| 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. |
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