Mori-Tanaka Homogenization Scheme Research Articles

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.

Dependency of critical damping on various parameters of tapered bidirectional graded circular plates rested on Hetenyi medium

In industrial machinery and automobiles the critical damping plates are used as shock absorbers to dampen system and return rest position, in the shortest period of time. The paper is focused on the effects of various parameters on critical damping of a complicated system including a tapered bidirectional graded circular plate subjected to radially variable in-plane pre-load and rested on visco-Hetenyi elastic medium with semi-rigid restraint. By employing a Chebyshev-Ritz method, dependency of critical damping is investigated for the first time on various parameters including bidirectional arbitrary material gradation, variable in-plane pre-load, Hetenyi elastic medium parameters, tapering, semi-rigid restraint and Poisson’s ratio. For the Chebyshev-Ritz method developed here, the proper scale factor and orthogonal shifted Chebyshev polynomials of the first kind without auxiliary function requirement are used. The conventional rule of mixture and Mori-Tanaka homogenization scheme are used to model transverse gradation. The characteristic equation is calculated by vanishing determinant of the total potential energy Hessian. An equivalent ABAQUS model is introduced to eliminate necessity of complicated modeling of original structure.

Micromechanics modelling of cement stabilised rammed earth

Rammed earth (RE) is a generic term for a number of construction materials that contain stabilised or non-stabilised earthen materials. RE materials are heterogeneous and their bulk performance depends on their mineralogical composition, physical texture, water content and mode of preparation. This category of building materials has recently been the subject of extensive studies, given the increasing interest in sustainable construction. This trend has highlighted not only the potential of RE, but also the lack of standards and guidelines that can promote its adoption by architects and builders. Predictive modelling techniques that faithfully forecast the mechanical behaviour of RE are also scarce. This study introduces a micromechanics approach and proposes a multi-scaling model to predict the mechanical behaviour of cement stabilised rammed earth (CSRE) in linear elasticity. The RE heterogeneity is addressed hierarchically at different scales separated by the sizes of their representative volume elements (RVE). Homogenisation procedures are applied to predict the material properties at each scale and investigate the overall macroscopic behaviour of earthen mixtures. The Mori-Tanaka homogenisation scheme is used to predict the linear elastic behaviour of CSRE by treating it as a porous composite. The theoretical results obtained by the application of various micro-mechanics-based models (Reuss-Voigt, Hashin-Strickman, Mori-Tanaka and Self-consistency) are assessed by comparison with the experimental data. The Mori-Tanaka homogenisation scheme proved to predict the material's macroscopic linear elastic behaviour fairly accurately while taking into account local heterogeneities.

Experimental characterization and micromechanical modeling of the elastic response of the human humerus under bending impact.

This paper investigates the characterization and numerical modeling of the elastic behavior of the human humerus bone using a recently developed micromechanical approach coupled to nanoindentation measurements. At first, standard three-point bending experiments were conducted under low static loading, using several humerus diaphysis in order to identify the apparent elastic modulus of the bone in static regime. Then, a drop tower impact experiment was used on the same set of humerus diaphysis specimens, in order to assess the elastic modulus in dynamic regime. These measurements will be used as reference bases for comparison purpose. The originality of this work, lies in the coupling between a two-phase micromechanical approach based on Mori-Tanaka homogenization scheme for cylindrical voids and nanoindentation measurements of the elastic modulus of the bone matrix phase. This model has been implemented using a user defined material subroutine VMAT in ABAQUS© Explicit code. The bone mechanical response prediction using the proposed methodology was validated against previous standard experimental data. Finally, it was shown that the numerical predictions are consistent with the physical measurements obtained on human humerus via the good estimation of the ultimate impact load.

Micromechanical modeling of ductile fracture of human humerus

This paper deals with the formulation, development and validation of a newly developed micromechanical-based model for the modeling of the nonlinear ductile fracture of human humerus. The originality of the present works concerns the coupling between the micromechanical formulation based on the Mori-Tanaka homogenization scheme for cylindrical voids and the Marigo nonlinear ductile damage model based on the porosity growth. The proposed model was implemented as a User Material UMAT within the explicit dynamic software LS-DYNA and validated by numerical and experimental analysis conducted by a drop tower impact of human humerus. The outcome of the proposed multi-scale model appears to correctly predict the general trends observed experimentally via the good estimation of the ultimate impact load and the fracture patterns of the human humerus.

