A new Kirchhoff plate model based on a modified couple stress theory

In this paper a new Kirchhoff plate model is developed for the static analysis of isotropic micro-plates with arbitrary shape based on a modified couple stress theory containing only one material length scale parameter which can capture the size effect. The proposed model is capable of handling plates with complex geometries and boundary conditions. From a detailed variational procedure the governing equilibrium equation of the micro-plate and the most general boundary conditions are derived, in terms of the deflection, using the principle of minimum potential energy. The resulting boundary value problem is of the fourth order (instead of existing gradient theories which is of the sixth order) and it is solved using the Method of Fundamental Solutions (MFS) which is a boundary-type meshless method. Several plates of various shapes, aspect and Poisson’s ratios are analyzed to illustrate the applicability of the developed micro-plate model and to reveal the differences between the current model and the classical plate model. Moreover, useful conclusions are drawn from the micron-scale response of this new Kirchhoff plate model.     

A non-classical Kirchhoff plate model incorporating microstructure,surface energy and foundation effects

A new non-classical Kirchhoff plate model is developed using a modified couple stress theory, a surface elasticity theory and a two-parameter elastic foundation model. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and the complete boundary conditions and provides a unified treatment of the microstructure, surface energy and foundation effects. The new plate model contains a material length scale parameter to account for the microstructure effect, three surface elastic constants to describe the surface energy effect, and two foundation moduli to represent the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the newly developed plate model includes the models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases and recovers the Bernoulli–Euler beam model incorporating the microstructure, surface energy and foundation effects. To illustrate the new model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection of the simply supported plate with or without the elastic foundation predicted by the current model is smaller than that predicted by the classical model. Also, it is observed that the difference in the deflection predicted by the new and classical plate models is very large when the plate thickness is sufficiently small, but it is diminishing with the increase of the plate thickness. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the elastic foundation is higher than that predicted by the classical plate model, and the difference is significant for very thin plates. These predicted trends of the size effect at the micron scale agree with those observed experimentally. In addition, it is shown both analytically and numerically that the presence of the elastic foundation reduces the plate deflection and increases the plate natural frequency, as expected.     

Damped vibration of a graphene sheet using a higher-order nonlocal strain-gradient Kirchhoff plate model

A higher-order nonlocal strain-gradient model is presented for the damped vibration analysis of single-layer graphene sheets (SLGSs) in hygrothermal environment. Based on Kirchhoff plate theory in conjunction with a higher-order (bi-Helmholtz) nonlocal strain gradient theory, the equations of motion are obtained using Hamilton's principle. The higher-order nonlocal strain gradient theory has lower- and higher-order nonlocal parameters and a material characteristic parameter. The presented model can reasonably interpret the softening effects of the SLGS, and indicates a reasonably good match with the experimental flexural frequencies. Finally, the roles of viscous and structural damping coefficients, small-scale parameters, hygrothermal environment and elastic foundation on the vibrational responses of SLGSs are studied in detail.     

Boundary element analysis of anisotropic Kirchhoff plates

In this paper, the radial integration method is used to obtain a boundary element formulation without any domain integral for general anisotropic plate bending problems. Two integral equations are used and the unknown variables are assumed to be constant along each boundary element. The domain integral which arises from a transversely applied load is exactly transformed into a boundary integral by a radial integration technique. Uniformly and linearly distributed loads are considered. Several computational examples concerning orthotropic and general anisotropic plate bending problems are presented. The results show good agreement with analytical and finite element results available in the literature.     

A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model

A new modified couple stress theory for anisotropic elasticity is proposed. This theory contains three material length scale parameters. Differing from the modified couple stress theory, the couple stress constitutive relationships are introduced for anisotropic elasticity, in which the curvature (rotation gradient) tensor is asymmetric and the couple stress moment tensor is symmetric. However, under isotropic case, this theory can be identical to modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The differences and relations of standard, modified and new modified couple stress theories are given herein. A detailed variational formulation is provided for this theory by using the principle of minimum total potential energy. Based on the new modified couple stress theory, composite laminated Kirchhoff plate models are developed in which new anisotropic constitutive relationships are defined. The First model contains two material length scale parameters, one related to fiber and the other related to matrix. The curvature tensor in this model is asymmetric; however, the couple stress moment tensor is symmetric. Under isotropic case, this theory can be identical to the modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The present model can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model of couple stress theory including only one material length scale parameter for modeling the cross-ply laminated Kirchhoff plate is suggested. Numerical results show that the proposed laminated Kirchhoff plate model can capture the scale effects of microstructures.     

