Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 3, pp. 755-768, Warsaw 2012
50th Anniversary of JTAM

CONSISTENT THEORIES OF ISOTROPIC AND ANISOTROPIC PLATES

Reinhold Kienzler, Patrick Schneider
Bremen Institute of Mechanical Engineering (BIME),

University of Bremen, Bremen, Germany

e-mail: rkienzler@uni-bremen.de; pasch@mechanik.uni-bremen.de

In this paper, the uniform-approximation technique in combination with the pseudo-
reduction technique is applied inorder toderiveconsistent theories for isotropicandanisotro-
pic plates. The approach is used to assess and validate the plate theories already established
in the literature. Further lines of research are indicated.

Key words: linear elasticity, consistent plate theories, uniform-approximation technique,
pseudo-reduction technique

1. Introduction

Plates belong to the class of thin plane structures. The two in-plane dimensions, characterized
by a scaling length parameter a, aremuch larger than the out-of-plane dimension, characterized
by a thickness parameter h

h

a
≪ 1 (1.1)

“Much smaller” than unity means in an engineering sense h/a < 1/10. In aircraft engineering
1/100 is usual; and in aerospace engineering 1/1000 is not unusual. But also structural elements
with a thickness-to-length ratio of 1/5 may still be considered as plates. Commonly, thin plane
structures loaded by in-plane forces are designated as discs, those loaded by out-of-plane forces
as plates. Generally, the disc and the plate problem are coupled.
It is convenient to introduce aplanemiddle surface,whichbisects thedisc or plate continuum

transversely. If the material under consideration exhibits at least a symmetry with respect to
one plane, i.e., monotropic material behavior, and if the material symmetry plane coincides
with the geometrical symmetry plane of the plate, i.e., the middle surface, the plate and the
disc problem are decoupled within the linear theory of elasticity and can, therefore, be treated
separately. In this contribution, wewill concentrate on thismost general case in linear elasticity:
a plate theory for monotropic materials, which are of increasing interest for, e.g., devices used
in microelectronics (generally, “smart materials”) built of single crystals.
Plate theories have, of course, a long and rich history. Interesting accounts on this subject

and further referencesmaybe found, e.g., inTodhunter andPearson (1960); Timoshenko (1983);
Szabó (1987).
Roughly speaking, aplate theory is the attempt tomodel the three-dimensional behavior of a

plate continuum by quantities that “live” on a plane surface. Thus plate theories are inherently
approximative. The development of plate theories may be classified into three branches. The
“engineering” branch relies on the application of the both admired and feared a priori assump-
tions. It depends on the intuition of engineers to construct a sound theory based on a set of
reasonable a priori assumptions. It is needless to say that this approach can not be systemized
and is prone to errors.



756 R. Kienzler, P. Schneider

The secondbranch is the so-called “direct approach”.Based on theCosserat theory, the plate
is modeled as a deformable initially plane surface with a set of deformable directors attached
to each point of the plane. In the special case with one director, the unknown functions are
the vector of displacements of the middle surface and the vector describing the deformations of
the director. Thus, such a theory contains six degrees of freedom at every point in the plane,
and as a consequence, six boundary conditions have to be imposed. The main problem in the
application of the direct approach consists in the establishment as well as in the identification of
the constitutive equations.A recent review of the direct approachwith an extended bibliography
is given in Altenbach et al. (2010).
The third branch, which will be followed here, is the “consistent approach”. All quantities

of interest, i.e., deformations, strains, stresses and loads are developed into series in the thick-
ness direction with respect to a suitable basis. The basis might bemonomials, scaled Legendre
polynomials or trigonometric functions. The series expansions can be truncated at different
orders, giving rise to “hierarchical” plate theories. In the following, wewill outline the “uniform-
approximation technique” in combination with a “pseudo reduction” of the resulting partial
differential equation system. Within the paper, we restrict ourselves for reasons of simplicity
and clarity to plates with constant thickness and homogeneous materials. The resulting plate
theories of different orders will be compared with already established theories.
Finally, the “asymptotic”method should bementioned, which can be used either within the

direct approach orwithin the consistent approach (cf., e.g., Goldenveizer et al. (1993)). Two sets
of governing plate equations are developed, one for the “interior” of the plate and the other one
for the “boundary layer” in combination with an asymptotic matching technique. Especially for
dynamic problems, where the characteristic in-plane dimension, i.e., the wave length λ, is much
smaller than the plane extension a, thismethod supplies accurate results and allows for reliable
error estimations (see, e.g., several contributions in Kienzler et al., 2004).

