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Flexomagneticity in buckled shear deformable hard-magnetic soft structures

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Abstract

This research work performs the first time exploring and addressing the flexomagnetic property in a shear deformable piezomagnetic structure. The strain gradient reveals flexomagneticity in a magnetization phenomenon of structures regardless of their atomic lattice is symmetrical or asymmetrical. It is assumed that a synchronous converse magnetization couples both piezomagnetic and flexomagnetic features into the material structure. The mathematical modeling begins with the Timoshenko beam model to find the governing equations and non-classical boundary conditions based on shear deformations. Flexomagneticity evolves at a small scale and dominant at micro/nanosize structures. Meanwhile, the well-known Eringen’s-type model of nonlocal strain gradient elasticity is integrated with the mathematical process to fulfill the scaling behavior. From the viewpoint of the solution, the displacement of the physical model after deformation is carried out as the analytical solution of the Galerkin weighted residual method (GWRM), helping us obtain the numerical outcomes on the basis of the simple end conditions. The best of our achievements display that considering shear deformation is essential for nanobeams with larger values of strain gradient parameter and small amounts of the nonlocal coefficient. Furthermore, we showed that the flexomagnetic (FM) effect brings about more noticeable shear deformations’ influence.

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Abbreviations

\(\varepsilon _{xx}\) :

Axial strain

\(\gamma _{xz}\) :

Shear strain

\(\eta _{xxz}\) :

Gradient of the axial elastic strain

\(C_{11}\) :

Elastic modulus

\(\sigma _{xx}\) :

Axial stress

\(\tau _{xz}\) :

Shear stress

\(f_{31}\) :

Component of the fourth-order flexomagnetic coefficients tensor

\(a_{33}\) :

Component of the second-order magnetic permeability tensor

\(q_{31}\) :

Component of the third-order piezomagnetic tensor

\(\xi _{xxz}\) :

Component of the higher-order hyper-stress tensor

\(B_{z}\) :

Magnetic flux

\(H_{z}\) :

Component of magnetic field

\(g_{31}\) :

Influence of the sixth-order gradient elasticity tensor

q :

Third-order piezomagnetic tensor

a :

Second-order magnetic permeability tensor

g :

sixth-order gradient elasticity tensor

C :

Fourth-order elasticity coefficient tensor

f :

Fourth-order flexomagnetic tensor

r :

Fifth-order tensor

\(u_{i}(i=1,3)\) :

Displacement in the x and z directions

u and w :

Axial and transverse displacements of the mid-plan

\(\phi \) :

Rotation of beam elements around the y axis

z :

Thickness coordinate

\(\psi \) :

External magnetic potential

\(\Psi \) :

Magnetic potential function

\(l\left( \mathrm{nm} \right) \) :

Strain gradient length scale parameter

\(\mu \left( \mathrm{nm} \right) ^{2}=\left( {e_{0} a} \right) ^{2}\) :

Nonlocal parameter

\(X_{m}\) :

Residue of the equations

\(k_{s}\) :

Shear correction factor

\(N_{x}\) :

Axial stress resultant

\(Q_{x}\) :

Shear stress resultant

\(M_{x}\) :

Moment stress resultant

\(T_{xxz}\) :

Hyper stress resultant

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Acknowledgements

V.A. Eremeyev acknowledges the support of the Government of the Russian Federation (contract No. 14.Z50.31.0046).

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Authors and Affiliations

  1. Department of Mechanics of Materials and Structures, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-233 Gdansk, Gdansk, Poland

    Mohammad Malikan & Victor A. Eremeyev

  2. Don State Technical University, Gagarina sq., 1, Rostov on Don, Russia, 344000

    Victor A. Eremeyev

  3. DICAAR, Università degli Studi di Cagliari, Via Marengo, 2, 09123, Cagliari, Italy

    Victor A. Eremeyev

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  1. Mohammad Malikan
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Correspondence to Mohammad Malikan.

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Malikan, M., Eremeyev, V.A. Flexomagneticity in buckled shear deformable hard-magnetic soft structures. Continuum Mech. Thermodyn. 34, 1–16 (2022). https://doi.org/10.1007/s00161-021-01034-y

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