On finite deformation of composites with periodic microstructure

By incorporating the periodic Green function, which is obtained from the instantaneous nominal moduli of the matrix, in the method of periodic structures (MPS), and utilizing the extremum principles of Hashin and Shtrikman (1962), the overall moduli of a rate-independent elastic-plastic matrix reiforced with periodic distribution of cylindrical inclusions are obtained. The MPS, itself, is re-examined by homogenizing the quasi-static rate equilibrium equations and utilizing a periodic stress-free velocity gradient. It is shown that by evaluating the instantaneous nominal moduli of the matrix based on the velocity gradient in the matrix adjacent to the inclusion, instead of the average velocity gradient in the matrix, a lower bound to the overall moduli can be obtained. The numerical example involves analysis of the growth of circular cylindrical cavities in an isotropically hardening matrix.     

Thermal expansion of composites

A theory is developed for the overall thermal expansion of a composite consisting of either spherical or long cylindrical inclusions of one material in a matrix of another. The strain field of a single inclusion consists of a uniform expansion and a short-range strain field. These two components are related by minimizing the elastic strain energy. To account for a dense array of inclusions, average properties of the mixture are used for the long-range field, but those of the matrix alone for the short-range field. The net dilatation is thus found for inclusions of mismatching volume; hence one finds a differential expression for the thermal expansion in terms of the volume fraction of inclusions, the individual thermal expansivities, the bulk moduli of inclusion and matrix, the shear modulus of the matrix, and, in the case of cylinders, the shear modulus of the inclusions. This expression is integrated over temperature; one accounts for plasticity by letting the shear modulus depend on the temperature and on the accumulated shear strain. A representative numerical example is given.Invited paper presented at the Ninth Symposium on Thermophysical Properties, June 24–27, 1985, Boulder, Colorado, U.S.A.     

Multiscale modeling of particle-modified polyethylene

A common practice in toughening of semicrystalline polymers is to blend them with second-phase rubber particles. A toughening mechanism has recently been suggested which considers a layer of transcrystallized material around well-dispersed particles. This layer has a reduced yield strength in certain preferentially oriented directions. A multiscale numerical model is used to investigate the effect of such a specific microstructural morphology on the mechanical behavior of voided systems. A polycrystalline model is used for high density polyethylene (HDPE) matrix material. The basic structural element in this model is a layered two-phase composite inclusion, comprising both a crystalline and an amorphous domain. The averaged fields of an aggregate of composite inclusions, having either a random or a preferential orientation, form the constitutive behavior of the polymeric matrix material. The anisotropy of material with preferential orientations is determined. The particle-dispersed system is described by finite element RVE models, with in each integration point an aggregate of composite inclusions. Transcrystallized orientations are found to have a limited effect on matrix shear yielding and alter the triaxial stress field. An hypothesized, flow-influenced, microstructure is shown to further improve material properties if loaded in the appropriate direction.     

Plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix

Based on the complex variable method, this paper addresses the plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is loaded by in-plane mechanical loads while the inclusions are applied by anti-plane electric loads at infinity. The general solutions are first derived for the complex potentials both in the matrix and inside the inclusions, and then numerical results are presented to show the effects of applied electric field, inclusion arrays and material properties on the electroelastic fields around the inclusions. It is shown that the inclusion arrays have a significant influence on the stress distribution at the interface between the matrix and piezoelectric inclusions.     

Anti-plane electro-elastic fields in an infinite matrix with N coated-piezoelectric inclusions

Electro-elastic fields in an infinite matrix with N coated-piezoelectric inclusions are studied based on the complex variable method. Based on the assumption that the coated inclusions are completely bounded the matrix which is subjected to remote anti-plane shears and in-plane electric fields, the general solutions for complex potentials are first derived for arbitrary arrays of inclusions. Then, the numerical results are obtained for special cases in which the inclusions are located in different ways. Discussions are made about the influence of the coating thickness, the array type of inclusions and the material properties on the interface stresses and electric displacements in the matrix. It is found that both the array types of inclusions and the Young’s modulus of the three-phase material system have remarkable effects on stresses, but not on electric displacements, while the piezoelectric constants of the coating and its thickness have significant effects on electric displacements, but not on stresses.     

