Generalized Continuum Theories Research Articles

Propagation of Transverse Waves in Elastic-Micropolar Porous Semispaces

Porous materials are widely used in the passive noise control field as sound absorbers. Conventional models of porous materials are assumed to have a rigid frame and show finite bulk elasticity. However, in the case of acoustical waves – characterized by high frequencies and small wavelengths – the effect of microstructure becomes significant. This effect of microstructure has resulted in the development of new types of waves, not found in the classical theory of elasticity. Generalized continuum theories include the construction of the linear theory of micropolar elasticity that consists of deformation and microrotation with six degrees of freedom, and hence can be used to study the acoustical characteristics of composites with a granular structure. In this study, we investigated transverse wave propagation and its reflection and transmission from a plane interface between two different elastic-micropolar porous interfaces in perfect contact. The micropolar porous composite was constructed using hollow glass microbubbles embedded in an epoxy matrix with six material constants that can be used as the acoustical absorbers. It was found that there are different wave types in a micropolar porous material for the incident SV (vertical transverse) or SH (horizontal transverse) wave. It was also found that these two coupled sets of transverse waves, when traveling with different velocities, are dominated by the critical value of microinertia, showing the influence of the micropolar porous characteristics.

Longitudinal Plane Wave Propagation in Elastic-Micropolar Porous Media

Porous materials are widely used in the passive noise control field as sound absorbers. Conventional models of porous materials are assumed to have a rigid frame and to satisfy finite bulk elasticity. However, it may be the case that when high-frequency sound is applied to porous materials for nanotechnology applications, the classical theory of elasticity cannot be satisfied. Generalized continuum theories, such as coupled stress theory and micropolar theory, have additional degrees of freedom compared with classical elasticity. It is hence assumed that such theories are applicable to composites with granular structure. In this study, two of the six material constants are obtained from experimental results on micropolar porous solids by Lakes. At the same time, a theoretical analysis concerning reflected and transmitted fields of an incident longitudinal plane wave propagating in elastic-micropolar porous media has been investigated. The results show that two fields are affected by micropolar porous characteristics. The boundary value problem of micro elasticity is also investigated in this paper.

Creep and recrystallization of large polycrystalline masses. I. General continuum theory

This is the first of a series of works on the continuum mechanics and thermodynamics of creep and recrystallization of large polycrystalline masses. The general continuum theory presented here is suited to mono- and multi-mineral rocks. It encompasses several symmetry groups (e.g. orthotropic and transversely isotropic) and diverse crystal classes of triclinic, monoclinic and rhombic systems, among others. The cornerstone of the current approach is the theory of mixtures with continuous diversity, which allows one to regard the polycrystal as a ‘mixture of lattice orientations’. Following this picture, balance equations of mass, linear momentum, lattice spin, energy, dislocations, and entropy are set forth to describe the response of the polycrystal (i.e. the ‘mixture’), as well as of a group of crystallites sharing the same lattice orientation ( viz . a ‘species’). The connection between the balance equations for a species and those for the mixture is established by homogenization rules, formulated for every field of the theory.

Continuum Description of the Cytoskeleton: Ring Formation in the Cell Cortex

Motivated by the formation of ringlike filament structures in the cortex of plant and animal cells, we study the dynamics of a two-dimensional layer of cytoskeletal filaments and motor proteins near a surface by a general continuum theory. As a result of active processes, dynamic patterns of filament orientation and density emerge via instabilities. We show that self-organization phenomena can lead to the formation of stationary and oscillating rings. We present state diagrams that reveal a rich scenario of asymptotic behaviors and discuss the role of boundary conditions.

Size effects in the mechanical behavior of cellular materials

Effective mechanical properties of cellular materials depend strongly on the specimen size to the cell size ratio. Experimental studies performed on aluminium foams show that under uniaxial compression, the stiffness of these materials falls below the corresponding bulk value, when the ratio of the specimen size to the cell size is small. Conversely, in the case of simple shear and indentation, the overall stiffness rises above the bulk value. Classical continuum theory, lacking a length scale, cannot explain this size dependent mechanical behaviour. One way to account for these size effects is to explicitly model the discrete cellular morphology. We performed shear, compression and bending tests using discrete models, for hexagonal (regular and irregular) microstructures. Even though discrete models give a very good agreement with the experiments, they are computationally expensive for complex microstructures, especially in three dimensions. To overcome this, one can use a generalized continuum theory, such as Cosserat continuum theory, which incorporates a material length scale. We fit the Cosserat elastic constants of the models by comparing the discrete calculations with the analytical Cosserat continuum solutions in terms of macroscopic properties. We critically address the limitations of the Cosserat continuum theory.

