Exact solution of Eshelby–Christensen problem in gradient elasticity for composites with spherical inclusions

The gradient elasticity theory is employed to solve exactly the problem of Eshelby–Christensen for filled composites with spherical inclusions across length scales. Relying on the fundamental symmetry considerations and using Lagrange’s variational formalism, we derive the governing relations of linear isotropic gradient elasticity. We demonstrate that to avoid spurious solutions, one should necessarily impose some additional symmetry restrictions on the operational strain gradient elastic constants that can be considered as a new correctness condition. By enforcing the “strain gradient” symmetry condition, we offer the variant of the correct applied one-parametric gradient theory of interfacial layer model. To solve the Eshelby–Christensen problem, we employ the generalized Eshelby’s integral representations for the gradient elasticity models that allow to formulate the closing equations in a self-consistent three-phase method, and we also use the generalized Papkovich–Neuber representation to determine the general form of the gradient solution and the structure of the scale effects. As a result, we obtained for the first time an exact solution of Eshelby–Christensen problem for composites reinforced with spherical inclusions in framework of the gradient interfacial layer model. There are known analogs of fundamental results for gradient models related to closed solution for composites with spherical inclusions obtained by R.M. Christensen and K.H. Lo in 1976. The obtained analytical solution of Eshelby–Christensen problem for correct gradient theory is used to determine the stress–strain state and the effective properties of dispersed composites. The analysis of the effect of scale factors is given; the error associated with the use of gradient theories that do not obey the proposed condition of correctness is estimated.     

Effective elastic modulus and micro-structure damage of particle-reinforced composites

A new analysis method of effective elastic modulus for composites has been developed by combining Eshelby’s equivalent inclusion method and self-consistent method. The equations obtained can describe the evolution of debonding damage of the composites with multi-phase particles and single-phase particles. Based on the incremental relation between particles and the matrix, the incremental constitutive relations of composite, matrix, particles and voids have been developed. Numerical analysis has been conducted for Ramburg–Qsgood function incorporating with equivalent elastic modulus obtained. The constitutive equation curves for different particle volume fractions can describe the influence of debonding damage on effective elastic modulus of the composites. Numerical results of the present study have a better agreement with the experimental results.     

Micromechanical analysis of nanocomposites using 3D voxel based material model

A computational study on the effect of nanocomposite structures on the elastic properties is carried out with the use of the 3D voxel based model of materials and the combined Voigt–Reuss method. A hierarchical voxel based model of a material reinforced by an array of exfoliated and intercalated nanoclay platelets surrounded by interphase layers is developed. With this model, the elastic properties of the interphase layer are estimated using the inverse analysis. The effects of aspect ratio, intercalation and orientation of nanoparticles on the elastic properties of the nanocomposites are analyzed. For modeling the damage in nanocomposites with intercalated structures, “four phase” model is suggested, in which the strength of “intrastack interphase” is lower than that of “outer” interphase around the nanoplatelets. Analyzing the effect of nanoreinforcement in the matrix on the failure probability of glass fibers in hybrid (hierarchical) composites, using the micromechanical voxel-based model of nanocomposites, it was observed that the nanoreinforcement in the matrix leads to slightly lower fiber failure probability.     

Three-dimensional elastodynamic solution for an arbitrary thick FGM rectangular plate resting on a two parameter viscoelastic foundation

A three-dimensional semi-analytic analysis based on the linear elasticity theory is offered to study the transient vibration characteristics of an arbitrarily thick, simply supported, functionally graded (FGM) rectangular plate, resting on a linear Winkler–Pasternak viscoelastic foundation, and subjected to general distributed driving forces of arbitrary temporal and spatial variations. The problem solution is obtained by adopting a laminate model in conjunction with the powerful state space solution technique involving a global transfer matrix and Durbin’s numerical Laplace inversion algorithm. Numerical calculations are carried out for the transient displacement and stress responses of aluminum-zirconia FGM square plates of selected thickness parameters and compositional gradients, resting on “soft” or “stiff” elastic foundations, under the action of moving transverse forces as well as uniformly distributed blast loads. Also, the response curves for the FGM plates are compared with those of equivalent bilaminate plates containing comparable total volume fractions of constituent materials. It is observed that the material gradient variation is substantially more influential on the dynamic stress concentrations induced across the plate thickness than on the displacement response of the inhomogeneous plates. In particular, the displacement response of the equivalent bilaminate plates can provide an accurate estimate for prediction of the dynamic response of the corresponding FGM plates, especially for thick plates resting on a stiff foundation. Limiting cases are considered and good agreements with the data available in the literature as well as with the computations made by using a commercial finite element package are obtained.     

