Shear Modulus Of Composites Research Articles

A Constitutive Law Modeling the Mixed Hardening Behavior of Particle-Reinforced Composites

Abstract A homogenized elasto-plastic constitutive law (including both constitutive equations and inversed constitutive equations) is proposed in an incremental form for the particle-reinforced composites based on the flow theory of plasticity and the asymptotic homogenization method. The constitutive law can be used to predict the mixed hardening behavior of particle-reinforced composites under arbitrary loading conditions if the uniaxial tension test curve of matrix materials is known. It is found that the constitutive law of particle-reinforced composites is similar in form to the law of matrix materials. There is a simple proportional relationship between the yield stress, the plastic modulus, and the deviatoric back stress of particle-reinforced composites and the corresponding parameters of matrix materials, which is equal to the ratio of the shear modulus of composites to the shear modulus of matrix materials. The tangent modulus of particle-reinforced composites can be calculated using a simple arithmetic formula according to the tangent modulus of matrix materials. A numerical algorithm is suggested.

Characterizing the curing behavior and high-temperature performance of epoxy-resin modified asphalts

This study was undertaken to quantify and analyze the microscopic morphology and curing reaction of epoxy asphalt (EA) modified with 2%, 5%, 10%, 20%, 30%, and 40% epoxy resin (ER) contents. The functional group changes and the ER dispersion before and after the curing reaction, at the micro level, were comparatively evaluated using the infrared spectroscopy and fluorescence microscopy. The influence of various ER contents on the high-temperature performance of EA was also analyzed and quantified, at macro level, using the dynamic shear rheometer (DSR). The research results showed that the particle size of the ER molecules after curing is more significant than before the curing reaction. When the ER content is low, the EA turned into a continuous phase. The peak area of the distinct functional groups in the EA was found to be linearly correlated with the quantitative content of the ER. The strain sweep results showed that the linear viscoelastic region of the EA shrinks after curing. The higher the ER content, the smaller the strain in the linear viscoelastic region. The frequency scanning results showed that the ER significantly increased the composite shear modulus (G*). The rutting factor (G*/Sin δ) showed that when the ER content exceeds 20%, the high-temperature deformation resistance of EA increases and vice versa. The test results of multiple stress creep recovery (MSCR) showed that the deformation recovery ability is significantly improved with increased ER contents after curing. As analyzed using the coefficient of variation (COV), the high-temperature DSR rheological indicated satisfactory statistical reliability with COV values less than 30%.

Molecular-dynamic modeling applied for analysis of composite wear resistance increasing as compared with the original polymer matrix

The influence of filling on the mechanical properties of polytetrafluoroethylene (PTFE) was inves- tigated by molecular dynamic modeling. Molecular models of PTFE and its composite F4K20 were built. Energy values of intermolecular interaction were determined, stiffness and flexibility matrices of PTFE and F4K20 were obtained. It was shown that energy of intermolecular interaction of F4K20 is approximately 15 times higher in comparison with energy of intermolecular interaction of PTFE. Calculation based on model- ing showed that the introduction of the filler leads to a significant increase in the composite shear modulus in comparison with the initial matrix, which may be the reason of wear resistance increasing of polymer compo- sites.

Surface functionalization and CNT coating induced improved interfacial interactions of carbon fiber with polypropylene matrix: A molecular dynamics study

An in-depth study of interfacial interactions between carbon fiber (CF) and polypropylene (PP) matrix was carried out through a series of pull-out simulations using molecular dynamics (MD). CF surface was functionalized by grafting separately with 2, 4, and 6 wt% of –OH, –COOH and NH2 groups to understand its impact on fiber–matrix interfacial properties. Besides CF grafting, the effect of carbon nanotube (CNT) presence on its surface, mimicking dip coating of CNTs over CF, was also studied. The addition of maleic anhydride-grafted-PP (MA-g-PP), was also investigated to assess its role in imparting interfacial interactions. Among the simulated composites, the one reinforced with 6 wt% –NH2 grafted CF displayed a significant ~126% increase in IFSS compared to the composite having pristine CF, while the other reinforced with CNT coated and 6 wt% –NH2 functionalized CF showed a ~112% increment. Composite with CNT coated CF alone exhibited IFSS ~35% higher than the composite with pristine CF. Furthermore, the Young’s and shear modulus of composites improved with both grafting percentage and CNT coating. Composite reinforced with CNT coated CF and CNT coated 6 wt% –NH2 grafted CF displayed ~58 and ~112% increase in Young’s modulus compared to composite strengthened with pristine CF.

