Anisotropic and Refined Plate Theories
Synonyms
Definitions
 heterogeneous

A solid is called heterogeneous if the material properties differ in different points of the solid, e.g., if the solid consists of different materials. If the material properties do not depend on the location in the solid it is called homogeneous.
 anisotropic

A material is called anisotropic, if the material properties are direction dependent, i.e., if two probes cut out of the solid with different orientations will react with different mechanical responses to the same load; otherwise, the material is called isotropic.
Anisotropic Materials in Practice
It is possible to manufacture solids (even with macroscopic dimensions) which are single crystals. These monocrystalline solids can be treated as a homogeneous continuum and have a natural anisotropy which stems from the crystal lattice. Wood, which behaves differently in fiber than transversally to the fiber direction, might serve as an example of a homogeneous, anisotropic material, where the anisotropy stems from a heterogeneity on a smaller (not resolved) scale. In technical applications, it is also common to treat even macroscopically heterogeneous materials as homogeneous anisotropic solids, like reinforced concrete or composite materials in general. If the material parameters of the components are known, effective parameters might be derived by analytical or numerical homogenization methods.
Voigt’s Notation for Anisotropic, Linear Elastic Material
Anisotropic linear elastic material is usually defined by stating the stiffness or compliance matrix of the Voigt notation of Hooke’s law.
The block interchange symmetry C_{ijkl} = C_{klij} of the stiffness tensor is equivalent to the existence of an elastic potential. Due to the other symmetries of the stiffness tensor, Hooke’s law (1) may be rewritten in a vectormatrix form, called the Voigt notation, where the block interchange symmetry of the stiffness tensor is equivalent to the symmetry of the stiffness matrix.
Engineering Constants
 E_{i}
Elastic or Young’s modulus
in x_{i}direction for uniaxial tension in x_{i}direction.
These are three independent constants.
 G_{ij}
Shear modulus
in the x_{i}x_{j}plane (i ≠ j). Hence the indices denote a plane, their order is irrelevant, i.e., G_{ij} = G_{ji}, and, therefore, there are three independent shear moduli.
 ν_{ij}
Poisson’s ratio
of negative normal strain (compression) in x_{j}direction to positive normal strain (tension) in x_{i}direction, i.e., \(\nu _{ij}=\frac {\varepsilon _{jj}}{\varepsilon _{ii}}\) (no summation convention), for uniaxial tension in x_{i}direction (i ≠ j). Due to the reciprocal relation \(\frac {\nu _{ij}}{E_i}=\frac {\nu _{ji}}{E_j}\), these are three independent constants.
 η_{i,jk}
Coefficient of mutual influence of the first kind
which is the ratio of normal strain in x_{i}direction to shear strain in the x_{j}x_{k}plane, i.e., \(\eta _{i,jk}=\frac {\varepsilon _{ii}}{\varepsilon _{jk}}\) (no summation convention), for simple shear stress in the x_{j}x_{k}plane. The associated coefficient of the second kind η_{jk,i} is the ratio of shear strain in the x_{j}x_{k}plane to normal strain in x_{i}direction for uniaxial tension in x_{i}direction. Due to the reciprocal relation \(\frac {\eta _{i,jk}}{G_{jk}}=\frac {\eta _{jk,i}}{E_i}\), these are nine independent constants.
 η_{ij,kl}
Chentsov coefficient
of shear strain in the x_{i}x_{j}plane to shear strain in the x_{k}x_{l}plane, i.e., \(\eta _{ij,kl}=\frac {\varepsilon _{ij}}{\varepsilon _{kl}}\), for simple shear stress in the x_{k}x_{l}plane (i ≠ j, k ≠ l, (i, j) ≠ (k, l)). Due to the reciprocal relation \(\frac {\eta _{ij,kl}}{G_{kl}}=\frac {\eta _{kl,ij}}{G_{ij}}\), these are three independent constants.
