Study of the effects of inner pressure and surface tension on the fibre drawing process with the aid of an analytical asymptotic fibre drawing model and the numerical solution of the full N.–St. equations

Micro-structured optical fibres (i.e. fibres that contain holes) have assumed a high profile in recent years and given rise to many novel optical devices. The problem of manufacturing such fibres by heating and then drawing a preform is considered for the case of annular capillaries. A fluid mechanics model suggested in the literature that uses asymptotic analysis based on the small aspect ratio of capillaries has been compared with the full 3D set of the N.–St. equations, for modelling the fabrication of capillaries. The final asymptotic equations, analysed in some asymptotic limits, are solved numerically and then compared with the N.–St. solutions, obtained with a commercial finite elements solver. These asymptotic limits give valuable practical information about the control parameters that influence the drawing process, taking into account the effects of surface tension and inner pressure, since those are of essential importance during drawing. It is shown that the asymptotic method delivers reliable results as long as the inner pressure does not exceed too high values.     

Analytical study of the interface in fibre-reinforced 2D composite material

Imperfect bonding between the constitutive components can greatly affect the properties of the composite structures.An asymptotic analysis of different types of imperfect interfaces arising in the problem of 2D fibrereinforced composite materials are proposed.The performed study is based on the asymptotic reduction of the governing biharmonic problem into two harmonic problems.All solutions are obtained in a closed analytical form.The obtained results can be used for the calculation of pull-out and pushout tests,as well as for the investigation of the fracture of composite materials.     

Asymptotic determination of effective elastic properties of composite materials with fibrous square-shaped inclusions

We propose an asymptotic approach for evaluating effective elastic properties of two-components periodic composite materials with fibrous inclusions. We start with a nontrivial expansion of the input elastic boundary value problem by ratios of elastic constants. This allows to simplify the governing equations to forms analogous to the transport problem. Then we apply an asymptotic homogenization method, coming from the original problem on a multi-connected domain to a so called cell problem, defined on a characterizing unit cell of the composite. If the inclusions' volume fraction tends to zero, the cell problem is solved by means of a boundary perturbation approach. When on the contrary the inclusions tend to touch each other we use an asymptotic expansion by non-dimensional distance between two neighbouring inclusions. Finally, the obtained “limiting” solutions are matched via two-point Padé approximants. As the results, we derive uniform analytical representations for effective elastic properties. Also local distributions of physical fields may be calculated. In some partial cases the proposed approach gives a possibility to establish a direct analogy between evaluations of effective elastic moduli and transport coefficients. As illustrative examples we consider transversally-orthotropic composite materials with fibres of square cross section and with square checkerboard structure. The obtained results are in good agreement with data of other authors.     

A generalization of Prandtl’s and Spencer’s solutions on axisymmetric viscous flow

New analytical solutions for axisymmetric deformation of a viscous hollow circular cylinder on a rigid fibre are given. One of the solutions generalizes the famous Prandtl’s solution for compression of a rigid perfectly plastic layer between two rough, parallel plates and the other is a modification of Spencer’s solution for compression of an axisymmetric rigid perfectly plastic layer on a rigid fibre. All equations are satisfied exactly whereas some boundary conditions are approximated in a standard manner. Special attention is devoted to frictional interface conditions since these conditions result in additional limitations of the applicability of the solution when compared to that based on a rigid perfectly plastic models. In particular, difficulties with the convergence of numerical solutions under certain conditions can be explained with the use of results obtained. Therefore, the solutions can serve as benchmark problems for verifying numerical codes. The solutions are also adopted to predict the brittle fracture of fibres by means of an approach used in previous studies and confirmed by experiment.     

Effective longitudinal shear moduli of periodic fibre-reinforced composites with radially-graded fibres

This paper presents a closed-form expression for the homogenized longitudinal shear moduli of a linear elastic composite material reinforced by long, parallel, radially-graded circular fibres with a periodic arrangement. An imperfect linear elastic fibre-matrix interface is allowed. The asymptotic homogenization method is adopted, and the relevant cell problem is addressed. Periodicity is enforced by resorting to the theory of Weierstrass elliptic functions. The equilibrium equation in the fibre domain is solved in closed form by applying the theory of hypergeometric functions, for new wide classes of grading profiles defined in terms of special functions. The effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions. A parametric analysis investigating the influence of microstructural and material features on the effective moduli is presented. The feasibility of mitigating the shear stress concentration in the composite by tuning the fibre grading profile is shown.     

A formalism for the solution of problems involving chemical reaction and concentration-dependent diffusion coefficients

By using a previously proposed novel integral transform approach, approximate analytical solutions for a certain class of concentration-dependent diffusion and reaction problems can be obtained. Three example problems of increasing complexity are presented to demonstrate the versatility of this technique. Perturbation techniques are also employed to determine asymptotic behaviour of the solutions in the limit of large or small values of some parameters.     