Non-local damage model based on displacement averaging

Continuum damage models describe the changes of material stiffness and strength, caused by the evolution of defects, in the framework of continuum mechanics. In many materials, a fast evolution of defects leads to stress–strain laws with softening, which creates serious mathematical and numerical problems. To regularize the model behaviour, various generalized continuum theories have been proposed. Integral-type non-local damage models are often based on weighted spatial averaging of a strain-like quantity. This paper explores an alternative formulation with averaging of the displacement field. Damage is assumed to be driven by the symmetric gradient of the non-local displacements. It is demonstrated that an exact equivalence between strain and displacement averaging can be achieved only in an unbounded medium. Around physical boundaries of the analysed body, both formulations differ and the non-local displacement model generates spurious damage in the boundary layers. The paper shows that this undesirable effect can be suppressed by an appropriate adjustment of the non-local weight function. Alternatively, an implicit gradient formulation could be used. Issues of algorithmic implementation, computational efficiency and smoothness of the resolved stress fields are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

The Mode III Crack Problem in Microstructured Solids Governed by Dipolar Gradient Elasticity: Static and Dynamic Analysis

This study aims at determining the elastic stress and displacement fields around a crack in a microstructured body under a remotely applied loading of the antiplane shear (mode III) type. The material microstructure is modeled through the Mindlin-Green-Rivlin dipolar gradient theory (or strain-gradient theory of grade two). A simple but yet rigorous version of this generalized continuum theory is taken here by considering an isotropic linear expression of the elastic strain-energy density in antiplane shearing that involves only two material constants (the shear modulus and the so-called gradient coefficient). In particular, the strain-energy density function, besides its dependence upon the standard strain terms, depends also on strain gradients. This expression derives from form II of Mindlin’s theory, a form that is appropriate for a gradient formulation with no couple-stress effects (in this case the strain-energy density function does not contain any rotation gradients). Here, both the formulation of the problem and the solution method are exact and lead to results for the near-tip field showing significant departure from the predictions of the classical fracture mechanics. In view of these results, it seems that the conventional fracture mechanics is inadequate to analyze crack problems in microstructured materials. Indeed, the present results suggest that the stress distribution ahead of the tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the classical results. The latter can be explained physically since materials with microstructure behave in a more rigid way (having increased stiffness) as compared to materials without microstructure (i.e., materials governed by classical continuum mechanics). The new formulation of the crack problem required also new extended definitions for the J-integral and the energy release rate. It is shown that these quantities can be determined through the use of distribution (generalized function) theory. The boundary value problem was attacked by both the asymptotic Williams technique and the exact Wiener-Hopf technique. Both static and time-harmonic dynamic analyses are provided.

Many-Field Theory for Crystals Containing Particles with Rotational Degrees of Freedom

We give a brief review of some generalized continuum theories applied to the crystals with complicated microscopic structure. Three different ways of generalization of the classical elasticity theory are discussed. One is the high-gradient theory, another is the micropolar type theory and the third one is the many-field theory. The importance of the first two types of theories has already been established, while the theory of the third type still has to be developed. With the use of 1D and 2D examples we show for each of these theories where they can be and should be applied, separately or in a combination.

Generalized continuum theory for ferroelectric thin films

A generalized Ginzburg-Landau-Devonshire free energy has been used to describe the influence of imperfect surface layers or interface layers in ferroelectric thin films. The natural boundary condition has been employed in solving the polarization profile, which can reflect a more realistic situation compared to the previous treatments of the same problem using the so-called extrapolation length. We show that the unscreened portion of the depolarization field makes the polarization distribution more uniform while reducing the amplitude of the effective polarization. The less perfect layer can lower the effective phase-transition temperature. We also used the model to study the effect of asymmetric boundary conditions, which will cause a shift of the hysteresis loop from the center point, similar to the effect of a bias field.