2. Uniform-approximation technique

Let us consider a plate continuum embedded in a cartesian coordinate system xi as depicted in
Fig. 1.

Fig. 1. Plate continuum

The plane x3 = 0 coincides with the plate middle surface, whereas x3 = −h/2 and
x3 =+h/2 are the upper and lower faces of the plate, respectively. The plate continuummay be
loaded by vertical tractions on the upper and lower face (represented by P− and P+ in Fig. 1,
respectively) and by vertical body forces f (not shown in Fig. 1).
With the characteristic in-plane dimension a, we introduce dimensionless coordinates as

ξα =
xα
a

ζ =
x3
a

α =1,2 (2.1)



Consistent theories of isotropic and anisotropic plates 757

and partial derivatives as

∂(·)

∂x1
=
1

a

∂(·)

∂ξ1
=
1

a
(·),1=

1

a
(·)′

∂(·)

∂x2
=
1

a

∂(·)

∂ξ2
=
1

a
(·),2=

1

a
(·)•

(2.2)

The displacements ui = ui(ξ
α,ζ) are expanded into series in the thickness direction ζ with

respect to a suitable basis. Within this contribution, we use a power series expansion, i.e., a
basis of monomic polynominals. One also could use unscaled or scaled Legendre polynomials
(Vekua, 1982; Rodionova et al., 1996; Schneider, 2010; Schneider et al., 2012), or trigonometric
functions. Thus

ui(ξα,ζ)= a
∞∑

ℓ=0

ℓui(ξ
α)ζℓ (2.3)

The index on the upper left-hand side of the generic symbol, here the dimensionless displa-
cement quantities ℓui, indicates the counting order of the series expansion. The displacement
Ansatz is inserted in the kinematical relations

εij =
1

2
(ui,j +uj,i) (2.4)

and by comparing equal coefficients of ζℓ, strain coefficients are introduced by

εij(ξα,ζ)=
∞∑

ℓ=0

ℓεij(ξα)ζ
ℓ (2.5)

In turn, we insert the strains into the strain-energy density W

W =
1

2
Eijkℓεijεkℓ (2.6)

(fourth-rank elasticity tensor Eijkℓ) and introduce the average of the strain energy W over the
height of the plate

W =

+ h
2a∫

−
h

2a

W(ξα,ζ) dζ = W(
ℓεij(ξα)) (2.7)

By means of the constitutive equations, we introduce stress resultants ℓmij as

ℓmij = a
ℓ+1 ∂W

∂ ℓεij
(2.8)

For conservative forces we introduce the average of the potential V of external forces as

V =−

+ h
2a∫

−
h

2a

fu3 dζ −P
+u3

(
+
h

2a

)
+P−u3

(
−
h

2a

)
(2.9)

and the load resultants ℓP as

ℓP =−
∂V

∂ ℓu3
(2.10)



758 R. Kienzler, P. Schneider

Finally, application of the virtual work theorem leads to the Euler-Lagrange equations and
the boundary conditions either written in stress resultants ℓmij (equilibrium conditions) or in
displacement quantities (Navier-Lamé equations). Based on Weierstrass’s approximation the-
orem, it can be shown (Schneider, 2010) that the complete set of equations is an exact repre-
sentation of the equations of the three-dimensional linear theory of elasticity.

The question arises, however, where to truncate the infinite series in order to obtain a trac-
table set of partial differential equations (PDEs).When considering (2.7) and (2.9), integration
over the thickness generates more or less automatically a plate parameter c2 defined as

c2 =
h2

12a2
≪ 1 (2.11)

such that the average of the strain-energy density and the stress resultants can schematically be
ordered by the powers of c2 as

W = Gh
{
(c2)0(·)+(c2)1(·)+(c2)2(·)+ . . .+ c2n(·)+ . . .

}

ℓmij = Gha
ℓ
{
(c2)0(·)+(c2)1(·)+(c2)2(·)+ . . .+c2n(·)+ . . .