Effective longitudinal shear moduli of periodic fibre-reinforced composites with functionally-graded fibre coatings

This paper presents a homogenization method for unidirectional periodic composite materials reinforced by circular fibres with functionally graded coating layers. The asymptotic homogenization method is adopted, and the relevant cell problem is addressed. Periodicity is enforced by resorting to the theory of Weierstrass elliptic functions. The equilibrium equation in the coating domain is solved in closed form by applying the theory of hypergeometric functions, for different choices of grading profiles. The effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions. The influence of microgeometry and grading parameters on the shear stress concentration at the coating/matrix interface is addressed, aimed at the composite optimization in regards to fatigue and debonding phenomena.     

An incremental variational approach to coupled thermo-mechanical problems in anelastic solids. Application to shape-memory alloys

We study the coupled thermo-mechanical problem that is obtained by combining generalized standard materials with Fourier’s law for heat conduction. The analysis is conducted in the framework of non-smooth mechanics in order to account for possible constraints on the state variables. This allows models of damage and phase-transformation to be included in the analysis. In view of performing numerical simulations, an incremental thermo-mechanical problem and corresponding variational principles are introduced. Conditions for existence of solutions to the incremental problem are discussed and compared with the isothermal case. The numerical implementation of the proposed approach is studied in detail. In particular, it is shown that the incremental thermo-mechanical problem can be recast as a concave maximization problem and ultimately amounts to solve a sequence of linear thermal problems and purely mechanical (i.e. at a prescribed temperature field) problems. Therefore, using the proposed approach, thermo-mechanical coupling can be implemented with low additional complexity compared to the isothermal case, while still relying on a sound mathematical framework. As an application, thermo-mechanical coupling in shape memory alloys is studied. The influence of the loading strain-rate on the phase transformation and on the overall stress–strain response is investigated, as well as the influence of the thermal boundary conditions. The numerical results obtained by the proposed approach are compared with numerical and experimental results from the literature.     

Endommagement d'un composite polypropylène renforcé par des fibres de verre courtes : approche expérimentale

To obtain good mechanical performance of the composite material, it is important to optimise the fibres ratio as well as the fibre/matrix interface quality which have influence on the damage. The main objective of this study is to determine the structural parameters influence on damage evolution concerning two types of polypropylene glass fibres composites. With a classical approach of damage mechanical theory which consists of load–unload tensile tests, acoustic emission permits to detect and follow damage mechanisms during loading. Fractographic analysis highlights the different assumptions and conclusions for this study.     

Analysis of the temperature field of cylindrical flow based on an on-the-average exact solution

The temperature field in a well is constructed on the basis of an on-the-average exact solution, which allows investigation of problems of subterranean thermodynamics and heat and mass transfer. The problem is represented in the form of a sequence of problems of a mixed type, whose solutions give corresponding asymptotic-expansion coefficients and the form of the remainder term and the functions taking into account the presence of the boundary layer, for which analytical solutions are also found. It is shown that the proposed modified asymptotic method provides vanishing of the solution of the averaged problem for the remainder term.     

Analysis of stress concentrations in unidirectional composites taking into account the tensile load in the matrix

A modified shear-lag model accounting for the effect of the tensile stiffness of the matrix is proposed for solving the stress redistribution due to the failure of fibers and matrix in unidirectionally fibre-reinforced composites. The advantages of this model are simple, reasonable and accurate by comparison with the other similar modified shear-lag models. It can be further extended to study the stress redistribution with interfacial damage between fibres and matrix. This paper quantitatively discusses the influence of the tensile stiffness ratio of matrix to fibre and of the fibre volume fraction on the stress concentration in the fibres and matrix adjacent to cut fibres and matrix, and suggests that the influence of the matrix stiffness on the stress concentration can be neglected when the matrix stiffness is low, such as polymer matrix composites, and the fibre volume fraction is high. For other cases such as ceramic and metal matrix composites, the tensile load of the matrix cannot be neglected in the shearlag analysis. The project supported by the Guangdong Provincial Natural Science Foundation of China.     

Progress in shakedown analysis with applications to composites

This paper reports on recent developments in the field of direct methods, in particular shakedown analysis (SA) from theoretical and numerical points of view. Emphasis is placed on problems connected with the failure of fibre-reinforced periodic composites under variable loads where special attention is paid to the problem of interface debonding between fibre and matrix materials. The approach is based on a local SA in a representative volume element of the composite and the use of averaging techniques to study the influence of each component (matrix, fibre and interface) on the macroscopic response of such composite. For numerical applications, the interior-point difference-of-convex-functions algorithm (IPDCA) is proposed as an efficient method for solving large-scale problems.     

Brush-like modes of fatigue fracture in unidirectional fibre composites

Fatigue fracture of unidirectional fibre composites under tension along the fibres is discussed with account of the interaction between various mechanisms of damage such as single and multiple fibre ruptures, matrix cracking, and matrix-fibre debonding. The case of brittle fibres and a comparatively weak and ductile matrix is considered that exposes non-conventional modes of fracture, named “brush-like” cracks. Growth of such cracks under cyclic quasistatic loading is studied, and the effect of various factors on the crack growth rate is investigated.     

