Equations of anisotropic elastodynamics in 3D quasicrystals as a symmetric hyperbolic system: Deriving the time-dependent fundamental solutions

The fundamental solution (FS) of the time-dependent differential equations of anisotropic elasticity in 3D quasicrystals are studied in the paper. Equations of the time-dependent differential equations of anisotropic elasticity in 3D quasicrystals are written in the form of a symmetric hyperbolic system of the first order. Using the Fourier transform with respect to the space variables and matrix transformations we obtain explicit formulae for Fourier images of the FS columns; finally, the FS is computed by the inverse Fourier transform. As a computational example applying the suggested approach FS components are computed for icosahedral QCs.     

Computation of the fundamental solution of electrodynamics for anisotropic materials

A new method for computation of the fundamental solution of electrodynamics for general anisotropic nondispersive materials is suggested. It consists of several steps: equations for each column of the fundamental matrix are reduced to a symmetric hyperbolic system; using the Fourier transform with respect to space variables and matrix transformations, formulae for Fourier images of the fundamental matrix columns are obtained; finally, the fundamental solution is computed by the inverse Fourier transform. Applying the suggested approach, the fundamental solution components are computed in general anisotropic media. Computational examples confirm robustness of the suggested method.     

Three dimensional elastodynamics of 2D quasicrystals: The derivation of the time-dependent fundamental solution

The time-dependent differential equations of elasticity for 2D quasicrystals with general structure of anisotropy (dodecagonal, octagonal, decagonal, pentagonal, hexagonal, triclinic) are considered in the paper. These equations are written in the form of a vector partial differential equation of the second order with symmetric matrix coefficients. The fundamental solution (matrix) is defined for this vector partial differential equation. A new method of the numerical computation of values of the fundamental solution is suggested. This method consists of the following: the Fourier transform with respect to space variables is applied to vector equation for the fundamental solution. The obtained vector ordinary differential equation has matrix coefficients depending on Fourier parameters. Using the matrix computations a solution of the vector ordinary differential equation is numerically computed. Finally, applying the inverse Fourier transform numerically we find the values of the fundamental solution. Computational examples confirm the robustness of the suggested method for 2D quasicrystals with arbitrary type of anisotropy.     

各向异性材料动态裂纹扩展特性和动态应力强度因子

基于各向异性材料力学,研究了无限大各向异性材料中Ⅲ型裂纹的动态扩展问题．裂纹尖端的应力和位移被表示为解析函数的形式,解析函数可以表达为幂级数的形式,幂级数的系数由边界条件确定．确定了Ⅲ型裂纹的动态应力强度因子的表达式,得到了裂纹尖端的应力分量、应变分量和位移分量．裂纹扩展特性由裂纹扩展速度M和参数alpha反映,裂纹扩展越快,裂纹尖端的应力分量和位移分量越大;参数alpha对裂纹尖端的应力分量和位移分量有重要影响．     

Analysis of post buckling behavior of circular plates with non-concentric hole using the Rayleigh–Ritz method

This study is conducted to determine the post buckling behavior of circular homogenous plates with non-concentric hole subjected to uniform radial loading using Rayleigh–Ritz method. In order to implement the method, a computer program has been developed and several numerical examples for different boundary conditions are presented to illustrate the scope and efficacy of the procedure. The integration is carried out in natural coordinates through a proper transformation. Consequently, the displacement fields respect to natural coordinates are expressed using the Hierarchical, Hermitian and Fourier series shape functions for interpolating the out-of-plane displacement field and Fourier series and Hierarchical, Lagrange shape functions for interpolating the in-plane displacement field of plate. The Kirchhoff theory is used to formulate the problem in buckling condition. Due to the asymmetry in geometry, the in-plane solution is required to find the stress distribution. Finally, the problem is formulated in post buckling condition using Von-Karman non-linear theory, and a proper Hookean displacement field is presented to analyze the post buckling behavior.     

An analytic method for the initial value problem of the electric field system in vertical inhomogeneous anisotropic media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem is reduced to an operator integral equation. After that the operator integral equation is solved by the method of successive approximations. Finally, a solution of the original initial value problem is found by the inverse Fourier transform.     

Longitudinal Shear of Layered Anisotropic Media with Band-Type Inhomogeneities

Using the method of jump functions, we solve the antiplane problem of elasticity theory for a stack of anisotropic strips containing plane band-type inhomogeneities. We model these inclusions by jumps of the stress vector and the derivative of the displacement vector at the middle surfaces. Applying the Fourier integral transformation, we obtain the dependence of the components of the stress tensor and displacement vector on the external load and unknown jump functions. Taking into account the conditions of the interaction between a thin inclusion and an anisotropic medium, we reduce the problem to a system of singular integral equations for the jump functions. A specific example is considered as well.     