Strengthening of poly (ethylene terephthalate) by filling with short glass fibers and multi-wall carbon nanotubes

The effect of hybrid filling of poly (ethylene terephthalate) (PET) with short glass fibers (GF) and multi-walled carbon nanotubes (MWCNTs) on the static and dynamic mechanical properties of composites, as well as the structural features and rheological behavior of melts were investigated. Compounding materials was carried out by the PET melt blending with the use of the reaction extrusion method. The studied composites contained from 15 to 60 wt. % GF, the concentration of MWCNTs was 0.5 and 1.6 wt. %. It was shown that the PET hybrid filling led to a synergistic increase in the strength, the elastic modulus and the dynamic shear modulus of composites under tension and bending in a wide temperature range. MWCNT additives intensified interfacial adhesion, had a nucleating effect on the crystallization of the binder, and increased its thermal stability.

The influence of the apparent shear modulus on the response of transversely isotropic soft materials under simple shear

The main purpose of this work is to investigate the mechanical behavior of transversely isotropic soft materials subjected to simple shear when the shear loading is applied perpendicular to the orientation of the fibers. The effect of the continuous fibers with fixed ends on the apparent shear modulus of composites was considered. In the experimental test, unidirectional composite specimens with distinct volume fractions of fibers were fabricated and subjected to simple shear deformations. A mathematical model based on a strain-energy function of two invariants was introduced to predict the shear response of composites at small and large deformations. In this model, a closed-form expression of the apparent shear modulus was proposed considering shear and bending contributions. The neat matrices and the fibers were characterized and all estimated properties were used as input data in order to predict the behavior of the soft composites. Supplementary experimental tests and numerical simulations were performed to verify the predictive model for the apparent shear modulus. A good agreement between the experimental/numerical data and the theoretical model was obtained for small and large deformations. Results showed that the proposed model is capable of predicting the influence of the height and the radius of gyration of the fibers on the apparent shear modulus, and consequently the shear response of the soft composites can be obtained.

Shear modulus of isotropic ferrogels

Abstract We present results of theoretical study of magnetorheological effect in ferrogels with magnetizable spherical particles chaotically distributed in a current gel. To avoid intuitive constructions with uncontrolled accuracy and adequacy, the analysis is done in the frames of the mathematically regular pair approximation. Our results demonstrate non monotonic increase of the composite shear modulus with the applied magnetic field. This effect is stronger for the systems with the soft gel, than for the relatively rigid ones.

A molecular dynamics study on efficient nanocomposite formation of styrene–butadiene rubber by incorporation of graphene

A molecular level study has been made on enhanced mechanical and tribological properties of graphene-reinforced styrene butadiene rubber (SBR) composites using molecular dynamics simulation technique. Constant strain method is applied to calculate the mechanical properties of developed structures. Two molecular level layer model one with SBR and another with graphene-reinforced SBR composites were developed, and shear loading was applied on the top and bottom Fe layers to study tribological properties, i.e., abrasion rate, and friction coefficient. The Young’s and shear modulus of composites with different graphene oxide volume fractions have been developed and studied. The 5 vol% addition of graphene into SBR matrix shows a significant increase in Young’s and shear modulus and hardness. By incorporation of the GO, 48 and 56% decrease in friction coefficient and abrasion rates of SBR polymer was observed, respectively. About 15% reduction in the RDF values of GO/SBR composites was obtained. The interaction energy between graphene oxide sheet and SBR matrix during the shear process has been obtained and discussed.

Effective shear modulus of solids reinforced by randomly oriented-/aligned-elliptic nanofibers in couple stress elasticity

Nowadays, by adding a small amount (about 0.5–5% by weight) of a desired nanomaterial to a matrix having certain properties one may design a multifunctional nanocomposites with a remarkably improved macroscopic properties of interest. The capability of conventional continuum theories in treating the problems of embedded ultra-small inhomogeneity with any of its dimensions comparable to the characteristic lengths of the involved constituent phases is questioned, mainly, on the grounds of the accuracy and the size effect. The micromechanical framework based on the Eshelby's ellipsoidal inclusion theory [1] which has been widely used to estimate the overall behavior of composites falls under the same category, as is size insensitive. In this work, effort is directed at the prediction of the macroscopic shear modulus of composites consisting of nano-/micro-size fibers of elliptic cross-sections via couple stress theory, a physically realistic theory that encompasses the size effect. To this end, the fundamental equations of couple stress elasticity in elliptic coordinates are derived and several fundamental elliptic inhomogeneity problems in plane couple stress elasticity are solved analytically. For the purpose of the application of these results to the study of the effective properties of the composites of interest, Mori and Tanaka theory [2] is first reformulated in the mathematical framework of couple stress theory. Subsequently, the overall shear modulus of solids reinforced by aligned as well as randomly oriented elliptic nanofibers will be predicted. The influences of the size, shape, orientation, rigidity, and intrinsic length of the reinforcing nanofibers as well as the effects of the characteristic length of the matrix on the effective shear modulus of the composite are addressed.