Size-dependent behaviour of functionally graded sandwich microbeams based on the modified strain gradient theory

This paper studies the bending, buckling and free vibration analyses of functionally graded (FG) sandwich microbeams using the third-order beam theory. The modified strain gradient theory (MSGT) with three material length scale parameters (MLSPs) is used to capture the size effect. The Mori-Tanaka homogenization scheme is employed to model the material distributions through the thickness. Finite element model is formulated to solve the problems. Verification studies for epoxy and FG microbeams are carried out to validate of the present model. Comparisons of the results of three different models such as MSGT, the modified couple stress theory (MCST) and classical continuum theory (CCT) are presented. Effects of small size, gradient index, shear deformation and boundary conditions on the structural behaviours of microbeams are investigated. Some new results of FG sandwich microbeams for both models (MCST and MSGT) are presented for the first time and can be used as benchmark in future studies.

Finite element model based on an efficient layerwise theory for dynamics and active vibration control of smart functionally graded beams

In this work, we present a two-noded efficient finite element (FE) model incorporating the layer-wise mechanics for the dynamics and active vibration control of smart functionally graded (FG) beams. The material properties in the FG beam are assumed to vary smoothly in the thickness direction according to power law variation. The effective properties are computed using Mori-Tanaka homogenization scheme. Electric potential profile in the electroded piezoelectric layers/patches is assumed quadratic across its thickness. The equations of motion are derived using extended Hamilton’s principle. Due to the complex algebraic expressions involved in the effective properties of FG system, the inertia and stiffness parameters are computed numerically using six point Gauss integration method. To fulfill the convergence requirements for weak integration of various energy terms in the variational formulation, the transverse displacement is interpolated with Hermite interpolation function possessing C1-continuity while the inplane displacement, shear rotation and quadratic component of the electric potential are interpolated with linear Lagrangian functions of C0-continuity. The equipotential condition of the electroded piezoelectric sensors and actuators is conveniently modelled using electric node concept. The control system is designed for constant gain velocity feedback (CGVF), and optimal LQR and LQG control strategies for a reduced order model using state space approach. The control performance is studied for single-input-single-output (SISO) and multi-input-multi-output (MIMO) configurations under various excitations. The effect of material inhomogeneity on stability/instability of the closed-loop response in the CGVF control has been discussed.

Postbuckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects

Herein, with the aid of the newly proposed theory of nonlocal strain gradient elasticity, the size-dependent nonlinear buckling and postbuckling behavior of microsized shells made of functionally graded material (FGM) and subjected to hydrostatic pressure is examined. As a consequence, the both nonlocality and strain gradient micro-size dependency are incorporated to an exponential shear deformation shell theory to construct a more comprehensive size-dependent shell model with a refined distribution of shear deformation. The Mori–Tanaka homogenization scheme is utilized to estimate the effective material properties of FGM nanoshells. After deduction of the non-classical governing differential equations via boundary layer theory of shell buckling, a perturbation-based solving process is employed to extract explicit expressions for nonlocal strain gradient stability paths of hydrostatic pressurized FGM microsized shells. It is observed that the nonlocality size effect causes to decrease the critical hydrostatic pressure and associated end-shortening of microsized shells, while the strain gradient size dependency leads to increase them. In addition, it is found that the influence of the internal strain gradient length scale parameter on the nonlinear instability characteristics of hydrostatic pressurized FGM microsized shells is a bit more than that of the nonlocal one.

