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Abstract
Based on the new structure recently established by Weng for the Mori-Tanaka theory, the effective elastic moduli of three types of composite containing transversely isotropic spheroidal inclusions are explicitly derived. For a multiphase composites with aligned, identically shaped inclusions, the derived moduli are believed to be generally reliable, where the three extreme cases involving circular fibers, spheres, and thin discs all lie on or within the respective Hashin-Shtrikman-Walpole bounds. For a multiphase aligned composite whose inclusion phases differ in shape, the M-T moduli tensor can lose its diagonal symmetry, which, for a hybrid composite containing fibers and another aligned spheroids, is found to be severest when the spheroids take the shape of thin discs, and tends to decrease as their aspect ratio increases. When the transversely isotropic spheroidal inclusions are randomly oriented in an isotropic matrix, the M-T moduli with spherical inclusions are shown to always lie on or within the isotropic Hashin-Shtrikman-Walpole bounds. Such a desired property however is not always assured with other inclusion shapes, where the needle and disc-like inclusions may cause the M-T moduli and Walpole's self-consistent estimates to lie outside the H-S-W bounds.
... K g is the bulk modulus of glass fiber. For transversely isotropic composites, the compliances tensor M can be written as [52] The longitudinal Young's modulus E 11 and Poisson's ratio 12 are given by [53] where k, l, n, m, p are Hill's notation [53]. ...
... As we shall deal with transversely isotropic materials, the stiffness tensors L can be written as [52]: ...
... When the glass fibers are randomly oriented in 3D space, the GRC system as a composite is macroscopically isotropic. Therefore, the tensor L can be written as [52] where Substituting Eq. (28) into Eq. (32), we obtain: ...
As a building material with economic potential and various beneficial properties, glass fiber reinforced concrete (GRC) has been rapidly promoted and applied in the field of civil construction. The coefficient of thermal expansion (CTE) of fiber-reinforced composites plays a crucial role in the design and analysis of such composites. Therefore, the thermal expansion coefficient of glass fiber reinforced concrete is the focus of this paper. A micromechanics model based on the Mori–Tanaka method is proposed to predict the overall thermal expansion coefficient of GRC. The model considers concrete as a two-phase composite material composed of cement mortar and aggregate. In the derivation process, concrete is treated as the equivalent matrix while glass fiber is treated as the inclusion phase. The present analysis takes into account the effects of glass fiber aspect ratio and reinforcement direction, as well as the volume fractions of aggregate and glass fiber. It is found that the glass fiber aspect ratio and reinforcement direction have a significant effect on the thermal expansion coefficient of concrete. The advantage of this model is that it provides a simple and accurate approach for predicting the CTE of glass fiber reinforced concrete. The developed model is validated by comparison with theoretical and experimental data in the literature.
... But on the other Unified mean-field modeling of viscous short-fiber suspensions and solid short-fiber reinforced composites hand, the symmetries of the effective stiffness tensor based on the Mori-Tanaka method are not ensured generally, as summarized by Kehrer et al. [47]. The symmetries are preserved for two-phase composites [37,48] and for multiphase composites with similar shaped and aligned inclusions [49,50]. The symmetries might be lost for differently shaped inclusions within multiphase composites [49][50][51]. ...
... The symmetries are preserved for two-phase composites [37,48] and for multiphase composites with similar shaped and aligned inclusions [49,50]. The symmetries might be lost for differently shaped inclusions within multiphase composites [49][50][51]. In addition, differently shaped anisotropic inclusions embedded in an isotropic matrix of a two-phase composite may violate the Hashin-Shtrikman bounds [50]. ...
... The symmetries might be lost for differently shaped inclusions within multiphase composites [49][50][51]. In addition, differently shaped anisotropic inclusions embedded in an isotropic matrix of a two-phase composite may violate the Hashin-Shtrikman bounds [50]. Kanaun and Levin [46] address the drawback of violated index symmetry regarding the Mori-Tanaka method and present a version of the above-mentioned effective field method avoiding this disadvantage. ...
Mean-field homogenization is an established and computationally efficient method estimating the effective linear elastic behavior of composites. In view of short-fiber reinforced materials, it is important to homogenize consistently during process simulation. This paper aims to comprehensively reflect and expand the basics of mean-field homogenization of anisotropic linear viscous properties and to show the parallelism to the anisotropic linear elastic properties. In particular, the Hill–Mandel condition, which is generally independent of a specific material behavior, is revisited in the context of boundary conditions for viscous suspensions. This study is limited to isothermal conditions, linear viscous and incompressible fiber suspensions and to linear elastic solid composites, both of which consisting of isotropic phases with phase-wise constant properties. In the context of homogenization of viscous properties, the fibers are considered as rigid bodies. Based on a chosen fiber orientation state, different mean-field models are compared with each other, all of which are formulated with respect to orientation averaging. Within a consistent mean-field modeling for both fluid suspensions and solid composites, all considered methods can be recommended to be applied for fiber volume fractions up to 10%. With respect to larger, industrial-relevant, fiber volume fractions up to 20%, the (two-step) Mori–Tanaka model and the lower Hashin–Shtrikman bound are well suited.
... The interphase is assumed to be isotopically elastic for simplicity. The temperature-dependent elastic moduli tensor of the coated graphene, L c , surrounded by an ultrathin creep-degraded interphase can be evaluated through the MT method (Qiu and Weng, 1990) L c (T) = L int (T) ...
... In the macro-scale geometrical setting, the overall graphene-based nanocomposite is constituted by the coated graphene nanoplatelets and the creep-damaged viscoplastic matrix. The temperature-dependent TD transformed bulk and shear moduli of the overall nanocomposite are written through the MT method (Qiu and Weng, 1990) ...
... in which S 1111 , S 2222 , S 3333 , S 2233 , S 3322 , S 2211 , S 3311 , S 1122 , S 1133 , S 2323 , S 1212 and S 1313 denote the components of Eshelby tensor for coated graphene embedded inside the temperature-dependent creep-degraded matrix (Qiu and Weng, 1990); ...
The ambient temperature at which creep deformation takes place is known to exert significant influence on the creep durability of graphene-based nanocomposites, but at present no theory could illustrate the underlying microstructural evolution to predict such a phenomenon. In this paper, a novel dual thermodynamics approach in conjunction with a time-temperature super-position principle (TTSP) is established to address this issue. First, temperature-dependent secant moduli are exclusively adopted as the unique homogenization variables shifted through TTSP, and then the time-and temperature-dependent effective stresses inside the matrix are calculated via the principle of equivalent work rate combined with the field-fluctuation method. Next, the dual irreversible thermodynamic processes, including the stress-induced creep damage inside the matrix and the temperature-dependent degradation at the interphase, are introduced through two independent sets of evolution equations. The predicted creep rupture strain and rupture time are calibrated with experiments over a wide range of temperature. It is demonstrated that the creep rupture strain increases with the rise of ambient temperature, while the creep rupture time decreases with it. The onset of creep-damage process also commences earlier at higher temperature. This research can provide a design guidance to assess the damage process and failure of low-dimensional nanocomposites at elevated temperature environment.
... To this end a bent composite thin plate is first constructed to describe the bending deformation and determine the volume change of the pressure sensor. In this step, Young's modulus of the sensor as a function of bending deformation and applied force will be derived, and we invoke the Mori-Tanaka method [25] and Qiu-Weng's theory [26] for an isotropic composite containing transversely isotropic randomly oriented ellipsoidal inclusions. Then the volume fraction changes of graphene and CNT phases during the bending process will be estimated. ...
... So, the issue can be treated as randomly oriented ellipsoidal inclusions in an isotropic matrix. By means of Mori-Tanaka's method [25], Qiu and Weng [26] developed an orientational scheme to calculate the effective elastic moduli of such composites. The effective moduli tensor of the entire three-phase composite holds the following expression ...
