Resonance methods for determining the complex shear moduli of orthotropic composites

A survey of various methods for determining the complex elasticity and shear moduli from the resonant frequencies of flexural and torsional vibrations of rectangular rods cut out from a plate of an orthotropic composite is presented. The errors in the computed values of dynamic shear moduli caused by inaccuracies in the experimental determination of resonance frequencies are examined. A new variant of the resonance method is developed, which permits one to find three complex shear moduli of a composite from the resonant frequencies and the damping of torsional vibrations of three rods oriented along three symmetry axes of the material. For computing the moduli in the case of an overdetermined system, an algorithm of nonlinear optimization based on the least-squares method is recommended. From the results obtained it follows that, for determining the interlaminar shear moduli with a necessary accuracy, the rods must be sufficiently thick. It is shown that a good agreement alone between calculated and experimental frequencies of flexural and torsional vibrations of rods does not ensure a reliable determination of the moduli of interlaminar shear if experiments are carried out on wide test specimens cut out from a thin plate. Recommendations are given for the choice of geometrical sizes of test specimens for resonance experiments. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 721–744, November–December, 2007.     

位于两不同正交各向异性半平面间张开型界面裂纹的性能分析

利用Schmidt方法分析了位于正交各向异性材料中的张开型界面裂纹问题．经富立叶变换使问题的求解转换为求解两对对偶积分方程,其中对偶积分方程的变量为裂纹面张开位移．最终获得了应力强度因子的数值解．与以前有关界面裂纹问题的解相比,没遇到数学上难以处理的应力振荡奇异性,裂纹尖端应力场的奇异性与均匀材料中裂纹尖端应力场的奇异性相同．同时当上下半平面材料相同时,可以得到其精确解．     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

Micromechanical Analysis of Inelastic Deformation of Unidirectional Fibrous Composites under Multiaxial and Shear Loadings

An algorithm is proposed for numerically solving nonlinear 3D problems of micromechanics of a unidirectionally reinforced composite with a regular structure. For the matrix, equations of the deformation theory of plasticity and relations of reduced rigidity in its failure zones are used, whereas the fibers are elastic and indestructible. According to the method of local approximation, fields of microstresses and microstrains are determined in a structural fragment containing nine periodic cells. Boundary conditions of the fragment correspond to an arbitrary combination of longitudinal, transverse, and shear microstresses occurring in the central part of the fragment. The solution to the nonlinear 3D problem is sought by the method of superposition with an iterational refinement based on the successive solution of an antiplane problem and a problem on a generalized plain strain state of the structural segment. Special features of the iteration procedure are considered. The calculated deformation diagrams and ultimate strengths of a unidirectional glass-epoxy composite are presented for several loading trajectories.     

Poisson方程差分格式迭代解法中一些最优参数的有理拟合方法(英文)

本文针对二维Poisson方程五点和九点差分格式,导出了求解这些格式的SOR方法中最优松弛因子与区域剖分数的有理拟合公式,给出了Jacobi结合Chebyshev加速方法中Jacobi迭代矩阵谱半径的有理拟合公式.实际计算表明这些公式计算效果良好.     

裂纹尖端场弹塑性分析的加权残数法及塑性应力强度因子的计算

本文以幂强化材料,平面应变情形为例,系统地提出了裂纹尖端场弹塑性分析的加权残数法,并根据此法,得出了裂纹尖端场的解析式弹塑性近似解.在此基础上.对整个裂纹区域,构造了弹塑性解叠加非线性有限元计算塑性应力强度因子的方法,从而为裂纹尖端场和整个裂纹体的分析和计算,提供了一个方法.     

多复变函数论中的一组偏微分方程的解的拓展性与解的中量性质

本文主要要讨论多复变函数论中的一组偏微分方程，讨论此方和组的解的拓展性及解的中量性质。     

Direct and inverse problems for prestressed functionally graded plates in the framework of the Timoshenko model

In this study, modelling and identification of prestress state in functionally graded plate within the framework of the Timoshenko theory are discussed. With the help of variational principles, statements of boundary problems for stationary vibration of inhomogeneous prestressed plates have been derived taking into account various factors of prestress state. The comparative analysis of classical and nonclassical models has been conducted. The effect of the prestress state factors on the solution characteristics has been estimated. New approaches to solving the inverse problems on a reconstruction of inhomogeneous prestress functions in a functionally graded plate have been proposed on the basis of derivation of reciprocity relations and iterative regularization. The results of numerical reconstruction experiments are presented; practical recommendations on a selection of frequency range for the purpose of getting the highest reconstruction accuracy are given.     

改进联盟结构的支付分配规则及其排序的TOPSIS方法

本文针对改进的联盟结构,在基于局中人个人理性与联盟结构集体理性实现的基础上,确立了一种支付分配规则,该规则成为联盟结构改进后保持稳定的一个必备条件,在此基础上,给出了对不同联盟结构优超关系进行排序的TOPSIS方法,从而可以对不同联盟结构的优劣进行区分比较．     

A new design of the elimination and choice translating reality method for multi-criteria group decision-making in an intuitionistic fuzzy environment

The purpose of this paper is to design a new extension of the ELECTRE, known as the elimination and choice translating reality method, for multi-criteria group decision-making problems based on intuitionistic fuzzy sets. This method is widely utilized when a set of alternatives should be identified and evaluated with respect to a set of conflicting criteria by reflecting decision makers’ (DMs’) preferences. However, handling the exact data and numerical measure is difficult to be precisely focused because the DMs’ judgments are often vague in real-life decision problems and applications. A more realistic and practical approach can be to use linguistic variables expressed in intuitionistic fuzzy numbers instead of numerical data to model DMs’ judgments and to describe the inputs in the ELECTRE method. The proposed intuitionsitic fuzzy ELECTRE utilizes the truth-membership function and non-truth-membership function to indicate the degrees of satisfiability and non-satisfiability of each alternative with respect to each criterion and the relative importance of each criterion, respectively. Then, a new discordance intuitionistic index is introduced, which is extended from the concept of the fuzzy distance measure. Outranking relations are defined by pairwise comparisons and a decision graph is depicted to determine which alternative is preferable, incomparable or indifferent in the intuitionistic fuzzy environment. Finally, a comprehensive sensitivity analysis is employed to further study regarding the impact of threshold values on the final evaluation, and a comparative analysis is demonstrated with an application example in flexible manufacturing systems between the proposed ELECTRE method and the existing intuitionistic fuzzy technique for order preference by similarity to ideal solution (IF-TOPSIS) method.     

A short note on the paper “Convergence of the TAGE iterative method for the system arisen from the cubic spline approximation for the solution of two-point BVPs with forcing function in integral form”, by Mohanty, Jain and Dhall

In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036-3047] and then correct it.     

A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

In this article, we study an explicit scheme for the solution of sine‐Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo‐spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth‐order Runge–Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo‐spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix–vector multiplications. Therefore, we propose a multidomain pseudo‐spectral method for the numerical simulation of the sine‐Gordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long‐time numerical behavior for the sine‐Gordon equation in one and two dimensions. Copyright © 2013 John Wiley & Sons, Ltd.     

A cubic spline approximation and application of TAGE iterative method for the solution of two point boundary value problems with forcing function in integral form

In this article, we report an efficient high order numerical method based on cubic spline approximation and application of alternating group explicit method for the solution of two point non-linear boundary value problems, whose forcing functions are in integral form, on a non-uniform mesh. The proposed method is applicable when the internal grid points of solution interval are odd in number. The proposed cubic spline method is also applicable to integro-differential equations having singularities. Computational results are given to demonstrate the utility of the method.