【作者】 杨涛；

【导师】 徐华；

【作者基本信息】 广西大学 ， 结构工程， 2020， 硕士

【摘要】 工程结构中的裂纹以及加载扩展后的裂纹通常以曲线型式存在,含曲线裂纹的功能梯度材料（Functionally Graded Material,简记FGM）结构因其材料属性的非均匀性,在复杂荷载作用产生的混合断裂模式下,同时求解其各裂尖应力强度因子（Stress Intensity Factors,简记SIFs）并保证高精性具有一定的难度,基于SIFs的FGM结构的裂纹扩展研究更为复杂。本文围绕含曲线裂纹FGM结构裂尖SIFs的求解和基于SIFs的裂纹扩展问题展开了以下研究:（1）因曲线裂纹不满足Williams级数位移场的边界条件,将裂尖奇异区曲线裂纹等效为折线裂纹,基于广义参数有限元法,详细推导了复杂荷载作用下均质材料结构裂尖SIFs分析的广义参数Williams单元（简记W单元）计算格式。空间任意荷载可分解为裂纹平面内与平面外荷载耦合作用,控制方程随结点自由度数的增加而增大,导致所需计算机内存随单元结点数呈指数级增长,出现计算缓慢甚至死机现象,将总刚进行压缩存储并结合改进的LU分解法可有效地对控制方程进行降阶处理,提高了计算效率,并同时获得所有裂尖SIFs值。W单元位移场中含有与SIFs直接相关的参数,可避免传统有限元法需要繁琐的后处理引入人为误差的缺陷。为含曲线裂纹FGM结构的断裂参数研究奠定了理论基础。（2）基于广义参数有限元法,考虑结构弹性模量呈单向梯度变化,利用分段指数模型对其进行分层处理,将随坐标变化的弹性模量E/剪切模量G代入弹性矩阵,建立了复杂荷载作用下含曲线裂纹FGM结构裂尖SIFs分析的非均质广义参数Williams单元（简记非均质W单元）。因材料属性的非均匀性,裂尖奇异区和外围常规区所有单元刚度矩阵均为材料参数的函数,奇异区中的单元尚需逐个单元进行广义参数单刚转换,总刚集成中同样存在矩阵爆炸问题,仍对总刚进行压缩存储和改进的LU分解法大大提高了计算效率且可直接确定FGM结构裂尖SIFs。分析表明:基于广义参数有限元法建立的非均质W单元结合改进的LU分解法求解含曲线裂纹FGM结构裂尖SIFs具有高精性,为准确模拟FGM结构裂纹扩展提供了新思路。（3）研究建立了裂纹扩展角θ₀与广义参数之间的关系表达式,并改进了最大周向应力准则。将Williams级数位移场代入最大周向应力准则,推导得到裂纹扩展角θ₀与广义参数之间的关系式以及改进的最大周向应力准则,采用本文建立的非均质W单元求解广义参数的值,代入裂纹扩展角表达式并判断裂纹是否开裂扩展,进一步预测裂纹的起裂荷载大小。假设裂纹沿直线等步长扩展,本文方法可避免普通有限元法多次求解裂尖SIFs造成的误差累积,较真实的模拟裂纹扩展路径。算例分析表明:可通过改变弹性模量分布形式、梯度大小及材料类型来改变裂纹的扩展方向及结构的起裂荷载,为含裂纹FGM结构裂纹扩展机理研究提供了一种新方法。更多还原

【Abstract】 When the direction of the complex load changes,the crack propagation angle changes,which makes the crack in the actual engineering usually behave as a curved crack.Due to the non-uniformity of the properties of Functionally Gradient Material（abbreviated as FGM）structures with cracks,the stress intensity factors（abbreviated as SIFs）of each tip of cracks are relatively difficult to be solved efficiently,precisely and simultaneously under complex loads.Therefore the results of researching on the crack propagation of FGM structure on the basis of SIFs becomes ever more comple.The research has been carried out focusing on the solution of the crack tip SIFs of FGM structure and the crack propagation problem based on the SIFs:（1）Because the boundary conditions of the Williams series displacement field can not be directly used for the curved crack,the curved crack in the singular region of the crack tip is equivalent to the broken line crack.Based on the finite element method with generalized degrees of freedom,the Williams element with generalized degrees of freedom（abbreviated as W element）calculation format for SIFs analysis of the crack tip of homogeneous material structure under complex load is derived in detail.The spatial arbitrary load can be decomposed into the coupling action of in-plane and out-of-plane loads,which makes the control equation increase with the increase of node degrees of freedom,resulting in the exponential growth of the required computer memory with the number of nodes of the element,resulting in the phenomenon of slow calculation or even crash.The control equation can be effectively reduced by compressing the total stiffness and combining the improved LU decomposition method.The calculation efficiency is improved and all SIFs values of crack tip are obtained at the same time.The displacement field of W element contains parameters directly related to SIFs,which can avoid the defect that the traditional finite element method needs tedious post-processing and human error.It lays a theoretical foundation for the study of fracture parameters of FGM structure with cracks（2）Considering the unidirectional gradient change of the elastic modulus of the structure,the elastic modulus E / shear modulus G with the coordinate change is taken into the elastic matrix by using the piecewise exponential method.Then based on the finite element method with generalized degrees of freedom,the heterogeneous W element with generalized degrees of freedom（abbreviated as heterogeneous W element）for SIFs analysis of the crack tip of FGM structure with curved crack under complex load is established.Because of the nonuniformity of the material properties,the stiffness matrix of all the elements in the singular region and the conventional region around the crack tip is a function of the material parameters.The generalized parameter single stiffness transformation needs to be carried out one by one.The matrix explosion problem also exists in the total stiffness integration.The improved LU decomposition method greatly improves the calculation efficiency and can directly determine the SIFs of the crack tip of FGM structure.The analysis shows that the solution of SIFs of FGM structure with curved crack based on heterogeneous W element and improved LU decomposition method is highly accurate and efficient,which provides a new idea for accurately simulating the crack propagation of FGM structure.（3）The relationship between the crack growth angle θ₀ and generalized parameters is established,and the improved maximum circumferential stress criterion is improved.After plugging the Williams series displacement field into maximum hoop stress criterion,the relationship between the crack growth angle θ₀ or the criterion of crack propagation and the generalized parameters is derived.The value of the generalized parameters,which is solved by using the heterogeneous W element established in this paper,is introduced to the expression between the crack growth angle θ₀ and the generalized parameters.If the crack propagates,the crack initiation load is predicted,and the crack is assumed to propagate in a straight line with equal steps.This method can avoid the error accumulation caused by the common finite element method solving SIFs at the crack tip many times,and simulate the crack propagation path more truly.The results show that the crack propagation direction and the crack initiation load can be changed by changing the distribution or the gradient of elastic modulus,or the type of material.A new method is provided for the study of crack propagation in FGM structure with crack.更多还原