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![Cantilever plate mid-surface mode shapes with L/h=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L/h = 10$$\end{document}, b/h=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b/h = 8$$\end{document}, and l/h=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l/h = 10$$\end{document} for three-dimensional RSM analysis: a first bending, b first twisting, c first lateral shearing, d second bending, e second twisting, f first lateral bending](publication/365602034/figure/fig7/AS:11431281126517577@1678762811785/Cantilever-plate-mid-surface-mode-shapes-with-L-h10documentclass12ptminimal.png)
Cantilever plate mid-surface mode shapes with L/h=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L/h = 10$$\end{document}, b/h=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b/h = 8$$\end{document}, and l/h=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l/h = 10$$\end{document} for three-dimensional RSM analysis: a first bending, b first twisting, c first lateral shearing, d second bending, e second twisting, f first lateral bending
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After deriving a novel Principle of Stationary Correlated Action for the general bi-anisotropic case of size-dependent materials governed by consistent couple stress theory, a Ritz Spline Method is created for free vibration analysis. The underlying formulation is then specialized for isotropic materials requiring the specification of only a single...
Citations
... -by-parts to the first two terms in equation(36) provides the following weak form for the 4 th order real-valued matter-wave equation: ...
Using a variational formulation, we show that Schrodinger's 4th-order, real-valued matter-wave equation which involves the spatial derivatives of the potential V(r), produces the precise eigenvalues of Schrodinger's 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues. Accordingly, the paper concludes that there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative (repelling) energy levels. Schrodinger's classical 2nd-order, complex-valued matter-wave equation which was constructed upon factoring the 4th-order, real-valued differential operator and retaining only one of the two conjugate complex operators is a simpler description of the matter-wave, since it does not involve the derivatives of the potential V(r), at the expense of missing the negative (repelling) energy levels.
... In the MCST and MSGT, only 1 and 3 additional material properties are introduced, respectively, in addition to the Lamé constants. Finite elements have been developed to obtain approximate solutions for these theories [18][19][20][21][22][23][24][25]. They have been widely utilized to analyze size effect in the elastic regime, with a particular focus on applications in functionally graded materials [26][27][28], composites [29,30], porous materials [31,32], and shape memory alloys [33]. ...
... where r represents the dilatation gradient and is expressed as a function of displacement gradient in a vector form as ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ϕ 11,1 ϕ 22,2 ϕ 33,3 ϕ 21,1 + ϕ 11,2 + ϕ 12,1 ϕ 31,1 + ϕ 11,3 + ϕ 13,1 ϕ 12,2 + ϕ 22,1 + ϕ 21,2 ϕ 32,2 + ϕ 22,3 + ϕ 23,2 ϕ 13,3 + ϕ 33,1 + ϕ 31,3 ϕ 23,3 + ϕ 33,2 + ϕ 32,3 ϕ 31,2 + ϕ 32,1 + ϕ 12,3 + ϕ 13,2 + ϕ 23,1 + ϕ 21 ϕ 13,2 + ϕ 12,3 − ϕ 31,2 − ϕ 32,1 ϕ 21,3 + ϕ 23,1 − ϕ 12,3 − ϕ 13,2 2ϕ 32,2 − ϕ 23,2 − ϕ 22,3 + ϕ 13,1 + ϕ 11,3 − 2ϕ 3,11 2ϕ 1,33 − ϕ 31,3 − ϕ 33,1 + ϕ 21,2 + ϕ 22,1 − 2ϕ 1,22 2ϕ 2,11 − ϕ 12,1 − ϕ 11,2 + ϕ 32,3 + ϕ 33,2 − 2ϕ 2,33 ...
Among various strain gradient theories, the modified couple stress theory, which introduces a single length scale parameter as an additional material property, has garnered significant interest owing to its simplified portrayal of the material behavior. In this study, we investigated whether a single length scale parameter is sufficient to predict the mechanical behavior under two different type of strain gradients: torsion and bending. L-shaped beams made of single crystal copper with thicknesses ranging from 2.4 μm to 9.1 μm were fabricated, and loads were applied using an indenter. The contributions of bending and torsion were controlled by adjusting the loading position. Through these experiments, we demonstrated the existence of elastic size effect of single crystalline materials under strain gradients. Specifically, size effect was observed in both bending and torsion, with a larger effect observed in cases closer to pure bending. Moreover, we report that the modified couple stress theory and the modified strain gradient theory are not applicable for simulating size effect under combined loading. This discovery highlights the necessity for the development of a new theory capable of adequately simulating size effect under the intricate loading scenarios encountered in practical applications.
... Numerous experimental studies have fully demonstrated the existence of size effect in the mechanical behavior of such small-scale structures and the classical continuum theory is incapable of incorporating this size dependence due to the lack of internal length scale material parameters. On the contrary, as a kind of high-order continuum theory, the couple stress theory (CST) has been proved to be effective in simulating the size-dependent mechanical behavior [4][5][6][7]. Compared with other high-order continuum theories, the advantage of the CST lies in its concise mathematical form and clear physical interpretation. ...
In this work, the stress function of the orthotropic plane strain problems in the modified couple stress theory (MCST) is first proposed and its polynomial analytical solutions are derived. Then, the stress function is used for developing a new 4-node membrane element for the MCST in that the C¹ continuity requirement is satisfied in weak sense using the penalty function method. In the element formulation, the stress function is adopted as the primary parameter for designing element’s stress and couple stress trial functions instead of assuming them directly. Therefore, the deduced stress and couple stress trial functions can satisfy both the equilibrium equations and deformation compatibility equations of the relevant problems a priori. Several numerical tests are examined and the results show that the element has good numerical accuracy and mesh distortion tolerance in simulating the size-dependent behaviors of small-scale orthotropic materials, proving that the usage of analytic trial function in the finite element implementation of the MCST can effectively improve element’s performance.