Stress analysis of multi-layered hollow anisotropic composite cylindrical structures using the homogenization method

This paper presents a general and efficient stress analysis strategy for hollow composite cylindrical structures consisting of multiple layers of different anisotropic materials subjected to different loads. Cylindrical material anisotropy and various loading conditions are considered in the stress analysis. The general stress solutions for homogenized hollow anisotropic cylinders subjected to pressure, axial force, torsion, shear and bending are presented with explicit formulations under typical force and displacement boundary conditions. The stresses and strains in a layer of the composite cylindrical structures are obtained from the solutions of homogenized hollow cylinders with effective material properties and discontinuous layer material properties. Effective axial, torsional, bending and coupling stiffness coefficients taking into account material anisotropy are also determined from the strain solutions for the hollow composite cylindrical structures. Examples show that the material anisotropy may have significant effects on the effective stiffness coefficients in some cases. The stress analysis method is demonstrated with an example of stress analysis of a 22-layer composite riser, and the results are compared with numerical solutions. This method is efficient for stress analysis of thin-walled or moderately thick-walled hollow composite cylindrical structures with various multiple layers of different materials or arbitrary fiber angles because no explicit interfacial continuity parameters are required. It provides an efficient and easy-to-use analysis tool for assessing hollow composite cylindrical structures in engineering applications.     

Anisotropic electrical properties of a eutectic InSb + MnSb composite

The temperature dependences of the Hall coefficient, electrical conductivity, and thermoelectric power for a eutectic InSb + MnSb composite have been studied in the temperature range from 80 to 700 K. Electron-microscopic results confirm that the system is in a two-phase state and consists of an InSb matrix and needle-like MnSb metallic inclusions. The inclusions are surrounded by interfacial zones ~0.3 μm in width. The observed anisotropy in the transport properties of the material is attributed to a short-circuiting effect of the metallic inclusions. Interpretation in terms of effective medium theory with allowance for the interfacial zones suggests that they make a significant contribution to the electrical conductivity anisotropy.     

Effective conductive and dielectric properties of matrix composites with inclusions of arbitrary shapes

The effective field method is applied to the calculation of overall dielectric permittivities, and electro- or thermo-conductivities of composite materials consisting of a homogeneous matrix and a set of isolated inclusions. The problem is reduced to the solution of the one particle problem for a typical inclusion subjected to a constant external field. An original numerical method is proposed for the solution of the one particle problem for an inclusion of an arbitrary shape. As an example, the effective dielectric properties of composites with cylindrical inclusions of various sizes and properties are calculated.     

Interactions between N circular cylindrical inclusions in a piezoelectric matrix

Summary This paper studies the interactions between N randomly-distributed cylindrical inclusions in a piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is subjected to an anti-plane shear stress and an in-plane electric field at infinity. Based on the complex variable method, the complex potentials in the matrix and inside the inclusions are first obtained in form of power series, and then approximate solutions for electroelastic fields are derived. Numerical examples are presented to discuss the influences of the inclusion array, inclusion size and inclusion properties on couple fields in the matrix and inclusions. Solutions for the case of an infinite piezoelectric matrix with N circular holes or an infinite elastic matrix containing N circular piezoelectric fibers can also be obtained as special cases of the present work. It is shown that the electroelastic field distribution in a piezoelectric material with multiple inclusions is significantly different from that in the case of a single inclusion.     

The over-all instantaneous properties of metal-matrix composites

A micromechanical analysis is presented for the determination of the instantaneous effective properties of metal-matrix composites, which consist of elastic fibers reinforcing elastic-viscoplastic matrices. At any stage of loading, the current behavior of the initially isotropic matrix exhibits anisotropy that is load-level- and history-dependent. It is shown that the knowledge of the local instantaneous properties of the inelastic-matrix and elastic-fiber constants provides, in conjunction with the micromechanical analysis, the over-all instantaneous stiffnesses of the metal-matrix composite. The method is illustrated for the prediction of the instantaneous properties of unidirectional and laminated composite materials.     