} (2.12)

(G is a characteristic measure of the stiffness, e.g., a shear modulus).

It turns out, that the terms in brackets do not differ in the order of magnitude. In order to
obtain a consistent plate theory it is demanded (see, e.g., Naghdi, 1972; Koiter and Simmonds,
1973;Loet al. 1977;Kienzler, 1982;Krätzig, 1989) thatall equationsareapproximateduniformly,
i.e., in a n-th order theory, all termsmultiplied by c2ℓ, ℓ ¬ n, have to be retained in all governing
equations and all terms multiplied by c2m, m > n have to be neglected. This procedure always
leads to countable many partial differential equations for countable many unknowns ℓui. The
resulting PDE system is symmetric and the number of equations coincides with the number
of unknowns. Since there might be easier equivalent PDE systems and since the classical plate
theories are also usually systems of at most two PDEs in two variables, we try to reduce the
number of unknowns andPDEsby an elimination process seeking for the easiest equivalentPDE
system (pseudo reduction).

3. Pseudo-reduction technique

The uniform-approximation argument, which is proposed for the derivation of the plate equ-
ations, was extended (Kienzler, 1982; 2002; 2004) to the requirement that is must be applied
also to all intermediate equations occurring during the elimination process. To this extent we
have to find a set of main variables, as few as possible, preferably one, and a main differential
equation system (the same number of equations as the number of main variables), which are
entirely formulated in main variables (for more details see Schneider and Kienzler, 2011). In
addition, one has to find a set of reduction differential equations, inwhich all non-main variables
are expressed in terms of themain variables. The original PDE systemmust be identically solved
by inserting the reduction differential equations, if themain variables are a solution to themain
differential equation system, i.e., the reduced PDE system and the original one are equivalent.

Since the PDEs are actually truncated power series in the characteristic parameter c2, mul-
tiplication or division by c2 would change the accuracy of the given equations. Therefore, every
product of different powers of the characteristic parameter with the same displacement coeffi-
cient has to be treated formally as an independent variable. This leads to an underdetermined
system of PDEs. The missing equations are generated by multiplying the original PDEs by c2

and neglecting again ensuing terms of c2m, m > n.



Consistent theories of isotropic and anisotropic plates 759

Thisprocedure can be followed “byhand”.For convenience, an algorithm for the automatisa-
tion of the pseudo reduction for PDE-systems arising from theuniform-approximation technique
has been implemented (Schneider and Kienzler, 2011).

4. Isotropic plates

Since theplate and thedisc problemaredecoupled (for at leastmonotropic plates), it is sufficient
to consider plate-displacement quantities only. Thus (2.3) becomes

uα = a(ψα+
3ψαζ

3+ 5ψαζ
5+ . . .)

u3 = a(w+
2wζ2+ 4wζ4+ . . .)

(4.1)

The classical deformation quantities, the displacement of the middle surface w and its rota-
tions ψα are not indexed by the counting index ℓ.
Working through the algebra indicated in paragraph 2, we arrive at a PDE system formally

sketched in Fig. 2. The symbolic operators Dn, which will be used also in the following, are of
the form

Dn =
n∑

i=0

bi
∂n

∂ξi1∂ξ
n−i
2

(4.2)

with coefficients bi which depend on elastic constants.
In an n-th order approximationwehave tobe sure, that all termsmultiplied by c2ℓ, ℓ ¬ nare

retained. For the series expansion of the displacements, it is important that enough coefficients
are considered. This is indicated in Fig. 2 by the bold lines for the zeroth-, first- and second-
order approximation. For a consistent second-order approximation, e.g., we have to take nine
displacement coefficients into account. From Fig. 2 it becomes obvious that it does not make
sense to add terms in the displacement series expansion in an arbitrary manner (Wang et al.,
2000). Either one uses 3(n +1) coefficients in the expansion or one does not get a consistent
n-th order approximation.
In the next step, we neglect in the 3(n +1)× 3(n +1) PDE system all terms that are

multiplied with c2m, m > n. The resulting system reveals a triangular structure.
In order to be a little more specific, the PDE operators for a second-order approximation,

for isotropic material behavior, are given explicitly in Fig. 3, with the following abbreviations

a1 =2
1−ν

1−2ν
a2 =

2ν

1−2ν
a3 =

1

1−2ν

a4 =
1−6ν

1−2ν
a5 =

1−10ν

1−2ν
a6 =

3−10ν

1−2ν

∆ =(·)′′+(·)•• ∆1 = a1(·)
′′+(·)•• ∆2 =(·)

