Two-dimensional Green's function of orthotropic three-phase material under a normal line force with application in the design of composite

Green's function of orthotropic three-phase material is an important and basic problem in the study of mechanics of materials. It is also the foundation of further theoretical researches and engineering applications. Most of adhesive structures in engineering can be well simulated by the mechanical model of orthotropic three-phase material, such as composite laminate, integrated circuit (IC) packaging, micro-electro-mechanical systems (MEMS) and biomedical materials, etc. In order to understand the mechanical properties of the adhesive structure, a two-dimensional Green's function of orthotropic three-phase material loaded with a normal line force is presented. Based on the Green's function proposed in this paper, the stress field of adhesive structure under arbitrary normal loadings can be obtained with superposition method. Besides, this Green's function is convenient to be used in further studies, because it is expressed explicitly in form of elementary functions. Numerical examples are proposed to study the mechanical properties of the adhesive structure in five difference aspects: (1) the distribution rule of stress fields of the adhesive structure; (2) the influence from fiber orientation of composite to the stress fields of the adhesive structure; (3) the influence from elastic modulus of adhesive layer to the stress transfer of the adhesive structure; (4) the influence from the thickness of adhesive layer to the stress transfer of the adhesive structure; (5) the reasonability of spring interface model.     

Structural elastic and creep models of a UD composite in longitudinal shear

The transient creep of a UD composite with a quadratic arrangement of elastic fibers of quadratic cross section is investigated. The deformational properties of the composite are determined from the known properties of its constituents. A structural model of the UD composite is developed, whose minimal elementary cell contains four elements. The stress-strain state of the elements is assumed homogeneous. Two types of basic and resolving governing equations of transient creep are deduced, which are based on static or kinematic assumptions. In each of the cases, a formula for the longitudinal elastic shear modulus of the composite is found. The stationary solutions of creep equations allow one to obtain formulas of the steady-state creep of the composite in a form similar to Norton’s law. Numerical calculations are also performed, and a comparison of the results with data given in the literature bears witness to the efficiency of the models developed and the solutions obtained. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 437–448, July–August, 2007.     

Imperfect interface effect for nano-composites accounting for fiber section shape under antiplane shear

This paper investigates the elastic responses of fibrous nano-composites with imperfectly bonded interface under longitudinal shear. The proposed imperfect interface model is the shear lag (or the spring layer) model; the presented nano interfacial stress model is the Gurtin–Murdoch surface/interface model; and the three-phase confocal elliptical cylinder model is the geometry model accounting for the fiber section shape. By virtue of the complex variable method, a generalized self-consistent method is employed to derive the closed from solution of the effective antiplane shear modulus of the fibrous nano-composites with imperfect interface. Five existing solutions can be regarded as the limit form the present analytic expression. The influences of the interface elastic constant, the interfacial imperfection parameter, the size of the elliptic section fiber, the fiber section aspect ratio, the fiber volume fraction and the fiber elastic property on the effective antiplane shear modulus of the nano-composites are discussed. Particularly, numerical results demonstrate that the interfacial elastic imperfection will always cause a significant reduction in the effective antiplane shear modulus; and the fiber interface stress effect on the effective modulus of the fibrous nano-composites will weaken with the interfacial imperfection increases.     

Modelling the mechanical behaviour of adhesively bonded and sintered hollow-sphere structures

The mechanical properties of periodic hollow-sphere structures are investigated numerically. Young’s modulus and the Poisson ratio are determined in order to describe their linearly elastic behaviour. The initial compressive yield strength is also calculated. The spheres are located at the nodes of a cubic primitive lattice. The cohesion is achieved by an adhesive concentrated in the minimum gap between neighbouring spheres. The geometry of the structure is discretized based on regular hexahedral elements. This approach is much more time-consuming, but it is important in order to achieve a more accurate simulation of the nonlinear behaviour (e.g., plasticity) of such materials. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 803–816, November–December, 2006.     

Torsional vibrations and dispersion of torsional waves in orthotropic composite rods

The Barr’s refined theory of torsional vibrations of isotropic rods of noncircular cross section is generalized for an orthotropic material. An analysis of natural frequencies of torsional vibration of free-free orthotropic prismatic rods of rectangular cross section is carried out with the help of an exact solution of the frequency equation. For orthotropic CFRP and GFRP rods, the improved theory, which takes into account the normal stresses and inertia forces in the axial direction, in some cases, predicts a noticeable raise in the natural frequencies compared with those following from the Saint-Venant classical theory. A good agreement is obtained between the experimental and calculated values of natural frequencies of torsional vibrations of rectangular quartz and fiber glass rods. The dispersion of torsional waves in an orthotropic quasi-homogeneous rod is considered. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 165–182, March–April, 2008.     

