Mori-Tanaka Homogenization Scheme Research Articles

A multiscale constitutive model of magnesium-shape memory alloy composite

In this work, a multiscale constitutive model is established to describe the deformation behaviors of magnesium-shape memory alloy (Mg-SMA) composite in a wide temperature range and reveal the strengthening mechanism of SMA reinforcement on Mg. The model is established at the grain scale firstly and gradually transited to the macroscopic scale by employing a newly developed three-level scale transition rule. At the grain scale, the thermodynamic-consistent constitutive models of Mg and SMA are, respectively, constructed by addressing different inelastic deformation mechanisms. The basal, prismatic, pyramidal, slip systems and extension twinning system are considered for the Mg phase, and the martensite transformation (MT) and austenitic plasticity are addressed for SMA reinforcement. Thermodynamic driving forces of each inelastic deformation mechanism are derived from the dissipative inequality and the constructed Gibbs free energies. At the polycrystalline scale, to evaluate the interactions among the grains and pores, and obtain the whole responses of the polycrystalline Mg and SMA, a thermo-mechanically coupled self-consistent homogenization scheme is employed. At the mesoscopic scale, a modified thermo-mechanically coupled Mori-Tanaka's homogenization scheme is adopted to evaluate the interaction between the Mg phase and SMA phase, and predict the whole responses for the representative volume element (RVE) of the composite. According to the geometrical features and mechanical loadings applied on the specimen, a hypothesis of homogeneous stress and strain fields at the macroscopic scale is adopted to achieve the scale transition from the RVE of the composite to the whole specimen. The capacity of the multiscale model is verified by comparing the predictions with the existing experimental data (Aydogmus, 2015). Moreover, the influences of characteristic information for the microstructures at different spatial scales on the deformation behaviors of the composite are predicted and discussed.

A new micromechanical damage model for quasi-brittle geomaterials with non-associated and state-dependent friction law

In this paper, we shall present a new micromechanical damage model for capturing the mechanical behaviors of quasi-brittle geomaterials under a wide range of compressive stress. Two essential coupled non-linear mechanisms are taken into account: plastic deformation related to frictional sliding along the microcrack surfaces, and microscopic damage induced by the initiation and propagation of microcracks. Within the framework of thermodynamics and Mori–Tanaka homogenization scheme, free energy and thermodynamic forces, including the local stress applied on the microcracks and damage driving force are deduced. The main novelty of this work lies in presenting a new physically based non-associated and state-dependent friction law to capture the plastic distortion, in which the friction coefficient is no longer a constant but a state-dependent variable varying with the surface asperities of microcracks. The relationship between the friction coefficient and mean local stress, as well as microscopic damage is subtly defined. Intriguingly, a macroscopic strength criterion is derived as an inherent part of the proposed model without rotation of the principal stress. Moreover, semi-analytical stress–strain–damage relationships of the proposed model can be derived under conventional triaxial compression. These up-scaling analyses provide a powerful tool for parameter calibration. For application, the semi-implicit return mapping algorithm with plasticity-damage decoupling correction procedure (SRM-PDDC) is adopted for numerical implementation. To demonstrate the performance of the micromechanical model, we compare the model predictions with experimental data of quasi-brittle rocks in literature as well as the results obtained by Zhu et al. (2016). Finally, the applicable condition of the SRM-PDDC algorithm is discussed in detail.

A micromechanical constitutive model of high-temperature shape memory alloys

In this work, a new micromechanical constitutive model is constructed for high-temperature shape memory alloys. Different from the classical constitutive models, the microstructural patterns, heterogeneities of stress and strain fields at the microscopic scale are addressed. In the proposed model, three different inelastic deformation mechanisms, i.e., reversible martensitic transformation, irreversible transformation-induced plasticity, viscoplasticity, as well as their complex interactions, are considered. In addition to the well-known austenite and martensite phases, a new austenite-martensite interface phase is introduced to address the high local stress induced by the martensitic transformation. The modified Mori-Tanaka's homogenization scheme and Hill's interfacial operator are employed to determine the non-uniform stress fields of the three phases. To satisfy the thermodynamic compatibility of the proposed model, the driving forces and evolution equations of the three inelastic deformation mechanisms are derived based on the constructed Helmholtz free energy and the fundamental laws of irreversible thermodynamics. Inheritances of the plastic deformation induced by moving austenite-martensite (A-M) interface during repeated martensitic transformation and its reverse are incorporated. Temperature evolution caused by internal heat production and heat exchange with the ambient medium is also addressed. To validate the capability of the new constructed model, the predicted results of creep behavior and stress-assisted two-way shape memory effect of NiTiPd high-temperature shape memory alloys are compared with the experimental ones. The proposed model provides a theoretical tool for simultaneously analyzing the deformation behaviors at the macroscopic scale and underlying mechanisms at the microscopic scale.