Special Kinds of Anisotropy
A general anisotropic material with 21 independent constants is called triclinic.
Plate Problems
Since the two subproblems are coupled for triclinic material, monoclinic material is the most general material anisotropy for which a plate problem can be defined.
Classical Theory of Monoclinic Plates
The classical theory of monoclinic plates treats homogeneous, thin plates of constant thickness h. It was mainly developed by Huber (1926, 1929) for the technically important special case of orthotropic material (where the planes of symmetry are given by the coordinate axes). An extension toward monoclinic material is straightforward. The probably most cited source for the derivation of the monoclinic theory is the classical book Lekhnitskii (1968), which was originally published in Russian language.
The theory is based on the same a priori assumptions as the KirchhoffLove plate theory, cf. Kirchhoff (1850), for isotropic material and is identical to this theory for the special case of isotropic material; hence it is considered a generalization of the KirchhoffLove theory toward anisotropic material behavior.
Laminate Theories
If the stress in a single ply of a laminate is of interest, one has to model the plate as an actual stack of anisotropic plates. Such theories are referred to as lamination or laminate theories. The classical lamination theory was derived by Jones (1975). It neglects transverse shear ε_{13} = ε_{23} = 0 in addition to the transversal normal strain ε_{33} = 0, and hence a simple twodimensional (inplane) treatment is possible.
Recent treatises about this twodimensional problem are often based on the socalled Stroh formalism which is treated in the books of Ting (1996) and Hwu (2016).
Refined Plate Theories
Plate theories are inherently approximative, but the approximation error of welldesigned theories converges to zero, if the thickness tends to zero. The aim of refined theories is to give a better approximation of the exact threedimensional theory; thus refined theories allow for the design of thicker plates than the classical theories (one says moderately thick plates).
Isotropic Refined Plate Theories
MindlinReissner Plate
More Recent Developments
The literature on refined isotropic plate theories is extensive, and still the field underlies substantial developments. One may distinguish three principle branches (or approaches) in the development of refined theories.
The engineering or classical approach introduces a priory assumptions, like kinematic assumptions, which may be motivated from experiments. Often shear correction factors are introduced which are intended to compensate intrinsic contradictions in the model.
The socalled direct approach immediately assumes a Cosseratcontinuum with a surface endowed with a set of deformable directors attached at each point. The approach does not necessarily involve a priori assumptions, but the formulation of constitutive relations requires known solutions of the socalled test problems. The resulting theory is crucially influenced by the choice of the test problems.
The consistent approach relies on series expansions of the elastic energy and does not necessarily introduce a priori assumptions either. However, the elastic energy has to be truncated in order to derive a tractable theory.
A survey on classical plate theories may be found in Szabó (1977). An overview about the direct approach is given in Altenbach et al. (2010). Often used refined models for isotropic plates were developed by Vekua (1985) and Reddy (2004).
Anisotropic Refined Plate Theories
Hence the problem of isotropic refined theories is still unsettled; there is far less literature on refined anisotropic theories.
A shear deformable (refined) plate theory for monoclinic material was derived by Ambartsumyan (1970). The theory extends the usual set of kinematic assumptions and, furthermore, assumes that the shear stress σ_{α3} distributions in thickness direction are proportional to given functions f_{α}. Furthermore, a semiinversion of Hooke’s law is used that assumes σ_{33} is given, which is finally calculated by integration of the threedimensional equilibrium equation σ_{i3,i} = 0 from the assumed shear stress distributions. The deviation of solutions from the classical theory increases with the ratio E_{i}∕G_{i3}.
Reddy (2004) introduced a first and thirdorder laminate theory. The order corresponds to the order of the polynomial ansatz used for the inplane deformations u_{α}, which is combined with the classical kinematic assumption u_{3} = w(x_{1}, x_{2}). The plane strain (ε_{33} = 0) kinematics are, furthermore, combined with the plane stress assumption σ_{33} = 0.
