Mechanics of Generalized Continua

Discursive historical perspective on the developments and ramifications of generalized continuum mechanics from its inception by the Cosserat brothers (Théorie des corps déformables. Hermann,

1909

) with their seminal work of 1909 to the most current developments and applications is presented. The point of view adopted is that generalization occurs through the successive abandonment of the basic working hypotheses of standard continuum mechanics of Cauchy, that is, the introduction of a rigidly rotating microstructure and

couple stresses

(Cosserat continua or

micropolar

bodies, nonsymmetric stresses), the introduction of a truly deformable microstructure (

micromorphic

bodies), “weak”

nonlocalization

with

gradient theories

and the notion of

hyperstresses

, and the introduction of characteristic lengths, “strong nonlocalization” with space functional constitutive equations and the loss of the Cauchy notion of stress, and finally giving up the Euclidean and even Riemannian material background. This evolution is paved by landmark papers and timely scientific gatherings (e.g., Freudenstadt, 1967; Udine, 1970, Warsaw, 1977).

The notions of semi-holonomic and quasi-holonomic Cosserat media are introduced and their differences outlined. Contrary to the classical holonomic and non-holonomic counterparts, the definition of semi- and quasi-holonomic media is not kinematic but constitutive. Possible applications include granular media embedded in a rigid matrix and colloidal suspensions in an ideal incompressible fluid.

The classical isotropic linear elastic material behavior is presented by two material parameters, e.g., the Young’s modulus and the Poisson’s ratio, while the Cosserat continuum is given by six material parameters. The latter continuum model can be the starting point for the deduction of the governing equations of the Cosserat plate theory via a through-the-thickness integration. In contrast, the basic equations of the Cosserat plate theory can be established applying the direct approach. It can be shown that both systems of equations are similar in the main terms. The assumed identity of both systems results in consistent stiffness parameters identification for the two-dimensional theory based on the direct approach and, in addition, in some constraints. Using the experimental results of Lakes, one can show in which cases the additional material properties coming from the tree-dimensional Cosserat material model have a significant influence on the stiffness parameters.

Modeling of particulate and layered materials (e.g., concrete and rocks, rock masses) by Cosserat continua involves characteristic internal lengths which can be commensurate with the microstructural size (the particle size or layer thickness). When fracture propagation in such materials is considered, the criterion of their growth is traditionally based on the parameters of the crack-tip stress singularities referring to the distances to the crack tip smaller than the characteristic lengths and hence smaller than the microstructural size. This contradicts the very notion of continuum modeling which refers to the scales higher than the microstructural size. We propose a resolution of this contradiction by considering an intermediate asymptotics corresponding to the distances from the crack tip larger than the microstructural sizes (the internal Cosserat lengths) but yet smaller than the crack length. The approach is demonstrated using examples of shear crack in particulate and bending crack in layer materials.

The micropolar model is certainly the best continuum-mechanical model to describe the collective behavior of molecules or rigid particles interacting via short-range forces and couples. We look at the necessary modifications of the original model for it to describe two unusual materials: nematic liquid crystals on the one hand and liquid-like granular materials on the other hand.

We describe a principle of bounded stiffness and show that bounded stiffness in torsion and bending implies a reduction of the curvature energy in linear isotropic Cosserat models leading to the so called conformal curvature case

$\frac{1}{2}\mu L_{c}^{2}\Vert{\operatorname{dev}\operatorname{sym}\nabla \operatorname{axl}\overline{A}}\Vert^{2}$

where

$\overline{A}\in\mathfrak{so}(3)$

is the Cosserat microrotation. Imposing bounded stiffness greatly facilitates the Cosserat parameter identification and allows a well-posed, stable determination of the one remaining length scale parameter

L

c

and the Cosserat couple modulus

μ

c

.

A generalized continuum approach is considered for structures in which a set of oscillators is attached to each material point of a linear elastic body. Dynamical consequences are given concerning traveling waves.

