Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Anisotropic and Refined Plate Theories

  • Patrick SchneiderEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_136-1

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.

A material is called elastic if there exists an elastic potential (energy), and it is called linear elastic if the stress-strain relation is linear, i.e., given by Hooke’s law
$$\displaystyle \begin{aligned} \sigma_{ij}=C_{ijkl}\varepsilon_{kl},\quad \text{or}\quad \varepsilon_{ij}=S_{ijkl}\sigma_{kl} {} \end{aligned} $$
(1)
in inverted form. Although the components of the forth-rank stiffness or elasticity tensor C are often denoted as elastic constants, they depend on the coordinates x of a material point in general. A linear elastic solid is homogeneous, if and only if C is a constant tensor. The inverse tensor of C is called the compliance tensor S = C−1 and defined via
$$\displaystyle \begin{aligned} S_{ijkl}C_{klrs}=C_{ijkl}S_{klrs}=\delta_{ir}\delta_{js}. {} \end{aligned} $$
(2)
Both tensors fulfill the symmetry relations,
$$\displaystyle \begin{aligned} T_{ijkl}=T_{jikl}=T_{ijlk}=T_{klij} {} \end{aligned} $$
(3)
where T may be C or S. These symmetry relations (3) are denoted by \(T_{( \underline {ij})( \underline {kl})}\) in the standard notation of representation theory.

The block interchange symmetry Cijkl = Cklij 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 vector-matrix form, called the Voigt notation, where the block interchange symmetry of the stiffness tensor is equivalent to the symmetry of the stiffness matrix.

To this end the stress vector and the strain vector are defined by
$$\displaystyle \begin{aligned} \begin{array}{rcl} \underline{\boldsymbol{\sigma}}&\displaystyle :=&\displaystyle \left[\sigma_{11},\sigma_{22},\sigma_{33},\sigma_{12},\sigma_{23},\sigma_{31}\right]^T,{} \end{array} \end{aligned} $$
(4)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \underline{\boldsymbol{\varepsilon}}&\displaystyle :=&\displaystyle \left[\varepsilon_{11},\varepsilon_{22},\varepsilon_{33},2\varepsilon_{12},2\varepsilon_{23},2\varepsilon_{31}\right]^T,{} \end{array} \end{aligned} $$
(5)
and the stiffness matrix
$$\displaystyle \begin{aligned} \underline{\underline{\boldsymbol{C}}}{:=}\left[ \begin{array}{cccccc} C_{{1111}}&C_{{1122}}&C_{{1133}}&C_{{1112}}&C_{{1123}}&C_{{1113}}\\ &C_{{2222}}&C_{{2233}}&C_{{2212}}&C_{{2223}}&C_{{2213}}\\ &&C_{{3333}}&C_{{3312}}&C_{{3323}}&C_{{3313}}\\ S&&&C_{{1212}}&C_{{1223}}&C_{{1213}}\\ &Y&&&C_{{2323}}&C_{{2313}}\\ &&M.&&&C_{{1313}}\end{array} \right] , \end{aligned} $$
(6)
and compliance matrix are defined by
$$\displaystyle \begin{aligned} \underline{\underline{\boldsymbol{S}}}:= \left[ \begin{array}{cccccc} S_{{1111}}&S_{{1122}}&S_{{1133}}&2S_{{1112}}&2S_{{1123}}&2S_{{1113}}\\ &S_{{2222}}&S_{{2233}}&2S_{{2212}}&2S_{{2223}}&2S_{{2213}}\\ &&S_{{3333}}&2S_{{3312}}&2S_{{3323}}&2S_{{3313}}\\ S&&&4S_{{1212}}&4S_{{1223}}&4S_{{1213}}\\ &Y&&&4S_{{2323}}&4S_{{2313}}\\ &&M.&&&4S_{{1313}}\end{array} \right]. \end{aligned} $$
(7)
(The order of the last three components of the stress and strain vector and hence of the last three columns and rows of the stiffness and compliance matrix are not consistent in the literature.) Voigt’s notation of Hooke’s law reads
$$\displaystyle \begin{aligned} \underline{\boldsymbol{\sigma}}=\underline{\underline{\boldsymbol{C}}}\,\underline{\boldsymbol{\varepsilon}},\quad \text{or}\quad\underline{\boldsymbol{\varepsilon}}=\underline{\underline{\boldsymbol{S}}}\,\underline{\boldsymbol{\sigma}}, \end{aligned} $$
(8)
in inverted form, where the right hand products are usual matrix-vector multiplications. The so-defined stiffness and compliance matrix are, in analogy to (2), the matrix inverse of each other
$$\displaystyle \begin{aligned} \underline{\underline{\boldsymbol{C}}}\,\underline{\underline{\boldsymbol{S}}} =\underline{\underline{\boldsymbol{S}}}\,\underline{\underline{\boldsymbol{C}}} =\underline{\underline{\boldsymbol{I}}}, {} \end{aligned} $$
(9)
where \( \underline { \underline {\boldsymbol {I}}}\) denotes the 6 × 6-matrix of unity. The elastic energy at a point is given by
$$\displaystyle \begin{aligned} \frac{1}{2}C_{ijrs}\varepsilon_{ij}\varepsilon_{rs} &=\frac{1}{2}\underline{\boldsymbol{\varepsilon}}^T\underline{\underline{\boldsymbol{C}}}\,\underline{\boldsymbol{\varepsilon}} =\frac{1}{2}\underline{\boldsymbol{\sigma}}^T\underline{\boldsymbol{\varepsilon}} =\frac{1}{2}\underline{\boldsymbol{\sigma}}^T\underline{\underline{\boldsymbol{S}}}\,\underline{\boldsymbol{\sigma}}\\ &=\frac{1}{2}S_{ijrs}\sigma_{ij}\sigma_{rs}. \end{aligned} $$

