A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli–Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli–Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.

A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

A new model for the bending of a Bernoulli–Euler beam is developed using a modified couple stress theory. A variational formulation based on the principle of minimum total potential energy is employed. The new model contains an internal material length scale parameter and can capture the size effect, unlike the classical Bernoulli–Euler beam model. The former reduces to the latter in the absence of the material length scale parameter. As a direct application of the new model, a cantilever beam problem is solved. It is found that the bending rigidity of the cantilever beam predicted by the newly developed model is larger than that predicted by the classical beam model. The difference between the deflections predicted by the two models is very significant when the beam thickness is small, but is diminishing with the increase of the beam thickness. A comparison shows that the predicted size effect agrees fairly well with that observed experimentally.

https://iopscience.iop.org/article/10.1088/0960-1317/16/11/015/pdf

Bernoulli–Euler beam model based on a modified couple stress theory

A microstructure-dependent nonlinear Euler–Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material are developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Karman geometric nonlinearity. The model contains a material length scale parameter that can capture the size effect in a functionally graded material, unlike the classical Euler–Bernoulli and Timoshenko beam theories. The influence of the parameter on static bending, vibration and buckling is investigated. The theoretical developments presented herein also serve to develop finite element models and determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on post-buckling response.

Microstructure-dependent couple stress theories of functionally graded beams

The size-dependent natural frequency of Bernoulli-Euler micro-beams

Crystalline piezoelectric dielectrics electrically polarize upon application of uniform mechanical strain. Inhomogeneous strain, however, locally breaks inversion symmetry and can potentially polarize even nonpiezoelectric (centrosymmetric) dielectrics. Flexoelectricity\char22{}the coupling of strain gradient to polarization\char22{}is expected to show a strong size dependency due to the scaling of strain gradients with structural feature size. In this study, using a combination of atomistic and theoretical approaches, we investigate the ``effective'' size-dependent piezoelectric and elastic behavior of inhomogeneously strained nonpiezoelectric and piezoelectric nanostructures. In particular, to obtain analytical results and tease out physical insights, we analyze a paradigmatic nanoscale cantilever beam. We find that in materials that are intrinsically piezoelectric, the flexoelectricity and piezoelectricity effects do not add linearly and exhibit a nonlinear interaction. The latter leads to a strong size-dependent enhancement of the apparent piezoelectric coefficient resulting in, for example, a ``giant'' 500% enhancement over bulk properties in $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$ for a beam thickness of $5\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$. Correspondingly, for nonpiezoelectric materials also, the enhancement is nontrivial (e.g., 80% for $5\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ size in paraelectric $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$ phase). Flexoelectricity also modifies the apparent elastic modulus of nanostructures, exhibiting an asymptotic scaling of $1∕{h}^{2}$, where $h$ is the characteristic feature size. Our major predictions are verified by quantum mechanically derived force-field-based molecular dynamics for two phases (cubic and tetragonal) of $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$.

Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect

A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions, thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a significant effect on material responses.

Variational formulation of a modified couple stress theory and its application to a simple shear problem

Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem

The problem of a pressurized thick-walled spherical shell is analytically solved using a simplified strain gradient elasticity theory. The closed-form solution derived contains a material length scale parameter and can account for microstructural effects, which qualitatively differs from Lame’s solution in classical elasticity. When the strain gradient effect (a measure of the underlying material microstructure) is not considered, the newly derived strain gradient elasticity solution reduces to Lame’s classical elasticity solution. To illustrate the new solution, a sample problem with specified geometrical parameters, pressure values and material properties is solved. The numerical results reveal that the magnitudes of both the radial and tangential stress components in the shell wall given by the current strain gradient solution are smaller than those given by Lame’s solution. Also, it is quantitatively shown that microstructural effects can be large and Lame’s solution may not be accurate for materials ...

Analytical Solution for a Pressurized Thick-Walled Spherical Shell Based on a Simplified Strain Gradient Elasticity Theory

A micromechanics model is developed for hexagonal honeycomb structures by using a modified couple stress theory. The formulation is based on structural analysis of repeating units and a homogenization procedure. Both bending and axial deformations of cell walls are considered here, unlike earlier models. The closed-form expressions for five effective in-plane elastic constants (including two Young's moduli, one shear modulus and two Poisson's ratios) of a honeycomb structure are derived in terms of the Young's modulus of the cell wall material and the slenderness (length-to-thickness) ratio of the cell wall. The formulas for two length scale parameters are obtained in terms of the slenderness ratio and thickness of the cell wall, thereby providing a direct measure of the microstructure of the honeycomb. The constitutive relations for the two-dimensional couple stress continuum equivalently representing the discrete honeycomb structure are provided using the five effective elastic constants and two length ...

S. K. Park

Papers

Bernoulli–Euler beam model based on a modified couple stress theory

Variational formulation of a modified couple stress theory and its application to a simple shear problem

Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem

Analytical Solution for a Pressurized Thick-Walled Spherical Shell Based on a Simplified Strain Gradient Elasticity Theory

Micromechanical Modeling of Honeycomb Structures Based on a Modified Couple Stress Theory