Anisotropy of hygrothermal damage in fiber/polymer composites: Effective elasticity measures and estimates

Measured elasticity moduli of a highly (68%) glass-fiber reinforced epoxy matrix for different amounts of fiber/matrix interface weakening and debonding, due to different hygrothermal ageing stages, are compared to estimated ones. Ultrasonic measurements provide seven of the nine elasticity moduli of the orthotropic material samples, including all the moduli significantly affected by damage. Theoretical estimates combine homogenization modeling techniques and Finite Element (FE) calculations, the latter when the effect of observed partial debonding on effective moduli is to be specified. These estimates are performed under different assumptions for the composite structure, with special attention to the existence of a fiber–matrix interphase. Analytical comparisons for the undamaged composite establish that matching US measurements with estimates cannot be obtained, regardless of the chosen model, without the assumption of an interphase layer of modified resin coating the fibers. This coating resin, when in relevant concentration with regard to literature data about fiber coating thickness, typically conserves the epoxy moduli transversally to the fiber orientation, while, in the fiber direction its moduli approach those of the fibers. The comparison of the US measurements on damaged samples to FE calculations assuming progressive one-directional debonding shows that most of the composite stiffness loss can preferentially be due to an initial interphase weakening, while the fiber/matrix debonding seems more likely delayed to long H-ageing times. This is consistent with physical interpretation of damage by water pooling through silane bridges bonding epoxy to glass. The calculations also provide the effective stiffness, at different damage stages, of the “Undamaged Equivalent Inhomogeneity” for this damaged inclusion type.     

Determination of sensitivity coefficients of the elastic effective properties for periodic fiber-reinforced composites using the response function method

Derivation and implementation of the homogenization method including determination of sensitivity gradients of the effective elasticity tensor using combined numerical-analytical approach are addressed in this paper. This is possible thanks to an application of the numerical response function together with the effective moduli method known from classical homogenization theory. Computational procedure is implemented using 4-noded quadrilateral plane strain finite elements (program MCCEFF) and the symbolic computations system MAPLE. The sensitivity coefficients are determined on the basis of partial derivatives of the homogenized elasticity tensor calculated using the response function method with respect to all composite components’ elastic characteristics. They are further separately subjected to normalization procedure for a final comparison with each other. Such an enriched homogenization procedure is tested on the periodic fiber-reinforced two component composites; the results of computational analysis are compared to the results of the central finite difference approach applied before. Computational methodology proposed here may be further successively applied not only in the context of homogenization method but also to extend various discrete computational techniques like boundary/finite element, finite difference and volumes together with various meshless methods.     

Effective properties of biopolymer composites: A three-phase finite element model

The effective Young’s modulus of starch–zein biopolymer composite is studied using a finite element model. This model handles the fact that starch–zein interface is not perfect in the sense that elasticity properties across the interface are altered. The motivation here is to predict the right modulus of elasticity near the interface for any zein content. In that way, the finite element model requires zein, starch and interphase properties to be implemented for the computation of the composite Young’s modulus as a function of zein content. The main variables are the interphase thickness and modulus. The comparison between the predicted and experimental results shows, firstly, that the effective composite properties are not correctly estimated using standard models with perfect interface hypothesis. Secondly, the three-phase model is able to suggest optimal interphase parameters in order to fit the experimental data obtained using three-point bending test. Finally, the optimal interphase parameters (thickness and modulus) vary as a function of zein content.     

Multiscale dynamic fracture analysis of composite materials using adaptive microstructure modeling

In this study, a computational framework is proposed to investigate multiscale dynamic fracture phenomena in materials with microstructures. The micro- and macro-scales of a composite material are integrated by introducing an adaptive microstructure representation. Then, the far and local fields are simultaneously computed using the equation of motion, which satisfies the boundary conditions between the two fields. Cohesive surface elements are dynamically inserted where and when needed, and the Park-Paulino-Roesler cohesive model is employed to approximate nonlinear fracture processes in a local field. A topology-based data structure is utilized to efficiently handle adjacency information during mesh modification events. The efficiency and validity of the proposed computational framework are demonstrated by checking the energy balances and comparing the results of the proposed computation with direct computations. Furthermore, the effects of microstructural properties, such as interfacial bonding strength and unit cell arrangement, on the dynamic fracture behavior are investigated. The computational results demonstrate that local crack patterns depend on the combination of microstructural properties such as unit cell arrangement and interfacial bonding strength; therefore, the microstructure of a material should be carefully considered for dynamic cohesive fracture investigations.     