Engineering Constants

A general anisotropic material can be described by the definition of 21 independent material constants, which may be chosen as the elements of the upper-right or lower-left triangular submatrix of either the stiffness or the compliance matrix. The material behavior might also be defined by the use of Engineering Constants which are associated with specific material tests and have, therefore, a direct physical meaning. The compliance matrix reads
$$\displaystyle \begin{aligned} \underline{\underline{\boldsymbol{S}}}= \left[ \begin{array}{cccccc} \frac{1}{E_1}&\frac{-\nu_{21}}{E_2}&\frac{-\nu_{31}}{E_3}&\frac{\eta_{1,12}}{G_{12}}&\frac{\eta_{1,23}}{G_{23}}&\frac{\eta_{1,13}}{G_{13}}\\ \noalign{\medskip}&\frac{1}{E_2}&\frac{-\nu_{32}}{E_3}&\frac{\eta_{2,12}}{G_{12}}&\frac{\eta_{2,23}}{G_{23}}&\frac{\eta_{2,13}}{G_{13}}\\ \noalign{\medskip}&&\frac{1}{E_3}&\frac{\eta_{3,12}}{G_{12}}&\frac{\eta_{3,23}}{G_{23}}&\frac{\eta_{3,13}}{G_{13}}\\ \noalign{\medskip}S&&&\frac{1}{G_{12}}&\frac{\eta_{12,23}}{G_{23}}&\frac{\eta_{12,13}}{G_{13}}\\ \noalign{\medskip}&Y&&&\frac{1}{G_{23}}&\frac{\eta_{23,13}}{G_{13}}\\ \noalign{\medskip}&&M.&&&\frac{1}{G_{13}}\end{array} \right], {} \end{aligned} $$
(10)
by the use of the following engineering constants:
Ei

Elastic or Young’s modulus

in xi-direction for uniaxial tension in xi-direction.

These are three independent constants.

Gij

Shear modulus

in the xi-xj-plane (i ≠ j). Hence the indices denote a plane, their order is irrelevant, i.e., Gij = Gji, and, therefore, there are three independent shear moduli.

