修正的梯度弹性理论及其损伤理论—有限元实现及其应用

发布时间：2018-02-28 10:22

本文关键词： 应变梯度内部特征长度尺寸效应网格依赖性损伤局部化　出处：《长沙理工大学》2015年硕士论文　论文类型：学位论文

【摘要】：材料的破坏是力学发展的世纪难题,梯度理论是解决材料破坏机理的必要途径。由于物体的均匀性假设不能精确满足,经典连续介质理论在描述物体微观结构特性起主导作用的破坏过程时会出现如“描述材料的应变/损伤局部化”,“描述材料的尺寸效应”,“数值模拟时的网格病态依赖性”等疑难。为了合理描述物体的力学行为,我们在本构方程中引入了应变梯度、内部特征长度向量和损伤,发展新的应变梯度弹性理论及其损伤理论。本文通过应变和应变梯度的比值定义一个材料内部特征长度向量;假设应变能密度是由应变张量和应变梯度张量决定。依据上述定义和假设,由应变能密度在初始状态的泰勒展开式推导出一种修正的梯度弹性理论(Modified Gradient Elasticity,MGE理论)。由于引入了应变梯度项,模型可以描述出现较大应变梯度时材料的强度和变形行为;在材料内部特征长度向量为零时,可退化成经典弹性理论。基于虚功原理和变分原理,建立了相应的小变形、准静态荷载情况下梯度弹性问题的有限元格式,编制MGE理论的有限元程序,数值算例验证了所编程序的正确性和MGE理论在内部特征长度向量为零时退化为经典弹性理论的结论。然后,利用MGE有限元程序对双材料剪切层问题、双材料拉伸边界层问题、裂纹尖端奇异性进行了数值模拟。结果显示:双材料剪切和拉伸边界层的厚度由内部特征长度决定;MGE理论能模拟边界层的尺寸效应并消除网格病态依赖性;消除了裂尖应变场的奇异性。最后,在MGE理论本构方程中引入损伤变量,建立了MGE损伤模型,模拟了土样单轴压缩的损伤局部化现象。结果显示:MGE损伤理论能消除数值结果的网格病态依赖性;能模拟岩土材料损伤局部化带的萌生、发展直到破坏的整个过程;损伤局部化剪切带的宽度与内部特征长度有关。
[Abstract]:The failure of materials is a difficult problem in the development of mechanics in the century. Gradient theory is the necessary way to solve the failure mechanism of materials. Classical continuum theory can be used to describe the failure process in which the microstructural characteristics of an object play a leading role, such as "describing strain / damage localization of material", "describing material size effect", "meshing morbid dependence" in numerical simulation. In order to reasonably describe the mechanical behavior of an object, The strain gradient, internal characteristic length vector and damage are introduced into the constitutive equation. A new strain gradient elastic theory and its damage theory are developed. In this paper, a material internal characteristic length vector is defined by the ratio of strain to strain gradient. It is assumed that the strain energy density is determined by strain Zhang Liang and strain gradient Zhang Liang. Based on the Taylor expansion of strain energy density in the initial state, a modified gradient Gradient elasticity theory is derived. With the introduction of the strain gradient term, the model can describe the strength and deformation behavior of the material with large strain gradient. When the characteristic length vector of material is 00:00, it can degenerate into classical elastic theory. Based on virtual work principle and variational principle, the finite element scheme of gradient elastic problem under small deformation and quasi-static load is established. The finite element program of MGE theory is compiled. Numerical examples verify the correctness of the program and the conclusion that the MGE theory degenerates into the classical elastic theory at 00:00 in the interior characteristic length vector. Then, the MGE finite element program is used to solve the problem of shearing layer in the bimaterial. Bimaterial stretching boundary layer problem, The numerical simulation of crack tip singularity shows that the thickness of the bimaterial shear and tensile boundary layer is determined by the internal characteristic length. The MGE theory can simulate the size effect of the boundary layer and eliminate the pathological dependence of the meshes. The singularity of the crack tip strain field is eliminated. Finally, the damage variable is introduced into the constitutive equation of MGE theory, and the MGE damage model is established. The damage localization of soil samples under uniaxial compression is simulated. The results show that the damage theory can eliminate the mesh-morbid dependence of numerical results, simulate the initiation and development of damage localization zones of geotechnical materials until the whole process of failure. The width of the damage localized shear band is related to the internal characteristic length.
【学位授予单位】：长沙理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TB301

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