Inhomogeneous Interphase Research Articles

Progressive failure analysis of unidirectional fiber-reinforced polymers with inhomogeneous interphase and randomly distributed fibers under transverse tensile loading

The effect of damage due to interfacial debonding on the post initial failure behavior of unidirectional fiber-reinforced polymers subjected to transverse tension was investigated using numerical homogenization techniques based on the finite element method. Calculations were performed for unit cells containing fibers distributed at random over the transverse cross-section with inhomogeneous interphase layers. The mechanism of progressive failure was examined at both a global and a local level. A detailed analysis of the proposed micromechanics model revealed that it is able correctly to simulate the evolution of damage and to explain the softening mechanism. It was found that the post initial failure behavior of unidirectional lamina under transverse tension is mainly controlled by the interface strength and the interphase stiffness. The present study showed that local fiber array irregularities are a significant contributor to matrix cracking through local stress concentrations and the occurrence of localization.

Effect of interfacial debonding on the failure behavior in a fiber-reinforced composite subjected to transverse tension

Abstract By using computational micromechanics, macroscopic stress–strain curves for a glass/epoxy lamina subjected to transverse tension were determined in this paper. To compute stress for given strain, a finite element model of a three-phased unit cell with the hexagonal symmetry was employed. Mixed mode debonding conditions between reinforcement and matrix were modeled by a bilinear cohesive law. A stress transfer between matrix and fiber was simulated by a inhomogeneous interphase. A detailed analysis of a debonding growth was also presented. Parametrical studies showed that a choice of values for cohesive parameters as well as debonding locations do have an influence on the macroscopic response of material. An ability of the proposed model to simulate the softening behavior of the material under transverse tension and to predict final failure was demonstrated.

Micromechanical modelling of the yield stress of polymer-particulate nanocomposites with an inhomogeneous interphase

In this work, a micromechanical model is proposed in order to predict the yield stress of polymer-based nanocomposites filled with spherical nanoparticles. The adopted approach integrates an inhomogeneous interphase around the nanoparticles. A characteristic length scale corresponding to the thickness of the interphase allows accounting for the particle size effects. The key role of particle size and features of inhomogeneous interphase on the overall initial yield surface of nanocomposites is theoretically studied.

A micro-mechanics model for composites reinforced by regularly distributed particles with an inhomogeneous interphase

Abstract Based on the Mori–Tanaka method, a micro-mechanics model is developed to study the effective elastic properties of composites reinforced by regularly distributed particles. The spatial distribution of particles is supposed to be cube symmetric in the three-dimensional space, and the corresponding finite element method (FEM) computation has been performed through a unit cell model. Additionally, particle interaction and distribution are simultaneously taken into account by using the strain Green’s function, and the specified strain Green’s function is determined by utilizing the necessary conditions of geometric symmetry. In order to analyze particle size effect on the effective properties of composites, the Double-inclusion configuration and related theory are introduced to describe the role of the interphase between the matrix and particles. Finally, the overall elastic properties of the composite with regularly distributed particles are described by three independent elastic constants expressed in the explicit form, and the accuracy of the developed model is verified by comparing with FEM results.

Approaches to Model Nanotube‐Reinforced Polymers using Micromechanics and Finite Elements

AbstractIn this work, the applicability of new analytical methods for modeling polymer nanocomposites is examined. The influence of an interphase between nanotube and matrix could be captured by this modeling strategy. However, it is shown that there are some difficulties in calculation of composites including extremely weak fibers with stiff coatings. Furthermore, finite element calculations are shown for comparison. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical studies on the effective shear modulus of particle reinforced composites with an inhomogeneous inter-phase

Numerical studies on the effective shear modulus of particle reinforced composites with an inhomogeneous inter-phase are performed. The influences of many parameters to the equivalent shear modulus of composites are carefully analyzed, including the inter-phase thickness, variation of interfacial properties, boundary conditions and volume fraction of particles, etc. Numerical results show that the Poisson ratio can be assumed as a constant across the whole inter-phase zone in the computation. The form of property variation across the inter-phase also greatly affects the effective shear modulus of composite. Numerical results predicted by the rigid boundary conditions are remarkably higher than those by the free boundary conditions and the exact solutions. The reasonability and exactness of the available models for predicting the effective shear modulus of composites are accessed by the numerical results in the present work.

