Tanaka Model Research Articles

CNT buckypaper/thermoplastic polyurethane composites with enhanced stiffness, strength and toughness

Abstract Carbon nanotube buckypaper/thermoplastic polyurethane elastomer composites were successfully fabricated. At certain polyurethane contents, the composites exhibited simultaneous improvements in stiffness (up to 6 GPa), strength (up to 120 MPa), ductility (up to 30%) and toughness (up to 36 MJ/m 3 ). The measured elastic modulus of the composites could be predicted by Mori–Tanaka model. The possible reinforcing mechanisms were discussed by Raman spectroscopy and SEM fractography. The results revealed that ductile polymers are very promising matrix in balancing the key mechanical properties of buckypaper, which are beyond the commonly used brittle thermosettings, e.g. epoxy resin.

A quadrilateral shallow shell element based on the third-order theory for functionally graded plates and shells and the inaccuracy of rule of mixtures

Abstract A four-node quadrilateral element is developed for the dynamic analysis of doubly curved functionally graded material (FGM) shallow shells, using the refined third order theory. Two micromechanics models, the Voigt's rule of mixtures (ROM) and the Mori–Tanaka model, are considered for computing the effective material properties at a point. The accuracy of the element is examined by comparing with various three dimensional elasticity and two dimensional (2D) analytical and finite element solutions available in the literature for static and free vibration responses of FGM plates and shells. It is shown that the present element, with the least number of degrees of freedom, achieves similar or better accuracy compared to other available 2D finite elements some of which are even based on higher order theories. Using this element, we also make a systematic assessment of the accuracy of the widely used ROM in predicting the behavior of FGM structures, for different values of the inhomogeneity parameter, and different geometrical parameters, boundary conditions, and material combinations. It is revealed that there can be very significant error in the deflection, stresses and natural frequencies predicted by the ROM, depending primarily on the inhomogeneity parameter and the difference in the material properties of the constituents.

Hydrogen bond and crystalline structure of the junction network in polyvinyl alcohol/dimethysulfoxide gels

The hydrogen bond and aggregation structures of polyvinyl alcohol (PVA) gels prepared with dimethylsulfoxide (DMSO) were investigated as were the the rheological properties. The variable temperature Fourier Transform Infrared (FTIR) and X-ray results of PVA-DMSO gels show that the hydrogen bond and crystalline structure are destroyed as temperature increases. In the Tanaka model, the value of ζ (the number of sequential units per chain in the junction) is 1.45 and s (the number of polymer chains in a single junction), which reveals the ability of solution to form gel, varied as a function of gel melting temperature T gm . The results of dynamic rheology show that the elastic modulus G′ is proportional to the gel concentration and inversely proportional to the temperature. It was found that gels formed from solutions with same viscosities showed different properties. This is because a gel with a higher polymerization degree has a lower value of s.

Multi-site micromechanics of composite materials with temperature-dependent constituents under small strain and finite thermal perturbation assumptions

This study presents mean-field based micromechanics models to predict the effective thermoelastic properties, namely, elasticities, thermal expansions and heat capacity, of thermoelastic composite materials with temperature-dependent constituents under finite temperature changes and small strain assumptions. First, the Helmholtz potential density for small strain finite thermoelasticity has been presented. The fundamental solution based on Green’s function of elasticity problem has been used to derive the general expressions for the elastic and thermal strains concentration tensors with temperature-dependent material properties. A family of mean-field based micromechanics models (multi-site Eshelby dilute model, Mori–Tanaka model and self-consistent model) has been presented. The models are general enough to account for the morphology and topology textures of the microstructure of a thermoelastic composite. Numerical examples based on the multi-site Mori–Tanaka model are used to quantify the differences of the effective properties based on the small strain finite thermoelasticity in comparison to the linear thermoelasticity. The predictions of the multi-site Mori–Tanaka micromechanics model are also compared to those of the VAMUCH, a finite element based micromechanics model.

