Spectral Chebyshev method coupled with a high order continuation for nonlinear bending and buckling analysis of functionally graded sandwich beams

Abstract

The main objective of the present work is to couple the spectral Chebyshev differential quadrature method (SCDQM) to the high order continuation method (HOCM) which was proposed in previous works with several discretization techniques. This new approach (SCDQM-HOCM) is proposed to analyze the nonlinear bending and buckling analysis of functionally graded sandwich beams.The originality of this work consists also to use a beam model which taken into account the nonlinear term neglected in several works of the literature. This term makes it possible to have the buckling and bending problems by using the classical Timoshenko model and to handle the boundary conditions that several works could not to take into account. A strong form of nonlinear equations is established based on the first order shear deformation theory of beams with the von-Kármán kinematic hypothesis. Regarding functionally graded materials (FGM), two typical types are investigated: sandwich beam with FGM faces and ceramic core (Type-A), and sandwich beam with FGM core and uniform faces (Type-B). The accuracy and efficiency of the SCDQM-HOCM compared with finite element method coupled with high order continuation method (FEM-HOCM) are illustrated on numerical examples of the FGM beams and then the FG sandwich beams. Furthermore, a parametric study is led to carry out the influence of different skin-core-skin thickness ratios, span-to-height ratio, and volume fraction on the bending and buckling behavior of FG sandwich beams subjected to different loadings and various boundary conditions. K E Y W O R D Sfirst order shear deformation theory, geometrically nonlinear Timoshenko FGM beam, high order continuation method, sandwich Timoshenko FGM beam, spectral Chebyshev differential quadrature method

Radial point interpolation‐Chebychev method with asymptotic numerical method for nonlinear buckling analysis of graphene oxide powder‐reinforced composite beams

“…However, if m is a non-zero value, then m Chebychev polynomials will be added to supplement the interpolation process. [20][21][22][23][24] The RBFs that can be used are: [20][21][22]…”

“…Using the Taylor series development in HOCM, the non-linear discrete Equation ( 21) can be transformed into a sequence of recursive linear equations with the same tangent operator. This allows expressing the unknowns of the problem as a Taylor series expansion truncated at order k with respect to the dimensionless scalar 𝜖 from a known starting solution, 22,23 as follows:…”

“…This could mean, for example, a transition from a stable equilibrium to an unstable one, or the emergence of new oscillatory behavior. 20,22,23,29,[31][32][33][34][35][36][37][38][39][40] Bifurcation analysis has applications in many areas of science and engineering. For example, in physics, bifurcation analysis can be used to understand the behavior of complex systems such as fluid flows.…”

“…In equation, it is noted that, if the value of $$$ m $$$ is equal to zero, then only pure radial functions will be utilized for interpolating the displacement function. However, if $$$ m $$$ is a non‐zero value, then $$$ m $$$ Chebychev polynomials will be added to supplement the interpolation process 20–24 . The RBFs that can be used are: 20–22 $$$$ \left\{\begin{array}{ll}R\left(r,c\right)=\sqrt{r^2+{\left(c{h}_x\right)}^2}& \mathrm{Classical}\ \mathrm{Multiquadrics}\ \left(\mathrm{CMQ}\right)\\ {}R\left(r,c\right)=\sqrt{1+{\left(c{h}_x\right)}^2{r}^2}& \mathrm{Normalized}\ \mathrm{Multiquadrics}\ \left(\mathrm{NMQ}\right)\\ {}R\left(r,k\right)={r}^k\mathit{\ln}(r)& \mathrm{Thin}\ \mathrm{Plate}\ \mathrm{Spline}\ \left(\mathrm{TPS}\right)\\ {}R\left(r,n\right)={e}^{-{(nr)}^2}& \mathrm{Gaussienne}\ \left(\mathrm{EXP}\right)\end{array}\right.$$…”

Efficient resolution of incompressible Navier–Stokes equations using a robust high‐order pseudo‐spectral approach

“…$$$$ \left\{\begin{array}{c}\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\\ {}u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{\partial p}{\partial x}+\frac{1}{\mathit{\operatorname{Re}}}\Delta u\\ {}u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{\partial p}{\partial y}+\frac{1}{\mathit{\operatorname{Re}}}\Delta v\end{array}\right., $$$$ where $$$ u $$$ and $$$ v $$$ are the velocity components in the $$$ x $$$‐ and $$$ y $$$‐directions respectively, $$$ p $$$ is the pressure and $$$ \mathit{\operatorname{Re}} $$$ is the nonnegative Reynolds number. The above nonlinear equations can be easily solved using several mesh‐free approaches that existing in the literature such as References 10,18,33,37‐40. Few papers have been published basing on a strong form of vorticity‐stream function formulation where the cited approaches find difficulty in managing hermit‐type bo...…”