3D Frequency domain BEM for solving dipolar gradient elastic problems

A three dimensional (3D) boundary element method (BEM) for treating time harmonic problems in linear elastic structures exhibiting microstructure effects is presented. These microstructural effects are taken into account with the aid of the dipolar gradient elastic theory, which is the simplest dynamic version of Mindlins generalized elastic theory. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The dipolar gradient frequency domain elastodynamic fundamental solution is explicitly derived and used to construct the boundary integral representation of the solution with the aid of a reciprocal integral identity. In addition to a boundary integral representation for the displacement, a boundary integral representation for its normal derivative is also necessary for the complete formulation of a well posed problem. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. The solution procedure is described in detail. A numerical example serves to illustrate the method and demonstrate its accuracy     

Deformation of composites with physically nonlinear matrix

Summary. A method and an algorithm for determining the effective deformational properties of granular material with a physically nonlinear matrix and linearly elastic inclusions are elaborated based on the stochastic differential equations of the physically nonlinear theory of elasticity. Their transformation to integral equations and the application of the method of conditional moments reduce the problem to a system of nonlinear algebraic equations, whose solution is constructed by the iteration method. The deformation diagrams as functions of the volume content of inclusions are investigated.     

On the capability of micromechanics models to capture the auxetic behavior of fibers/particles reinforced composite materials

This work investigates the possibility to predict the auxetic behavior of composites consisting of non-auxetic phases by means of micromechanical models based on Eshelby’s inclusion concept. Two specific microstructures have been considered: (i) the three-layered hollow-cored fibers-reinforced composite and (ii) a microstructure imitating the re-entrant honeycomb micro-architecture. The micromechanical analysis is based on kinematic integral equations as a formal solution of the inhomogeneous material problem. The interaction tensors between the inhomogeneities are computed thanks to the Fourier’s transform. The material anisotropy due to the morphological and topological textures of the inhomogeneities was taken into account thanks to the multi-site approximation of these tensors. In both cases, the numerical results show that auxetic behavior cannot be captured by such models at least in the case of elastic and isotropic phases. This conclusion is supported by corresponding finite element investigations of the second microstructure that indicate that auxetic behavior can be recovered by introducing joints between inclusions. Otherwise, favorable issues are only expected with auxetic components.     

Elastic bending analysis of bilayered beams containing a gradient layer by an alternative two-variable method

Aifantis’s strain gradient elasticity theories and Zhang’s two-variable method are used to study elastic bending problems of bilayered micro-cantilever beams, containing a gradient layer, subjected to a transverse concentrated load. The differential element method is used to obtain differential governing equations. The variational method is employed to overcome the difficulty in deriving nonlocal natural boundary conditions, which could not be automatically fulfilled in gradient theories, not like that in classical theories. Then the differential governing equations subjected to the related boundary conditions are solved analytically to obtain the deformation field, which could be degenerated to that in classical elasticity theories. The gradient parameters of epoxy polymeric resin and copper single crystals in the present model are provided by fitting Lam’s and Demir’s experiments. The influences of length and layer thickness on normalized deflection and effective rigidity are discussed in a representative case of a Cu/epoxy polymeric resin beam. Results show that size effect makes the effective rigidity vary more prominently with shorter beam length or larger layer thickness. For given materials, although size effect exists, classical elasticity theories are still valid in some particular combination of three geometric parameters: beam length, upper and lower layer thicknesses.     

A boundary element method for solving 3D static gradient elastic problems with surface energy

 A boundary element methodology is developed for the static analysis of three-dimensional bodies exhibiting a linear elastic material behavior coupled with microstructural effects. These microstructural effects are taken into account with the aid of a simple strain gradient elastic theory with surface energy. A variational statement is established to determine all possible classical and non-classical (due to gradient with surface energy terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution with surface energy is explicitly derived and used to construct the boundary integral equations of the problem with the aid of the reciprocal theorem valid for the case of gradient elasticity with surface energy. It turns out that for a well posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. All the kernels in the integral equations are explicitly provided. The numerical implementation and solution procedure are provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. Advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved. Two numerical examples are presented to illustrate the method and demonstrate its merits. Received: 9 November 2001 / Accepted: 20 June 2002 The first and third authors gratefully acknowledge the support of the Karatheodory program for basic research offered by the University of Patras.     