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

The propagation of non-linear elastic anti-plane shear waves in a unidirectional fibre-reinforced composite material is studied. A model of structural non-linearity is considered, for which the non-linear behaviour of the composite solid is caused by imperfect bonding at the “fibre–matrix” interface. A macroscopic wave equation accounting for the effects of non-linearity and dispersion is derived using the higher-order asymptotic homogenisation method. Explicit analytical solutions for stationary non-linear strain waves are obtained. This type of non-linearity has a crucial influence on the wave propagation mode: for soft non-linearity, localised shock (kink) waves are developed, while for hard non-linearity localised bell-shaped waves appear. Numerical results are presented and the areas of practical applicability of linear and non-linear, long- and short-wave approaches are discussed.     

Dynamics of fibres in a turbulent flow field – A particle-level simulation technique

A particle-level simulation technique has been developed for modelling the flow of fibres in a turbulent flow field. A single fibre is conceived here as a chain of segments, thus enabling the model fibre to have all the degrees of freedom (translation, rotation, bending and twisting) needed to realistically reproduce the dynamics of real fibres. Equations of motion are solved for each segment, accounting for the interaction forces with the fluid, the contact forces with other fibres and the forces that maintain integrity of the fibre.The motion of the fluid is resolved as a combination of 3D mean flow velocities obtained from a CFD code and fluctuating turbulent velocities derived from the Langevin equation. A case of homogeneous turbulence is treated in this paper.The results obtained show that fibre flocs in air-fibre flows can be created even when attractive forces are not present. In such a case, contacts between fibres, properties of an individual fibre (such as flexibility and equilibrium shapes) and properties of the flow of the carrying fluid are shown to govern the physics behind formation and breaking up of fibre flocs. Highly irregular fibre shapes and stiff fibres lead to strong flocculation.The modelling framework applied in this work aims at making possible a numerical model applicable for designing processes involving transport of fibres by air at industrial scale.     

Generalized solutions of transient thermal shock problem with bounded boundaries

In this work, the generalized thermoelastic solutions with bounded boundaries for the transient shock problem are proposed by an asymptotic method. The governing equations are taken in the context of the generalized thermoelasticity with one relaxation time (L–S theory). The general solutions for any set of boundary conditions are obtained in the physical domain by the Laplace transform techniques. The corresponding asymptotic solutions for a thin plate with finite thickness, subjected to different sudden temperature rises in its two boundaries, are obtained by means of the limit theorem of Laplace transform. In the context of these asymptotic solutions, two specific problems with different boundary conditions have been conducted. The distributions of displacement, temperature and stresses, as well as the propagations, intersections and reflections of two elastic waves, named as thermoelastic wave and thermal wave separately, are obtained and plotted. These results are agreed with the results obtained in the existing literatures.     

Load transfer in fibre-reinforced composites with viscoelastic matrix: an analytical study

In A fibre-reinforced 2D composite material with elastic fibres and viscoelastic, isotropic matrix is studied. Starting from the solution of a reference-problem with elastic matrix material the elastic matrix parameters are substituted by their viscoelastic correspondents in the Laplace domain. For simplification the time-dependent solution is approximated by using limiting value theorems that give information about the time-dependent solution for t → 0 and t → ∞. Then the method of asymptotically equivalent functions is used and illustrated with examples of a steel fibre in a PMMA matrix. The analytical solutions are compared with their numerical counterparts. In summary it can be stated that this paper is a further contribution to the vast literature about the application of the correspondence principle to the solution of special problems of the linear viscoelasticity.     

Analytical bending solutions of free orthotropic rectangular thin plates under arbitrary loading

This paper presents the analytical solutions for the bending of orthotropic rectangular thin plates by the double finite integral transform, which, as an effective tool in solving plate problems, should have received attention. As a representative and difficult problem in the theory of plates, free plates’ bending is successfully solved to demonstrate the accuracy of the method by comparing the present analytical solutions with those from the literature as well as those by the finite element method. With the proper integral transform kernels, the proposed solution procedure is applicable to the bending of orthotropic rectangular plates with all combinations of simply supported, clamped and free boundary conditions, which serves as an elegant approach to analytical solutions of plate bending problems.     

Effect of pore shape on effective porothermoelastic properties of isotropic rocks

The present work is devoted to the determination of the macroscopic poroelastic and porothermoelastic properties of geomaterials or rock-like composites constituted by an isotropic matrix with embedded ellipsoidal inhomogeneities and/or pores randomly oriented. By considering the solution of a single ellipsoidal inhomogeneity in the homogenization problem it is possible to observe the significant influence of the shape of inhomogeneities on the effective porothermoelastic properties. In the particular case of microscopic and macroscopic isotropic behaviors, a closed form solution based on analytical integrate of the Eshelby solution for the single ellipsoidal inhomogeneity can be obtained for the randomly oriented distribution. This result completes the well known solutions in the limiting cases of spherical and penny shape inhomogeneities. Based on recent works on porous rock-like composites such as shales or argillites, an application of the developed solution to a two-level microporomechanics model is presented. The microporosity in homogenized at the first level, and multiple solid mineral phase inclusions are added at the second level. The overall porothermoelastic coefficients are estimated in the particular context of heterogeneous solid matrix. Numerical results are presented for data representative of isotropic rock-like composites.