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On generalized Cosserat-type theories of plates and
shells: a short review and bibliography
Johannes Altenbach, Holm Altenbach, Victor A. Eremeyev
To cite this version:
Johannes Altenbach, Holm Altenbach, Victor A. Eremeyev. On generalized Cosserat-type theories of
plates and shells: a short review and bibliography. Archive of Applied Mechanics, Springer Verlag,
2010, 80 (1), pp.73-92. �hal-00827365�
74 J. Altenbach et al.
and rotations (and by analogy of forces and couples or force and moment stresses) is stated, see, e.g., [307].
Historically the first scientist, who obtained similar results, was L. Euler. Discussing one of Langrange’s papers
he established that the foundations of Mechanics are based on two principles: the principle of momentum and
the principle of moment of momentum. Both principles results in the Eulerian laws of motion [307]. In [214]
is given the following comment: the independence of the principle of moment of momentum, which is a
generalization of the static equilibrium of the moments, was established by Jacob Bernoulli (1686) one year
before Newton’s laws (1687). It must be noted that Newton’s law allows a satisfying description of the motion
of material points. If the continuum is presented by material particles with arbitrary shape this is not enough.
Hence the mechanics of generalized continua such as Cosserat continuum which differs from the classical
(or Cauchy-type) continuum mechanics is recognized as a branch of mechanics with origins in the seventeenth
century, but the first serious theoretical discussions have been started in the mid of the nineteenth century. On
the other hand, it is of current and emerging interest of mechanicians, physicians, materials scientists as well
as engineers since the limits and possibilities of such theories are not fully known, which is a serious constraint
for applications.
At the same time another tendency in Mechanics can be observed. Since Langrange’s Mécanique analytique
(1788) summarizing the state of the art in Mechanics at the end of the eighteenth century Mechanics is split into
two branches: the mathematical one and the engineering one. As a result the mathematical branch is developed
in a more axiomatic direction while the engineering branch is focussed on technical applications. Examples
of the axiomatic approach are, for example, G. Herglotz’ Lectures [143] or Cosserats’ monograph [54]. The
necessity to establish an axiomatic foundation of Mechanics was pointed out by D. Hilbert in 1900 at the 2nd
International Congress of Mathematicians in Paris
1
(the 6th Hilbert problem: establishment of the axiomatic
structure of Physics, especially Mechanics). The solution of this problem is nontrivial. During the first half
of the twentieth century only some scientist worked on this problem. One of them, G. Hamel, published his
results from 1908 in [140](seealso[141]). Since the mid fifties of the last century the interest to the axiomatic
approaches and Cosserats’ ideas is growing again and many publications have appeared. One of the initiators
of this direction was W. Noll (see [51] among others).
In the micropolar theory, besides ordinary stresses, couple stresses are introduced, see, for example, [308,
309]. The deformation of the micropolar continuum may be described by the position vector
r
r and the three
orthonormal vectors d
i
, i = 1, 2, 3, so-called directors, which model the translations and the orientation
changes of the material particles. Sometimes the properly orthogonal tensor Q is used for the description of the
rotations. In the case of small strains the deformation of the micropolar medium is defined by two independent
fields: the displacement vector u and the microrotation vector ϑ. The linear Cosserat theory was developed in
the original papers by Günther [137], Aero and Kuvshinskii [3,4], Toupin [304], Mindlin and Tiersten [203],
Koiter [161], Palmov [234,235], Eringen [90,92], Schaefer [278], Ie¸san [146], etc. Let us mention here the
books [68,95,98,147,169,232,295], where many references to other papers can be found. The problems of
the micropolar continuum at finite deformations are considered by Grioli [135,136], Toupin [305], Kafadar
and Eringen [153,154], Stojanovi´c[296], Besdo [24], Reissner [261,262,265], and Shkutin [286], see also
[34,67,95,200,219,228,252,294,321].
W
ithin the framework of the Cosserat continuum many problems are solved which demonstrate the qualita-
tiveand quantitativedifference from the solutions based on the classical Cauchycontinuum model. In particular,
in the monographs [95,98,200,232] wave processes in micropolar continua are investigated. The acceleration
waves in nonlinear elastic micropolar media are considered in [153]. A generalization is presented in [198],
where acceleration waves in elastic and viscoelastic micropolar media are studied. The relation between the
existence of acceleration waves and the condition of strong ellipticity of the equilibrium equations is established
in [15,74]. The theory of dislocations and disclinations is presented in [328,331,333]. In [79] the theory of
superposed small deformations on a large deformation of an elastic micropolar continuum is developed. Three
ways of introducing of the Lagrangian strain measures as well as an extended review are given in [245,246].
Variational problems in the micropolar continuum are investigated in [228,229,294].
The main problem of any micropolar theory is the establishment of the constitutive equations. For exam-
ple, this problem is not discussed in Cosserats’ original monograph [54], and this was a reason that the ideas
were not recognized by many researchers. But even in the case when this problem is solved in a satisfying
manner another problem can be stated: the identification of the material parameters. For example, in the lin-
ear micropolar elasticity of isotropic solids one needs six material parameters while only two Lamé moduli
are needed in the classical elasticity. The experimental identification of the elastic moduli is discussed in
1
http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/rede.html and http://en.wikipedia.org/wiki/Hilbert.