Representative volume element for composites reinforced by spatially randomly distributed discontinuous fibers and its applications

Combining with the random sequential absorption (RSA) algorithms, the method entitled Fiber Growth Method is presented for generating the representative volume element (RVE) with the spatially randomly distributed discontinuous fibers and with a relatively high fiber volume fraction (∼25%). The formulation of the Axes Distance Method for checking the overlapping of fibers in the RVEs is discussed. Within the framework of elasticity, the generated RVEs are analyzed to obtain the effective mechanical properties of composites reinforced by spatially randomly distributed discontinuous fibers using the numerical homogenization technique based on the finite element analysis (FEA). The validation is performed by comparing the Bulk modulus and Shear modulus predicted by the FEA against those obtained from the traditional equations based on the Halpin–Tsai estimation. FEA results show that due to the spatial random distribution of fibers, composites behaves homogeneously isotropic at the macroscopic scale. Bulk moduli and Shear moduli of composites increase with the fiber volume fraction increasing while the Poisson’s ratios of composites decrease.

Low-temperature internal friction of fiber composites with a copper matrix

The temperature dependences of the internal friction and shear modulus of composites with a copper matrix reinforced with fibers of different functional purposes (Cu, Fe, Al, NbTi) have been studied in the temperature range of 100–300 K. The internal friction demonstrates a peak located in a narrow temperature range and caused by a thermally activated relaxation process. It has been shown that its behavior is due to a number of technological factors whose combination determines the strength of adhesive bond of the composite components. The observed fact makes it possible to use the internal friction as a nondestructive method of controlling the adhesion strength of a fiber-matrix compound.

The effective propagation constants of SH wave in composites reinforced by dispersive parallel nanofibers

In the present paper, the effective propagation constants of elastic SH waves in composites with randomly distributed parallel cylindrical nanofibers are studied. The surface stress effects are considered based on the surface elasticity theory and non-classical interfacial conditions between the nanofiber and the host are derived. The scattering waves from individual nanofibers embedded in an infinite elastic host are obtained by the plane wave expansion method. The scattering waves from all fibers are summed up to obtain the multiple scattering waves. The interactions among random dispersive nanofibers are taken into account by the effective field approximation. The effective propagation constants are obtained by the configurational average of the multiple scattering waves. The effective speed and attenuation of the averaged wave and the associated dynamical effective shear modulus of composites are numerically calculated. Based on the numerical results, the size effects of the nanofibers on the effective propagation constants and the effective modulus are discussed.

Effects of higher-order interface stresses on the elastic states of two-dimensional composites

We propose a simple model to simulate higher-order interface stresses along the interface between two neighboring media in two dimensions. The interface behavior is modeled from a thin interphase of constant thickness by taking a proper limit process. In the formulation the deformation of the thin interphase is approximated by the Kirchhoff–Love assumption of thin shell. To incorporate the higher-order interface stresses, we consider the bending effects resulting from the non-uniform surface stress across the layer thickness. The stress equilibrium conditions is fulfilled by consideration of balance for forces as well as stress couples. Depending on the difference in stiffness and length scales of the interphase, we show that the interfaces can be classified into four different types. This findings, upon suitable definitions of material parameters, agree with a rigorous asymptotic analysis proposed by Benveniste and Miloh [Benveniste, Y., Miloh, T., 2001. Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech. Mater. 33 309–323]. To illustrate the higher-order effects, we derive analytically the stress concentration factor of an infinite plate containing a circular cavity with interface stresses of different orders subjected to a remote transverse shear loading. The closed-form expressions show how the orders of interface stresses influence the concentration factor in a successive manner. In addition, we examine the effective shear modulus of composites with circular inclusions with higher-order interface effects. The effective transverse shear modulus is derived based on the generalized self-consistent method.

Elliptic inhomogeneities and inclusions in anti-plane couple stress elasticity with application to nano-composites

It is well-known that classical continuum theory has certain deficiencies in predicting material’s behavior at the micro- and nanoscales, where the size effect is not negligible. Higher order continuum theories introduce new material constants into the formulation, making the interpretation of the size effect possible. One famous version of these theories is the couple stress theory, invoked to study the anti-plane problems of the elliptic inhomogeneities and inclusions in the present work. The formulation in elliptic coordinates leads to an exact series solution involving Mathieu functions. Subsequently, the elastic fields of a single inhomogeneity in conjunction with the Mori–Tanaka theory is employed to estimate the overall anti-plane shear moduli of composites with uni-directional elliptic cylindrical fibers. The dependence of the anti-plane elastic moduli on several important physical parameters such as size, aspect ratio and rigidity of the fiber, the characteristic length of the constituents, and the orientation of the reinforcements is analyzed. Based on the available data in the literature, certain nano-composite models have been proposed and their overall behavior estimated using the present theory.

