Ali Fallah

Free Vibration Analysis of Symmetrically Laminated Fully Clamped Skew Plates Using Extended Kantorovich Method

In this paper, free vibration analysis of thin symmetrically laminated skew plates with fully clamped edges is investigated. The governing differential equation for skew plate which is a fourth order partial differential equation (PDE) is obtained by transforming the differential equation in Cartesian coordinates into skew coordinates. Based on the multi-term extended Kantorovich method (MTEKM) an efficient and accurate approximate closed-form solution is presented for the governing PDE. Application of the MTEKM reduces the governing PDE to a dual set of ordinary differential equations. These sets of equations are then solved with infinite power series solution, in an iterative manner until convergence was achieved. Results of this study show the fast rate of convergence of the MTEKM. Usually two or three iterations are enough to obtain reasonably accurate results. The frequency parameters of laminated composite plates are obtained for different skew angles and lay-up configuration for different composites laminates skew plates. Comparisons have been made with the available results in the literature which show the accuracy and efficiency of the method.

Micromechanics and constitutive modeling of connective soft tissues

In this paper, a micromechanical model for connective soft tissues based on the available histological evidences is developed. The proposed model constituents i.e. collagen fibers and ground matrix are considered as hyperelastic materials. The matrix material is assumed to be isotropic Neo-Hookean while the collagen fibers are considered to be transversely isotropic hyperelastic. In order to take into account the effects of tissue structure in lower scales on the macroscopic behavior of tissue, a strain energy density function (SEDF) is developed for collagen fibers based on tissue hierarchical structure. Macroscopic response and properties of tissue are obtained using the numerical homogenization method with the help of ABAQUS software. The periodic boundary conditions and the proposed constitutive models are implemented into ABAQUS using the DISP and the UMAT subroutines, respectively. The existence of the solution and stable material behavior of proposed constitutive model for collagen fibers are investigated based on the poly-convexity condition. Results of the presented micromechanics model for connective tissues are compared and validated with available experimental data. Effects of geometrical and material parameters variation at microscale on macroscopic mechanical behavior of tissues are investigated. The results show that decrease in collagen content of the connective tissues like the tendon due to diseases leads 20% more stretch than healthy tissue under the same load which can results in connective tissue malfunction and hypermobility in joints.

Nonlinear Initial Value Ordinary Differential Equations

Ordinary frequently occur as mathematical models in many fields of science, engineering, differential equations (ODEs) and economy. It is rarely that ODEs have closed form analytical solutions, so it is common to seek approximate solutions by means of numerical methods. One of the most useful methods for the solution of ODEs is multi-step methods. Although the existing multi-step methods such as Adams–Moulton are accurate and useful, they also have their own limitations such as instability at large step sizes or weak performance in the case of stiff ODEs. Thus, multi-step methods that show better behavior compared to the existing methods are preferred, because they decrease the computational effort needed to achieve results with the desired order of accuracy.

Micromechanical modeling of rate-dependent behavior of Connective tissues

In this paper, a constitutive and micromechanical model for prediction of rate-dependent behavior of connective tissues (CTs) is presented. Connective tissues are considered as nonlinear viscoelastic material. The rate-dependent behavior of CTs is incorporated into model using the well-known quasi-linear viscoelasticity (QLV) theory. A planar wavy representative volume element (RVE) is considered based on the tissue microstructure histological evidences. The presented model parameters are identified based on the available experiments in the literature. The presented constitutive model introduced to ABAQUS by means of UMAT subroutine. Results show that, monotonic uniaxial test predictions of the presented model at different strain rates for rat tail tendon (RTT) and human patellar tendon (HPT) are in good agreement with experimental data. Results of incremental stress-relaxation test are also presented to investigate both instantaneous and viscoelastic behavior of connective tissues.

