【作者】 王敏；

【导师】 易志坚；

【作者基本信息】 重庆交通大学 ， 桥梁与隧道工程， 2023， 博士

【摘要】 许多工程学科,如土木工程、机械工程、航空航天工程等中的力学基础理论,既是力学问题,也是工程问题。工程中存在大量的对称性问题。对称性问题最典型的特征就是具有对称线,对称线附近的区域往往是相关问题最具代表性的区域,最大或最小应力、塑性区尺寸往往都在对称线上,如果能够对对称线附近的弹-塑性场和弹塑性边界进行解析求解,在很大程度上就解决了这一类问题的核心和关键。然而,弹-塑性分析遇到的最大困难在于塑性理论的偏微分方程难以解析求解,工程中绝大多数对称性问题都难以求得弹-塑性解析解。本文提出基于泰勒级数的弹塑性问题的对称线分析方法,利用弹-塑性应力场在对称线附近泰勒级数的奇偶性,有效减少了泰勒级数幂次项的数量,进而能够在对称线附近将求解偏微分方程的问题转化为求解常微分方程,精确得出塑性应力场在对称线附近泰勒级数形式的通解,破解了偏微分方程解析求解的难题,进而能对理想弹塑性平面和反平面对称性问题进行弹-塑性解析分析,得出对称线附近足够精确的弹-塑性场和弹塑性边界。本文的主要内容和结果如下:(1)将对称线问题划分为3类:Ⅰ类、Ⅱ类和Ⅲ类,根据3类对称性问题塑性应力场在对称线附近泰勒级数的奇偶性,将求解偏微分方程的问题转化为求解常微分方程,分别得出3类问题的塑性应力场在对称线附近泰勒级数形式的通解。(2)根据3类对称性问题的弹塑性边界在对称线附近的泰勒级数表达式,分别在直角坐标和极坐标下,得出弹塑性边界上任意一点的单位法向量、对称线附近弹性应力场和塑性应力场在弹塑性边界上的表达式,以及弹性应力场和塑性应力场在弹塑性边界上的匹配表达式。(3)针对远场均匀受剪的经典Ⅲ型理想弹塑性材料裂纹无限板问题进行了弹-塑性分析,通过塑性区应力场与弹性区应力场在弹塑性边界上的匹配,得出其在对称线附近的弹-塑性场和弹塑性边界,并对结果进行了分析,验证了对称线分析方法与裂纹线场分析方法的一致性。(4)针对过去一直没有弹-塑性解析结果的Ⅱ类对称性问题——受面内剪切的理想弹塑性材料中心圆孔无限板问题,分别在平面应力和平面应变下进行弹-塑性解析分析,通过塑性区应力场与弹性区应力场在弹塑性边界上的匹配,得到了两种情况下的弹-塑性场和弹塑性边界,并分别对两种情况下的弹性极限荷载、塑性极限荷载和塑性区的发展变化进行了分析和比较。(5)针对两个过去一直没有获得弹-塑性解析结果的Ⅰ类对称性问题——理想弹塑性材料半平面体受集中力问题和受单向均匀拉伸的中心圆孔板问题进行弹-塑性解析分析,通过塑性区应力场与弹性区应力场在弹塑性边界上的匹配,分别得到了两个问题的弹-塑性场和弹塑性边界。结果表明,受集中力作用的半平面体,无论集中力大小如何,都存在塑性区(因集中力为点荷载)。受单向均匀拉伸的中心圆孔板有两条对称线,对过圆孔圆心,垂直于受拉方向的轴为对称轴的远场受均布拉伸圆孔板,当塑性区长度r_p刚好达到圆孔边缘时,对应的弹性区对称线上的拉应力σ_y刚好达到屈服强度2k,拉伸荷载为q为(2k)/3,是问题的弹性极限荷载。当q→2k时,此时,r/p→∞,整块板都进入塑性区,达到塑性极限荷载。而在过圆孔圆心,平行于受拉方向的对称线上,当拉伸荷载达到塑性极限荷载时,塑性区才开始出现,对称线上的其它地方不会产生塑性屈服。(6)本文解析结果的正确性通过了三种方法的验证,(1)对过去已有相应的解析结果的情形,本文得到的解析结果与线场分析方法的结果和用其他方法得到的解析结果相比较,结果完全一致;(2)对过去没有求得相应的解析结果的情形,本文通过在不同坐标下比较,结果完全一致;(3)通过有限元数值计算,与本文解析结果进行比较,规律的一致性相同,相对误差小。(7)在工程应用方面,本文采用对称线分析方法,对一些工程上常见的、典型的对称性问题,包括远场受均匀拉伸的有限宽圆孔板、受扭转的矩形截面直杆、受扭转的圆截面直杆和受均布外压的圆环进行了弹-塑性分析,得到四种问题在对称线附近的弹-塑性场和弹塑性边界,得到了塑性区尺寸、弹性极限荷载、塑性极限荷载等可实际用于工程的解析公式。与具体的工程实例相比,本文分析的是工程中常见的、典型的对称性问题,结果具有更一般、更普遍的意义,为此类问题在工程上的分析和实际应用提供了解析解答,也为更多无解析结果的、工程中常见的、典型的对称性问题提供了弹塑性解析思路和方法。更多还原