νij

Poisson’s ratio

of negative normal strain (compression) in xj-direction to positive normal strain (tension) in xi-direction, i.e., \(\nu _{ij}=\frac {-\varepsilon _{jj}}{\varepsilon _{ii}}\) (no summation convention), for uniaxial tension in xi-direction (i ≠ j). Due to the reciprocal relation \(\frac {\nu _{ij}}{E_i}=\frac {\nu _{ji}}{E_j}\), these are three independent constants.

ηi,jk

Coefficient of mutual influence of the first kind

which is the ratio of normal strain in xi-direction to shear strain in the xj-xk-plane, i.e., \(\eta _{i,jk}=\frac {\varepsilon _{ii}}{\varepsilon _{jk}}\) (no summation convention), for simple shear stress in the xj-xk-plane. The associated coefficient of the second kind ηjk,i is the ratio of shear strain in the xj-xk-plane to normal strain in xi-direction for uniaxial tension in xi-direction. Due to the reciprocal relation \(\frac {\eta _{i,jk}}{G_{jk}}=\frac {\eta _{jk,i}}{E_i}\), these are nine independent constants.

ηij,kl

Chentsov coefficient

of shear strain in the xi-xj-plane to shear strain in the xk-xl-plane, i.e., \(\eta _{ij,kl}=\frac {\varepsilon _{ij}}{\varepsilon _{kl}}\), for simple shear stress in the xk-xl-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.

The number of elastic constants may be further reduced, if a specific material possesses certain symmetries. If two probes cut out symmetrically to the x1-x2-plane show the same material behavior, the material is called monoclinic with respect to the x1-x2-plane. Hence the stiffness tensor must be invariant with respect to the inversion of the x3-direction; therefore, any stiffness component with an uneven number of tensor indices that equal 3 has to vanish and only 13 independent material parameters remain:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \underline{\underline{\boldsymbol{C}}}&\displaystyle =&\displaystyle \left[ \begin{array}{cccccc} C_{{1111}}&\displaystyle C_{{1122}}&\displaystyle C_{{1133}}&\displaystyle C_{{1112}}&\displaystyle 0&\displaystyle 0\\ \noalign{\medskip}&\displaystyle C_{{2222}}&\displaystyle C_{{2233}}&\displaystyle C_{{2212}}&\displaystyle 0&\displaystyle 0\\ \noalign{\medskip}&\displaystyle &\displaystyle C_{{3333}}&\displaystyle C_{{3312}}&\displaystyle 0&\displaystyle 0\\ \noalign{\medskip}S&\displaystyle &\displaystyle &\displaystyle C_{{1212}}&\displaystyle 0&\displaystyle 0\\ \noalign{\medskip}&\displaystyle Y&\displaystyle &\displaystyle &\displaystyle C_{{2323}}&\displaystyle C_{{2313}}\\ \noalign{\medskip}&\displaystyle &\displaystyle M.&\displaystyle &\displaystyle &\displaystyle C_{{1313}}\end{array} \right]. \end{array} \end{aligned} $$
If the material has two planes of symmetry, which are both orthogonal to coordinate axes, the third plane, which is orthogonal to the remaining axis, is automatically a plane of reflectional symmetry, too. Such a material is called orthotropic (or rhombic). The components
$$\displaystyle \begin{aligned} C_{1112}=C_{2212}=C_{3312}=C_{2313}=0 \end{aligned} $$
(11)
vanish in addition, and only nine independent material parameters remain.
If the material behavior is invariant with respect to any rotation, the material is isotropic (and especially orthotropic) and only two material parameters remain, e.g., Young’s modulus E and Poisson’s ratio ν,
$$\displaystyle \begin{aligned} C_{{1111}}&=C_{{2222}}=C_{{3333}}\\ &={\frac { \left( 1-\nu \right) E}{ \left( 1+\nu \right) \left( 1-2\,\nu \right) }},\\ C_{{1122}}&=C_{{1133}}=C_{{2233}}\\ &={\frac {\nu\,E}{ \left( 1+\nu \right) \left( 1-2\,\nu \right)}},\\ C_{{1212}}&=C_{{2323}}=C_{{1313}}=\frac{1}{2}\left(C_{{1111}}-C_{{1122}}\right)\\ &=G=\frac{E}{2\,(1+\nu)}.{} \end{aligned} $$
(12)

