Determination of hourglass coefficients in the theory of a Cosserat point for nonlinear elastic beams

The theory of a Cosserat point has recently been used [Int. J. Solids Struct. 38 (2001) 4395] to formulate the numerical solution of problems of nonlinear elastic beams. In that theory the constitutive equations for inhomogeneous elastic deformations included undetermined constants associated with hourglass modes which can occur due to nonuniform cross-sectional extension and nonuniform torsion. The objective of this paper is to determine these hourglass coefficients by matching exact solutions of pure bending and pure torsion applied in different directions on each of the surfaces of the element. It is shown that the resulting constitutive equations in the Cosserat theory do not exhibit unphysical stiffness increases due to thinness of the beam, mesh refinement or incompressibility that are present in the associated Bubnov–Galerkin formulation. Also, example problems of a bar hanging under its own weight and a bar attached to a spinning rigid hub are analyzed.     

A constrained theory of a Cosserat point for the numerical solution of dynamic problems of non-linear elastic rods with rigid cross-sections

A constrained theory of a Cosserat point has been developed for the numerical solution of non-linear elastic rods. The cross-sections of the rod element are constrained to remain rigid but tangential shear deformations and axial extension are admitted. As opposed to the more general theory with deformable cross-sections, the kinetic coupling equations in the numerical formulation of the constrained theory are expressed in terms of the simple physical quantities of force and mechanical moment applied to the common ends of neighboring elements. Also, in contrast with standard finite element methods, the Cosserat element uses a direct approach to the development of constitutive equations. Specifically the kinetic quantities are determined by algebraic expressions which are obtained by derivatives of a strain energy function. Most importantly, no integration is needed over the element region. A number of example problems have been considered which indicate that the constrained Cosserat element can be used to model large deformation dynamic response of non-linear elastic rods.     

Nonlinear thickness-shear vibration of an infinite piezoelectric plate with flexoelectricity based on the method of multiple scales

This paper presents a nonlinear thickness-shear vibration model for onedimensional infinite piezoelectric plate with flexoelectricity and geometric nonlinearity. The constitutive equations with flexoelectricity and governing equations are derived from the Gibbs energy density function and variational principle. The displacement adopted here is assumed to be antisymmetric through the thickness due to the thickness-shear vibration mode. Only the shear strain gradient through the thickness is considered in the present model. With geometric nonlinearity, the governing equations are converted into differential equations as the function of time by the Galerkin method. The method of multiple scales is employed to obtain the solution to the nonlinear governing equation with first order approximation. Numerical results show that the nonlinear thickness-shear vibration of piezoelectric plate is size dependent, and the flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro-size thin plates. The geometric nonlinearity also affects the thickness-shear vibration frequencies greatly. The results show that flexoelectricity and geometric nonlinearity cannot be ignored in design of accurate high-frequency piezoelectric devices.     

A New Stable Bubble Element for Incompressible Fluid Flow Based on a Mixed Petrov–Galerkin Finite Element Formulation

A new mixed Petrov–Galerkin formulation employing the MINI element with a non-confirming bubble function for an incompressible media governed by the Stokes equations, which is equivalent to the stabilized finite element by P ¹-P ¹ approximation, is proposed. The new formulation possesses better stability properties than the conventional Bubnov–Galerkin formulation employing the MINI element. In this aspect, the stabilizing effect of this formulation is evaluated by a stabilizing parameter determined by both shapes of the trial and the weighting bubble functions.     

Finite element formulation of slender structures with shear deformation based on the Cosserat theory

This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.     

Basic Equations of Kelvin's Medium and Analogy with Ferromagnets

A nonlinear elastic Cosserat continuum of special kind whose particles have dynamical spins (Kelvin's medium) is considered. The basic equations for this medium are obtained on the basis of fundamental mechanical laws. Various complete sets of strain tensors are discussed. The analogies between Kelvin's medium and other types of media are discussed. First, we show the analogy between constitutive equations for Kelvin's medium and for elastic shells. The main difference is caused by the dimensionality of the media. Then, for a special case of fast proper rotation we establish the analogy between the dynamic equations of Kelvin's medium and elastic ferromagnetic saturated insulators. The most general way to take into account the coupling between magnetic and elastic subsystems is presented. All results can be interpreted in terms of a mechanical medium as well as in terms of ferromagnets. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Double-Strand Elastic Rod Theory

Motivated by applications in the modeling of deformations of the DNA double helix, we construct a continuum mechanics model of two elastically interacting elastic strands. The two strands are described in terms of averaged, or macroscopic, variables plus an additional small, internal or microscopic, perturbation. We call this composite structure a birod. The balance laws for the macroscopic configuration variables of the birod can be cast in the form of a classic Cosserat rod model with coupling to the internal balance laws through the constitutive relations. The internal balance laws for the microstructure variables also take a mathematical form analogous to that for a Cosserat rod, but with coupling to the macroscopic system through terms corresponding to distributed force and couple loads.     