【Abstract】 The basic theories of mechanics in engineering science,such as civil engineering,mechanical engineering,aerospace engineering,are concerned with both mechanical and engineering problems.A large number of symmetry problems exist in engineering structures.A typical feature of this type of problems is the presence of symmetric lines.The symmetric line is usually the most critical area where the maximum or minimum stress,maximum or minimum plastic zone size locates.Once the elastic-plastic field near the symmetric line is obtained,it in most cases solves the key of the problems in elastic-plastic analysis.However,the biggest difficulty encountered in elastic-plastic analysis is that the partial differential equations of the theory of plasticity can hardly be solved analytically.For the vast majority of engineering symmetry problems,elastic-plastic analytical solutions are still waiting to be obtained.The symmetric line analysis method is applied by this study for elastic-plastic analysis of symmetry problems based on their Taylor series.The symmetric line analysis method makes use of the parity nature of the elastic-plastic stress field in the Taylor series near the symmetric line,thus effectively reduces the number of power terms of the Taylor series,and then transform the problem of solving partial differential equations into ordinary ones near the symmetric line,and finally obtain the general Taylor series solution of the plastic field near the symmetric line.Accordingly,elastic-plastic analytical analysis on the elastic-perfectly plastic plane and anti-plane symmetry problems can be conducted and sufficiently precise elastic-plastic fields and elastic-plastic boundaries near the symmetrical line can be obtained.The main content of this study is as follows.(1)Divide the elastic-perfectly plastic symmetry problems into three categories:Type Ⅰ,Type Ⅱ and Type Ⅲ problems.Based on the parity nature of the plastic stress field in Taylor series form near the symmetric line,the problem of solving partial differential equations in the theory of plasticity is turned into solving ordinary ones,so the general Taylor series solutions of the plastic stress fields near the symmetric lines are obtained for the three types of problems respectively.(2)Based on the Taylor series expansions of the elastic-plastic boundary near the symmetric line,the unit normal vector of any point on the elastic-plastic boundary,the expressions of the elastic and plastic stress fields near the symmetric line on the elastic-plastic boundary,and the matching equations of the elastic and plastic stress fields on the elastic-plastic boundary are obtained for the three types of symmetry problems respectively.(3)Elastic-plastic analytical analysis was conducted on a classical Mode Ⅲ crack under uniform shear in the far field.By matching the plastic stress field with the elastic stress field on the elastic-plastic boundary,the elastic-plastic field and elastic-plastic boundary near the symmetric line was obtained,and the results were analyzed,which verifies the consistency between the symmetric line analysis method and the crack line field analysis method.(4)Elastic-plastic analytical analysis was conducted on a Type Ⅱ problem—a plate with a central circular hole subjected to in-plane shear,for which elastic-plastic analytical solution has not been obtained.Analysis was done under the plane stress condition and plane strain condition respectively.By matching the elastic stress field and the plastic stress field on the elastic-plastic boundary,the elastic-plastic fields and elastic-plastic boundaries near the symmetric line were obtained for both conditions.The elastic limit,plastic limit and the plasticity development trend under two different conditions were analyzed and compared.(5)Elastic-plastic analytical analysis was conducted on two Type Ⅰ problems--a half-plane body subjected to a concentrated force and a plate with a circular hole subjected to uniaxial tension in the far field,for both of which elastic-plastic analytical solutions have not been obtained before.By matching the elastic stress field and the plastic stress field on the elastic-plastic boundary,the elastic-plastic fields and elastic-plastic boundaries near the symmetric line were obtained for the two problems.The results indicate that for the half-plane body subjected to a concentrated force,regardless of the magnitude of the concentrated force,there always exists a plastic zone(as the concentrated force is a point load).The plate with a circular hole subjected to uniaxial tension in the far field has two symmetric lines.For the one with the symmetric line passing through the center of the circular hole and perpendicular to the tensile direction,when the plastic zone length r_p reaches the edge of the circular hole,the tensile stress σ_y on the symmetric line reaches the elastic limit 2k,and the corresponding tensile loadis q is 2k/3.When the tensile load q→2k,the plastic zone length r_p→∞,which indicates that the whole symmetric line of the plate enters plasticity and the plastic limit is reached.For the other one with symmetric line passing through the center of the circular hole and parallel to the tensile direction,plasticity emerges only at the edge of the circular hole on the symmetric line when the tensile load reaches the plastic yield limit,and plastic yield will not occur elsewhere on the symmetrical line.(6)The correctness of the analytical results in this study was verified by three ways.(1)For the cases where corresponding analytical results have been obtained in the past,the analytical results obtained in this study were compared and verified with them,including the results by the crack line field analysis and other methods.(2)For the cases where no corresponding analytical results are available yet,the results under different coordinates were derived and compared,and the results show that they are consistent;(3)Comparison between the finite element results and the analytical results in this study show that the results are consistent with small differences.(7)In terms of the engineering application of the symmetric line method,elastic-plastic analysis was done on four typical engineering symmetry problems,i.e.,a finite-width plate with a central circular hole subjected to tension,a rectangular cross-section bar subjected to torsion,a circular cross-section bar subjected to torsion and a circular cross-section cylinder or disc subjected to compression.The elastic-plastic fields and elastic-plastic boundaries near the symmetric line were obtained for all the four problems.Compared with other specific engineering examples,the symmetry problems analyzed in this study are common and typical examples and thus the results have more general significance.They provide analytical solutions for the similar problems in engineering analysis and practical applications,and also provide ideas and methods for solving other engineering elastic-plastic symmetry problems without analytical results so far.更多还原