Order Continuum Theory Research Articles

Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro-plates

This article introduces new methods for static and free vibration analyses of functionally graded annular and circular micro-plates, which can take into account spatial variation of the length scale parameter. The underlying higher order continuum theory behind the proposed approaches is the modified couple stress theory. A unified way of expressing the displacement field is adopted so as to produce numerical results for three different plate theories, which are Kirchhoff plate theory (KPT), Mindlin plate theory (MPT), and third-order shear deformation theory (TSDT). Governing partial differential equations and corresponding boundary conditions are obtained following the variational approach and the Hamilton's principle. Derived systems of differential equations are solved numerically by utilizing the differential quadrature method (DQM). Comparisons to the results available in the literature demonstrate the high level of accuracy of the numerical results generated through the developed methods. Extensive analyses are presented in order to illustrate the influences of various geometric and material parameters upon static deformation profiles, stresses, and natural vibration frequencies. In particular, the length scale parameter ratio -which defines the length scale parameter variation profile-is shown to possess a profound impact on both static and dynamic behaviors of functionally graded annular and circular micro-plates.

Single crystal plasticity with bend–twist modes

In this work a formulation is proposed and computationally implemented for rate dependent single crystal plasticity, which incorporates plastic bend–twist modes that arise from dislocation density based poly-slip mechanisms. The formulation makes use of higher order continuum theory and may be viewed as a generalized micromechanics model. The formulation is then linked to the burgers and Nye tensors, showing how their material rates are derivable from a newly proposed third-rank tensor Λp, which incorporates a crystallographic description of bend–twist plasticity through selectable slip-system level constitutive laws. A simple three-dimensional explicit finite element implementation is outlined and employed in three simulations: (a) bi-crystal bending; (b) tension on a notched single crystal; and (c) the large compression of a microstructure to induce the plastic buckling of secondary phases. All simulation are transient, for computational expediency. The results shed light on the physics resulting from dynamic inhomogeneous plastic deformation.

Scattering of an anti-plane shear wave by an embedded cylindrical micro-/nano-fiber within couple stress theory with micro inertia

The inadequacy of traditional theory of elasticity in describing such a phenomenon as dispersion associated to a propagating wave with wavelength comparable to the intrinsic length of the medium of interest is well-known. Moreover, under certain circumstances it is incapable of capturing all the propagating waves. A remedy to such dilemmas is the employment of the more accurate higher order continuum theories which give rise to the appearance of at least one new characteristic length in the formulation. The experimental evidences as well as lattice dynamic analysis suggest that, although higher order continuum theories result in some improvements, but cannot fully overcome the above-mentioned dilemmas, unless the micro inertia term is included in the formulations. The current work addresses the elastodynamic fields of an anti-plane shear wave scattered by a micro-/nano-fiber embedded in an infinite matrix using couple stress theory with micro inertia term. Moreover, the formulations pertinent to the cases where the incident wave strikes an embedded micro-/nano-size circular cavity or a rigid immovable micro-/nano-fiber are also obtained. Within this theory, the appearance of a new length scale, so-called “dynamic characteristic length” stems from consideration of the micro inertia, which gives rise to physically realistic dispersion relations with characteristic resembling those observed in experiments. The effects of two different types of boundary conditions for the cases of elastic and rigid immovable fiber encountered within the present theory are discussed. By using this theory, the corresponding analytical expressions of the elastodynamic fields, total and differential scattering cross-sections, and the dynamic stress concentration factor are presented and their dependence on the characteristic lengths and frequency are examined. It has been shown that the effect of micro inertia term is more noticeable in higher frequencies.

Three-dimensional free vibration of arbitrarily shaped laminated micro-plates with sliding interfaces within couple stress theory

Free vibration of laminated micro-plates with arbitrary geometry and boundary conditions consisting of several micro-layers with free sliding/frictional sliding/perfect interfaces is of interest. The inter-layer bond in the direction normal to the interfaces is perfect, and thus the corresponding displacement component is continuous across the interfaces. The mentioned in-plane interface conditions may be realized by varying the stiffness of the sliding inter-layer spring of vanishing thickness. For free sliding and perfect interface conditions the stiffness→0 and ∞, respectively. Couple stress theory as a useful higher order continuum theory is utilized to formulate the problem. Subsequently, the corresponding Hamiltonian is presented. The three components of the displacement field in each layer are given by three distinct expressions; each expression consists of the product of a base function and three dimensional polynomials with unknown coefficients. The base functions are chosen according to the geometries of the system and the corresponding layer and to satisfy the homogeneous essential boundary conditions. For the enforcement of the displacement continuity in the direction normal to the interfaces, an appropriate series with unknown coefficients pertinent to each layer is added. The influences of the level of inter-layer imperfection and couple stress on the dynamical characteristics including the angular frequencies of free vibration and the corresponding mode shapes are addressed.

