Von Karman Strain Research Articles

Galerkin’s Method and Multivariate Newton’s Method for the Nonlinear Deformation of the Magneto-Electro-Elastic Bi-Layered Laminates

In this paper, mathematical modelling for the large deformation of a magneto-electro-elastic rectangular bi-layered laminate with general boundary conditions is presented. Constitutive equations involving the magneto-electro-elastic (MEE) material properties are introduced, Maxwell equations accounts for the electric and magnetic effects are also utilized. First-order shear deformation theory (FSDT) considering the von Karman nonlinear strain is adopted, and the plain strain/stress assumption applicable for thin plate analysis is used. A rather compact set of governing equations related to kinematical variables, electric/magnetic potentials and the Airy stress function is obtained as a consequence of the preliminary condensation for the electro-magnetic state to the plate kinematics. Semi-analytic solution for a bi-layered BaTiO3-CoFe2O4 laminate with specified boundary conditions subjected to various external applied loads is performed. By employing the Bubnov-Galerkin method, the set of nonlinear partial differential equations is transformed to a set of third-order nonlinear algebraic equations for the static deformation due to applied load. Numerical results are carried out by using the multivariate Newton's method with respect to various volume fractions indicating the volume ratio between piezoelectric BaTiO3 layer and piezomagnetic CoFe2O4. From the result, the nonlinearity of the von Karman strain appears to enhance system rigidity as smaller deformations will be detected when external load is applied. Also, some other interesting results are obtained which could be useful to future analysis and design of multiphase composite plates.

Vibration analysis of graphene-reinforced porous aluminum-based variable-walled thickness sandwich joined conical-conical panel with elastic boundary conditions using differential quadrature method

In this paper, a unified solution is proposed to investigate the vibration characteristics of the variable-walled thickness graphene-reinforced porous aluminum-based (GRPA) sandwich joined conical-conical panel (JCCP) with the arbitrary elastic support boundary conditions by using the differential quadrature method (DQM). The two surfaces of the sandwich JCCP are made of metallic aluminum and the central core layer is the GRPA. The core thickness of each conical plate varies linearly with its generatrix. Three types of the graphene distributions and two types of porosity distributions are considered along the core thickness direction. Based on the first-order shear deformation theory (FSDT), von-Karman strain displacement relationship, constitutive relationship and Hamiltonian principle, the partial differential governing equations of motion are obtained for the variable wall thickness GRPA sandwich JCCP. Using the DQM, the dynamic equation is discretized into the ordinary differential equation. The matrix of the characteristic equation is analyzed to solve the frequencies and mode shapes of the variable wall thickness GRPA sandwich JCCP. The effects of the spring stiffness, boundary conditions, graphene distributions, porosity distributions and geometric parameters on the vibration properties are studied for the variable wall thickness GRPA sandwich JCCP with two interesting elastic supported boundary conditions. At the same time, this article provides a useful approach for studying the arbitrary boundary coupled plate and shell structures with the variable wall thickness.

Application of von-Karman model for laminated composite singly curved shells with non-uniform boundary conditions

The industrial applications of singly curved shells significantly increased in modern engineering structures and several of those have non-uniform boundary conditions. A detailed relative performance study for the non-uniformly restricted shells can popularize their use further among practicing engineers. The literature review confirms that there is dearth of research studies in this area. This paper aims to fill up that void. A finite element code of C0 continuity combining von-Karman strains, first order shear deformation theory and eight noded curved elements is proposed. Wide variety of shell problems are solved for different boundary conditions, laminations, radii of curvatures, thickness, and plan dimensions. Relative nondimensional deflections and fundamental frequencies are studied for varying boundaries. It is concluded that although the performances of shells are comparable for different boundaries, the shells with CCCC edges offer better performances. These shells should have longer span along the arch direction, a symmetric 16°/-16°/-16°/16° laminate, side/thickness = 100, and radius of curvature/plan = 0.75 to achieve minimum deflection and maximum fundamental frequency.

