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Mathematical Modeling of Bending of a Thin Orthotropic Rectangular Plate Based on Couple Stress Theory.

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Title: Mathematical Modeling of Bending of a Thin Orthotropic Rectangular Plate Based on Couple Stress Theory.
Authors: Germider, O. V.1 (AUTHOR) o.germider@narfu.ru, Popov, V. N.1 (AUTHOR) v.popov@narfu.ru
Source: Computational Mathematics & Mathematical Physics. Apr2026, Vol. 66 Issue 4, p732-748. 17p.
Subjects: Rectangular plates (Engineering), Chebyshev polynomials, Structural plates, Deformations (Mechanics), Collocation methods, Strains & stresses (Mechanics), Mathematical models
Abstract: A two-parameter mathematical model is developed for studying the strain state of a thin orthotropic nanoplate of constant thickness subjected to an arbitrary transverse load. Assuming that the deformation conditions are isothermal and the displacements of the plate points are small compared to the plate thickness, a differential equation for the nanoplate deflection is derived from the minimum total free energy condition, taking into account the curvature tensor components under small rotation at the microlevel. A solution to this equation for a rectangular nanoplate is constructed with Chebyshev polynomials of the first kind used as a basis in a Hilbert function space. The coefficients in the expansion of the approximating function in terms of these polynomials are found using the collocation method. The roots of Chebyshev polynomials of the first kind are used as collocation points. The error of the constructed solution is estimated in the norm of the Banach space of essentially bounded functions. The plate deflection under constant and distributed loading for which there is an analytical solution is computed and analyzed depending on nonlocal length scale parameters. [ABSTRACT FROM AUTHOR]
Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Mathematical Modeling of Bending of a Thin Orthotropic Rectangular Plate Based on Couple Stress Theory.
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  Data: <searchLink fieldCode="JN" term="%22Computational+Mathematics+%26+Mathematical+Physics%22">Computational Mathematics & Mathematical Physics</searchLink>. Apr2026, Vol. 66 Issue 4, p732-748. 17p.
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  Data: <searchLink fieldCode="DE" term="%22Rectangular+plates+%28Engineering%29%22">Rectangular plates (Engineering)</searchLink><br /><searchLink fieldCode="DE" term="%22Chebyshev+polynomials%22">Chebyshev polynomials</searchLink><br /><searchLink fieldCode="DE" term="%22Structural+plates%22">Structural plates</searchLink><br /><searchLink fieldCode="DE" term="%22Deformations+%28Mechanics%29%22">Deformations (Mechanics)</searchLink><br /><searchLink fieldCode="DE" term="%22Collocation+methods%22">Collocation methods</searchLink><br /><searchLink fieldCode="DE" term="%22Strains+%26+stresses+%28Mechanics%29%22">Strains & stresses (Mechanics)</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+models%22">Mathematical models</searchLink>
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  Data: A two-parameter mathematical model is developed for studying the strain state of a thin orthotropic nanoplate of constant thickness subjected to an arbitrary transverse load. Assuming that the deformation conditions are isothermal and the displacements of the plate points are small compared to the plate thickness, a differential equation for the nanoplate deflection is derived from the minimum total free energy condition, taking into account the curvature tensor components under small rotation at the microlevel. A solution to this equation for a rectangular nanoplate is constructed with Chebyshev polynomials of the first kind used as a basis in a Hilbert function space. The coefficients in the expansion of the approximating function in terms of these polynomials are found using the collocation method. The roots of Chebyshev polynomials of the first kind are used as collocation points. The error of the constructed solution is estimated in the norm of the Banach space of essentially bounded functions. The plate deflection under constant and distributed loading for which there is an analytical solution is computed and analyzed depending on nonlocal length scale parameters. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1134/S0965542526700065
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      – Code: eng
        Text: English
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        Type: general
      – SubjectFull: Chebyshev polynomials
        Type: general
      – SubjectFull: Structural plates
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      – SubjectFull: Deformations (Mechanics)
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      – SubjectFull: Collocation methods
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      – SubjectFull: Strains & stresses (Mechanics)
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      – TitleFull: Mathematical Modeling of Bending of a Thin Orthotropic Rectangular Plate Based on Couple Stress Theory.
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              M: 04
              Text: Apr2026
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              Y: 2026
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