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Second-order linear plate theories: Partial differential equations, stress resultants and displacements
Accepted Manuscript
Second-order linear plate theories: Partial differential equations, stress resultants and displacements R. Kienzler , P. Schneider PII: DOI: Reference:
S0020-7683(17)30004-5 10.1016/j.ijsolstr.2017.01.004 SAS 9418
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
17 October 2016 1 December 2016 3 January 2017
Please cite this article as: R. Kienzler , P. Schneider , Second-order linear plate theories: Partial differential equations, stress resultants and displacements, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.01.004
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ACCEPTED MANUSCRIPT Second-order linear plate theories: Partial differential equations, stress resultants and displacements R. Kienzler, P. Schneider Bremen Institute of Mechanical Engineering (bime) University of Bremen, Department of Production Engineering Am Biologischen Garten 2, 28359 Bremen, Germany [email protected], [email protected]
uniform-approximation technique, pseudo-reduction approach, consistent plate theory, second-order theory, stress resultants, displacements
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Keywords
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Abstract By combining the uniform-approximation technique with the pseudo-reduction approach, a consistent second-order plate theory is developed with recourse to neither kinematical assumptions nor to shear-correction factors by truncation of the elastic energy. The governing partial differential equations and the expressions for the stress resultants are compared with those of other authors. Free coefficients of the resulting displacement “ansatz” are determined a posteriori, in order to satisfy three-dimensional boundary conditions and local equilibrium equations.
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1 Introduction Plates are thin plane structures loaded transversally to their midplanes. The characteristic thickness dimension h is much smaller than the characteristic in-plane dimension a , h a . Plate theories are inherently approximative in that they attempt to describe the actual threedimensional plate continuum by quantities that are defined on a surface. Generally, three different categories of derivation techniques for plate theories may be distinguished. The classical or engineering approach starts with a set of kinematical a priori assumptions for the stress and displacement distributions in thickness direction. Either transvers shear strains are neglected, or their influence is considered by the introduction of shear-correction factors. A historical survey on classical plate theories may be found, e.g., in [1].
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Following the direct approach, all quantities “live” on a Cosserat-type surface endowed with a set of deformable directors attached at each point of the plane. Despite of the mathematical elegance, the main drawback of the method lies in the problem of establishing constitutive relations. Material parameters are identified by comparisons with a set of solutions of known test problems. The choice of test problems has a crucial influence on the resulting theory. An excellent overview over the theories relying on the direct approach is given in [2] with an extended bibliography. Within the present paper, we will follow the consistent approach introduced in the pioneering treatise [3] by Naghdi. Starting from the three-dimensional theory of elasticity, the approach uses abstract Fourier-series expansions in thickness direction with respect to a suitable basis to achieve a dimension reduction. The basis might be monomials, scaled Legendre polynomials or trigonometric functions. After introducing proper non-dimensional quantities and integrating over the plate thickness, the dimensionless plate parameter c2 h2 (12a 2 ) evolves quite naturally, which is a small quantity in plate theories. The elastic potential, i.e., the sum of the strain-energy and the potential of external forces, both per unit of plate area, appear as power series in this plate parameter. This infinite series can be truncated at different
ACCEPTED MANUSCRIPT orders, giving rise to hierarchical plate theories. The idea of the uniform-approximation technique for an Nth-order theory is to consider all terms multiplied by c 2 n , n N and omit all terms of the order O(c 2 m ), m N . The associated Euler-Lagrange equations of a variational problem corresponding to the Navier-Lamé formulation result in 3( N 1) partial differential equations (PDEs) for 3( N 1) unknown displacement coefficients “living” on the two-dimensional midplane of the plate, cf. [4].
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It has been shown [5] that the infinite two-dimensional PDE system ( N ) corresponding to the untruncated elastic potential is equivalent to the three-dimensional equilibrium equations of linear elasticity and in a forthcoming paper, we will give an a priori estimate for the approximation error that results from truncating the energy series.
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During the pseudo-reduction of the PDE system, i.e., elimination of unknowns, terms of the order O(c 2 m ), m N , i.e., O(c 2( N 1) ) are consequently also neglected (pseudo-reduction method, for details see [6]). The procedure results in plate-differential equations which are derived without any recourse to either kinematical a-priori assumption or shear-correction factors. We compare the plate-differential equations and the resulting stress resultants with those of other authors. As it turns out, not all displacement coefficients are determined uniquely by the reduction equations. We use this freedom to satisfy the three-dimensional boundary conditions for the stresses at the upper and lower plate faces and the local stress-equilibrium conditions within the plate continuum.
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For the sake of simplicity, we treat plates with constant thickness of homogeneous isotropic materials. In the conclusion, we give some indications for plates of anisotropic materials.
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2 Uniform approximation technique Let us assume that the midplane of the plate occupies a bounded domain A 2 of the Cartesian ( x1 , x2 ) -plane, whereas the plate continuum extends by h 2 in the x3 -direction. The plate thickness h is much smaller than the characteristic in-plane dimension a . The loads P and P are applied at the upper and lower faces, respectively, and point in x3 direction as indicated in Fig. 1. Volume forces f in x3 -direction, e.g., due to gravity, may also be applied. A further specification of the loading conditions will be discussed later.
Fig. 1: Plate continuum We introduce the dimensionless coordinates
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x x , 3 , 1, 2 , a a
(1)
and differentiation with respect to these dimensionless coordinates by the comma notation () () a (), , 1, 2 . x
(2)
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Since due to isotropy, the plate (transverse loading) and disc (in-plane loading) problems are decoupled, we restrict ourselves to plate-like displacements, i.e., displacements that are odd functions with respect to in the in-plane component functions and even functions in the outof-plane component function, cf. [7], and make use of the MacLaurin-series expansion (i.e., we choose a basis of mononic polynomials. Legendre polynomials will be discussed later).
u a 1 3 3 5 5 ... ,
(3)
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u3 a 0 w 2 w 2 4 w 4 ... .
The displacement ansatz is inserted into the kinematic relations
ij
1 ui , j u j ,i 0 ij 1 ij 2 ij 2 ..., 2
(4)
1 Eijk ij k 2
(5)
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W
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and in turn, the strain-energy density per unit of volume
W , d Gh () c () c () ... ,
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W
h 2a
PT
(fourth-rank elasticity tensor Eijk with constant components; the summation convention is implied for repeated indices in the same summand, latin indices have the range 1,2,3, greek indices have the range 1,2) can be computed and integrated over the thickness
2
4
(6)
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h 2a
where G is a characteristic measure of the stiffness, e.g., the shear modulus, and c 2 h2 12a 2 is the characteristic plate parameter introduced in Section 1. Proceeding similarly with the constitutive relation
ij
W , ij
(7)
we may introduce stress resultants of the plate by mij a
1
W Gha () c 2 () c 4 () .... ij
(8)
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The terms () in (6) and (8) turn out to be of the same order of magnitude. Thus the strain energy W per unit of plate area and the stress resultants appear as power-series expansion in c2 . Finally, we obtain for the potential of external forces
V
h 2a
h 2a
h h f u3 d P u3 , P u3 , , 2a 2a
and may introduce load resultants by
P
(9)
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V . u3
(10)
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The external loads are splitted into symmetrical and antisymmetrical parts with respect to the middle surface, cf. Fig. 2,
1 f ( , ) f ( , ) , f a 1 f ( , ) f ( , ) , 2 2 1 1 Ps P P , Pa P P . 2 2
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fs
Fig. 2: Splitting of the tractions applied to the plate faces
h into symmetric and 2
antisymmetric parts The antisymmetric loads belong to the disc-load case (“squeezing” of the plate) and are not considered here. Indeed, the symmetric parts of the loads in x3 -directions lead to displacements of type (3), cf. [8].
ACCEPTED MANUSCRIPT h only, the load 2a resultants are not independent of each other. With P P P 2P s we find
If we further restrict ourselves to loads applied through the plate faces at
0
P aP,
2
P 3ac2 P,
4
P 9ac4 P,...
(11)
by insertion of the series expansion for u3 , cf. (3). Thus, also V appears as a power-law expansion in c 2
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V aP 0 w 3c 2 2 w 9c 4 4 w ... .
It turns out [4, 7] that P itself is of order c 2 P O c2 .
(12)
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The idea of consistent plate theories is to approximate all governing equations uniformly (cf., e.g., [3, 9-12]). This is ensured, as long as all quantities, i.e., the coefficients and w of the power-law series (3), contributing to the elastic potential W V will be considered up to the order c 2 n , n N whereas all terms of order c 2 m , m N will be neglected.