Representation of Solutions of the Deflection Equilibrium Equations for Thick Transversely Isotropic Plates

A system of equilibrium equations for the deflection of thick transversely isotropic plates with constant thicknesses is derived by expanding the stress and displacement components as Fourier series in Legendre polynomials of the thickness coordinate. A method for constructing a general solution of this system is described. It is made up of three types of solutions: biharmonic, potential, and rotational.     

Elastic stress analysis of blunt V-notches under mixed mode loading by considering higher order terms

This paper presents the in-plane asymptotic displacement and stress fields for blunt V-notched components based on Kolosov–Muskhelishvili's approach. In the first part, the displacement and stress components in the polar coordinate system are determined by choosing appropriate complex potential functions. In order to construct the notch geometry, the Neuber's mapping relation is utilized. Then, the notch boundary conditions are imposed to calculate the free parameters of the stress distribution. Eventually, the stress and displacement components are calculated in the Cartesian and polar coordinates in the forms of series expansion. In the second part, the coefficients of series expansions are computed by using the least square method (LSM). The blunt V-notched Brazilian disk (BV-BD) specimen under mixed mode loading is used as an example to verify the proposed procedure. The stress components in arbitrary distances and directions are determined for different blunt V-notches in order to evaluate the accuracy of the calculated stress series solutions and their associated coefficients. The numerical results indicate that a single-term solution can lead to considerable errors, and to achieve good accuracy in the stress field calculation, one should take account of at least three terms in the stress series solution.     

Asymptotic solutions of coupled dynamic problems of thermoelasticity for isotropic plates

The asymptotic method of solving boundary-value problems of the theory of elasticity for anisotropic strips and plates is used to solve coupled dynamic problems of thermoelasticity for plates, on the faces of which the values of the temperature function and the values of the components of the displacement vector or the conditions of the mixed problem of the theory of elasticity are specified. Recurrence formulae are derived for determining the components of the displacement vector, the stress tensor and for the temperature field variation function of the plate.     

Piecewise-linearized methods for single degree-of-freedom problems

A piecewise linearization method based on the Taylor series expansion of the nonlinearities and forcing with respect to time, displacement and velocity for the study of smooth single degree-of-freedom problems, is presented. The method provides piecewise analytical solutions which are smooth everywhere, is second-order accurate in time and yields explicit finite difference formulae for the displacement and velocity. The method is applied to nine single degree-of-freedom problems and its accuracy is assessed in terms of the displacement, velocity and energy as functions of the time step, and its results are compared with those of piecewise linearization methods that use Taylor series expansion of the forcing and nonlinearities with respect to time. It is shown that, for nonlinear problems with unknown free frequency and damping, the linearization method presented here is more accurate and robust than linearization techniques based on Taylor series expansions with respect to time. For linear problems with oscillatory forcing, linearization methods that employ fourth-order expansions in time are more accurate than the linearization method proposed here provided that the time step is sufficiently small.     

对称迭层矩形板的平面应力分析

常用的对称迭层板为各向异性板．根据平面应力问题的基本方程精确地用应力函数解法求得了各向异性板的一般解析解．推导出平面内应力和位移的一般公式,其中积分常数由边界条件来决定．一般解包括三角函数和双曲函数组成的解,它能满足4个边为任意边界条件的问题．还有代数多项式解,它能满足4个角的边界条件．因此一般解可用以求解任意边界条件下的平面应力问题．以4边承受均匀法向和切向载荷以及非均匀法向载荷的对称迭层方板为例,进行了计算和分析．     

极限分析的广义界限定理

本文研究了非连续流动场中,刚塑性介质极限分析完全解的界限问题.提出了一个包括界面条件及间断面条件在内的混合边值问题的广义变分原理,建立了极限载荷乘子的变分解析公式.并证明了一个新的界限定理,其中的场变量将不再受到屈服条件、不可压缩条件等约束的限制.此定理的推论给出了变分解与完全解之间的关系.初步应用表明,对于简单选取的场变量,由本文公式可以得到准确解的较佳界限值,结果具有较好的稳定性.     

Static analysis of rectangular nano-plate using three-dimensional theory of elasticity

Analytical solution for bending of a simply supported rectangular graphene sheets based on three dimensional theory of elasticity, is studied employing non-local continuum mechanics. By applying the Fourier series solution to the both displacement and stress field along the in-plane rectangular coordinates direction, and to the governing equation and constitutive relations, the three-dimensional governing equations in term of displacement components can be obtained. Closed form solution for the bending behavior of nano-plate is obtained by exerting the surface tractions on the state equations. To validate the accuracy and convergence of the present approach, numerical results are presented and compared with the results available in the open literature. Effect of non-local parameter, aspect ratio, thickness to length ratio and half wave numbers in the bending behavior of plate are examined. Furthermore, these results may also serve as benchmark to further results into the two-dimensional plate theories.     