We revisit the problem of identification of effects of independent rotations and reconstruction of the parameters of Cosserat continuum. The effect of rotational degrees of freedom leads to the appearance of additional shear and rotational waves; however, only at high frequencies corresponding to the wave lengths comparable to the microstructural sizes. Such waves are difficult to detect. Rotations also affect the conventional s-waves (shear-rotational waves) making them dispersive for all frequencies. We considered propagation of planar waves in 2D granulate materials consisting of parallel cylindrical particles connected by tensile, shear and rotational springs and arranged in either square or hexagonal patterns. We deduced the s-wave dispersion relationships for both cases and determined the main terms of asymptotics of low frequency; they were found to be quadratic in the frequency. These terms, alongside with the zero order terms, can be determined by fitting of the theoretically determined frequency dependence of the phase shift between the sent and received wave to its experimentally determined counterpart. This opens a way for direct experimental identification of Cosserat effects in granulate materials.

Definitions of the Lagrangian stretch and wryness tensors in the non-linear Cosserat continuum are discussed applying three different methods. The resulting unique strain measures have several distinguishing features and are called the natural ones. They are expressed through the translation vector and either the rotation tensor or various finite rotation vector fields. The relation of the natural strain measures to those proposed in the representative literature is reviewed.

Motivated by the need to construct models of slender elastic media that are versatile enough to accommodated non-linear phenomena under dynamical evolution, an overview is presented of recent practical applications of simple Cosserat theory. This theory offers a methodology for modeling non-linear continua that is physically accurate and amenable to controlled numerical approximation. By contrast to linear models, where non-linearities are sacrificed to produce a tractable theory, large deformations are within the range of validity of simple Cosserat models. The geometry of slender and shell-like bodies is exploited to produce a theory that contains as few degrees of freedom as is physically reasonable. In certain regimes it is possible to include fluid-structure interactions in Cosserat rod theory in order to model, for example, drill-string dynamics, undersea riser dynamics and cable-stayed bridges in light wind-rain conditions. The formalism also lends itself to computationally efficient, effective models of microscopic carbon nanotubes and macroscopic gravitational antennae.

In order to investigate the properties of microstructured materials, the underlying heterogeneous material is commonly replaced by a homogeneous material involving additional degrees of freedom. Making use of an appropriate homogenization methodology, the present contribution compares deformation states predicted by the homogenization technique to the deformation state within a reference solution. The results indicate on what terms the predicted deformation modes can be clearly interpreted from the physical point of view.

Micromorphic theory envisions a material body as a continuous collection of deformable particles; each possesses finite size and inner structure. It may be considered as the most successful top-down formulation of a two-level continuum model, in which the deformation is expressed as a sum of macroscopic continuous deformation and microscopic deformation of the inner structure. To enlarge the domain of applicability of the micromorphic theory, starting from many-body dynamics, we took a bottom-up approach to formulate a generalized continuum field theory in which a crystalline material is viewed as a continuous collection of lattice points while embedded within each lattice point is a group of discrete atoms. In this work, atomistic definitions and the corresponding field representations of fundamental physical quantities are introduced. The balance laws and the constitutive relations are obtained through the atomistic formulation, which naturally leads to a generalized continuum field theory. It is identical to molecular dynamics at atomic scale and can be reduced to classical continuum field theory at macroscopic scale.

A nonlinear theory of microscopic and macroscopic strains is developed for the case of large inhomogeneous relative displacements of two sublattices making up a complex crystal lattice. The standard linear theory of acoustic and optical oscillations of a complex lattice is generalized, taking into account new additive principle of internal translational symmetry—relative shear of two sublattices leaving deformation energy invariant. As a result, the force interaction between the sublattices is characterized by a nonlinear periodic force of its mutual displacements. The theory describes large microdisplacements due to bifurcation transitions of atoms into neighboring cells. As a result, the theory predicts defect formations, switching interatomic bonds, phase transitions, formation of nanoclasters, etc. Some examples of resolutions of nonlinear equations of equilibrium are presented.

This paper deals with the dynamic behavior of periodic reticulated beams and materials. Through the homogenization method of periodic discrete media the macro-behavior is derived at the leading order. With a systematic use of scaling, the analysis is performed on the archetypical case of symmetric unbraced framed cells. Such cells can present a high contrast between shear and compression deformability, conversely to “massive” media. This opens the possibility of enriched local kinematics involving phenomena of global rotation, inner deformation or inner resonance, according to studied configuration and frequency range.

