A non-classical theory of elastic dielectrics incorporating couple stress and quadrupole effects: part I – reconsideration of curvature-based flexoelectricity theory

A new non-classical theory of elastic dielectrics is developed using the couple stress and electric field gradient theories that incorporates the couple stress, quadrupole and curvature-based flexoelectric effects. The couple stress theory and an extended Gauss’s law for elastic dielectrics with quadrupole polarization are applied to derive the constitutive relations of this new theory through energy conservation. The governing equations and the complete boundary conditions are simultaneously obtained through a variational formulation based on the Gibbs-type variational principle. The constitutive relations of general anisotropic and isotropic materials with the corresponding independent material constants are also provided, respectively. To illustrate the newly proposed theory and to show the flexoelectric effect in isotropic materials, one pure bending problem of a simply supported beam is analytically solved by directly applying the formulas derived. The analytical results reveal that the flexoelectric effect is present in isotropic materials. In addition, the incorporation of both the couple stress and flexoelectric effects always leads to increased values of the beam bending stiffness.

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Nonlinear Analysis of Microswitches Considering Nonclassical Theory

International Journal of Applied Mechanics ◽

10.1142/s1758825117501137 ◽

2017 ◽

Vol 09 (08) ◽

pp. 1750113 ◽

Cited By ~ 4

Author(s):

S. Hakamiha ◽

M. Mojahedi

Keyword(s):

Classical Theory ◽

Couple Stress ◽

Numerical Solutions ◽

Couple Stress Theory ◽

Modified Couple Stress Theory ◽

Stress Theory ◽

Size Dependent ◽

Modified Couple Stress ◽

Dependent Model ◽

Analytical Approaches

This paper introduces a new nonlinear model for microswitches based on the modified couple stress theory. The microswitch includes a microbeam which is connected to the clamped support from one side and attached to an electrostatically driven proof mass with a large gap from the other side. The microswitch operates in the pull-in instability with large deformation. The effects of fringing field and large curvature as well as size dependency are considered in the modeling. With regard to the size-dependent model, the equations of motion are obtained using Hamilton’s principle and solved by both numerical and analytical approaches. Consequently, dynamic pull-in instability is investigated based on the analytical and numerical solutions for dynamic conditions. The results depict that the dynamic deflection predicted by the modified couple stress theory is smaller than that obtained by the classical theory. The classical theory underestimates the pull-in instability voltage of the microswitches especially when the beam’s thickness is in the order of material length scale parameter. Furthermore, it is shown that neglecting nonlinearity due to large deflection leads to significant errors in the pull-in instability of the microstructures and these errors are calculated. The novelty of this paper is to provide a nonlinear size-dependent model for microswitches and to investigate the nonlinearity and instability of microswitches based on this model using the analytical and numerical methods.

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An Assessment of Couple Stress Theories

10.20944/preprints201811.0236.v1 ◽

2018 ◽

Cited By ~ 1

Author(s):

Ali R. Hadjesfandiari ◽

Gary F. Dargush

Keyword(s):

Stress Tensor ◽

Continuum Mechanics ◽

Couple Stress ◽

Couple Stress Theory ◽

Modified Couple Stress Theory ◽

Two Dimensional ◽

Stress Theory ◽

Size Dependent ◽

Modified Couple Stress ◽

Beam Bending

In this paper, we examine the mathematical and physical consistencies of the three primary couple stress theories: original Mindlin-Tiersten-Koiter couple stress theory (MTK-CST), modified couple stress theory (M-CST) and consistent couple stress theory (C-CST).  As has been known for many years, MTK-CST suffers from some fundamental inconsistencies, such as the indeterminacy of the couple-stress tensor.  Therefore, despite the fact that MTK-CST has a fundamental position in the evolution of size-dependent continuum mechanics, it is not a reliable theory within continuum mechanics, for example, in developing new size-dependent multi-physics formulations.  We also observe that M-CST not only inherits all inconsistences from the original MTK-CST, but also suffers from new additional inconsistencies, such as the introduction of a new non-physical governing equation.  These inconsistencies refute the claim of those who state that the couple-stress tensor may be chosen symmetric.  Therefore, the apparent success of MTK-CST and M-CST in describing a size-effect for some problems, such as two-dimensional plate and beam bending, is not enough to justify these theories as suitable for general cases.  In fact, the symmetric couple-stresses in M-CST create torsional or anticlastic deformation, not bending.  On the other hand, C-CST, with a skew-symmetric couple-stress tensor, is the consistent continuum mechanics suitable for solving different size-dependent solid, fluid and multi-physics problems. 

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Size-dependent dynamic behavior of microcantilevers under suddenly applied DC voltage

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406213490376 ◽

2013 ◽

Vol 228 (5) ◽

pp. 896-906 ◽

Cited By ~ 14

Author(s):

Masoud Rahaeifard ◽

Mohammad Taghi Ahmadian ◽

Keikhosrow Firoozbakhsh

Keyword(s):

Dynamic Behavior ◽

Classical Theory ◽

Couple Stress ◽

Multiple Scales ◽

Couple Stress Theory ◽

Modified Couple Stress Theory ◽

Stress Theory ◽

Dc Voltage ◽

Size Dependent ◽

Modified Couple Stress

This paper investigates the dynamic behavior of microcantilevers under suddenly applied DC voltage based on the modified couple stress theory. The cantilever is modeled based on the Euler–Bernoulli beam theory and equation of motion is derived using Hamilton’s principle. Both analytical and numerical methods are utilized to predict the dynamic behavior of the microbeam. Multiple scales method is used for analytical analysis and the numerical approach is based on a hybrid finite element/finite difference method. The results of the modified couple stress theory are compared with those from the literature as well as the results predicted by the classical theory. It is shown that the modified couple stress theory predicts size-dependent normalized dynamic behavior for the microbeam while according to the classical theory the normalized behavior of the microbeam is independent of its size. When the thickness of the beam is in order of its material length scale, the difference between the results given by the modified couple stress theory and those predicted by the classical theory is considerable. As the beam thickness increases, the results of the modified couple stress theory converge to those of the classical theory.

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