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For a second-order approximation ( N 2) , the Euler-Lagrange equations of the associate variational problem result in the system of partial differential equations given explicitly in Tab.1. With Young’s modulus E and Poisson’s ratio v , the following abbreviations have been used
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E 2v , , 2(1 v) 1 2v () (), ,
(13)
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G
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( ) ,
1 , . 1 2v
Tab. 1: PDE system for a consistent second-order approximation. w
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0
1
2
w
3
0
0
(d )
0
0
0
0
( e)
0
0
0
0
(f)
c2
3c 2 (),
1 :
(),
1 c 2
(1 2 )c 2 (),
2w :
c2
(1 2 )c 2 (),
9 4 c 4(2 )c 2 5 9 (3 2 ) c 4 (), 5 9 8(2 ) c 4 5
9 3c 2 c 4 5 9 (3 2 ) c 4 (), 5 81 4 c 5
0
9 4 c 5
5
9c 4 (),
9c 4
RHS
0
(),
4w :
5
9c 4 (),
9 3c 2 c 4 5 9 (1 4 ) c 4 (), 5
w
9 4 c 5 9 (1 4 ) c 4 (), 5 9 8(2 ) c 4 5
0w :
3 : 3c 2 (),
4
9c 4 0
a P Gh 0
a 2 3c P Gh
(a) (b) ( c)
ACCEPTED MANUSCRIPT It may be noted that dividing the equations by the common factors c 2 or c 4 is not admissible, since all equations are accurate but terms of order O(c 6 ) . Therefore, the division would alter the accuracy, if we do not add additional terms that are already neglected and in general would also include summands in new (i.e., higher-order) displacement coefficients. Dividing by the common factors 3, 9/5 or 9 is also not advisable since this would disturb the symmetry of the PDE system (Tab. 1). It may be further noted, that the RHS of equation (e) in Tab. 1 is set to zero, since due to (12) and (11) the load term would be of order O(c 6 ) . Before we start with the reduction of the equations in Tab. 1, we introduce new variables
3 , ,
3
3 1 , ,
1
5 , ;
5
3 3 , ,
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1 , ,
1
(14)
3 5 ,
3
5
( ijk is the completely skew symmetric permutation tensor).
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Applying the divergence operator to equations (b), (d) and (f) on the one hand, and application of the rotary operator to the same equations on the other, with (14) leads to two, decoupled PDE systems: one formulated in 0 w, 2 w, 4 w, 1 , 3 and 5 and another one formulated in 1 , 3 and 5 depicted in Tab. 2 and 3. Tab. 2: PDE system for 0 w , 2 w , 4 w , 1 , 3 and 5 . w
1
1
c2
1 (2 )c 2
c2
(1 2 )c 2
3c 2
9 3c 2 (2 ) c 4 5
9 4 c 5
9 (1 4 ) c 4 5
3
w
5
3c 2
9 4 c 5
9c 4
(1 2 )c 2
9 3c 2 (2 ) c 4 5
9 (1 4 ) c 4 5
9c 4
9 4(2 )c 2 c 4 5
9 (3 2 ) c 4 5
1
3
M
w
PT
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2
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0
9c 4
(2 )
72 4 c 5
0
AC
9c 4
9 (3 2 ) c 4 5
Tab. 3: PDE system for 1 , 3 and 5 .
1
3
5
3 3c 2 1 c 2 9c 4 5 3 27 3c 2 1 c 2 c2 0 5 5 1 c2
9c 4
0
0
RHS 0
(a)
0
(b)
0
(c)
5
4
(2 )
72 4 c 5
0
RHS
a P Gh 0
a 2 3c P Gh
(a ) (b) (c )
81 4 c 5
0
0
0
(d )
0
0
0
0
(e)
0
0
0
0
(f)
ACCEPTED MANUSCRIPT Both PDE systems are, again, symmetric, if we replace by , say, in Tab 2 for this purpose only.
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3 Pseudo-reduction method The uniform-approximation argument that is proposed for the derivation of the plate equations was extended [11, 7, 13] as to apply 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, and a main differential equation system, that are entirely formulated in main variables. In addition, one has to find a set of reduction differential equations, in which all non-main variables are expressed in terms of the main variables. The original PDE system must be identically solved by inserting the reduction differential equations and omitting all terms of order O(c 2( N 1) ) , if the main variables are a solution of the main differential equation system, i.e., the reduced PDE system and the original are equivalent despite terms of the order O(c 2( N 1) ) (for more details see [6]).
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Since the PDEs are actually truncated power series in the characteristic plate parameter c 2 , multiplication or division by powers of c 2 would change the accuracy of the given equations. Therefore, every product of different powers of the characteristic parameter with the same displacement coefficient 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 c 2 and neglecting again the arising terms of order O(c 2( N 1) ) . Note that the equations generated by multiplications by powers of c 2 are formulated in terms of the original displacement coefficients of Tab. 1, whereas equations generated by divisions would in general contain terms in additional displacement coefficients.
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Before we proceed along this line, let us recall that we already used tacitly a model assumption every time we differentiated a power series of the type of equations (6) and (8). It has been assumed throughout and will be assumed in the following that (15)
PT
if f ( w, ) O(c k ) f ( w, ) O(c k ) ,
i.e., differentiation with respect to the dimensionless variables does not change the order of magnitude of a quantity. This assumption has to be changed when treating the terms 1 , 3
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and 5 , what was discussed in detail in [5,16]. The quantity 1 introduced in (14) is given by
1 2,1 1 1,2 .
(16)
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1
Employing the normal hypothesis of the classical Kirchhoff-Love-type theory, 1 0 w, , this term vanishes or is negligible as of higher order of magnitude, i.e.,
O(c 2 ) .
(17)
In this sense, the quantity 1 measures the deviation from the normal hypothesis and may therefore be regarded as a measure of the transverse-shear deformation. From Reissner’s theory we know that 1 is a fast decaying function and describes edge effects. It is qualitatively of the appearance [14]
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1
f e c ,e
1
2 c
.
(18)
This leads to the necessary second modeling assumption, cf. [5]: if g ( ) O(c k ) c2 g ( ) O(c k ) .
(19)
c 4 3
1 2v 4 0 c w O (c 6 ); 6 1 v
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Choosing 0 w and 1 as main variables and observing these two modeling assumptions, (15, 19), pseudo reduction of the equations of Tab. 2 is straight forward and yields the following reduction differential equations: 9 1 4v 2 0 3 31 18v v 2 4 1 3c 2 3 9c 4 5 c 4 4 w 0 w c w c 0 w 2 5 2 1 v 20 (1 v) 3 1 6v a 2 c P O(c 6 ), 8 1 v Gh 3 4 0 c 2 1 c 2 0 w c w O (c 6 ), 1 v c 4 1 c 4 0 w O (c 6 );
(20)
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18 3 1 10v v 2 4 0 1 v 2 0 c2 2 w c2 4 w c w c w 5 20 (1 v) 2 2 1 v 3 1 2v a 2 c P O (c 6 ), 8 1 v Gh 1 v 4 0 c4 2 w c w O (c 6 ). 2 1 v
PT
Note that multiplying (20.a) by c 2 and neglecting terms of O(c 6 ) delivers
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c 2 1 3c 4 3 c 2 0 w
1 4v 4 0 c w O(c6 ) 2 1 v
AC
i.e., the same result as if c 2 1 and 3c 4 3 would be replaced by (20.b) and (20.d), respectively. Thus, the reduction equations are inherently consistent. Substituting equations (20) into the equations of Tab 2, it turns out that (b-f) are satisfied identically, whereas (a) yields 3 4v 2 3 v 2 K 0 w c 0 w a3 P c P 5 1 v 2 1 v
(21)
with the plate stiffness Ghc 2 a 2 Eh3 . K 2 1 v 12(1 v 2 )
(22)
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Neglecting all terms of O(c 4 ) (note that K a 2 (22) contains the factor c 2 and P (12) is of order O(c 2 ) ) results in the classical Kirchhoff-Love plate-differential equation w
a3 P . K
(23)
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Thus the classical plate theory of Kirchhoff may be regarded as a consistent first-order approximation including boundary conditions involving Kirchhoff’s effective shear forces [7]. Multiplication of (21) with c 2 and neglection of higher-order terms leads to c 2 K 0 w c 2a3 P O(c6 ) .
(24)
3 8 3v 2 K 0 w a3 1 c P. 10 1 v
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With (24) and (15), the second term of the left-hand side of (21) may be replaced by 3 4v 2 3 c a P leading finally to our first main differential equation 5 1 v (25)
c 4 3 c 2 1 O(c 6 ),
(26)
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c 4 1 0 O(c 6 ).
c2 (5 9 )(5 27 ) 1 O(c 6 ), 15
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3c 2 3 9c 4 5
1
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Since the differential equation system in Tab 3 is homogeneous, we have some freedom to reduce the equations. For an arbitrary chosen constant we obtain
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With it, (c) of Tab 3 is satisfied and both (a) and (b) deliver the same main differential equation 3 c 4 1 0 O(c 6 ). 5 27
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c 2 1
In earlier papers [4, 13, 15], we did not use this freedom but employed 3 c 2 1 c 2 1 0 O(c 6 ). 2
(27) 1 and obtained 9
(28)
The proper choice of will be given shortly. Before we can calculate the stress resultants, we need, in addition, the corresponding reduction differential equations derived from the original differential system (Tab. 1). With (20), (26) and (27) we find
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9 5
3c 2 3 9c 4 5 c 4 4 w, c 2 2 w
1
18 2 4 c w , 5
2 2 0 2 v(5 27 ) 2 3 8 3v ac 2 1 1 c w, c 3 , P, O(c 6 ), 5 1 v 10 1 v Gh 1 v (29)
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3 2 0 3 v 4 c 2 1 c 2 1 c w, c 3 1 , O(c 6 ), 2 1 v 1 v 1 2v 0 v c 4 3 c 4 w, 3 1 , O(c 6 ). 6 1 v 1 v
Rendering also these reduction equations consistent, we proceed similar as in the paragraph following equation (20). We multiply (29.a) by c 2 and replace the ensuing quantities by (29.b) and (29.c) leading to
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2 5 (5 27 ) 1 . 5 54
It may be mentioned that the corresponding differential equation system for anisotropic plates does not allow for a free choice of an arbitrary constant since the equation system is inhomogeneous. This will be dealt with in a forthcoming paper [16].
