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Investigation of the static behavior of a micropolar functionally graded plate using two finite element methods in Hilbert space and differential transformation method

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Abstract

The main purpose of this paper is to analyze the bending of a micropolar functional graded plate depending on the splitting parameter, with two finite element methods in Hilbert space and the semi-analytical method of differential transformations, the advantages and disadvantages of using these two methods. The problem is developed using enhanced mathematical methods and kinematic assumption that the forces are defined based on the existing boundary conditions and using the internal energy of the plate, which are extracted by a splitting parameter. The splitting parameter is the optimal value for the minimization of the elastic energy which increases the degree of freedom and controls the force applied to the plate. The variation of the separator operating parameters is investigated by changing the degree of functionally graded material. Finally, the effects of the changes in the behavior of the plate are examined by changing the degree of functionally graded materials, which is defined along with the thickness of the plates. The influence of plate length and thickness, as well as the degree of FG material, are investigated on the kinematic variables of a clamped plate. Furthermore, the superiority of the various solution methods is examined. Because the two methods converge, the DTM method provided more efficient results in terms of processing time and expenditure as well as using the analytical equation to obtain precise solutions.

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Abbreviations

E :

Elasticity modulus

Z :

Thickness coordinate

h :

Thickness of the plate

\(P_{p}\) :

Material properties of polyurethane foam

\(P_{s}\) :

Material properties of synthetic foam

\(\sigma _{ij}\) :

Cauchy stress tensor

\(\mu _{ij}\) :

Couple-stress tensor

\(\varepsilon _{ijk}\) :

Levi-Civita tensor

\(\rho \) :

Density

f :

Force on the plate

\(u_{i}\) :

Displacement vector

\(l_{i}\) :

Body coupling

j :

Microinertia

\(\phi \) :

Microrotation vector

\(\gamma _{ij}\) :

Micropolar strain tensor

\(\chi _{ij}\) :

Micropolar torsion-rotation tensor

\(\alpha ,\beta ,\gamma ,\varepsilon \) :

The asymmetric elasticity constants

\(\lambda ,\mu \) :

Lamé parameters

G :

The boundary of the middle surface

\(\psi _{1},\psi _{2}\) :

The rotation of the middle plate

\(W+W^{*}\) :

Vertical deflection

\(\frac{5}{4}\Omega _{\alpha }^{0}+{\hat{\Omega }}_{\alpha }\) :

Indicates microrotation

\(\Omega _{3}\) :

Rate of change of the microrotation

\(\sigma ^{t}\) :

Cauchy stress at the top of the plate

\(\mu ^{t}\) :

Couple stress at the top of the plate

\(\sigma ^{b}\) :

Cauchy stress at the bottom of the plate

\(\mu ^{b}\) :

Couple stress at the bottom of the plate

p :

Bending pressure

\(\eta \) :

Splitting parameter

\(p_{1},p_{1}\) :

Bending pressure is divided by \(\eta \)

\(\Theta \) :

HPR function

\({\mathcal {U}}\) :

Set of kinematic variables

S :

Set of micropolar plate stress

\(M_{11},M_{22}\) :

Bending moments

\(M_{12},M_{21}\) :

Twisting moments

\(Q_{\alpha }\) :

Shear force

\(Q_{\alpha }^{*},{\hat{Q}}_{\alpha }\) :

Transverse shear force

\(R_{11},R_{22},R_{11}^{*},R_{22}^{*}\) :

Micropolar bending moments

\(R_{12},R_{21},R_{12}^{*},R_{21}^{*}\) :

Micropolar twisting moments

\(S_{\alpha }^{*}\) :

Micropolar couple moments

\({\mathcal {E}}\) :

The set of micropolar plate strain

L :

Operator matrix

\({\mathcal {F}}\) :

Force vector

\({\mathcal {W}}^{\eta }\) :

Density of the work

\(H^{1}(B_{0})\) :

The standard function in Hilbert space

\(\nu \) :

Poisson’s ratio

\(l_{b}\) :

Characteristic length of twisting

\(l_{t}\) :

Characteristic length of twisting

N :

The couplings number

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Authors and Affiliations

  1. Department of Mechanical Engineering, Urmia University of Technology, Urmia, Iran

    Ahad Ahmadpour fard, Shirko Faroughi & Leyla Abbasiniyan

  2. Department of Mathematics, Urmia University, Urmia, 165, Iran

    Parviz Darania

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Ahmadpour fard, A., Faroughi, S., Darania, P. et al. Investigation of the static behavior of a micropolar functionally graded plate using two finite element methods in Hilbert space and differential transformation method. Acta Mech 233, 4441–4466 (2022). https://doi.org/10.1007/s00707-022-03325-0

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