Explicit solution of the frictional contact problem of anisotropic materials indented by a moving stamp with a triangular or parabolic profile

The frictional contact problem of anisotropic materials under a moving rigid stamp is solved exactly. Inside the contact region, the Coulomb friction law is applied. Both Galilean transformation and Fourier transform are employed to get the appropriate fundamental solutions, which can lead to real solutions of physical quantities no matter whether the eigenvalues are real or complex. The complicated mixed boundary value problem is converted to singular integral equations of the second kind, which are solved analytically in terms of elementary functions for either a triangular or a parabolic stamp. Explicit formulae of surface stresses are obtained. Numerical analyses are performed in detail to reveal the surface damage mechanism. It is also found that in the frictionally moving contact problem, the friction coefficient has a more important role than the moving velocity.     

Mixed boundary-value problems of thermoelasticity for anisotropic-in-plan inhomogeneous toroidal shells

Problems of thermoelasticity for an anisotropic-in-plan inhomogeneous thin toroidal shell are solved by asymptotic integration of the equations of the three-dimensional problem of the theory of an anisotropic inhomogeneous solid for various boundary conditions. Recurrence formulae are derived for the components of the asymmetric stress tensor and the displacement vector. An example is given.     

The Effect of Inhomogeneity of Elastic Properties on the Stress State in Composite Cylindrical Panels

The effect of inhomogeneity of elastic properties in the circumferential direction on the distribution of stress and displacement fields in orthotropic cylindrical panels is studied. The mechanical properties of the panels and the load acting on them are constant in the axial direction, which makes it possible to neglect the influence of the curvilinear ends. From the initial relations of a three-dimensional problem of the elasticity theory of inhomogeneous anisotropic bodies, a resolving system of partial differential equations is obtained, whose solution is presented in the form of truncated Fourier series, so that the conditions of free support of the rectilinear ends are satisfied. This allows us to separate the variables and to get a system of ordinary high-order differential equations, which is integrated by a stable numerical method. The problem on the stress-strain state of an orthotropic composite panel with a varying relative volume content of reinforcing elements in the circumferential direction is solved. The effect of the change in the reinforcement density on the stresses and displacements of the panel is studied.     

Analytical solution of piezolaminated rectangular plates with arbitrary clamped/simply-supported boundary conditions under thermo-electro-mechanical loadings

This paper extends an analytical method for static analysis of general cross-ply piezolaminated rectangular plates with any combination of clamped/simply-supported boundary conditions under uncoupled thermo-electro-mechanical loadings. This method is based on the novel superposition method and the first-order shear deformation theory (FSDT). The FSDT enables this expanded method to consider the effect of shear deformation of the plate. The process of applying electrical and thermal resultants causes some advantages due to its simplicity and less computational process. In this analysis displacement components are written in terms of unknown force and moment resultants. Using Fourier series for displacement components, mechanical, thermal, and/or electrical stress resultants, the complex governing differential equations of the plate are reduced to a set of linear algebraic equations with non-trivial solution. The obtained equations may be solved analytically to determine the unknown stress resultants. Several examples are proposed, and their obtained numerical results are compared with those available in the literature to verify the convergence, high accuracy, and the capability of the present method to analyze the static behavior of piezolaminated plates. It is found that there is high agreement between the present results with those obtained by other investigators.     

Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function

A new method is presented for Fourier decomposition of the Helmholtz Green function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Green function are split into their half advanced + half retarded and half advanced–half retarded components, and closed form solutions for these components are then obtained in terms of a Horn function and a Kampé de Fériet function respectively. Series solutions for the Fourier coefficients are given in terms of associated Legendre functions, Bessel and Hankel functions and a hypergeometric function. These series are derived either from the closed form 2-dimensional hypergeometric solutions or from an integral representation, or from both. A simple closed form far-field solution for the general Fourier coefficient is derived from the Hankel series. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented. Fourth order ordinary differential equations for the Fourier coefficients are also given and discussed briefly.     

An analytical solution for sectional estimation of stress and displacement fields around a shallow tunnel

In this paper, we present a solution for the sectional estimation of the stress and displacement fields of a liner and a geomaterial near a shallow buried tunnel. The complex variable method and the discrete Fourier transform are employed in the analytical derivation. The initial boundary conditions are the nonexistence of external loads acting on the liner interior and a Fourier curve obtained by fitting the displacement along the liner interior. Stress and displacement are continuous along the excavation line. In the verification, the stress and displacement fields within the excavation line obtained by the analytical solution agree well with the numerical results. Accuracy decreases as the spatial distance between the geomaterial and the excavation line increases, and the 5% error zones are defined. The accuracy of the solution would fall greatly out of these error zones. In another word, the analytical solution is not a general solution for the whole liner–geomaterial entity, but could serve as a quick sectional estimation, and is suitable for a variety of external loads on the ground surface.