基于非局部增强梯度模型的混凝土断裂损伤研究
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摘要
混凝土材料由于微裂纹的存在和材料本身的非均匀性,使得其断裂失效非常复杂。损伤力学通过引入一个场变量,能够对混凝土中存在的微缺陷或微裂纹及其形成与发展直至混凝土的断裂失效进行统一描述,其对裂纹的连续性描述特别适宜于数值模拟。然而,由于损伤的局部性和损伤场光滑性相矛盾,基于标准损伤模型的有限元解严重依赖于空间离散,随着网格越来越密,损伤局部化区域体积趋于零。非局部损伤模型和梯度损伤模型通过在本构关系中引入空间相互作用,较为有效地消除了在损伤的数值分析中病态的网格依赖性。
本文首先对国内外混凝土的断裂和损伤的研究进行了综述,并对在混凝土的断裂和损伤的有限元数值分析中的病态网格依赖性的解决途径进行了总结。
本文的重点之一是对非局部模型进行了研究。基于混凝土的微结构的不均匀性,在传统的非局部模型中引入光滑的权函数代替隐含的均匀分布的权,构造了一种新的非局部损伤模型。并分别用B-Spline函数和Gauss函数作为权函数求解非局部模型中的非局部等效应变,用带有I型中心穿透裂纹的受拉伸板作为计算模型进行了数值计算。在此基础上,运用非局部损伤模型对混凝土的应变软化和损伤的局部化进行了有限元数值模拟,并与局部损伤模型的数值计算结果进行了比较。证实了带有光滑权函数的非局部损伤模型能较为有效地消除有限元数值分析中的病态网格的依赖性,且能模拟混凝土的应变软化和损伤的局部化现象。
本文的另一个重点是研究了梯度损伤模型。首先分析了非局部损伤模型在有限元实现过程中的缺陷,对非局部等效应变进行泰勒展开,得到显式梯度损伤模型与隐式梯度损伤模型。并通过推导隐式梯度模型的控制方程,证明了隐式梯度模型的控制方程在整个损伤过程中将保持类型不变。讨论了隐式梯度模型与非局部模型的关系。其次分析了隐式梯度模型的边界条件,在此基础上,分析了受均布载荷作用下平面应力裂纹混凝土板裂尖的应变场,提出了在隐式梯度模型中采用数值外插法求解裂尖的非局部等效应变,以解决复杂内边界的损伤场光滑问题。在空间复杂应力状态下,引入空间有效损伤矩阵,构造一种基于隐式梯度公式的各向异性的非局部损伤模型。最后讨论了隐式梯度模型的有限元实现,提出了在有限元实现过程中插值形函数的选取原则,并分别对一个均匀受拉的I型中心穿透裂纹的板的损伤和三点弯曲梁的损伤进行了数值模拟。
最后,本文对隐式梯度模型在高周疲劳损伤中的应用进行了研究,提出了一种新的计算疲劳寿命问题的计算方法——基于非局部增强梯度损伤模型的疲劳损伤算法。并以一单侧带缺口的拉杆为例,将数值计算结果和Paris公式计算结果进行了比较。
The failure of concrete is very complicated as the micro-cracks of concrete existence and non-uniformity of the material itself. Damage mechanics by introducing a set of field variables can describe not only the existence and the growth of the micro-cracks or micro-defects of concrete but also the failure and fracture of concrete. The very fact that damage mechanics use a continuous representation of crack renders them particularly suitable for numerical simulations. However the localization of damage is in contradiction with the supposed smoothness of the damage field and thus the finite element solution based on damage model depend heavily on the spatial dispersion. As denser mesh, the growth of damage tends to localize in the smaller possible volume. Nonlocal damage mode and the gradient-enhanced damage model by introducing the space interaction into the constitutive relations can overcome the pathological mesh dependence when damage of concrete is simulated.
The domestic and foreign research of fracture and damage of concrete are reviewed according to extensive literature and then the solutions to the mesh dependence in the finite element numerical analysis for fracture and damage of concrete are summarized in this paper.
One of the focuses of this paper is to research nonlocal damage model. Based on the non-uniform of micro structural of concrete, the smooth weight function is introduced to substitute for the piecewise uniform weighting which has been implicitly assumed in the classical non-local model and therefore a new non-local damage model is built. B-Spline and Gauss functions are used as weight functions to find solutions to nonlocal equivalent strain in the nonlocal damage model. A board with mode -I crack is used a calculation model for numerical calculation. Furthermore, nonlocal damage model and finite element numerical analysis are used for concrete strain softening and damage localization. Its result is compared with the result of numerical results of local damage model. The analytical results prove that nonlocal damage model with smooth weight function can effectively eliminate the pathological mesh dependence in the finite element numerical analysis.
Another focus of this paper is to research gradient damage model. Firstly, the faults of implementation of finite element of non local damage model are analyzed, and by expanding nonlocal equivalent strain into Taylor series, explicit gradient model and implicit gradient model is constructed. Control equations of implicit gradient model remain elliptic during the whole damage process, which is proved by derivation of control equations of the implicit gradient model. The relationship of implicit gradient model and nonlocal model is discussed. Secondly, Boundary conditions of implicit gradient models are analyzed. The numerical extrapolated method for calculation of nonlocal equivalent strain at the crack tip of concrete board under uniformly distributed loads is presented while the strain field was discussed and the supposed smoothness of the damage field is ensured in the complicated internal boundary. In the complicated space stress, effective damage matrix is introduced and anisotropic nonlocal damage model based on the implicit gradient formula is constructed. Finally, finite element implementation of the implicit gradient model is discussed and the defining principle of the interpolations functions is studied during the finite element implementation. The model is applied to numerical simulation of damage of a concrete board with mode -I crack and a three-point bending concrete beam.