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Summing up this chapter, the pseudo-reduction method leads to two main differential equations
PT
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3 8 3v 2 K 0 w a 3 1 c P 10 1 v 6 c 2 1 c 2 1 0 5
AC
CE
and to the eight reduction differential equations
(30)
ACCEPTED MANUSCRIPT 9 18 2 4 c w , 5 5 2 2 0 v 4 3 8 3v ac 2 1 c w, c 3 1 , P, , 1 v 10 1 v Gh 1 v 3 2 0 3 v 4 c 2 1 c 2 1 c w, c 3 1 , , 2 1 v 1 v 1 2v 0 v c 4 3 c 4 w, 3 1 , , 6 1 v 1 v 18 3 1 10v v 2 4 0 1 v 2 0 c2 2 w c2 4 w c w c w 5 20 (1 v) 2 2 1 v 3 1 2v a 2 c P O (c 6 ), 8 1 v Gh 1 v 4 0 c4 2 w c w O (c 6 ); 2 1 v 25 1 3c 2 3 9c 4 5 c 2 1 O (c 6 ) 36 5 c 4 3 c 2 1 O (c 6 ) 54 4 c O (c 6 )
3c 2 3 9c 4 5 c 4 4 w, c 2 2 w
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1
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(31)
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With that, all necessary equations are provided to calculate the stress resultants.
mij a
h 2a
PT
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4 Stress resultants The stress resultants can either be calculated by the constitutive relations (8) in explicit form or by the usual definition
1
ij
d ,
(32)
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h 2a
AC
where (5), (4) and (3) have to be inserted. Due to the special plate-displacement ansatz (3), only plate-stress resultants occur whereas the corresponding disc-stress resultants vanish. We obtain the following relations [7]
M : 1m Ghac 2 1 , 1 , ( 1 , 2 2 w)
1
9 c 2 3 , 3 , ( 3 , 4 4 w) O(c 6 ), 5 9 3 M : 3m Gha3 c 4 1 , 1 , ( 1 , 2 2 w) O(c6 ), 5 2 1 6 m O(c ), 2;
1
9 m33 Ghac 2 2(2 ) 2 w 1 , c 2 4(2 ) 4 w 3 , O(c 6 ) 5
(33)
ACCEPTED MANUSCRIPT 9 m33 Gha3 c 4 2(2 ) 2 w 1 , O(c 6 ), 5 2 1 m33 O(c6 ), 2;
3
(34)
9 Q : 0 m 3 Gh ( o w, 1 ) c 2 ( 2 w, 3 3 ) c 4 ( 4 w, 5 5 ) O(c 6 ), 5 9 2 Q : 2 m 3 Gha 2c 2 ( 0 w, 1 ) c 2 ( 2 w, 3 3 ) O(c 6 ), 5 9 4 Q : 4 m 3 Gha 4 c 4 0 w, 1 O(c 6 ), 5 2 6 m 3 O(c ), 2.
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0
(35)
The equations of equilibrium for a second-order consistent theory written in stress resultants are given by
(36)
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M
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10 Q , P, a 11 M , 0Q 0, a 12 Q , 2 1m33 3a 2 c 2 P, a 13 M , 3 2Q 0, a 14 Q , 4 3 m33 9a 4 c 4 P O (c 6 ). a
PT
On inserting (33) - (35) into (36), the governing PDE system of Tab. 1 is recovered, i.e., the equations of (36) and Tab. 1 are equivalent. The infinite system of equilibrium equations
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1 m j , a 1 a
1
m3 j c Pj ,
AC
which contain the disc equations as well, is equivalent to the exact problem of the threedimensional elasticity, cf. [5, 14]. Now we insert the reduction differential equations (31) into the relations for the stress resultants (33) - (35) in order to express them exclusively by the main variables w and to obtain [15] K 1 3 8 3v 2 0 Q 2 0 w, v 3 1 , c aP, O(c 6 ), a 2 10 1 v M
1
K 3 8v 2 c (1 v) 0 w, v 0 w, 1 a 10 1 v 3 6 v 2 2 v c 2 3 1 , 3 1 , c a P O (c 6 ); , 5 5 1 v
(37)
ACCEPTED MANUSCRIPT 6 m33 c 2 a 2 P O(c 6 ), 5 3 m33 O(c 6 ), 3 3 2 Q Kc 2 0 w, O(c 6 ) c 2 a 2 0Q O (c 6 ), 5 5 4 6 Q O(c ),
1
3
(38)
9 9 M Ka c 2 (1 v) 0 w, v 0 w, O(c 6 ) c 2 a 2 1M O (c 6 ). 5 5
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Thus it turns out that within the second-order approximation, the higher-order moments and shear forces are expressible by their classical counterparts, whereas the non-classical stress resultants m33 , that neither have a resulting moment nor a resulting shear force, are either directly determined by the external load or to be neglected.
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On substitution of (38) it is seen that the equilibrium equations (36.c-e) are satisfied identically whereas substitution of (37) into (36.a and b) leads to
a3 3 8 3v 2 c P 1 K 10 1 v
10 Q , P a
0 w
11 M , 0Q a
6 c 1 c 2 1 , 0. 5 2
(39)
ED
M
Equation (39.a) is identical to our main differential equation (30.a). By integration of (39.b) and setting the integration constants to zero, the second main differential equation (30.b) is recovered.
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The variational principle also delivers boundary conditions [15] for the prescribed stress resultants M and Q or the displacement quantities and w along the boundary of the plate’s midplane as
CE
M n 1M n
AC
Q n 0Q n
or
3 8v
6 v
1 c 2 0 w, 3 c 2 1 , , 10 1 v 5 1 v
(40) or
3 v 2 0 w 1 c w . 10 1 v
The quantities w and may be interpreted as energetic means over the plate thickness. It is
handsome to replace the displacements 0 w and 1 by their means. We introduce therefore
ACCEPTED MANUSCRIPT 3 v 2 0 c 2 wK c 2 1 c w O (c 6 ) 10 1 v c 4 w K c 4 0 w O (c 6 ) 3 8v 2 0 6 v c 2 K c 2 1 c w, 3 c 2 1 , O(c 6 ) 5 1 v 10 1 v 2 K 21 6 c c O (c )
(41)
K c 4 , c 4 1 , O(c 6 ).
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The factor c 2 is applied in (41) because only the terms c 2 0 w, c 2 1 and c 2 1 and their derivatives appear in the differential equations (30, 31) and in the equations for the stress resultants (37, 38). If the factor c 2 in (41) had been omitted, further terms of order O(c 4 ) would have been to be added in the relations.
K a
12 K c 2 (1 v) w, vw,K 1 5(1 v) 3 v c 2 3 , K 3 ,K , 5 6 v 2 2 c a P O(c 6 ), 5 1 v
(42)
M
M K
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For the moments we obtain
ED
and two equivalent variants for the transverse-shear forces 6 v 2 12 1 c 2 w, K v 3 ,K c aP, O (c 6 ), 1 2 5(1 v) 5 1 v K 1 6 2v 2 QK 2 w, K v 3 ,K c aP, O(c 6 ). a 2 5 1 v K a2
(43)
CE
PT
QK
The equilibrium equations now become with (43.b)
AC
1 K Q , P a
wK
a3 6 2 v 2 c P , 1 K 5 1 v
(44)
and with (42) and (43.a) (see the comment after equation (39))
1 M K, QK a
6 c 2 1 c 2 K 0 , 5
and the boundary conditions are
(45)
ACCEPTED MANUSCRIPT M n M K n Q n QK n
or * K ,
(46)
or w wK .
With (41-46) we obtain a most compact representation of the consistent second-order plate theory.
u a 1 3 3 5 5 O(c 7 ) , u3 a 0 w 2 w 2 4 w 4 6 w 6 O(c8 ) .