Influence of inhomogeneous interphase on thermal stresses in fiber-reinforced composites

The effects of an inhomogeneous interphase on thermal response of fiber-reinforced composites are investigated in this paper. The work is based on a linear variation of Young's modulus and thermal expansion coefficient of the interphase and the assumption of generalized plane strain. An accurate analytical approach is presented to determine thermal stresses in composites reinforced with isotropic fibers containing an inhomogeneous interphase. How the inhomogeneous interphase, and Young's modulus and thermal expansion coefficient of the matrix affect thermal response of the composites is examined. It is found that the inhomogeneous interphase causes different stress distribution from that of the homogeneous interphase. Raising Young's modulus and thermal expansion coefficient of the matrix obviously increases the maximum radial, circumferential and axial stresses in all the constituents of the composites.

A two way particle mapping for calculation of the shear modulus of a spherical inclusion composite with inhomogeneous interphase

Based on the Mori--Tanaka method and a replacement scheme, a pair of coupled first order differential equations which model the shear modulus of a particulate composite with inhomogeneous interphase are derived. However, the results derived are not exact since the Mori--Tanaka method is not exact for the shear problem. An improved model is therefore proposed which utilises the generalised self consistent scheme for a spherical inclusion that is surrounded by a hypothetical homogeneous interphase layer. To find the properties of this hypothetical interphase layer a mapping of a homogeneous particle onto a two phase composite is utilised. The results are then presented for a simple power law profile and are shown to be consistent with the conclusions of Shen and Li [Int. J. Solids and Struct., 40, 2003, 1393--1409].

A two-way particle mapping for calculation of the effective dielectric response of graded spherical composites

This work is a study of the dielectric properties of composites where each embedded spherical inclusion is surrounded by an inhomogeneous interphase. This is an extension of work done by Lombardo and Ding [Lombardo N, Ding Y. Effect of inhomogeneous interphase on the bulk modulus of a composite containing spherical inclusions. In: Proceedings of the 4th Australasian congress on applied mechanics; 2005] on the bulk modulus of composites. Here we derive a coupled pair of differential equations using the replacement method [Hill R. Theory of mechanical properties of fibre-strengthened materials: I. Elastic behaviour. J Mech Phys Solids 1964:12;199] as the foundation. These differential equations model the dielectric properties of composites for a wide class of functions describing the interphase inhomogeneity and are shown to coincide with the equation obtained using DEDA [Yu KW, Gu GQ, Huang JP. Dielectric response of spherical particles of graded materials. cond-mat/0211532; 2002]. It is also shown how the model may be combined with the model of Vo and Shi [Vo HT, Shi FG. Towards model-based engineering of optoelectronic packaging materials: dielectric constant modeling. Microelectron J 2002:33;409] for the dielectric constant. The differential equations are solved for three different profiles and the result of the combined models is compared to experimental data. All three profiles compared excellently to the experimental data.

Thermomechanical analysis of elastic–plastic fibrous composites comprising an inhomogeneous interphase

An analytical approach is presented to investigate thermomechanical response of composites consisting of a transversely isotropic fiber, an inhomogeneous interphase and an elastic–plastic matrix. Using the existing cubic variation to describe the continuous change of the material properties of the interphase and dividing the interphase into a number of subdomains, the continuously varying material properties of the interphase are approximated by the constant ones of these subdomains, and the deformations and stresses of the interphase are described with the same formulae as those of transversely isotropic fibers. The analytical expressions of elastic–plastic deformations and stresses of the matrix are obtained from the basic equations of axisymmetric problems in elasticity, the assumption of generalized plane strain, the linear strain–hardening stress–plastic strain relation, Tresca’s yield condition, the associated flow rule and impressibility of plastic deformation. The boundary conditions of the composites and the continuities of the radial displacement and stress between different components are used to determine all the unknown constants and the obtained analytical solution is applied to thermomechanical analysis of the composites. The effects of the inhomogeneity of the interphase, and the plasticity and material properties of the matrix on the thermomechanical response of the composites are investigated.