′′+a1(·)
••

(4.3)

The right-hand sides are given for the special case where the external forces are applied only
through tractions P+ and P− at the lower and upper plate, respectively (see Fig. 1).With this
assumption, the theorywill not distinguish between upper- and lower-face load applications.We
introduce the resultant load per unit of area as

P = P++P− (4.4)

and the load resultants defined in (2.10) follow to be 0P = P, 2P =3P, 4P =9P.
The symbol G is now the shear modulus and is given for isotropic materials in terms of

Young’s modulus E and Poisson’s ratio ν as

G =
E

2(1+ν)
(4.5)



7
6
0

R
.
K
ien
zler,
P
.
S
ch
n
eid
er

w ψ1 ψ2
2w 3ψ1

3ψ2
4w 5ψ1

5ψ1
6w 7ψ1

7ψ2 · · · RHS

δw D2 D1 D1 c
2D2 c

2D1 c
2D1 c

4D2 c
4D1 c

4D1 c
6D2 c

6D1 c
6D1 · · ·

0Pc0

δψ1 1+ c
2D2 c

2D2 c
2D1 c

2+ c4D2 c
4D2 c

4D1 c
4+ c6D2 c

6D2 c
6D1 c

6+ c8D2 c
8D2 · · · 0

δψ2 1+ c
2D2 c

2D1 c
4D2 c

2+ c4D2 c
4D1 c

6D2 c
4+ c6D2 c

6D1 c
8D2 c

6+ c8D2 · · · 0

δ 2w c2+ c4D2 c
4D1 c

4D1 c
4+ c6D2 c

6D1 c
6D1 c

6+ c8D2 c
8D1 c

8D1 · · ·
2Pc2

δ 3ψ1 c
4+ c6D2 c

6D2 c
6D1 c

4+ c6D2 c
6D2 c

6D1 c
6+ c8D2 c

8D2 · · · 0

δ 3ψ2 c
4+ c6D2 c

6D1 c
8D2 c

6+ c8D2 c
8D1 c

10D2 c
8+ c10D2 · · · 0

δ4w c6+c8D2 c
8D1 c

8D1 c
8+ c10D2 c

10D1 c
10D1 · · ·

4Pc4

δ5ψ1 symm. c
8+c10D2 c

10D2 c
10D1 c

10+ c12D2 c
12D2 · · · 0

δ5ψ2 c
8+ c10D2 c

10D1 c
12D2 c

10+ c12D2 · · · 0

δ6w c10+ c12D2 c
12D1 c

12D1 · · · 6Pc6

δ7ψ1 c
12+ c14D2 c

14D2 · · · 0

δ7ψ2 c
12+ c14D2 · · · 0

... · · ·
...

Fig. 2. Schematic sketch of the PDE system



C
o
n
siste
n
t
th
eo
r
ie
s
o
f
iso
tro
p
ic
a
n
d
a
n
iso
tro
p
ic
p
la
te
s

7
6
1

w ψ1 ψ2
2w 3ψ1

3ψ2
4w 5ψ1

5ψ2 RHS

δw ∆ (·)′ (·)• c2∆ 3c2(·)′ 3c2(·)•
9

5
c4∆ 9c4(·)′ 9c4(·)• −

a

Gh
P

δψ1 (·)
′ 1− c2∆1 −c

2a3(·)
′• c2a4(·)

′ 3c2−
9

5
c4∆1 −

9

5
c4a3(·)

′•
9

5
c4a5(·)

′ 9c4 0 0

δψ2 (·)
• −c2a3(·)

′• 1−c2∆2 c
2a4(·)• −

9

5
c4a3(·)

′• 3c2−
9

5
c4∆2

9

5
c4a5(·)

• 0 9c4 0

δ2w c2∆ c2a4(·)
′ c2a4(·)

• −c24a1+
9

5
c4∆

9

5
c4a6(·)

′
9

5
c4a6(·)