Elastic properties of two-dimensional two-phase composites with isotropic phases

Formal series of powers of Fourier coefficients for the effective elastic constants of a heterogeneous material (Herring’s series) are considered. It is demonstrated that, on their basis, all the known exact solutions of an elastic problem for a two-dimensional two-phase composite can be found. It is also shown how a full renormalization of the series for the inverse bulk modulus can be carried out. A general expression for Young’s modulus is deduced, leading to considerable simplifications in some special cases. All results have been obtained without any restrictions on the Fourier coefficients of local parameters of the composite.     

Finite-element investigation of the stress-strain state of an inhomogeneous rectangular plate

With the use of the finite-element method, the generalized plane stressed state of a rectangle of isotropic functionally gradient materials under the action of normal load is investigated. A finite-element model is constructed by the Bubnov–Galerkin method. The domain of the body is split into rectangular gradient elements that take into account dependences of Young’s modulus and Poisson’s ratios on coordinates. Numerical calculations are performed for the case where Young’s modulus is a polynomial function. The influence of the material gradientness and the sizes of the rectangle on its stress-strain state is analyzed.     

Nonlinear finite element analysis of orthotropic and prestressed membrane structures

A new methodology for the geometrically nonlinear analysis of orthotropic membrane structures using triangular finite elements is presented. The approach is based on writing the constitutive equations in the principal fiber orientation of the material. A direct consequence of the fiber orientation strategy is the possibility to analyze initially out-of-plane prestressed membrane structures. An algorithm to model wrinkling behavior is also described. Examples of application to a number of membrane structures are presented.     

Modelling of composite plates including damage

The paper is concerned with the modelling and numerical simulation of fibre-composite plates in the nonlinear range due to large strains and damage. The layer-wise approach is applied. Each layer is treated as elastic-brittle and assumed to be orthotropic in the local material coordinate system. The appearance of damage is controlled according to the failure criteria [1,2,3,4]. When the failure condition is satisfied, the mechanical properties of the material are modified appropriately, depending on the type of damage (fibre breakage, matrix crack, fibre-matrix shear). We have programmed the model as a user subroutine within the ABAQUS environment and carried out a number of numerical simulations. The obtained numerical results are compared with the experimental data available in the literature [3]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical homogenization of composite structures with rhombic fiber arrangements

A numerical procedure is developed to determine effective material properties of unidirectional fiber reinforced composites with rhombic fiber arrangements. With the assumption of a periodic micro structure a representative volume element (RVE) is considered, where the phases have isotropic or transversely isotropic material characterizations. The interface between the phases is treated as perfect. The procedure handles the primary non-rectangular periodicity with homogenization techniques based on finite element models. Due to appropriate boundary conditions applied to the RVE elastic effective coefficients are derived. Six different boundary condition states are required to get all coefficients of the stiffness tensor. Results are listed and compared with other publications and good agreements are shown. Furthermore new results are presented, which exhibit the orthotropic behavior of such composites caused by the rhombic fiber arrangement. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

New heuristics for the maximum diversity problem

The maximum diversity problem (MDP) consists of identifying, in a population, a subset of elements, characterized by a set of attributes, that present the most diverse characteristics among the elements of the subset. The identification of such solution is an NP-hard problem. Some heuristics are available to obtain approximate solutions for this problem. In this paper, we propose different GRASP heuristics for the MDP, using distinct construction procedures and including a path-relinking technique. Performance comparison among related work and the proposed heuristics is provided. Experimental results show that the new GRASP heuristics are quite robust and are able to find high-quality solutions in reasonable computational times. G.C. Silva’s work sponsored by CAPES MSc scholarship. L.S. Ochi’s work sponsored by CNPq research grants 304103/2003-9 and 550059/2005-9. S.L. Martins’s work sponsored by CNPq research grant 475124/03-0. A. Plastino’s work sponsored by CNPq research grants 300879/00-8 and 475124/03-0.     

hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.     

Investigation of the three-dimensional micromechanical behavior of woven-fabric composites

A three-dimensional representative volume-element model is presented to study the micromechanical behavior of woven-fabric composites. The effects of the fiber undulation zone and the fiber braid angle on the elastic modulus of the composites are taken into account in the unit cell. Based on isostrain and isostress assumptions, a standard homogenization procedure is used to calculate the effective elastic properties of woven-fabric composites, and all the final stiffness components are expressed in an explicit form. The results obtained by the model considered agree with published experimental results very well. The relationship between the geometric parameters, such as fiber width, thickness, volume fraction, etc., and the macromechanical behavior of the composites can be obtained by this model. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 2, pp. 209–220, April–May, 2006.     

