Research On The Application And Development Of Complex Variable Method In Fracture Mechanics(1909-2019)

Based on the classical Griffth(A.A.Griffth,1893-1963)theory,Fracture mechanics which is an important branch of solid mechanics,was developed in the early 20 th century and gradually formed in the 1950 s.Takes the crack as the main research object,Fracture mechanics analyzes the distribution state of stress under stress,thus exploring the fracture criterion and the law of crack's growth.Fracture mechanics is derived from production practice,therefore,has widespread applications in construction engineering,aerospace,transportation,machinery manufacturing,bioengineering and many other fields.With its further study,the complex variable method is popular to many researchers because of its complete theoretical system.The theory of complex function can be traced back to the early 20 th century,when it was basically developed by France's Cauchy(A.L.Cauchy,1789-1857),Germany's Riemann(B.Riemann,1826-1866)and Germany's Weierstrass(K.T.W.Weierstrass,1815-1897).And because of its perfect content system,the theory of complex function has laid a solid foundation for the application of complex variable method in fracture mechanics.In 1909,the Russia scientist,Kolosov(G.V.Kolosov,1867-1936)effectively solved the mechanical problems with the theory of complex variable function.In 1933,Mushailishvili(??????? ???????? ????????????,1891-1976)further systematized the work of Kolosov,and comprehensively studied the application of the complex method in the plane elastic theory.The introduction of this method not only enriched the method of solving mechanical problems,but also laid the foundation for its application in fracture mechanics.In 1957,Irwin(G.R.Irwin,1907-1998)proposed the energy release rate,marking the establishment of linear elastic fracture theory.Consequently,the complex variable method was naturally applied to the field of fracture mechanics,and began to play its unique advantages.Although there are abundant research results so far on the application of complex variable method in fracture mechanics,most of which focus on the specific application process,and almost no literature on systematic research from the perspective of history.Thus,this paper starts from the perspective of the history of mathematics,on the basis of reviewing a large number of documents at home and abroad,makes a systematic analysis and summary of the application of the complex variable method in fracture mechanics.During the research,mainly uses document analysis,historical research and comparative analysis and so on.This research has practical importance both in theory and practice for further understanding the development of fracture mechanics and even predicting its future development direction.The main work of this paper is as follows:1.This paper focuses on the history of the formation and development about fracture mechanics,introduces the important contributions and influences made by Englis(C.E.Englis,1875-1952),Griffith,Orowan(E.Orowan,1901-1989)and Irwin in the formation of fracture mechanics.Meanwhile,the work done by some of the Chinese scholars in this respect and its impact on the development of fracture mechanics are studied.2.The application of complex variable method in fracture mechanics is traced.This paper describes the pioneering work done by Kolosov and Mushailishvili,and it is pointed out that although fracture mechanics was not fully produced at that time,their research results provide necessary theoretical support for the application of complex variable method in fracture mechanics,and also lay a foundation for its future development.3.The application of complex variable method in fracture mechanics in the middle and late 20 th century(1950-1990)is studied.Through summing up along with analysis,this paper discusses the summary and development of the complex variable method by A.H.England,Tang Limin and Lu Jianke in China in detail.Their work reflects how the complex variable method developed during that period.4.The development of complex variable method in fracture mechanics after 1990 s is analyzed.In this period,the application range of complex variable method is extended from classical material to new material,and conformal mapping is extended from rational function to irrational function.The achievements and impact of Fan Tianyou's group on the complex variable method of fracture mechanics are mainly introduced.5.The application and development of complex variable method in solid and piezoelectric quasicrystals are studied.Inspired by the existing literature,this paper uses the complex variable method to discuss the elastic field of tow-dimensional quasicrystals of point group 10 under the action of straight dislocation and linear force,and also to research the elastic problem of one-dimensional hexagonal piezoelectric quasicrystals with moving screw dislocation.It is found that the application and development of complex variable method in fracture mechanics have the following characteristics: 1.Its development follows the overall law from slow to fast and from point to surface;2.Uneven geographical distribution and lack of international communication in the early stage;3.Since the 21 st century,the depth and breadth of application have been increasing,and the discipline integration has been further strengthened;4.Chinese scholars have made important contributions to the application and development of complex transformation methods.