... The elastic moduli and electrical conductivity are selected as the two unique homogenization parameters for this electromechanically coupled homogenization model. The straindependent elastic moduli and conductivity of coated MWCNTs with loading-dependent tunneling effect, L c and χ (c) i (ε x ), are evaluated through the Mori-Tanaka method [44] L c = L int ...
... Similarly, the strain-dependent bulk and shear moduli of MWCNT/ epoxy nanocomposite sensors, κ mod e (ε x ) and μ mod e (ε x ), can be calculated at a specific strain loading via the Mori-Tanaka method [44] κ mod e (ε x ) = ...
Remarkable discrepancy between the tensile and compressive sensing performance has been experimentally observed in highly sensitive strain sensors, but at present no existing micromechanics-based theory can explain such a difference. To this end, we will investigate the tensile and compressive gauge factors of MWCNT/epoxy nanocomposite sensors in a unified fashion via an electromechanically coupled homogenization scheme. The discrepancy between the tensile and compressive sensing performance is analytically revealed by the loading-dependent tunneling effect assisted with a unified strain-dependent tunneling distance between the adjacent MWCNTs. The predicted gauge factor and resistance change ratio of MWCNT/epoxy nanocomposite sensors are calibrated with two independent sets of experimental data ranging from the compressive to tensile loadings. The tensile gauge factor of MWCNT/epoxy nanocomposite sensors is revealed to be higher than the corresponding compressive one. The absolute values of tensile and compressive resistance change ratio both enhance with the MWCNT waviness, the strain loading and the height of potential barrier between adjacent MWCNTs. In addition, a low MWCNT volume concentration and MWCNT aspect ratio can attain the high tensile and compressive gauge factors for MWCNT/epoxy nanocomposite sensors. The present research can contribute to the accurate design, fabrication, and optimization of highly sensitive strain sensors.
... Considering concrete as a multiphase composite material, several researchers (Kuster and Toksöz, 1974;Qiu and Weng, 1990;Yaman, 2002;Yaman et al., 2002;Wang and LI, 2005; have applied the equivalent inclusion theory of mesomechanics (equivalent representative body, as shown in Figure 3), considering the different factors discussed in the calculation formula of the concrete elastic modulus. ...
... Based on the Mori-Tanaka theory (Qiu and Weng, 1990), Wang et al. considered the effects of water viscosity, late hydration of cement, and pore geometry on the elastic modulus of wet concrete. They used the modified differential decomposition method of mesomechanics to decompose unsaturated concrete into saturated concrete equivalent medium and dry pores and assumed that the elastic properties of the unsaturated concrete depended on the water content, pore geometry, and elasticity of the matrix concrete. ...
The service performance of concrete structures in a water environment differs from that in a normal environment. An accurate evaluation of the mechanical properties and service status of wet concrete is related to the reliable design and safe operation of concrete structures, e.g., hydraulics, marine engineering, bridges, and tunnels. To promote the application of new and high-performance concrete to complex water environments and grasp the future development trend, the research progress on the service performance of concrete in water environments was reviewed worldwide. Starting from the internal water content of concrete, the existing research is combed, the influence of water content, water pressure, and loading rate on the static and dynamic characteristics of concrete in a water environment is summarized, and the influence mechanism is analyzed. The literature review demonstrates that the static compressive strength of wet concrete is lower than that of dry concrete; however, the elastic modulus improves. With an increase in the strain rate, the compressive strength of wet and dry concretes improves, and the rate sensitivity of the wet concrete is greater than that of the normal concrete. Pore structure characteristics mainly affect the static strength of the wet concrete. The improvement in the dynamic strength of the wet concrete is caused by the combined effect of the concrete rate sensitivity, “Stefan” effect, and water pressure.
... Weng (Weng 1984(Weng , 1990) a remanié l'estimateur MT sous une forme identique à la structure des bornes de Hashin-Shtrikman. Avec Qiu (Qiu et Weng 1990) l'a appliqué à des inclusions sphéroïdales multiples isotropes transverses (alignées et orientées aléatoirement) aboutissant à un tenseur effectif explicite mais de forme complexe exprimé selon une écriture proposé par Walpole (Walpole 1966a) (qui faciliterait l'identification de la symétrie matérielle, par exemple écrire le tenseur dans une base conforme à l'écriture de Hill pour les matériaux isotropes transverses). (Chen, Dvorak, et Benveniste 1992) ont appliqué MT pour un composite renforcé avec des fibres/plaquettes alignées ou aléatoirement orientées. ...
... (Chen, Dvorak, et Benveniste 1992) ont appliqué MT pour un composite renforcé avec des fibres/plaquettes alignées ou aléatoirement orientées. Ils ont obtenu un tenseur de rigidité effectif difficile à interpréter en termes des cinq constantes indépendantes de l'IT comme l'ont montré (Qiu et Weng 1990). (Benveniste, Dvorak, et Chen 1991) ont examiné les tenseurs effectifs donnés par les trois estimateurs (Schéma dilue, MT et SCS). ...
Ce travail porte sur le développement d’un nouvel estimateur à champs moyens de type Eshelby, appelé GEEE (General Explicit Eshelby type Estimator), pour évaluer les tenseurs effectifs de rigidité, de conductivité thermique et de dilatation thermique d’inclusions ellipsoïdales multicouches et multiphases, présentant un nombre de couches et des symétries matérielles quelconques. Cet estimateur se présente sous la forme d’une équation tensorielle explicite pour une inclusion ellipsoïdale à deux phases. Il se généralise aux inclusions à N phases (N>2) par une procédure récursive de balayage et remplacement. Pour valider numériquement GEEE et montrer ses capacités prédictives, il a été comparé à plusieurs estimateurs micromécaniques à champs moyens classiques de la littérature, de types Eshelby et non Eshelby, implémentés dans Matlab au même titre que GEEE. Ce dernier a également été comparé à des calculs d’homogénéisation par éléments finis à champs complets, réalisés sur des composites renforcés avec des inclusions enrobées, distribuées selon un arrangement cubique ou hexagonal. A l’occasion des simulations numériques effectuées, les effets du contraste entre phases, de leurs fractions volumiques, et de leurs rapports de forme ont été examinés. Enfin, la capacité prédictive de GEEE en élasticité, thermique et thermoélasticité a été démontrée en reproduisant correctement des mesures expérimentales auxquelles il a été confronté.
... The CNT nanofiller is regarded as a high aspect ellipsoid. Taking the original CNT nanofillers to be homogeneously dispersed and randomly oriented in the creep-damaged matrix, the effective bulk and shear moduli in the transformed domain can be written in terms of the formulae derived by Qiu and Weng (1990) ...
... 1212 ) -in Hill and Walpole notations -is the Eshelby S-tensor of CNT nanofiller surrounded by the interphase, which can be obtained from the literature (Qiu and Weng, 1990); c int is the volume concentration of the ultrathin interphase in the coated CNT nanofiller ...
Effective viscoplastic creep strain Creep damage and creep rupture Irreversible thermodynamics A B S T R A C T While the linear viscoelastic characteristics of polymer composites have been extensively studied, the nonlinear creep with progressive damage leading to the ultimate creep rupture in CNT-based viscoplastic nanocomposites still remains a challenging issue. In this paper, a unified approach is presented to study the whole lifetime of creep process including deformation, damage, and creep rupture. First, based on the concept of secant viscosity, a linear viscoelastic comparison composite is introduced to mimic the nonlinear viscoplastic nanocomposite, with the linear part further formulated in the Laplace transformed domain. Then, both progressive degradation of the interphase and creep damage of the polymer matrix are presented. A joint field-fluctuation method and work-rate equivalence is subsequently utilized to calculate the effective Mises stress and hydrostatic stress of the viscoplastic matrix. The principle of irreversible thermody-namics is then invoked to derive the thermodynamic driving force to characterize the evolution of creep damage and rupture process. After the theory is developed, it is demonstrated that the predicted effective creep strains of multi-walled CNT/polypropylene nanocomposites are calibrated by the experiments spanning over the whole three stages of primary creep, secondary creep, and tertiary creep. Both creep rupture time and rupture strain of the overall nanocomposite are found to increase with CNT volume concentration. It is concluded that population of CNT nanofillers can remarkably enhance the creep rupture resistance of CNT/polymer viscoplastic nanocomposites.