An approach is proposed that allows formulating self-consistent problem that includes equations of the material’s dynamics and conditions of its damage. It is shown that damage in the material introduces a frequency-dependent damping and dispersion of the phase velocity of ultrasonic acoustic waves that allows estimating damage using the acoustic method. Applied field of deformations leads to the accumulation of damage. A kinetic equation is obtained, whose analysis shows that damage grows exponentially. The parameters of the system for which accumulation of damage can be considered as linear are estimated.

In this paper, the volumetric density of the Lagrangian of a second-order isotropic gradient continuum is critically examined. This density is first derived from a cubic lattice using an implicit continualization procedure. Then, using the derived Lagrangian, an equation of motion of the continuum as well as expressions for the standard and second-order stresses are obtained. It is shown that these expressions are contradictory. To obtain non-contradictory expressions for the stresses, the Lagrangian density is trimmed using the so-called null-Lagrangians that do not affect the equation of motion but influence the expressions for the stresses. This results in a unique non-contradictory expression for the Lagrangian of the continuum.

Essentially nonlinear proper structural model for continua with internal structure is analyzed using the localized strain wave solutions of the governing equations for macro- and micro-strains. The intervals of the velocity are found when either compression or tensile localized strain waves exist. It is obtained that simultaneous existence of compression and tensile macro-strain waves is impossible in contrast to the micro-strains responsible for the lattice defects. Also, it is shown that similar profiles of the macro-strain solitary waves may be accompanied by distinct profiles of the micro-strain waves. Generation of the bell-shaped defects in the lattice is studied numerically that allows us to describe structural deviations caused by the dynamical loading due to the localized macro-strain wave propagation.

Cecchi and Sab homogenization method (Cecchi and Sab in Int. J. Solids Struct. 44(18–19):6055–6079,

2007

) for the derivation of the effective Reissner–Mindlin shear moduli of a periodic plate is applied to sandwich panels including chevron pattern. Comparison with existing bounds (Lebée and Sab in Int. J. Solids Struct.,

2010

) and full 3D finite element computation validates the method. Finally, the skins effect on transverse shear stiffness is put forward.

We present a three-phase model describing wave propagation phenomena in residual-saturated porous media. The model consists of a continuous non-wetting phase and a discontinuous wetting phase and is an extension of classical biphasic (Biot-type) models. The model includes resonance effects of single liquid bridges or liquid clusters with miscellaneous eigenfrequencies taking into account a visco-elastic restoring force (pinned oscillations and/or sliding motion of the contact line). For the quasi-static limit case, i.e.,

ω

↦

0, the results of the model are identical with the phase velocity obtained with the well-known Gassmann–Wood limit.

A brief discussion of some current generalized continuum mechanics theories of elasticity and plasticity is provided. Attention is focused on works directly or indirectly motivated by the initial gradient models proposed by the author which, in turn, rest on ideas pioneered by Maxwell and van der Waals for fluid-like bodies but within a solid mechanics framework in the spirit of the celebrated monograph of brothers Cosserat published a quarter of a century later. The work of Cosserat, being dormant for half a century, ignited in the 1960s a plethora of generalized elasticity theories by the founders of modern continuum mechanics, as described in the treatises of Truesdell and Toupin and Truesdell and Noll. But it was not until another quarter of a century later that the interest in generalized continuum mechanics theories of elasticity and plasticity was revived, partly due to the aforementioned robust gradient models introduced and elaborated upon by the author and his co-workers in relation to some unresolved material mechanics and material physics issues; namely, the elimination of elastic singularities from dislocation lines and crack tips, the interpretation of size effects, and the description of dislocation patterns and spatial features of shear bands. This modest contribution is not aiming at a detailed account and/or critical review of the current state-of-the-art in the field. It only aims at a brief account of selected recent developments with some clarification on difficult points that have not been adequately considered or still remain somewhat obscure (origin and form of gradient terms, boundary conditions, thermodynamic potentials).