Thermoelastic and infrared-thermography methods for surface strains in cracked orthotropic composite materials

Two quantitative thermoelastic strain analysis (TSA) experimental methods are proposed to determine the surface strain fields in mechanically loaded orthotropic materials using the spatial distribution of temperature gradient measured from the surface. Cyclic loadings are applied to orthotropic composite specimens to achieve adiabatic conditions. The small change in surface temperatures that resulted from the change in the elastic strain energy is measured using a high sensitivity infrared (IR) camera that is synchronized with the applied loading. The first method is applied for layered orthotropic composites with a coat layer made of isotropic or in-plane transversely isotropic material. In this case, one material parameter (pre-calibrated from the surface) is required to map the strain invariant to the temperature gradients. The proposed method can be used together with Lekhnitskii’s elasticity solution to quantify the full strain field and determine mixed-mode stress intensity factors (SIFs) for crack tips in composite plates subjected to off-axis loading. The second method is formulated for orthotropic layers without a coat and it requires thermo-mechanical calibrations for two material parameters aligned with the material axes. The virtual crack closure technique (VCCT), Lekhnitskii’s and Savin’s elasticity solutions, and finite element (FE) analyses are used for demonstrations and validations of the second experimental method. The SIFs from the TSA methods are very sensitive to the uncertainty in the location of the crack tip and the unknown inelastic or damage zone size around the crack tip. The two experimental methods are effective in generating the strain fields around notched and other FRP composites.     

Relations between compliances of inhomogeneities having the same shape but different elastic constants

The contributions of inhomogeneities having the same shape but different elastic constants, to the overall elastic properties are interrelated. The utility of these relations lies, in particular, in the possibility to extend available results for pores or rigid inclusions to inhomogeneities of arbitrary elastic properties. The relations are exact for ellipsoids and approximate for non-ellipsoidal shapes. The constructed approximation also constitutes approximate connection between the first Eshelby’s problem (the eigenstrain problem) and the second one (the inhomogeneity problem), for non-ellipsoidal shapes. It also yields approximate formulas for the contribution of a non-ellipsoidal inhomogeneity to effective elastic properties.     

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.     

A Strip-Yield algorithm for the analysis of closure evaluation near the crack tip

“Plasticity-induced crack closure” phenomenon is the leading mechanism of different effects (R-ratio, overload retardation, … ) acting on crack growth rate in many metallic materials. Experimental tests are carried out to quantify the physical phenomenon, while Strip-Yield analytical models have been developed for predicting life of components. In the present work, an additional module to be applied to a Strip-Yield model is proposed in order to derive the strains near the crack tip. Particularly, the module is based on the Westergaard’s elastic complex potential. The presented algorithm allowed us to obtain the correlation between “local compliance” experimental results and the corresponding Strip-Yield analyses. This method can be taken as a semi-analytical procedure for calibrating the constraint factor, i.e., the most delicate parameter for Strip-Yield models.     

Effect of nanotube geometry on the elastic properties of nanocomposites

Many analytical models replace carbon nanotubes with “effective fibers” to bridge the gap between the nano and micro-scales and allow for the calculation of the elastic properties of nanocomposites using micromechanics. Although curvature of nanotubes can have a direct impact on these properties, it is typically ignored. In this work, the nanotube geometry in 3D is included in the calculation of the elastic properties of a modified effective fiber. The strain energy of the nanotube and the effective fiber are calculated using Castligiano’s theorem and constraints imposed by the matrix on the deformation are taken into consideration. Model results are compared to results from archived literature, and a reasonable agreement is observed. Results show that the effect of nanotube curvature on reducing the modulus of the effective fiber is not limited to in-plane curvature but also to curvature in 3D. The impact of the nanotube curvature on the elastic properties of nanocomposites is studied utilizing the modified fiber model and the approach developed by Mori–Tanaka. Analytical results show that for a low weight fraction of nanotubes the effect of curvature seems to be minor and as the weight fraction increases, the effect of nanotube curvature becomes critical.     

On the effective elastic properties of matrix composites: Combining the effective field method and numerical solutions for cell elements with multiple inhomogeneities

The work is devoted to calculation of effective elastic constants of homogeneous materials containing random or regular sets of isolated inclusions. Our approach combines the self-consistent effective field method with the numerical solution of the elasticity problem for a typical cell. The method also allows analysis of detailed elastic fields in the composites. By the numerical solution of the elasticity problem for a cell, integral equations for the stress field are used. Discretization of these equations is carried out by Gaussian approximating functions. For such functions, elements of the matrix of the discretized problem are calculated in explicit analytical forms. If the lattice of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure. The matrix-vector products with such matrices may be carried out by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Results are given for 2D-media with regular and random sets of circular inclusions, and compared with existing exact solutions.     

