Micromechanical Scheme Research Articles

Technique of rock thermal conductivity evaluation on core cuttings and non-consolidated rocks

The paper focuses on the development of a new technique to evaluate thermal conductivity of rock core cuttings and non-consolidated rocks. Very often consolidated cores cannot be recovered even during the well drilling in high porous and/or fractured reservoirs. At the same time reliable data on rock thermal properties are required for basin and petroleum system modeling, hydrodynamic modeling of hydrocarbon producing with thermal methods of enhanced oil recovery, interpretation of temperature logging data, heat flow density determination, etc. The proposed methodology is based on preparation of the synthetic composite specimens consisting of rock particles mixed with material-fillers (wax, water, air), measurements of the thermal conductivity of these specimens and evaluation of the thermal properties of the particles solving inverse problem of homogenization. The thermal conductivity is measured using the optical scanning technique which is a rapid, non-destructive, contactless methodology providing high accuracy (providing evaluation of thermal conductivity with uncertainty not more than ± 6% at confidence level of 0.95). The results of measurements are used as the effective properties in various homogenization schemes to evaluate properties of the inhomogeneities. To validate the approach, we calculated effective thermal properties using Mori-Tanaka-Benveniste and Maxwell micromechanical schemes and compared analytical predictions against experimental data. Our results show a good correspondence between micromechanical approximations and experimental measurements with average absolute error no more than 4%. Inverse homogenization problem was formulated to reconstruct thermal properties of considered rock cuttings from the known properties of remaining constituents and measured effective properties. Approximations based on inverse Mori-Tanaka-Beneveniste and Maxwell schemes are compared against experimental data measured on solid rock specimens. Satisfactory agreement between inverse solutions and experimental data were observed for both MTB and Mawell schemes with with average absolute error no more than 6%. The sensitivity analysis of the results to the shape of inhomogeneities corresponding to the entrapped air was performed.

On micromechanical modeling of orthotropic solids with parallel cracks

A detailed comparison of the accuracy of several popular micromechanical schemes utilized for prediction of the effective elastic properties of materials with parallel cracks is presented. In particular, the non-interaction, Mori–Tanaka, differential and self-consistent schemes are compared against the direct finite element simulations. The latter are performed on the periodic representative volume elements containing 30 strongly oblate spheroids representing the penny-shaped cracks. This work extends the existent results to a more general class of matrix materials – orthotropic materials, which requires the ability to calculate the Eshelby tensor for an ellipsoid in non-isotropic matrix. In addition to the implementation of the integration procedure used for the Eshelby tensor calculation, this work also presents a variation of the Random Sequential Adsorption algorithm modified for periodic structures.Analysis of the results indicates that in the case of parallel nearly flat cracks (strongly oblate spheroids) the overall out-of-plane moduli are best predicted by the differential scheme. On the other hand, Mori–Tanaka scheme should be used for estimation of the in-plane moduli. It also appears that as the cracks are inflated from strongly oblate spheroids to slightly deformed spheres, the best choice of the micromechanical scheme for the out-of-plane properties gradually shifts towards Mori–Tanaka.

Stability analysis of variable stiffness composite laminated cylindrical panels containing delamination

A finite strip method is formulated in order to extract the stability characteristics of laminated composite cylindrical panels containing embedded square delamination region. The material properties are assumed to vary along the panel axial length of any lamina according to a linear fiber orientation variation. The effects of existence of a square delamination region are studied on the stability characteristics of the curved panel. The friction effects, contact conditions, and delamination growth phenomenon at delaminated interfaces and edges are overlooked. The natural frequencies of the panel have been investigated based on two composite micromechanical schemes. The dynamic behavior of the delaminated panel has been extracted through using an enhanced B-spline finite strip method formulated based on the principle of virtual work. A third-order Reddy-type shear deformation theory is utilized in order to bring the effects of moderately thick laminate in to account besides a Koiter–Sanders strain–displacement theory of shallow shells. The effects of material volume fractions, layup, micromechanical model, delamination size, and location on dynamic stability characteristics of the structure are studied. In order to demonstrate the capabilities of the developed method in predicting the structural dynamic behavior, some representing results are obtained and compared with those available in the literature.