Nonlinear forced vibration of a nanobeam resting on Winkler-Pasternak elastic foundation and subjected to a mechanical impact

The nonlinear governing equations of nanobeam taking into consideration its curvature, resting on an elastic Winkler-Pasternak foundation and based on non-local Euler-Bernoulli beam theory is analyzed. The equation of motion and the boundary conditions are modeled within the framework of a simple supported nanobeam which accounts the presence of a mechanical impact and nonlinear von-Karman strain. The resulting nonlinear differential equations are reduced to only one differential equation which is studied by means of the Optimal Auxiliary Functions Method (OAFM). An explicit analytical solution is proposed for a complex problem. The main quality of our technique consists in the existence of some auxiliary functions derived from the expressions of the solution for the initial linear equation and the form of nonlinear term calculated from the above solution of the linear equation. The convergence-control parameters present in the auxiliary functions are evaluated by a rigorous mathematical procedure. The obtained solutions are in very good agreement with the numerical solution.

Nonlinear Thermal/Mechanical Buckling of Orthotropic Annular/Circular Nanoplate with the Nonlocal Strain Gradient Model.

This article presents the nonlinear investigation of the thermal and mechanical buckling of orthotropic annular/circular single-layer/bilayer nanoplate with the Pasternak and Winkler elastic foundations based on the nonlocal strain gradient theory. The stability equations of the graphene plate are derived using higher-order shear deformation theory (HSDT) and first-order shear deformation theory (FSDT) considering nonlinear von Karman strains. Furthermore, this paper analyses the nonlinear thermal and mechanical buckling of the orthotropic bilayer annular/circular nanoplate. HSDT provides an appropriate distribution for shear stress in the thickness direction, removes the limitation of the FSDT, and provides proper precision without using a shear correction coefficient. To solve the stability equations, the differential quadratic method (DQM) is employed. Additionally, for validation, the results are checked with available papers. The effects of strain gradient coefficient, nonlocal parameter, boundary conditions, elastic foundations, and geometric dimensions are studied on the results of the nondimensional buckling loads. Finally, an equation is proposed in which the thermal buckling results can be obtained from mechanical results (or vice versa).

Geometrically nonlinear shape sensing of anisotropic composite beam structure using iFEM algorithm and third-order shear deformation theory

The inverse finite element method (iFEM) has been used to achieve the shape sensing of small displacements based on linear elastic theory. However, with the development of smart structure technology, the formulation is not suitable for the anisotropic composite structures with large deformation in practical engineering. Therefore, a nonlinear iFEM algorithm is proposed to monitor the linear and nonlinear deformation of the anisotropic composite beams. The formulation not only involves the effect of orientation of composite fibers on strain distribution into the shape sensing model, but also accounts for the effect of shear deformation without any requirement of shear correction factor. Initially, the third-order shear deformation theory (TSDT) is reviewed along with deriving the nonlinear strain field based on von-Karman strain theory. Considering the problem that the couple term of the shear strain and bending strain is unmeasurable, the relationship between shear and bending displacements is established according to the derived constitutive equations. Then, the proposed nonlinear iFEM method reconstructs the deformed structural shape, where isogeometric analysis (IGA) approach is used to construct the displacement functions and the experimental section strains are calculated from the discretized surface strains. Finally, several examples are solved to verify the proposed methodology. Numerical results demonstrate that the nonlinear iFEM algorithm can improve the reconstruction accuracy by 4% with respect to the linear iFEM method for beam structures. Hence, the proposed approach can be used as a viable tool to predict nonlinear deformation of composite structure.

