08 拉伸硬化参数${h^ + }/{\rm{MPa} } $ 2,733 泊松比ν 0.20 控制拉伸应力-应变曲线形状参数${\alpha _u} $ 4.14 初始损伤d0 0.063 控制拉伸应力-应变曲线形状参数a 0.05 单轴抗拉
强度$f_0^ + /{\rm{MPa}} $3.47 − − 注：h+表示单轴拉伸饱和应力，用来控制单轴拉伸时硬化曲线的形状。
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表 3 单轴单调压缩试验所用材料参数表
Table 3. Material properties used for the monotonic uniaxial compressive test
材料参数 数值 材料参数 数值 初始弹性模量${E_0}/{\rm{GPa}}$ 31.70 峰值应力$f_{\rm u}^ -$对应应变$\varepsilon _{\rm u}^ - /( \times {10^{ - 3} })$ −1.61 初始未损伤弹性模量${ {\bar E}_0}/{\rm{GPa}}$ 33.83 极限应变$\varepsilon _{\rm f}^ - /( \times {10^{ - 3} })$ −7.23 泊松比ν 0.20 压缩硬化参数Q−/MPa 27.58 初始损伤d0 0.063 压缩硬化参数b− 1,655 单轴抗压屈服强度$f_0^ - /{\rm{MPa}}$ −11.74 控制压缩应力-应变曲线形状参数c1 3.00 单轴受压峰值应力$f_{\rm u}^ - /{\rm{MPa} }$ −29.13 控制压缩应力-应变曲线形状参数c2 −1.96 注：Q−和b−分别代表单轴压缩饱和应力及饱和率，用来控制单轴压缩时硬化曲线的形状。
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表 4 单轴单调拉伸加载试验所用材料参数表
Table 4. Material parameters used for the monotonic uniaxial tensile test
材料参数 数值 材料参数 数值 初始弹性模量E0/GPa 31.00 抗拉强度$f_0^ + $对应应变$\varepsilon _0^ + /( \times {10^{ - 4}}) $ 1.24 初始未损伤弹性模量${\bar E_0}/{\rm{GPa}}$ 33.08 拉伸硬化参数${h^ + }/{\rm{MPa}}$ 14,600 泊松比ν 0.20 控制拉伸应力-应变
曲线形状参数${\alpha _{\rm u}}$4.20 初始损伤d0 0.063 控制拉伸应力-应变
曲线形状参数a0.05 单轴抗拉强度$f_0^ + /{\rm{MPa}}$ 3.84 − − 注：表示h+单轴拉伸饱和应力，用来控制单轴拉伸时硬化曲线的形状。
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表 5 双轴压缩加载试验所用材料参数表
Table 5. Material constants used for the biaxial compressive test
材料参数 数值 材料参数 数值 初始弹性模量${E_0}/{\rm{GPa}}$ 32.00 单轴抗压屈服强度$f_0^ - /{\rm{MPa }}$ −15.82 初始未损伤弹性模量${ {\bar E}_0}/{\rm{GPa}}$ 34.15 单轴受压峰值应力$f_{\rm u}^ - /{\rm{MPa} }$ −33.50 泊松比ν 0.20 峰值应力$f_{\rm u}^ -$对应应变$\varepsilon _{\rm u}^ - /( \times {10^{ - 3} })$ −1.77 初始损伤d0 0.063 极限应变$\varepsilon _{\rm f}^ - /( \times {10^{ - 3} })$ −6.81 单轴抗拉强度$f_0^ + /{\rm{MPa}}$ 2.88 压缩硬化参数${Q^ - }/{\rm{MPa}}$ 39.14 抗拉强度$f_0^ + $对应应变$\varepsilon _0^ + /( \times {10^{ - 4}}) $ 0.91 压缩硬化参数b− 875.97 拉伸硬化参数h+/MPa 13,175 控制压缩应力-应变曲线形状参数c1 3.59 控制拉伸应力-应变曲线形状参数${\alpha _{\rm u}}$ 1.07 控制压缩应力-应变曲线形状参数c2 −5.05 控制拉伸应力-应变曲线形状参数a 0.05 − − 注：h+表示单轴拉伸饱和应力，用来控制单轴拉伸时硬化曲线的形状；Q−和b−分别代表单轴压缩饱和应力及饱和率，用来控制单轴压缩时硬化曲线的形状。
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表 6 双边开口混凝土梁试件断裂数值模拟材料参数
Table 6. Material parameters used for the DEN specimen fracture simulation
材料参数 数值 材料参数 数值 初始弹性模量${E_0}/{\rm{GPa}} $ 30.00 单轴抗压屈服强度$f_0^ - /{\rm{MPa }} $ −20.00 初始未损伤弹性模量${ {\bar E}_0}/{\rm{GPa}} $ 32.02 单轴受压峰值应力$f_{\rm u}^ - /{\rm{MPa} }$ −46.60 泊松比ν 0.20 峰值应力$f_{\rm u}^ -$对应应变$\varepsilon _{\rm u}^ - /( \times {10^{ - 3} })$ −2.31 初始损伤d0 0.063 极限应变$\varepsilon _{\rm f}^ - /( \times {10^{ - 3} })$ −17.73 单轴抗拉强度$f_0^ + /{\rm{MPa}} $ 3.44 压缩硬化参数${Q^ - }/{\rm{MPa}} $ 60.98 抗拉强度$f_0^ + $对应应变$\varepsilon _0^ + /( \times {10^{ - 4}}) $ 1.19 压缩硬化参数b− 837.00 拉伸硬化参数h+/MPa 13,552 控制压缩应力-应变曲线形状参数c1 23.35 控制拉伸应力-应变曲线形状参数${\alpha _{\rm u}}$ 3.43 控制压缩应力-应变曲线形状参数c2 −29.78 控制拉伸应力-应变曲线形状参数a 0.05 − − 注：h+表示单轴拉伸饱和应力，用来控制单轴拉伸时硬化曲线的形状；Q−和b−分别代表单轴压缩饱和应力及饱和率，用来控制单轴压缩时硬化曲线的形状。
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表 7 CDP模型计算时所用混凝土弹塑性参数
Table 7. Elasto-plastic mechanic parameters of concrete used by CDP model
材料参数 数值 材料参数 数值 弹性模量E/GPa 32.0 流动势偏心率e 0.1 泊松比ν 0.2 粘性耗散系数$\mu $ 0.0005 双轴极限抗压强度与单轴抗压强度之比fb0/fc0 1.16 拉伸子午面上与压缩
子午面上的第二应力
不变量之比K0.6667 膨胀角$\psi $/(°) 30.0 − −
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表 8 CDP模型计算时所用混凝土损伤参数
Table 8. Damage parameters of concrete used by CDP model
压缩屈服
强度/MPa非弹性
应变/(×10−3)压损伤
因子拉伸屈服
强度/MPa非弹性
应变/(×10−3)拉损伤
因子20.00 0.00 0.00 2.49 0.00 0.00 31.67 0.10 0.06 1.43 0.12 0.11 38.70 0.21 0.09 0.94 0.23 0.26 43.83 0.37 0.14 0.67 0.34 0.41 45.92 0.74 0.24 0.46 0.54 0.59 43.35 1.16 0.33 0.38 0.67 0.67 32.17 2.30 0.51 0.30 0.90 0.78 19.08 3.88 0.69 0.26 1.10 0.86 14.54 4.80 0.77 0.18 1.78 0.94 7.10 8.12 0.91 0.16 2.12 0.95 4.41 11.66 0.97 0.14 2.50 0.96
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