... The CNT nanofiller is regarded as a high aspect ellipsoid. Taking the original CNT nanofillers to be homogeneously dispersed and randomly oriented in the creep-damaged matrix, the effective bulk and shear moduli in the transformed domain can be written in terms of the formulae derived by Qiu and Weng (1990) ...
... 1212 ) -in Hill and Walpole notations -is the Eshelby S-tensor of CNT nanofiller surrounded by the interphase, which can be obtained from the literature (Qiu and Weng, 1990); c int is the volume concentration of the ultrathin interphase in the coated CNT nanofiller ...
While the linear viscoelastic characteristics of polymer composites have been extensively studied, the nonlinear creep with progressive damage leading to the ultimate creep rupture in CNT-based viscoplastic nanocomposites still remains a challenging issue. In this paper, a unified approach is presented to study the whole lifetime of creep process including deformation, damage, and creep rupture. First, based on the concept of secant viscosity, a linear viscoelastic comparison composite is introduced to mimic the nonlinear viscoplastic nanocomposite, with the linear part further formulated in the Laplace transformed domain. Then, both progressive degradation of the interphase and creep damage of the polymer matrix are presented. A joint field-fluctuation method and work-rate equivalence is subsequently utilized to calculate the effective Mises stress and hydrostatic stress of the viscoplastic matrix. The principle of irreversible thermodynamics is then invoked to derive the thermodynamic driving force to characterize the evolution of creep damage and rupture process. After the theory is developed, it is demonstrated that the predicted effective creep strains of multi-walled CNT/polypropylene nanocomposites are calibrated by the experiments spanning over the whole three stages of primary creep, secondary creep, and tertiary creep. Both creep rupture time and rupture strain of the overall nanocomposite are found to increase with CNT volume concentration. It is concluded that population of CNT nanofillers can remarkably enhance the creep rupture resistance of CNT/polymer viscoplastic nanocomposites.
... Then, the coated MWCNT is constituted by the hollow MWCNT and ultrathin interphase with decoration-dependent interface effects. The EM properties of coated MWCNT are derived via the Mori-Tanaka method (Qiu and Weng, 1990) with the decoration-dependent functional interface effects in Eqs. (5), (8) and (10) ...
... The overall viscoelastic MWCNT/polyethylene nanocomposite stress sensor is composed by the polymer matrix and the coated MWCNTs while considering the timeand stress-dependent interface effects. The effective complex bulk and shear moduli of overall nanocomposite sensor, κ TD e and μ TD e , are obtained via the Mori-Tanaka method [43] ...
... The explicit expressions of S (σ ) int are given as [57] S (σ ) 1111 ...
The sensitivity of randomly distributed multi-walled carbon nanotube (MWCNT)/polymer nanocomposite sensors has been extensively investigated, while that of aligned nanocomposite sensors is to be explored. In this paper, a novel hybrid computational scheme is presented for the transversely isotropic sensing behaviors of aligned MWCNT/polymer nanocomposite sensors combining micromechanics and finite element simulations. Specifically, the strain-dependent tunneling effect and multi-scale simulation of underlying heterogeneous microstructures are analyzed cooperatively to quantitatively illustrate the electromechanical coupling phenomenon of strain sensors. First, the effective elastic and electric properties of coated MWCNT with the weak interface connection are calculated by the Mori–Tanaka method on the microscopic scale. Then, the mesoscopic representative volume element (RVE) is established by coated MWCNTs as inclusions and polymer as the matrix. The tunneling effect and electric damage process are implemented with a proposed strain-dependent tunneling distance. Next, the macroscopic strain sensing behaviors of homogenized RVEs are evaluated with finite element simulations. The predicted resistance change ratio and sensitivity of aligned MWCNT/polymer nanocomposite sensors are both consistent with the experimental data over a wide loading range. This research has demonstrated the high sensing performance of aligned MWCNT/polymer nanocomposite sensors along the preferred direction.
... Benvenitne et al. [184] showed that a predicted effective stiffness tensor satisfies the symmetric condition only for a two-phase (matrix and inhomogeneity) composite and a multi-phase composite whose inclusions have a similar shape and the same orientation. Similar findings were also mentioned in the literature [185][186][187]. To overcome this problem, a stepping scheme was proposed by Yang Q-S et al. [188] and this method can be applied to n number of inclusions and can be extended to any micromechanical approaches. ...
The influence of void-type manufacturing defects on the mechanical properties of textile composites was investigated both by experimental characterization and by multiscale modeling. In particular, voids characteristics such as not only void volume fraction but also its size, shape, and distribution have been characterized for textile composites and their effect on the mechanical properties have been analyzed. Several textile composite plates were fabricated by the resin transfer molding (RTM) process where 3D interlock glass textile reinforcement was impregnated by epoxy resin under a constant injection pressure to generate different types of voids. A series of mechanical tests were performed to examine the dependency of tensile modulus and strength of composites on the total void volume fraction, intra & inter-yarn void volume fraction, and their geometrical characteristics. Microscopy observations were performed to obtain the local information about fibers (diameter and distribution), and intra-yarn voids (radius, aspect ratio and distribution). Based on these results, a novel algorithm was proposed to generate the statistically equivalent representative volume element (RVE) containing voids. Moreover, the effect of void morphology, diameter and spatial distribution (homogeneous, random and clustering) on the homogenized properties of the yarns was also investigated by the finite element method. X-ray micro-computed tomography was employed to extract the real meso-scale geometry and inter-yarn voids. Subsequently, this data was utilized to create a numerical model at meso-scale RVE and used to predict the elastic properties of composites containing voids. A parametric study using a multiscale numerical method was proposed to investigate the effect of each void characteristic, i.e. volume fraction, size, shape, distribution, and location on the elastic properties of composites. Thus, the proposed multiscale method allows establishing a correlation between the void defects at different scales and the mechanical properties of textile composites.
... Another shortcoming of the MTM was revealed by application of the method to media with several different types of heterogeneities, e.g., cracks and inclusions or even two crack families of different orientations. It was noted by many authors (see e.g., Qui and Weng 1990;Ferrari 1991;Liu and Huang 2014) that application of the MTM to such "hybrid" composites results in tensors of effective elastic stiffness that are not symmetric with respect to pairs of tensor indices. The latter is in contradiction with some basic principles of thermodynamics. ...
Anisotropic elastic materials containing multiple elliptical cracks of different orientation distributions are considered using the (self-consistent) effective field method. The method allows, in particular, accounting for mutual positions of cracks. The derived effective elastic stiffness tensor applies to a broad range of the crack density. Group velocity surfaces (wave surfaces) of cracked materials are constructed. It is shown that, for the isotropic materials with parallel penny shaped cracks, the wave surfaces are close to ellipsoidal. A method of solution of the inverse problem and determination of fracture parameters from acoustical data (group velocities of quasi-longitudinal and quasi-transverse waves of various directions in damaged rock materials) is proposed. Examples of application of the method are presented, and its accuracy is assessed. The results can be used for determination of crack density in rock materials by acoustical methods.
... To determine the effective elastic properties of the CNT fiber reinforced laminate, various micromechanical models are available in literature. Among them, Mori-Tanaka (Qiu and Weng 1990;Arani et al. 2011;Liu and Huang 2014) and rule of mixture (ROM) (Shen 2009;Rafiee et al. 2013;Kiani 2016a) are extensively used by researchers. In this study, the effective properties of the CNT reinforced composite are evaluated using Mori-Tanaka (MT) model, because of its many advantages over the rule of mixture (ROM). ...