Stabilized Bubble Function Method for Shallow Water Long Wave Equation

A relationship between the stabilized bubble function method and the stabilized finite element method is shown in this paper. The Petrov–Galerkin formulation with bubble function, i.e. a stabilized bubble function method, is proposed for the shallow water long wave equation. The Petrov–Galerkin formulation with the bubble function formulation possesses better stability than the Bubnov–Galerkin formulation with the bubble function.     

A Helical Cauchy-Born Rule for Special Cosserat Rod Modeling of Nano and Continuum Rods

We present a novel scheme to derive nonlinearly elastic constitutive laws for special Cosserat rod modeling of nano and continuum rods. We first construct a 6-parameter (corresponding to the six strains in the theory of special Cosserat rods) family of helical rod configurations subjected to uniform strain along their arc-length. The uniformity in strain then enables us to deduce the constitutive laws by just solving the warping of the helical rod’s cross-section (smallest repeating cell for nanorods) but under certain constraints. The constraints are shown to be critical in the absence of which, the 6-parameter family reduces to a well known 2-parameter family of uniform helical equilibria. An explicit formula for the 6-parameter helical map is derived which maps atoms in the repeating cell of a nanorod to their images for the purpose of repeating cell energy minimization. A scheme for the passage from nano to continuum scale is also presented to derive the constitutive laws of a continuum rod via atomistic calculations of nanorods. The bending, twisting, stretching and shearing stiffnesses of diamond nanorods and carbon nanotubes are computed to demonstrate our theory. We show that our scheme is more general and accurate than existing schemes allowing us to deduce shearing stiffness and several coupling stiffnesses of a nanorod for the first time.     

Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method: DAVSS-G/DG

Accurate and robust finite element methods for computing flows with differential constitutive equations require approximation methods that numerically preserve the ellipticity of the saddle point problem formed by the momentum and continuity equations and give numerically stable and accurate solutions to the hyperbolic constitutive equation. We present a new finite element formulation based on the synthesis of three ideas: the discrete adaptive splitting method for preserving the ellipticity of the momentum/continuity pair (the DAVSS formulation), independent interpolation of the components of the velocity gradient tensor (DAVSS-G), and application of the discontinuous Galerkin (DG) method for solving the constitutive equation. We call the method DAVSS-G/DG. The DAVSS-G/DG method is compared with several other methods for flow past a cylinder in a channel with the Oldroyd-B and Giesekus constitutive models. Results using the Streamline Upwind Petrov–Galerkin method (SUPG) show that introducing the adaptive splitting increases considerably the range of Deborah number (De) for convergence of the calculations over the well established EVSS-G formulation. When both formulations converge, the DAVSS-G and DEVSS-G methods give comparable results. Introducing the DG method for solution of the constitutive equation extends further the region of convergence without sacrificing accuracy. Calculations with the Oldroyd-B model are only limited by approximation of the almost singular gradients of the axial normal stress that develop near the rear stagnation point on the cylinder. These gradients are reduced in calculations with the Giesekus model. Calculations using the Giesekus model with the DAVSS-G/DG method can be continued to extremely large De and converge with mesh refinement.     

A micromorphic electromagnetic theory

This work is concerned with the determination of both macroscopic and microscopic deformations, motions, stresses, as well as electromagnetic fields developed in the material body due to external loads of thermal, mechanical, and electromagnetic origins. The balance laws of mass, microinertia, linear momentum, moment of momentum, energy, and entropy for microcontinuum are integrated with the Maxwell’s equations. The constitutive theory is constructed. The finite element formulation of micromorphic electromagnetic physics is also presented. The physical meanings of various terms in the constitutive equations are discussed.     

Elastic buckling of I-beams under linear moment gradient

This paper investigates the elastic lateral–torsional buckling of I-beams under linear moment gradient that very precisely incorporates the effects of moment gradient and various end restraints. The elastic critical buckling moments are obtained independently by using: (1) the Bubnov–Galerkin method and (2) the finite element method. The present formula of the moment gradient correction factor cannot satisfactorily predict the buckling capacities of doubly symmetric and monosymmetric I-beams with various end restraints. We propose alternative equations for evaluating the moment gradient correction factor, considering end restraint conditions.     