... Kong et al. [22] studied the dynamic characteristics of the microbeam based on the modified couple stress theory and obtained the enhancement behavior of the natural frequency. Ye et al. [23] investigated the size dependency of the static bending of a bilayer microbeam, and the size effect on beam stiffness is significant when the beam height is on orders of the equivalent length scale parameter. Furthermore, the size-dependent natural frequencies of fluid-conveying microtubes [24], the size-dependent nonlinear dynamics of a microbeam [25], the size-dependent buckling behavior of microtubules [26], the size-dependent resonant frequencies and sensitivities of AFM microcantilevers [27], the size-dependent couple stress natural frequency analysis for two-and three-dimensional problems [28], the size-dependent parametrization of active vibration control for periodic piezoelectric microplate [29] have been investigated based on the modified couple stress theory. ...
... Equations (23)(24)(25) represent the classical Timoshenko beam model. If no axial deformation is considered (i.e., u(x,t) 0) which can be further reduced to that formulated in Hutchinson [44] and to that summarized in Wang [45] for static bending if u( ...
In this work, we discuss size-dependent and microinertia effects on the static and dynamic performances of a microscale model based on the microinertia-based modified couple stress theory, a non-classical continuum theory capable of capturing the behavior of size dependence and frequency dispersion characteristics. In the framework of the variational statement, a microscale structure model is developed and the governing equations of equilibrium as well as all boundary conditions for statics and dynamics are reformulated. The developed theory is imposed to tackle microstructure-dependent Timoshenko beam model in two distinct scale parameters: the material length scale parameter is utilized to determine the size dependence and the microinertia length scale parameter is employed to describe the higher-order microrotation relation. The generally valid closed-form analytic expressions are obtained and suitable for various formats of boundaries and mechanical loads. As case studies, the predicted trends agree with those observed within the framework of the modified couple stress theory. Results indicate that the material microlength scale parameter strengthens the static deformations, while the microinertia length scale parameter weakens the dynamic frequencies. In addition, boundary conditions are also an important aspect in statics and dynamics as well as the mechanical response predicted by non-classical continuum theories.
... Rectangular plate is a typical structural unit in the engineering fields of roads, bridges and buildings, etc. Due to the inhomogeneous deformation of the structural boundary of the rectangular plate and other reasons, there is an in-plane initial stress in the structure under in-plane loads, which changes its original dynamic properties and makes the actual dynamic properties of the structure difficult to predict [1][2][3][4]. Most researchers currently use experimental or finite element simulation methods to study its dynamic properties [5][6][7][8]. ...
... The proposed method is found to be fast converging, accurate and universal. The plane stress equation is an equilibrium differential equation based on the two-dimensional elastic theory, and the consistency equation [8] is as follows: ...
... where the relation between the admissible function ϕ and stresses in each direction [8], i.e., ...
Stresses are generated in plates under initial loads, which couple with the subsequent transverse deformation to affect the buckling and transverse vibration characteristics of plates. There exists no exact solution method for the stress field of a rectangular plate under arbitrary in-plane loads. In this paper, the stress field of a rectangular plate under arbitrary in-plane loads is solved based on the principle of minimum potential energy. The stress function is decomposed into homogeneous and special solutions. The homogeneous solution is represented by the Chebyshev polynomial while the special solution is expanded by the Fourier series. Then, the Ritz method is used to analyze the transverse vibration and buckling characteristics of the rectangular plate. The product of the boundary function and Chebyshev polynomial is used to build the vibration mode function. The present results are verified by comparing to existing studies and those obtained from finite element method (FEM). The effects of the magnitude and distribution of in-plane boundary stresses and boundary conditions on the dynamic characteristics and stability of the rectangular plate are analyzed. The method used in the paper has demonstrated improved convergence and accuracy with good universality.
... Neff et al. [33] reviewed some fundamental misunderstandings in the indeterminate couple stress model. In recent years, the couple stress theories have been applied to many researches, including elastic plates [34,35] and other different fields [36][37][38][39]. ...
In this paper, materials with nonlocal properties are considered as a continuum model composed of micro-elements with certain volumes. Based on this hypothesis, the deformation and corresponding energy of micro-structure system are studied in detail, and the equivalent governing equations in simplified form are given. In the framework of micro-structure system, a simplified deformation gradient theory (SDG) with two length-scale parameters is obtained by defining the micro-strain and micro-rotation of elements specifically from the perspective of deformation, which has definite physical significance. The generalized strain energy is introduced in the SDG, which gives a new explanation of elastic moduli, and the nonlocal effect parameter is defined to capture nonlocal properties of materials quantitatively. Under certain micro-deformation assumptions, the SDG can be degenerated into couple stress theory, strain gradient theory and classical continuum theory. The nonlocal deformation consists of two branches: one is the macro-deformation of non-uniform materials and the other is the micro-deformation of materials with micro-structures. For the macro-tension of particle reinforced composites, the SDG successfully verifies and predicts the approximatively linear relationship between elastic moduli and particle sizes on the micron scale. Moreover, the theoretical solution to nonlocal micro-torsion based on the SDG agrees well with the experiment results and also predicts the torsion stiffness of cylinder at smaller diameters.
