PAMM · Proc. Appl. Math. Mech. 11, 273 – 274 (2011) / DOI 10.1002/pamm.201110128

A consistent second-order plate theory for monotropic material

Patrick Schneider1,∗ and Reinhold Kienzler1,∗∗

1 Department of Production Engineering, University of Bremen, Am Biologischen Garten 2, D-28359 Bremen, Germany

Mathematical homogenization (or averaging) of composite materials, such as fibre laminates, often leads to non-isotropichomogenized (averaged) materials. Especially the upcoming importance of these materials increases the need for accuratemechanical models of non-isotropic shell-like structures. We present a second-order (or: Reissner-type) theory for the elasticdeformation of a plate with constant thickness for a homogeneous monotropic material. It is equivalent to Kirchhoff’s platetheory as a first-order theory for the special case of isotropy and, furthermore, shear-deformable and equivalent to R. Kienzler’stheory as a second-order theory for isotropy, which implies further equivalences to established shear-deformable theories,especially the Reissner-Mindlin theory and Zhilin’s plate theory. Details of the derivation of the theory will be published in aforthcoming paper [3].

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

For an arbitrary three-dimensional body Ω ⊂ R3, which is assumed to be a bounded region, with the boundary regularity∂Ω ∈ C0,1 (this assumption allows to treat basically every body in engineering applications) and a boundary decompositionof the type ∂Ω = ∂Ω0 ∪ ∂ΩN and ∂Ω0 ∩ ∂ΩN = ∅, with ∂Ω0 6= ∅ relatively open and ∂ΩN relatively open, one canproof that the three-dimensional weak formulation of the mixed-boundary-value problem of linear elasticity (II) has a uniquesolution under very weak assumptions for the regularity of the given data, i.e., the prescribed displacement field u0 on ∂Ω0, theprescribed traction-vector field g on ∂ΩN and the prescribed field of body force f . If we assume that the component functionsof the fourth-order elasticity tensor Eijrs : Ω −→ R are continuous or piecewise constant and fulfill the symmetry relationsEijrs = Ejirs, Eijrs = Eijsr and Eijrs = Ersij , so that in addition the associated 6 × 6-elasticity tensor is symmetricpositive definite for all x ∈ Ω (which is a basic modeling assumption in continuum solid mechanics), we get:

Theorem: Existence and uniqueness of the weak solution of the three-dimensional linear elasticity problemLet u0 ∈

[W 1/2,2(∂Ω0)

]3, g ∈

[W 1/2,2(∂ΩN )

]3, f ∈ [L2(Ω)]

3 and let

B(u, v) :=

∫Ω

Eijrsur|svi|j dV, F (v) :=

∫Ω

fivi dV +

∫∂ΩN

givi dA and Π(u) :=1

2B(u, u)− F (u).

Then there exists an u0 ∈ X that fulfills the weak displacement-boundary condition Su0(x) = u0(x) for almost all x ∈ ∂Ω0

(with respect to the corresponding Lebesgue measure) and the problems

(I) Find: u ∈ X0 : ∀v ∈ X0 \ u : Π(u+ u0) < Π(v + u0)

(II) Find: u ∈ X0∀v ∈ X0 : B(u+ u0, v) = F (v)

are equivalent and have a unique solution.

Here S : W 1,2(Ω) −→ W 1/2,2(∂Ω) denotes the trace operator, X :=[W 1,2(Ω)

]3and X0 := v ∈ X|∀i ∈ 1, 2, 3 :

Svi = 0 on ∂Ω0. Problem (I) corresponds to the "principal of minimal potential energy" in a weak setting, which is usuallythe starting point of the derivation of consistent theories (e.g. in [1]). We use problem (II) as a starting point instead.

We now assume our body to be a plate of constant thickness h. Therefore, let the mid-plane A ⊂ R2 be a bounded regionwith ∂A ∈ C0,1 and Ω :=

(ξ1, ξ2, ξ3) ∈ R3| (ξ1, ξ2) ∈ A, ξ3 ∈

(−h2 ,

h2

). For the boundary decomposition we assume

∂A = Γ0 ∪ ΓN and Γ0 ∩ ΓN = ∅ with Γ0 6= ∅, ΓN relatively open, ∂Ω0 := Γ0 ×(−h2 ,

h2

)and ∂ΩN := ΓN ×

(−h2 ,

h2

)∪

A ×−h2∪ A ×

h2

. Furthermore, we introduce the plate parameter c := h√

12a, where a is a characteristic in-plane

length of the plate, e.g., the diameter of A. The plate parameter is a dimensionless constant that characterizes the relativethickness of the plate and is, therefore, assumed to be small c 1. We use it to introduce a orthogonal polynomial basisbn : R −→ R|n ∈ N ∪ 0 by bn(ξ) :=

√(2n+ 1)cnpn(ξ/

√3c), where pn is the n-th Legendre polynomial. The basis

polynomials fulfill the orthogonality relation∫ h/2a−h/2a b

n(ξ)bm(ξ)dξ = ha c

2nδnm for all n,m ∈ N∪0. On the one hand, thisrelation assures the existence of series-expansions by the theory of abstract Fourier series. On the other hand, the fact that theL2-norm of the basis polynomials decreases very fast for increasing n is in the later the key for the application of the truncationapproach of the uniform-approximation technique (compare [1]). In the following, we assume a homogeneous material, i.e.,Eijkl = const.. Furthermore, we assume that our solution reaches the regularity u ∈

