A Method Of Hamiltonian System For Fracture Problems And Its Applications

With the development of science and technology, multifunctional materials and smart materials drew more and more attention of the designers. The magneto-electro-elastic material is one of these popular materials. Based on the properties of these materials, many intelligent structures and products are applied in the engineering structure. With the aid of magneto-electro-mechanical energy conversion, many instruments and facilities are designed and extensively used in petroleum industry, chemical Industry, aerospace industry, military affairs, manufacturing and nuclear industry. When subjected to coupled mechanical, magnetic, electric and thermal loads in service, these instruments and facilities are fail prematurely due to some defects arising during their manufacturing processes, i.e. fatigue crack. Therefore, it is of great importance to study the fracture behaviors of there composites and understand the cracking mechanism.Studying and understanding the cracking mechanism of these materials and structures is useful to optimize the structural design and manufacturing standards. Moreover, it is pertinent to minimize the catastrophic failures for enhanced performance in fracture and wear resistance. However, the nature of the problem requires a systematic examination of the interplays among the electro-magneto-mechanical-thermal effects and working condition. Although some theoretical and experimental studies have been performed, many related challenges have not been fully resolved and need to be refine and improved, particularly in the fracture behaviors of the magneto-electro-elastic materials. In view of these literatures, it can be seen that all of the methods used the governing equation derived previously in Lagrangian sense involving only one kind of variables in terms of the energy. Since highter-order differential equations are not conducive to numerical solution methods, such elimination will cause problems in numerical analysis. Fortunately, Academician Zhong Wanxie developed an analytical symplectic approach for some basic problems in elasticity and in applied mechanics. It is a new concept and method which provids a research platform based on Hamiltonian system. Under the leadership of Academician Zhong, his associates have extendeded the method to many areas and directions. Some research results have been published and provide a basic technique for the dissertation. In this dissertation, we study the singularities and intensity factors systematically for the edge-crack elastic, piezoelectric and magneto-electro-elastic materials. With the aid of symplectic expansion and symplectic adjoint orthogonality among the eigenfunctions, the analytic expressions for both of the intensity factors and dual variables are obtained. It overcomes the the defects of classical semi-inverse methods, and it is rational and systematic with a clearly defined, step-by-step derivation procedure. The present work provides a better understanding for fracture problem. The conclusions are listed below:1. Analytic stress intensity factors for two-and three-dimensional problemsA Hamiltonian system is introduced by the energy method. The displacements and stresses are proved to be conjugating (dual) to each other. The eigensolutions for fracture problems are solved from the Hamilton equations based on the mixed variables. These eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions. In the symplectic space, the solution consists of two parts:zero eigenvalue solutions and non-zero eigenvalue solution. All the Saint Venant solutions have been identified as the zero-eigenvalue solutions and the Saint Venant solutions represent the average physical. The non-zero-eigenvalue solutions corresponding to effects which are coverd by the Saint-Venant principle, i.e. the local boundary layer effects. The plane problems are regarded as a breakthrough, the symplectic method is extended to the space problems. Based on symplectic adjoint orthogonality among the eigensolutions, the analytical solutions are obtained and can be expanded in terms of the symplectic eigensolutions. The stress intensity factors and T-stresses are identified to be the coefficients of certain eigenfunctions. The coefficients of the series are determined from the boundary conditions and the relationship of symplectic adjoint orthogonality. Thus, Modeâ… ,â…¡,â…¢stress intensity factors are obtained simultaneously. It is a direct and effective method. In addition, a boundary integral technology is developed for the non-circular domains, semi-analytical or numerical results can be obtained. Moreover, the present work provides a way to solve the dynamic problems. These work has been published in the Engineering Fracture Mechanics (2009, 76(12):1866-1882), International Journal of Mechanical Sciences (2010,52(7):892-903) and Journal of Sound and Vibration (2011,330:1005-1017).2. Stress/electric intensity factors and singularities analysis for the edge-crack piezoelectric materialsThe Hamiltonian formalism is used to analyze singularities