This chapter discusses the stress gradient theories of Eringen and Aifantis in terms of their original formulations, their differing dispersion properties in dynamics, and their finite-element implementation. A proposed combination of the two provides a dynamically consistent gradient enrichment while avoiding implementation difficulties.

This work aims at drawing the attention of mechanicians interested in the development of extended continuum theories on the unresolved issue of the physical interpretation of the additional boundary conditions introduced by 2nd-grade models. We discuss this issue in the context of the linearized theory of elasticity as an appropriate platform for discussion. Apart from lineal densities of edge-forces, 2nd-grade models allow for the prescription of force-like quantities energy-conjugated to the gradient of the velocity field on the boundary. Previous works proposed reductionistic interpretations, treating 2nd-grade models as particular cases of continua with affine microstructure; from the latter one can deduce field equations reminiscent of 2nd-grade models, either in the “low-frequency, medium wavelength” limit or by constraining the microstructural degrees of freedom to the gradient of the velocity field. The interpretation we propose here is based on the concept of

ortho-fiber

, and has the merit of achieving a simple physical interpretation of the boundary conditions without recourse to extensive algebra or the need to invoke microstructure.

In the paper, a rigorous continuous media model with conserved dislocations is developed. The important feature of the newly developed classification is a new kinematic interpretation of dislocations, which reflects the connection of dislocations with distortion, change in volume (porosity), and free forming. Our model generalizes those previously derived by Mindlin, Cosserat, Toupin, Aero–Kuvshinskii and so on, and refines some assertions of these models from the point of view of the account of adhesive interactions.

An overview on dislocations in the framework of different types of generalized continua is given. We consider Cosserat elasticity, gradient Cosserat elasticity, strain gradient elasticity, and dislocation gauge theory. We review similarities and differences between these generalized continuum theories. In fact, we discuss the constitutive relations for the linear and isotropic case. Moreover, we demonstrate how the characteristic length scales are given in terms of the material parameters. Also, we discuss the mathematical solutions of the elastic fields of a screw dislocation.

In this chapter, a rigorous analysis is given as a reference in which dislocations are treated as discrete entities. Then the transition towards a crystalline continuum is made in a number of steps by subsequent averaging along and perpendicular to the slip direction. This procedure eliminates the short-range dislocation interactions. Based on the considered idealized configuration, a back-stress term is derived which allows the conventional theory to predict finite-size pile-ups and turns out to be virtually identical to the result obtained by means of statistical arguments.

The distributed dislocation technique proved to be in the past an effective approach in studying crack problems within classical elasticity. The present work aims at extending this technique in studying crack problems within standard couple-stress elasticity (or Cosserat elasticity with constrained rotations), i.e., within a theory accounting for effects of microstructure. This extension is not an obvious one since rotations and couple-stresses are involved in the theory employed to analyze the crack problems. Here, the technique is introduced to study the case of a Mode I crack. Due to the nature of the boundary conditions that arise in couple-stress elasticity, the crack is modeled by a continuous distribution of climb dislocations and wedge disclinations that create both standard stresses and couple stresses in the body. In particular, it is shown that the Mode I case is governed by a system of coupled singular integral equations with both Cauchy and logarithmic kernels. The numerical solution of this system shows that a cracked solid governed by couple-stress elasticity behaves in a more rigid way (having increased stiffness) as compared to a solid governed by classical elasticity.

The theory of a Cosserat Point is a special continuum theory that characterizes the motion of a small material region which can be modeled as a point with finite volume. This theory has been used to develop a 3-D eight-noded brick Cosserat Point Element (CPE) to formulate the numerical solution of dynamical problems in finite elasticity. The kinematics of the CPE are characterized by eight director vectors which are functions of time only. Also, the kinetics of the CPE are characterized by balance laws which include: conservation of mass, balances of linear and angular momentum, as well as balances of director momentum. The main difference between the standard Bubnov–Galerkin and the Cosserat approaches is the way that they each develop constitutive equations. In the direct Cosserat approach, the kinetic quantities are given by derivatives of a strain energy function that models the CPE as a structure and that characterizes resistance to all models of deformation. A generalized strain energy function has been developed which yields a CPE that is truly a robust user friendly element for nonlinear elasticity that can be used with confidence for 3-D problems as well as for problems of thin shells and rods.