Flexural studies on asymmetric hybrid Kevlar fabric/epoxy composites

For composites reinforced with Kevlar fabrics, the method of asymmetric hybridization is employed for the improvement of flexural properties such as maximum fibre yield stress and modulus of elasticity in bending. Calculations based on the elastic-plastic analysis are used to assess the shift in the neutral axis during bending, and the bimaterial beam model is invoked to estimate the arrangement and replacement of Kevlar fibres by carbon fibres in the compression face, for two relative fibre orientations. Flexural properties of the bimaterial are compared with those of unmodified Kevlar/epoxy composite for three different loading rates. Scanning electron microscopic examination of the fracture features is discussed.     

Boundary collocation method for multiple defect interactions in an anisotropic finite region

The modified mapping collocation method is extended for the solution of plane problems of anisotropic elasticity in the presence of multiple defects in the form of holes, cracks, and inclusions under general loading conditions. The approach is applied to examine the stress and strain fields in an anisotropic finite region including an elliptical and a circular hole, an elliptical flexible inclusion, and a line crack. It can be readily incorporated into micro-mechanics models, capturing the relative importance of the matrix, the fiber/matrix interface, and reinforcement geometry and arrangement while estimating the effective elastic properties of composite materials. The accuracy and robustness of this method is established through comparison with results obtained from finite element analysis.     

Numerical assessment of an anisotropic porous metal plasticity model

The objective of this paper is to perform numerical assessment of a micromechanical model of porous metal plasticity developed previously by the authors. First, upper bound estimates for the yield loci are computed using homogenization and limit analysis of a spheroidal representative volume element containing a confocal spheroidal void, neglecting elasticity. Unlike in the development of the analytical model, the computational limit analysis is performed without recourse to approximations so that the obtained yield loci are rigorous upper bounds for the true criterion. Next, the model’s macroscopic dilatancy at incipient plastic flow is compared against that of the numerical limit analysis approach. Finally, finite-element calculations, with elasticity included, are presented for transversely isotropic porous unit-cells loaded axisymmetrically. The effective stress–strain response as well as evolution of the unit-cell porosity and void aspect ratio are compared with theoretical predictions.     

Two-dimensional and axisymmetric unit cell models in the analysis of composite materials

Unit cell models have been widely used for investigating fracture mechanisms and mechanical properties of composite materials assuming periodically arrangement of inclusions in matrix. It is desirable to clarify the geometrical parameters controlling the mechanical properties of composites because they usually contain randomly distributed particulate. To begin with a tractable problem this paper focuses on the effective Young’s modulus E of heterogeneous materials. Then, the effect of shape and arrangement of inclusions on E is considered by the application of FEM through examining three types of unit cell models assuming 2D and 3D arrays of inclusions. It is found that the projected area fraction and volume fraction of inclusions are two major parameters controlling effective elastic modulus of inclusions.     

Effective properties of layered magneto-electro-elastic composites

A micro-mechanics model is established to evaluate the effective properties of a layered composite with piezoelectric and piezomagnetic components. In the approach the mechanical and electromagnetic continuity conditions across an interface of two dissimilar media is incorporated, decomposing those quantities such as the stress/strain, electric displacement, electric field intensity, magnetic induction, and magnetic field intensity into two orthogonal complementary parts. Accordingly, the linear coupling effect between elasticity, electricity and magnetism of the composite is derived. Numerical results for a BaTiO₃–CoFe₂O₄ composite with 2-2 connectivity are obtained using the model, and some interesting results are discussed.     

Morphological study of metal/dielectric composite films with various spatial distributions of metal particles

Morphological properties of metal/dielectric composite films consisting of individual metal particles in a dielectric matrix were studied by methods of computational physics. The simulated composite structures were prepared by hard-sphere and soft-sphere models with given spatial distribution of objects ranging from totally random, through various degrees of order up to anisotropic in certain directions. The images of sections of generated composite films were processed using the two strongest morphological methods - of “Voronoi tessellation” and “covariance function” with the aim to analyse the spatial distribution of particles in the composite film and to reveal the possible irregularities in this distribution. The results show that both the methods can effectively help to quantify the degree of particles arrangement as well as the anisotropy of structures studied.     

Review Micromechanical testing of bone trabeculae - potentials and limitations

The mechanical properties of bone are studied mostly for reasons related to skeletal pathology. However, bone is also very interesting from a material science perspective because it is a natural hierarchical composite material. The mechanical properties of bone depend on both the structural arrangement and the properties of the constituting materials, namely the organic polymer collagen and the inorganic salt apatite. While the mechanical properties of bone samples at the macroscopic scale are measured routinely, mechanical tests on micrometer-sized specimens are still at development stage. In this paper, protocols for measuring the elasticity of cancellous bone trabeculae are reviewed. The published values for the elastic modulus of trabeculae vary between 1 GPa and 15 GPa. Reasons for this broad range of values may be located in the intrinsic difficulties of preparing, handling, and testing inhomogeneous, anisotropic and asymmetric micro-samples. We discuss the major error sources in existing testing procedures and suggest potential strategies to enhance their performance.     

