Optimal Design of a Circular Plate with Rectilinear Anisotropy

We investigate the problem of optimization of the basic frequency of free oscillations of a nonorthotropic ring plate with rectilinear anisotropy that is undergoing shear. We introduce a small physical parameter characterizing the anisotropy properties of the plate material. By assuming that the thickness of the plate varies within relatively small limits, we apply the small-parameter method to solve the optimization problem, which allows us to obtain an approximate analytic solution of the problem and an asymptotic estimate of the maximum value of the basic frequency.     

Comparative analysis of two algorithms for numerical solution of nonlinear static problems for multilayer anisotropic shells of revolution. 1. Account of transverse shear

The discussion focuses on two numerical algorithms for solving the nonlinear static problems of multilayer composite shells of revolution, namely the algorithm based on the discrete orthogonalization method and the algorithm based on the finite element method with a local linear approximation in the meridian direction. The material of each layer of the shell is assumed to be linearly elastic and anisotropic (nonorthotropic). A feature of this approach is that the displacements of the face surfaces of the shell are chosen as unknown functions, i.e., the functions which allows us to formulate the kinematic boundary conditions on these surfaces. As an example, a cross-ply cylindrical shell subjected to uniform axisymmetric tension is considered. It is shown that the algorithms elaborated correctly describe the local distribution of the stress tensor over the shell thickness without an expensive software based on the 3D anisotropic theory of elasticity.Tambov State Technical University, Tambov, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 347–358, May–June, 1999.     

Approximate torsion calculations for anisotropic members with a cross section in the form of an elongated profile

An approximate method is proposed for solving the problem of the free torsion of a prismatic anisotropic member with a narrow cross section. The material is assumed to be homogeneous with a special form of anisotropy (nonorthotropic). An arbitrary elongated section and special cases are considered. Experimental data are presented for DSP-B wood-resin laminate and the limits of applicability of the method are established.Mekhanika Polimerov, Vol. 3, No. 2, pp. 312–317, 1967     

Redistribution of finite elastic strains after the formation of inclusions. Approximate analytical solution

A class of two-dimensional static problems of the stress-strain state of non-linearly elastic bodies, in which domains with different elastic properties (inclusions) arise after preloading, is considered. Problems are formulated and solved using the theory of the repeated superposition of finite strains. The mechanical properties of the initial material and the material of the inclusions are described by Murnaghan-type or Mooney-type constitutive relations. Two ways of specifying the constitutive relations for the material of an inclusion are considered: when there are inherent strains in this material and when there are not. Approximate analytical methods are used for the solution.     

Longitudinal and Transverse Waves in Finite Elastic Strain. Hadamard and Green Materials

This work is concerned with the propagation of purely longitudinaland purely transverse waves in homogeneously deformed isotropicelastic materials. Two types of compressible material are alsodiscussed. A Hadamard material, so called by John in the hyperelasticcase, is one in which longitudinal waves may propagate in everydirection when the material is homogeneously deformed. A secondmaterial, called a "Green material" is introduced. In it twotransverse waves can propagate in every direction when the materialis homogeneously deformed. It is seen that a Mooney materialis the only isotropic incompressible elastic material in whichtwo transverse waves can propagate in every direction when itis homogeneously deformed, while the pressure stays constantthroughout the material. The propagation of finite amplitudewaves in these materials is discussed. Finally, it is shownthat the only motions which can be maintained in all homogeneouscompressible elastic Hadamard materials under the action ofsurface forces alone, are necessarily homogeneous and accelerationless.     

Mean and full field homogenization of artificial long fiber reinforced thermoset polymers

Based on two artificial microstructures representing a long fiber reinforced thermoset material, the effective linear elastic material properties are calculated by both a mean and a full field homogenization method. Concerning the mean field method, the effective elastic material properties are approximated using the homogenization scheme by Mori and Tanaka, formulated explicitly in terms of orientation averages. This allows to use orienation tensors of 2nd and 4th order describing the orientation information on the micro level. The full field method is based on the fast Fourier transformation (FFT), for which the effective material properties are determined by volume averaging. The comparison between both methods show good agreements, the deviations are in the range between 2% and 12%. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Description of the elastic deformation and degradation of elastic properties of dispersedly failing isotropic materials

A model is put forward for describing the elastic deformation of a quasi-homogeneous isotropic material capable of accumulating scattered microdamages during loading, which eventually leads to its total failure. The degree of damage of the material at a point is characterized by a centrosymmetric scalar function on the unit sphere, named the damage function, whose values depend on a dimensionless equivalent stress. This function is approximated by a fourth-rank tensor, which is used to construct a constitutive relation between stresses and strains in a differential form. By way of example, the elastic deformation of concrete and the degradation of its linearly elastic properties are described, and the basic three-dimensional sections of the corresponding strength surface are constructed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 2, pp. 193–208, March–April, 2006.     