Strain-gradient theories have been used to model a variety of problems (such as elastic deformation, fracture behavior and plasticity) where size effect is of importance. Their use with the finite element method, however, has the drawback that specially designed elements are needed to obtain correct results.

This work presents an overview of the use of elements with

C

1

continuous interpolation for strain-gradient models, using gradient elasticity as an example. After showing how the

C

1

requirement arises and giving details concerning the implementation of specific elements, a theoretical comparison is made between elements based on this approach and elements resulting from the use of some alternative formulations.

For the numerical solution of gradient elasticity, the appearance of strain gradients in the weak form of the equilibrium equation leads to the need for

C

1

-continuous discretization methods. In the present work, the performances of a variety of

C

1

-continuous elements as well as the

C

1

Natural Element Method are investigated for the application to nonlinear gradient elasticity. In terms of subparametric triangular elements the Argyris, Hsieh–Clough–Tocher and Powell–Sabin split elements are utilized. As an isoparametric quadrilateral element, the Bogner–Fox–Schmidt element is used. All these methods are applied to two different numerical examples and the convergence behavior with respect to the

L

2

,

H

1

and

H

2

error norms is examined.

Electro-active polymers (EAP) recently attracted much interest because, upon electrical loading, EAP exhibit a large amount of deformation while sustaining large forces. On the other hand, generalized continuum frameworks are relevant to electro-mechanical coupled problems as they naturally incorporate couples or higher order stresses and can describe scale effects. Here, we want to adopt a strain gradient approach based on the generalized continuum framework as formulated in (Sansour in J. Phys. IV, Proc. 8:341–348,

1998

; Sansour and Skatulla in Geomech. Geoeng. 2:3–15,

2007

) and extend it to encompass the electro-mechanically coupled behavior of EAP. A new aspect of the electro-mechanical formulation relates to the multiplicative decomposition of the deformation gradient, well known from plasticity, into a purely elastic part and a further part which relates to the electric field. The formulation is elegant, makes for clarity and is numerically efficient. A numerical example of coupled large deformations is presented as well.

The generalization of the Hamilton’s and Onsager’s variational principles for dissipative hydrodynamical systems is represented in terms of the mechanical and the heat displacement fields. A system of equations for these fields is derived from the extreme condition for action with a Lagrangian in the form of the difference between the kinetic and the free energies minus the time integral of the dissipation function. The generalized hydrodynamic equation system is then evaluated on the basis of the generalized variational principle. At low frequencies, this system corresponds to the traditional Navier–Stokes equation system, and in the high frequency limit it describes propagation of acoustical and heat modes with the finite propagation velocities.

Beyond the usual 5-field theory (the basic fields are the mass density, velocity, internal energy), additional variables are needed for the unique description of complex media. Beside the conventional method of introducing additional fields by their balances, another procedure, the mesoscopic theory, is here discussed and applied to Cosserat continua.

This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media which are specified by a mass (or spatial) fractal dimension

D

, a surface fractal dimension

d

, and a resolution length scale

R

. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a theory based on a dimensional regularization, in which

D

is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving

D

,

d

and

R

. The formulation allows a generalization of the principles of virtual work, and virtual stresses, which, in turn, allow us to extend the extremum and variational theorems of elasticity and plasticity, as well as handle flows in fractal porous media. In all the cases, the derived relations depend explicitly on

D

,

d

and

R

, and, upon setting

D

=3 and

d

=2, reduce to conventional forms of governing equations for continuous media with Euclidean geometries. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure making it capable of grasping local material anisotropy. The product measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress, as noted in condensed matter physics. On this basis, a framework of micropolar mechanics of fractal media is developed.

In the present, paper we aim at constructing three variants of general mathematical models of micropolar elastic electro-conducting and non-ferromagnetic shells and plates: one with independent fields of transition and rotation, one with constraint rotation and one with “small shift rigidity”. The construction is based on the asymptotic method.