Estimating the mechanical properties of hydrating blended cementitious materials: An investigation based on micromechanics

The hydration model of Parrot & Killoh (1984) has been extended to blended cements and coupled to a micromechanical scheme similar to that of Pichler & Hellmich (2011) to estimate the Young modulus and the compressive strength of cementitious materials as a function of time. A finite aspect ratio of 7 is introduced to describe the shape of the hydrates and improve the estimate of the early age strength by the micromechanical scheme. Furthermore, accounting for the stress fluctuations in the cement paste partly explains the fact that the compressive strength of a concrete can be lower than that of its cement paste. Finally, the estimated physical properties are compared to numerous experimental measurements from the literature and new experimental measurements on blended cement pastes featuring significant weight fractions of limestone filler, fly ash or silica fume. It is shown that the present model slightly overestimates the dilution effect.

Axial postbuckling analysis of multilayer functionally graded composite nanoplates reinforced with GPLs based on nonlocal strain gradient theory

In this paper, a new size-dependent inhomogeneous plate model is constructed to analyze the nonlinear buckling and postbuckling characteristics of multilayer functionally graded composite nanoplates reinforced with graphene platelet (GPL) nanofillers under axial compressive load. To this purpose, the nonlocal strain gradient theory of elasticity is implemented into a refined hyperbolic shear deformation plate theory. The mechanical properties of multilayer graphene platelet-reinforced composite (GPLRC) nanoplates are evaluated based upon the Halpin-Tsai micromechanical scheme. The weight fraction of randomly dispersed GPLs remain constant in each individual layer, which results in U-GPLRC nanoplate, or changes layerwise in accordance with three different functionally graded patterns, which make X-GPLRC, O-GPLRC and A-GPLRC nanoplates. Via a two-stepped perturbation technique, explicit analytical expressions for nonlocal strain gradient stability paths are established for layerwise functionally graded GPLRC nanoplates. It is demonstrated that both the nonlocal and strain gradient size dependencies are more significant for multilayer GPLRC nanoplates filling by GPL nanofillers with higher length-to-thickness and width-to-thickness ratios.

Effective thermal expansion coefficient of a sintered glass–eucryptite composite

The paper focuses on a glass–ceramic composite consisting of lead borate glass matrix containing isolated eucryptite granules (hereafter called inhomogeneities) having negative thermal expansion coefficient. We discuss the dependence of the overall thermal expansion on the size of the particles and the temperature of sintering as well as the process of the diffusion of lithium from eucryptite into the glass matrix. The latter leads to the formation of interphase zones between the matrix and inhomogeneities. The overall thermal expansion coefficient is measured experimentally and modeled using various micromechanical schemes. The best agreement for the entire range of variation of the eucryptite volume fraction is provided by Maxwell homogenization scheme.

A nonlocal strain gradient hyperbolic shear deformable shell model for radial postbuckling analysis of functionally graded multilayer GPLRC nanoshells

The present study addresses the size dependency in nonlinear instability of functionally graded multilayer graphene platelet-reinforced composite (GPLRC) nanoshells under hydrostatic pressure including jointly the nonlocal elastic and strain gradients stress fields. For this objective, the new unconventional continuum theory namely as nonlocal strain gradient theory of elasticity is utilized within the framework of a refined hyperbolic shear deformation shell theory. Via stacking up a number of individual layers, the graphene platelet (GPL) nanofillers are distributed uniformly and three different functionally graded patterns based upon a layerwise change of the GPL weight fraction through the shell thickness direction. The effective material properties corresponding to uniform (U-GPLRC) and X-GPLRC, O-GPLRC, A-GPLRC functionally graded patterns of dispersion are extracted by Halpin-Tsai micromechanical scheme. By employing jointly the boundary layer theory of shell buckling and a two-stepped perturbation technique, explicit analytical expressions are achieved for nonlocal strain gradient stability curves of functionally graded multilayer GPLRC nanoshells. It is indicated that by increasing the value of GPL weight fraction for the U-GPLRC and O-GPLRC nanoshells, the significance of the both nonlocal and strain gradient size dependencies reduces, while for the X-GPLRC and A-GPLRC nanoshells, it increases.

