Zhenhuan Zhou

Hamiltonian Approach to Analytical Thermal Stress Intensity Factors—Part 2 Thermal Stress Intensity Factor

An analytic method to determine the thermal intensity factors of finite domains in two-dimensional heat conduction in the vicinity of a crack is introduced. The method first separates the variables of the heat conduction equations in polar coordinates using the Hamiltonian formalism and finds the symplectic eigenfunctions analytically. The coefficients of the symplectic series are determined from the boundary conditions. The thermal intensity factor is given by the first coefficient of the antisymmetric part of the series and no post-processing is required.

Rigorous buckling analysis of size-dependent functionally graded cylindrical nanoshells

This paper presents new analytical solutions for buckling of carbon nanotubes (CNTs) and functionally graded (FG) cylindrical nanoshells subjected to compressive and thermal loads. The model applies Eringens nonlocal differential constitutive relation to describe the size-dependence of nanoshells. Based on Reddys higher-order shear deformation theory, governing equations are established and solved by separating the variables. The analysis first re-examines the classical buckling of single-walled CNTs. Accurate solutions are established, and it is found that the buckling stress decreases drastically when the nonlocal parameter reaches a certain value. For CNTs with constant wall-thickness, the buckling stress eventually decreases with enhanced size effect. By comparing with CNTs molecular dynamic simulations, the obtained nonlocal parameters are much smaller than those proposed previously. Subsequently, FG cylindrical nanoshells are analyzed, and it is concluded that similar behavior that has been observ...

Finite-Element Discretized Symplectic Method for Steady-State Heat Conduction with Singularities in Composite Structures

A new-finite element discretized symplectic method for solving the steady-state heat conduction problem with singularities in composite structures is presented. The model with a singularity is divided into two regions, near and far fields, and meshed using conventional finite elements. In the near field, the temperature and heat flux densities are expanded in exact symplectic eigensolutions. After a matrix transformation, the unknowns in the near field are transformed to coefficients of the symplectic series, while those in the far field are as usual. The exact local solutions for temperature and heat flux densities are obtained simultaneously without any post-processing.

Hamiltonian analysis of a magnetoelectroelastic notch in a mode III singularity

The stress intensity factor (SIF) of a multi-material magnetoelectroelastic wedge in anti-plane deformation is analytically determined by the symplectic method. The Lagrangian equations in configuration variables alone are transformed to Hamiltonian equations in dual variables (configuration and momentum) which allow the use of the method of separation of variables. The solutions of the Hamiltonian equations can be expanded analytically in terms of the symplectic eigenfunctions with coefficients to be determined by the boundary conditions. For the wedge problem, the pairs of anti-plane displacements and shear stresses, electric fields and electric displacements, and magnetic fields and magnetic inductions are proved to be the dual (momentum) variables of the configuration variables. The singularity orders depend directly on the first few eigenvalues whose real parts are less than one but greater than zero. Numerical results for various conditions show the variations of the singularity orders. In particular, special behaviors of the order of the singularity for some special wedge angles are noted.

A New Analytical Approach for Free Vibration, Buckling and Forced Vibration of Rectangular Nanoplates Based on Nonlocal Elasticity Theory

In this paper, an analytical Hamiltonian-based model for the dynamic analysis of rectangular nanoplates is proposed using the Kirchhoff plate theory and Eringen’s nonlocal theory. In a symplectic s...

Evaluation of mode III interface cracks in magnetoelectroelastic bimaterials by symplectic expansion

The mode III interface crack between two dissimilar magnetoelectroelastic materials is studied by means of symplectic expansion. Four crack surface electromagnetic assumptions, that is, electrically and magnetically permeable, electrically permeable and magnetically impermeable, electrically impermeable and magnetically permeable, and electrically and magnetically impermeable, are taken into account. In the symplectic space, stress, electric displacement, and magnetic induction are found to be the dual variables of the displacement, electric potential, and magnetic potential. The Hamiltonian governing equations that allow the use of the method of separation of variables are formulated by employing the variational principle of mixed energy. Expressions for the mechanical, electric, and magnetic fields are derived explicitly in the series of symplectic eigensolutions with coefficients to be determined by the boundary conditions, and so are the fracture parameters, such as field intensity factors and energy release rates for dissimilar bimaterials. Numerical results are provided to verify the validity of the present method and reveal the influences of the material constants and boundary conditions.