[W 2,2(Ω)

]3and that u0 ∈

[W 2,2(Ω)

]3is given. For the Dirichlet problem (∂ΩN = ∅), the regularity of the solution u can be guarantied by the results of the

∗ Corresponding author: email pasch@mechanik.uni-bremen.de∗∗ email rkienzler@uni-bremen.de

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

274 Section 4: Structural mechanics

regularity theory, if we assume sufficiently smooth data. Since no such results exist for the (in praxis very important) mixedboundary-value problem treated here, we are forced to directly assume the desired regularity. In order to arrive at an equivalenttwo-dimensional formulation, we transform the weak problem (II) from the original coordinates ξ1, ξ2, ξ3 to dimensionlesscoordinates x = x1, y = x2, z = x3 by xi := ξi/a, make intensive use of the specific geometry of the plate, develop u andv into series of type ui(x, y, z) = a

∑∞k=0 u

ki (x, y)bk(z) and use the variational lemma to gain a formulation (III) which is

entirely formulated in the Fourier-coefficients uki ∈ W 2,2(A) of the displacement field u. If we introduce load resultants P liby P li (x, y) :=

√2l + 1

(gi(x, y, h2a

)+ (−1)lgi

(x, y, −h2a

))+ hclf li (x, y), where the f li are the Fourier coefficients of the

prescribed body force fi(x, y, z) =∑∞l=0 f

li (x, y)bl(z), we finally get an exact two-dimensional formulation (III).

Theorem: The equivalent two-dimensional problem for a homogeneous plate with constant thicknessLet u0 ∈

[W 2,2(Ω)

]3, g ∈

[W 1/2,2(∂ΩN )

]3, f ∈ [L2(Ω)]

3, u ∈[W 2,2(Ω)

]3and E be constant. Then:

The problems (I), (II) and (III):Find u ∈ X with: ∀i ∈ 1, 2, 3 ∀l ∈ N ∪ 0 :

f.a.a. (x, y) ∈ A :

a2h

∞∑k=0

[Eiβrαu

kr,αβc

k+l

1 if k = l0 otherwise

− 1

6

√(2k + 1)(2l + 1)Ei3r3u

krck+l−2

k2 + k if k + l even and 0 ≤ k ≤ ll2 + l if k + l even and l < k

0 if k + l odd

+

√(2k + 1)(2l + 1)

3ck+l−1

−Ei3rαukr,α if k + l odd and 0 ≤ k ≤ lEiβr3u

kr,β if k + l odd and l < k

0 if k + l even

= −a3clP li .

f.a.a. (x, y) ∈ Γ0 : c2luli =1

h

∫ h/2a

−h/2au0ib

l dz

f.a.a. (x, y) ∈ ΓN : Eiαrβnαc2lulr,β + Eiαr3nα

√2l + 1

3

∞∑n=0

√2l + 4n+ 3c2l+2nul+2n+1

r =a

h

∫ h/2a

−h/2agib

l dz

are equivalent and have a unique solution.This exact representation is not treatable in practice because it has countably many field equations for countably many

unknown displacement coefficients uki . Since the field equations of (III) and also the force boundary conditions are powerseries in the squared plate parameter c2, the truncation approach of the uniform-approximation technique is to neglect all termswith factors of powers of c2 higher than a certain truncation-power. We decide to drop all terms of magnitude O(c6), whatmakes the resulting theory a so-called second-order theory. The resulting PDE system consists of 18 PDEs for 18 unknowndisplacement coefficients. If we insert the material law for a monotropic material with respect to the mid-plane, the systemdecouples into two systems of 9 second-order PDEs for 9 displacement coefficients. One problem can be identified to be theplate problem, while the other one is the disk problem. Monotropy is the most general material model leading to a decoupledplate problem. The PDE system of the plate problem can finally be further (pseudo)-reduced to a system of two coupled PDEsin the mid-plane displacement w := u0

3 and ψ := u12,1 − u1

1,2, a measure for the shear influence:

c2D41w + c4D6

2w + c4D43ψ = P 0

3 + c2D24P

03 + c2D2

5P23 +O(c6), c4D4

3w + c2D06ψ + c4D2

7ψ = 0 +O(c6),

where eachDni is some linear operator of differentials of the n-th orderDn

i (•) =∑nj=0 aij

∂n(•)∂xn−j∂yj and each aij is a constant

only depending on the material parameters Eklrs. The specific PDEs and the corresponding boundary conditions are omittedhere due to page limitations. For details we refer to [2] and [3]. For the special case of an isotropic material the equationsabove read

K∆∆w = a3

(P 0

3 +c2

1− ν

[2

5(ν − 6)∆P 0

3 +ν√5

∆P 23

])+O(c6), c2

(ψ − 6

5c2∆ψ

)= 0 +O(c6).

One can show that these PDEs are equivalent to Kirchhoff’s PDE as a first-order theory and, furthermore, equivalent toR. Kienzler’s theory as a second-order theory, which implies (see [1]) further equivalences to established shear-deformabletheories, especially the Reissner-Mindlin theory and Zhilin’s plate theory.

References[1] R. Kienzler, On consistent plate theories, Archive of Applied Mechanics 72, 229–247 (2002).[2] P. Schneider, Eine konsistente Plattentheorie zweiter Ordnung für monotropes Material, diploma thesis, Universität Bremen, 2010.[3] P. Schneider, R. Kienzler, M. Böhm, Modeling of consistent second-order plate theories for monotropic materials, in review for

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