Effective elastic moduli and stress concentrator factors in random structure aligned fiber composites

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically uniform random set of aligned fibers. Effective elastic moduli as well as the stress concentrator factors in the components are estimated. The micromechanical approach is based on the Green’s function technique as well as on the generalization of the “multiparticle effective field method” (MEFM, see for references, Buryachenko [1]). The refined version of the MEFM takes into account the variation of the effective fields acting on each pair of fibers. The dependence of effective elastic moduli and stress concentrator factors on the radial distribution function of the fiber locations is analyzed. Received: October 20, 2004     

Orientational Design of Anisotropic Materials Using the Hill and Tsai–Wu Strength Criteria

Linearly elastic two- and three-dimensional orthotropic materials are considered. The problems of optimal material orientation are studied in the cases of the Hill and Tsai–Wu strength criteria. The necessary optimality conditions are derived for a 3D orthotropic material. In the case of a 2D orthotropic material, an analytical solution is obtained. An analysis of global and local extrema is presented.     

Deformation of composites with arbitrarily oriented orthotropic fibers under matrix microdamages

In the present work, a model of nonlinear deformation of stochastic composites under microdamaging is developed for the case of a composite with orthotropic inclusions, when microdefects are accumulated in the matrix. The composite is treated as an isotropic matrix strengthened by triaxial arbitrarily oriented ellipsoidal inclusions with orthotropic symmetry of the elastic properties. It is assumed that the process of loading leads to accumulation of damage in the matrix. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic modules in the case of orthotropic components are taken as basic relations. The fracture criterion is specified as the limiting value of the intensity of average shear stresses acting in the intact part of the material. On the basis of the analytic and numerical approach, we propose an algorithm for the determination of nonlinear deformation properties of the investigated material. The nonlinearity of composite deformations is caused by the finiteness of deformations. By using the numerical solution, the nonlinear stress–strain diagrams are predicted and discussed for an orthotropic composite material for various cases of orientation of inclusions in the matrix.     

Determination of the shear modulus of glass-reinforced plastics by means of a four-pin frame

The conditions of deformation of an orthotropic sheet of glass-reinforced plastic in a rigid four-pin frame are considered from the standpoint of the theory of elasticity of an anisotropic body. Possible cases of the direct experimental determination of the shear modulus are examined with reference to the structure of the reinforcing fabric and the orientation of the axes of elastic symmetry with respect to the edges of the frame.Leningrad. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 799–802, September–October, 1968.     

Investigation of the modulus of elasticity of carbon fibers

A simplified analysis of the dependence of the elastic properties and electrical conductivity of a carbon fiber on its orientation index is presented. The experimentally confirmed inverse correlation between the electrical resistivity and the speed of sound makes it possible to calculate the modulus of elasticity of carbon fibers from data on the bulk resistivity.Moscow Aviation Technological Institute. Translated from Mekhanika Polimerov, No. 5, pp. 846–850, September–October, 1971.     

Effect of Coulomb friction on the fiber/matrix debonding and matrix cracking in unidirectional fiber-reinforced brittle-matrix composites

A model for a macroscopic crack transverse to bridging fibers is developed based upon the Coulomb friction law, instead of the hypothesis of a constant frictional shear stress usually assumed in fiber/matrix debonding and matrix cracking analyses. The Lamé formulation, together with the Coulomb friction law, is adopted to determine the elastic states of fiber/matrix stress transfer through a frictionally constrained interface in the debonded region, and a modified shear lag model is used to evaluate the elastic responses in the bonded region. By treating the debonding process as a particular problem of crack propagation along the interface, the fracture mechanics approach is adopted to formulate a debonding criterion allowing one to determine the debonding length. By using the energy balance approach, the critical stress for propagating a semi-infinite fiber-bridged crack in a unidirectional fiber-reinforced composite is formulated in terms of friction coefficient and debonding toughness. The critical stress for matrix cracking and the corresponding stress distributions calculated by the present Coulomb friction model is compared with those predicted by the models of constant frictional shear stress. The effect of Poisson contraction caused by the stress re distribution between the fiber and matrix on the matrix cracking mechanics is shown and discussed in the present analysis. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 171–190, March–April, 2007.