Numerical modeling of the elastic behavior of fiber-reinforced composites with inhomogeneous interphases

This paper describes a numerical approach for modeling the micromechanical behavior and macroscopic properties of multi-phase fiber-reinforced composites with inhomogeneous interphases. The interphases are modeled as functionally graded elastic layers with the Young’s modulus and Poisson’s ratio varying in the radial direction. In general, the fibers can have different elastic properties and sizes and can, if desired, be randomly distributed. The approach is based on the numerical solution of a complex boundary integral equation in which the boundary parameters are expressed in terms of complex Fourier series. All the integration can be done analytically and thus the method allows for accurate calculation of the elastic fields anywhere within the material, including inside the fibers and interphases. Explicit expressions for the effective elastic constants can be obtained from general relations between the average stresses and strains. Numerical experiments and comparisons with the numerical benchmark results available in the literature have demonstrated the versatility, accuracy, and efficiency of the presented approach.

Effect of an inhomogeneous interphase on the thermal expansion coefficient of a particulate composite

This work is a study of the thermal properties of composite materials containing isotropic spherical inclusions surrounded by an inhomogeneous interphase, embedded in an isotropic matrix. By modelling the thermal and mechanical properties of the interphase as continuous radial functions, a replacement technique is used to derive a differential equation which models the thermal expansion coefficient of the composite. The results are generalised for a wide class of functions describing the inhomogeneous properties of the interphase. When the mechanical properties of the interphase are described by a power law profile a closed form solution is obtained for various thermal expansion profiles. The results are displayed graphically against various other theoretical models and with experimental results. A comparison of the results for different values of the problem parameters is also presented. The present model showed good agreement with the experimental results of Holliday and Robinson [Holliday L, Robinson J. Review: the thermal expansion of composites based on polymers. J Mater Sci 1973;8:301–11].

Homogenization of a fibre/sphere with an inhomogeneous interphase for the effective elastic moduli of composites

An effective interphase model (EIM) and a uniform replacement model (URM) are proposed to study the effect of an inhomogeneous interphase with varying elastic properties in the radial direction on the effective elastic moduli of composites reinforced by fibres/spheres. The central idea of these models is to convert a fibre/sphere with its interphase into a two-phase or homogeneous fibre/sphere. Then, the strain energy changes can be obtained using the three-phase model or Eshelby's solution. Detailed comparisons with the finite-element method (FEM) results of the strain energy changes for various possible material combinations of fibre/sphere, interphase and matrix are carried out to check the validity of the two models. Moreover, the other two existing models, the uniform interphase model (UIM) and differential replacement model (DRM), are compared with the new ones. It is shown that the validity of these analytical models depends on the material combinations. The EIM can be valid for general cases, while the simple URM is only valid for some cases. The validity ranges of the two existing models lie between those of the two new ones. Finally, the expressions of the effective elastic moduli of composites involving an inhomogeneous interphase are given by combining these models and the Mori–Tanaka method. The application of these expressions is illustrated through three examples and further comparisons with FEM results are also given.

Analytical solutions for elastostatic problems of particle-and fiber-reinforced composites with inhomogeneous interphases

By transforming the governing equations for displacement components into Riccati equations, analytical solutions for displacements, strains and stresses for Representive Volume Elements (RVEs) of particle- and fiber-reinforced composites containing inhomogenous interphases were obtained. The analytical solutions derived here are new and general for power-law variations of the elastic moduli of the inhomogeneous interphases. Given a power exponent, analytical expressions for the bulk moduli of the composites with inhomogenous interphases can be obtained. By changing the power exponent and the coefficients of the power terms, the solutions derived here can be applied to inhomogeneous interphases with many different property profiles. The results show that the modulus variation and the thickness of the inhomogeneous interphase have great effect on the bulk moduli of the composites. The particle will exhibit a sort of “size effect”, if there is an interphase.

A unified numerical approach for thermal analysis of transversely isotropic fiber-reinforced composites containing inhomogeneous interphase

This paper presents a general method to tackle thermal analysis of transversely isotropic fiber-reinforced composites comprising an inhomogeneous interphase with arbitrary variations of material properties. Based on the assumption of generalized plane strain, the governing equation describing the deformations and stresses of the composites subjected to a thermal loading is derived from the theory of elasticity and solved efficiently. With the proposed method, the effects of different variations of material properties and different volume fractions of the interphase on the stresses in the composites are investigated. They strongly influence the circumferential and axial stresses in the interphase, but make slight differences to these stresses in the fiber and matrix.