When lateral and direct shear loads arise on precast concrete sandwich panels, the structural response depends on the ability of the panel to develop composite action. The wythe’s span and thickness, concrete mix constituents and strength, insulation and nature of the shear connectors all play a significant role in determining the load capacity and failure modes. This chapter will report on thermally conductive and non-conductive shear connectors and how they interact with the wythes and insulation in providing load resistance to both flexural and direct shear loading. In particular, the post-cracking behaviour prior to pull-out, shear or buckling failure is observed to vary considerably and needs to be understood in order to successfully design systems that are fit for purpose.
... To determine the effective elastic properties of the CNT fiber reinforced laminate, various micromechanical models are available in literature. Among them, Mori-Tanaka (Qiu and Weng 1990;Arani et al. 2011;Liu and Huang 2014) and rule of mixture (ROM) (Shen 2009;Rafiee et al. 2013;Kiani 2016a) are extensively used by researchers. In this study, the effective properties of the CNT reinforced composite are evaluated using Mori-Tanaka (MT) model, because of its many advantages over the rule of mixture (ROM). ...
Buckling analysis of thick isotropic plates subjected to uniaxial and biaxial in-plane forces is presented using the 5th order shear deformation theory. The 5th order shear deformation theory considers both transverse shear deformation and transverse normal strain deformation effects. The assumed displacement field accounts for non-linear variation of in-plane displacements as well as transverse displacement through the plate thickness. The condition of zero transverse shear stresses on the upper and lower surface of plate is satisfied. Hence, the present formulation does not require any shear correction factor generally associated with the first order shear deformation theory (FSDT). Numerical results for buckling analysis include the effects of side to thickness ratio and plate aspect ratio for simply supported isotropic plates. The results of present theory are compared with classical plate theory (CPT), first order shear deformation theory (FSDT), higher order shear deformation theory (HSDT) and trigonometric shear deformation theory (TSDT).
... For the sake of simplicity, the CNT is assumed as an equivalent solid cylinder fiber (Kundalwal, 2018). Thus, the Eshelby tensor for the cylindrical inclusion of carbon fiber and CNT in the polymer matrix can be explicitly written as (Qiu and Weng, 1990): ...
The present work deals with the study of active constrained layer damping treatment of multiscale carbon nanotube-based hybrid carbon fiber-reinforced composite plates. The distinctive feature of novel multiscale hybrid carbon fiber-reinforced composites is that the wavy/straight carbon nanotubes are distributed uniformly in the matrix phase of hybrid carbon fiber-reinforced composites, and the waviness of carbon nanotubes is considered to be coplanar with two mutually orthogonal planes. Firstly, the effective elastic properties of hybrid carbon fiber-reinforced composites are estimated using the Mori-Tanaka method. The outcomes of the Mori-Tanaka method suggest that the transverse effective elastic properties of hybrid carbon fiber-reinforced composites containing wavy carbon nanotubes are better than those of hybrid carbon fiber-reinforced composites with straight carbon nanotubes. Secondly, a finite element model using the layer-wise first-order shear deformation theory is developed to study the damping performance of hybrid carbon fiber-reinforced composite plates integrated with the active constrained layer damping treatment. The constraining layer of active constrained layer damping treatment is a 1–3 piezoelectric composite layer with vertically/obliquely-oriented piezo-fibers. The piezo-fibers make an angle [Formula: see text] with the vertical plane of a layer coplanar with the xz- or yz-plane. The effect of the orientation of piezo-fibers angle on the control authority of active constrained layer damping treatment patches is investigated. Our results reveal that the performance of multiscale hybrid carbon fiber-reinforced composite plates has improved due to the incorporation of wavy carbon nanotubes, and the orientation angle of piezo-fibers has a major impact on the control authority of active constrained layer damping patches. The proposed multiscale composite with the combination of wavy carbon nanotubes and active constrained layer damping patches can be actively used in the aerospace and transport industries to control the mechanical vibrations of structures.
... Each coefficient of B is a function of all five coefficients of A. Taking the orientation average (Advani and Tucker III, 1987) of B, e.g., using Eq. (16), leads to (Benveniste, 1987) Various aspects of the orientation averaging Mori-Tanaka approximation are discussed in Brylka (2017), Weng (1990) and Qiu and Weng (1990). Nevertheless, a comprehensive summary of the basic equations for the special case of homogeneous fiber lengths, i.e., an isotropic fiber length distribution, is given hereafter. ...
A comprehensive study on the influence of planar fourth-order fiber orientation tensors on effective linear elastic stiffnesses predicted by orientation averaging mean field homogenization is given. Fiber orientation states of sheet molding compound (SMC) are identified to be in most cases approximately planar. In the planar case, all possible fourth-order fiber orientation tensors are given by a minimal invariant set of structurally differing planar fourth-order fiber orientation tensors. This set defines a three-dimensional body and forms the basis for a comprehensive study on the influence of a fiber orientation distribution in terms of a fourth-order tensor on homogenized stiffnesses. The methodology of this study is the main contribution of this work and can be adopted to analyse the orientation dependence of any quantity which is a function of a planar fourth-order fiber orientation tensor. At specific points inside the set of planar fiber orientation tensors, effective stiffnesses are calculated with selected mean field homogenization schemes. These schemes are based on orientation averaging of transversely isotropic elasticity tensors following Advani and Tucker (1987), which is explicitly recast as linear invariant composition in the fiber orientation tensors of second and fourth order of Kanatani third kind. A maximum entropy reconstruction of a fiber orientation distribution function based on leading fiber orientation tensors, enables a new numerical formulation of the Advani and Tucker average for the special planar case. Polar plots of Young’s modulus and generalized bulk modulus obtained by selected homogenization schemes are arranged on two-dimensional slices within the body of admissible fiber orientation tensors, visualizing the influence of the orientation tensor on the stiffness tensor. The orientation-dependence of the generalized bulk modulus differs significantly between selected homogenizations. Restrictions on the effective anisotropic material response caused by orthotropy of closure approximations are discussed.
... The readers can find the components of the Eshelby tensor in Refs. [47,48]. Using equation (5) and equation (6), it is possible to predict the total stiffness tensor of the composite and extract the compliance tensor S as ...
This paper presents a computationally efficient multi-scale analytical framework for predicting the effective elastic response of short fiber reinforced polymers (SFRPs) under triaxial and flexural loading conditions where the details of microstructure such as core/shell thickness, volume fraction distribution, fiber misalignment and fiber length variation are objectively taken into account. To this end, the mean-field homogenization and finite element approaches are compared to calculate the elastic response of SFRPs at the microscopic level while the orientation averaging approach is used to address the effects of fiber misalignment. The obtained mechanical behavior is then linked to an enhanced laminate theory to predict the effective triaxial and bending macrostructural behavior considering the core/shell effects and variation of volume fraction through the thickness. Using the second-order homogenization technique, the numerical validation of the proposed analytical approach is investigated based on the micro- and meso-scale analyses. Furthermore, the potential of the proposed strategy is demonstrated for hybrid composites. Finally, the accuracy of the suggested model is thoroughly studied using the available experimental tests in literature where the statistical information about the details of SFRP microstructures is presented.
... This fact, coupled with the assumption of needle shaped inclusions, allows the expression of the entire homogenisation scheme in closed form, thus further reducing computational complexity. Application of the same homogenisation scheme in three dimensions is identical to the presented process, with only Eshelby's tensor S assuming different size and values [23] and the stiffness tensors for three dimensional elasticity needing to be adopted. In such an approach a third inclusion group, oriented in the z axis, can also be included while maintaining the closed form of the scheme. ...
A homogenisation scheme based on inclusion modelling is coupled with constitutive laws for damage and implemented in a finite element model for the simulation of concrete and reinforcement bar damage in reinforced concrete structures. The scheme is employed for simulating the behaviour of evenly distributed reinforcement and adapted for the simulation of zones with concentrated reinforcement in structural members.
The model is validated against experimental tests from the literature carried out on reinforced concrete members subjected to bending and direct tension. The model captures the main characteristics of the behaviour of and damage in the constituent materials of reinforced concrete without resorting to individual meshing of the embedded bars and with very low computational cost.