Spin fluids and hyperfluids

The general theory of continua with microstructure reviewed in [2] (and, in particular, the theory of gyrocontinua studied in [1]) is adapted to apply to perfect spin fluids and hyperuids [10]. Obvious changes are needed due to the prevailing interest for solids in [1] and to some extent in [2]; here, in addition, the conservative character of the continuum is exploited. Finally, the roles of metric, coframe, connection, torsion and curvature in fluids are explored. .

Generalized Continuum Theories: Application to Stress Analysis in Bone*

Bone is a heterogeneous material with microstructural features. Continuum models of bone on the basis of classical elasticity ignore microstructure-related scale effects on the macroscopic mechanical properties. Consequently, these models do not provide a complete description of the mechanical behavior when the microstructural size of bone approaches the macroscopic length scale. Such effects are most pronounced near bone–implant interfaces and in areas of high strain gradients. This issue is investigated here by studying generalized continuum mechanics theories which account for the influence of microstructure-related scale effects on the macroscopic properties of bone. Furthermore, a two-dimensional finite element on the basis of the micropolar continuum theory is summarized. As a simple illustrative example, a bone–prosthesis configuration is analyzed. Results show that the stress concentrations in bone near the bone–prosthesis interface are substantially smaller with micropolar theory compared to classical theory of elasticity.

Smectic Liquid Crystals with Compressible Elastic Layers

Smectic liquid crystals partake some of the essential properties of both solids and fluids, in that they possess microstructure both of the material and local type (the nematic microstructure and the lamellae). Balance equations for smectics are derived from the general theory of continua with microstructure and tested against a very simple mode of movement: Couette type displacement of a cell confined between two parallel planes with strong planar anchoring conditions at the boundary.

Anti-plane shear Lamb's problem treated by gradient elasticity with surface energy

The consideration of higher-order gradient effects in a classical elastodynamic problem is explored in this paper. The problem is the anti-plane shear analogue of the well-known Lamb's problem. It involves the time-harmonic loading of a half-space by a single concentrated anti-plane shear line force applied on the half-space surface. The classical solution of this problem based on standard linear elasticity was first given by J.D. Achenbach and predicts a logarithmically unbounded displacement at the point of application of the load. The latter formulation involves a Helmholtz equation for the out-of-plane displacement subjected to a traction boundary condition. Here, the generalized continuum theory of gradient elasticity with surface energy leads to a fourth-order PDE under traction and double-traction boundary conditions. This theory assumes a form of the strain-energy density containing, in addition to the standard linear-elasticity terms, strain-gradient and surface-energy terms. The present solution, in some contrast with the classical one, predicts bounded displacements everywhere. This may have important implications for more general contact problems and the Boundary-Integral-Equation Method.

A phenomenological model of twinning based on dual reference structures

A large class of solids, including certain metals and polymers, when subject to significant deformations undergo microstructural changes that engender a response that is distinctly different from that before which the microstructural changes occurred. Such microstructural changes usually lead to inelastic response. This paper is devoted to the study of one such microstructural change, attributed to deformation induced twinning, from a macroscopic point of view, within the framework of a general continuum theory originally developed for polymers by Rajagopal and Wineman [1] and modified appropriately to model crystalline materials by Rajagopal and Srinivasa [2,3]. In this paper, we elucidate the intricate interplay between the storage and dissipation of energy due to deformation and their influence on the propagation and arrest of twinning. The onset of twinning is determined purely by energy considerations. We show that the entire constitutive structure of the material can be reduced to the specification of three scalar functions to model “quasi equilibriated twinning”: the Helmholtz free energy potential ψ, the rate of dissipation function ϵ and the activation function g. For the dynamical case (when inertial effects cannot be ignored), an additional constitutive function for the kinetic energy associated with the process of twinning must be specified.

Generalized diffusion theory of hydrodynamical particle migration in suspensions. Part 1: The case of equal densities

The general continuum theory has been developed for two-phase flows of fluid with deformable particles, where the micro-deformation of particles and the relative motion between phases have been taken into account [1-3]. This paper is concerned with using the simplest model from developed general theory for modeling of particle migration in suspensions- one of the most important and complicated aspects of particle-liquid two-phase flows, that has been observed and studied by many authors. For this purpose it is considered the motion of Newtonian fluid-rotating rigid spherical particles two-phase continuum with specialized nonlinear constitutive equations, when the particle and fluid have equal densities. The obtained equation system has been used for studying quantitatively particle migration problem in the circular Couette flow.