Extended mean-field homogenization of unidirectional piezoelectric nanocomposites with generalized Gurtin-Murdoch interfaces

This paper presents for the first time an extended Mori-Tanaka approach aimed at identifying the little-explored piezoelectric response of unidirectional nanoporous composites with energetic surfaces. The interface is simulated using the generalized Gurtin-Murdoch coherent interface model considering nonvanishing in-plane surface stress and surface electric displacement. The analytical solutions for Eshelby’s inhomogeneity problems are obtained by solving four systems of equations, generated from the known macroscopic electromechanical loading conditions. The dilute concentration tensors for the fiber and energetic surface are evaluated exactly for coupled electromechanical fields, which are then utilized in the extended multiphysics Mori-Tanaka homogenization scheme. The reliability and accuracy of the extended theory are demonstrated through an extensive comparison with the predictions of the composite cylinder assemblage (CCA), the generalized self-consistent method (GSCM), as well as the finite-element (FE) homogenization technique. New results generated in this work demonstrate that, except for the transverse shear modulus, the extended Mori-Tanaka, CCA, and FE provide indistinguishable results under axisymmetric, axial shear, and electric loading, even at high volume fractions and small pore sizes where the surface piezoelectric effects are significant.

Experimental Investigation and Micromechanical Modeling of Hard Rock in Protective Seam Considering Damage–Friction Coupling Effect

The hard rock in the protective coal seam of the Pingdingshan Mine in China is a typical quasi-brittle material exhibiting complex mechanical characteristics. According to available research on the mechanical property, the inelastic deformation and development of damage are considered related with crack initiation and propagation, which are main causes of the material degradation. In the present study, an original experimental investigation on the rock sample of the Pingdingshan coal mine is firstly carried out to obtain the basic mechanical responses in a conventional triaxial compression test. Based on the homogenization method and thermodynamic theory, a damage–friction coupled model is proposed to simulate the non-linear mechanical behavior. In the framework of micromechanics, the hard rock in a protective coal seam is viewed as a heterogeneous material composed of a homogeneous solid matrix and a large number of randomly distributed microcracks, leading to a Representative Elementary Volume (REV), i.e., the matrix–cracks system. By the use of the Mori–Tanaka homogenization scheme, the effective elastic properties of cracked material are obtained within the framework of micromechanics. The expression of free energy on the characteristic unitary is derived by homogenization methods and the pairwise thermodynamic forces associated with the inelastic strain and damage variables. The local stress tensor is decomposed to hydrostatic and deviatoric parts, and the effective tangent stiffness tensor is derived by considering both the plastic yield law and a specific damage criterion. The associated generalized Coulomb friction criterion and damage criterion are introduced to describe the evolution of inelastic strain and damage, respectively. Prepeak and postpeak triaxial response analysis is carried out by coupled damage–friction analysis to obtain analytical expressions for rock strength and to clarify the basic characteristics of the damage resistance function. Finally, by the use of the returning mapping procedure, the proposed damage–friction constitutive model is applied to simulate the deformation of Pingdingshan hard rock in triaxial compression with respect to different confining pressures. It is observed that the numerical results are in good agreement with the experimental data, which can verify the accuracy and show the obvious advantages of the micromechanic-based model.

Superelasticity degradation of NiTi shape memory alloy in wide ranges of temperature and loading level: Experimental observation and micromechanical constitutive model