On semi‐analytical probabilistic finite element method for homogenization of the periodic fiber‐reinforced composites

The main aim of this paper is a development of the semi‐analytical probabilistic version of the finite element method (FEM) related to the homogenization problem. This approach is based on the global version of the response function method and symbolic integral calculation of basic probabilistic moments of the homogenized tensor and is applied in conjunction with the effective modules method. It originates from the generalized stochastic perturbation‐based FEM, where Taylor expansion with random parameters is not necessary now and is simply replaced with the integration of the response functions. The hybrid computational implementation of the system MAPLE with homogenization‐oriented FEM code MCCEFF is invented to provide probabilistic analysis of the homogenized elasticity tensor for the periodic fiber‐reinforced composites. Although numerical illustration deals with a homogenization of a composite with material properties defined as Gaussian random variables, other composite parameters as well as other probabilistic distributions may be taken into account. The methodology is independent of the boundary value problem considered and may be useful for general numerical solutions using finite or boundary elements, finite differences or volumes as well as for meshless numerical strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Multi-scale modeling, stress and failure analyses of 3-D woven composites

The very complex, multi-level hierarchical construction of textile composites and their structural components commonly manifests via significant property variation even at the macro-level. The concept of a “meso-volume” (introduced by this author in early 1990s) is consistently applied in this work to 3-D stress/strain and failure analyses of 3-D woven composites at several levels of structural hierarchy. The meso-volume is defined as homogeneous, anisotropic block of composite material with effective elastic properties determined through volumetrically averaged 3-D stress and strain fields computed at a lower (“finer”) level of structural hierarchy and application of generalized Hooke’s law to the averaged fields. The meso-volume can represent a relatively large, homogenized section of a composite structural component, a lamina in laminated composite structure, a homogenized assembly of several textile composite unit cells, a single homogenized unit cell, a resin-impregnated yarn, a single carbon fiber, even a carbon nanotube assembly. When composed together, distinct meso-volumes constitute a 3-D Mosaic model at the respective hierarchy level. A multi-scale methodology presented in this paper first illustrates 3-D stress/strain analysis of the Mosaic unidirectional composite, computation of its effective elastic properties and their further use in 3-D stress/strain analysis of the Mosaic model of 3-D woven composite Unit Cell. The obtained 3-D stress/strain fields are then volumetrically averaged within the Unit Cell, and its effective elastic properties are computed. The predicted effective elastic properties of 3-D woven composite are compared with experimental data and show very good agreement. Further, those effective elastic properties are used in 3-D simulations of three-point bending tests of 3-D woven composite; theoretical predictions for central deflection show excellent agreement with experimental data. Finally, a 3-D progressive failure analysis of generic 3-D Mosaic structure is developed using ultimate strain criterion and illustrated on the 3-D woven composite Unit Cell. The predicted strength values are compared to experimental results. The presented comparisons of theoretical and experimental results validate the adequacy and accuracy of the developed material models, mathematical algorithms, and computational tools.     

Effect of carbon nanotube orientation on the mechanical properties of nanocomposites

Carbon nanotubes (CNTs) have been regarded as ideal reinforcements of high-performance composites with enormous applications. In this paper, nano-structure is modeled as a linearly elastic composite medium, which consists of a homogeneous matrix having hexagonal representative volume elements (RVEs) and homogeneous cylindrical nanotubes with various inclination angles. Effects of inclined carbon nanotubes on mechanical properties are investigated for nano-composites using 3-D hexagonal representative volume element (RVE) with short and straight CNTs. The CNT is modeled as a continuum hollow cylindrical shape elastic material with different angles. The effect of the inclination of the CNT and its parameters is studied. Numerical equations are used to extract the effective material properties for the hexagonal RVE under axial as well as lateral loading conditions. The computational results indicated that elastic modulus of nano-composite is remarkably dependent on the orientation of the dispersed SWNTs. It is observed that the inclination significantly reduces the effective Young’s modulus of elasticity under an axial stretch. When compared with lateral loading case, effective reinforcement is found better in axial loading case. The effective moduli are very sensitive to the inclination and this sensitivity decreases with the increase of the waviness. In the case of short CNTs, increasing trend is observed up to a specific value of waviness index. It is also found from the simulation results that geometry of RVE does not have much significance on stiffness of nano-structures. The results obtained for straight CNTs are consistent with ERM results for hexagonal RVEs, which validate the proposed model results.     

Prediction of the macroscopic properties of elastoplastic porous materials

Results are presented from the solution of a problem of the mechanics of composite materials which involves determination of the yield point, plastic strains, and elasticity tensor of an elastoplastic medium with elliptical pores. The dependences of these properties on pore concentration are obtained on the basis of physically substantiated assumptions and the effective field method developed previously by the authors.Translated from Problemy Prochnosti, No. 1, pp. 36–39, January, 1991.     

