Calculation Of Effective Elastic Moduli Research Articles

An anisotropic micromechanical model for calculation of effective elastic moduli of Ni-based single crystal superalloys

An anisotropic micromechanical model based on Mori-Tanaka method is developed to calculate the effective elastic moduli of Ni-based single crystal superalloys. In the micromechanical model, the γ precipitate with very high volume fraction is regarded as matrix, γ phase is divided into three parts as three different kinds of inclusions, and the actual cubic structure and orthogonal anisotropy properties of γ phase and γ ′ precipitate are taken into account. Based on this anisotropic micromechanical model, the effective elastic moduli of Ni-based single crystal superalloys composite materials is obtained, and the influences of volume fraction and elastic constants of γ ′ precipitate on the effective elastic moduli are also discussed. The results provide useful information for understanding mechanical behavior of composite materials in Ni-based single crystal superalloys and other anisotropic polygonal inclusion problem.

Determination of hoop direction effective elastic moduli of non-circular profile, fiber reinforced polymer composite sewer liner pipes from lateral ring compression tests

A new material property determination method is presented for the calculation of effective elastic moduli of non-circular ring specimens cut from filament wound oval profile polymer composite sewer liner pipes. The hoop direction elastic moduli was determined using the test results obtained from ring compression tests, which is a very basic setup, and requires no special equipment. Calculations were executed for many different oval profiles, and diagrams were constructed, from which the cross section dependent Ceff constants can be taken. The new method was validated by the comparison of tests and finite element analysis results. The calculation method and the diagrams are essential design tools for engineers, and a big step forward in sizing non-circular profile liner pipes.

Boundary element method for calculation of effective elastic moduli in 3D linear elasticity

AbstractA boundary integral formulation of the linear elasticity problem for a multi‐component composite is given. The fast BEM solver based on the adaptive cross approximation is then obtained by the data‐sparse representation of the resulting Galerkin matrices. The solver is used to obtain effective elastic moduli of fibre and particle reinforced composites in three dimensions by means of the strain energy equivalence principle. Copyright © 2009 John Wiley & Sons, Ltd.

Treatment of Elastic and Plastic Anisotropy of Polycrystalline Materials with Texture

One of Professor Bunge's great contributions to materials science is his continuous effort to promote the use of quantitative texture analysis, not in the least in a series of applications which are of high interest to materials engineering. In this, Professor Bunge has often drawn the attention to the great possibilities offered by the harmonic method. This is mainly due to two powerful formulas based on C-coefficients, one for the calculation of the average of a crystal orientation-dependent property, and one for re-expressing an ODF in a rotated reference frame. The application of these formulas often allows to greatly reduce the required calculation time. All this will be illustrated by means of several examples of the implementation of texture-based anisotropy in engineering applications: - the calculation of r-values of sheet material; - the calculation of yield loci of textured materials; - the calculation of effective elastic moduli of textured materials; - the determination of residual stresses in textured materials on the basis of diffraction data.

Numerical calculation of effective elastic moduli for incompressible porous material

Incompressible elastic material with a periodic system of pores is considered. Processes are studied with a typical length which is much more than the typical diameter of pores and the typical distance between pores. Porous material behaves as a certain “effective” material without pores in such processes. The method of calculation of effective moduli based on mathematical homogenization theory is described. The estimates for the effective moduli are proved. The results of numerical calculations of effective moduli for materials with spherical and cubic pores are presented. The dependence of the effective moduli on the volume fracture of pores is investigated. The explicit formulae for effective coefficients are deduced. Comparison with the effective moduli for compressible materials is performed.

Calculation of Effective Elastic Moduli of Textured Materials

Effective elastic constants of polycrystalline materials were determined with a recently developed method. This method bases on the modelation of the actual material by a cluster of 100 to 500 single crystals. In the present version of the scheme parallelepipeds are used. The ODF was calculated with the series expansion method. The transformation of this ODF into a finite sum of single orientations permits to assign any grain an individual orientation.Reliable results for the effective elastic moduli of textured materials are reported. They lie always within the bounds of Voigt and Reuss. The very high anisotropic substances, e.g. shape-memory-alloys, show a significant deviation from the Hill values.

Calculation of effective elastic moduli of polycrystalline materials including nontextured samples and fiber textures.

Calculation of effective elastic moduli of polycrystalline materials including nontextured samples and fiber textures.