Nonlinear static/dynamic behaviors of composite curved panels with piezoelectric patches in supersonic airflow

The nonlinear aeroelastic static deflection and dynamic response of the composite curved panel with rectilinear- and curvilinear-fiber lay-up configurations under supersonic flow are investigated. The geometrical nonlinear effect of the composite curved panel is depicted by the Von-Karman strain and the aerodynamic load is incorporated through the nonlinear third-order piston theory. The Newton-Raphson method is employed to determine the static deflection of the composite curved panel and the four-order Runge-Kutta method is for the dynamic response, respectively. After examining the accuracy of the established mechanical model through numerical comparisons with existing literatures, the parametric studies focusing on panel curvature, high-order piston aerodynamic force, fiber angle orientation, temperature rise, locations and applied voltages of the piezoelectric patch on nonlinear static deformation and dynamic response are investigated. The results are deliberated in detail, and it is observed that the panel curvature and fiber angle orientation significantly affect the aeroelastic static deflection and dynamic response of the curved panel under thermal environment. Moreover, high-order aerodynamic force enlarges the static deflection and dynamic response of the curved panels with a large curvature whereas the impact of applied voltages of the piezoelectric patch will be weaker.

Nonlocal Strain Gradient Model for the Nonlinear Static Analysis of a Circular/Annular Nanoplate.

A nonlinear static analysis of a circular/annular nanoplate on the Winkler-Pasternak elastic foundation based on the nonlocal strain gradient theory is presented in the paper. The governing equations of the graphene plate are derived using first-order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT) with nonlinear von Karman strains. The article analyses a bilayer circular/annular nanoplate on the Winkler-Pasternak elastic foundation. HSDT while providing a suitable distribution of shear stress along the thickness of the FSDT plate, eliminating the defects of the FSDT and providing good accuracy without using a shear correction factor. To solve the governing equations of the present study, the differential quadratic method (DQM) has been used. Moreover, to validate numerical solutions, the results were compared with the results from other papers. Finally, the effect of the nonlocal coefficient, strain gradient parameter, geometric dimensions, boundary conditions, and foundation elasticity on maximum non-dimensional deflection are investigated. In addition, the deflection results obtained by HSDT have been compared with the results of FSDT, and the importance of using higher-order models has been investigated. From the results, it can be observed that both strain gradient and nonlocal parameters have significant effects on reducing or increasing the dimensionless maximum deflection of the nanoplate. In addition, it is observed that by increasing load values, the importance of considering both strain gradient and nonlocal coefficients in the bending analysis of nanoplates is highlighted. Furthermore, replacing a bilayer nanoplate (considering van der Waals forces between layers) with a single-layer nanoplate (which has the same equivalent thickness as the bilayer nanoplate) is not possible when attempting to obtain exact deflection results, especially when reducing the stiffness of elastic foundations (or in higher bending loads). In addition, the single-layer nanoplate underestimates the deflection results compared to the bilayer nanoplate. Because performing the experiment at the nanoscale is difficult and molecular dynamics simulation is also time-consuming, the potential application of the present study can be expected for the analysis, design, and development of nanoscale devices, such as circular gate transistors, etc.

Nonlinear thermo-acoustic response and fatigue prediction of three-dimensional braided composite panels in supersonic flow

In this paper, a novel aerodynamic-acoustic coupling model to investigate the influence of aeroelastic effects on the thermo-acoustic response of three-dimensional, four-directional braided composite panel is proposed, and the fatigue life under the combined aerodynamic load and acoustic load is discussed by using a rainflow cycle counting method. Assuming that the fibers are transversely isotropic and the matrix is isotropic, the fiber inclination model was used to predict the effective stiffness matrices of the composite panels. The piston theory and Gaussian random acoustic excitation were used to simulate the aerodynamic load and acoustic load, the thermal load is assumed to be a uniform distribution in steady state, and the material properties are assumed to be independent of temperature. Based on the von-Karman strain–displacement relation, the governing equations of the panel were formulated, and the response solutions of the four sides simply supported panel were obtained using Galerkin method and Runge-Kutta method. The aeroelastic effect on the thermos-acoustic response of the braided composite panel was investigated in time and frequency domains, and the life of different load conditions was predicted. The results show that the aeroelastic not only transforms the movement form, but also affects fatigue life of the panel.