Prediction of the Elastic Properties of the Clay-Rice Straw Composite by Numerical Homogenization Technique using Digimat

This paper deals with numerical modelling of the elastic properties of the rice straw-clay composite. This composite material is currently attracting interest because of its low impact on the environment but also for economic reasons. Benefiting from the advantages of a promising building material, straw reinforced composites attract more attention. This work allowed to model the effect of the straw inclusions on the elastic properties of the composite by a numerical approach considering the aspect ratio, the volume fraction, the orientation and the distribution of the straw inclusions. The Mori-Tanaka homogenization scheme for the RVEs is done using Digimat-MF software and the 3D composite microstructure models are generated by the Digimat-FE software. This made it possible to calculate the effective elastic properties of the clay-rice straw composites by numerical simulation. An empirical linear correlation between the volume fraction of the inclusions and the Young's modulus has been proposed. The results obtained can help to better control the formulation of the composite in developing countries.

Prediction of elastic anisotropic thermo-dependent properties of discontinuous fiber-reinforced composites

This study focuses on the micromechanical prediction of temperature-dependent elastic properties of a composite made of a polypropylene matrix reinforced with discontinuous glass fibers. Firstly, an experimental investigation of the mechanical behavior is presented. Specimen are cut from injection-molded rectangular plates using a pattern based on fiber orientation. The microstructure is investigated by X-ray tomography at the specimen center and an average orientation tensor is calculated. Tensile tests are performed over a temperature range from ambient temperature to 85℃ and dispersion of mechanical properties is rather low; moreover, they are representative of the ones measured out of an industrial injected part. Then, the evolution of elastic properties with orientation and temperature is analyzed and compared with numerical predictions calculated with Mori–Tanaka homogenization scheme.

Stress wave propagation in a through-thickness functionally graded adhesive layer

In this study, an improved mathematical model is presented to investigate the stress wave propagation in two circular cylinders bonded with a functionally graded adhesive layer. In the proposed model, the spatial derivatives of mechanical properties are included in the governing equations of the wave propagation. Also, the finite-difference method was used for the solution of the governing equations and boundary conditions. The Mori-Tanaka homogenization scheme was employed to evaluate the through-thickness mechanical properties of the adhesive layer. The effects of the spatial derivatives of the local mechanical properties and the through-thickness material composition variation in the adhesive layer were examined in detail. The presence of the material spatial derivatives in the governing equations mitigated the stress and displacement levels as well as axial and radial wave speeds.

Modeling of transversely isotropic properties of CNT-polymer composites using meso-scale FEM approach

The CNT based nano-composites usually behave as anisotropic materials. Elastic properties vary differently in global as well as material direction. This paper proposes an efficient computational methodology to predict effective orthotropic elastic properties of nanocomposites at diverse constituent conditions. The effective orthotropic material properties have been presented with the consideration of all possible conditions of composite matrix like elastic, elastoplastic and interfacial behavior. Mori-Tanaka (MT) homogenization scheme has been implemented with finite element method (FEM) approach to predict the effective material properties of nanocomposites. In the presented study, CNTs are aligned and uniformly distributed throughout the composite matrix. The proposed computational methodology has been validated with available literature and further extended to investigate the effect of the diverse behavior of composite matrix. From the obtained numerical results, it has been seen that the matrix and filler interface significantly affect the effective elastic strength of polymer composites.

Mathematical Models of Homogenization for a Rammed Earth Blend Made of Crushed Limestone and Cement in Linear Micro-poro-elasticity

Rammed earth (RE) designates natural building materials that can contain stabilized or non-stabilized soils or crushed stone blends. RE materials are non-homogenous and their performance depends on their compositions, grain size distributions, porosity, and mode of preparation. This category of building materials has recently been subject of extensive studies, due to the increasing interest in sustainable construction, which have highlighted the potential and limits of its large uptake. Although it has been demonstrated that RE can be treated as multi-component materials [1] a full characterization based on micro-structural considerations is still lacking. Full understanding of the role of porosity in the overall material performance has not been addressed. However, extensive literature [2, 3, 4, 5] is available that can describe the overall mechanical behaviour of similar materials, based on the principle continuum micromechanics. This study applies the Mori-Tanaka homogenization scheme to characterize the linear elastic behaviour of cement-stabilized crushed-limestone-based material by treating it as a porous composite. The composite comprises a solid phase consisting of limestone grains bonded together with hydrated cement bridges and a fluid phases consisting of different families of pores. The microstructure of rammed earth samples has been observed using an optical microscope. The obtained images have been collected and processed to obtain the pore sizes and pore size distribution. Finally, these statistics along with the Mori-Tanaka homogenization scheme have been used to determine the overall elastic properties that depend on a damage parameter enclosing the pore sizes, density, and texture