• −
72

5
c4a1 0 0 −3c

2 a

Gh
P

δ 3ψ1 3c
2(·)′ 3c2−

9

5
c4∆1 −

9

5
c4a3(·)

′•
9

5
c4a6(·)

′
81

5
c4 0 0 0 0 0

δ 3ψ2 3c
2(·)• −

9

5
c4a3(·)

′• 3c2−
9

5
c4∆2

9

5
c4a6(·)

• 0
81

5
c4 0 0 0 0

δ4w
9

5
c4∆

9

5
c4a5(·)

′
9

5
c4a5(·)

• −
72

5
c4a1 0 0 0 0 0 −9c

4 a

Gh
P

δ5ψ1 9c
4(·)′ 9c4 0 0 0 0 0 0 0 0

δ5ψ2 9c
4(·)• 0 9c4 0 0 0 0 0 0 0

Fig. 3. PDE system for a consistent second-order approximation



762 R. Kienzler, P. Schneider

Zeroth-order approximation

Retaining only terms without any c2-factor, we obtain as the PDE system

∆w+ψ′1+ψ
•

2 +O(c
2)=−

Pa

Gh

w′+ψ1+O(c
2)= 0 w•+ψ2+O(c

2)= 0
(4.6)

It becomes obvious that the PDE system is only satisfied, if P itself is of the order c2

P = O(c2) (4.7)

and the rigid body transformations

w = α0+α1ξ1+a2ξ2

ψ1 =−α1 ψ2 =−α2 α0,α1,α2 = const
(4.8)

are solutions to the system (4.6). If the plate is rigidly supported, we have the trivial solution

w ≡ 0 (4.9)

First-order approximation

We neglect all terms multiplied by c4 in the table in Fig. 3, choose w as the main variable
and

∆w+ψ′1+ψ
•

2 + c
2(∆2w+33ψ′1+3

3ψ•2)+O(c
4)=−

Pa

Gh
(4.10)

as the main PDE. The five additional equations of Fig. 3 have to be used for the reduction

w′+ψ1+ c
2(−a1ψ

′′

1 −ψ
••

2 −a3ψ
′•

2 +a4
2w′+33ψ1)+O(c

4)= 0 (a)

w•+ψ2+ c
2(−a3ψ

′•

1 −ψ
′′

2 −a1ψ
••

2 +a4
2w•+33ψ2)+O(c

4)= 0 (b)

c2(∆w+a4(ψ
′

1+ψ
•

2)−4a1
2w)+0(c4)= O(c4) (c)

c2(3w′+3ψ1)+O(c
4)= 0 (d)

c2(3w•+3ψ2)+O(c
4)= 0 (e)

(4.11)

Starting the pseudo reduction, we immediately obtain from (4.11)(d),(e)

(d) : c2ψ1 =−c
2w′+O(c4)

(e) : c2ψ2 =−c
2w•+O(c4)

(4.12)

It follows that the deviation fromKirchhoff’s normal hypothesis is of O(c4). Inserting (4.12)
into (4.11)(c) leads to

(c) : c2 2w = c2
ν

2(1−ν)
∆w+O(c4) (4.13)

Hence, also the deviation from plane stress is of O(c4), i.e., the influence of σ33 on the
equations of the first-order approximation is of O(c4) and thus negligible. Note, however, that
the transverse displacement u3 (see Eq. (4.1)) is not constant over the plate thickness

u3 = a
(
w+ ζ2

ν

2(1−ν)
∆w
)
6= u3(ξ1,ξ2) ε33 6=0 (4.14)



Consistent theories of isotropic and anisotropic plates 763

Proceeding further, we obtain by insertion of (4.12) and (4.13) into (4.11)(a),(b)

(a) : ψ1+3c
2 3ψ1 =−w

′− c2
4+v

2(1−ν)
∆w′+O(c4)

(b) : ψ2+3c
2 3ψ2 =−w

•− c2
4+ν

2(1−ν)
∆w•+O(c4)

(4.15)

Inserting (4.13) and (4.15) into the main PDE (4.10) results in the classical Kirchhoff-plate
PDE

∆∆w =
Pa3

K
with K =

Eh3

12(1−ν2)
(4.16)