In the end, research is performed on the implicit gradient model in the high-cycle fatigue damage and a new method—fatigue damage algorithm based on nonlocal strengthened gradient damage model is presented for calculating the fatigue life. The numerical calculation result of bar with a single lateral is compared with the result of Paris formula.
本文首先对国内外混凝土的断裂和损伤的研究进行了综述,并对在混凝土的断裂和损伤的有限元数值分析中的病态网格依赖性的解决途径进行了总结。
本文的重点之一是对非局部模型进行了研究。基于混凝土的微结构的不均匀性,在传统的非局部模型中引入光滑的权函数代替隐含的均匀分布的权,构造了一种新的非局部损伤模型。并分别用B-Spline函数和Gauss函数作为权函数求解非局部模型中的非局部等效应变,用带有I型中心穿透裂纹的受拉伸板作为计算模型进行了数值计算。在此基础上,运用非局部损伤模型对混凝土的应变软化和损伤的局部化进行了有限元数值模拟,并与局部损伤模型的数值计算结果进行了比较。证实了带有光滑权函数的非局部损伤模型能较为有效地消除有限元数值分析中的病态网格的依赖性,且能模拟混凝土的应变软化和损伤的局部化现象。
本文的另一个重点是研究了梯度损伤模型。首先分析了非局部损伤模型在有限元实现过程中的缺陷,对非局部等效应变进行泰勒展开,得到显式梯度损伤模型与隐式梯度损伤模型。并通过推导隐式梯度模型的控制方程,证明了隐式梯度模型的控制方程在整个损伤过程中将保持类型不变。讨论了隐式梯度模型与非局部模型的关系。其次分析了隐式梯度模型的边界条件,在此基础上,分析了受均布载荷作用下平面应力裂纹混凝土板裂尖的应变场,提出了在隐式梯度模型中采用数值外插法求解裂尖的非局部等效应变,以解决复杂内边界的损伤场光滑问题。在空间复杂应力状态下,引入空间有效损伤矩阵,构造一种基于隐式梯度公式的各向异性的非局部损伤模型。最后讨论了隐式梯度模型的有限元实现,提出了在有限元实现过程中插值形函数的选取原则,并分别对一个均匀受拉的I型中心穿透裂纹的板的损伤和三点弯曲梁的损伤进行了数值模拟。
最后,本文对隐式梯度模型在高周疲劳损伤中的应用进行了研究,提出了一种新的计算疲劳寿命问题的计算方法——基于非局部增强梯度损伤模型的疲劳损伤算法。并以一单侧带缺口的拉杆为例,将数值计算结果和Paris公式计算结果进行了比较。
The failure of concrete is very complicated as the micro-cracks of concrete existence and non-uniformity of the material itself. Damage mechanics by introducing a set of field variables can describe not only the existence and the growth of the micro-cracks or micro-defects of concrete but also the failure and fracture of concrete. The very fact that damage mechanics use a continuous representation of crack renders them particularly suitable for numerical simulations. However the localization of damage is in contradiction with the supposed smoothness of the damage field and thus the finite element solution based on damage model depend heavily on the spatial dispersion. As denser mesh, the growth of damage tends to localize in the smaller possible volume. Nonlocal damage mode and the gradient-enhanced damage model by introducing the space interaction into the constitutive relations can overcome the pathological mesh dependence when damage of concrete is simulated.
The domestic and foreign research of fracture and damage of concrete are reviewed according to extensive literature and then the solutions to the mesh dependence in the finite element numerical analysis for fracture and damage of concrete are summarized in this paper.
One of the focuses of this paper is to research nonlocal damage model. Based on the non-uniform of micro structural of concrete, the smooth weight function is introduced to substitute for the piecewise uniform weighting which has been implicitly assumed in the classical non-local model and therefore a new non-local damage model is built. B-Spline and Gauss functions are used as weight functions to find solutions to nonlocal equivalent strain in the nonlocal damage model. A board with mode -I crack is used a calculation model for numerical calculation. Furthermore, nonlocal damage model and finite element numerical analysis are used for concrete strain softening and damage localization. Its result is compared with the result of numerical results of local damage model. The analytical results prove that nonlocal damage model with smooth weight function can effectively eliminate the pathological mesh dependence in the finite element numerical analysis.
Another focus of this paper is to research gradient damage model. Firstly, the faults of implementation of finite element of non local damage model are analyzed, and by expanding nonlocal equivalent strain into Taylor series, explicit gradient model and implicit gradient model is constructed. Control equations of implicit gradient model remain elliptic during the whole damage process, which is proved by derivation of control equations of the implicit gradient model. The relationship of implicit gradient model and nonlocal model is discussed. Secondly, Boundary conditions of implicit gradient models are analyzed. The numerical extrapolated method for calculation of nonlocal equivalent strain at the crack tip of concrete board under uniformly distributed loads is presented while the strain field was discussed and the supposed smoothness of the damage field is ensured in the complicated internal boundary. In the complicated space stress, effective damage matrix is introduced and anisotropic nonlocal damage model based on the implicit gradient formula is constructed. Finally, finite element implementation of the implicit gradient model is discussed and the defining principle of the interpolations functions is studied during the finite element implementation. The model is applied to numerical simulation of damage of a concrete board with mode -I crack and a three-point bending concrete beam.
In the end, research is performed on the implicit gradient model in the high-cycle fatigue damage and a new method—fatigue damage algorithm based on nonlocal strengthened gradient damage model is presented for calculating the fatigue life. The numerical calculation result of bar with a single lateral is compared with the result of Paris formula.
引文
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