... where the non-zero components of the Eshelby tensor for the equivalent fiber phase, , and fiber-matrix interphase, , are calculated as follows (Qiu & Weng, 1990 ...
This paper presents a Mori-Tanaka-based statistical methodology to predict the effective Young modulus of carbon nanotubes (CNTs)-reinforced composites considering three variables: weight content, reinforcement dispersion and orientation. Last two variables are quantified by two parameters, namely, free-path distance between nano-reinforcements and orientation angle regarding the loading direction. To validate the present methodology, samples of multi-walled CNTs (MWCNTs)-reinforced polyvinyl alcohol (PVA)-matrix composite were manufactured by mixing solution. The MWCNT/PVA Young modulus was measured by nano-indentation, while the MWCNTs Young modulus was quantified by micro-Raman spectroscopy. Both stretched and unstretched composite specimens were fabricated. Transmission electron microscopy (TEM) and in-plane image analysis were used to obtain fitting coefficients of log-normal frequency distribution functions for the free-path distance and orientation angle. It was evidenced that numerical results fit well to measured values of effective Young modulus of MWCNTs and MWCNT/PVA, with exception of some particular cases where significant differences were found. Microstructural heterogeneities, cluster formation, polymer chains alignment, errors associated with the dispersion, orientation and mechanical characterization procedures, as well as idealization and statistical errors, were identified as possible causes of these differences. Finally, using the proposed methodology and the dispersion and orientation distribution functions experimentally obtained, the effective Young modulus is estimated for three kinds of thermoplastic matrices (polyvinyl alcohol, polyethylene ketone, and ultra-high molecular weight polyethylene) with different kinds of nanotubes (single wall, double wall, and multi-walled), at different weight contents, finding the superior mechanical performance for double-walled CNTs-reinforced composites and the lower one for multi-walled CNTs-reinforced ones.
... We note in passing that application of the Mori-Tanaka theory to the present two microgeometries would not lead to any asymmetry in the predicted effective moduli. Such a condition may arise when there is simultaneous presence of two or more inclusion shapes (or two or more values of p in a given problem), as discussed by Qiu and Weng [19]. ...
In contrast to the traditional study of composites containing ellipsoidal inclusions, we highlight some calculated results for the effective moduli when the inclusion shape can be described by the superspherical equation, , such that when p = 2 it reduces to a sphere and when p → ∞ it becomes a perfect cube. We consider the cases of both aligned and randomly oriented superspherical inclusions with isotropic, cubic, and transversely isotropic properties, and show how the shape parameter, p, affects the overall moduli of the composites during the spherical to cuboidal transition.
... The system as shown in Fig. l(b) involves two kinds of inclusions--one is the bonded and the other the debonded--with identical shape. This makes the Mori and Tanaka theory [3] suitable for applications, as the issue of asymmetry in the predicted overall moduli involving two kinds of inclusions with different shapes will not arise (see Weng [4] and Qiu and Weng [5]). Detailed exposition of the theory for a general multiphase composite can be found in Weng [6]. ...
A micromechanical theory is developed to examine the progressive debonding process of a brittle matrix composite containing aligned oblate inclusions under a high state of triaxial tension. Complete debonding is taken to be the debonding mode under such a high triaxial loading and its debonding strength is assumed to be governed by a Weibull probability function in terms of the hydrostatic tension of the inclusions. The micromechanical theory provides the required hydrostatic tensile stress at a given stage of debonding for a given inclusion shape and concentration. This allows one to calculate the debonding process progressively as the applied stress increases. The resulting stress-strain relation of the progressively debonded system is found to start out with that of the perfectly bonded composite, then deviates from it and eventually approaches the stress-strain curve of the corresponding porous material. It is further revealed that debonding with spherical inclusions is completed faster than with thin discs and it also occurs faster at a lower volume concentration of inclusions. The loss of stiffness of the transversely isotropic composite is also established as a function of inclusion shape and concentration for all five independent moduli; these moduli are seen to decrease gradually in the initial stage, then drop sharply while progressively debonding and finally level off again as all inclusions become debonded.
Carbon nanotube (CNT) has fostered research as a promising nanomaterial for a variety of applications due to its exceptional mechanical, optical, and electrical characteristics. The present article proposes a novel and comprehensive micromechanical framework to assess the viscoelastic properties of a multiscale CNT-reinforced two-dimensional (2D) woven hybrid composite. It also focuses on demonstrating the utilisation of the proposed micromechanics in the dynamic analysis of shell structure. First, the detailed constructional attributes of the proposed trans-scale composite material system are described in detail. Then, according to the nature of the constructional feature, mathematical modelling of each constituent phase or building block’s material properties is established to evaluate the homogenised viscoelastic properties of the proposed composite material system. To highlight the novelty of this study, the viscoelastic characteristics of the modified matrix are developed using the micromechanics method of Mori–Tanaka (MT) in combination with the weak viscoelastic interphase (WI) theory. In the entire micromechanical framework, the CNTs are considered to be randomly oriented. The strength of the material (SOM) approach is used to establish mathematical frameworks for the viscoelastic characteristics of yarns, whereas the unit cell method (UCM) is used to determine the viscoelastic properties of the representative unit cell (RUC). Different numerical results have been obtained by varying the CNT composition, interface conditions, agglomeration, carbon fibre volume percentage, excitation frequency, and temperature. The influences of geometrical parameters like yarn thickness, width, and the gap length to yarn width ratio on the viscoelasticity of such composite material systems are also explored. The current study also addresses the issue of resultant anisotropic viscoelastic properties due to the use of dissimilar yarn thickness. The results of this micromechanical analysis provide valuable insights into the viscoelastic properties of the proposed composite material system and suggest its potential applications in vibration damping. To demonstrate the application of developed novel micromechanics in vibration analysis, as one of the main contributions, comprehensive numerical experiments are conducted on a shell panel. The results show a significant reduction in vibration amplitudes compared to traditional composite materials in the frequency response and transient response analyses. To focus on the aspect of micromechanical behaviour on dynamic response and for the purpose of brevity, only linear strain displacement relationships are considered for dynamic analysis. These insights could inform future research and development in the field of composite materials.
The direct current (DC) and alternative current (AC) electromechanically coupled phenomena have been reported in carbon nanotube (CNT)‐based nanocomposite sensors. In this contribution, a unified micromechanics‐based model is established for the DC and AC strain sensors. The electric damage and volume change of nanocomposite are considered to be responsible for the electromechanically coupled effects in CNT‐based nanocomposite sensors. The predicted DC resistance change ratio, AC dielectric loss change ratio and corresponding strain sensitivity factors of CNT‐based nanocomposite sensors are all consistent with the experimental results. High strain sensitivity is achieved for CNT‐based nanocomposite sensors with a low CNT‐content. This study confirms the advantage of adopting CNT‐based nanocomposite sensors via the dielectric loss over the electric resistance. The present electromechanically coupled homogenization theory can be utilized to rapidly determine the macroscopic DC and AC sensing performance by choosing a specific set of microstructural parameters, and further simplify the design process of CNT‐based nanocomposite sensors.
The paper is devoted to simulation of the effective elastic properties of rock materials containing two substantially different types of defects simultaneously: random sets of ellipsoidal pores (cavities) and elliptical cracks. For solution of the homogenization problem and calculation of the effective elastic properties of such materials, the self-consistent effective field method is used. For composites with one type of heterogeneities, the method coincides with the Mori–Tanaka method. The method allows deriving analytical expressions for the effective elastic stiffness tensors of the materials containing pores and cracks of various scales. These tensors have correct symmetry with respect to tensor indices and provide physically reasonable values of the effective elastic constants in wide regions of porosity and crack density. The materials with pores and cracks of substantially different sizes and of close sizes are considered. The method predicts substantially different effective elastic constants of the materials with these microstructures. Comparison of the components of the effective elastic stiffness tensors for the materials with various values of porosity and crack density are presented. Wave surfaces of acoustical waves in the materials containing pores and cracks are constructed. It is shown that for materials with spherical pores and circular cracks, the shapes of these surfaces are close to ellipsoidal. The results of the paper can be used for determination of fracture level in damaged rock materials by acoustical methods.