On inspection (Kienzler, 2002), all stress resultants and also the boundary conditions involving
Kirchhoff’s “Ersatz-shear forces” are recovered. ThusKirchhoff’s plate theory turns out to be a
fully consistent first-order approximation of the three-dimensional linear theory of elasticity! In
our approach, a priori hypotheses are not necessary since the plane stress andnormal hypotheses
area posteriori results. Itmay furtherbenoted that thevariables 3ψ1 and

3ψ2 arenotfixedbythe
pseudo-reduction procedure. Neither the result for the governing PDE nor the stress resultants
and boundary conditions are effected by the specific choice of 3ψα. Theymay arbitrarily be set
equal to zero. If we choose instead

3ψα =
1

2

2−ν

1−ν
∆w,α (4.17)

a parabolic shear-stress distribution can be embedded into the theory a posteriori, e.g.

τα3 =
G∆w,α
1−ν

(ζ2−3c2)=
3

2

Qα
h

(
1−
4x23
h2

)
(4.18)

(transverse shear resultant Qα). This is the classical “Dübel formula”.

Thus, assuming a priori distributions of shear stresses τα3 or normal stresses σ33 over the
thickness of the plate are not necessary for the establishment of a consistent theory, theymight
even be contra productive.

Finally, we may note that a linear representation of the displacements with respect to the
thickness direction involvingmerely w and ψα cannotnot lead to a theorywhich is in accordance
with the consistent approximation without posing additionally a priori assumptions. A linear
approximation would lead to the following result

∆∆w =
Pa3

K

1−2ν

(1−ν)2
(4.19)

Equation (4.19) coincides with Vekua’s (Vekua, 1982) approximation of the order N = 1.
Vekua has established a hierarchical structure of plate (and shell) theories, counting the order of
the approximation differently as the order of the polynomials considered in the series expansion.
His second-order approximation leads to the correct result (4.16). Higher-order theories are only
sketched and are not worked out explicitly.

For the following, we observe that the quantity

c2ψ := c2(ψ′2−ψ
•

1)= c
2 rot ~ψ = O(c4) (4.20)

is of the order c4. ψ may be regarded as a measure of the transverse-shear deformation.



764 R. Kienzler, P. Schneider

Second-order approximation

For the second-order approximation, we choose w and ψ (4.20) as independent variables.
The pseudo-reduction procedure of the full PDE-system shown in Fig. 3 has been dealt with
in detail in Kienzler (2002, 2004), Bose and Kienzler (2006) and will not be repeated here. We
restrict ourselves to the discussion of the resultingmain PDEs after pseudo reduction

K∆∆w = a3
(
P −

3

10

8−3ν

1−ν
c2∆P

)
c2
(
ψ−
3

2
c2∆ψ

)
=0 (4.21)

Note thatno shear correction factor has been introduced.The stress resultants andboundary
conditions are given explicitly in, e.g., Kienzler (2004). As in the first-order approximation, the
second-order approach contains displacement parameters which are not fixed by the reduced go-
verning equations (4w, 5ψα, . . .). Theyprovide enough freedomto fulfill the boundary conditions
at the upper and lower faces of the plate a posteriori

τα3
(
ζ =±

h

2a

)
=0 σ33

(
ζ =±

h

2a

)
=

{
+P+

−P−
(4.22)

The existing theories for shear deformable plates deliver similarPDEs.For instance,Reissner
(1945) developed a theory based on several a priori assumptions and arrived finally at the PDE

∆∆wR =
a3

K

(
P −

1

ΓR

2−ν

1−ν
c2∆P

)
ΓR =

5

6
(4.23)

which is quite similar to (4.21) (shear correction factor ΓR introduced by Reissner). It has to
be observed, however, that the transverse displacement wR is differently defined than w used
here, namely by

QαwR =

+h
2∫

−
h

2

τα3u3 dx3 (4.24)

It has been shown (Kienzler, 2002) that wR as given in (4.24) is related to w by

wR = w+ c
2 3ν

10(1−ν)
∆w+O(c4) (4.25)

Inserting (4.25) and (4.23)(b) into (4.23)(a), equation (4.21)(a) is recovered. Thus Reissner’s
plate equation (4.23) is equivalent to (4.21) within the second-order approximation.
Zhilin (1992) advanced a similar plate equation as

∆∆wZ =
a3

K

[
P −

2

1−ν

( 1
ΓZ
−
ν

2

)
c2∆P

)
ΓZ =

5

6−ν
(4.26)