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5 Displacements We return to our power-series expansion of the displacements (3)
(47)
u a 1 0 1 2 c2 1 4 c4
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Although no integration over the plate thickness has been performed yet, we use the symbol h h , i.e. 2 3c2 . Thus 2 is of the order of c2 O(cn ), since the range of is 2a 2a and a misunderstanding of the notation is not apprehended. In the following, we will consider as many terms in the series expansion as it turns out to be necessary during the following calculation. Guided by the reduction equations (31), the individual displacement coefficients may be represented in power series of c2 themselves, leading to the ansatz
M
3 3 0 3 2 c2 5 5 0 O(c7 ) ,
u3 a 0w0 0w2 c2 0 w4 c4 0 w6 c6
(48)
w w c w c 2
0
2
2 2
2
4 4
ED
2
4 4 w0 4 w2 c2 6 6 w0 O(c8 ) ,
CE
PT
where the new double indexed quantities are terms depending merely on E, v, wK , K and derivatives of the latter. First, we introduce the definition of wK . With (41.a) and (41.b) we have
AC
w0c2 0w0c2 0w2c4 O(c6 ) wK c2
3 4 v c wK O(c6 ). 10 1 v
Thus we can identify w0 wK , 3 v 0 2 w wK . 10 1 v 0
(49)
Before preceding further we have to transfer the reduction equations (31) to wK and K with the result
ACCEPTED MANUSCRIPT
0
9 18 w, 1 3c2 3 9c4 5 c4 4w, c2 2w c2 4w , 5 5 v 2v 2 12 c2 w,K 3 ,K c4 w,K O (c6 ), 2 1 v 1 v 5 (1 v)
c2 1 c4 1
3 10 v K 3 v c2 w,K c4 w, 3 ,K O(c6 ), 2 1 v 10 1 v 4 K 6 c w, O(c ),
(50)
1 2 v K 1 v c4 w, 3 ,K O(c6 ), 6 1 v 6 1 v 18 v 3 1 1 c2 2 w c2 4 w c2 wK c4 wK O (c6 ), 2 5 5 (1 v) 2 1 v 1 v c4 2 w c4 wK O (c6 ). 2 1 v
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c4 3
AN US
It is easy to prove, that on insertion of (50) into the PDE system depicted in Tab. 1, the main differential equations (44) and (45) follow from (a) and (b), whereas (c) - (f) are satisfied identically. From (50), the following ansatz coefficients are identified
(50.d): (50.c): (50.b):
M
0 w,K ,
1
3 10 v K 3 v w, 3 ,K , 10 1 v 2 1 v 3 9 4v 9 1 4 w,K 0w,4 4w,0 3 3 2 9 5 0 . 2 5 (1 v) 5
2
1
CE
(50.a):
w0
ED
(50.e):
1 v wK , 2 1 v 3 1 18 2 2 w wK 4w0 , 2 5 (1 v) 5 1 2v K 1 v 3 0 w, 3 ,K , 6 1 v 6 1 v 2
PT
(50.f):
AC
Replacing the identified quantities (49) and (51) in (48) yields
(51)
ACCEPTED MANUSCRIPT 3 10 v K 3 v u a w,K c2 w, 3 ,K 2 1 v 10 1 v 9 3 9 4v c4 w,K 0w,4 4w,0 3 3 2 9 5 0 2 5 5 (1 v) 1 2 v K 1 v 2 w, 3 , c2 3 2 4 5 0 O(c6 ) , 6 1 v 6 1 v
(52)
4 4 w0 c2 4 w2 6 6 w0 O (c8 ) .
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3 v u3 a wK c2 wK c4 0w4 c6 0w6 10 1 v 1 v 18 3 1 2 wK c2 wK 4w0 c4 2w4 2 5 5 (1 v) 2 1 v
AN US
It turns out that not all displacement coefficients are fixed individually by the reduction differential equations. We will use this freedom to satisfy the traction-boundary conditions along the plate faces and the local equilibrium conditions in the following. To this extent, we calculate the stresses. For isotropic material, Hooke’s law leads to
1 E 1 v u , u3,3 u , u , a 1 v 2 1 2v
E 1 v2
ED
1 E (1 v)u3,3 vu , a (1 v)(1 2v)
PT
33
M
1 K K 2 2 K K (1 v) w, vw v 9c 3 , 3 , , 12 2 3 2 1 v 6 11v v2 v 1 v 4 0 K K c w 24 w (1 v)(10 v)w, 10 1 v 1 2v 1 2v 1 v v(1 v) 4 0 2 (2 v) w,K wK 24 w O(c4 ) , 6 1 2v 1 2v
E 3 2 2 K 24 0 c 4 10v v w 24(1 v) w (1 v )(1 2v) 10 1 2 v(2 v)wK 24(1 v)2 4w0 6 9 3 v(9 v) c4 wK v(1 v) 0w4 v (1 v ) 4w0 5 5 1 v
AC
CE
2
2(1 v)2 2 w4 3v(1 v) 3 2 , 9v(1 v) 5 0 ,
2 2 c2 4 1 v 4w2 v(1 v ) 3 2 ,
4 6 1 v 6 w0 v(1 v ) 5 0 , O (c6 ) ,
(53)
ACCEPTED MANUSCRIPT 1 E u3, u ,3 a 2(1 v)
3
E 2 3 K 3 c w, v 3 ,K 2 1 v 2 4 1 1 2 w,K v 3 ,K 2 4 9 3 9 3 9 4v c4 w,K (1 v) 4w,0 (1 v) 3 2 (1 v) 5 0 10 2 2 10 1 v 9 3 3 1 2 c2 w,K (1 v) 4w,0 (1 v) 3 2 5 2 10 1 v 5 1 4 1 v 4 w,0 (1 v) 5 0 O(c6 ) . 2 2
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It may be mentioned that on insertion of the stresses (53) into the formula for the stress K K resultants (32), the moments M (42) and Q (43) are recovered (with the use of (24) and (41)) and the relations for the non-classical (higher-order) stress resultants m33 , Q and
M
M (38) are confirmed. The so-far undetermined displacement coefficients do not contribute anything to the stress resultants of a second-order approximation. They may be chosen arbitrarily (e.g., set to zero) without changing the main PDEs (44) and (45) from which wK and K are determined uniquely when the boundary conditions (46) are taken into account.
h 2a
E 0 O ( c4 ) . 2 1 v
PT
3
ED
Therefore, the following determination of the additional displacement coefficients is a complete a posteriori procedure! It may be mentioned further that the shear stresses 3 satisfy already the traction boundary conditions of a first-order approximation
CE
In order to satisfy the traction boundary conditions for the normal stresses 33 within a firstorder approximation, merely 4w0 has to be determined
AC
33
h 2a
1 1 1 v P O(c4 ) 4w0 wK 2 24 1 v
(54)
leading to
33
E 1 v2
1 2 2 K 4 c 9 2 w O(c ) . c 6
Thus with (49), (51) and (54) (and setting 0w4 , 2w4 , 4w2 , 6w0 , 32 and 5 0 to zero), a consistent second-order plate theory may be established which, however, satisfies the traction boundary conditions only up to the order of O(c4 ) .
ACCEPTED MANUSCRIPT
Note that with (54), the inplane stresses (53.a) are completely determined. The traction-boundary conditions within a second-order approximation deliver
33
h 2a
h 2a
0 O ( c6 )
P O ( c6 ) 2
2
w4
2
3
4
1 3 2v w,K 6 5 0 , 2 5 (1 v)
w0
1 1 v wK , 24 1 v
(55)
3 32 7v v3 1 v wK 0w4 6 4w2 27 6w0 . 3 80 (1 v) 2 1 v
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3
Finally, we adopt the local homogeneous equilibrium conditions and find
4
E 1 (1 v)(1 2v) 1 1 5 0 0 O(c6 ) 6w0 wK , , 2 2 1 v 1440 (1 v) 12 1 v
AN US
3, 33,3
1 17 18v v2 2v3 3 1 5 0 wK , ; 3 160 (1 v) 4 1 v E 1 3v 1 v 0 O(c4 ) 5 0 w,K 3 ,K . 2 1 v 120 1 v 120 1 v
w2
, 3 ,3
(56)
ED
M
Two coefficients, which have no influence on the stress distribution, 0 w4 and 0 w6 , remain undetermined. They may be kept for cosmetical reasons or for comparison with results of other authors, which we, however, are not aware of. In summing up, the series expansion for the displacements read
2 1 1 4 2 36(25 13 v ) 6 15 12 v v (3 v )(1 v ) w,K 2 2 4 120 (1 v) c c
CE
PT
1 c2 2 K 4 0 4 u a w,K 9(10 v ) 5(2 v ) w, c w, 30 1 v c2
1 v 2 2 1 v 4 2 4 c 9 2 3 ,K c 9 6 2 4 3 ,K O (c6 ) , 6 1 v c 120 1 v c c
AC
1 vc2 2 K u3 a wK 3 5 2 w 10 1 v c 2 4 1 1 2 2 c4 0w4 18 5 v 5 1 v wK 2 2 4 120 (1 v) c c 2 2 1 v 1 1 25 13v c6 0 w6 0w4 54 2 2 1 v c 720 (1 v)2 1 v c2 4 6 2 K 8 9 10 v v (2 v )(1 v ) w O ( c ) . c4 c6
(57)
ACCEPTED MANUSCRIPT This displacement ansatz delivers the stresses E (1 v)w,K vwK 1 v2 1 5v c2 9(10 v)w,K 9v wK 30 1 v 2 5 2 (2 v)w,K vwK c 2 1 c2 v 9 2 3 ,K 3 ,K , O (c4 ) , 12 c E 1 2 2 33 c 9 wK 2 2 1 v 6 c
3
1 4 25 13v 2 5 3v 4 c 9 6 2 4 wK O (c6 ) , 60 1 v c 1 v c
E 1 2 2 K 3 2 c w, 1 v2 2 c
(58)
AN US
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1 4 25 13v 2 c 3 5 2 w,K 30 1 v c
1 2 2 c v 3 60 K c2 3 5 2 K , 120 c
M
h and the local 2a three-dimensional equilibrium conditions within a consistent second-order approximation. As long as wK and K are determined from (44)-(46) the stress distribution within the plate follows from (58).
PT
ED
which satisfy the traction-boundary conditions along the plate faces
CE
We hardly believe that neither (57) nor (58) could be estimated, even not by an educated guess, a priori.
AC
6 Comparison with other theories The equations (42-46) will be used to compare the proposed theory with those of other authors. 6.1 Schneider’s approach In the paper of Schneider et al. [5], the same uniform-approximation technique in combination with the pseudo-reduction method was applied for anisotropic plates but using - instead of monomials - suitably scaled Legendre polynomials as basis for the Fourier-series expansion in thickness direction. Further details may be found in [14]. The coefficients of the two series expansions are thus differently defined. As has been shown in [5, 14], the list of corresponding quantities is given by
ACCEPTED MANUSCRIPT 1 v 2 0 1 v 2 K c 2 wS c 2 1 c w c 2 1 c w 5 1 v 2 1 v 4 S 40 4 K c w c wc w
(59)
c c c , 2
S
21
2
K
c S c 4 1 c 4 K . 4
CR IP T
Note that wS and S are geometric means, whereas wK and K are energetic means. The terms c2 wS (59. a) and c2 wK (41.a), therefore, differ from each other. However, the geometric mean and the energetic mean c2 S and c2 K , respectively, coincide within a consistent second-order approximation, cf. (59.c).