Stress wave propagation in adhesively bonded functionally graded cylinders: an improved model

This study presents an improved mathematical model to analyse the stress wave propagation in adhesively bonded functionally graded (FG) circular cylinders (butt joint) under an axial impulsive load. The volume fractions of the material constituents in the upper and lower cylinders were functionally tailored through the thickness of each cylinder using a power-law. The effective material properties of both cylinders, which are made of aluminum (Al) and silicon carbide (SiC), at any point were predicted by using the Mori–Tanaka homogenization scheme. In this improved model, the governing equations of the wave propagation include the spatial derivatives of local mechanical properties and were discretized by means of the finite difference method. The influence of these spatial derivatives and the compositional gradient exponent on the displacement and stress distributions of the joint was investigated. The material composition variations of both cylinders affected the displacement and stress fields whereas the compositional gradient exponent had a minor effect. The stress concentrations were alleviated in time, the displacement and stress distributions/variations around/along the upper and lower cylinder-adhesive interfaces were significantly affected by the adhesive layer. The spatial derivatives also affected the temporal histories of the displacement and stress components evaluated at the selected critical points of the upper cylinder, adhesive layer and lower cylinder. The consideration of the spatial local material derivatives provided a more accurate mathematical model of wave propagations through the graded layered structures.

Surface effects on nonlinear vibration and buckling analysis ofembedded FG nanoplates via refined HOSDPT inhygrothermal environment considering physical neutralsurface position

In this paper the hygro-thermo-mechanical vibration and buckling behavior of embedded FG nanoplates are investigated. The Eringen\'s and Gurtin-Murdoch theories are applied to study the small scale and surface effects on frequencies and critical buckling loads. The effective material properties are modeled using Mori-Tanaka homogenization scheme. On the base of RPT and HSDPT plate theories, the Hamilton\' s principle is employed to derive governing equations. Using iterative and GDQ methods the governing equations are solved and the influence of different parameters on natural frequencies and critical buckling loads are studied.

Micromechanical method for effective piezoelectric properties and electromechanical fields in multi-coated long fiber composites

This paper proposes a micromechanical framework for identifying the macroscopic behavior of multi-coated long fiber composites, as well as the average electromechanical microscopic fields of all phases (matrix, fibers, coating layers), generated upon known macroscopic conditions. The work aims at developing a unified micromechanical approach that provides an analytical solution standing for non-coated and multi-coated long fiber composites with transversely isotropic piezoelectric behavior. The proposed method solves specific boundary value problems and utilizes the Mori-Tanaka homogenization scheme, in which the dilute strain and electric field concentration tensors are obtained analytically with the help of an extended composite cylinders method that accounts for coupled electromechanical fields. The capabilities of this homogenization strategy are illustrated with the help of numerical examples, and comparisons with known solutions from the literature for non-coated and coated fiber piezoelectric composites are provided.

Stress wave propagation in a functionally graded adhesive layer between two identical cylinders