The normalized interval regression model with outlier detection and its real-world application to house pricing problems

Tanaka and his colleagues initially proposed fuzzy linear regression models in 1982. From then on, fuzzy regression analysis has been widely studied by many researchers. Since Tanaka's model uses the inclusion relationship between the given data and the estimated data, it was pointed out that Tanaka's model is sensitive to outliers. How to deal with the outlier problem has been a very important issue and attracted much attention from fuzzy regression researchers. Given crisp multiple independent variables and single interval dependent variable, we introduce the normalized upper and lower interval regression models and propose two outlier detection approaches for them. The dual problem of the normalized upper interval regression model is used to find out potential outliers. For the normalized lower interval regression model, the existence of the outliers makes it infeasible. The relaxation approach is proposed to find out the outliers. By deleting the outliers, we can find out the possibilistic functional relationship of the major data. A case study involving a house pricing problem is analyzed in detail. Household size, loan ratio and annual household income are identified as the independent variables and acceptable purchase price is the interval dependent variable. With the data collected from the survey in Shanghai, the normalized upper and lower interval regression models are built with deleting the outliers. The proposed models provide managerial insights into the real estate market and important policy implications in regulating urban land development.

Thermo-mechanical properties of a piezoelectric polyimide carbon nanotube composite: Assessment of composite theories

In this work, we have characterized the thermomechanical properties of carbon nanotube based piezoelectric polymer nanocomposite using a hybrid force field for all atomistic molecular dynamic simulations. In addition, applicability of some of the well-known micromechanics composite theory in estimating carbon nanotube based polymer nanocomposite properties were assessed. We found that the primary reason for the strengthening effect of a nanocomposite with incorporation of a nanotube is the carbon–carbon bond and angle strength. The orientation of the nanotube in the polymer matrix is key to its reinforcement effect on a nanocomposite. We also observed that a perfect axial orientation does result in improving the axial modulus, but in the radial direction any strengthening for such a unidirectional composite does not seem possible without any bonding at the interface between the filler and the matrix material. The self-consistent field theory was found to be the closest with the atomistic simulation results for predicting mechanical properties of a polymer nanocomposite. The Halpin–Tsai model also demonstrated reasonable capability in predicting the strengthening effect. Mori–Tanaka model, however, underestimated the strengthening effect of carbon nanotube on the polymer matrix. It was also found that the existing composite theories are better at estimating low weight percentage nanotube strengthening effect.

A comprehensive validation of analytical homogenization models: The case of ellipsoidal particles reinforced composites

This paper presents a rigorous and exhaustive evaluation of the analytical homogenization models accuracy for the case of randomly distributed and oriented ellipsoidal fibers reinforced composites. Artificial random microstructures were generated using a molecular dynamics (MD) algorithm. Numerical effective properties were computed using a Fast Fourier Transforms (FFT) based technique. The numerical predictions were compared to those of the analytical models for a wide range of phases mechanical properties, fibers volume fractions and aspect ratios. The validation campaign involved a rigorous Representative Volume Element (RVE) determination process and approximately, 66,000 simulations were performed. The results show that the analytical models accuracy is more sensitive to the mechanical properties contrasts than to the fibers volume fraction and aspect ratio. Among all the studied models, Lielens’ model remains the most accurate, especially for high contrasts and volume fractions. For high aspect ratio fibers, Lielens’s model and Beveniste’s interpretation of the Mori–Tanaka model provide similar estimates, especially when predicting the effective shear modulus. In this case, the latter could be an alternative to Lielens’ model, especially for composites where the fibers are not completely stiffer than the matrix. All conclusions of this study apply to both prolate and oblate ellipsoidal fibers.

The relation between particle density and static elastic moduli of lightweight expanded clay aggregates

Lightweight expanded clay aggregates are one type of artificial lightweight aggregates (LWAs) which have widespread application. They are produced in a rotary kiln. Output aggregates of kiln often contain a wide grading curve. The micromechanics method with some simplifications has been used for determining the elastic modulus of the different sizes of these aggregates. In general, the smaller the LWA particle size, the higher is the density. The effect of the density and size of aggregates on elastic moduli are investigated in this research. Two groups of quaternary lightweight expanded clay aggregates are used, with each group produced under the following identical conditions: particle density within the range of 480 to 1100kg/m3 and the domain sizes from 4 to 14mm. The elastic modulus of expanded clay aggregates were determined by the micromechanics method and using the Mori–Tanaka model based on the elastic properties of mortar matrix and lightweight concrete which were determined experimentally. Based on results the elastic modulus of aggregates ranged from 0.6 to 6.3GPa and there is a linear relation between the elastic modulus and the particle density of lightweight expanded clay aggregates. Their elastic modulus exponentially decreases with the increase in particle size.