1 12 7v 2 K wS a 3 1 c P, 5 1 v 6 c 2 1 c 2 S 0, 5 and the stress resultants are given as
1 12 7v 2 S 1 S c aP, , w, v 3 , 2 5 1 v K 12 v 2 1 c (1 v) w,S vw,S a 5(1 v) 3 6 v 2 2 v c 2 ( 3 S , 3 S , ), c a P . 5 5 1 v
(60)
M
K a2
(61)
PT
ED
QS S M
AN US
When the load resultants are also defined correspondingly, the differential equations for the special case of isotropic materials follow to be
CE
The equilibrium equations
1 S Q , P. a 1 S M , QS 0 a
AC
(62)
lead to (60). Use is made again (compare (24)) of the relation c2 K wS c2 a3 P O(c6 ) . Just by insertion of (59) into (60) and (61), a complete equivalence is established with the corresponding equations (25), (37) and (40) or with (44)-(46). Although the boundary condition written in wS 1 v 2 S w 1 c w 5 1 v
,
(63)
ACCEPTED MANUSCRIPT may have not such a concise appearance as that written in wK , cf. (46. b), both conditions are equivalent by insertion of (59.a). 6.2 Reissner’s approach
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Although Mindlin [17] investigated the influence of rotatory inertia and introduced a shear correction factor of 2 12 , his approach has much in common with Reissner’s to the extent that the theory is frequently referred to as Reissner-Mindlin-plate theory. Reissner’s approach (cf. [18, 19] and later papers) belongs to the class of classical approaches relying on a set of a priori assumptions. In the following we transpose Reissner’s notation to dimensionless coordinates and dimensionless displacement quantities in order to compare his theory with ours. First, he introduces an energetic mean of the transverse displacement [7] 3 v 2 0 ! K c w w 10 1 v
(64)
AN US
wRs 0 w
and an energetically motivated shear correction factor Rs
5 6
(65)
2 5 0.82 are quite similar) and arrives 0.83 and 12 6 finally at the differential equation for the transverse displacement as
M
(note that the numerical values of
a3 1 2v 2 w 1 c P K Rs 1 v
ED
Rs
(66)
PT
a3 6 2 v 2 1 c P. K 5 1 v
CE
Equation (66) coincides with (44), and Reissner’s engineering intuition is remarkable in choosing a set of partly self-contradicting a priori assumptions to arrive at a consistent theory.
AC
In order to introduce stress resultants, Reissner splits the displacements and the transverse shear forces into homogeneous and particular parts by
wRs wHRs wPRs
QRs QRsH QRsP
(67)
and introduces a shear-force potential by
K 1 wHRs, 3 , 2 a a K 6 2v 2 2 wPRs, c aP, . a 5 1 v
QRsH Rs P
Q
(68)
ACCEPTED MANUSCRIPT Combining (68. a and b) yields QRs
K 1 6 2v 2 w,Rs 3 , c aP, . 2 a a 5 1 v
(69)
The moments are defined by
K Rs (1 v) w,Rs vw, a 6 6 v 2 2 ac 2 QRs, QRs, c a P a . 5 5 1 v
Rs M
This leads with (69) after some rearrangements to Rs M
K 12 Rs c 2 (1 v) w,Rs vw, 1 a 5(1 v)
CR IP T
(70)
(71)
AN US
6 6 v 2 2 c 2 ( 3 , 3 , ), c a P O(c 6 ). 5 5 1 v
M
Note that in deriving (71) use was made of c4 P O(c6 ) and c2 K wRs c2 a3 P O(c6 ) . The 1 1 Rs Rs equilibrium equation QRs, P delivers (66) and M , Q 0 leads to Reissner’s a a second equation 6 5
(72)
ED
c 2 0.
K1 v K . a 2
(73)
CE
PT
On comparing (69, 71) with (43.b, 42), respectively, a one-to-one relation is established between and as
AC
Thus it turns out that Reissner’s theory fully coincides with the theory proposed here up to terms of order O(c 6 ) . 6.3 Marguerre/Hencky’s approach As in the classical Kirchhoff theory, Marguerre [20] and Hencky [21] assume that straight fibres vertical to the plate midplane remain straight and do not change length, but release the normal hypothesis, i.e., 0 w and 1 are independent variables. Referring back to Reissner [18], the transverse displacement wM is understood as energetic mean. Marguerre uses, in contrary to Hencky, the shear correction factor 5 6 . The main difference to Reissner’s theory is the assumption of plane-stress conditions in the plate plane. Thus the stress resultants are immediately given by
ACCEPTED MANUSCRIPT K 1 v M , M, v M, , a 2 5 Eh QM GhS M w,M with GhS . 12 1 v M M
(74)
With the abbreviation
M 3M,
(75)
CR IP T
and the equilibrium equations (36.a and b), Marguerre obtains, finally, two PDEs
12 K wM a3 1 c2 P , 5(1 v) 6 M c2 M 0 5
12
6 v
AN US
and the “reduction equation”
(76)
12
aP
M 1 c2 w,M c23 ,M c2 , . 5 1 v 5(1 v) GhS 5(1 v)
M
The stress resultants follow in turn as
K 12 1 c2 (1 v) w,M vw,M a 5(1 v) 3 v c2 3 ,M 3 ,M , O(c6 ), 5 K 1 12 QM 2 w,M v 3 ,M c2 aP, O (c6 ). a 2 5(1 v)
(77)
PT
ED
M M
CE
Hencky’s equations are obtained by simply setting the factor 6 5 equal to 1, where ever it appears.
AC
Clearly, wM and M may be identified with wK and K , respectively. The mere difference between (76.a), (77) and (42)-(45) consists in the missing factor 5 v 2 2 caP 6 1 v
in (76.a), (77.a) and (77.b), which is attributed to the plane stress assumption. The advantage of a straight-forward derivation of the theory is counterbalanced by a resulting non-consistent approach. 6.4 Zhilin’s approach
ACCEPTED MANUSCRIPT A quite different approach is proposed by Zhilin [22]. From the beginning, he introduces geometrical averages of the transverse displacement w and the rotation , such that
c 2 wZ c 2 wS ,
(78)
c 2 Z c 2 S c 2 K .
Z
5 . 6v
CR IP T
In order to omit additional algebraic manipulations, it appears to be most appropriate to compare Zhilin’s theory with Schneider’s. A shear correction factor Z is introduced as (79)
Zhilin’s plate differential equations are given finally as
AN US
2 1 v 2 K wZ a 3 1 c P 1 v z 2 1 12 7v 2 a3 1 c P, 5 1 v 6v 2 c F 0. 1 5
(80)
M
Equation (80.a) coincides with (60.a). The quantity F will be discussed and identified in the following.
ED
The transverse-shear forces and the bending and twisting moments are defined in the following way
K 1 v 2 2 z z z c a P . (1 v) , , v , a 2 1 v
(81)
CE
z M
PT
Qz Gh z w,z z ,
AC
For z , Zhilin applies a Helmholtz decomposition in the plane involving the two (scalar) displacement potentials and F as
z , 3 F, .
(82)
Thus
z , , 3 z , F . From (82) with (20.c), (59.b) and (80.b) it follows immediately that
(83)
ACCEPTED MANUSCRIPT c 4 wS c 4 O(c 6 ), 5 c 2 S c 2 F F O(c 6 ). 6v
(84)
From further equations of Zhilin’s derivation it follows that
(85)
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2 6v 2 z c 2 c 2 1 c w O(c 6 ), 5 1 v 2 2 6 v 2 z 1 12 7v c a 1 c w P O(c 6 ). 5 1 v Gh z 5 1 v
As a consequence the stress resultants read after elimination of , and higher-order terms as
K 1 12 7v 2 w,Z Gh Z 3 F, c aP O (c 6 ), 2 a 5 1 v K 2 6v 2 1 c (1 v) w,Z vw,Z a 5 1 v K1 (1 v)( 3 F, 3 F, ), a 2 v 2 2 c a P O (c 6 ). 1 v
(86)
M
Z M
AN US
QZ
ED
Due to (78.a), the following conclusion may be drawn from a comparison between (61) and (86)
QZ QS O(c 6 ), M
(87)
CE
c 4 ,S
v c2 M K w,S a 3 P O(c 6 ), 5 a 5 1 v 2 c F, O(c 6 ). 6 v S
PT
Z
AC
Due to (60.b), (84.b) and (87.c) do not contradict each other. Obviously, it is not possible to obtain (87.c) from (84.b) by differentiating (84.b) twice and multiplying by c2 , because our second modeling assumption (19) must be observed. Instead an equation similar to (60.b) has 1 Z to be invoked. Firstly, we observe that the equilibrium conditions Q , P and a 1 Z M , QZ 0 lead to the equations (80). Secondly, whereas the transverse shear resultants a of both approaches coincide, the moments exhibit a difference in the O(c 4 ) -terms. This additional term vanishes identically when substituted into the moment equilibrium condition S and may thus be interpreted as an added equilibrium group to M . It is expected that the S
Z
difference between M and M is rather insignificant. The resulting difference in the stress distribution over the thickness of the plate might be compensated by a proper choice of the
ACCEPTED MANUSCRIPT free displacement-ansatz coefficients (cf. chapter 5). Despite of this deviation, Zhilin’s and the consistent second-order approach lead to the same differential equations and may be regarded as equivalent within the order of O(c 6 ) . Again, the engineering intuition which guided Zhilin is admirable. 6.5 Ambartsumyan’s approach
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Ambartsumyan [23] uses a displacement ansatz, which is reversely constructed from a priori assumptions concerning the distribution of shear stresses, resulting in an intrinsically overdetermination of the system of the equations of the linear theory of elasticity. This is compensated by a semi-inversion of Hooke’s law, which assumes 33 to be given. In fact, since Hooke’s law for the stresses 13 and 23 is decoupled from the other stresses (this holds even for monotropic materials), these two stresses can be computed in accordance with the semi-inversion without the knowledge of 33 . This determines 33 afterwards by the corresponding equilibrium condition, which in turn determines all other stresses by use of the semi-inverted Hooke’s law.