First-principles approach for effective elastic-moduli calculation: Application to continuous fractal structure.

We present a new first-principles approach to the calculation of effective elastic moduli of composites based on the iterative solution of the Dyson equation for elastic wave scatterings in inhomogeneous media. Application of the method to the examination of elastic scaling behavior of a continuous fractal shows that the moduli associated with different elastic eigenmodes exhibit distinct exponents in scaling with the size of the system.

First-principle approach to the calculation of elastic moduli for arbitrary periodic composites

We present a first-principle approach to the calculation of effective elastic moduli for arbitrary periodic composites. The method is based on the iterative solution of the inhomogeneous elastic wave equation in wave-vector space. By using Fourier coefficients of the periodic system as structural inputs, the present approach offers the advantage of circumventing the need for explicit boundary-conditions matching across interfaces. As a result, it can handle complex unit-cell geometries just as easily as simple cell geometries. We illustrate the application of this method by calculating the effective moduli of (1) a three-dimensional porous frame composed of a simple cubic array of fused solid spheres, and (2) a periodic two-component composite.

Graded and Perfect Disorder in Random Media Elasticity

The basic problem considered concerns the calculation of effective elastic moduli of a random elastic medium such as a polycrystalline aggregate or a disordered composite, given some statistical information about its microstructure. A rigorous formal general solution is obtained as a Neumann-type series of multiple integrals which contain the statistical information in terms of n-point correlation functions of increasing order n. The integrals can be calculated provided the medium is highly disordered. To describe the disorder in a quantitative fashion, the concept of graded and perfect (=maximum) disorder is introduced. Rigorous bounds of nth order are derived for the effective elastic moduli of random media which are classified as disordered of grade n. The effective elastic moduli of the perfectly disordered medium concide in rigor with the well-known self-consistent moduli.

The First Pressure Derivative of the Shear Modulus of Porous Materials

Summary A general theory for the calculation of the second order effective elastic moduli of porous materials in which the porosity is in the form of isolated cavities is presented. The particular case of spherical cavities distributed randomly within an isotropic matrix in such a manner that the material is macroscopically isotropic is then considered in detail and an expression for the first pressure derivative of the effective shear modulus of such a material is obtained correct to first order in the porosity. 1. Introduction In a paper by Walton (1973), hereafter referred to as Paper I, the first pressure derivative of the effective bulk modulus of a porous material was calculated. The particular porous medium considered was that of a homogeneous isotropic matrix containing a dilute distribution of spherical cavities, not necessarily of the same size but such that the total porosity c (that is, the ratio of cavity volume to total volume) is so small that terms of order c2 may be neglected in comparison with unity. Furthermore, the distribution was assumed to be random and such that the material is macroscopically homogeneous and isotropic. The aim of the present paper is to extend the method used in Paper I to the calculation of the first pressure derivative of the effective shear modulus of such a material. 2. Second order effective moduli The method is based on considerations of the overall constitutive law and, in the spirit of Hill (1963), the problem of the calculation of the effective elastic moduli of porous materials may be formulated as follows. The model to be considered is that of a large volume V of some porous material subjected to a uniform strain in its outer boundary. The matrix material is assumed both perfectly elastic and homogeneous, although not necessarily isotropic. The porosity, on the other hand, is assumed to be in the form of isolated cavities distributed throughout the matrix in such a manner that the material is macroscopically homogeneous, although not necessarily isotropic. Finally, there is no restriction at this stage on the size of the porosity c. With the 9-vectors S and D denoting the nominal stress and displacement gradient respectively and with superscripts (m) and (c) referring respectively to the solid matrix and the cavities, the constitutive law for the matrix material may be written, correct to second order in D(m),

Calculation of effective elastic moduli of composite materials with multiphase interactions taken into consideration

The effective tensors of elastic moduli and pliability were calculated by renormalizing the equations of equilibrium and incompatibility. All multiphase interactions were taken into consideration, but the calculation was done approximately by using only the singular parts of derivatives of Green's functions, which is equivalent to the assumption of uniformity of the random components of stress and strain fields within grain boundaries. Solutions derived by two methods are reconciled, making it possible to determine the position of the elastic moduli of shear and of bulk K between the Hashin-Shtrikman bounds. Particular cases in which only the bulk modulus is not uniform, as well as those of composite materials with all of their constituents having the same ratio K=4μ/3 are considered.