Nonlinear wave propagation in damaged hysteretic materials using a frequency domain-based PM space formulation

In this work, a new and simple numerical approach to simulate nonlinear wave propagation in purely hysteretic elastic solids is presented. Conversely to classical time discretization method, which fully integrates the nonlinear equation of motion, this method utilizes a first-order approximation of the nonlinear strain in order to separate linear and nonlinear contributions. The problem for the nonlinear displacements is then posed as a linear one in which the solid is enforced with nonlinear forces derived from the linear strain. In this manner, a frequency analysis can be easily conducted, leading directly to a well-known frequency spectrum for the nonlinear strain. A mesoscale approach known as Preisach–Mayergoyz space (PM space) is used for the chacterization of the nonlinear elastic region of the solid. A meshless element free Galerkin method is implemented for the discretized equations of motion. Nevertheless, a mesh-based method can be still used as well without loss of generality. Results are presented for bidimensional isotropic plates both in plane stress and in plane strain subjected to harmonic monotone excitation.     

A justification of nonlinear properly invariant plate theories

A single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame-invariance properties of three-dimensional finite elasticity. By a successive scaling of the external loading on the three-dimensional body, the nonlinear membrane theory, the nonlinear inextensional theory and the von Kármán equations are derived as the leading-order terms in the asymptotic expansion of finite elasticity. The governing equations of the nonlinear inextensional theory are of particular interest where 1) plane-strain kinematics and plane-stress constitutive equations are derived simultaneously from the asymptotic analysis, 2) the theory can be phrased as a minimization problem over the space of isometric deformations of a surface, and 3) the local equilibrium equations are identical to those arising in the one-director Cosserat shell model. Furthermore, it can be concluded that with a regular, single-scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading-order term of an asymptotic analysis.     

A parallel monolithic algorithm for the numerical simulation of large‐scale fluid structure interaction problems

A novel parallel monolithic algorithm has been developed for the numerical simulation of large‐scale fluid structure interaction problems. The governing incompressible Navier–Stokes equations for the fluid domain are discretized using the arbitrary Lagrangian–Eulerian formulation‐based side‐centered unstructured finite volume method. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant–Kirchhoff material, and the classical Galerkin finite element method is used to discretize the governing equations in a Lagrangian frame. A special attention is given to construct an algorithm with exact total fluid volume conservation while obeying both the global and the local discrete geometric conservation law. The resulting large‐scale algebraic nonlinear equations are multiplied with an upper triangular right preconditioner that results in a scaled discrete Laplacian instead of a zero block in the original system. Then, a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned sub‐domains is utilized for the modified system. The accuracy and performance of the proposed algorithm are verified for the several benchmark problems including a pressure pulse in a flexible circular tube, a flag interacting with an incompressible viscous flow, and so on. John Wiley & Sons, Ltd.     

NONLINEAR DYNAMIC ANALYSIS OF A LAMINATED HYBRID COMPOSITE PLATE SUBJECTED TO TIME-DEPENDENT EXTERNAL PULSES

Nonlinear dynamic responses of a laminated hybrid composite plate subjected to time-dependent pulses are investigated. Dynamic equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von Kármán large deflection theory of thin plates. Approximate solutions for a clamped plate are assumed for the space domain. The single term approximation functions are selected by considering the nonlinear static deformation of plate obtained using the finite element method. The Galerkin Method is used to obtain the nonlinear differential equations in the time domain and a MATLAB software code is written to solve nonlinear coupled equations by using the Newmark Method. The results of approximate-numerical analysis are obtained and compared with the finite element results. Transient loading conditions considered include blast, sine, rectangular, and triangular pulses. A parametric study is conducted considering the effects of peak pressure, aspect ratio, fiber orientation and thicknesses.     

Parallel folding in multilayered structures

A recent two-layer nonlinear model for parallel folding is extended to a full multilayer formulation for n identical layers. The general behaviour of the multilayer is characterized using a sinusoidal Galerkin approximation substituted into the linearized potential energy function and equations for the wavelength and critical load are found. Comparisons with experiments conducted on layers of paper constrained between two sheets of foam rubber show that the trend of the load-amplitude plots is in good agreement with that predicted. Excellent correlation is also achieved between the wavelengths calculated by the linearized buckling analysis and those seen experimentally.