Application of nonlocal theory in dynamic pull-in analysis of electrostatically actuated micro and nano beams

One of the most important phenomena related to electrically actuated micro and nano electromechanical systems (MEMS\NEMS) is dynamic pull-in instability which occurs when the electrical attraction and beam inertia forces are more than elastic restoring force of the beam. According to failure of classical mechanics constitutive equations in prediction of dynamic behavior of small size systems, nonlocal theory is implemented here to analyze the dynamic pull-in behavior. Equation of motion of an electrostatically actuated micro to nano scale doubly clamped beam is rewritten using differential form of nonlocal theory constitutive equation. To analyze the nonlocal effect equation of motion is nondimentionalized. Governing partial differential equation is transformed to an ordinary differential equation using the Galerkin decomposition method and then is solved implementing differential quadrature method (DQM). Change of dynamic pull-in voltage with respect to size change is investigated. Results indicate as the beam length decreases dynamic pull-in voltage increases due to nonlocal effect and the difference with clasical mechanics results is up to 20% for nano beams.Copyright

Nonlinear Free Vibration of Nanobeams With Surface Effects Considerations

In this paper, simple analytical expressions are presented for geometrically non-linear vibration analysis of thin nanobeams with both simply supported and clamped boundary conditions. Gurtin-Murdoch surface elasticity together with Euler-Bernoulli beam theory is used to obtain the governing equations of motions of the nanobeam with surface effects consideration. The governing nonlinear partial differential equation is reduced to a single nonlinear ordinary differential equation using Galerkin technique. He’s variational approach is employed to obtain analytical solution for the resulted nonlinear governing equation. The effects of different parameters such as vibration amplitude, boundary conditions, and beam dimensions on the natural frequencies of nanobeams are investigated and results are presented for future studies.Copyright

Large Amplitude Thermo-Mechanical Vibration Analysis of Asymmetrically Laminated Composite Beams

In this paper, simple analytical expression is presented for large amplitude thermo-mechanical free vibration analysis of asymmetrically laminated composite beams. Euler-Bernoulli assumptions together with Von Karmans strain-displacement relation are employed to derive the nonlinear governing partial differential equation (PDE) of motion. Hes variational method is employed to obtain a simple and efficient approximate closed form solution of the nonlinear governing equation. Comparison between results of the present work and those available in literature shows the accuracy of presented technique. Some new results for the nonlinear natural frequencies of the laminated beams such as the effect of vibration amplitude, lay-up configuration and thermal loading are presented for future references.

Analytical Solutions for Generalized Duffing Equation

Structures like beams, plates, shells, and sectors have a key role in different fields of science and engineering. Reliable design of structures requires detailed understanding of their behavior in different service conditions including statics, dynamics, and vibration. For instance, dynamics and vibration characteristics of structures such as natural frequencies, mode shapes, and damping ratios are among interesting properties. Furthermore, in most severe conditions, linear theories are not enough to accurately capture real structural behavior. Thus, investigation of nonlinear and/or large amplitude vibration of structures which is normally governed by system of nonlinear differential equations is necessary.

Micromechanical Modeling of Connective Soft Tissues

In this paper, a physically motivated micromechanical model for connective soft tissues like tendon and ligament is presented. A representative volume element (RVE) is introduced based on the crimped pattern of collagen fiber in the tissue. In order to investigate the macroscopic behavior of tissue, numerical homogenization and appropriate periodic boundary conditions are used. Neglecting the effects of nano scale structures and properties on tissues macroscopic behavior leads to linear transversely isotropic model for collagen fibers. Comparison of obtained result with available experimental one, shows that this assumption leads to inaccurate results. Including the effects of nano scale structures into the presented model leads to a nonlinear transversely isotropic constitutive model for collagen fibers which leads to results that are in good agreement with experimental results.Copyright

Nonlinear Free Vibration of Piezoelectric Functionally Graded Beams in Thermal Environment

Nonlinear free vibration analysis of piezoelectric functionally graded (PFG) beams in thermal environment with any boundary conditions are investigated. The nonlinear governing Equation of motion is derived based on the Euler-Bernoulli beam theory and Von Karman’s strain-displacement relation. Simple analytical expression is presented for the nonlinear natural frequency using energy balanced method (EBM). Results are validated and compared with available results in the literature. Effects of different parameters such as vibration amplitude, boundary conditions, material inhomogenity, electrical and thermal loading on the nonlinear behavior of PFG beams are presented.Copyright