Finite element analysis of Saint‐Venant end effects in micropolar elastic solids

Distributions of stress and strain in composite and cellular materials can differ significantly from the predictions of classical elasticity. For example, concentration of stress and strain around holes and cracks is consistently less than classical predictions. Generalized continuum theories such as micropolar (Cosserat) elasticity offer improved predictive power. In this article Saint‐Venant end effects for self equilibrated external forces in micropolar solids are investigated in two dimensions. A two dimensional finite element analysis is used which takes into account the extra degrees of freedom, to treat the problem of localized end loads acting upon a strip. The rate of decay of strain energy becomes slower in a two dimensional strip as the micropolar characteristic length l is increased (for l sufficiently less than the strip width). For the strip geometry a Cosserat solid exhibits slower stress decay than a classical solid.

Laboratory experiments on thin-walled tubes at large finite strain: Symmetry, coaxiality, rigid body rotation, and the role of invariants, for the applied stress σ = RTRT, the cauchy stress σ∗ = [IIIV]−1FTRT, and the left cauchy-green stretch tensor V = FRT

From laboratory measurements extending back to the mid-1960s, a frame indifferent general continuum theory for large finite plastic strain in ordered solids has been developed. Its foundations are based on a stress σ that replaces the Cauchy stress σ∗ in the applicable constitutive statements. With W the work per unit volume in the underformed reference configuration, a work function W = traceσV, established from experiment, is consistent with an internal constraint on the left Cauchy-Green stretch tensor, trace V = 3, in whose presence simple shear becomes inadmissible. Compatible with this internal constraint, the decrease in volume during deformation is measured. Also shown in the laboratory is the fact that even when determined just prior to catastrophic collapse, the measured rigid body rotation is too small to play other than a negligible role. In this article, laboratory experimental results are presented which reveal that the Cauchy stress σ∗, the stress σ of the present theory, and the left Cauchy-Green stretch tensor V, are all coaxial and symmetric. Moreover, for the approximation R = I, the second invariant T of the deviatoric stress S and the second invariant Γ of the finite strain E = V − I provide a universal parabolic rule that applies for stress paths of arbitrary composition and direction. This general nonlinear theory of finite strain plasticity during loading is shown here to square in detail with a recently published general nonlinear theory of finite strain elasticity derived from the analytical foundations of rational mechanics by Beatty and Hayes [1992].

Theory of two-phase flow of fluid with rigid ellipsoidal particles

In the paper [1] the general continuum theory has been developed for two phase of fluid with deformable particles of arbitrary form, where the micro deformation of the particles and the relative motion between phases are taken into account. The extended theory of irrational flow of fluid caused by a moving deformable body has been used to obtain the general expressions for the generalized induced mass tensors.

A Continuum Theory for Smectic C Liquid Crystals

Abstract Our aim in this paper is to present a rather general non-linear continuum theory that may prove useful for the interpretation of observations and experiments upon smectic C liquid crystals. In its present formulation our theory is constrained to exclude variations in the layer spacing thickness and also changes in tilt with respect to the layer normal. De Gennes1 discusses a theory of this type for smectic A liquid crystals, but for these liquids applications of such a constrained theory are somewhat more restricted than for smectic C materials, and consequently our interest focuses on the latter. Throughout, much of our motivation and reasoning stem from experience with nematic theory, details of which are available in, for example, the reviews by Ericksen2 and Leslie3 as well as the books by de Gennes1 and Chandrasekhar.4

Experimental Micro Mechanics Methods for Conventional and Negative Poisson’s Ratio Cellular Solids as Cosserat Continua

Continuum representations of micromechanical phenomena in structured materials are described, with emphasis on cellular solids. These phenomena are interpreted in light of Cosserat elasticity, a generalized continuum theory which admits degrees of freedom not present in classical elasticity. These are the rotation of points in the material, and a couple per unit area or couple stress. Experimental work in this area is reviewed, and other interpretation schemes are discussed. The applicability of Cosserat elasticity to cellular solids and fibrous composite materials is considered as is the application of related generalized continuum theories. New experimental results are presented for foam materials with negative Poisson’s ratios.