AN US
By converting Ambartsumyan’s notation to ours (which is by-far not an easy task and will be dealt with in a forthcoming paper) and introducing isotropy, the equations [23], (II. 3.3-5) or (II. 5.5-7), are rewritten as
, aP,
K 6 c2 A w (1 v) , , , a2 5 1 v
(88)
M
h3 h3 and have been replaced by 1 and 2 , respectively. The 12 12 quantities and , cf. (II. 2.18, 19), have the dimensions N m4 ).
ED
(The ansatz coefficients
PT
Similar to (14), we introduce
, ,
CE
3 ,
(89)
AC
and obtain from (88) aP,
K 6 c2 A w (2 v), (1 v) 3 , . , a2 5 1 v
(90)
Applying the divergence operator to (90.b) yields in combination with (90.a)
K 6 2v 2 w A c aP . 2 a 5 1 v
Obviously, is of the order O(c 2 ) and thus
(91)
ACCEPTED MANUSCRIPT c2 ac2 P O(c6 ).
(92)
Invoking our first modeling assumption (15) the term c 2 may be replaced by ac 2 P and (91) becomes 6 2v 2 K w A a3 1 c P. 5 1 v
(93)
We return to (90.b). With (92), we have
K 6 2v 2 6 w,A c aP, c 2 3 , . 2 a 5 1 v 5
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If we identify w A with wK or wRs (which is by no means justified by Ambartsumyan’s displacement ansatz nor does he mention that w A is to be understood as an energetic mean of the transverse displacement), (93) coincides with (44).
(94)
AN US
Now, after differentiation with respect to and multiplication by c 2 , it turns out that the second term of the right-hand side of (94) is of order O(c 6 ) and has to be neglected, whereas the third term has due to our second modeling assumption (19) to be retained leading to K 2 A 6 4 c w, c 3 , O(c 6 ). a2 5
(95)
M
c 2 ,
QA ,
ED
The stress resultants can be calculated by insertion of equations [23], either (II. 2.38-42) or (II. 5.11-14) as
CE
PT
K (1 v) w,A vw,A a 6 1 c2a (1 v)( , , ) v , . 5 1 v
A M
Using (94), (95) and (92), we find
AC
K 6 2v 2 6 w,A c aP, c 2 3 , , 2 a 5 1 v 5 K 12 1 c 2 (1 v) w,A vw, a 5(1 v)
QA
A M
(96)
6 v 2 2 36 c a P ac 4 3 , 3 , , . 5 1 v 25
By comparison of (96.a and b) with (43.b) and (42), respectively, the quantity is identified uniquely as
ACCEPTED MANUSCRIPT
c2
5 v Ghc 2 K . 6 1 v
The equilibrium conditions
(97)
1 A 1 A A Q , P and M , Q 0 in combination with (96) yield a a
6 2v 2 K w A a 3 1 c P 5 1 v 6 c 2 1 c 2 0. 5
CR IP T
(98)
AN US
With w A wK and (97), the differential equations (98.a and b) and (44), (45) coincide and the stress resultants (96.a and b) and (43.b and 42) are identical, respectively. Thus it turns out that Ambartsumyan’s theory could be interpreted as a consistent second-order plate theory, provided that w A is identified as energetic mean of the transverse displacement w , which was not introduced by Ambartsumyan. In [23], Ambartsumyan treats also several assumptions and simplifications. One of them is to neglect the normal stress 33 in thickness direction ( A1 A2 ( A3 ) 0, cf. [23], chapter II.5). The resulting equations coincide exactly with those of Marguerre, (76) and (77) and are, therefore, not dealt with in detail.
M
6.6 Vekua’s approach
PT
ED
Hierarchical shell theories have been established by Vekua [24, 25]. The approximation order of this theory is defined by the maximum order of polynomials (Legendre polynomials there) which are taken into account, i.e., the first-order theory considers a linear distribution of the displacement quantities with respect to the plate thickness, the second-order theory allows for a quadratic distribution, etc. Specializing the shell equations to plates by neglecting all contributions due to the curvature tensor, we obtain for Vekua’s first-order theory 1 2v . (1 v)2
(99)
CE
K wV 1 a3 P
AC
This is not astonishing since a linear displacement approximation leads to the consistent Kirchhoff equation only, if additionally the plane-stress assumption and the normal hypothesis would be invoked [7]. The quadratic displacement ansatz then leads to the consistent first-order result K wV 2 a3 P.
(100)
The same results (99) and (100) have been obtained by Preusser [26] by using a linear and a quadratic displacement ansatz, respectively. From the cubic displacement ansatz onwards, her higher-order theories coincide - after pseudo reduction - with (25), cf. [7]. Higher-order theories are not further elaborated in Vekua’s book [25]. It would be interesting to compare a fifth-order Vekua theory with the consistent second-order approach. It is
ACCEPTED MANUSCRIPT conjectured that both theories coincide, if additional terms of order O(c 6 ) are neglected within Vekua’s approach (pseudo-reduction approach). 6.7 Reddy’s third-order shear theory
Ry M
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In order to improve Reissner’s plate theory (which is called in Reddy’s nomenclature a firstorder shear theory) Reddy and coworkers [27, 28] propose a third-order shear-deformable plate theory. By introducing an ansatz for the displacements involving cubic polynomials in thickness direction with the three unknowns 0 w and 1 , and assuming further plane-stress conditions in the middle plane of the plate, he defines stress resultants as h 2
x dx 3
3
M ,
h 2
PRy
h 2
x dx 3 3
3
M ,
3
h 2
QRy
h 2
dx 3
Q ,
3
RRy
h 2
2 3 3
x dx3
2
Q ,
ED
h 2
(101)
M
h 2
AN US
and establishes the following equations of equilibrium
PT
1 Ry 4 Ry 4 Ry Q 2 R 2 2 P , P, a h 3 h a ,
(102)
CE
1 Ry 4 Ry Ry 4 Ry M 2 P Q 2 R 0. a 3h h ,
AC
Insertion of (101) into (102) leads to
5c 2 4c 2 15 a 1 0 w 1 1 , P 8 Gh 28(1 v) 7(1 v) 4c 2 17 1 v 1 0 w, 1 c 2 1 , , 0. 1 14 1 v 7(1 v)
(103)
We did not introduce specific symbols since 0 w and 1 coincide in our’s and Reddy’s notation provided that dimensionless coordinates and displacements (1) - (3) are used. Solving for 0 w delivers
ACCEPTED MANUSCRIPT 1 17 K 1 c 2 0 w a3 1 c 2 P. 35(1 v) 7(1 v)
(104)
Equation (104) is identical to (7.4.24) in combination with (7.4.23) in [28] despite of terms of the order O(c 6 ) . By use of (24) we arrive finally at
12 K 0 w a3 1 c 2 P. 5(1 v)
(105)
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Application of the operator rot to (103.b) yields with 1 3 1 , 17 c 2 1 c 2 1 0. 14
(106)
AN US
It may be mentioned that the factor 17c 2 14 in (106) is equal to 17h2 168a 2 which has already been reported by Lewinski [29]. Equation (105) coincides with Marguerre’s (non-consistent) plate theory [20],cf.(76.a), which is derived, however, in a much more straight-forward manner. On comparing (106) with (30.b), it is observed that the factors and
6 1.20 5
M
17 1.21 14
(107)
ED
are quite similar. Anyway, it turns out that Reddy’s third-order shear theory is not a consistent second-order plate theory and does not improve the plate theory of Reissner.
1 (36.d) 0. 9c 2 a 2
CE
(36.b)
PT
The origin of the discrepancy may be explained by the following reasoning. The second equilibrium equation (102.b) may be regarded simply as a combination of (108)
AC
Dividing an equation by c 2 , however, is only justified, if higher-order terms are taken into account additionally (see the discussion in Section 3). Therefore, terms of order c 2 are missing in (106) and cause the discrepancy with (30.b). The same argument applies to (102.a) since it may be interpreted as
36.a
1 1 (36.d), . 9c 2 a 2 a
(109)
Since plane-stress conditions and a constant displacement ansatz for w are assumed, a higher-order load term is not present on the right-hand side of (103.a). By replacing P in (103.a) - in view of (20.f) and (21) - by
ACCEPTED MANUSCRIPT v 2 P 1 c P, 1 v
(110)
and by choosing 9 10 , equations (105) and (25) would coincide. The deviation between (106) and (30.b), however, would remain. 7 Concluding remarks
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By expanding the displacements in thickness directions into an abstract Fourier series with respect to a suitable basis and integrating over the plate thickness, the densities of both the elastic strain energy W and the potential of external forces V per unit of midplane area of the plate appear as power series in the plate parameter c2 h2 (12a 2 ) . By truncating the series expansion in such a way that all terms appearing in W V up to the order c 2 n , n N , are retained and all terms of the order c 2 m , m N are omitted, an Nth -order consistent plate theory is established, where the governing field equations and the associated boundary conditions follow from a variational principle, which delivers 3( N 1) partial differential equations (PDEs) for 3( N 1) unknown displacement coefficients “living” on the twodimensional midplane of the plate. Involving the pseudo-reduction method, the system of PDEs may be reduced to a system of 3( N 1) PDEs of higher (differential) order in main unknowns. In this way, a hierarchy of consistent Nth -order plate theories is established.
