Von Karman Strain Research Articles

The Deflection of the Shape Memory Alloy Composite Beams Using Finite Element Method

In this study, shape memory alloy (SMA) wires were embedded within a matrix to become a composite beam. Active shape control property of the SMA that controls the deflection of the composite beam was studied using finite element method (FEM). The values of recovery stresses of the SMA were determined using the Brinson’s model. A non-linear finite element formulation that is based on the von Karman strain and the Timoshenko’s beam theory was developed for this purpose. The Newton-Raphson method was then used to solve the formulation to obtain the transverse deflection of the beam. It was found that the effect of moment recovery on the beam deflection is significant where it is possible that the deflection of the beam due to external loading can be greatly suppressed.

Evaluation of homogenized effective properties for FGM panels in aero-thermal environments

Abstract Based on homogenization methods, effective properties of functionally graded (FG) panels are investigated for the two types of models such as the power law (P-) and the sigmoid (S-) FGM functions. For heterogeneous micromechanical model, the effective material properties of the structures are obtained using Voigt rule (VR) and Mori–Tanaka method (MTM). The governing equations are derived by the first-order shear deformation plate theory with von Karman strain–displacement relationships. And then, the effective properties of the FGMs are evaluated by VR and MTM in detail. The results show that the characteristics of FGM structures are determined dominantly according to the constituents. In order to investigate the macroscopic behaviors, Newmark time integration method is applied to analyze the aero-thermoelastic behaviors of the models based on the first-order piston theory. Also, nonlinear dynamic behaviors of the panels are studied under supersonic aerodynamic force.

Large post-buckling behavior of Timoshenko beams under axial compression loads

Large post-buckling behavior of Timoshenko beams subjected to non-follower axial compression loads are studied in this paper by using the total Lagrangian Timoshenko beam element approximation. Two types of support conditions for the beams are considered. In the case of beams subjected to compression loads, load rise causes compressible forces end therefore buckling and post-buckling phenomena occurs. It is known that post-buckling problems are geometrically nonlinear problems. The considered highly non-linear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. The beams considered in numerical examples are made of lower-Carbon Steel. In the study, the relationships between deflections, rotational angles, critical buckling loads, post-buckling configuration, Cauchy stress of the beams and load rising are illustrated in detail in post-buckling case.

Nonlinear dynamics of SMA-fiber-reinforced composite beams subjected to a primary/secondary-resonance excitation

In this paper, nonlinear free vibration and primary/secondary resonance analyses of shape memory alloy (SMA) fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up are investigated. The simplified Brinson constitutive model and cosine phase transformation kinetics are utilized to simulate the behavior of the SMA materials and calculate the recovery stress. In order to predict the behavior of the smart laminated beam, Euler–Bernoulli beam theory and nonlinear von-Karman strain field are employed. Two types of micromechanical models, namely Voigt and Reuss models are considered. The Galerkin procedure together with the elliptic function and multi timescales method is adopted to obtain analytical solutions for the nonlinear free vibration and primary/secondary response phenomena. Numerical results reveal that some of the geometrical and physical parameters such as the SMA volume fraction, the amount of prestrain in the SMA fiber, orientation of composite fiber, vibration amplitude and temperature are important factors affecting the free vibration characteristic in the pre/post-buckled region, and primary and secondary resonance of the laminated beams reinforced with SMA fibers. The analytical solutions and results are reported for the first time and can serve as benchmark for researchers to validate their numerical and analytical methods in the future.

Energy harvesting from limit cycle oscillation of a cantilever plate in low subsonic flow by ionic polymer metal composite

In this work, energy harvesting from limit cycle oscillation of low aspect ratio rectangular cantilever plates in low subsonic flow by ionic polymer metal composite is considered. The plate is modeled in according to classic plate theory with Von-Karman strain–displacement relations for modeling large deflections due to midplane stretching. Aerodynamics flow is modeled based on modified vortex lattice aerodynamics model. Linear and nonlinear aeroelastic characteristics of the considered model are accurately examined and the effects of ionic polymer metal composite energy harvesting properties on flutter margin and limit cycle oscillation amplitudes are investigated. The obtained results show that output power of [Formula: see text] is generated when the flow velocity reaches to [Formula: see text] , and position of ionic polymer metal composite on the plate has great effect on the amount of power produced. The extraction of energy from plate vibration in conjunction with large strain characteristics of ionic polymer metal composite materials produces static deflection of plate. This static deflection produces stiffness hardening of the entire system, accordingly reduces the amplitude of limit cycle oscillation. Obtained results show that ionic polymer metal composite actuator has more influence on limit cycle oscillation of plate, while its effect on flutter instability is negligible.