Graphene agglomeration tends to develop with the increase of graphene loading. In this article, we present a unified model that first considers the evolution of graphene agglomeration and then incorporates it into the calculation of electrical and mechanical properties of agglomerated graphene/polymer nanocomposites. In the evolution of graphene agglomerates, a modified nanoparticle distance in terms of yield strength is introduced, while in the calculation of composite properties, a two‐scale framework that consists of the graphene‐rich agglomerates and the remainder as the graphene‐poor region is constructed. In electrical conduction, electron tunneling is modeled through Simmons formula and its difference with the widely used Cauchy function is compared. We highlight that both Simmons and Cauchy functions could well describe the interfacial tunneling activity, but the former is physics‐based while the latter is statistics‐based. In the calculation of nonlinear elastoplastic response, a field‐fluctuation method is adopted. We also demonstrated how the presented model gives rise to experimentally consistent electrical and mechanical properties of graphene/polypropylene nanocomposites, and how filler agglomeration hinders the performance of the materials.
The current research aims to study the influence of hybridisation on the tensile and fracture surface morphology of plain weave glass fibre/epoxy composites. The polymer resin is primarily filled with inorganic particles such as silicon carbide and alumina, and the corresponding modified resin is termed as effective matrix. This effective matrix is further reinforced by plain weave glass fibre to fabricate the hybrid composite. Elastic properties of the composites are estimated after developing Micro-mechanical-model-based formulation considering different analytical model like Mori-Tanaka, Paul, Ishai-Cohen model along with equivalent modulus theory. Experiments are conducted to estimate the elastic constants and post-tensile fractured surface morphology of these composites. The tensile modulus predicted using Mori-Tanaka and equivalent modulus theory matches well with the experiments and found to be in good convergence compared to the other models. Detailed analyses of the fractured surface of the tensile samples express the positive effect of hybridisation on the tensile and failure behaviour. It is found that the tensile strength and elastic modulus of the composites increase with an increase in particulate content.
Modelling and simulation techniques are now considered an essential practice for the materials industry. In order to gain insight into factors that can affect the final properties of a particular composite system, several design criteria need to be considered to create a more realistic composite behaviour. Such criteria would include the geometrical shape of reinforcements, interfacial bonding strength, types of reinforcement materials, and analytical formula utilised. In addition, multiscale modelling and simulation of composite materials require the consideration of various length and time scales, especially nano-composites. In terms of applicability, not all approaches can be used or would be effective in all situations to deliver reasonable results. A fair degree of accuracy and efficiency needs to be achieved when conducting multiscale modelling of any composite type. The selection of a competent model depends on its ability to provide rational results and precise estimations while requiring a short or moderate console execution time at a low computational cost. In this paper, various multiscale techniques of heterogeneous materials are critically reviewed in the context of modelling effective properties of nano- and conventional composites. The models are developed considering the frameworks of hierarchical type approaches and concurrent methods. Multiscale techniques are classified into three categories, namely sequential, parallel and synergistic methods. Subdivisions under those three major categories will be further explained. Interfacial models representing the interphase zone between inclusions and the matrix material are also discussed and categorised into interphase and interface models. Examples of modelling techniques are implemented for further illustrations. Comprehensive, detailed schematic charts are created to summarise and compare the different modelling approaches. The archival value of this paper is to provide fast and straightforward decision-making guidance based on the required criteria for which each model would be applied.
In this paper, making use of Eshelby tensor for the elastoplastic medium, will be presented an approach for the strain concentration tensor for materials that present spherical or spheroidal pores. The pores are embedded in a homogeneous isotropic matrix which presents an elastoplastic behavior, governed by the von Mises yield criterion and isotropic hardening. In this context, it will also be analyzed the behavior of a porous material through the influence of the extraction of the isotropic part of the anisotropic operator. The effective properties of the material in each incremental step are obtained through the micromechanical Mori-Tanaka's model. Through elastoplastic strain concentration tensor for aligned or misaligned pores, it was observed that the pore direction represents a greater influence on the elastoplastic behavior of the material studied than aspect ratio (length divided by pore diameter). This approach has the advantage of being easily incorporated into conventional mean-field homogenization (MFH) schemes for the case where inclusion stiffness is neglected and presents different geometries or directions in 3D space.
Remarkable discrepancy between the tensile and compressive sensing performance has been experimentally observed in highly sensitive strain sensors, but at present no existing micromechanics-based theory can explain such a difference. To this end, we will investigate the tensile and compressive gauge factors of MWCNT/epoxy nanocomposite sensors in a unified fashion via an electromechanically coupled homogenization scheme. The discrepancy between the tensile and compressive sensing performance is analytically revealed by the loading-dependent tunneling effect assisted with a unified strain-dependent tunneling distance between the adjacent MWCNTs. The predicted gauge factor and resistance change ratio of MWCNT/epoxy nanocomposite sensors are calibrated with two independent sets of experimental data ranging from the compressive to tensile loadings. The tensile gauge factor of MWCNT/epoxy nanocomposite sensors is revealed to be higher than the corresponding compressive one. The absolute values of tensile and compressive resistance change ratio both enhance with the MWCNT waviness, the strain loading and the height of potential barrier between adjacent MWCNTs. In addition, a low MWCNT volume concentration and MWCNT aspect ratio can attain the high tensile and compressive gauge factors for MWCNT/epoxy nanocomposite sensors. The present research can contribute to the accurate design, fabrication, and optimization of highly sensitive strain sensors.
The ambient temperature at which creep deformation takes place is known to exert significant influence on the creep durability of graphene-based nanocomposites, but at present no theory could illustrate the underlying microstructural evolution to predict such a phenomenon. In this paper, a novel dual thermodynamics approach in conjunction with a time-temperature superposition principle (TTSP) is established to address this issue. First, temperature-dependent secant moduli are exclusively adopted as the unique homogenization variables shifted through TTSP, and then the time- and temperature-dependent effective stresses inside the matrix are calculated via the principle of equivalent work rate combined with the field-fluctuation method. Next, the dual irreversible thermodynamic processes, including the stress-induced creep damage inside the matrix and the temperature-dependent degradation at the interphase, are introduced through two independent sets of evolution equations. The predicted creep rupture strain and rupture time are calibrated with experiments over a wide range of temperature. It is demonstrated that the creep rupture strain increases with the rise of ambient temperature, while the creep rupture time decreases with it. The onset of creep-damage process also commences earlier at higher temperature. This research can provide a design guidance to assess the damage process and failure of low-dimensional nanocomposites at elevated temperature environment.
The microstructure of 3D printed composites is inherently different than traditional composites due to the manufacturing process. The differences influence morphological characteristics such as the contour of the cross-section of the fiber and alter the macroscopic behavior of 3D printed parts. This article investigates the microstructural morphology of 3D printed nylon reinforced with continuous carbon fibers and the effect of microstructural irregularities on the macrostructural elastic response through stochastic homogenization modeling. The contour of the carbon fibers is extracted from scanning electron microscopy (SEM) micrographs available in the literature and used to generate realistic and ideal (ellipsoidal, circular) contours in a stochastic manner using a new methodology. Furthermore, a novel method is introduced to generate single- and multiple-fiber representative volume elements (RVEs) in finite element (FE) software for the approximation of the effective elastic properties. To minimize the computational effort associated with the full numerical modeling of multiple-fiber RVEs, a novel semi-analytical approach is demonstrated based on the numerical estimation of stiffness contribution tensors and the implementation of analytical effective field methods. The results of the numerical and semi-analytical models are compared with analytical models and exhibit a good agreement.