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

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

Resonance methods for determining the complex shear moduli of orthotropic composites

A survey of various methods for determining the complex elasticity and shear moduli from the resonant frequencies of flexural and torsional vibrations of rectangular rods cut out from a plate of an orthotropic composite is presented. The errors in the computed values of dynamic shear moduli caused by inaccuracies in the experimental determination of resonance frequencies are examined. A new variant of the resonance method is developed, which permits one to find three complex shear moduli of a composite from the resonant frequencies and the damping of torsional vibrations of three rods oriented along three symmetry axes of the material. For computing the moduli in the case of an overdetermined system, an algorithm of nonlinear optimization based on the least-squares method is recommended. From the results obtained it follows that, for determining the interlaminar shear moduli with a necessary accuracy, the rods must be sufficiently thick. It is shown that a good agreement alone between calculated and experimental frequencies of flexural and torsional vibrations of rods does not ensure a reliable determination of the moduli of interlaminar shear if experiments are carried out on wide test specimens cut out from a thin plate. Recommendations are given for the choice of geometrical sizes of test specimens for resonance experiments.

海島構造を有する高分子系複合材料の巨視的損失係数のマイクロメカニックス解析(S07-4 複合材料II,S07 メゾスケールにおける先進材料の力学と強度・機能評価)

Macroscopic loss factor and shear modulus of a sea-island type polymer-based composite is calculated by using the equivalent inclusion method combined with the Mori-Tanaka theorem. The influences of the loss factor and the shear modulus of the particle on the macroscopic loss factors and shear modulus of the composite are examined under the shear applied stress. Consequently, an aspect ratio of the particles influences on both of the macroscopic loss factor and the shear modulus of the composite. Moreover, it is found that an optimum range of shear modulus of the particles exists for the maximum value of the macroscopic loss factor of the composite in the case of any aspect ratio of the particles.

A rigorous analytical method for doubly periodic cylindrical inclusions under longitudinal shear and its application

An infinite elastic solid containing a doubly periodic parallelogrammic array of cylindrical inclusions under longitudinal shear is studied. A rigorous and effective analytical method for exact solution is developed by using Eshelby’s equivalent inclusion concept integrated with the new results from the doubly quasi-periodic Riemann boundary value problems. Numerical results show the dependence of the stress concentrations in such heterogeneous materials on the periodic microstructure parameters. The overall longitudinal shear modulus of composites with periodic distributed fibers is also studied. Several problems of practical importance, such as those of doubly periodic holes or rigid inclusions, singly periodic inclusions and single inclusion, are solved or resolved as special cases. The present method can provide benchmark results for other numerical and approximate methods.

Damping and Stiffness of Particulate Sic–Insn Composite

Metal matrix composites of silicon carbide particles in indium–tin alloy were fabricated with the aim of achieving a high value of the product of stiffness and viscoelastic damping tan, without excess density. Stiffness and viscoelastic damping were measured over a wide range of frequency. For monodisperse 40% by volume SiC, and for hierarchical 60% by volume SiC the composite damping increased compared with the matrix at frequencies above 100 Hz. Composite shear modulus was almost a factor two greater than matrix for 40% and a factor of four greater than that of matrix for 60%. The product of stiffness and damping exceedsthat of well-known materials including polymer damping layers. Hashin–Shtrikman analysis modelled the observed stiffness increase. The damping increase at higher frequency cannot be accounted for by a purely mechanical composite model; it is attributed to thermoelastic coupling and an increase in matrix dislocations during fabrication.

In Situ Matrix Shear Response Using Torsional Test Data of Fiber Reinforced Unidirectional Polymer Composites

Abstract The in situ shear response of the matrix in polymer matrix composites (PMC) has been studied. Torsion tests were performed on solid cylinders of unidirectional glass fiber reinforced/vinylester and unidirectional carbon fiber reinforced/vinylester composites. The composite specimens were subjected to a uniform rate of twist. From the composite stress-strain curve, a plot of tangent shear modulus vs shear strain was derived. Then, using the Halpin-Tsai equations, the in situ matrix shear modulus was determined. The in situ matrix properties obtained from glass/vinylester and carbon/vinylester composites were found to be different. In addition, the properties of the in situ matrix were found to be a function of fiber volume fraction and the elastic properties of the reinforcing fiber. The behavior of the in situ matrix as a function of the fiber volume fraction was explained by using a three cylinder interphase model. The validity of the interphase model in predicting the composite shear modulus was studied by comparison of results against a conventional 2 cylinder model.