An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain‐gradient theory of elasticity

An advanced boundary element method (BEM) for solving two‐ (2D) and three‐dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form‐II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Ultra-strength materials

Recent experiments on nanostructured materials, such as nanoparticles, nanowires, nanotubes, nanopillars, thin films, and nanocrystals have revealed a host of “ultra-strength” phenomena, defined by stresses in a material component generally rising up to a significant fraction of its ideal strength - the highest achievable stress of a defect-free crystal at zero temperature. While conventional materials deform or fracture at sample-wide stresses far below the ideal strength, rapid development of nanotechnology has brought about a need to understand ultra-strength phenomena, as nanoscale materials apparently have a larger dynamic range of sustainable stress (“strength”) than conventional materials. Ultra-strength phenomena not only have to do with the shape stability and deformation kinetics of a component, but also the tuning of its physical and chemical properties by stress. Reaching ultra-strength enables “elastic strain engineering”, where by controlling the elastic strain field one achieves desired electronic, magnetic, optical, phononic, catalytic, etc. properties in the component, imparting a new meaning to Feynman’s statement “there’s plenty of room at the bottom”. This article presents an overview of the principal deformation mechanisms of ultra-strength materials. The fundamental defect processes that initiate and sustain plastic flow and fracture are described, and the mechanics and physics of both displacive and diffusive mechanisms are reviewed. The effects of temperature, strain rate and sample size are discussed. Important unresolved issues are identified.     

Creep behavior of pultruded GFRP elements – Part 2: Analytical study

This paper presents an analytical study about the viscoelastic time-dependent (creep) behavior of pultruded GFRP elements made of polyester and E-glass fibers. Experimental results reported in Part 1 are firstly used for material characterization by means of empirical and phenomenological formulations – a good general agreement is obtained using the following analytical models: (i) Findley’s power law, (ii) Bruger–Kelvin model and (iii) Prony–Dirichlet series. Based on accelerated characterization methodology – Time-Stress Superposition Principle (TSSP) coupled with Findley’s law, for a reference stress of 20% of the material ultimate stress, an elastic deformation increase of 30% is obtained after 50,000 h. The creep parameters and deformation estimated by using the Findley’s model derivations indicate a consistent prediction of time-dependent deformation and viscoelastic properties of the two types of elements analysed – laminates and beam. A straightforward formulation to predict the time-dependent elastic modulus is applied, showing that the flexural stiffness should be reduced by 25% of its initial value after 1-year and as much as 50% after 50-years. Similarly, the power law coupled to Euler’s classical beam theory suggests a reasonable adaptability to the creep phenomenon in the linear regime and proved to provide accurate predictions for deflections under flexural loading up to 40% of the ultimate strength. After 50 years, under normal service load level (1/3 of the failure load), the total creep deflection will attain almost twice the initial deflection. If taking into account the shear deformation (Timoshenko’s postulated) of the full-size element with “effective” stiffness properties such estimate is reduced nearly 25%.     

A fundamental solution for SH-waves in a class of inhomogeneous anisotropic media

In the present work we derive a fundamental solution for SH waves in a class of inhomogeneous anisotropic media. The derivation is accomplished in terms of a “transmutation” formula. The time-harmonic fundamental solution is also obtained and a non-propagating (‘evanescent’) mode is identified for sources with lower frequencies than the critical one. We show that the same transmutation formula can be used to derive the fundamental solution for inhomogeneous media with linear velocity variation as presented by Watanabe and Payton [K. Watanabe, R.G. Payton, Green’s function and its non-wave nature for SH-wave in inhomogeneous elastic solid, Int. J. Eng. Sci. 42 (2004) 2087-2106].     

Size-dependent effective dynamic properties of unidirectional nanocomposites with interface energy effects

This article studies the size effect on wave propagation characteristics of plane longitudinal and transverse elastic waves in a two-phase nanocomposite consisting of transversely isotropic and unidirectionally oriented identical cylindrical nanofibers embedded in a transversely isotropic homogeneous matrix. The surface elasticity theory is employed to incorporate the interfacial stress effects. The effect of random distribution of nanofibers in the composite medium is taken into account via a generalized self-consistent multiple scattering model. The phase velocities and attenuations of longitudinal and shear waves along with the associated dynamic effective elastic constants are calculated for a wide range of frequencies and fiber concentrations. The numerical results reveal that interface elasticity at nanometer length scales can significantly alter the overall dynamic mechanical properties of nanofiber-reinforced composites. Limiting cases are considered and excellent agreements with solutions available in the literature have been obtained.