Nonlinear vibration of shear deformable FGM cylindrical panels resting on elastic foundations in thermal environments

This paper investigates the large amplitude vibration behavior of a shear deformable FGM cylindrical panel resting on elastic foundations in thermal environments. Two kinds of micromechanics models, namely, Voigt model and Mori–Tanaka model, are considered. The motion equations are based on a higher order shear deformation shell theory that includes shell panel-foundation interaction. The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent. The equations of motion are solved by a two step perturbation technique to determine the nonlinear frequencies of the FGM cylindrical panel. Detailed parametric studies are carried out to investigate effects of volume fraction index, temperature variation, panel curvature ratio, foundation stiffness and in-plane boundary conditions on nonlinear vibration behaviors of FGM cylindrical panels. The results confirm that in most cases Voigt model and Mori–Tanaka model have the same accuracy for predicting the vibration characteristics of FGM cylindrical panels.

Determination of Parameter Value Domain for Macro-micro Coupled Viscoplastic Constitutive Model Consider Dynamic Recrystallization

Parameter identification of macro-micro coupled constitutive model is generally taken through inverse analysis. To make the identified result with high confidence degree, the reasonable parameter value domain must be given. Based on the physical mechanism, a procedure is proposed. The procedure can be applied to determine the value domain of the parameter, which is involved in a macro-micro coupled constitutive model considering dynamic recrystallizaton. To improve the robustness of the parameter identification result, some parts of the model is revised and the steady flow stress is described by Tanaka model; a six-step method for the determination of the parameter value domain is given based on the physical mechanism involved in the model; the hot compression deformation of low alloy steel 300M is carried out under different combination of temperature and strain rate, the datum on true stress-strain curve, recrystallized volume fraction, recrystallized grain size, unrecrystallized grain size, maximum grain size difference are obtained, and the steady flow stress in unrecrystallized zone and recrystallized zone is determined, based on the true stress-strain curves. Based on the proposed procedure and the corresponding experimental data, the reasonable parameter ranges are given. The research provides a foundation for parameter identification of macro-micro coupled constitutive model.

Average electro-mechanical properties and responses of active composites

This study deals with the overall electro-mechanical response of randomly positioned spherical particles reinforced piezoelectric composites. Different composites comprising of linearly elastic and piezoelectric constituents were studied. For the piezoelectric constituent, both linear and nonlinear electro-mechanical coupling behaviors were considered. Numerical representative volume elements (RVEs) were generated and finite element (FE) method was used in order to compute overall electro-mechanical response of the RVEs. The electro-mechanical predictions of the RVEs were compared against those of Mori–Tanaka, self-consistent and simplified unit-cell micromechanical models. A new first moment secant linearization was introduced in order to perform the homogenization of the nonlinearly piezoelectric composites followed by iteration in order to minimize errors (residual) from the linearization. For all boundary conditions, including nonlinear response, simulated in this work, the predictions given by the Mori–Tanaka and UC models were reasonably close to the ones of the RVE cases. Finally the RVEs were modified to examine the linear and nonlinear electro-mechanical responses of piezoelectric ceramics with pores. Depending on the prescribed boundary conditions, the existence of pores could significantly alter the electro-mechanical response of piezoelectric ceramics.

Micromechanical models for the effective time-dependent and nonlinear electromechanical responses of piezoelectric composites

This study introduces micromechanical models for analyzing the overall electromechanical responses of piezoelectric composites comprising polarized piezoelectric ceramics and polymeric constituents. The polarized piezoelectric ceramics can experience nonlinear electromechanical responses due to an application of large electric fields, while the polymer exhibits viscoelastic response. Thus, the piezoelectric composites can experience significant time-dependent and nonlinear electromechanical coupling behaviors. Two micromechanical models are considered: the Mori–Tanaka and unit-cell models. Linearized micromechanical relations are first defined for obtaining the overall responses of the piezoelectric composites followed by iterative schemes in order to correct errors from linearizing the nonlinear responses. Numerical results are presented for two composite systems, that is, piezoelectric unidirectional fiber with circular/square cross section and spherical/cubic particle inhomogeneities embedded in a polymeric matrix. The linear electromechanical responses from the two micromechanical models are compared with the experimental data available in the literature. Parametric studies are performed in order to examine the effect of inhomogeneity geometry and compositions and prescribed boundary conditions on the overall time-dependent and nonlinear electromechanical responses of the composites.