(shear correction factor ΓZ introduced, by Zhilin), whereas his transverse displacement is defi-
ned as

hwZ =
1

a

+h
2∫

−
h

2

u3 dx3 (4.27)

It follows with (Kienzler, 2004) that

wZ = w+ c
2 ν

2(1−ν)
∆w+O(c4) (4.28)



Consistent theories of isotropic and anisotropic plates 765

and again, Zhilin’s theory is equivalent to (4.21) and is thus a consistent second-order plate
theory. Reissner’s and Zhilin’s engineering intuition is admirable by all means.
Ambartsumyan (1970) concentrates in his work on anisotropic plates. Using his equations

within the framework of the uniform-approximation technique and introducing the special case
of isotropy we arrive at (see a forthcoming report)

∆∆wA =
a3

K

(
P −

3

10

8−3ν

1−ν

12−6ν

12−7ν
c2∆P

)
wA = w (4.29)

The difference in comparison to (4.21) is quite small since the factor α =(12−6ν)/(12−7ν)
is not far from unity (ν =0 : α =1; ν =0.5 : α =1.06), at least for isotropy. For anisotropic
plates, the difference might bemore pronounced.
The reason for Ambartsumyan’s approach being not consistent is that his modeling does

not pay respect to all equations of linear elasticity. His displacement Ansatz, which is reverse-
ly constructed from a priori assumptions concerning the distribution of shear stresses, results
in an intrinsic over determination of the system of the equations of linear elasticity. This is
compensated very tricky by a semi-inversion of Hooke’s law (Ambartsumyan, 1970, Eq. 2.2.3-
-2.2.7), which assumes σ33 to be given. In fact, since Hooke’s law for the stresses σ13 and σ23
is decoupled from the other stresses even for a monotropic material (cf. Eq. (5.2)), these two
stresses can be computed in accordance with the semi-inversion without the knowledge of σ33.
This determines σ33 afterwards by the corresponding equilibrium condition, which in turn de-
termines all other stresses by the semi-inverted Hooke’s law. However, Hooke’s law for σ33 is
not at all taken into account by this approach and is violated by terms of order O(c4).

5. Monotropic plates

When treatingmaterials with anisotropic behavior, it is common to introduce a “vector-matrix”
notation for the generalized Hooke’s law (cf., e.g., Altenbach et al., 1998)




σ11
σ22
σ33
σ12
σ23
σ31




=




E11 E12 E13 E14 E15 E16
E22 E23 E24 E25 E26

E33 E34 E35 E36
sym. E44 E45 E46

E55 E56
E66







ε11
ε22
ε33
2ε12
2ε23
2ε31




=




E1111 E1122 E1133 E1112 E1123 E1131
E2222 E2233 E2212 E2223 E2231

E3333 E3312 E3323 E3331
sym. E1212 E1223 E1231

E2323 E2331
E3131







ε11
ε22
ε33
2ε12
2ε23
2ε31




(5.1)

Equation (5.1) provides a unique correlation between the “matrix” coefficients EAB
(A,B = 1,2, . . . ,6) and the tensor coefficients Eijkℓ (i,j,k,ℓ = 1,2,3), which is necessary if
coordinate transformations have to be carried out. If the plane of material symmetry coincides
with the x3 =0 plane, some of the coefficients have to vanish yielding a reducedmatrix

[EAB] =




E11 E12 E13 E14 0 0
E22 E23 E24 0 0

E33 E34 0 0
sym. E44 0 0

E55 E56
E66




(5.2)



766 R. Kienzler, P. Schneider

The consistent first-order approximation leads after pseudo reduction to a Kirchhoff-type PDE
(see (4.16)(a)) in w as (Schneider, 2010; Schneider et al., 2012)

c2
{
(E33E11−E

2
13)w

IV +4(E14E33−E34E13)w
′′′•

+2[E12E33−E13E23+2(E44E33−E
2
34)]w

′′••

+4(E24E33−E34E23)w
′••• +(E33E22−E

2
23)w

••••

}
= E33

a

h
P

(5.3)

This equation is attributed to Huber (1929).