ABSTRACTThis study investigates the stress wave propagation in circular aluminum cylinders bonded with a functionally graded adhesive layer subjected to an axial impulsive load. The adhesive joint consists of two identical (aluminum) cylinders and a functionally graded adhesive layer. The volume fractions of the two constituents: aluminum and epoxy in the adhesive layer were functionally tailored through the adhesive thickness by obeying a power-law. Therefore, the effective material properties at any point in the adhesive layer were predicted by the Mori-Tanaka homogenization scheme. The governing equations of the wave propagation in the joint were discretized by means of the finite difference method. The influence of the compositional gradient exponent on the displacement and stress distributions of the joint was examined. It was observed that changing the material composition of the adhesive layer had an evident effect on the displacement and stress levels, especially in the lower cylinder. On the contrary, the influence of the compositional gradient exponent was found to be minor on the displacement and stress distributions. The displacement and stress distributions were also investigated along the upper and lower cylinder-adhesive interfaces. Accordingly, with increasing the ductility of the adhesive layer the waves transmitted to the lower cylinder caused lower displacement levels. The normal stresses become peak at the bottom corners of the upper and lower cylinder-adhesive interfaces whereas the shear stresses concentrate in the middle region of the interfaces. In addition, the temporal variations of the displacement and stress components were evaluated at some critical points of the adhesive and lower cylinder. The compositional gradient exponent played an important role on the displacement and stress levels as well as the wave speeds in the adhesive and lower cylinder rather than in the upper cylinder. The stresses in the joints were observed to be alleviated by employing a functionally graded adhesive layer.

Micromechanical Modeling of DP600 and DP800 Steels Plastic Behavior Based on the Mori-Tanaka Homogenization Method

The properties of the dual-phase steels are attributed to the chemical composition, type, size, amount, and spatial distribution of different phases that can be obtained during thermomechanical treatments. In this way, modeling of the mechanical behavior of the dual-phase steel constituents, namely, ferrite and martensite, is crucial to the numerical simulation of sheet metal forming processes mainly to forecast the residual stresses per phase. In this work, the microstructure of as-received DP600 and DP800 cold rolled steel sheets with 1.2 mm nominal thickness were firstly characterized by means of scanning electron microscopy technique. The grain sizes and volume fractions of ferrite and martensite phases were obtained by means of digital image analysis. The Mori-Tanaka homogenization scheme was implemented in the finite element code ABAQUS assuming linear isotropic elasticity and isotropic work-hardening behavior for both ferrite (matrix) and martensite (inclusion) phases. The numerical predictions obtained with the Mori-Tanaka homogenization scheme for the macroscopic uniaxial tensile behavior are in good agreement with the experimental curves of both dual-phase steels.

Buckling and free vibration analysis of functionally graded sandwich micro-beams resting on elastic foundation by using nonlocal strain gradient theory in conjunction with higher order shear theories under thermal effect

Based on the nonlocal strain gradient theory (NLSGT), and various higher order shear deformation beam theories a formulation for buckling and free vibration of size dependent functionally graded sandwich micro-beams resting on two parameter elastic foundation including Winkler and Pasternak shear layer springs with thermal effects is presented. The sandwich FG micro-beams are assumed to be formed with homogenous ceramic core and ceramic-metal FG skins. According to the Mori-Tanaka homogenization scheme and the classical rule of mixture the material properties of the FG part of the sandwich size dependent beam changes continuously through the thickness of the beam. Equations of motion and the associated boundary conditions are derived via Hamilton's principle. Static buckling loads and natural frequencies are obtained by using generalized differential quadrature method (GDQM) for size dependent sandwich FG beam with different boundary conditions. As original contributions to the literature, the effects of the nonlocal parameter (ea), the length scale parameter (lm), aspect ratio (L/h), gradient index (k), different cross-section shapes, temperature change (ΔT)and stiffnesses of Winkler and shear layer springs (KW, KS respectively) on the buckling and free vibration of the sandwich FG micro-beam are investigated, reported and discussed in detail. To verify the present formulation present results (buckling and free vibration) are compared with the previously published results. Good agreement is observed between the present solutions and the previously published results.

Static Response of Functionally Graded Plates using Higher Order Theories

This work presents an assessment of higher order theories for the static response of functionally graded plates. The governing equations, variationally consistent boundary conditions, force and moment resultants of higher order shear deformable theories are derived using principle of virtual work. Material properties are assumed to vary smoothly in the thickness direction with power law variation of ceramic volume fraction. The effective material properties of FG material are obtained using rule of mixture as well as Mori-Tanaka homogenization scheme. Analytical solution using a double Fourier series expansion is obtained for simply supported plates. The computer program has been developed in MATLAB and the results are validated by comparing with the results available in the literature. The values of deflection, inplane and transverse stresses are presented for different theories, span to thickness ratio and inhomogeneity parameter.