Vibration analysis of axially moving line supported functionally graded plates with temperature-dependent properties

Vibration analysis of axially moving functionally graded plates with internal line supports and temperature-dependent properties is investigated using harmonic differential quadrature method. The plate is subjected to static in-plane forces while out-of-plane loading is dynamic. Stability of an axially moving plate, traveling at a constant velocity between different supports and experiencing small transverse vibrations are considered. The series of internal rigid line supports parallel to the plate edges are considered together with various arbitrary combinations of boundary conditions. Material properties of the plate are assumed temperature-dependent which is a non-linear function of temperature and differ continuously through thickness according to a power-law distribution of the volume fractions of the plate constituents. Two types of micromechanical models, namely, the Voigt and Mori–Tanaka models are considered. Based on the classical plate theory, the governing equations are obtained for functionally graded plate using the Hamilton’s principle. In the frame of a general dynamic analysis, it is shown that the onset of instability takes place in the form of divergence. The plate may experience divergence or flutter instability at a super critical velocity. Results for dynamic analysis of isotropic and laminated plates are validated with available data in the existing literature, which show excellent agreement. Furthermore, some new results are presented for vibration analysis of functionally graded material plates to study effects of the location of line supports, material properties, volume fraction, temperature, loading, aspect ratio and speed.

Prediction of elastic properties in polymer–clay nanocomposites: Analytical homogenization methods and 3D finite element modeling

Accurate predictive models can support the exploitation of Polymer–Clay Nanocomposites (PCN) in their evergrowing applications. The purpose of this paper is to validate commonly used analytical micromechanical models for the prediction of PCN’s elastic properties with the help of 3D periodic Finite Element (FE) simulations of the same microstructures, considered as reference data. The effect of the interphase was taken into account in a two-step homogenization procedure that exploits the effective particle concept. The predictions of a range of procedures relying on the different micromechanical models (i.e. Mori–Tanaka, self-consistent and Lielens’s) were tested. An elaborate series of FE simulations was performed to determine Representative Volume Elements (RVEs). In addition, a simplified procedure to guide the definition of the RVE based on statistical and transverse isotropy symmetry criteria was developed. The predicted elastic properties of PCN were studied as a function of the thickness and the elastic properties of the formed interphase around the nanoclay platelets. It was found that the Mori–Tanaka model was the most reliable method for the simulated cases. Furthermore, numerical model predictions were compared with experimental results extracted from the literature for aligned exfoliated Nylon-6 Montmorillonite nanocomposites. Finally, a comparison between the parametric study results and the experimental data was used to estimate the properties and the thickness of the interphase.

Free vibration analysis of two-dimensional functionally graded cylindrical shells

In this paper, the free vibration of a two-dimensional functionally graded circular cylindrical shell is analyzed. The equations of motion are based on the Love’s first approximation classical shell theory. The spatial derivatives of the equations of motion and boundary conditions are discretized by the methods of generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ). Two kinds of micromechanics models, viz. Voigt and Mori–Tanaka models are used to describe the material properties. To validate the results, comparisons are made with the solutions for FG cylindrical shells available in the literature. The results of this study show that the natural frequency of the material can be modified in order to meet the expected results through manipulation of the constituent volume fractions. A comprehensive comparison is then drawn between ordinary and 2-D FG cylindrical shells.

Warpage analysis of a micro-molded parts prepared with liquid crystalline polymer based composites

The purpose of this study is to understand the warpage behavior of a micro-injection molded part induced by anisotropy of the material. Microstructural anisotropy was characterized to understand warpage behavior of liquid crystalline polymer based composites. A new definition of the microstructure was introduced along the thickness of a molded part, i.e., “skin–shear–core” layer was proposed depending on the orientation of LCP molecules and fillers. The microstructural anisotropy of LCP composites was verified by using WAXD, CT, and SEM analyses. The anisotropic elastic modulus and coefficient of thermal expansion (CTE) of micro-injection molded parts were investigated by using a new numerical method with the modified Mori–Tanaka (M–T) model. The interaction coefficient, CI, and coefficient of thermal expansion (CTE) were the most important factors determining the warpage of a micro-injection molded part made of an anisotropic material. The control of warpage due to the orientation of glass fibers was investigated by simulation.