Nonlinear instability of axially loaded functionally graded multilayer graphene platelet-reinforced nanoshells based on nonlocal strain gradient elasticity theory

With the aid of a more comprehensive size-dependent continuum elasticity theory, the nonlinear instability of functionally graded multilayer graphene platelet-reinforced composites (GPLRC) nanoshells under axial compressive load is examined. To accomplish this end, the newly proposed theory of elasticity namely as nonlocal strain gradient elasticity theory is implemented into a refined hyperbolic shear deformation shell theory to establish a more accurate size-dependent shell model. The graphene platelets (GPLs) are supposed to be randomly oriented with uniform and three different functionally graded dispersions relevant to each layer as the weight fraction of GPL varies layerwise through the shell thickness direction. In accordance with the Halpin–Tsai micromechanical scheme, the effective material properties are achieved corresponding to uniform (U-GPLRC) and X-GPLRC, O-GPLRC, A-GPLRC functionally graded patterns of dispersion. The boundary layer theory of shell buckling and a two-stepped perturbation solving process are employed jointly to capture explicit analytical expressions for nonlocal strain gradient stability curves of axially loaded functionally graded GPLRC nanoshells. Among different patterns of GPL distribution, it is observed that for both nonlocality and strain gradient size dependencies, the maximum and minimum size effects on the critical buckling loads are corresponding to X-GPLRC and O-GPLRC nanoshells, respectively.

Towards prediction of springback in deep drawing using a micromechanical modeling scheme

Abstract Deep drawing is one of the most commonly used sheet metal forming processes, which can produce metal parts at a high rate. One of the major problems in deep drawing is springback, which is mainly elastic deformation occurring when the tool is removed. The focus of this work is the prediction of springback in deep drawing for DC04 steel using a micromechanical modeling scheme. A novel method is used for the characterization of material that leads to cyclic stress-strain curve. Simulations are performed with the Yoshida Uemori (YU) model for the prediction of springback for a U draw-bend geometry. The maximum deviation between the geometries of experiment and the springback simulation for hat geometry is 2.2 mm. It is shown that this micromechanical modeling scheme allows us to relate the influence of the microstructure to the springback prediction.

Multiscale modelling of hybrid glass fibres reinforced graphene platelets polyamide PA6 matrix composites for crashworthiness applications

This work investigates the crashworthiness response for a hierarchical modelling of hybrid composite material consisting of short-glass fibres reinforced graphene platelets polyamide PA6 matrix. A multiscale approach, using both mean-field homogenisation and finite element FE techniques, is employed to derive the overall response. Graphene is considered as platelets GPL embedded within an elasto plastic matrix phase. The 2-phases composite response is, therefore, computed under the Mori-Tanaka micromechanics scheme by accounting for the GPL spatial orientation. The modelling of hybrid 3-phases composites consists of a hierarchical double-scale approach, which includes the 2-phases GPL/polymer composite and short glass fibres as reinforcements. Numerical characterisations involving tensile, compression, fracture toughness and Charpy impact tests enable the determination of damage/failure thresholds for crashworthiness applications. The full crash box is simulated by implementing the constitutive 3-phases composite using a user-defined Digimat/LS-DYNA linkage. Numerical results, which are compared to those from conventional steel and glass fibres composites, show the contribution of the GPL volume fraction in the improvement of the specific energy absorption SEA.

Constitutive relationship of irradiated metallic materials by Eshelby formalism and micro-mechanical scheme

In this paper, a theoretical framework including the micro-mechanical method and crystal plasticity model is developed to characterize the macroscopic mechanical behaviors of irradiated metallic materials. At the microscale, a rate-dependent crystal plasticity model is applied to capture the effects of irradiation-induced defects and helium (He) bubbles for individual grains. Then, the micro-mechanical method is adopted to establish the relationship between microscopic individual grains and macroscopic polycrystals. The developed theoretical framework is applied to simulate the mechanical behaviors of neutron irradiated polycrystalline Cu and He implanted Cu/Nb nano-metallic-multilayers (NMMs), and the numerical results match well with corresponding experimental data.