In this work, cyclic tension-unloading tests with different peak strains (10%-18%) in the temperature range from 313 K to 393 K are performed to investigate the effects of temperature and loading level on the cyclic deformation behavior of NiTi shape memory alloy (SMA) wires. Experimental results demonstrate that the superelasticity degradation occurs during the cyclic deformation process, and such phenomenon aggravates with the rise of ambient temperature and loading level, which can be attributed to the complex interactions among the martensite transformation (MT), austenite plasticity (AP), martensite plasticity (MP) and the transformation-induced plasticity (TRIP). Then, a micromechanical cyclic constitutive model is proposed based on the framework of irreversible thermodynamics and Eshelby's inclusion theory. The martensite (M) phase is treated as mobile inclusions with alterable eigenstrain embedded in the austenite (A) phase matrix. The volume fractions of the M-phase, A-phase, and A-M interface-phase are introduced. The non-uniform stress fields in the three phases are estimated by employing the Mori-Tanaka's homogenization scheme and interfacial operator. The driving forces of the four types of the inelastic deformation mechanisms, MT, AP, MP, and TRIP are derived based on the proposed new Helmholtz free energy, instantaneous growth hypothesis of M domains, and the energy dissipation inequality. The inheritances of the plastic deformation induced by the movement of A-M interfaces during the repeated MT and its reverse are incorporated. Finally, to validate the predictive capability of the proposed model, the predicted results for the superelasticity degradation of NiTi SMA with various peak strains at different ambient temperatures are compared with the experimental ones. Moreover, the dominant plastic deformation mechanisms and the effects of AP, MP, TRIP on the cyclic deformation of NiTi SMA are discussed.

Geometric nonlinear transient analysis of mechanically and thermally shocked functionally graded shell panels

The present paper investigates the von-Karman geometrically nonlinear structural responses of ceramic-metal functionally graded shell panels under the effect of the cases of mechanical and thermal shocks on their ceramic-rich surface. The constituents in the FGM are temperature-dependent and are graded as per the recently proposed four-parameter power laws while their effective properties in the shells are obtained using the Mori-Tanaka homogenization scheme. The transient thermoelastic problem is solved by importing the temporally varying temperature distribution from the nonlinear 1D heat equation as thermal loads into the FE code developed for thin and moderately thick shells based on higher-order shear deformation theory. The nonlinearity in the heat equation is linearized using Picard method in each Crank-Nicholson time-step while the linearization step in the geometric nonlinear problem uses Newton-Raphson method in each Newmark time-step. The new results presented in the manuscript study the effect of geometric nonlinearity on the responses of FG shell panels and the effect of different parameters of four-parameter FG laws. It was observed that including geometric nonlinearity significantly affects the responses of the shell panel structures when the loading direction, either mechanical or thermal, along with the gradation direction is reversed.

Static Analysis of Skew Functionally Graded Plate Using Novel Shear Deformation Theory.

In this article, the static response of a functionally graded material (FGM) plate is studied via hybrid higher-order shear deformation theory which uses hyperbolic and polynomial shape functions and includes the effect of thickness stretching. The composition of the plate comprises metallic and ceramic phases. The ceramic volume fraction varies gradually along with the thickness following the power law. The mechanical properties of the FGM plate are determined by the rule of mixtures and the Mori–Tanaka homogenization scheme. The displacement fields are defined to satisfy the requirement of traction-free boundary conditions at the bottom and top surfaces of the plate surface removing the need for determination of shear correction factor. A C0 continuity FE model is developed for the present mathematical model. Nine-node isoparametric elements with eight nodal unknowns at each node are developed. The present model comparison with existing literature is completed and found to be coherent. Inhouse MATLAB code is developed for the present work. Sinusoidal and uniformly distributed loading is analyzed in the present work. The parametric study is undertaken to explore the effect of the side-to-thickness ratio, aspect ratio, thickness, and volume fraction index on stresses and transverse displacements.

Calibration of the PA6 Short-Fiber Reinforced Material Model for 10% to 30% Carbon Mass Fraction Mechanical Characteristic Prediction.

Short-fiber reinforced composites are widely used for the mass production of high-resistance products with complex shapes. Efficient structural design requires consideration of plasticity and anisotropy. This paper presents a method for the calibration of a general material model for stress–strain curve prediction for short-fiber reinforced composites with different fiber mass fractions. A Mori–Tanaka homogenization scheme and the J2 plasticity model with layered defined fiber orientation were used. The hardening laws: power, exponential, and exponential and linear were compared. The models were calibrated using experimental results for melt front, orientation tensor analysis, fiber length, and diameter and tension according to ISO 527-2, for samples of PA6 which were either non-reinforced, or reinforced with 10%, 15%, 20%, and 30% carbon fiber mass fractions. The novelty of this study lies in the transition from the strain–stress space to the strain–stress–fiber fraction space in the approximation of the material model parameters. We found it necessary to significantly reduce the fiber aspect ratio for the correct prediction of the mechanical characteristics of a composite using the Mori–Tanaka scheme. This deviation was caused by the ideal solution of ellipsoidal inclusion in this homogenization scheme. The calculated strength limits using Tsai–Hill failure criteria, based on strain, could be used as a first approximation for failure prediction.

