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Finite element formulation of metal foam microbeams via modified strain gradient theory

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Abstract

Size-dependent behaviours of metal foam microbeams with three different porosity distribution models are studied in this paper. Based on the finite element model, a normal and shear deformation theory has been employed for the first time to investigate their structural behaviours by using modified strain gradient theory and considering the effects of variable material length scale parameter. The equations of motion and boundary conditions of system are derived from Hamilton’s principle. Finite element models are presented for the computation of deflections, vibration frequencies and buckling loads of the metal foam microbeams. The verification of proposed models is carried out with a comparison of the numerical results available in the literature. Calculations using the different parameters reveal the effects of the porosity parameters (distribution and coefficient), small size, boundary conditions and Poisson’s ratio on the displacements, frequencies and buckling loads of metal foam microbeams. Some benchmark results of these structures for both models (modified couple stress theory and modified strain gradient theory with constant and variable material length scale parameter) and with/without Poison’s effect are provided for future study.

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Abbreviations

IGA:

Iso-geometric analysis

FSDT:

First order shear deformation theory

CST:

Classical shell theory

CCT:

Classical continuum theory

MCST:

Modified couple stress theory

MSGT:

Modified strain gradient theory

NSGT:

Nonlocal strain gradient theory

CNT:

Carbon nanotube

MLSP:

Material length scale parameter

NSDT:

Normal and shear deformation theory

FEM:

Finite element method

L, b, h :

Geometry of beam

UPD:

Uniform porosity distribution

NUPD1:

Non-uniform porosity distribution 1

NUPD2:

Non-uniform porosity distribution 2

\(E\) :

Young’s modulus

\(\rho \) :

Mass density

\( \ell \) :

MLSP

\({E}_{\text{max}}\) :

Maximum E

\({E}_{\text{min}}\) :

Minimum E

\({\rho }_{\text{max}}\) :

Maximum \(\rho \)

\({e}_{0}\) :

Porosity parameter

\({e}_{m}\) :

Porosity parameter

\({\ell}_{\text{max}}\) :

Maximum MLSP

\(\mathcal{U},V,K\) :

Strain energy, external work and kinetic energy

\({\sigma }_{ij},{m}_{ij}, {p}_{i},{\tau }_{ijk}\) :

Stress and modified strain gradient stress components

\({\varepsilon }_{ij},\) \({\chi }_{ij},\) \({\gamma }_{i}, {\eta }_{ijk}\) :

Strain and symmetric curvature, dilatation gradient and deviatoric stretch gradient tensors

\({u}_{1},{u}_{2},{u}_{3}\) :

Displacements in the 1, 2 and 3 directions of an arbitrary point

\({\delta }_{ij}\) :

Kronecker delta

\({e}_{ijk}\) :

Permutation symbol

\(\nu \) :

Poisson’s ratio

\(\mathcal{V}\) :

Volume of the body, which can be decomposed to the cross-sectional area \(A=bxh\) and the length of the domain L

\({ \ell }_{0}\),\({ \ell }_{1}\) and \({ \ell }_{2}\) :

MLSPs of modified stress tensors

\(u, {w}_{b}, {w}_{s}\) and \({w}_{z}\) :

In-plane displacement and bending, shear and thickness stretching displacements

\({f}_{1}\left(z\right), {f}_{2}\left(z\right)\) and \({f}_{3}\left(z\right)\) :

Shape function describing the contribution of the bending, shear and thickness stretching displacements across the thickness

TBT:

Third-order beam theory

\({Q}_{ij}\) :

Elastic constants

\(q\) :

Uniformly distributed load

\({N}_{0}\) :

Axial load

\({I}_{0},{I}_{1},{I}_{2},{J}_{1},{J}_{2},{J}_{3},{K}_{1},{K}_{2}\) :

Inertial constant coefficients

\(\omega \) :

Natural frequency

\({\varphi }_{j}\) :

FEM shape function

\(\Pi \) :

Total energy

\({[K}_{kl}]\), \({[M}_{kl}]\), \({[G}_{kl}]\) and \({F}_{k}\) :

FEM matrices

BC:

Boundary condition

DMD (\(\overline{w })\) :

Dimensionless mid-span deflection

DFF (\(\overline{\uplambda })\) :

Dimensionless fundamental frequency

DCBL (\({\overline{N} }_{cr})\) :

Dimensionless critical buckling load

SBT:

Sinusoidal beam theory

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Authors and Affiliations

  1. Faculty of Engineering and Natural Sciences, Mechanical Engineering, Istinye University, Istanbul, Turkey

    Armagan Karamanli

  2. School of Engineering and Mathematical Sciences, La Trobe University, Bundoora, VIC, 3086, Australia

    Thuc P. Vo

  3. Research Center for Interneural Computing, China Medical University, 40402, Taichung, Taiwan

    Omer Civalek

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  1. Armagan Karamanli
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Karamanli, A., Vo, T.P. & Civalek, O. Finite element formulation of metal foam microbeams via modified strain gradient theory. Engineering with Computers 39, 751–772 (2023). https://doi.org/10.1007/s00366-022-01666-x

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