On generalized Cosserat-type theories of plates and shells 75
[115,116,175,177,204,239], see also the data in [95,98] and the websites by P.Neff
2
and R.Lakes.
3
Another
approach to the determination of the micropolar moduli is based on the various homogenization procedures,
see, for example, [47,72,178,179,291,318].
Many publications on the Cosserat continuum are published in the sixties of the last century (in [295]
are presented 400 references). In 1969, H. Lippmann has published his famous paper A Cosserat Theory of
Plastic Flow [184]. He based his theoretical approach on [131,278] and described large strains ignoring the
elastic part (rigid-plastic behavior). A motivation for this approach was that the Poynting-effect discussed
by Swift in [298] can be presented. Later this approach has been continued by D. Besdo [24]. A possi-
ble application is discussed in [23,32]. H. Lippmann continued his investigations in the Cosserat plasticity,
see [63,155,187,188,310–312]. Up to now there are many publications on the plastic Cosserat continuum,
among them DeBorst [57], Ehlers [69,70], Forest [103], Forest et al. [104–108,282,287], Grammenoudis
and Tsakmakis [120–123], Neff [218], Neff and Chełmi´nski [37,38,221,223], Steinmann [292,293], see also
[55,64,65,71,99,100,145,157,166,181,190,225,226,238,266,273,277,283,284].
Another field of interest of
H. Lippmann was rock mechanics [185,186]. This field of applied mechanics is closely related to the Cosserat
plasticity, see for example [1,2,36,196].
The Cosserat model is used to describe solid materials with a complex microstructure like soils, polycrys-
talline and composite materials, granular and powder-like materials, see [35,56,60,69,71,72,117,142,164,
165,197,205,218,224,297,299–301,306,314,324,325], nanostructures [150,149], porous media and foams
[59,61,62,66,69,70,175–177],
and even bones [102,177,239], as well as electromagnetic and ferromagnetic
media, see [97,133,199] among others. Starting from the papers by Aero et al. [5] and Eringen [91] the micro-
polar continuum is applied to model magnetic liquids, polymer suspensions, liquid crystals, and other types
of fluids with microstructure, see, for example, [7,14,94,323,332] and the books of Eringen [96], Migoun
and Prokhorenko [201] among others. Let us note that the Cosserat approach may be used as a base for the
construction of special finite elements [269,271] or as the special regularization procedure.
Since the paper of Ericksen and Truesdell [89] the Cosserat model has found applications in construction of
various generalized models for beams, plates, and shells. Within the framework of the direct approach applied
in [89], the shell is modeled as a deformable surface at each point of which a set of deformable directors is
attached. Hence, in general the deformation of a shell is described by the position vector r and p directors dd
i
,
i = 1,... p. This approach is developed in the original papers by Ericksen [82,83,86,87], Green and Naghdi
[124–130], Green et al. [132], Naghdi [209], Naghdi et al. [210–212] and DaSilva and Tsai [58]. This variant
of the shell theory is also named Cosserat shell theory or the theory of Cosserat surfaces. Some criticism con-
cerning the direct approach in general and especially the theory of Cosserat surfaces exists, see, for example,
[289]. But the theory is developed successfully and there are various applications, see [18,25–31,40,48–
50,101,119,138,139,148,156,168,173,174,216,220,253,254,270,272,313] among others. In particular, the
theory of symmetry of the constitutive equations is developed in [84,85,206,207]. Finite element formulations
of the Cosserat shell theory are presented in [39,152,167,274,322]. Let us mention only the fundamental books
by Naghdi [208], Rubin [269]
,andAntman[19] where the theory of Cosserat shells is presented.
The theory of Cosserat shells contains as a special case the linear theory of Cosserat plates. This theory
is mostly formulated with the help of the introduction of one deformable director [124,125]. A variant with
various directors is discussed, for example, in [251]. Applications of the Cosserat surface theory to sandwich
plates are given in [119,194,195]. In the case of the theory of Cosserat plates with one director the unknown
functions are the vector of displacements of the surface, representing the plate, and the vector describing the
deformations of the director. Thus one assumes that such theory contains six degrees of freedom, and as a
consequence one has to establish six boundary conditions. In the case of Cosserat shells it is also possible to
describe the thickness changes. So one can conclude that in this case for each material point six degrees of
freedom are assumed: three translational degrees of freedom, two rotational degrees of freedom describing the
rotations about the directors and one degree of freedom which is related to the thickness changes.
Independently Eringen has formulated a linear theory of micropolar plates in [93], see also the monograph
[95]. The two-dimensional equations of this theory are deduced with the help of the independent integration
over the thickness of both the first and the second Euler laws of motion of the linear elastic micropolar con-
tinuum. The theories of the zeroth and the first order are presented applying a special linear approximation
of the displacement and the microrotation fields. Eringen’s theory is based on eight unknowns: the averaged
displacements, the averaged macrorotations of the cross-sections and the averaged microrotations. This means
that one has to introduce eight boundary conditions. The static boundary conditions in Eringen’s plate theory
2
http://www.mathematik.tu-darmstadt.de/fbereiche/analysis/pde/staff/neff/patrizio/Home.html.