This study has focused on the instability behavior of the laminated functionally graded CNT reinforced (FG-CNTRC) plate under different types of the non-uniform dynamic in-plane loads. The application of the different non-uniform loads would resemble with the actual operation of the structures. The kinematics of the FG-CNTRC has been modelled based on the Reddy’s third order shear deformation theory. The developed in-plane stresses may not be uniform when the structure is subjected to the non-uniform harmonic edge loads. Hence, the evaluation of the in-plane stresses become necessary prior to the calculation of the critical buckling load of the FG-CNTRC plate. These in-plane stresses have been calculated using the stress recovery procedure implemented in finite element method. Once the critical buckling load of the structure determined, the differential equation of motion of the plate converted into Mathieu type differential equation and solved as the general eigenvalue problem as suggested by Bolotin. The efficacy and flexibility of the present formulation has been validated with the available solution for different geometric parameter of the FG-CNTRC plate. Further, various parametric studies on dynamic instability behavior of the FG-CNTRC plate have presented by considering different parameters like span-thickness ratio, CNT gradation through thickness, aspect ratio, edge constraint, different type of edge load etc.
Low-dimensional nanocarbon reinforcements in multifunctional composites have neither defect-free conjugate structures nor efficient interfacial properties in general. Moreover, synthesizing polymer nanocomposites involves intrinsic and extrinsic structural defects, which undoubtedly degrade the properties of the nanocarbons. However, several advantages and even useful applications of such structural defects have pioneered the defect engineering of nanocarbon structures. Here, we assessed the reciprocity of the degradation of single-layer graphene and the tailoring of the interfacial load transfer by structural defects. For this purpose, we used a molecular dynamics simulation and its hierarchical bridging to a mean-field micro-mechanics model for linear elastic behavior. Crystallographic defects and oxygen functionalization were considered the representative intrinsic and extrinsic defects, respectively. Despite the notably degraded elasticity of the nanocomposites involving direct load bearing by the defected graphene, those readily governed by the interfacial load transfer were significantly improved. The contributions of both types of defects to the interface were verified using the local stress concentration on the embedded graphene during the mechanical loading of nanocomposites and an additional mode I and II decohesion test. Lastly, the critical aspect ratio of graphene that maximizes the effects of the defects on the interface stiffness of the nanocomposites was derived for defect engineering.
This work is concerned with the characterization of load-dependent dielectric loss and sensitivity analysis for CNT-based nanocomposite sensors (CNCSs) under AC loading. To this end, an electromechanically coupled microstructural theory is developed from the bottom up to quantitatively predict their overall dielectric loss change ratio and strain-sensitivity factor. In the theory, various categories of load-dependent functional interface effects, such as strain- and filler-dependent electron hopping and dielectric relaxation, are incorporated into it. The electric damage process under mechanical load is characterized through the principle of irreversible thermodynamics. The outcome is a microstructure-based coupled theory whose predictions of AC dielectric loss change ratio can be directly calibrated with the experimental data of MWCNT/PVDF nanocomposite sensor over a broad strain loading range from 0 to 10% and a wide frequency spectrum from 5 kHz to 500 kHz, where PVDF is a shape memory polymer. The theory further demonstrates the advantage of demarcating CNCSs via the dielectric loss over the traditional electric resistance. It can be used to rapidly determine the macroscopic dielectric loss change ratio by choosing a specific CNT volume concentration and AC working frequency, and further simplify the design procedure of highly sensitive strain sensors.
Covalent grafting between carbon nanotubes (CNTs) and a polymer matrix is the most efficient way to tailor the intrinsically weak interface in nanocomposites. To understand the grafted structure-to-improved property relationship of nanocomposites, however, the degradation of grafted CNT and the properties of surrounding interphase zone should be accounted for in constitutive model. In this study, the reciprocity of the interface, interphase, and elasticity of CNTs depending on the covalent grafting between the CNT and a polyethylene terephthalate (PET) matrix were studied through molecular dynamics simulations and a mean-field micromechanical interface/interphase model. The replacement stiffness method was used to determine the effect of the tailored interface on the overall elasticity of a nanocomposite in micromechanics. The elasticity of the CNTs and the nanocomposites was determined from molecular mechanics and molecular dynamics simulations, respectively. Despite the degraded elasticity of the nanotubes, clear improvements in the transverse modulus and shear moduli were observed in the covalently grafted nanocomposites. The elasticity of the interphase was determined in terms of the number of covalent grafting and the interfacial compliance using a two-step inverse micromechanical analysis. Regardless of the number of covalent grafting addressed, the elastic moduli of the interphase were always larger than those of a neat PET matrix. Furthermore, the accuracy of the proposed multiscale micromechanical interface/interphase model at various volume fractions of CNTs was validated.
Among the numerous methods in micromechanics mentioned in Chap. 1 (see Willis, J Appl Mech, 50:1202–1209, 1983), in most details, we will consider in this chapter the self-consistent methods based on some mathematical approximations for solving the classical GIE for statistically homogeneous CMs analyzed in Chap. 7. The notion of an effective field in which each particle is located is a basic concept of such powerful methods in micromechanics as the methods of self-consistent fields, effective fields, and its generalization a multiparticle effective field method (MEFM) which includes as the particular cases the methods of the effective medium, Mori-Tanaka, conditional momenta, differential, and Maxwell. Numerical examples for estimations of effective moduli and stress concentration factors are considered for the matrix CMs containing spheroidal inclusions, noncanonical fibers with prescribed random orientations, polifractional spherical inclusions, and clusters of spheroidal inclusions.
The key tools of random structure composite materials (CMs) such as Green’s function and related tensors are considered in this chapter for the problems of static and dynamic elasticity, conductivity, and thermoelectroelasticity. Inhomogeneity in an elastic medium is considered for thermoelastic problem with the interface boundary operators. One presents the theorem of polynomial conservation (the p-property) and Eshelby tensors and related tensors in a special basis for ellipsoidal inclusion. Coated ellipsoidal inclusion and imperfect interface models with estimation of elastic moduli of fictitious ellipsoidal homogeneous inclusion are considered. Related problems (conductivity, scattering of elastic waves, and piezoelectric problem) for ellipsoidal inhomogeneity in an infinite medium are also analyzed.
Additively manufactured parts of thermoplastic materials using the Fused Filament Fabrication (FFF) technique express transversely isotropic characteristics. Those characteristics are related to the FFF technique which leads to the formation of particular bonds and voids between printed filaments, both of which are investigated in this study. Tensile testing of printed Nylon samples reveals that the transverse isotropy can be possibly caused solely by the bond formation. Micro-CT images aid the visualization of long, convex and concave voids extending parallelly to the printing direction. Finally, novel analytical techniques are presented for the accurate modeling of the void geometry considering both concave and convex shapes and illustrate a very good agreement with Finite Element models.
The 3D-printed composites manufactured by the fused filament fabrication technique contain a substantial amount of voids as a result of the layerwise deposition process. The voids obtain convex or concave shapes which influence the macrostructural response and must be captured by homogenization approaches. However, the majority of approaches that consider the real shape of voids are based on computationally intensive numerical models that are time-consuming. This article presents a novel methodology to convert the real shape of voids to ideal supercylindrical one, which can be defined using only one parameter, the concavity parameter, thus drastically reducing the computational effort. The methodology is stochastically implemented using a non-intrusive uncertainty quantification technique for analytical, semi-analytical, and numerical homogenization approaches to describe the effects of voids on the effective elastic properties of 3D-printed thermoplastics. The stochastic modeling is applied for acrylonitrile butadiene styrene (ABS) and the results are compared to previous experiments and computationally intensive models and illustrate a good overall prediction of the effective elastic properties.
A new method, called the polarization approximation (PA), appears as an interesting technique to calibrate the effective properties of composite materials. Constructed from the minimum energy principle, the popularization approximation is a potential alternative to the popular Mori-Tanka approximation (MTA), and has been derived as an approximation of the macroscopic moduli as well as the microscopic fields. In the literature, MTA has been applied to various homogenization problems of a wide range of composite materials though sometimes, the results lack robust accuracy. It is shown in this paper that PA is more reliable than MTA. The similarity and differences between PA and MTA when applying to elastic modulus will be pointed out using some types of heterogeneous microstructures, which have isotropic macroscopic elastic moduli in 2D or 3D. Results from other methods, such as experiments, the numerical unit cell method and the Hashin-Shtrikman bounds will also be presented as well for comparisons.