用张量导数法求横观各向同性弹性材料的弹性张量的一般形式

用对张量函数求导的方法导出了横观各向同性材料和各向同性材料的弹性张量的一般形式与应力-应变关系式.从推导过程可更清楚地看出为什么横观各向同性材料和各向同性材料分别有五个和两个独立的弹性常数,即材料有几个独立的弹性常数是由其应变能函数的形式所决定的.     

Displacement potential solution of stiffened composite struts subjected to eccentric loading

The elastic behavior of stiffened struts of orthotropic composite material is investigated under an eccentric loading using displacement potential formulation. Solutions are obtained for different aspect ratios of the strut where the fibers are assumed to be directed along the axis of the struts. Solutions of stresses and displacements of boron/epoxy composite struts are presented numerically in the form of graphs. The effects of material orthotropy and different stiffeners on the elastic field are also investigated. Finally, some of the results of the present solution are compared with the corresponding finite element predictions, which, in turn, establishes the soundness as well as reliability of the present analysis of the composite struts.     

Modelling of wave phenomena in the Zahorski material based on modified library for ADINA software

This paper presents numerical modelling of wave phenomena in simple elastic structures such as rods and shields made of hyperelastic Zahorski material. The main difference between the Zahorski material, which is an elastic material in the Green sense, and the commonly used Mooney–Rivlin material lies in the non-linear term including the constant C₃. Consequently, qualitative and quantitative differences are observed compared to the Mooney–Rivlin material, for example in the values of effective stresses. The extension to the ADINA software developed by the author, which helps create 2D and 3D libraries, significantly facilitates modelling of the Zahorski material. The modification can be used for comparison of wave phenomena that are observed during the propagation of disturbances in the Mooney–Rivlin and Zahorski materials. It should be emphasised that the Zahorski material behaves much better at high strains during the analysis of incompressible rubber and rubber-like hyperelastic materials and can be used in various fields of science wherever the model of Mooney–Rivlin material is successfully applied. The results of numerical computations for both Mooney–Rivlin and Zahorski materials were presented in a graphical form and compared in order to illustrate the differences.     

Dispersion of elastic waves in a micropolar metamaterial plate with periodical arranged resonators

The wave propagation in a micropolar elastic metamaterial is investigated in this paper. The elastic metamaterial is composed of the micropolar elastic host material and the periodically arranged local resonators. Compared with the classical elastic metamaterial, the micropolar elastic metamaterial has more material parameters that can be elaborately designed to manipulate the elastic wave propagation. By introducing additional displacement fields, a multi-displacement continuum model of the micropolar elastic metamaterial is presented to characterize the resonance behavior of the resonators and the microstructure effects of the unit cell. According to this continuum model, two independent wave systems exist: one is a longitudinal system and the other is a shear and rotation coupled transversal system. The dispersive curves and band gaps of the longitudinal and transversal systems are numerically discussed and the influences of the resonators are mainly considered.     

An indifferent constitutive law in finite elasticity

The concepts of material frame-indifference and material symmetry group with respect to isotropic scalar functions, as represented by energy functions, are discussed. An energy function for a structured heterogeneous (transversal isotropic) medium in large elastic deformations, which is known to satisfy the Ponyting’s effect [1], is highlighted. It is shown that the constitutive relation due to this energy function is material frame-indifferent.     

Effective moduli of elasticity of a composite composed of anisotropic layers

Relations are obtained for the effective moduli of elasticity and Poisson's ratios of a laminated fiber-reinforced composite, each layer of which has at least orthorhombic symmetry. The elastic properties of the composite in terms of the elastic constants of the layer are expressed exactly, and the elastic constants of the individual layer in terms of the values for the fiber and the matrix are expressed approximately. Two approximations are considered: one corresponds to the Hashin-Shtrikman variational approach, while in the second the comparison material is assigned elastic properties equal to the Voigt or Reuss means of the values for each layer. A numerical example is worked for the combination boron fibers-epoxy resin. The results of the calculation are compared with the exact solution of the problem for a composite composed of alternating layers of boron and epoxy resin.     

