The Symplectic Method On Fracture Analysis Of Structure With Crack And Weak Interface

The advanced composite materials play an important role in the national development because of the development of materials science and manufacturing processes. The functional materials which possess two or more characteristics of power-electricity-magnetic-heat-sound and light etc., such as polymer-based composite materials, piezoelectric materials, magnetic materials and photovoltaic etc. are widely used in aerospace, construction industry, electronic industry and medical equipment fields and any other fields. Wherein the laminate structure which bonded by many single plates with polymer is a typical structure, debonding becomes the main failure mode of such structure. Therefore, the study of viscoelastic fracture and electro-magnetic coupling problems is necessary.The current research methods of fracture problem are analytical method (complex potential method, integral transform method, weight function method, etc.) and numerical methods (finite element method, boundary element method, mesh free method, etc.). Analytical research methods are mainly attributed to the solving of higher order partial differential or integral equation. They reduce the number of variables by means of improving the order of control differential equations, but the solving of high-order differential equations is difficult, and this is the limitation of the Lagrangian system. In this case, the Hamiltonian system is employed in this paper, the symplectic method which reduce the order of differential equations by increasing the number of variables and high performance computers are used to solve the low order differential equations.In this paper, the composite interfacial fracture problem is seleced as the research object, the symplectic method is used to analyze the fracture problems of magnetoelectroelastic materials with weak interface, the main works are as follows:(1) An analytical method for the plane and anti-plane fracture problems of viscoelastic material is presented. Firstly, the viscoelastic fracture problem in the time domain should be transformed into frequency domain by Laplace transform. Secondly, the Hamiltonian canonical equations are established by the Lagrangian and Hamiltonian function, hence, the original problem could be taken as the problem about symplectic eigenvalues and eigensolutions. Lastly, the conjugate and orthogonal relationship between eigensolutions and expansion theorem are used to solve the complete solution, then, the analytical expression of the physical field and the singularity of the crack tip are obtained, as well as analytical stress intensity factors and J integral. Numerical results show the convergence and effectiveness of the method, and the unique phenomenon of stress relaxation and creep of the viscoelastic material is obvious, specially, the temperature load is sensitive to the viscoelastic fracture parameters.(2) Introduce the weak interface model into plane fracture problem of elastic material, and the symplectic method is employed to solve this issue. The bonded issue of composite materials is approximately seen as weak connection problem, and the spring model is adopted to describe the weak interface condition. Firstly, the symplectic solutions form of two elastic materials are derived. Secondly, the symplectic eigenvalues and eigensolutions in two materials are determined by the weak interface conditions, so the eigensolutions in whole region are composed by that of two regions, and they meet the conjugate orthogonal relationship in whole region. Lastly, make use of the expansion theorem, outer boundary conditions and conjugate orthogonal relationship, the singularity within the crack tip, stress distribution, generalized stress intensity factors can be obtained directly in analytical form. The results of the classic problem show well agreement with the existing results, and the method described herein is reasonable and has high precision.(3) Introduce the weak interface model into anti-plane fracture problem of magnetoelectro-electric materials, and the symplectic method is used to solve the multi-field coupling issue. Firstly, the dual variable (stress, electric displacement and magnetic induction intensity) of the original variable (displacement, electric potential and magnetic potential) are determined by means of the energy method and Legendre transform, then the issue is transformed into Hamiltonian system. Secondly, the spring model is adopted to describe the weak interface condition, together with different mechanical, electrical and magnetic crack surface conditions, the eigensolution vectors and eigenvalues in whole region are obtained, and the conjugate orthogonal relationship between eigensolutions exist also. Lastly, make use of the outer boundary conditions and the conjugate orthogonal relationship, the analytical expressions and fracture parameters of the problem solution are determined. Numerical results show that there are two categories of the generalized intensity factors because of the presence of a weak interface, and the weak interface parameter affects the magnitude of the intensity factor while the crack surface conditions determine that the generalized intensity factors appear or disappear.