General Third-order Theory Research Articles

A general third-order theory of functionally graded plates with modified couple stress effect and the von Kármán nonlinearity: theory and finite element analysis

Finite element analysis of functionally graded plates based on a general third-order shear deformation plate theory with a modified couple stress effect and the von Karman nonlinearity is carried out to bring out the effects of couple stress, geometric nonlinearity and power-law variation of the material composition through the plate thickness on the bending deflections of plates. The theory requires no shear correction factors. The principle of virtual displacements is utilized to develop a nonlinear finite element model. The finite element model requires C1 continuity of all dependent variables. The microstructural effects are captured using a length scale parameter via the modified couple stress theory. The variation of two-constituent material is assumed through the thickness direction according to a power-law distribution. Numerical results are presented for static bending problems of rectangular plates with various boundary conditions to bring out the parametric effects of the power-law index and length scale parameter on the load–deflection characteristics of plates with various boundary conditions.

Third-order likelihood-based inference for the log-normal regression model

This paper examines the general third-order theory to the log-normal regression model. The interest parameter is its conditional mean. For inference, traditional first-order approximations need large sample sizes and normal-like distributions. Some specific third-order methods need the explicit forms of the nuisance parameter and ancillary statistic, which are quite complicated. Note that this general third-order theory can be applied to any continuous models with standard asymptotic properties. It only needs the log-likelihood function. With small sample settings, the simulation studies for confidence intervals of the conditional mean illustrate that the general third-order theory is much superior to the traditional first-order methods.

Nonlinear viscoelastic analysis of orthotropic beams using a general third-order theory

The displacement based finite element model of a general third-order beam theory is developed to study the quasi-static behavior of viscoelastic rectangular orthotropic beams. The mechanical properties are considered to be linear viscoelastic in nature with a scope to undergo von Kármán nonlinear geometric deformations. A differential constitutive law is developed for an orthotropic linear viscoelastic beam under the assumptions of plane-stress. The fully discretized finite element equations are obtained by approximating the convolution integrals using a trapezoidal rule. A two-point recurrence scheme is developed that necessitates storage of data from the previous time step only, and not from the entire deformation history. Full integration is used to evaluate all the stiffness terms using spectral/hp lagrange polynomials. The Newton iterative scheme is employed to enhance the rate of convergence of the nonlinear finite element equations. Numerical examples are presented to study the viscoelastic phenomena like creep, cyclic creep and recovery for thick and thin beams using classical mechanical analogues like generalized n-parameter Kelvin-Voigt solids and Maxwell solids.

A nonlinear modified couple stress-based third-order theory of functionally graded plates

In this paper a general nonlinear third-order plate theory that accounts for (a) geometric nonlinearity, (b) microstructure-dependent size effects, and (c) two-constituent material variation through the plate thickness (i.e., functionally graded material plates) is presented using the principle of virtual displacements. A detailed derivation of the equations of motion, using Hamilton’s principle, is presented, and it is based on a modified couple stress theory, power-law variation of the material through the thickness, and the von Kármán nonlinear strains. The modified couple stress theory includes a material length scale parameter that can capture the size effect in a functionally graded material. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity, microstructure dependent size effect, and material gradation through the thickness are specialized to classical and shear deformation plate theories available in the literature. The theory presented herein also can be used to develop finite element models and determine the effect of the geometric nonlinearity, microstructure-dependent size effects, and material grading through the thickness on bending and postbuckling response of elastic plates.

A General Nonlinear Third-Order Theory of Functionally Graded Plates

A general third-order plate theory that accounts for geometric nonlinearity and twoconstituent material variation through the plate thickness (i.e., functionally graded plates) is presented using the dynamic version of the principle of virtual displacements. The formulation is based on power-law variation of the material through the thickness and the von Karman nonlinear strains. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity and material gradation through the thickness are specialized to the existing classical and shear deformation plate theories in the literature. The theoretical developments presented herein can be used to develop finite element models and determine the effect of the geometric nonlinearity and material grading through the thickness on the bending, vibration, and buckling and postbuckling response of elastic plates.

A nonlinear higher order shear deformation shell element for dynamic explicit analysis:: Part I. Formulation and finite element equations

This work presents the finite element formulation of a higher order shear deformation shell element for nonlinear dynamic analysis with explicit time integration scheme. A corotational approach is combined with the velocity strain equations of a general third-order theory in the formulation of a four-noded quadrilateral element with selectively reduced integration. A bilinear isoparametric formulation is utilized in the shell plane resulting in nine degrees of freedom per node. The formulation requires C 0 continuity for the nodal variables. The finite element implementation of the new element in a general explicit finite element code is described in details, including boundary conditions and nodal mass calculation. A simple formula for the explicit time integration critical time step of the higher order element is developed. The described element is capable of correctly representing the through thickness distribution of the transverse shear, which makes it suitable for composite and sandwich shells analysis. In addition, the developed shell can be used for better representation of plastic flow through thickness in isotropic materials. It has been added to the element library of the nonlinear explicit finite element code DYNA3D. Its performance has been evaluated through a series of standard shell verification test problems, which show great promise for many applications. The results are presented in Part II of the present work.