Polarization gratings using ferronematics—An elastic continuum theory

In our paper we apply the elastic continuum theory in order to describe the working of a polarization grating device based on the ferronematic liquid crystal and driven by magnetical field. Solving the Euler–Lagrange equations we obtain analytical expressions for the critical thickness and threshold magnetic field in strong and weak anchoring conditions. We discuss the influence of the ferroparticle presence and properties on the behavior of a device based on the ferronematic liquid crystal polarization gratings. We study also the dynamic behavior of this kind of device and we find the expression for switching on and off times. Our study may be useful in designing magneto-optical devices which include nanoparticles or microprocessed surfaces.

Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory

In this paper, Modified couple stress theory is developed for third-order shear deformation functionally graded (FG) micro beam. Classical Continuum theories are not able to capture size effects while higher order continuum theories consider material length scale parameters to improve the results in micro scales by interpreting size dependencies. By applying Hamilton’s principle, governing equations, boundary and initial conditions are derived for a FG micro beam. It is assumed that properties of the FG micro beam follow a power law form through thickness. In addition, Poisson’s ratio is assumed constant in the current model. Bending and free vibration of the simply supported FG micro beam is analyzed to illustrate size effects. It is observed that if the thickness of the beam is in the order of the material length scale parameter, size effects are more considerable. It is shown that size dependencies of FG micro beams are quite different from isotropic homogeneous beams as they have maximum and minimum size effects for certain values of power index of material distribution and Poisson’s ratio. Moreover, it is shown that the results obtained by the present model deviates significantly from those in which Poisson's effect is neglected. Also, the effect of Poisson’s ratio on the deflection and natural frequency based on modified couple stress theory is proven to be different from Classical one. The numerical results for simply supported thin beams show that the first natural frequency estimated by the current model is higher than the classical one. In addition, it is observed that size effect is more significant for higher vibration modes.

Investigation of Cosserat material parameters obtained by homogenization of a Cauchy continuum

AbstractIf a higher order continuum theory is used in a numerical simulation, the material parameters have to be derived. The experimental determination is laborious and sometimes impossible. Alternatively homogenization methods can be used for the numerical identification of overall material parameters. A short introduction to the micropolar continuum and the homogenization approach is followed by the discussion of the identified properties. Therefore some parameter studies are presented. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Elliptic inhomogeneities and inclusions in anti-plane couple stress elasticity with application to nano-composites

It is well-known that classical continuum theory has certain deficiencies in predicting material’s behavior at the micro- and nanoscales, where the size effect is not negligible. Higher order continuum theories introduce new material constants into the formulation, making the interpretation of the size effect possible. One famous version of these theories is the couple stress theory, invoked to study the anti-plane problems of the elliptic inhomogeneities and inclusions in the present work. The formulation in elliptic coordinates leads to an exact series solution involving Mathieu functions. Subsequently, the elastic fields of a single inhomogeneity in conjunction with the Mori–Tanaka theory is employed to estimate the overall anti-plane shear moduli of composites with uni-directional elliptic cylindrical fibers. The dependence of the anti-plane elastic moduli on several important physical parameters such as size, aspect ratio and rigidity of the fiber, the characteristic length of the constituents, and the orientation of the reinforcements is analyzed. Based on the available data in the literature, certain nano-composite models have been proposed and their overall behavior estimated using the present theory.

Finite element (FE) simulation of textile‐reinforced composites using a COSSERAT Element

AbstractThe efficient simulation of textile reinforced composites requires reliable material parameters describing the macroscopic material behavior. Due to the micro–structure, decoupled tensile and bending stiffnesses are observed in experimental investigations. This motivates the use of higher order continuum theories. For numerical simulation of this material behavior, the formulation and application of a volume element based on the Cosserat continuum theory is presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical homogenization of foam‐like structures based on the FE2‐approach

AbstractThe computation of foam–like structures is still a topic of research. There are two basic approaches: the microscopic model where the foam–like structure is entirely resolved by a discretization (e.g. with Timoshenko beams) on a micro level, and the macroscopic approach which is based on a higher order continuum theory. A combination of both of them is the FE2‐approach where the mechanical parameters of the macroscopic scale are obtained by solving a Dirichlet boundary value problem for a representative microstructure at each integration point.In this contribution, we present a two–dimensional geometrically nonlinear FE2‐framework of first order (classical continuum theories on both scales) where the microstructures are discretized by continuum finite elements based on the p‐version.The p‐version elements have turned out to be highly efficient for many problems in structural mechanics. Further, a continuum–based approach affords two additional advantages: the formulation of geometrical and material nonlinearities is easier, and there is no problem when dealing with thicker beam–like structures.In our numerical example we will investigate a simple macroscopic shear test. Both the macroscopic load displacement behavior and the evolving anisotropy of the microstructures will be discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An atomistic study of nanosprings