Error estimates for mixed finite element approximations of nonlinear problems

The finite element methods have proved a very effective tool for the numerical solutions of nonlinear problems arising in elasticity and other related engineering sciences. Relative to linear elliptic theory, little is known about the accuracy and convergence properties of mixed finite element approximation of nonlinear elliptic boundary value problems. The nonlinear problems are much more complicated, since each problem has to be treated individually. This is one of the reasons that there is no unified and general theory for the nonlinear problems. In this paper, the application of the mixed finite element method to a highly nonlinear Dirichlet problem, which arises in the field of oceanography and elasticity is studied and new results involving the error estimates are derived. In fact, some of the results and methods to be described in this paper may be extended to more complicated problems or problems with other boundary conditions. As a special case, we obtain the well known error estimates for the corresponding linear and mildly nonlinear elliptic boundary value problems.     

The effects of surface elasticity and surface tension on the transverse overall elastic behavior of unidirectional nano-composites

The effects of surface elasticity and surface tension on the transverse overall behavior of unidirectional nano-scale fiber-reinforced composites are studied. The interfaces between the nano-fibers and the matrix are regarded as material surfaces described by the Gurtin and Murdoch model. The analysis is based on the equivalent inhomogeneity technique. In this technique, the effective elastic properties of the material are deduced from the analysis of a small cluster of fibers embedded into an infinite plane. All interactions between the inhomogeneities in the cluster are precisely accounted for. The results related to the effects of surface elasticity are compared with those provided by the modified generalized self-consistent method, which only indirectly accounts for the interactions between the inhomogeneities. New results related to the effects of surface tension are presented. Although the approach employed is applicable to all transversely isotropic composites, in this paper we consider only a hexagonal arrangement of circular cylindrical fibers.     

Effects of nanotube helical angle on mechanical properties of carbon nanotube reinforced polymer composites

Carbon nanotubes (CNTs) possess exceptional mechanical properties and are therefore suitable candidates for use as reinforcements in composite materials. The CNTs, however, form complicated shapes and do not usually appear as straight reinforcements when introduced in polymer matrices. This results in a decrease in nanotube effectiveness in enhancing the matrix mechanical properties. In this paper, theory of elasticity of anisotropic materials and finite element method (FEM) are used to investigate the effects of CNT helical angle on effective mechanical properties of nanocomposites. Helical nanotubes with different helical angles are modeled to investigate the effects of nanotube helical angle on nanocomposite effective mechanical properties. In addition, the results of models consisting of helical nanotubes are compared with the effective mechanical properties of nanocomposites reinforced with straight nanotubes. Ultimately, the effects of helical CNT volume fraction on nanocomposite longitudinal modulus are investigated.     

Predicting the mechanical properties of ordinary concrete and nano-silica concrete using micromechanical methods

By combining several materials with specific mechanical properties, new materials with unknown mechanical properties are obtained. Various experiments are required to determine the mechanical properties of the produced composite materials. Since conducting experiment processes is costly and time-consuming, comprehensive studies have been conducted in recent years to solve the problem. Fortunately, it is possible to easily predict the mechanical properties of composite materials without the need to construct them, by inspecting their constituent’s properties using micromechanical methods. Although various micromechanical methods have been presented so far, few of them yielded precise predictions of the properties of composite materials. Therefore, selecting a method suitable to predict the properties of composite materials is of much importance. In this study, some micromechanical approaches, including Hirsch, Hansen, Bache, Cavento, Mori–Tanaka, Eshelby, self-consistent, effective interface and double-inclusion models, were employed for the estimation of elasticity modulus and Poisson’s ratio of ordinary and nanomaterial concretes. The results obtained from the micromechanical methods were compared to those obtained from experimental tests. The obtained numerical results showed that Bache’s model is the most accurate micromechanics model for predicting the elastic mechanical properties of ordinary and nanomaterial concretes.     

A feed-back approach to error control in finite element methods: application to linear elasticity

Recently a refined approach to error control in finite element (FE) discretisations has been proposed, Becker and Rannacher (1995b), (1996), which uses weighted a posteriori error estimates derived via duality arguments. The conventional strategies for mesh refinement in FE models of problems from elasticity theory are mostly based on a posteriori error estimates in the energy norm. Such estimates reflect the approximation properties of the finite element ansatz by local interpolation constants while the stability properties of the continuous model enter through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in cases where the domain consists of very heterogeneous materials and for the computation of local quantities, e.g., point values or contour integrals. This deficiency is cured by using certain local norms of the dual solution directly as weights multiplying the local residuals of the computed solution. In general, these weights have to be evaluated numerically in the course of the refinement process, yielding almost optimal meshes for various kinds of error measures. This feed-back approach is developed here for primal as well as mixed FE discretisations of the fundamental problem in linear elasticity.