Computation of the effective mechanical properties including nonclassical moduli of 2.5D and 3D interlocks by micromechanical approaches

Micromechanical schemes based on the discrete homogenization method are developed to predict the effective mechanical properties of 3D dry textiles, including the bending moduli, and accounting for internal scale effects by the consideration of additional rotational degrees of freedom at the mesoscopic level. The unit of repeat identified for a given interlock is represented as a set of structural beam like elements that defines the representative volume element (RVE), with periodicity boundary conditions. The complex interactions between yarns within the RVE lead to a non affine motion of the yarns responsible for internal scale effects, which are accounted for by a micropolar anisotropic continuum obtained by the discrete homogenization method. The Cosserat moduli are identified together with the associated internal bending lengths versus the microstructural geometrical and mechanical parameters that characterize the internal organization of yarns within each preform. A satisfactory agreement is obtained between the effective classical and nonclassical moduli predicted by homogenization and those computed by finite element simulations.

The Eshelby, Hill, Moment and Concentration Tensors for Ellipsoidal Inhomogeneities in the Newtonian Potential Problem and Linear Elastostatics

One of the most cited papers in Applied Mechanics is the work of Eshelby from 1957 who showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in an unbounded (in all directions) homogeneous isotropic host would feel uniform strains and stresses when uniform strains or tractions are applied in the far-field. Of specific importance is the uniformity of Eshelby’s tensor $\mathbf{S}$ . Following Eshelby’s seminal work, a vast literature has been generated using and developing Eshelby’s result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor $\mathbf{P}=\mathbf{S}\mathbf{D}^{0}$ arises, where $\mathbf{D}^{0}$ is the host medium compliance tensor. The tensor $\mathbf{P}$ is associated with Hill and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc. and in the vector problems of elastostatics. Knowledge of the Hill and Eshelby tensors permit a number of interesting aspects to be studied associated with inhomogeneity problems and more generally for inhomogeneous media. Micromechanical methods established mainly over the last half-century have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill tensor, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases. Of specific interest is that a number of important limits of the ellipsoidal inhomogeneity can be taken in order to be employed in predictions of the effective properties of, for example, layered media and fibre reinforced composites and also to the cases when voids and cracks are present. In the main, results for the Hill and Eshelby tensors are distributed over a wide range of articles and books, using different notation and terminology and so it is often difficult to extract the necessary information for the tensor that one requires. The case of an anisotropic host phase is also frequently non-trivial due to the requirement of the associated Green’s tensor. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled. A scaling argument leads to the derivation of the Eshelby tensor for potential problems where the host phase is at most orthotropic, without the requirement of using the anisotropic Green’s function. The Concentration tensor $\boldsymbol{\mathcal{A}}$ linking interior fields to those imposed in the far-field is derived for a wide variety of problems. These results can therefore be used in the various micromechanical schemes. Directly related to the tensors of Eshelby and Hill is the so-called Moment tensor $\mathbf{M}$ . As well as arising in the literature on micromechanics, this tensor is important in the vast area of research associated with inverse problems and specifically with the problem of identifying an object inside some domain given the application of a specific set of boundary conditions. Due to its fundamental importance and direct link to the Eshelby and Hill tensors, here we state the connection between $\mathbf{M}, \mathbf{P}$ and $\mathbf{S}$ in an effort to ensure that the work is of use to as wide a community as possible. Both tensor and matrix formulations are considered and contrasted throughout. Appendices give various details that illustrate the implementation of both approaches.