Deflection of sandwich plates in terms of corresponding Kirchhoff plate solutions

Summary This study presents exact relationships between the deflections of isotropic sandwich plates and their corresponding Kirchhoff plates. The governing equilibrium equations for the sandwich plates are derived on the basis of the Reissner-Mindlin shear deformation plate theory. The considered plates are either (i) simply supported, of general polygonal shape and under any transverse loading condition or (ii) simply supported and clamped circular plates under axisymmetric loading. As the relationships are exact under the assumptions used in the plate theories, one may obtain exact deflection solutions of sandwich plates if the Kirchhoff plate solutions are exact. The relationships should also be useful for the development of approximate formulas for plates with other shapes, boundary and loading conditions, and may serve to check numerical deflection values computed from sandwich plate analysis software.     

Finite element approximation of field dislocation mechanics

A tool for studying links between continuum plasticity and dislocation theory within a field framework is presented. A finite element implementation of the geometrically linear version of a recently proposed theory of field dislocation mechanics (J. Mech. Phys. Solids 49 (2001) 761; Proc. Roy. Soc. 459 (2003) 1343; J. Mech. Phys. Solids 52 (2004) 301) represents the main idea behind the tool. The constitutive ingredients of the theory under consideration are simply elasticity and a specification of dislocation velocity and nucleation. The set of equations to be approximated are non-standard in the context of solid mechanics applications. It comprises the standard second-order equilibrium equations, a first-order div-curl system for the elastic incompatibility, and a first-order, wave-propagative system for the evolution of dislocation density. The latter two sets of equations require special treatment as the standard Galerkin method is not adequate, and are solved utilizing a least-squares finite element strategy. The implementation is validated against analytical results of the classical elastic theory of dislocations and analytical results of the theory itself. Elastic stress fields of dislocation distributions in generally anisotropic media of finite extent, deviation from elastic response, yield-drop, and back-stress are shown to be natural consequences of the model. The development of inhomogeneity, from homogeneous initial conditions and boundary conditions corresponding to homogeneous deformation in conventional plasticity, is also demonstrated. To our knowledge, this work represents the first computational implementation of a theory of dislocation mechanics where no analytical results, singular solutions in particular, are required to formulate the implementation. In particular, a part of the work is the first finite element implementation of Kröner's linear elastic theory of continuously distributed dislocations in its full generality.     

Effects of inplane constraints and curvature on composite plate behavior

An investigation into the behavior of unsymmetrically laminated anisotropic plates is carried out. Using the basic assumptions of linear plate theory and strain-displacement relations modified to account for small initial curvature, energy formulations are established and the Ritz method utilized to obtain solutions for static deflections and natural frequencies of vibration. The boundary condition with particular inplane restraints and the particular stacking arrangement are shown to have important effects on plate response when bending-extensional coupling is present. Small initial curvature is shown to greatly increase natural frequencies and may obscure the effects of bending-extensional coupling due to nonsymmetrical layering. The increase in lateral stiffness due to curvature is also shown to depend heavily on fiber orientation, orthotropicity and boundary conditions.     

Finite element approximation for single-phase multicomponent flows

With the objective of performing computational simulations of mixture problems, this work employed a mechanical modeling of single-phase incompressible multicomponent flows able to deal with geometric and material non-linearities. This model is based on conservative laws of mass and momentum in continuum mechanics, assuming the mixture as a superposition of a number of continuous media, each one with a variable volume fraction field. Using appropriate hypothesis, the model was formulated as a set of non-linear partial differential equations and associated boundary conditions. This set was approximated by a stabilized finite element method, based on a Galerkin/least-squares scheme, in order to circumvent Babu ka–Brezzi condition and to generate stable approximations even in highly advective situations. Some preliminary numerical results have been obtained for non-Newtonian axial injections in backward facing step incompressible flows.     

A new type of plate bending element

Based on a two-field generalized variational principle, a new type of arbitrary quadrilateral plate bending element, with four nodes and with the effect of transverse shear deformation taken into account, is proposed. The element is applicable to a wide range of plate thickness and an explicit expression of stiffness matrix can be obtained. Therefore, it possesses the distinguished features of general applicability, high precision and less computer time.