The Motion of a Heavy Cylinder over the Surface of an Anisotropic Elastic Solid

The response of an anisotropic elastic half-space to a heavycylinder moving steadily over its surface is examined. The analysisholds for the most general anisotropy in which no symmetry elementsof the material are assumed.     

Stress intensity factors for three cracks at the interfaces of a graded layer bonding two different materials

We consider two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer in which there is one crack at the lower interface between the elastic layer and the lower half-plane and two cracks at the upper interface between the elastic layer and the upper half-plane. The stress intensity factors for these three cracks are solved for when tension is applied perpendicular to the interface cracks. The material properties of the bonding layer vary continuously between those of the lower half-plane and those of the upper half-plane. The differences in the crack surface displacements are expanded in a series of functions that are zero outside the cracks. The unknown coefficients in the series are solved by the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors are calculated numerically for selected crack configurations.     

Aerated autoclaved concrete: Stochastic structure model and elastic properties

Aerated autoclaved concrete (AAC) is a modern and important construction material, whose elastic properties are primarily defined by its porosity. The possibility to predict elastic properties of AAC based on the voids distribution is very important. The report describes simulations of the mechanical properties of AAC, based on a stochastic-geometric model of its structure. The model is the well-known “cherry-pit” model, which presents a random system of partially overlapping spheres. In the mechanical analysis the solid phase is approximated by a network model with the help of the so-called radical tessellation with respect to the hard spheres of the “cherry-pit” model. The network edges are modelled in ANSYS as 3D beams. In this approach, the discretized elements (the edges) have in distinction to FE calculations with small polyhedral same dimension as the air voids and so the numerical costs can be drastically reduced. The FE simulations calculate the elastic constants and energy concentrations, which are responsible for the material failures, in large samples. Comparisons with fracture tests showed good matching between simulations and experiments. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非均匀法向荷载下半空间的二阶弹性效应问题

本文提供各向同性弹性半空间，在非均匀分布法向荷载下，二阶弹性效应的一个封闭形式解，运用积分变换方法，讨论了按Ｈｅｒｔｚ规律分布的荷载情形；导出了不可压缩各向同性弹性材料的极限解；算出了上述二阶弹性材料问题在ｚ方向的位移和法向应力数值。我们发现，与线弹性情形相比较，在二阶弹性材料中相应位移增大而法向应力减小。     

Numerical Estimation of the Elastic Properties of Thin-Walled Structures Manufactured from Short-Fiber-Reinforced Thermoplastics

A model which allows us to estimate the elastic properties of thin-walled structures manufactured by injection molding is presented. The starting step is the numerical prediction of the microstructure of a short-fiber-reinforced composite developed during the filling stage of the manufacturing process. For this purpose, the Moldflow Plastic Insight^® commercial program is used. As a result of simulating the filling process, a second-rank orientation tensor characterizing the microstructure of the material is obtained. The elastic properties of the prepared material locally depend on the orientational distribution of fibers. The constitutive equation is formulated by means of orientational averaging for a given orientation tensor. The tensor of elastic material properties is computed and translated into the format for a stress-strain analysis based on the ANSYSÒ finite-element code. The numerical procedure and the convergence of results are discussed for a thin strip, a rectangular plate, and a shell of revolution. The influence of manufacturing conditions on the stress-strain state of statically loaded thin-walled elements is illustrated.     

Strength and reliability of cylindrical shells prepared by the method of continuous filament winding

Conclusions 1. Strength calculation of shells whose material has a scatter in elastic and strength characteristics should be carried out using statistical methods. A deterministic calculation using values of elastic and strength characteristics leads to appreciable errors in the estimate of bearing ability.2. The problems of estimating the strength and the reliability of constructions from laminar composites are closely interconnected.3. Physicomechanical characteristics of a material which have been obtained in tests of unidirectional specimens make it possible to describe the behavior of a construction on the condition that the structure of the shell layer material is reduced to the structure of the specimen material.4. Allowing for the scale effect plays an important role in estimating the reliability and bearing ability of heavy-duty shells for large-size pressure vessels.Translated from Mekhanika Polimerov, No. 3, pp. 443–451, May–June, 1978.