Molecular statics method incorporating minimum energy concept was employed to study the one-dimensional copper nanospring with faced-center-cubic crystal structure. Various geometric sizes (wire diameter, radius, pitch), numbers of turns and crystal orientations of nanosprings were systematically modeled to investigate the size dependence of elastic properties. It was observed that as the wire diameter increases and the radius and number of turns decrease, the nanospring stiffness would increase irrespective of the crystal orientations. Moreover, the elastic constants of nanosprings would become larger while the pitches become smaller for almost all the crystal orientations. Also the simulation results were compared to the predictions based on continuum theory in order to clarify whether the classical theory could apply to nanosprings.

Influence of Nonassociativity on Localization and Failure in Geomechanics Based on Gradient Elastoplasticity

We present results of a parametric study on shearband formation in geomaterials, examining the influence of material dilatancy on shearband properties. The study is based on a new higher order continuum theory, called gradient elastoplasticity, developed to model localization of deformation. We present results for two problems with different initial stress fields: The biaxial test and the hollow cylinder test. We show that nonassociativity promotes bifurcation at lower deformation, leading to localized deformation and more abrupt final failure. Both the inclination and thickness of the calculated shearbands decrease with increasing nonassociativity. Gradient elastoplasticity is capable of reproducing the experimentally observed localized failure modes, including details such as reorientation of shearband close to a free boundary.

Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams

An anisotropic compressible plasticity model is incorporated into the framework of the micromorphic continuum theory in order to describe some size effects observed in ductile nickel foams. This continuum model reproduces the fact that the presence of a machined hole in a foam plate does not affect its mechanical response when the hole size becomes comparable to the cell size of the material. Finite element simulations are compared to strain field measurements in nickel foam plates with a machined hole for different hole sizes, in order to identify the characteristic length of the model. Based on a simple ductile damage law, the model is then shown to be able to account for the strong anisotropy of the initiation of crack propagation in central crack panels made of nickel foams under mode I loading conditions.

A continuum theory of soils: Viewed as peculiar immiscible mixtures

In this work, we construct an atypical continuum theory in order to describe the behaviour of soils: they have a particulate structure and fluids can seep through pores, mechanically interacting with the solid skeleton. We use a model of immiscible mixtures consisting of a porous elastic solid with very large lacunae (a continuum with ellipsoidal microstructure) partially, or totally, filled by a classical fluid inclusion.

Some Applications of Higher Order Continuum Theories in Rock Mechanics

Some Applications of Higher Order Continuum Theories in Rock Mechanics

Monte Carlo lattice simulations of the elastic behaviour of nematic liquid crystals

A pairwise additive potential, which approximately reproduces the free energy density for the elastic deformations of a nematic liquid crystal, originally proposed by Gruhn and Hess, has been investigated by simulating the three Freedericksz transitions as well as that of the Schadt-Helfrich cell. The pair potential depends on the three elastic constants K1, K2 and K3 for the splay, twist and bend deformations, respectively. The results of the simulations are compared with the analytical solutions obtained from continuum theory in order to test the accuracy of the model potential at a quantitative level. This comparison is also made for different temperatures to explore the influence of director fluctuations on the elastic behaviour.

The effective moduli of short-fiber composites

A first order continuum theory with microstructure is developed for aligned short-fiber composites. The fibrous material is modeled by a triply periodic array of rectangular parallelpiped elastic fibres which are embedded in an elastic matrix. A proper reduction of the theory, by which the microstructure variables are eliminated, yields the effective moduli of the short-fiber composite. The overall elastic constants of the three specific situations of long-fiber composites, particulate composites and periodically bilaminated media, are obtained as special cases. The reliability of the predicted effective moduli is verified by numerous comparisons with available experimental results in various cases.

A continuum theory for fiber-reinforced elastic-viscoplastic composites

A higher order continuum theory with microstructure is derived for the modeling of the 3-dimensional motion of fiber-reinforced composites in which both the matrix and fibers constituents are assumed to be elastic-viscoplastic work-hardening materials. The fibers are unidirectional with rectangular cross section and are imbedded in the matrix in the form of a doubly periodic array. The derivation of the theory is systematic and can be applied to various types of non-elastic composites to the desired degree of expansion. An appropriate reduction of the theory gives the average behavior of the viscoplastic composite in the form of effective rate-dependent stress-strain curves. In the special case of perfectly elastic constituents the reduction gives the approximate effective moduli of the composite.