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It is shown that the rather classical Kirchhoff-Love plate theory is a consistent first-order ( N 1) approximation resulting in one ( 1) fourth-order PDE formulated in the transverse displacement w .
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The consistent second-order ( N 2) approximation results in a system of two ( 2) PDEs. For isotropic materials, we obtain one PDE of fourth order in the transverse displacement w and one PDE of second order for the transverse-shear-deformation measure , which describes fast decaying edge effects of the plate. In order to treat this quantity properly, a special modeling assumption had to be invoked.
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Once a solution of the boundary value problem given by the two main PDEs of the secondorder approximation is obtained only a subset of the coefficients of the series expansion for the displacements is determined uniquely by the reduction equations gained by the pseudoreduction procedure. The remaining coefficients are determined a posteriori from the traction boundary conditions along the plate faces at h 2a and from the local (threedimensional) equilibrium conditions. In this way, the stress distribution within the plate continuum can be calculated up to the same order of accuracy, i.e., beside terms of order O (c 6 ) . The PDE system, the stress resultants and the boundary conditions have been compared with those of other authors. Within a consistent second-order approximation, Reissner’s plate theory coincides completely with our approach. A consistent improvement of Reissner’s plate theory is only possible within a uniform third-order approximation, i.e., if all terms of the order O(c 6 ) would have been retained. This implies a series expansion of the displacements including seventh-order polynomials in (cf. eq. (3)). Other theories treated in the paper
ACCEPTED MANUSCRIPT show more or less pronounced differences. The origin of the discrepancies are discussed in some detail. For anisotropic plates, the PDEs for w and are coupled and may be given in the following schematic form [4, 5, 14, 16]
c D
c 4 D6 w c 4 D 4
4
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a 1 c 2 D 2 P, h 4 2 4 c D 4 w c D0 c D2 0. 2
The symbolic operator Dn is given by n
Dn bi i 0
n , 1i 2n i
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where the coefficients bi , which are different in the operators with and without tilde, depend on the elastic constants and are given explicitly in [5, 14, 16].
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The first-order approximation leads after pseudo reduction to a Kirchhoff-type PDE for anisotropic plates attributed to Huber [30]. The equations of the second-order approximation coincide with those of Ambartsumyan [23], provided that the transvers displacement w A is (in contrary to his original work) interpreted as an energetic mean and under the condition that none of his suggested additional simplifications are introduced. Details will be given in a forthcoming paper.
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The uniform-approximation technique in combination with the pseudo-reduction technique has been used to derive material conservation laws within a consistent second-order plate theory [15]. It has also been applied to shells [11].
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Recently, the same technique was applied to beams. Although merely one-dimensional, beam theory is more advanced than plate theory because the series expansion has to be performed with respect to two directions. Thus two parameters have to be considered, a height and a width parameter, which results in a more complex pseudo-reduction problem. First results of this investigation are given in [8].
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Plates with variable stiffness distributions in thickness direction like functionally graded plates and plates with variable thickness are the subject of ongoing research. Acknowledgment The research work of R.K. has been partly funded by a Distinguished Visiting Fellowship of the Royal Academy of Engineering. The generous support is greatly acknowledged. References [1] Szabo, I. (1987) Geschichte der mechanischen Prinzipien, Birkhäuser, Basel [2] Altenbach, J., Altenbach, H., Eremeyev, V. A. (2010) On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Archive of Applied Mechanics 80, 73-92 [3] Naghdi, P. M. (1972) The theory of plates and shells. In: Flügge, S. (Ed.), Handbuch der Physik, Springer, Berlin VI, A2, 425-646
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[10] [11]
[12]
[13]
[14] [15] [16]
[17]
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[18]
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[9]
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[8]
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[6]
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[5]
Kienzler, R., Schneider P. (2012) Consistent theories of isotropic and anisotropic plates. Journal of Theoretical and Applied Mechanics 50, 755-768 Schneider, P., Kienzler, R., Böhm, M. (2014) Modeling of consistent second-order plate theories for anisotropic materials. Zeitschrift für Angewandte Mathematik und Mechanik 94, 21-42 Schneider, P., Kienzler, R. (2011) An algorithm for the automatization of pseudo reductions of PDE systems arising from the uniform-approximation technique. In: Altenbach, H., Eremeyev, V. A. (Eds.) Shell-like structures; non-classical theories and applications. Advanced Structured Materials 15, Springer, Berlin, 377-390 Kienzler, R. (2002) On consistent plate theories. Archive of Applied Mechanics 72, 229-247 Schneider, P., Kienzler, R. (2015) On exact rod/beam/shaft theories and the coupling among them due to arbitrary material anisotropies. International Journal of Solids and Structures 56-57, 265-279 Koiter, W. T., Simmonds, J. G. (1973) Foundations of shell theories, In: Becker, E., Mikhailov, G. K. (Eds.) Theoretical and Applied Mechanics. Proceedings of the 13th International Congress of Theoretical and Applied Mechanics, 150-176 Lo, K. H., Christensen, R. M., Wu, E. M. (1977) A higher-order theory of plate deformation, Part I: Homogeneous plates. Journal of Applied Mechanics 44, 663-668 Kienzler, R. (1982) Eine Erweiterung der klassischen Schalentheorie; der Einfluss von Dickenverzerrungen und Querschnittsverwölbungen. Ingenieur-Archiv 52, 311-322; Extended version in: PhD-Thesis, Technische Hochschule Darmstadt, Darmstadt (1980) Krätzig, W. B. (1989) On the structure of consistent plate theories, In: Koiter, W. T., Mikhailov, G. K. (Eds.) Proceedings of the 3rd IUTAM Symposium on Shell Theory. North-Holland, Amsterdam, 353-368 Kienzler, R. (2004) On consistent second-order plate theories. In: Kienzler, R., Altenbach, H., Ott, I. (Eds.) Theories of plates and shells: critical review and new applications. Springer, Berlin, 85-96 Schneider, P. (2010) Eine konsistente Plattentheorie zweiter Ordnung für monotropes Material. Diploma thesis, University of Bremen, Bremen Bose, D. K., Kienzler, R. (2006) On material conservation laws for a consistent plate theory. Archive of Applied Mechanics 75, 606-617 Schneider, P., Kienzler, R. (2016) A Reissner-type plate theory for monoclinic material derived by extending the uniform approximation technique by orthogonal tensor decompostions of the nth-order gradient. Submitted Mindlin, R. D. (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 18, 31-38 Reissner, E. (1944) On the theory of bending of elastic plates. Journal of Mathematical Physics 23, 184-191 Reissner, E. (1945) On the effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 12, 69-77 Marguerre, K., Wörnle, H. T. (1975) Elastische Platten. Mannheim, BI Wissenschaftsverlag Hencky, H. (1947) Über die Berücksichtigung der Schubverzerrungen in ebenen Platten. Ingenieur Archiv 16 (1947) Zhilin, P. (1992) On the Poisson and Kirchhoff plate theories from the point of view of the modern plate theory (In Russian). Izvesta Akademii Nauk Rossii, Mekhanica Averdogotela, Moscow 3, 48-64 Ambartsumyan, S. A. (1970) Theory of anisotropic plates. Progress in Material Science Series II, Technonic, Stamford, Conn.
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Vekua, I. N. (1955) On one method of calculating prismatic shells (In Russia). Trudy Tbilis. Mat. Inst. 21, 191-259 Vekua, I. N. (1982) Shell theory: General methods of construction. Pitman, Boston et al. Preusser, G. (1984) Eine systematische Herleitung verbesserter Plattengleichungen. Archive of Applied Mechanics 54, 51-61; Extended version in: PhD-Thesis, Technische Hochschule Darmstadt, Darmstadt (1982) Reddy, J. N. (1984) A refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 22, 881-896 Wang, C. M., Reddy, J. N., Lee, K. H. (2000) Shear deformable beams and plates. Elsevier, Amsterdam et al. Lewinski, T. (1987) On refined plate models based on kinematical assumptions. Ingenieur-Archiv 57, 133-146 Huber, M. T. (1929) Probleme der Statik technisch wichtiger orthotroper Platten. Gebetner and Woff, Warsaw
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Second-order linear plate theories: Partial differential equations, stress resultants and displacements R. Kienzler , P. Schneider PII: DOI: Reference:
S0020-7683(17)30004-5 10.1016/j.ijsolstr.2017.01.004 SAS 9418
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
17 October 2016 1 December 2016 3 January 2017
Please cite this article as: R. Kienzler , P. Schneider , Second-order linear plate theories: Partial differential equations, stress resultants and displacements, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.01.004
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ACCEPTED MANUSCRIPT Second-order linear plate theories: Partial differential equations, stress resultants and displacements R. Kienzler, P. Schneider Bremen Institute of Mechanical Engineering (bime) University of Bremen, Department of Production Engineering Am Biologischen Garten 2, 28359 Bremen, Germany [email protected], [email protected]
uniform-approximation technique, pseudo-reduction approach, consistent plate theory, second-order theory, stress resultants, displacements
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Keywords
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Abstract By combining the uniform-approximation technique with the pseudo-reduction approach, a consistent second-order plate theory is developed with recourse to neither kinematical assumptions nor to shear-correction factors by truncation of the elastic energy. The governing partial differential equations and the expressions for the stress resultants are compared with those of other authors. Free coefficients of the resulting displacement “ansatz” are determined a posteriori, in order to satisfy three-dimensional boundary conditions and local equilibrium equations.