A Semi-Analytical Approach for the Linearized Vibration of Clamped Beams with the Effect of Static and Thermal Load

The geometric nonlinearity due to static and thermal load can significantly alter the vibration response of structures. This study presents a semi-analytical approach to illustrate the nonlinear vibration of clamped-clamped beams under static and thermal loads. The von Karman strain and Hamilton’s principle are employed to derive the nonlinear static equilibrium equation and nonlinear governing equation. The vibration equation’s coefficient is variable. The transfer-matrix method and local homogenization are used to solve the equation. The proposed method’s accuracy is validated by commercial software and literature. The numerical results indicate that uniform stress caused by thermal load only reduces the structural mode frequencies. The geometric nonlinearity of the structural static deformation affects both the mode frequencies and mode shapes. And the mode shapes cannot be approximated by harmonic functions. When the static deformation is significant, the structure’s local RMS response is substantially affected. The combined loads have a more significant impact on the acceleration response than the superposition of individual load effects.

Nonlinear dynamic response of bimorph piezoelectric energy harvester with functionally graded porous core

This study deals with piezoelectric energy harvesters carrying the magnet at the free end while a magnifier connected them to the base. The harvester consists of a cantilever beam with Functionally Graded (FG) porous foam core and two piezoelectric faces while taking into account the von-Karman strains and the magnetic interaction of two magnets. The governing equation has been developed using Hamilton’s principle and reduced-order via the Galerkin method. Frequency response of vibrations and voltage derived using harmonic balance method and the adjusted model has been confirmed by parametric studies of the lumped parameter model. A parametric study is also performed to expose the effects of porosity coefficient and pattern, magnifier ratios, and excitation level on the responses. Results show that by correct selection of the magnifier parameters, the proposed harvester can provide a higher output over a wider frequency band. Also, the core porosity increases the flexibility of the beam and raised the voltage by a factor of 1.8.

Nonlinear vibration of electro-rheological sandwich plates, coupled to quiescent fluid

The current paper aims to study the free vibration analysis of a sandwich plate in contact with quiescent fluid. The structure is made of an electro-rheological (ER) fluid core, and two laminated composite skins. The fluid is ideal, interacting partially with a plate, and fluid potential velocity is derived, satisfying continuity equation. Higher-order shear deformable theory is employed for the displacement field, and nonlinear Von-Karman strains are taken into account. After calculating the structural strain energy and kinetic energies of the fluid and plate, Galerkin approach and harmonic balance method are adopted to obtain natural frequencies of the system. Next, the accuracy and reliability of the developed model are confirmed, and then a detailed parametric study is conducted to examine the effects of different parameters such as geometrical dimensions of plate, fluid parameters and electric field strength on vibrational characteristics of the structure. The novelty of the presented model focuses on the nonlinear analysis of an ER sandwich panel coupled to ideal fluid, and the obtained results can guide the future design of composite ER panels.

Buckling and post-buckling of variable stiffness plates with cutouts by a single-domain Ritz method

Structural components with variable stiffness can provide better performances with respect to classical ones and offer an enlarged design space for their optimization. They are attractive candidates for advanced lightweight structural applications whose functionalities often impose the presence of cutouts that requires accurate and effective analysis for their design. In the present work, a single-domain Ritz formulation is proposed, implemented and validated for the analysis of buckling and post-buckling behaviour of variable stiffness plates with cutouts. The plate model is based on the first-order shear deformation theory with nonlinear von Karman strain–displacement relationships. The plate generalized displacements are approximated with trial functions built as products of one-dimensional Legendre orthogonal polynomials. The non linear governing equations system is then deduced from the stationarity of the energy functional; the involved matrices are numerically computed by a special integration algorithm based on the implicit description of the cutout via suitable level-set functions. The formulation has been implemented in a computer code which has been used to validate the method through comparison with literature solutions for variable angle tow laminates with circular cutouts. Several investigations on buckling and post-buckling behaviour of variable angle tow composite plates with cutouts having different shapes and dimensions are then presented to illustrate the approach capabilities, provide benchmark results and point out features and design opportunities of the variable stiffness concept for the buckling and post-buckling design of advanced lightweight structures.