for the edge-crack piezoelectric materials. A space coordinate is defined as the longitudinal direction via an appropriate variable transformation to simulate the "time coordinate". With the aid of variational principle and the Lagrangian function which consists of elastic potential energy and piezoelectric energy, we generalized displacements (longitudinal direction displacement and electrical potential function) and stresses (shear stresses and electric displacements) as primary unknowns will result in a complete set of eigensolutions ensuring convergence and will give the generalized stresses directly as dual variables. Based on the properties of Hamilton system and modern mathematical tools, analysis of the edge-crack piezoelectric materials is preformed. The singularities and intensity factors for a permeable or impermeable crack in piezoelectric material are obtained. Furthermore, the influence factors are discussed. The results show that the electric field intensity factors for the electrically permeable crack is always of the zero value, or the electric field has no singularity at the crack tip. The strain intensity factors become independent of the material constants, it depends on the edge loading conditions only. The stress and the electric displacement intensity factors can be represented by a combination of material constants and the generalized displacement intensity factors. This work is published in International Journal of Solids and Structures (2009,46(20): 3577-3586).3. The study of coupling intensity factors for the edge-crack in magneto-electro-elastic mediaA Hamilton system is established for the edge-crack in magneto-electro-elastic media for studying the fracture behaviors. It can be reduced to the anti-plane problem. In symplectic space, it can be proved that the displacements and stresses, electric potentials and electric displacements, magnetic potentails and magnetic induction functions are conjugating (dual) variables respectively. It is convenient to solve the mixed boundary conditions with the aid of the mixed variables which consist of the original variables and dual variables. Using the exited eigensolutions, the solutions of stress, electric displacement and magnetic induction intensity factors are reduced to the solutions of a set of linear algebraic equations. Both of the permeable and impermeable electromagnetic boundary conditions at the crack surfaces are adopted and discussed. Some resultes and a closed form solution for-the anti-plane fracture problem of magneto-electro-elastic materials are obtained. The results show that the intensity factors can be obtained by the terms associated with the eigenvalue solutions having the coefficients of 1/2;The generalized stress variables at the crack tip show the traditional square root singularities and can be represented by a combination of material constants and the generalized displacement intensity factors; the field variables, which can permeate the crack surfaces, produce no singularities or their corresponding intensity factors always equal to zero. These work has been published in the Engineering Fracture Mechanics (77(16),2010, 3157-3173) and Computers & Structures (2011,89:631-645).4. Analytic stress intensity factors for the steady and transient thermoelasticityThe equations of thermal thermal conduction and thermoelasticity are first rewritten in Hamiltonian form where the variables are separable in spatial coordinates. Our study will be considered in two parts:At first, a generalized Hamilton system will be introduced to the heat conduction problem and the temperature function will be represented by a symplectic series analytically. Using the existed solution, the temperature function for both of the steady and transient thermal conducitons will be obtained. Then, a set of inhomogeneous Hamiltion equations and corresponding boundary conditions are obtained by the temperature function which obtained from the first part. For the symplectic approach, the radial coordinate is defined as the longitudinal direction via an appropriate variable transformation to simulate the "time coordinate", so that it raises a new way to solve the problems. With the assumption, the eigensolutions are solved which inclued the general solution and particular solution of the inhomogeneous Hamiltion equations. The following conclusions can be drawn from the analysis of the problem:The singular order at the crack-tip is -0.5; the value of stress intensity factor can be represented by the combination of the first coefficient of the non-zero eigensolutions and the series of temperature functions; the distribution of radial stresses shows that the high stresses are always occurred nearby the center of the crack and have exponentially decaying distributions. It should be pointed out that the value of thermal stress intensity factor decreases as the length of crack increases. It is an important feature for engineering design and evaluations of fatigue life. Based on these results, two papers have been published in the Journal of Thermal Stresses (2010,33(3):262-278; 2010,33(3): 279-301).