Stacking sequence optimization for maximum strengths of laminated composite plates using genetic algorithm and isogeometric analysis

Abstract In this paper, an optimal procedure based on genetic algorithm and NURBS-based finite element isogeometric analysis (IGA) is proposed for seeking maximum load-carrying capacity of imperfect laminated composite plates with respect to a specified displacement. Nonlinear behavior of plates including bending, buckling and postbuckling are investigated. Fiber orientations are the design variables. Governing equations are derived in the framework of first-order shear deformation theory (FSDT) associated with von-Karman strain theory for large deformation. Isogeometric analysis is known as a robust numerical method in dealing with laminated structures due to the smoothness, high continuity and the reduction of total degrees-of-freedom (DOFs). Modified Riks method is employed for nonlinear analysis. The arc-length scheme simultaneously adjusts displacement and load so that accelerates the analysis procedure. Computational time is significantly reduced based on the cooperation of IGA and Riks algorithm. Numerical examples including 4-ply and quasi symmetric laminates are investigated to show the efficiency and flexibility of the proposed approach.

Active flutter and aerothermal postbuckling control for nonlinear composite laminated panels in supersonic airflow

In this article, active flutter and aerothermal postbuckling control of the nonlinear composite laminated panel with large deformation in supersonic airflow using the piezoelectric material are investigated. The distributed piezoelectric actuator and sensor pairs are bonded on the top and bottom surfaces of the laminated panel. The von Karman strain–displacement relation is used in the structural modeling. Unsteady aerodynamic pressure is evaluated by the supersonic piston theory. Hamilton’s principle and the assumed mode method are applied to formulate the equation of motion of the structural system. A discrete linear-quadratic-Gaussian controller is designed in this study to conduct the active flutter and postbuckling control. The proportional feedback control method is also applied in designing the controller. Time domain responses of the structural system are computed using the Runge–Kutta method. The control effects of the two control methods are compared. The influences of the placements of the piezoelectric actuator and sensor pairs on the control effects are also investigated. This study is helpful for the active aeroelastic design of the nonlinear structural system.

Postbuckling of laminated composite plates using NURBS-based isogeometric analysis

In this paper, postbuckling behavior of laminated composite plates is investigated using NURBS-based isogeometric analysis (IGA). A Green strain tensor with small rotations is proposed. Both von-Karman strain tensor and the proposed strain tensor are formulated. Governing equations are derived in the framework of first-order shear deformation theory (FSDT). Finite elements based on NUBRS show the robustness in performing a laminate with intentional imperfections as well as deformed mode shapes. Quadratic NURBS elements (Q9Nurbs) are used to construct physical meshes in C 1 continuity. Newton–Raphson scheme is adjusted to geometric imperfections and employed for non-linear analysis procedure. In numerical examples, the postbuckling paths of laminated composite plates in sense of von-Karman strain theory and the proposed strain theory are presented. The obtained results of isotropic, symmetric, antisymmetric angle-ply and cross-ply laminates are verified with those available in literature.

A new deformation monitoring method for a flexible variable camber wing based on fiber Bragg grating sensors

This work presented an indirect method to monitor the deformation of the flexible variable camber wing using attached fiber Bragg gratings. To measure the transverse strains resulting from deformation of flexible variable camber wing, two groups of fiber Bragg grating sensors were attached on both upper and lower surfaces of the metal sheet, which was used to replace the traditional hinges. When the flexible variable camber wing was actuated by the lower surface’s pneumatic artificial muscle actuators, the upper surface would undergo the tensile deformation. A redshift could be observed from the reflective wavelength of the fiber Bragg grating sensors located on the upper surface. On the contrary, a blueshift could also be observed from the lower surface due to the compressive deformation. The comparison of the results with those obtained from strain gauges demonstrated the reliability of fiber Bragg grating sensors. Then, the vertical displacement and deflection angle of the flexible variable camber wing were obtained through the von Karman strain–displacement relation. A finite element model of the flexible variable camber wing was developed to simulate the variation of strain and displacement, which is caused by deformation of the metal sheet. A good agreement could be seen from the vertical displacements among the fiber Bragg grating strain sensors, finite element model, and direct measurements with laser range-finders.