Thermal buckling and post-buckling analyses of rotating Timoshenko microbeams reinforced with graphene platelet

A comprehensive study is established to investigate on the thermal buckling instability and the subsequent post-critical deflection of a rotating nanocomposite microbeam reinforced with graphene platelet. The main scope of the current research is shedding light on the effectiveness of reinforcing a rotating microbeam with graphene platelet subjected to a constant temperature development. The Timoshenko beam theory comes along the von-Karman strain displacement relations to extract the governing equations of motion in accordance with the modified couple stress theory. The modified Halpin–Tsai micromechanical model is implemented to determine the practical elasticity modulus of the nanocomposite beam. The Ritz technique is accompanied by the Chebyshev orthonormal polynomial set to discretize the weak form of the nonlinear equations of motion. The Newton–Raphson technique is applied to the discretized static nonlinear equations of motion to compute the static deformation due to the centrifugal force. On the basis of two effective algorithms the nonlinear equations of motion are attacked to find out the critical buckling temperature change as well as the succeeding post-critical deflection. It is observed that adjoining the graphene phase in the form of an X-pattern promotes the static strength of a rotating microbeam against the buckling instability. However, in contrast, for stationary microbeams and microbeams that rotate with a velocity below a threshold speed adjoining the graphene platelet reinforcement in the form of an O-Pattern not only does not strengthen the microbeam but also weakens it against the buckling instability. Moreover, the buckling modeshape of rotating simply-clamped microbeams is more impressed by the graphene phase.

Nonlinear Thermoelastic Numerical Frequency Analysis and Experimental Verification of Cutout Abided Laminated Shallow Shell Structure

Abstract The cutout and temperature loading influences on the nonlinear frequencies of the laminated shell structures are predicted numerically using two different types of geometrical nonlinear strain-displacement relationships to count the large deformation. The displacement of any generic point on the structural panel is derived using the third-order shear deformation theory (TSDT). Moreover, the direct iterative method has been adopted to obtain the nonlinear eigenvalues in conjunction with the isoparametric finite element (FE) steps. The present analysis includes the effect of temperature and the temperature-dependent composite elastic properties on the thermoelastic frequencies. This study intends to establish the Green-Lagrange type of nonlinear strain's efficacy in computing the nonlinear frequency of layered structure with and without cutout instead of von-Karman strain kinematics. The numerical model's validity has been established by comparing the results to previously published results. In addition, experimentally obtained fundamental frequency values of a few modes are compared to numerical proposed numerical results under the thermal loading. Finally, the effects of cutout (shape and size) and the associated structural geometrical parameters on the nonlinear thermal frequency responses of the laminated structure are expressed in the final output form.

Nonlinear effect of Low-velocity impact on tapered laminated composite structures using spline finite strip method

In present paper, the behavior of tapered composite plate under low-velocity impact was studied. The tapered plates were used in many structures according to their stiffness to weight ratio, while, the behavior of such plates did not evaluate under impact load in the literatures. Geometrically nonlinear Von-Karman strain was taken into account with respect to large deformations. The spline finite strip method (SFSM) was used to predict the impact behavior of composite laminates. The numerical fourth-order Runge-Kutta integration technique was used to solve the governing differential equations. The displacement field was defined according to the refined plate theory. The four most common internal taper configurations were considered. Local indentation was modeled with the help of contact Hertz’s laws. The governing differential equation was written based on the principal of virtual displacement and Hamilton’s principle for dynamic systems. The obtained results on eccentric impact showed that contact force and indentation are increased when indenter contact near supports.

A refined hyperbolic shear deformation theory for nonlinear bending and vibration isogeometric analysis of laminated composite plates

In this paper, a refined hyperbolic tangent higher order shear deformation theory is developed. Assuming that the shear function is parameter dependent, the parameters are determined by the inverse method. The refined theory is assessed by using the NURBS based isogeometric analysis for the geometric linear and nonlinear bending and free vibration problems of laminated composite plates. The nonlinearity of the plates is based on the von-Karman strain assumptions. Numerical examples show that the refined hyperbolic tangent shear deformation theory combined with IGA has high accuracy in both linear and geometric nonlinear analysis of laminated composite plates. The effects of plate length–thickness ratio, modulus ratio and stacking sequence on the static bending and free vibration behaviors of laminated plate are also discussed.