For orthotropic material (E14 = E24 = E34 = E56 = 0), equation (5.3) contains only even
derivatives in ξ1 and ξ2. If thematerial is transversally isotropicwith x3 as the axis of rotational
symmetry of the material, (5.3) reduces to

∆∆w =
Pa3

c2
(
E11−

E2
13

E33

) =:
Pa3

K̂
(5.4)

Assuming isotropic material behaviour we finally arrive at (4.16).

A check of the literature reveals that all anisotropic first-order plate theories (at least known
to the authors) are consistent, since a linear displacementAnsatzwith the usual set ofKirchhoff-
type a priori hypotheses is equivalent to the consistent first-order approximation.

In a symbolic notation (see (4.2)), equation (5.3) may be written as

c2D4w =
a

h
P (5.5)

Note that the coefficients bi of the symbolic notation (4.2) are given explicitly in (5.3).

The second-order approximation combined with the pseudo-reduction procedure is complex
and involvesmanyalgebraicmanipulations.Finally, in symbolic notation (4.2), the reducedmain
PDE system is given as

(c2D4+c
4D6)w+ c

4D̃4ψ =
a

h
(1+ c2D̃2)P

c4D̃4w+(c
2D0+ c

4D2)ψ =0
(5.6)

The operators with tilde differ from the operators without tilde only in that the constant
coefficients bi and b̃i are different. In Schneider (2010), the coefficients are given in detail. To
our best knowledge, (5.6) is the only consistent second-order approximation for monotropic
plates. The coefficients of the symbolic operators in (5.6) have a complicated but well defined
appearance depending on EAB. As soon as a specificmaterial is chosen they are just “numbers”.
Within the framework of the consistent second-order approximation, the differential equation
problem is of the sixth-order in ξ1 and ξ2 giving rise to three boundary conditions at each
boundary.

Again, for orthotropicmaterials, theoperators Dn contain only evenderivatives in ξ1 and ξ2.
For transversally isotropic materials, the PDE system (5.6) becomes uncoupled and resembles
system (4.21). For isotropic materials, (4.21) is recovered.

6. Concluding remarks

The messages from the consistent n-th order plate theory in combination with the pseudo-
reduction technique are as follows:



Consistent theories of isotropic and anisotropic plates 767

• If you want to apply a refined theory, i.e., an improved Kirchhoff-plate theory, including
shear deformation, warping of the cross section and transverse normal stress effects, use
one of the consistent second-order approximations.

• There is no way to improve a consistent plate theory by adding in an unsystematic way
single terms in the displacement Ansatz or by assuming specific stress distributions in the
thickness direction.

• If youwant to improve the consistent second-order theory (thismightbeuseful for dynamic
plate problems, where the characteristic in-plane length becomes the wave length λ ≪ a)
use a consistent third-order approach with all terms of O(c6) included.

The uniform-approximation techniquewas quite helpful to derivematerial conservation laws
within a consistent-plate theory (Bose and Kienzler, 2006). It has also been applied to shells
(Kienzler, 1982).

Recently, the same technique has beenapplied to beams.At first glance, it is astonishing that
the beam theory is more advanced than plate theory. The reason is that the series expansion
has to be performed in two directions and two parameters have to be considered, a height and a
width parameter, which results in amuchmore complex pseudo-reduction problem. The results
of this investigation will be dealt with in a forthcoming paper.

The proposed plate theorymay be extended to introduce externalmoment loadings, variable
plate thickness and geometrically non-linear effects. This is the subject of ongoing research.

References

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Berechnung von Scheiben und Platten, Springer, Berlin

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of plates and shells: a short review and bibliography,Archive of Applied Mechanics, 80, 73-92

3. Ambartsumyan S.A., 1970,Theory of Anisotropic Plates, Progress inMaterial Science Series II,
Technonic, Stamford, Conn.

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Spójne teorie izotropowych i anizotropowych płyt

Streszczenie

Wpracy omówionometodę jednorodnej aproksymacji połączonej z techniką pseudo-redukcji zastoso-
wanądo sformułowania spójnych teorii płyt izotropowych i anizotropowych.Zaprezentowanąmetodologię
wykorzystano do oceny i weryfikacji teorii już istniejących i dobrze znanych z literatury.Wskazano także
dalsze kierunki badań.

Manuscript received February 20, 2012; accepted for print March 26, 2012