The Asymptotic Expansion Solutions And The Singular Finite Elements For Fracture Analysis Of Reissner Plate Bending Problems

As an important component subjected to transverse loads,plate has found wide application in many engineering fields.However,local stress singularities often occur in a plate resulting from cracks or V-shaped holes in the processes of manufacture and operation,and the plate might be subject to severe fracture damage.Hence,fracture analysis of plate bending problems has always been one of the most important aspects in fracture mechanics.The finite element method(FEM)is a common numerical method used in fracture analysis.but dense meshes are required near the singular points in order to ensure the accuracy of solutions when the conventional FEM is used to deal with the problem of stress singularity.Obviously,this can lead to low computational efficiency.Therefore,improving the accuracy and computational efficiency for the fracture analysis of plate bending problems with local stress singularities is an interesting research topic which is of great practical value in engineering applications.The bending problem of plates and shells is actually a simplified analysis model of three-dimensional problem,and there are a great variety of plate/shell theories in the literature according to different basic assumptions.On the bending fracture problems of cracked plates,Kirchhoff plate theory has been used in most of the earlier research.However,there exist certain theoretical shortcomings when using Kirchhoff plate theory to analyze fracture problems of plate bending.This is because only the bending moment which is one of the three physically nature boundary conditions along the free edge can be satisfied strictly in Kirchhoff plate theory,while the concept of equivalent shear force is employed to treat the transverse shear force and twisting moment.The Kirchhoff boundary condition for transverse shear resultant yields the angular distribution of the leading singular stress term near the crack tip depending on the Poisson's ratio,which does not conform to the general three-dimensional solutions.In order to overcome the mentioned shortcomings,Reissner's plate theory was employed to analyze the bending fracture problems of cracked plates.Unlike Kirchhoffs theory,Reissner's theory takes the transverse shear deformation into account.This can accommodate all the three physically natural boundary conditions along the crack surfaces strictly.Hence,Reissner's theory is more suitable to be applied for the analysis of the plate bending problem in fracture mechanics.On the basis of Reissner's plate theory,the bending problems of homogeneous plate with V-shape notches,bi-material plate containing an inclined crack terminating at the interface and bi-material plate with interface cracks were analyzed in both theoretical analysis and finite element numerical calculation.The contents of this doctoral dissertation are as follows:The generalized displacement fields near the tip of V-notch in a Reissner plate were restudied adopting William's eigenfunction series expansion method.The explicit expressions of the second and third expansion terms in the eigensolutions of the displacement fields were presented further in order to improve precision.Furthermore,an important eigensolution that was missed in the previous literatures was found.Meanwhile it was found that the eigensolution becomes infinite for some special vertex angles of the notch,which is a paradox.The cases of the paradox were studied,and the corresponding bounded solutions were explained by the Jordan form solution and specified in explicit form.In addition,the first four order terms of Jordan form eigensolution corresponding to the eigenvalue which is a double root were also derived and specified in an explicit form.A novel singular finite element was constructed by using the complete bounded expressions of the generalized displacement fields near the V-notch tip.Combining with conventional finite elements,the novel element was applied to solve the Reissner plate bending problem with V-shaped notches.Numerical results showed that the singular element has some excellent properties:(1)the singular element has high precision,because its displacement model performs higher order asymptotic expressions of displacement fields near the V-notch tip,which can well depict the characteristic of singular stress fields;(2)the singular element can be directly connected with the conventional finite elements and easily integrated into existing finite element program since it is a displacement mode element;(3)in post-processing,all the fracture parameters,for example stress intensity factors,can be calculated directly and analytically because the displacement and stress fields in the singular element have explicit expressions;(4)the element size can be taken in a large reasonable range,which can ensure that the singular element has excellent numerical stability.The corresponding eigenequation was derived by using the eigenfunction expansion method for an arbitrarily inclined crack terminating at the interface of two bonded dissimilar material plates.The dominant eigenvalues that determine the singularity orders of the in-plane stresses and transverse shear stress at the crack tip were evaluated numerically from the eigenequation.The influences of the bi-material parameters and the crack inclination angle on the in-plane stresses and transverse shear stress singularity orders were discussed.Specifically,it was proved that the transverse shear stress singularity order is a completely monotonic function of the bi-material parameter and the inclination angle.For interface crack problem,the explicit expressions of the first two order asymptotic displacement fields around the tip of an interface crack were derived further using William's eigenfunction series expansion method.A novel singular finite element was constructed by taking advantage of the explicit expressions of the asymptotic displacement fields around the tip of an interface crack.The singular element is a displacement mode element,which can well depict the characteristic of displacements and singular internal force fields around the tip of an interface crack.Combining with conventional finite elements,the novel element was applied to solve the bi-material plate bending problem with interfece cracks.Numerical results showed that-the singular element has good calculating precision.Furthermore,the singular element,like the conventional ones,can directly participate in the accumulation of the total stiffness matrix during the analysis.This feature can ensure an excellent compatibility.In post-processing,all the fracture parameters and stress distribution near the tip of the interface crack can be calculated directly and analytically.Those calculations indicate the present method is convenient.The achievements in this doctoral dissertation not only enrich the theory system of fracture analysis of plate bending,but also improve the activity of finite element method for solving plate bending problems in fracture mechanics.