A viscoelastic constitutive model for shape memory polymer composites: Micromechanical modeling, numerical implementation and application in 4D printing

In this work, a novel micromechanics-based thermo-viscoelastic constitutive model for shape memory polymer composites (SMPCs) is proposed and applied to four-dimensional (4D) printed SMPCs. The multi-branch constitutive model is used to simulate the time- and temperature-dependent mechanical behavior of the shape memory polymer matrix. According to the elastic–viscoelastic correspondence principle, the equivalent viscoelastic stiffness tensor of the composite is obtained in the micromechanics framework of energy-based effective strain theory and the Mori-Tanaka homogenization scheme, in which a two-parameter interfacial damage model is adopted to consider the displacement discontinuity and traction continuity conditions at the interface between the inclusion and the matrix. The numerical integration scheme for the three-dimensional (3D) viscoelastic constitutive model of SMPCs is presented, and the finite element application is implemented by the user material subroutine UMAT of ABAQUS. After identifying the model parameters with experimental data, the tensile stress-strain curves and stress relaxation phenomena of 4D printed unidirectional SMPCs are successfully described by theoretical simulations. Besides, theoretical simulations also adequately predict the shape memory behavior of 4D printed composites and complex members.

Micromechanical elastoplastic limit analysis of in-plane bending of Functionally Graded Pipe elbows

Pipe elbows, also known as pipe bends, are important components of a piping system. These curved elements are very flexible when compared to their straight counterparts, and are often responsible for reducing the reaction forces and moments within the piping system. However, due to large deformations, the elbows are more likely to exceed their stress limits. The effect of in-plane bending on plastic limit loads of Functionally Graded (FG) pipe bends is presented using three-dimensional non-linear finite element analysis and a micromechanical constitutive model. The effective properties of the material are estimated by the Mori–Tanaka homogenization scheme and the TTO plasticity model. The analysis was carried out for different loading cases, geometries and volume fraction distribution. The results showed that an FG elbow has a very distinct behavior comparatively to a similar homogeneous elbow made out of metal, specially in terms of limit moments and stress distribution. Finally, an analytical model to estimate the limit moment of FG elbows is presented.

Effects of MgSO4 on Calcium-Silicate-Hydrate

Abstract Magnesium sulfate (MgSO4) is a salt that has the potential to damage portland cement-based materials. This paper argues that the mechanism of damage is dependent on the concentration of MgSO4 and reports the results of the change in the micromechanical properties of calcium-silicate-hydrate (C-S-H) when exposed to different MgSO4 concentrations . Different concentrations were selected starting from the concentration found in seawater and increasing up to 20 g/l, which is the concentration considered for accelerated tests. The micromechanical properties of C-S-H were probed using nanoindentation; X-ray diffraction measurements were also performed on C-S-H specimens. A Mori-Tanaka homogenization scheme was employed to upscale these results, yielding two homogenization levels. At low MgSO4 concentrations (2.2 and 4 g/l) the formation of brucite and gypsum crystals in the pore solution contributes to the overall magnitude of the elastic modulus of the specimen, while at higher concentrations of MgSO4 (10 and 20 g/l), decalcification of C-S-H is observed, which results in the degradation of C-S-H micromechanical properties.

Multi-objective optimization for lightweight design of bi-directional functionally graded beams for maximum frequency and buckling load

This paper proposes a multi-objective particle swarm optimization (MOPSO) algorithm integrated with a semi-analytical solution to solve optimization problems of lightweight design of bi-directional functionally graded (BDFG) beams for maximum fundamental frequency and/or maximum critical buckling load. The MOPSO algorithm is utilized to search for Pareto-optimal solutions that present the best trade-off optimal material distributions to achieve the required objectives. The proposed semi-analytical solution replaces the time-consuming numerical methods in computing the objective functions during the optimization process. The material volume fraction is described in both longitudinal and thickness directions by a 10-parameter trigonometric function. These parameters are taken as the design variables of the optimization problem. The effective material properties of the BDFG beam are estimated according to the Mori–Tanaka homogenization scheme. The governing variable-coefficients differential equations of BDFG beams under different boundary conditions are derived based on Euler-Bernoulli beam theory then solved by the proposed semi-analytical method for the fundamental frequency and buckling load. The accuracy, efficiency, and applicability of the proposed method are demonstrated through several multi-objective optimization problems. The predicted optimal results are compared with those of other methods to investigate the reliability of the proposed method. The optimization results show that the proposed 10-parameter function provides flexible material profiles and gives designers a powerful tool for optimal material distributions. More importantly, the obtained optimal material distribution is essential for the manufacture of BDFG beams.