Non-linear elastic moduli of Graphene sheet-reinforced polymer composites

• The linear and non-linear elastic moduli of the Graphene sheet are computed based on the modified Morse potential. • The Eshelby’s tensor is obtained for a rectangular aspect ratio. • The non-linear stiffness tensor of the Graphene sheet-reinforced polymer composite is derived using the Mori–Tanaka micro-mechanics scheme. • The anisotropic degree is analysed for different aspect ratio and mass fractions. The non-linear elastic moduli of the Graphene sheet-reinforced polymer composite are investigated using a combined molecular mechanics theory and continuum homogenisation tools. Under uni-axial loading, the linear and non-linear constitutive equations of the Graphene sheet are derived from a Taylor series expansion in powers of strains. Based on the modified Morse potential, the elastic moduli and Poisson’s ratio are obtained for the Graphene sheet leading to the derivation of the non-linear stiffness tensor. For homogenisation purpose, the strain concentration tensor is computed by the means of the irreducible decomposition of the Eshelby’s tensor for an arbitrary domain. Therefore, a mathematical expression of the averaged Eshelby’s tensor for a rectangular shape is obtained for the Graphene sheet. Under the Mori–Tanaka micro-mechanics scheme, the effective non-linear behaviour is predicted for various micro-parameters such as the aspect ratio and mass fractions. Numerical results highlight the effect of such micro-parameters on the anisotropic degree of the composite.

Determination of the effective conductivity of composites with spherical and spheroidal anisotropic particles and imperfect interfaces

The present work aims to develop a micromechanical scheme to estimate the effective conductivity of isotropic particulate multiphase composites with imperfect interfaces between the matrix and inhomogeneity phases. The interface can be either highly conducting or resistive, the inhomogeneities can be spherical or spheroidal, and the matrix phase is isotropic but the inhomogeneity phases can be anisotropic. By applying the generalized Mori–Tanaka (GMT) model based on the morphologically representative pattern (MRP) approach and by using the solution results obtained for auxiliary problems accounting for imperfect interfaces, the closed-form expression for the effective conductivity is obtained. Numerical results are provided to illustrate the dependence of the effective conductivity on the size and shape of inhomogeneities and to compare the estimations with the relevant upper and lower bounds.

Locally-exact homogenization theory for transversely isotropic unidirectional composites

The locally-exact homogenization theory for unidirectional composites with square periodicity and isotropic phases proposed by Drago and Pindera [18] is extended to architectures with hexagonal symmetry and transversely isotropic phases. The theory employs Fourier series representation for the displacement fields in the fiber and matrix phases in the cylindrical coordinate system that satisfies exactly the equilibrium equations and continuity conditions in the unit cell's interior. The inseparable exterior problem involves satisfaction of periodicity conditions for the hexagonal unit cell geometry demonstrated herein to be readily achievable using the previously introduced balanced variational principle for square geometries. This variational principle plays a key role in the employed unit cell solution, ensuring rapid convergence of the Fourier series coefficients with relatively few harmonic terms, yielding converged homogenized moduli and local stress fields with little computational effort. The solution's stability is illustrated using the dilute case which is shown to reduce to the Eshelby solution regardless of the employed number of harmonic terms. Comparison with published results and predictions of a finite-volume based homogenization in a wide fiber volume range and different fiber/matrix modulus contrast validates the approach's accuracy, and its utility is demonstrated through rapid local stress recovery in a multi-scale application. This extension completes the development of the theory for three important classes of unidirectional reinforcement arrays, thereby providing an efficient alternative to finite-element based homogenization techniques or approximate micromechanical schemes, as well as an efficient standard against which other methods may be compared.

Overall thermo-magneto-electro-elastic properties of multiferroics composite materials with arbitrary heterogeneities spatial distributions

This study extends our recent work (Koutsawa, 2014) to multiferroics composite materials with temperature-independent properties. The key ingredient in this development is the computation of the so-called interaction tensors between the multiferroics inhomogeneities which are for the first time presented here for the magneto-electro-elastic coupling. The present mean field based micromechanics model is able to account for the morphological and topological textures of the multiferroics heterogeneities. Numerical examples based on the multiple site Mori–Tanaka model are used to quantify the prediction capabilities of the proposed micromechanics scheme. The predictions of the micromechanics model are compared to those of the variational asymptotic method for unit-cell homogenization (VAMUCH), a finite element based micromechanics approach. There is a good quantitative agreement between the mean field model predictions and numerical results obtained with VAMUCH. The model is then applied to three typical packing arrangements of particulate thermo-magneto-electro-elastic composites, namely, Body Centered Cubic (BCC), Face Centered Cubic (FCC) and Simple Cubic (SC) to demonstrate its capability to account for the topological texture of multiferroics composite materials microstructures.