Nonlinear vibration of a nonlocal functionally graded beam on fractional visco-Pasternak foundation

This paper investigates the nonlinear dynamic behavior of a nonlocal functionally graded Euler–Bernoulli beam resting on a fractional visco-Pasternak foundation and subjected to harmonic loads. The proposed model captures both, nonlocal parameter considering the elastic stress gradient field and a material length scale parameter considering the strain gradient stress field. Additionally, the von Karman strain–displacement relation is used to describe the nonlinear geometrical beam behavior. The power-law model is utilized to represent the material variations across the thickness direction of the functionally graded beam. The following steps are conducted in this research study. At first, the governing equation of motion is derived using Hamilton’s principle and then reduced to the nonlinear fractional-order differential equation through the single-mode Galerkin approximation. The methodology to determine steady-state amplitude–frequency responses via incremental harmonic balance method and continuation technique is presented. The obtained periodic solutions are verified against the perturbation multiple scales method for the weakly nonlinear case and numerical integration Newmark method in the case of strong nonlinearity. It has been shown that the application of the incremental harmonic balance method in the analysis of nonlocal strain gradient theory-based structures can lead to more reliable studies for strongly nonlinear systems. In the parametric study, it is shown that, on the one hand, parameters of the visco-Pasternak foundation and power-law index remarkable affect the amplitudes responses. On the contrary, the nonlocal and the length-scale parameters are having a small influence on the amplitude–frequency response. Finally, the effects of the fractional derivative order on the system’s damping are displayed at time response diagrams and subsequently discussed.

Dynamic Stability Analysis of Size-Dependent Viscoelastic/Piezoelectric Nano-Beam

This paper analyzes the dynamic stability of an isotropic viscoelastic Euler–Bernoulli nano-beam using piezoelectric materials. For this purpose, the size-dependent theory was used in the framework of the modified couple stress theory (MCST) for piezoelectric materials. In order to capture the geometrical nonlinearity, the von Karman strain displacement relation was applied. Hamilton’s principle was also employed to obtain the governing equations. Furthermore, the Galerkin method was used in order to convert the governing partial differential equations (PDEs) to a nonlinear second-order ordinary differential one. Dynamic stability analysis was performed and the effects of such parameters as viscoelastic coefficients, size effect, and piezoelectric coefficient were investigated. The results showed that in this system, saddle points, central points, Hopf bifurcation points, and fork bifurcation points could be created, and the phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, and homoclinic orbits.

Nonlinear Flutter Analysis of Porous Functionally Graded Plate in Yawed Hypersonic Flow

An aerothermoelastic analysis of functionally graded plate containing porosities in ‎yawed hypersonic flows is investigated. Due to some incorrect manufacturing processes, two ‎different types of porosity, namely, even and uneven distributions are taken into account. The ‎third-order piston theory is utilized to estimate the unsteady aerodynamic pressure induced by the ‎hypersonic airflow. The material properties of a plate are assumed to vary across the thickness ‎direction according to a simple power law. Based on classical plate theory, the motion equations are ‎developed with geometric nonlinearity taking into consideration of von Karman strains. The ‎formulations are established based on Hamilton’s principle while the generalized differential ‎quadrature method is employed to solve the nonlinear aerothermoelastic equations. Due to lower ‎computational efforts‎‏, ‏the method of generalized differential quadrature (GDQM) is used to obtain ‎accurate results. Moreover, the assumed mode method along with the Runge-Kutta integration ‎algorithm is used as a solution method. The reliability and precision of the obtained results are ‎validated by published literature. Then, the influence of porosity distribution, porosity coefficient, ‎and yawed flow angle are discussed in detail. In general, this paper shows that even porosity ‎distribution would have a more destabilizing effect compared with the ‎uneven porous model.‎ And also, for both porosity distributions, the chaotic behavior appears in higher top surface ‎temperature but even porosity distribution has a profound effect on chaotic motion.‎