Micromechanical modeling of magnetoelectroelastic composite materials with multicoated inclusions and functionally graded interphases

In this article, micromechanical modeling of magnetoelectroelastic composites with multicoated inclusions and functionally graded interphases are elaborated. The integral equation taking into account the continuously varying interphase properties as well as the multifunctional coating effects is introduced based on Green’s tensors and interfacial operators. Magnetoelectroelastic composites with functionally graded interphases are analyzed, and the effective properties are derived. Based on the Mori–Tanaka, Self-Consistent, and Incremental Self-Consistent models, the numerically predicted effective properties of magnetoelectroelastic composites are presented with respect to the volume fractions, shapes of the multicoated inclusions, and the thickness of the coatings. The multicoating and functionally graded interphase concepts can be used to optimize the effective properties of multifunctional composites. This can be used to design new multifunctional composite materials with higher coupling coefficients.

Green's function approach to the nonlinear transient heat transfer analysis of functionally graded materials

This paper presents a Green's function approach to the solution of nonlinear transient heat transfer problem in a functionally graded material (FGM) object for the purpose of both the numerical analysis accuracy and computational efficiency. The Green's functions of transient heat transfer problem for a short hollow cylinder with power law functionally graded materials have been obtained analytically. The generalized expressions of the effective material properties in FGM are simplified based on micromechanical models (Voigt model and Mori–Tanaka model). The nonlinearity resulting from the temperature-dependent properties is resolved using the artificial parameter method. Two examples have been given to compare the effectiveness of the proposed Green's function approach with the finite element method (FEM), and the accuracy is satisfactory. This approach can become a cost-effective and accurate method for FGM structure design and risk assessment.

Manufacturing process improvement and mechanical modelling of multiwalled carbon nanotube/epoxy composites

Multiwalled carbon nanotube (MWNT) reinforced epoxy resin composites were fabricated and characterised. Several process variables were investigated using design of experiments. The MWNTs (0·5 wt-%) were dispersed in Epon 862 epoxy resin under various sonication conditions. Young’s modulus, energy to failure, glass transition temperature Tg and storage modulus E′ were assessed. The first three were selected in a design of experiment optimisation study. The results indicated that as the sonication intensity and the duration of sonication were increased, the material response of the MWNT/epoxy composite, specifically Young’s modulus, energy and Tg, was enhanced. The fracture surfaces of the composites were examined using scanning electron microscopy. Improved dispersion was observed in samples fabricated with increased sonication intensities. The Mori–Tanaka model was used to predict the mechanical properties of the MWNT/epoxy composites and was found to be in reasonable agreement with experimental data. Nonetheless, the experimental results yielded slightly inferior properties to those from the model prediction.

Estimating salt intake in a Caucasian population: can spot urine substitute 24-hour urine samples?

A simple and valid alternative for 24-hour urine collection to estimate populational 24-hour urinary sodium excretion would be desirable for monitoring sodium intake in populations. To assess the validity of the predicted 24-hour urinary sodium excretion using spot urine and two different prediction methods in a Danish population. Overall, 473 Danish individuals provided a para-aminobenzoic acid-validated complete 24-hour urine collection and a spot urine sample. Data were collected in the DanThyr study (248 women aged 25-30 years and 60-65 years) and the Inter99 study (102 men and 113 women aged 30-60 years), respectively. The measured 24-hour urine sodium excretion was compared with the predicted 24-hour sodium excretion from a causal urine specimen, using both the Tanaka prediction method and a prediction model developed in a Danish population. The measured 24-hour sodium excretion (median, 5th to 95th percentile) was men 195 (110 to 360) and women 139 (61 to 258), whereas the predicted 24-hour sodium excretion for the Tanaka model was men 171 (117 to 222) and women 153 (92 to 228) and for the Danish model was men 207 (146 to 258); women 134 (103 to 163). The Spearman correlation between predicted and measured 24-hour sodium excretion was 0.39 and 0.49 for the Tanaka and the Danish model, respectively. For both prediction models, the proportion of individuals classified in the same or adjacent quintile was 74% for men and 64% for women. Both prediction models gave a reasonable classification of individuals according to their sodium excretion. However, the median daily sodium intake was estimated more precisely by the Danish model, especially among men.