Plate Problems

The three-dimensional problem of a plane, linear elastic solid of constant thickness which is loaded by transversal tractions (in x3-direction) on the top-side \(P_3^-\) and on the bottom-side \(P_3^+\), decouples into two independent subproblems, if the material is homogeneous and monoclinic with respect to the midplane (x1-x2-plane). The plate (or bending) problem is driven by the symmetric part of the load, i.e., by \(P_3^{+\text{ sym}}\) and \(P_3^{-\text{ sym}}\) defined by
$$\displaystyle \begin{aligned} P_3^{+\text{ sym}}(x_\alpha)&=P_3^{-\text{ sym}}(x_\alpha)=\frac{q(x_\alpha)}{2}\\ &=\frac{P_3^-(x_\alpha)+P_3^+(x_\alpha)}{2}, {} \end{aligned} $$
(13)
cf. Fig. 1, and results in a lateral displacement u3 of the midplane in x3-direction, cf. e.g., Friedrichs and Dressler (1961).
Fig. 1

Decomposition of the transversal tractions at an arbitrary point (x1, x2)

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 Kirchhoff-Love 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 Kirchhoff-Love theory toward anisotropic material behavior.

Like the Kirchhoff theory, the theory combines the classical set of the kinematic assumptions
$$\displaystyle \begin{aligned} u_3=w(x_\alpha),\ u_\alpha=-x_3 \frac{\partial w(x_\alpha)}{\partial x_\alpha}, {} \end{aligned} $$
(14)
which leads to plane strain ε33 = 0, with the plane stress assumption σ33 = 0. Both assumptions contradict each other and achieve acceptable accuracy of the solution only if the plate is sufficiently thin.
Solving the plane stress assumption 0 = σ33 = C33rsεrs for ε33 and insertion into Hooke’s law leads to
$$\displaystyle \begin{aligned} \sigma_{ij}=\underbrace{\left(C_{ijrs}-\frac{C_{ij33}C_{rs33}}{C_{3333}}\right)}_{=:C^*_{ijrs}}\varepsilon_{rs}, {} \end{aligned} $$
(15)
which is Hooke’s law with reduced stiffness C. In Voigt’s notation, the lower 2 × 2-block-diagonal submatrix of \( \underline { \underline {\boldsymbol {C}^*}}\) equals those of \( \underline { \underline {\boldsymbol {C}}}\). \( \underline { \underline {\boldsymbol {C}^*}}\) has a vanishing third column (and row) in Voigt notation. Hence the matrix is not invertible, but the 5 × 5 matrix obtained by removing the vanishing column and row is the inverse matrix of the 5 × 5 matrix obtained by removing the third column and row of the original compliance matrix \( \underline { \underline {\boldsymbol {S}}}\), which delivers an alternate approach to derive (15) by incorporating plane stress into the inverted Hooke’s law (1).
ε13 = ε23 = 0 follows from (14). In turn σ13 = σ23 = 0 would follow from Hooke’s law (1), (8), or (15). Instead, one integrates the in-plane three-dimensional equilibrium equation’s σ,i = 0 to derive nonvanishing shear stress distributions for σ13 and σ23 from the components σαβ obtained by (15). This means in combination with ε13 = ε23 = 0 that we have infinite shear-moduli, i.e., shear-rigidity. Incorporating the conditions σ13 = σ23 = 0 at the top and bottom x3 = ±h∕2, the shear stress distributions are fixed, and in turn we obtain the stress resultants:
$$\displaystyle \begin{aligned} M_{\alpha\beta}&=-\frac{h^3}{12}\,C^*_{\alpha\beta\gamma\delta}\, w_{, \gamma\delta},\\ Q_\alpha&=-\frac{h^3}{12}\,C^*_{\alpha\beta\gamma\delta}\, w_{, \beta\gamma\delta}. {} \end{aligned} $$
(16)
Since the derivation of the equilibrium equations in terms of stress resultants for the Kirchhoff plate is independent of the material law, they could be adopted one-on-one. Elimination of the shear forces and insertion of the moments leads, like in the isotropic case, to the differential equation of bending
$$\displaystyle \begin{aligned} -M_{\alpha\beta ,\alpha\beta} =\quad\frac{h^3}{12}\,C^*_{\alpha\beta\gamma\delta}\, w_{, \alpha\beta\gamma\delta} =q, {} \end{aligned} $$
(17)
which simplifies to the Kirchhoff equation for isotropic material (12). Also, the boundary conditions, including the definition of effective shear forces \(Q^*_\alpha \), equal those of the isotropic case.