Interphase regions that form in heterogeneous materials through various underlying mechanisms such as poor mechanical or chemical adherence, roughness, and coating, play a crucial role in the response of the medium. A well- established strategy to capture a finite-thickness interphase behavior is to replace it with a zero-thickness interface model characterized by its own displacement and/or traction jumps, resulting in different interface models. The contributions to date dealing with interfaces commonly assume that the interface is located in the middle of its corresponding interphase. We revisit this assumption and introduce a universal interface model, wherein a unifying approach to the homogenization of heterogeneous materials embedding interfaces between their constituents is developed. The proposed novel interface model is universal in the sense that it can recover any of the classical interface models. Next, via incorporating this universal interface model into homogenization, we develop bounds and estimates for the overall moduli of fiber-reinforced and particle-reinforced composites as functions of the interface position and properties. Furthermore, we elaborate on the computational implications of this interface model. Finally, we carry out a comprehensive numerical study to highlight the influence of interface position, stiffness ratio and interface parameters on the overall properties of composites, where an excellent agreement between the analytical and computational results is observed. The developed interface-enhanced homogenization framework also successfully captures size effects, which are immediately relevant to emerging applications of nano-composites due their pronounced interface effects at small scales.
Mori-Tanaka's theory with the general anisotropic constituents has been recast into a new form and it is shown that this form bears an identical structure to that developed by Walpole for the bounds. The equivalent polarization stress and strain in the former theory are exactly those chosen by Hashin-Shtrikman and Walpole and the average stress and strain of the matrix phase are equal to the image stress and strain imposed on the approximate fields by Walpole to meet the required boundary conditions. The consequence is that the effective moduli of the composite containing either aligned or randomly-oriented, identically shaped ellipsoidal inclusions always have the same expressions as those of the H-S-W bounds, only with the latter's comparison material identified as the matrix phase and Eshelby's tensor interpreted according to the appropriate inclusion shape. This connection allows one to draw a line of important conclusions regarding the predictions of the M-T theory, and it also points to the conditions where this theory can always be applied safely without ever violating the bounds and where such an application might be less reliable.
This chapter presents theoretical foundations about elastic behavior of composite materials. A composite material is a heterogeneous solid continuum that bonds together a number of discrete homogeneous continua, each of which has a well-defined sharp boundary. It is understood that the bonding at the interfaces (and the continuity of each region) remains intact in the present circumstances where the entire mixture is to be placed in an equilibrated state of infinitesimal elastic strain by external loads and constraints. Each separate homogeneous region has its characteristic tensor of elastic moduli (in the stress–strain relation), which when anisotropic reflects a particular alignment of the crystallographic axes relative to the fixed Cartesian ones. A single phase consists of all those regions that share the same tensor of elastic moduli and perhaps also the same alignment of a geometrical shape and of crystallographic axes. This chapter presents an analysis of tensors and elastic behavior. The elastic field of a composite body is discussed. The overall elastic behavior of a composite body is also explained in detail.
The effect of variable fiber aspect ratio on the thermo-mechanical properties of aligned short fiber composites is analytically studied by use of the Eshelby's equivalent inclusion method. Then, the thermo-mechanical properties of the composite predicted for a given density function are compared with those obtained for the mean value. Based on this comparison, the validity of using the mean value is discussed. As a demonstration, the results on two types of a short fiber composite are presented, SiC whisker/Aluminum and Carbon fiber/Epoxy composites.
It is supposed that a region within an isotropic elastic solid undergoes a spontaneous change of form which, if the surrounding material were absent, would be some prescribed homogeneous deformation. Because of the presence of the surrounding material stresses will be present both inside and outside the region. The resulting elastic field may be found very simply with the help of a sequence of imaginary cutting, straining and welding operations. In particular, if the region is an ellipsoid the strain inside it is uniform and may be expressed in terms of tabulated elliptic integrals. In this case a further problem may be solved. An ellipsoidal region in an infinite medium has elastic constants different from those of the rest of the material; how does the presence of this inhomogeneity disturb an applied stress-field uniform at large distances? It is shown that to answer several questions of physical or engineering interest it is necessary to know only the relatively simple elastic field inside the ellipsoid.
This paper examines the influence of aspect ratio α, from zero to infinity, on the effective elastic moduli of a transversely isotropic composite. The reinforcing inclusions, which could be flakes or short fibers, are assumed to be spheroidal and unidirectionally aligned. Of the five independent elastic constants, the longitudinal Young's modulus E11 and in-plane shear modulus μ12 appear to increase with increasing aspect ratio, while the transverse Young's modulus E22, out-plane shear modulus μ23, and plane-strain bulk modulus K23, generally decrease. It is further noted that E11 is more sensitive to α when α > 1 but the others are more so when α < 1. The present analysis was carried out by the combination of Eshelby's and Mori-Tanaka's theories of inclusions.
Variational principles in the linear theory of elasticity, involving the elastic polarization tensor, have been applied to the derivation of upper and lower bounds for the effective elastic moduli of quasi-isotropic and quasi-homogeneous multiphase materials of arbitrary phase geometry. When the ratios between the different phase moduli are not too large the bounds derived are close enough to provide a good estimate for the effective moduli. Comparison of theoretical and experimental results for a two-phase alloy showed good agreement.
The overall elastic moduli of solid composite materials are bounded by employing extremum principles. Composites that are homogeneous and isotropic in a statistical sense, but with otherwise arbitrary geometry, are considered. When each of the constitutive phases is uniform and isotropic, bounds are established in detail in terms of the elastic moduli and relative volumes of the phases.
Having noted an important role of image stress in work hardening of dispersion hardened materials, (1,3) the present paper discusses a method of calculating the average internal stress in the matrix of a material containing inclusions with transformation strain. It is shown that the average stress in the matrix is uniform throughout the material and independent of the position of the domain where the average treatment is carried out. It is also shown that the actual stress in the matrix is the average stress plus the locally fluctuating stress, the average of which vanishes in the matrix. Average elastic energy is also considered by taking into account the effects of the interaction among the inclusions and of the presence of the free boundary.
The main overall elastic moduli of fibre composites with transversely isotropic phases are shown to be connected by simple universal relations which are independent of the geometry at given concentration. Exact values of the moduli themselves are found when the phases have equal transverse rigidities. Otherwise, upper and lower bounds are obtained in terms of phase properties and concentrations. It is proved that these are the best possible without taking account of the detailed geometry.
The internal inhomogeneities of stress and strain in an arbitrarily deformed aggregate of elasto-plastic crystals are evaluated theoretically. A tensor constitutive law of a general kind is assumed for the individual crystals. The implied mechanical properties of the aggregate as a whole are estimated by means of a self-consistent model akin to one used by Hershey (1954), Kröner (1958, 1961) and Budiansky and Wu (1962), but differing in significant respects.
The overall elastic moduli of some solid composite materials are evaluated, first by bounding them precisely, and secondly by a ‘self-consistent’ estimate. Transversely isotropic inclusions of ‘needle’ and ‘disc’ shapes are particularly considered, at both random and aligned orientations, and at arbitrary volume concentration.
The method of Part I is applied in detail to bound the overall elastic moduli of solid composite materials whose phases are uniform but may be arbitrarily anisotropic. The geometry is assumed such that a composite appears homogeneous and isotropic on average; otherwise it is arbitrary. The bounds are expressed in terms of the individual elastic moduli and relative volumes of the constitutive phases.
The Plane strain bulk modulus and the two shear moduli of multiphase transversely isotropic fibre reinforced materials of arbitrary transverse phase geometry, are bounded from above and below in terms of phase moduli and phase volume fractions. Particular attention is given to the important special case of two-phase fibre reinforced materials.
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