A multiscale magneto-thermo-mechanically coupled model for ultra-low-field induced magneto-elastocaloric effect in magnetostrictive-shape memory alloy composite system

Recent experimental results show that apparent magneto-elastocaloric effect can be induced in the magnetostrictive-shape memory alloy composite system under an ultra-low magnetic field. In this paper, a multiscale theoretical model is constructed to predict the magneto-thermo-mechanically coupled response of such a composite system by considering the multi-field interactions among the heterogeneous constituent elements in the grain, polycrystalline aggregate and macroscopic scales. In the grain scale, for the magnetostrictive alloy (MEA), the fluctuations of stress and magnetic fields caused by the interactions among the domains are addressed by the self-consistent homogenization scheme. Adopting a probabilistic domain switching criterion based energetic analysis, a constitutive model of MEA is established. For the shape memory alloy (SMA), a crystal plasticity based thermo-mechanically coupled constitutive model is constructed in the framework of irreversible thermodynamics. The interactions among austenite phase and martensite variants are considered by the Mori-Tanaka's homogenization scheme. Thermodynamic driving force for martensite transformation and the internal heat production originated from transformation latent heat and inelastic deformation dissipation are derived from the dissipative inequality and the conservation of energy, respectively. In the polycrystalline aggregate scale, to estimate the interactions among the grains and predict the overall responses of the polycrystalline aggregates of MEA and SMA, a unified incremental magneto-thermo-mechanically coupled self-consistent homogenization scheme is developed. In the macroscopic scale, the magneto-thermo-mechanical interaction equations among the MEA rod, SMA cuboid and Al alloy frame in the composite system are derived by considering the conditions of deformation compatibility, force balance and thermodynamic equilibrium. The capability of the proposed multiscale model to describe the magneto-elastocaloric effect of MEA-SMA composite system is validated by comparing the predictions with the existing experimental data. Moreover, the influences of geometric dimension, pre-load, frame's stiffness and crystallographic orientation on the magneto-elastocaloric effect of the composite system are discussed. The proposed model provides a theoretical guidance for the optimization design of solid-state refrigeration devices in both the microscopic and macroscopic scales.

Tailoring the thermal conductivity of functional cementitious composites with micro core-shell particles: A multiscale homogenization study

Micro core-shell particles have been highlighted in developing energy-efficient functional cementitious composites (FCCs). In this paper, a multiscale model is established to predict the thermal property of cementitious matrices containing micro core-shell particles. Based on the composite material theory, the interfacial thermal conductance and inclusion size distribution are introduced to the Mori-Tanaka homogenization scheme. Following the scale division of FCCs, the effective thermal conductivity is homogenized step by step from the cement paste level to micro-encapsulated cementitious composites level. In what follows, the coupling effects of composition, microstructural and physical aspects are quantified through the model parametrization. As validated by experimental results, accurate numerical predictions can be obtained with the variations of w/c ratio, hydration degree, and mix proportion. Numerical parameter analysis further reveals that the thermal conductivity and volume fractions of composition phases dominate the maximum thermal design range of FCCs, while the size-dependent interfacial thermal conductance matters more when the thermal conductivity of shell material is lower than that of the core material. The findings provide a theoretical basis for the thermal design of multi-phase composites, and contribute to the rational utilization of micro particles to achieve desirable thermal properties of FCCs.

A multiscale four-layer finite element model to predict the effects of collagen fibers on skin behavior under tension.