The influence of nonmetallic inclusions on the mechanical properties of steel: A review

The literature regarding the influence of nonmetallic inclusions on the mechanical properties of steel is reviewed, with critical comments on various studies. A brief discussion of inclusion rating methods and a synopsis of the effects of applied stress on inclusions in an isotropic, elastic matrix are presented. The parameters considered are tensile strength, impact strength, reduction of area, fatigue properties and fracture toughness. It is concluded that in many applications, the type of inclusions are more important than the total content and as matrix strength increases, the notch effect of inclusions becomes more significant. Also, mechanical properties can be influenced by any one or a combination of the following inclusion parameters; shape, size, quantity, interspacing, distribution, orientation, interfacial strength, and physical properties relative to the matrix.     

Investigation of anisotropy problems in sheet metal forming using finite element method

This paper discusses the results of the numerical study of rectangular cup drawing of steel sheets using finite element methods. To be able to verify the results of the numerical solutions, an experimental study was done where the material behavior under deformation was analyzed. A 3D parametric finite element (FE) model was built using the commercial FE-package ABAQUS/Standard. ABAQUS allows analyzing physical models of real processes putting special emphasis on geometrical non-linearities caused by large deformations, material non-linearities and complex friction conditions. Friction properties of the deep drawing quality steel sheet were determined by using the pin-on-disc tribometer. The results show that the friction coefficient depends on the measured angle from the rolling direction and corresponds to the surface topography. A quadratic Hill anisotropic yield criterion was compared with von Mises yield criterion having isotropic hardening. The sensitivity of constitutive laws to the initial data characterizing material behavior is also presented. It is found out that plastic anisotropy of the matrix in ductile sheet metal has influence on deformation behavior of the material. When the material and friction anisotropy are taken into account in the finite element analysis, this approach gives better approximate numerical results for real processes.     

Influence of the inclusions distribution on the effective properties of heterogeneous media

In this paper we investigate the effective conductivity of composite materials by means of the homogenization method. We concentrate on composites with circular or elliptic cylindrical inclusions. In particular, we are interested in the effect of the distribution of the cylinders in the continuous material on the effective properties. We compare rectangular and hexagonal distributions with random distributions for different volume fractions of the inclusions. We also study the effect of the number of inclusions in each periodic cell for the random structure as well as shape influence of the elliptical inclusions.     

The interaction of mode I crack with multi-inclusions in a three-phase model

An approximately close form solution has been developed for mode I crack interacting with multi-inclusions in composite materials. The crack-tip stress intensity factor is evaluated in a three-phase model, which combines the present knowledge that the inclusions only in the immediate neighborhood of the crack-tip have strong effect on the stress intensity factor and that the far inclusions have an overall effects which can be estimated by effective properties of the composites. As validated by numerical examples, the solution has good accuracy for a wide range of the modulus ratios between the inclusion and matrix material.     

Mechanical and swelling properties of polystyrene-polyolefin blends

Differential scanning calorimetry and swelling characterization have been extended to low density polyethylene-polystyrene blends. The thermal data reveal high incompatibility between the two components in the blend. Swelling measurements in the state of quasi-equilibrium show a marked anisotropy of the cylindrical extruded specimens. Mechanical measurements in the tensile mode were carried out at room temperature on blends of the same atactic polystyrene with each of four polyolefins with increasing side-chains featured in a previous work. As the composition is varied from the pure polystyrene to the pure polyolefin the stress-strain curve changes gradually from one exhibiting brittle fracture to one showing increasingly ductile yield. At a critical concentration that ranges generally from 30 up to 50% polystyrene there is clearly an inversion of phases. Below such value the blend consists of a polyolefin matrix with polystyrene fibrils oriented in the direction of extrusion, as inferred by the swelling data. Beyond this critical composition the blends consist of a glassy polystyrene matrix with polyolefin inclusions. In the latter case the anisotropy is due to the cylindrical shape of the entire specimen.     