A finite element formulation for large deflection of multilayered magneto-electro-elastic plates

Abstract An original finite element formulation for the analysis of large deflections in magneto-electro-elastic multilayered plates is presented. The formulation is based on an equivalent single-layer model in which first order shear deformation theory with von Karman strains and quasi-static behavior for the electric and magnetic fields are assumed. To obtain the plate model, the electro-magnetic state is firstly determined and condensed to the mechanical primary variables, namely the generalized displacements. In turn, this result is used to obtain laminate effective stiffness coefficients that allow to express the plate mechanical stress resultants in terms of the generalized displacements and applied electro-magnetic boundary conditions only, taking the magneto-electro-elastic couplings into account. The weak form of the mechanical equilibrium equations is then written and used to express the finite element matrices. Numerical results obtained by implementing an isoparametric four-node MITC finite element are presented to validate the approach and show its features and potentiality. The proposed modeling strategy and the corresponding finite element formulation can take advantage of the solution tools available for the mechanics of classical multilayered plates leading to the straightforward integration of magneto-electro-elastic plate finite elements into existing codes.

An analytical approach for nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams

Abstract In this article, large amplitude vibration and thermal post-buckling of shape memory alloy (SMA) fiber reinforced hybrid composite beams with symmetric and asymmetric lay-up are analytically investigated. To predict the behavior of the smart laminated beam, the Euler–Bernoulli beam theory and the nonlinear von-Karman strain field are employed. Also, one-dimensional Brinson SMA model is utilized to calculate the recovery stress of SMA fibers in the case of restrained strain. Nonlinear governing equations of motion are derived via the Hamilton principle. Using an analytical approach based on the Galerkin procedure together with the simple harmonic motion assumption, a closed-form solution is obtained for the thermal post-buckling and nonlinear free vibration analysis of SMA fiber reinforced hybrid composite beams. Due to lack of any results on the free vibration and thermal stability of SMA fiber reinforced composite beams, the results obtained from the present solution for laminated composite beams without SMA fiber are compared with counterpart data in the open literature, which validate the present solution. Then, a set of parametric study is carried out to show the influence of SMA volume fraction, amount of prestrain in the SMA fiber, orientation of composite fiber, SMA-reinforced layer thickness to total thickness ratio, location of SMA layer, vibration amplitude, boundary conditions and temperature on the vibration characteristic of the laminated beam reinforced with SMA in the pre- and post-buckled domains.

Dynamic model for free vibration and response analysis of rotating beams

Abstract A new modeling method for the vibration analysis of a rotating cantilever beam is proposed and compared to the other modeling methods. Applying the Hamilton principle, the equations for the axial, chordwise and flapwise motions are derived. During the derivation of these equations, the nonlinear von Karman strain and the corresponding linear stress are used to consider the stiffening effect due to rotation. The derived equations of motion obtained from the proposed method are analytically and numerically compared to equations from the previous modeling methods. Computations of the natural frequencies and time responses show that the described equations of motion are more reliable than the equations of the other modeling methods.

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.

Large-amplitude vibrations of nanowires with dissipative surface stress effects

In this study, nonlinear free vibrations of doubly clamped Euler–Bernoulli nanowires (NWs) have been considered. The von Karman strain–displacement relationships along with the classic Zener model are implemented to derive the nonlinear differential equation of the flexural motion of NW. Nonlinear natural frequencies are calculated using the computer package Mathematica. The effects of size-dependent surface dissipation, mode numbers, and amplitude of vibrations on the nonlinear natural frequencies are investigated. It is shown that the surface dissipation effect on the normalized nonlinear natural frequencies depends on the amplitudes of vibrations. Also, comparisons are made with the results published in previous studies.

Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations

Abstract In the present study, thermal buckling and post-buckling analysis of Functionally Graded Material (FGM) Timoshenko beams resting on a non-linear elastic foundation are examined. Thermal and mechanical properties of the FGM media are considered to be functions of both temperature and position. Theory of Timoshenko beam combined with von-Karman's strain–displacement relations are applied in virtual work principle to obtain the system of non-linear equilibrium equations. Different types of boundary conditions such as clamped, simply supported, and rolled edges are assumed for edge supports. Generalized Differential Quadrature Method (GDQM) is employed to discrete the equilibrium equations in space domain. Post-buckling equilibrium paths are depicted for different values of the power law index, non-linear elastic foundation parameters, boundary conditions, thermal loading type, and slenderness ratio. It is found that depending on the boundary conditions and the type of thermal loading, the response of the structure may be of the bifurcation-type or unique stable path.

