On generalized Cosserat-type theories of plates and shells: a short review and bibliography
Summary (1 min read)
4 Theories of shells and plates by reduction of the three-dimensional micropolar continuum
- The authors have mentioned in the Introduction that there are some approaches based on the reduction of the threedimensional Cosserat continuum to the two-dimensional equations.
- These two-dimensional theories inherit some Cosserat-type properties from the three-dimensional continuum.
- Here the authors discuss briefly the governing equations of this theory in the case of equilibrium.
4.3 Relation to the other theories
- It is obvious that Eringen's plate theory is not coincide with the linear variant of Cosserat surface as well as with the linear variant of micropolar shell theories discussed above.
- In contrast to the Cosserat surface theory Eringen's micropolar plates theory take into account the drilling moment but can not take into account the force dipoles.
- On the other hand, this theory differs from the 6-parametric shell theory because it has only six degrees of freedom.
- In the first order micropolar plate theory by Eringen the two-dimensional couple stress tensor splits into two independent parts, i.e., the force-stress and moment-stress resultants.
5 Conclusion
- In this paper the authors briefly present some applications of the Cosserat continuum and the contribution of Professor Horst Lippmann.
- Further the authors restrict ourselves by two-dimensional Cosserat-type theories, i.e., the Cosseratbased theories of plates and shells.
- The authors present the most popular in the literature theories: the Cosserat-surface shell model, the micropolar or 6-parametric shell model, and the Eringen's micropolar plate theory.
- In this paper the authors also attempt to collect the corresponding papers and monographs related to the considered theories.
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"On generalized Cosserat-type theori..." refers background or methods in this paper
...Now we have thousands of papers and monographs on the Reissner’s and Mindlin’s approach, see the reviews [134,213] among others....
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...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....
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...αs is similar to the shear correction factor introduced in the plate theory by Reissner [256] (αs = 5/6) and by Mindlin [202] (αs = π2/12), see also [134]....
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...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], Ieşan [146], etc....
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...(20) In [44] the following relations for the elastic moduli appearing in (19) are used α1 = Cν, α2 = 0, α3 = C(1 − ν), α4 = αsC(1 − ν), β1 = Dν, β2 = 0, β3 = D(1 − ν), β4 = αt D(1 − ν), (21) C = Eh 1 − ν2 , D = Eh3 12(1 − ν2) , where E and ν are the Young’s modulus and the Poisson’s ratio of the bulk material, respectively, αs and αt are dimensionless coefficients, while h is the shell thickness. αs is similar to the shear correction factor introduced in the plate theory by Reissner [256] (αs = 5/6) and by Mindlin [202] (αs = π2/12), see also [134]. αt plays the same role for the moment stresses....
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3,504 citations
"On generalized Cosserat-type theori..." refers background in this paper
...Starting with the pioneering contributions of [180,255], new theories are established systematically....
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3,018 citations
"On generalized Cosserat-type theori..." refers background in this paper
...In the micropolar theory, besides ordinary stresses, couple stresses are introduced, see, for example, [308, 309]....
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