Modeling of effective material properties of pyrolytic carbon with different texture degrees by homogenization method

The elastic properties of pyrolytic carbon material as a function of texture degree were calculated by means of a homogenization method. The material microstructure is modeled as a system of graphite crystals (inclusions) embedded in an infinite homogeneous matrix with unknown effective (overall) parameters. The texture degrees of carbon planes extracted from the experimental selected-area electron diffraction patterns as well as size of coherent domains extracted from high resolution transmission electron microscopy images have been used as reference points for modeling of material properties. The experimental diffraction curves exhibiting a good fitting with the Gauss density function have been used to simulate the spatial orientation of inclusions. After that the overall elasticity tensor is calculated and the influence of the texture degree of pyrolytic carbon material on the engineering elastic parameters is studied.     

Integral Equation Method for Evaluation of Eddy-Current Impedance of a Rectangular,Near Surface Crack Inside a Cylindrical Hole

An integral equation method for solving the eddy-current nondestructive evaluation problem of a flat, rectangular, near surface crack inside of a cylindrical hole in a conducting material is presented. The method involves expanding the Green’s tensor, the incoming field, and the jump in electric potential over the crack in suitable basis functions. Here, plane waves, cylindrical waves, and basis functions related to the Chebyshev polynomials, are used. The way of discretization in this method leads to a formulation where the scattering is defined by a scattering matrix, independent of the incoming field. This presents an advantage, when conducting numerical simulations, since the scattering matrix does not have to be recalculated for every probe position. The numerical calculations are straightforward to perform and model predictions are compared with finite element results.     

Modeling carbon nanotube reinforced composite materials with the cylindrical method of cells

Micromechanics modeling, utilizing a cylindrical method of cells (CMOC) model, is employed to obtain the effective mechanical properties of an elastic transversely isotropic, isothermal material system consisting of a hollow carbon nanotube (CNT) embedded in an isotropic polymeric material matrix. It is shown that weak interfacial bonding between the CNT and polymeric matrix, which is characteristic of this type of material system, can be modeled with the CMOC. Numerical solutions of the effective independent material constants are obtained, based upon appropriate values of the properties of the carbon nanotube and epoxy matrix. The numerical results are presented graphically and compared with corresponding classical closed‐form solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Identification of effective properties of particle reinforced composite materials

For the determination of effective elastic properties an energy averaging procedure has been used for particle reinforced composite materials. This procedure is based on finite element calculations of the deformation energy of a characteristic volume element. The proposed approach allows the determination of effective properties of particle reinforced composite with acceptable precision. The calculated effective properties of the composite are found in range between upper and lower Hashin-Shtrikman bounds. The averaging elastic properties of the composite depend on the properties of the particles, matrix volume fraction of the particles and some parameters taking into account the influence of the interphase between matrix and particles. These dependencies can be presented by simple analytical functions approximatically. An identification procedure basing on numerical experiments allows the estimation of the unknown approximation parameters. The obtained functions describe precisely the numerical data for any relationship between material constituents.     

A time domain direct boundary integral method for a viscoelastic plane with circular holes and elastic inclusions

This paper considers the problem of an infinite, isotropic viscoelastic plane containing an arbitrary number of randomly distributed, non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size. All inclusions are assumed to be perfectly bonded to the material matrix but the elastic properties of the inclusions can be different from one another. The Kelvin model is employed to simulate the viscoelastic plane. The numerical approach combines a direct boundary integral method for a similar problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described in [Crouch SL, Mogilevskaya SG. On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries. Int J Numer Methods Eng 2003;58:537–578], and a time-marching strategy for viscoelastic material analysis described in [Mesquita AD, Coda HB, Boundary integral equation method for general viscoelastic analysis. Int J Solids Struct 2002;39:2643–2664]. Several numerical examples are given to verify the approach. For benchmark problems with one inclusion, results are compared with the analytical solution obtained using the correspondence principle and analytical Laplace transform inversion. For an example with two holes and two inclusions, results are compared with numerical solutions obtained by commercial finite element software—ANSYS. Benchmark results for a more complicated example with 25 inclusions are also given.