Finite Element Discretized Symplectic Method For Fracture Analysis Of Interface V-notched Structures

With the continuous improvement of China's comprehensive national strength,China's manufacturing industry is gradually moving to the forefront of the world.In "13th Five-Year Plan",it's clearly pointed out that China will implenent high-end equipment innovation and development projects at this stage.In the high-end equipment manufacturing process,many complex structures and materials will inevitably be involved,such as complex plate structures in marine engineering equipment and electromagnetic composite materials in intelligent manufacturing equipment.Due to the discontinuity of the material and structure at the interface,cracks are generated at the interface during the manufacture or use of the device.and gradually evolve into a V-shaped slit.Therefore.it is of great practical significance to study the materials and structures containing V-shaped notches and improve the fracture resistance of equipment.In addition,unlike the traditional interfacial crack problem(the angle of the defect is fixed to zero degrees).the stress singularity and stress field distribution at the tip of the notch are highly correlated with the geometry of the incision.The existing literature has not yet proposed an effective analysis method for the fracture problem of a finite-size interface V-shaped notched structure.Therefore,it is of great theoretical significance to propose a theoretical method suitable for V-notched structural fracture analysis and to develop relevant theories.In summary,this paper proposes a finite element discretized symplectic method for the analysis and evaluation of the fracture behavior of multi-material bending plates with interface V-shaped incisions and bimaterial piezoelectric and electromagnetic materials subjected to anti-plane loading.The meta-method,which can accurately calculate the fracture parameters characterizing the singularity of the stress field at the tip of the V-notch,and directly obtain the analytical expression of the physics near the tip of the notch.The main research work of this paper is as follows:(1)The Hamilton system of multi-material plates under bending is derived.And the finite element discretized symplectic method is proposed for the fracture analysis of plate with interface V-shaped notch.The fracture parameters of the notch and the analytic expression of the singular physics near the tip are obtained directly.Starting from the basic equations of the bending plate problem,and derives the dual control equation of the original problem in the Hamiltonian system by introducing the dual variable and Hamiltonian variational principle.Therefore,the problem is transformed into the eigenvalue and eigensolution problem under the symplectic space,and the general solution form of the basic unknown expressed in the form of the series expansion is directly obtained by the separation variable method.Secondly,according to the interface connection conditions and coordinate transformation relationship of adjacent material regions,the relationship between the undetermined coefficients in the unknown solution of each material region in the global coordinate system is established.the eigenvalues and eigensolutions are obtained by combining the free boundary conditions of the notch surfaces.Furthermore.an analytical solution of the physical field of the multi-material plate bending problem is obtained.Again,the overall structure is discrete using the Kirchhoff theoretical unit,and the multi-material plate structure containing the V-shaped slit is divided into two types of regions.a near field containing the tip of the slit and a far field away from the tip of the slit.In the near field,the analytical expression of the notch tip is obtained as a global interpolation function,and a large number of node unknowns in the near field are converted into a small number of prine eigensolution coefficients.At the same time,keep the unknowns of-the nodes in the far field unchanged.Thus,a symplectic discrete finite element 1ethod formula suitable for fracture analysis of multi-material plates containing V-shaped slits is obtained.Finally,the fracture parameters of the V-shaped incisions in this finite-size structure and the analytical expressions of the singular physics near the tip of the notch can be directly obtained by combining the specific external boundary conditions.The results show that the number of eigensolutions which contain stress singularity varies with number and property ratio of materials and the geometry of structural.The first two eigenvalues with smaller modes perform as two cases:two different real eigenvalues and a pair of conjugate complex eigenvalues.Besides,the interfaces near the axis of symmetry in the plate are prone to tensile-type fracture.and the interfaces away from the symmetry axis are prone to slip-type fracture.(2)A Hamiltonian solution system for the fracture problem of piezoelectric/magnetoelectroelastic structures with interface V-shaped incisions under anti?plane loading is established,and the finite elelent discretized symplectic method for the calculation of fracture parameters of such materials is obtained.Unlike traditional elastic materials,piezoelectric/magnetoelectroelastic materials are multi-field coupling materials,and it is not possible to use the elastic variables to obtain basic variables to establish a Hamiltonian system.In order to solve this problem.the Legendre transform is used to obtain the basic unknowns in the piezoelectric/magnetoelectroelastic material fracture problem.firstly.The z?direction displacement and generalized shear force,potential and generalized electric displacement?magnetic potential and generalized magnetic induction are the dual variable.The corresponding Hamiltonian function is derived by using the basic variables and the Lagrangian function of the problem,and then the governing equation in the Hamiltonian system can be obtained.Secondly,according to the interface connection conditions between materials and the free boundary conditions of the notch surface,the eigenvalues and eigensolutions of the piezoelectric/magnetoelectroelastic structure with interface V-shaped notches are derived.the analytical solution can be obtained.Thirdly,the finite element formula of the anti-plane fracture analysis of piezoelectric/magnetoelectroelastic materials is derived.The goble structure is meshed and divided into near field and far field.In the near field.the nodes' coordinates are substituted into the analytical solutions of displacement,and the relationship between the nodes'displacenents in near-field and the coefficient of synplectic series is established.Then the formula of finite element discretized symplectic?nethod can be obtained.Finally.the fracture parameters about elastic field.electric field and magnetic field,and the singular stress field and electromagnetic field near the tip of the notch are obtained by combining the outer boundary conditions of the structure.Numerical examples show that the stress singularity of the tip of the notch is independent of the properties of the material,and is only inversely proportional to the angle of the notch.The intensity factors increase with the increase of length and angle of the notch.For piezoelectric structures,the higher the structural symmetry.the greater the stress intensity factor and energy release rate,and the smaller the electrical displacement intensity factor.For magnetoelectroelastic structures.the higher symmetry of structure,the larger the stress intensity factor,the electric displacement intensity factor and the energy release rate,and the smaller the magnetic induction intensity factor.