Recently a shear deformable monoclinic plate theory based on an a priori assumptionfree consistent approach which allows for an a priori error estimation of the resulting theory was developed in Schneider et al. (2014) and Schneider and Kienzler (2017). For its derivation, the elastic energy is truncated at a fixed power (which defines the order of approximation) of a characteristic parameter that describes the relative thinness of the plate. While the firstorder theory is the classical theory of monoclinic plates, the secondorder theory is an extension of the MindlinReissner theory and is equivalent to the theory for isotropic material. A comparison to other refined plate theories may be found in Schneider and Kienzler (2014).
CrossReferences
References
 Altenbach J, Altenbach H, Eremeyev V (2010) On generalized cosserattype theories of plates and shells: a short review and bibliography. Arch Appl Mech 80(1):73–92. https://doi.org/10.1007/s0041900903653 CrossRefGoogle Scholar
 Ambartsumyan SA (1970) Theory of anisotropic plates: strength, stability, vibration. Technomic Publishing Company, StamfordGoogle Scholar
 Friedrichs K, Dressler R (1961) A boundarylayer theory for elastic plates. Commun Pure Appl Math 14(1):1–33. https://doi.org/10.1002/cpa.3160140102 MathSciNetCrossRefGoogle Scholar
 Huber MT (1926) Einige Anwendungen der Biegungstheorie orthotroper Platten. ZAMM – Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 6(3):228–231. https://doi.org/10.1002/zamm.19260060306 CrossRefGoogle Scholar
 Huber MT (1929) Probleme der Statik technisch wichtiger orthotroper Platten. Gebethner & Wolff, WarsawGoogle Scholar
 Hwu C (2016) Anisotropic elastic plates, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
 Jones RM (1975) Mechanics of composite materials, vol 193. Scripta Book Company, Washington, DCGoogle Scholar
 Kirchhoff GR (1850) Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für die reine und angewandte Mathematik 39:51–88. http://eudml.org/doc/147439 CrossRefGoogle Scholar
 Lekhnitskii S (1968) Anisotropic plates. Gordon and Breach, Cooper Station, New YorkGoogle Scholar
 Mindlin RD (1951) Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18:31–38zbMATHGoogle Scholar
 Reddy JN (2004) Mechanics of laminated composite plates and shells: theory and analysis. CRC press, LondonCrossRefGoogle Scholar
 Reissner E (1944) On the theory of bending of elastic plates. J Math Phys 23:184–191MathSciNetCrossRefGoogle Scholar
 Schneider P, Kienzler R (2014) Comparison of various linear plate theories in the light of a consistent secondorder approximation. Math Mech Solids 20(7):871–882. https://doi.org/10.1177/1081286514554352 MathSciNetCrossRefGoogle Scholar
 Schneider P, Kienzler R (2017) A Reissnertype plate theory for monoclinic material derived by extending the uniformapproximation technique by orthogonal tensor decompositions of nthorder gradients. Meccanica 52(9):2143–2167. https://doi.org/10.1007/s1101201605731 MathSciNetCrossRefGoogle Scholar
 Schneider P, Kienzler R, Böhm M (2014) Modeling of consistent secondorder plate theories for anisotropic materials. ZAMM – Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 94(1–2):21–42. https://doi.org/10.1002/zamm.201100033 MathSciNetCrossRefGoogle Scholar
 Szabó I (1977) Geschichte der Mechanischen Prinzipien. Birkhäsuser, Basel. https://doi.org/10.1007/9783034892889 CrossRefGoogle Scholar
 Ting T (1996) Anisotropic elasticity: theory and applications, vol 45. Oxford University Press, New YorkzbMATHGoogle Scholar
 Vekua I (1985) Shell theory: general methods of construction. Monographs, advanced texts and surveys in pure and applied mathematics. Wiley, New YorkGoogle Scholar