Interfacial and mechanical properties of epoxy nanocomposites using different multiscale modeling schemes

In this study, we investigate the interfacial and mechanical properties of carbon nanotube (CNT) reinforced epoxy composite. The work carried out in two stages. In the first, we conducted molecular dynamics (MD) simulations to determine the atomic-level interfacial and mechanical properties of the transversely isotropic representative volume element (RVE) comprised of CNT-epoxy composite. In the second, the Mori–Tanaka micromechanics scheme was used to scale up the mechanical properties of the atomic structure to the microscale level. The work was further extended and used atomistic-based continuum (ABC) multiscale modeling technique, which makes use of constitutive relations derived solely from interatomic potentials to model the same system. Interestingly, the results of our comparative investigation reveals that (i) the ABC technique and MD simulation provide almost identical predictions for the atomic-level interfacial and mechanical properties of the nanocomposite, (ii) both models predict comparable bulk mechanical properties of the nanocomposite containing randomly dispersed CNTs, and (iii) they also reveal that a higher degree of orthotropy of the nanoscale representative fiber significantly influences the bulk mechanical properties of the nanocomposite.

Homogenization of fibrous piezoelectric composites with general imperfect interfaces under anti-plane mechanical and in-plane electrical loadings

The present paper aims mainly to estimate the size-dependent effective properties of fibrous piezoelectric composites with general imperfect interfaces. The interface model used states that the displacement, traction, electric potential, and normal electric displacement all suffer jumps across an interface. In addition, it can degenerate into the well-known special ones by employing appropriate high-contrast interfacial parameters. To achieve our objective, an auxiliary inhomogeneity problem of a circular fiber embedded in an infinite cylindrical reference phase via general imperfect interface under anti-plane mechanical and in-plane electrical boundary conditions is analytically solved. This solution allows us to apply the well-known micromechanical schemes such as the dilute, Mori–Tanaka to obtain the closed-form expressions for the size-dependent overall properties of composites under consideration. Some numerical examples are provided for illustrating the features of the obtained general results.

Incremental mean-fields micromechanics scheme for non-linear response of ductile damaged composite materials

This work is concerned with the modeling of ductile damage behavior in composite materials by the means of the Incremental Micromechanics Scheme (IMS) as Mean-Fields Homogenization (MFH) technique. Indeed, IMS is known for its capability to overcome the well-known accuracy restrictions of the Mori–Tanaka (MT) and Self-Consistent (SC) schemes when a high volume fraction of heterogeneities or/and a high contrast between phases properties is reached. This micromechanics formalism is based on the Eshelby’s inclusion concept. The kinematic equation of Dederichs and Zeller (1973) is used as formal solution of the heterogeneous material problem. The nonlinear behavior of the composite is addressed in a general framework based on the kinematic hardening of Lemaıˆtre–Chaboche’s ductile damage model. Thus a classical J2 plasticity that accounts for the damage evolution within the microstructure is implemented. The time discretization of all rate relations is solved through a generalized mid-point rule that yields to an anisotropic consistent (algorithmic) tangent modulus. To avoid a stiffer macroscopic stress–strain response, an isotropization procedure is adopted during the computation of the Eshelby tensor involved in the IMS modeling. From a computational aspect, the non linear response of the composite is obtained through two interdependent loops: inner and outer. In the inner loop, the IMS determines the global strain concentration tensor that is passed to the outer loop. Then, the macroscopic stress–strain response is derived using an iterative algorithm based on the Hill-type incremental formulation. Numerical results are obtained considering several heterogeneous materials such as Metal Matrix Composites (MMCs) as well as Carbon fibers reinforced Epoxy Matrix Composites. The model’s predictions are compared in most of the cases, with experimental data and predictions obtained from MT-based modeling in the open literature.