Effect of an inhomogeneous interphase zone on the bulk modulus and conductivity of a particulate composite

A model is presented of a particulate composite containing spherical inclusions, each of which are surrounded by a localized region in which the elastic moduli vary smoothly with radius. This region may represent an interphase zone in a composite, or the transition zone around an aggregate particle in concrete, for example. An exact solution is derived for the displacements and stresses around a single inclusion in an infinite matrix, subjected to a far-field hydrostatic compression, and is then used to derive an approximate expression for the effective bulk modulus of a material containing a random dispersion of these inclusions. The analogous conductivity (thermal, electrical, etc.) problem is then discussed, and it is shown that the expression for the normalized effective conductivity corresponds exactly to that for the normalized effective bulk modulus, if the Poisson ratios of both phases are set to zero.

Effective moduli of particle-filled composite with inhomogeneous interphase: Part II – mapping method and evaluation

In this part of the two-part paper, a method that maps a particle together with the inhomogeneous interphase into an effective homogeneous particle is used to predict the effective moduli of particle-filled composites with an inhomogeneous interphase. The effective bulk modulus of the equivalent particle can be obtained in closed-form expressions for a wide variety of variations of the elastic properties of the interphase. The effective shear modulus is obtained by two different models: a differential scheme based on the Hashin–Shtrikman lower bound and an average approximation using the average value of the elastic properties of the interphase. This mapping method can be used along with many micromechanics approaches to evaluate the effective moduli of particle-filled composites with an inhomogeneous interphase. In this part, the effective moduli of a composite obtained using the generalized self-consistent method (GSCM) and the third-order approximation (TOA) of Torquato are compared with the bounds presented in Part I. It is found that the effective bulk and shear moduli are within the bounds for the considered case.

Effective moduli of particle-filled composite with inhomogeneous interphase: Part I – bounds

This part of the two-part paper reports calculations for the effective moduli of a particle-filled composite with the properties of the interphase continuously varying in the radial direction. First, the difficulties in obtaining the explicit expressions of the effective moduli are explained. Then by using the generalized self-consistent method (GSCM) and by introducing an inhomogeneous comparison material with a microstructure different from that of the real material, explicit expressions of the upper and lower bounds of the effective moduli for the composite are obtained. These bounds are found to be very close for the considered case. The comparison of the explicit bounds with the numerical results shows that the present solutions are satisfactory.

Effective elastic moduli of composites reinforced by particle or fiber with an inhomogeneous interphase

The solution of the strain energy change of an infinite matrix due to the presence of one spherical particle or cylindrical fiber surrounded by an inhomogeneous interphase is the basis of solving effective elastic moduli of corresponding composites based on various micromechanics models. In order to find out the strain energy change, the composite sphere or cylinder, i.e., the spherical particle or cylindrical fiber together with its interphase, is replaced by an effective homogeneous particle or fiber. Independent governing differential equations for each modulus of the effective particle or fiber are derived by extending the replacement method [J. Mech. Phys. Solids 12 (1964) 199]. As far as the strain energy changes of the infinite matrix subjected to various far-field stress systems are concerned, the present model is simple. Meanwhile, FEM analysis is carried out for a verification, which shows that the model can lead to rather accurate results for most practical interphases. Besides, to check the validity of the model further when the interactions among composite cylinders exist, the two problems of an infinite matrix containing two composite cylinders and the effective moduli of composites with the equilateral triangular distribution of composite cylinders are analyzed using FEM. The FEM results show that the model is still rather accurate, especially for the case of interphase properties varying between those of fiber and matrix. Therefore, composite spheres or cylinders are assumed as the effective homogeneous particles or fibers and simple expressions of the effective moduli of composites containing the composite spheres or cylinders are obtained. Furthermore, the present model is compared with some existing models that are based on very complicated derivations.

Interphase characterization in polymer composites - monitoring of interphasial behavior in dependence on the mode of loading

Single fiber model composites consisting of epoxy resin matrix and differently sized glass fibers were investigated using pull-out tests, scanning electron microscopy (SEM), scanning force microscopy (SFM) and single fiber dynamic load test (SFDL). The inhomogeneous stress distribution along the embedded fiber length could be visualized by monitoring. SEM images showed either cohesive fracture or adhesive failure on pulled-out fibers with different sizings. The crack initiation and propagation were detected randomly and multiply distributed as the inhomogeneous interphase itself and depending strongly on the fiber-matrix model combination. The meniscus region acts as a material inhomogeneity and its appearence depends on the surface free energies of fiber and matrix and on the curing conditions of the resin. SFM in force modulation mode has visualized different interphase thicknesses and gradients of local stiffness. The SFDL test has been shown as a worthful tool for the comprehensive determination of fiber-matrix interaction.