Thermo-mechanical characteristics and stability boundaries of antenna structures in supersonic flows

Abstract In a supersonic airflow, a smart skin antenna model is analyzed to investigate the thermo-mechanical characteristics and stability boundaries of the structure. The skin is modeled as a multi-layer sandwich structure with a dielectric polymer. Especially, the polymer layer is embedded on the middle surface of the face sheets similar to the honeycomb core to perform the role of antenna or radar. In order to formulate the structural model including the geometric non-linearity, first-order shear deformation theory of plate and von Karman strain–displacement relations are adopted in this work. Also, aerodynamic pressure is obtained by the first-order piston theory for the aerodynamic. To get the numerical results, finite element method is used and Newton–Raphson iterative method is applied to solve the nonlinear equations of motion. For the validation of this work, present results are compared with previous data. And then, thermo-mechanical characteristics of the skin structures are fully discussed, especially. Additionally, effects of sizes and various shapes of dielectric layer relative to the enclosure layer are analyzed.

Flexural stress analysis of uniform slender functionally graded material beams using non-linear finite element method

Functionally graded material (FGM) typically consists of two constituent materials combined together with a particular distribution. A non-linear flexural stress analysis of through-thickness functionally graded uniform slender beam, subjected to a uniformly distributed load, is studied using the versatile finite element method based on Euler–Bernoulli beam hypothesis. The von-Karman strain–displacement relations are used to account for geometric non-linearity. Simply supported and clamped FGM beams with axially immovable ends are considered. Governing non-linear equations are obtained using the principle of virtual work. Numerical results are provided to show the effect of boundary conditions and volume fraction exponent on the non-linear structural behaviour, in terms of the strains and stresses, of the FGM beams, for the first time. A shift in the neutral axis, from the mid-thickness of the beam, is observed due to the large transverse deflections, for the homogenous as well as the FGM beams. The throu...

Nonlinear modal interaction in rotating composite Timoshenko beams

Abstract Nonlinear free vibration analysis of rotating composite Timoshenko beams featuring internal resonance is studied in this paper. Three nonlinear coupled equations of motion for flapping, shear and axial motions, are based on the assumptions of Timoshenko theory and the nonlinear von Karman strain–displacement relationships. Due to the small magnitude of low-order flapping/shear frequencies ratio, usually a big gap exists between the aforementioned frequencies especially in isotropic beams, while this gap reduces for composite beams which exhibit more characteristics of Timoshenko beams. Different material, geometrical and operational parameters effects on the frequencies of rotating beams are investigated. For the first time the possibility of internal resonance occurrence between low-order flapping and shear modes is proved. The direct multiple scales method is implemented for construction of the flapping nonlinear normal modes. Results for the flapping nonlinear normal modes are validated via comparison with the results of the fourth-order Runge–Kutta method. Depend on the amplitude ratios besides the nearness of the frequencies of the interacting modes, four bifurcation regions are defined through the nonlinear normal mode stability analysis; (a) one stable coupled mode, (b) two stable and one unstable coupled modes, (c) three stable coupled modes and (d) one stable coupled mode.

Nonlinear free vibration analysis of rotating composite Timoshenko beams

Abstract Linear and nonlinear free vibrations of rotating composite Timoshenko beams are investigated in this paper. The formulation is based on the assumptions of Timoshenko beam theory and the nonlinear von-Karman strain–displacement relationships. The updated equations of motion about the prestressed configuration are obtained from the three-coupled partial differential equations of motion. The differential transform method is implemented to find the prestressed configuration due to centrifugal forces. The Galerkin discretization approach is applied to the linearized updated equations of motion to determine the linear normal modes and the associated natural frequencies of the rotating composite Timoshenko beam. The rotation speed and number of layers variation effects on the flapping natural frequencies are studied and the necessity of employing the Timoshenko beam model for the composite beams is illustrated. The direct multiple scales method is applied to the three-coupled nonlinear updated partial differential equations for the nonlinear vibration study especially on the flapping backbone curves. The number of layers variation effects is investigated on the three lowest flapping backbone curves. The present results are validated with the previous literature results.

USING VON KARMAN NONLINEAR DISPLACEMENT FUNCTIONS IN THE FINITE ELEMENT ANALYSIS OF FUNCTIONALLY GRADED COLUMN

This paper focuses on deriving the exact displacement fields of a functionally graded column (FGC) subjected to mechanical and thermal loads under the assumptions of the Timoshenko beam theory and von Karman strains. Valid only when an axial load is present, the obtained displacement fields are, therefore, not applicable in pure bending analysis. These displacement fields are used as the interpolation functions for formulating static finite element equations whose DOFs are arbitrary constants rather than nodal displacements. The element is super-convergent in von Karman nonlinear analysis, and its superiority over beam elements formulated by using linear exact displacement fields is shown in two examples: (1) the buckling analysis of a stepped FGC, and (2) an FGC subjected to a thermal load. Also investigated is the efficiency of using von Karman nonlinear displacement fields as the interpolation functions for analyzing cases in which the level of nonlinearity is higher than that of the von Karman strain.