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 ε13 = ε23 = 0 in addition to the transversal normal strain ε33 = 0, and hence a simple two-dimensional (in-plane) treatment is possible.

Recent treatises about this two-dimensional problem are often based on the so-called 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 well-designed theories converges to zero, if the thickness tends to zero. The aim of refined theories is to give a better approximation of the exact three-dimensional theory; thus refined theories allow for the design of thicker plates than the classical theories (one says moderately thick plates).

Isotropic Refined Plate Theories

Mindlin-Reissner Plate

The most established refined plate theory treats isotropic material and was developed by Reissner (1944) and Mindlin (1951) independently of each other. Their theories coincide in most (but not all) aspects and were thus merged in the subsequent literature and are referred to as the Mindlin-Reissner plate theory. The theory drops the normal hypothesis of the Kirchhoff theory, i.e., ϕα ≠ − w,α, and includes shear deformation effects. The shear stresses σα3 = G(w,α + ϕα) arising from Hook’s law are constant over the thickness, which contradicts the three-dimensional boundary conditions. Thus a so-called shear-correction factor κ is introduced, which is often set to κ = 5∕6. (The value is obtained by comparing the elastic energies of the constant distribution and a (more realistic) parabolic distribution.) Beside a fourth-order partial differential equation (PDE) which tends to the one of the Kirchhoff-Love theory for shear rigidity (Gκh →), the model comprises a second PDE:
$$\displaystyle \begin{aligned} K\varDelta\varDelta w=q-\frac{K}{G\kappa h}\varDelta q,\quad\frac{12\kappa}{h^2}\varPsi-\varDelta\varPsi=0, {} \end{aligned} $$
(18)
for the quantity Ψ := ϕ2,1 − ϕ1,2, which measures the deviation from the normal-hypothesis. Hence it is possible to prescribe all three physical meaningful quantities independent of each other at each boundary and no effective shear forces need to be introduced. The solutions for Ψ are singular perturbation solutions that decline exponentially from the boundary and hence only govern the behavior in small boundary layers along the edges. Therefore, the classical Kirchhoff-Love theory is considered a suitable approximation of (only) the interior solution, i.e., away from the boundary.

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 so-called direct approach immediately assumes a Cosserat-continuum 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 so-called 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 σα3 distributions in thickness direction are proportional to given functions fα. Furthermore, a semi-inversion of Hooke’s law is used that assumes σ33 is given, which is finally calculated by integration of the three-dimensional equilibrium equation σi3,i = 0 from the assumed shear stress distributions. The deviation of solutions from the classical theory increases with the ratio EiGi3.

Reddy (2004) introduced a first- and third-order laminate theory. The order corresponds to the order of the polynomial ansatz used for the in-plane deformations uα, which is combined with the classical kinematic assumption u3 = w(x1, x2). The plane strain (ε33 = 0) kinematics are, furthermore, combined with the plane stress assumption σ33 = 0.

Recently a shear deformable monoclinic plate theory based on an a priori assumption-free 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 first-order theory is the classical theory of monoclinic plates, the second-order theory is an extension of the Mindlin-Reissner 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).

Cross-References

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Institute for Lightweight Construction and Design (KLuB)Technische Universität DarmstadtDarmstadtGermany

Section editors and affiliations

  • Karam Sab
    • 1
  1. 1.Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTARUniversité Paris-EstMarne-la-ValléeFrance