3
http://silver.neep.wisc.edu/~lakes/home.html.
76 J. Altenbach et al.
cannot be presented as forces and moments at the boundaries like in the Kirchhoff-type theories [302]. From
the point of view of the direct approach Eringen’s micropolar plate is a deformable surface with eight degrees
of freedom. Eringen’s approach is widely discussed, for example, in [20,33,52,163,170–172,233,250,279–
281,303,316,317] and in the monograph [95].
The theories of plates and shells and the theories based on the reduction of the three-dimensional equations
of the micropolar continuum are presented in several publications. In [16,17,118,151,259,263] various aver-
aging procedures in the thickness direction together with the approximation of the displacements and rotations
or the force and moment stresses in the thickness direction are applied. As a result, one gets different numbers
of unknowns and the number of two-dimensional equilibrium equations differs. For example, Reissner [263]
presented a generalized linear theory of shells containing nine equilibrium equations. In addition, Reissner
worked out the two-dimensional theory of a sandwich plate with a core having the properties of the Cosserat
continuum [260]. The variants of the micropolar plate theory based on the asymptotic methods are developed
in [6,73,215,275,276]. The nonlinear theory of elastic shells derived from the pseudo-Cosserat continuum is
considered in [22]. The linear theory of micropolar plates is discussed in [11] where the discussion on the
reduction procedure is given. The -convergence based approach to the Cosserat-type of theory of plates and
shells is discussed in [216,217,222].
In both the Cosserat’s and the Eringen’s micropolar plate theories one has additional kinematic vari-
ables—the rotations. It should be noted that in the theories of plates and shells the rotations are introduced
as independent kinematic variables before the Cosserat theory was established. The term “angle of rotation”
is introduced in Kirchhoff’s theory too—but the rotations are expressed by the displacement field. After the
pioneering work of Kirchhoff [160] thousands of publications are presented, which try to give the foundations
and the methods of deduction of the equations of the Kirchhoff–Love theory, but also of improvements, see,
for example, [45,46,110–114,158,159,227,249] among others. Considering sandwich structures with a soft
core Reissner worked out a theory by taking into account the transverse shear which was ignored by Kirchhoff
[256–258,264]. Similar governing equations (only some effects are not included) were derived by Mindlin
introducing additional degrees of freedom for the points of the midplane [202]. The order of the system of the
partial differential static equilibrium equations in the case of Reissner-type theories is equal to ten. That means,
that the number of boundary equations is equal to five. In the theories of Reissner and Mindlin only two angles
of rotations are independent of the displacements, and the transverse shear can be taken into account. The
third angle of rotation (rotation about the normal to the surface, so-called drill rotation) is not considered as
an independent variable. In Reissner’s theory the static boundary conditions are equivalent to the introduction
of distributed forces and moments on the contour, the last one has no components in the normal direction. The
Reissner’s plate as well as the Kirchhoff’s plate are not able to react on the distributed moments on the sur-
faces or boundaries which are directed along the surface normal (so-called drilling moments). That means that
Reissner’s plate is modeled by a material surface each point having five degrees of freedom (three translation
and two rotations) while Kirchhoff’s plate is a material surface each point of which has only three degrees
of freedom (three translations). The original Kirchhoff’s plate theory has only one degree of freedom (the
deflection). Now we have thousands of papers and monographs on the Reissner’s and Mindlin’s approach, see
the reviews [134,213] among others.
In the last decades the so-called higher order theories are very popular. Starting with the pioneering con-
tributions of [180,255], new theories are established systematically. If one discusses higher order theories in
the point of view of the direct approach one assumes deformable surfaces with additional degrees of freedom.
For example, the third order theory presented in [315] can be regarded as a theory with seven degrees of free-
dom including rotations of the plate cross-sections. Let us mention also the papers [21,109,144,167,241,290]
where the rotations in shells are considered, while an extensive discussion of the application of the rotations
in Continuum Mechanics is given in [244].
The direct approach in the theory of shells based on Cosserats’ ideas is applied also in [326]. In contrast to
[89], the shells are regarded as deformable surfaces with material points at which three directors are prescribed.
The directors have the following properties: they are orthogonal unit vectors. The deformations of the shell
are presented by a position vector and a properly orthogonal tensor. This variant of the shell and plate theories
based on the direct approach is developed and continued, for example, in [75,80,81,236,237,285,286,327–
330]. It must be noted that this variant is very similar to the one presented within the general nonlinear theory
of shells discussed in the monographs of Libai and Simmonds [183], and Chró´scielewski et al. [44], see also
[41–43,76–78,162,182,189,191–193,243,247,
248,288,289]. The two-dimensional equilibrium equations
givenin[44,183,243] one gets by the exact integration over the thickness of the equations of motion of a