Human skin is a complex multilayered multiscale material that exhibits nonlinear and anisotropic mechanical behavior. It has been reported that its macroscopic behavior in terms of progression of wrinkles induced by aging is strongly dependent on its microscopic composition in terms of collagen fibers in the dermis layer. In the present work, a multiscale four-layer 2D finite element model of the skin was developed and implemented in Matlab code. The focus here was to investigate the effects of dermal collagen on the macroscopic mechanical behavior of the skin. The skin was modeled by a continuum model composed of four layers: the Stratum Corneum, the epidermis, the dermis, and the hypodermis. The geometry of the different layers of the skin was represented in a 2D model with their respective thicknesses and material properties taken from literature data. The macroscopic behavior of the dermis was modeled with a nonlinear multiscale approach based on a multiscale elastic model of collagen structure going from cross-linked molecules to the collagen fiber, combined with a Mori-Tanaka homogenization scheme. The model includes the nonlinear elasticity of the collagen fiber density, the fiber radius, the undulation, and the fiber orientation. An axial tension was applied incrementally to the lateral surfaces of the skin model. A parametric study was performed in order to investigate the effect of the collagen constituents on the macroscopic skin mechanical behavior in terms of the predicted macroscopic stress-strain curve of the skin. The results of the FE computations under uniaxial tension showed that the different layers undergo different strains, leading to a difference in the transversal deformation at the top surface. In addition, the parametric study revealed a strong correlation between macroscopic skin elasticity and its collagen structure.

Micro-Macroscopic coupled modeling for the prediction of synergistic improvement on the thermal conductivity of boron nitride and multi-walled carbon nanotube reinforced composites

The use of hybrid fillers consisting of phycochemically different nanoparticles has been widely investigated to realize synergistic improvement of thermal conduction behaviors. The enhancement of thermal conductivity of composites is often limited by the thermal contact resistance of the reinforcing particles. We prepare a series of hybrid nanofiller reinforced composites composed of two-dimensional boron nitrides and one-dimensional multiwalled carbon nanotubes to investigate the interfacial thermal resistances of the composites by a finite element method. The interfacial thermal resistances calculated by the finite element modeling are implemented into a Mori–Tanaka homogenization scheme to predict the thermal conductivities of the composites with perfect and imperfect interfaces. The prediction by the Mori–Tanaka method was in good agreement with the finite element simulation and the experimental results.

A Quasi 3D Modified Strain Gradient Formulation for Static Bending of Functionally Graded Micro beams Resting on Winkler-Pasternak Elastic Foundation

This paper presents the bending analysis of simply supported functionally graded (FG) size dependent beams based on modified strain gradient theory. The shear and normal deformations are considered in displacement field according to hyperbolic shear deformation theory. Governing equations and corresponding boundary conditions for FG micro beam are derived utilizing principle of minimum total potential energy. Mori–Tanaka homogenization scheme and the classical rule of mixture are used for prediction of material properties through the thickness. Effects of Winkler-Pasternak elastic foundation parameters are studied for different side to thickness ratios. Effects of different aspect ratios, elastic foundation parameters, power law gradient indexes and different loading conditions are investigated. The efficiency and accuracy of present model is demonstrated by comparing to the existing results in especial cases.

Thermal wave propagation in a functionally graded annular fin with fixed base

Thermal wave propagation and efficiency analysis of thin axially symmetric functionally graded annular fins with a fixed base boundary condition are examined. All material properties are graded along the radial direction using the Mori–Tanaka homogenization scheme to design a realistic and physically meaningful graded material model and create a smooth distribution of elastic and thermal material properties. The novelty of this research is handling a more realistic annular fin, which has sufficient strength in practice and significantly reduces weight, using two different metal materials with the elasticity-based Mori–Tanaka model. These conditions result in two-point linear boundary value problems that are numerically solved by the pseudospectral Chebyshev method. The temperature distributions, thermal stresses and the efficiency in the functionally graded annular fin are compared with fins made of steel only and aluminium only.

Exact third-order static and free vibration analyses of functionally graded porous curved beam

An exact solution based on refined third-order theory (TOT) has been presented for functionally graded porous curved beams having deep curvature. The displacement field of the refined TOT is derived by imposing the shear free conditions at the outer and inner surfaces of curved beams. The properties of the two phase composite are tailored according the power law rule and the effective properties are computed using Mori-Tanaka homogenization scheme. The equations of motion as well as consistent boundary conditions are derived using the Hamilton's principle. The curved beam stiffness coefficients (A, B, D) are obtained numerically using six-point Gauss integration scheme without compromising the accuracy due to deepness (1+z/R) terms. The porosity has been modeled assuming symmetric (even) as well as asymmetric (uneven) distributions across the cross section of curved beam. The programming has been performed in MATLAB and is validated with the results available in the literature as well as 2D finite element model developed in ABAQUS. The effect of inclusion of 1+z/R terms is studied for deflection, stresses and natural frequencies for FG curved beams of different radii of curvature. Results presented in this work will be useful for comparison of future studies.