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1 Introduction Plates are thin plane structures loaded transversally to their midplanes. The characteristic thickness dimension h is much smaller than the characteristic in-plane dimension a , h a . Plate theories are inherently approximative in that they attempt to describe the actual threedimensional plate continuum by quantities that are defined on a surface. Generally, three different categories of derivation techniques for plate theories may be distinguished. The classical or engineering approach starts with a set of kinematical a priori assumptions for the stress and displacement distributions in thickness direction. Either transvers shear strains are neglected, or their influence is considered by the introduction of shear-correction factors. A historical survey on classical plate theories may be found, e.g., in [1].
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Following the direct approach, all quantities “live” on a Cosserat-type surface endowed with a set of deformable directors attached at each point of the plane. Despite of the mathematical elegance, the main drawback of the method lies in the problem of establishing constitutive relations. Material parameters are identified by comparisons with a set of solutions of known test problems. The choice of test problems has a crucial influence on the resulting theory. An excellent overview over the theories relying on the direct approach is given in [2] with an extended bibliography. Within the present paper, we will follow the consistent approach introduced in the pioneering treatise [3] by Naghdi. Starting from the three-dimensional theory of elasticity, the approach uses abstract Fourier-series expansions in thickness direction with respect to a suitable basis to achieve a dimension reduction. The basis might be monomials, scaled Legendre polynomials or trigonometric functions. After introducing proper non-dimensional quantities and integrating over the plate thickness, the dimensionless plate parameter c2 h2 (12a 2 ) evolves quite naturally, which is a small quantity in plate theories. The elastic potential, i.e., the sum of the strain-energy and the potential of external forces, both per unit of plate area, appear as power series in this plate parameter. This infinite series can be truncated at different
ACCEPTED MANUSCRIPT orders, giving rise to hierarchical plate theories. The idea of the uniform-approximation technique for an Nth-order theory is to consider all terms multiplied by c 2 n , n N and omit all terms of the order O(c 2 m ), m N . The associated Euler-Lagrange equations of a variational problem corresponding to the Navier-Lamé formulation result in 3( N 1) partial differential equations (PDEs) for 3( N 1) unknown displacement coefficients “living” on the two-dimensional midplane of the plate, cf. [4].
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It has been shown [5] that the infinite two-dimensional PDE system ( N ) corresponding to the untruncated elastic potential is equivalent to the three-dimensional equilibrium equations of linear elasticity and in a forthcoming paper, we will give an a priori estimate for the approximation error that results from truncating the energy series.
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During the pseudo-reduction of the PDE system, i.e., elimination of unknowns, terms of the order O(c 2 m ), m N , i.e., O(c 2( N 1) ) are consequently also neglected (pseudo-reduction method, for details see [6]). The procedure results in plate-differential equations which are derived without any recourse to either kinematical a-priori assumption or shear-correction factors. We compare the plate-differential equations and the resulting stress resultants with those of other authors. As it turns out, not all displacement coefficients are determined uniquely by the reduction equations. We use this freedom to satisfy the three-dimensional boundary conditions for the stresses at the upper and lower plate faces and the local stress-equilibrium conditions within the plate continuum.
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For the sake of simplicity, we treat plates with constant thickness of homogeneous isotropic materials. In the conclusion, we give some indications for plates of anisotropic materials.
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2 Uniform approximation technique Let us assume that the midplane of the plate occupies a bounded domain A 2 of the Cartesian ( x1 , x2 ) -plane, whereas the plate continuum extends by h 2 in the x3 -direction. The plate thickness h is much smaller than the characteristic in-plane dimension a . The loads P and P are applied at the upper and lower faces, respectively, and point in x3 direction as indicated in Fig. 1. Volume forces f in x3 -direction, e.g., due to gravity, may also be applied. A further specification of the loading conditions will be discussed later.
Fig. 1: Plate continuum We introduce the dimensionless coordinates
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x x , 3 , 1, 2 , a a
(1)
and differentiation with respect to these dimensionless coordinates by the comma notation () () a (), , 1, 2 . x
(2)
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Since due to isotropy, the plate (transverse loading) and disc (in-plane loading) problems are decoupled, we restrict ourselves to plate-like displacements, i.e., displacements that are odd functions with respect to in the in-plane component functions and even functions in the outof-plane component function, cf. [7], and make use of the MacLaurin-series expansion (i.e., we choose a basis of mononic polynomials. Legendre polynomials will be discussed later).
u a 1 3 3 5 5 ... ,
(3)
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u3 a 0 w 2 w 2 4 w 4 ... .
The displacement ansatz is inserted into the kinematic relations
ij
1 ui , j u j ,i 0 ij 1 ij 2 ij 2 ..., 2
(4)
1 Eijk ij k 2
(5)
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W
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and in turn, the strain-energy density per unit of volume
W , d Gh () c () c () ... ,
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W
h 2a
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(fourth-rank elasticity tensor Eijk with constant components; the summation convention is implied for repeated indices in the same summand, latin indices have the range 1,2,3, greek indices have the range 1,2) can be computed and integrated over the thickness
2
4
(6)
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h 2a
where G is a characteristic measure of the stiffness, e.g., the shear modulus, and c 2 h2 12a 2 is the characteristic plate parameter introduced in Section 1. Proceeding similarly with the constitutive relation
ij
W , ij
(7)
we may introduce stress resultants of the plate by mij a
1
W Gha () c 2 () c 4 () .... ij
(8)
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The terms () in (6) and (8) turn out to be of the same order of magnitude. Thus the strain energy W per unit of plate area and the stress resultants appear as power-series expansion in c2 . Finally, we obtain for the potential of external forces
V
h 2a
h 2a
h h f u3 d P u3 , P u3 , , 2a 2a
and may introduce load resultants by
P
(9)
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V . u3
(10)
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The external loads are splitted into symmetrical and antisymmetrical parts with respect to the middle surface, cf. Fig. 2,
1 f ( , ) f ( , ) , f a 1 f ( , ) f ( , ) , 2 2 1 1 Ps P P , Pa P P . 2 2
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fs
Fig. 2: Splitting of the tractions applied to the plate faces
h into symmetric and 2
antisymmetric parts The antisymmetric loads belong to the disc-load case (“squeezing” of the plate) and are not considered here. Indeed, the symmetric parts of the loads in x3 -directions lead to displacements of type (3), cf. [8].
ACCEPTED MANUSCRIPT h only, the load 2a resultants are not independent of each other. With P P P 2P s we find
If we further restrict ourselves to loads applied through the plate faces at
0
P aP,
2
P 3ac2 P,
4
P 9ac4 P,...
(11)
by insertion of the series expansion for u3 , cf. (3). Thus, also V appears as a power-law expansion in c 2
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V aP 0 w 3c 2 2 w 9c 4 4 w ... .
It turns out [4, 7] that P itself is of order c 2 P O c2 .
(12)
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The idea of consistent plate theories is to approximate all governing equations uniformly (cf., e.g., [3, 9-12]). This is ensured, as long as all quantities, i.e., the coefficients and w of the power-law series (3), contributing to the elastic potential W V will be considered up to the order c 2 n , n N whereas all terms of order c 2 m , m N will be neglected.
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For a second-order approximation ( N 2) , the Euler-Lagrange equations of the associate variational problem result in the system of partial differential equations given explicitly in Tab.1. With Young’s modulus E and Poisson’s ratio v , the following abbreviations have been used
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E 2v , , 2(1 v) 1 2v () (), ,
(13)
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G
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( ) ,
1 , . 1 2v
Tab. 1: PDE system for a consistent second-order approximation. w
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0
1
2
w
3
0
0
(d )
0
0
0
0
( e)
0
0
0
0
(f)
c2
3c 2 (),
1 :
(),
1 c 2
(1 2 )c 2 (),
2w :
c2
(1 2 )c 2 (),
9 4 c 4(2 )c 2 5 9 (3 2 ) c 4 (), 5 9 8(2 ) c 4 5
9 3c 2 c 4 5 9 (3 2 ) c 4 (), 5 81 4 c 5
0
9 4 c 5
5
9c 4 (),
9c 4
RHS
0
(),
4w :
5
9c 4 (),
9 3c 2 c 4 5 9 (1 4 ) c 4 (), 5
w
9 4 c 5 9 (1 4 ) c 4 (), 5 9 8(2 ) c 4 5
0w :
3 : 3c 2 (),
4
9c 4 0
a P Gh 0
a 2 3c P Gh
(a) (b) ( c)
ACCEPTED MANUSCRIPT It may be noted that dividing the equations by the common factors c 2 or c 4 is not admissible, since all equations are accurate but terms of order O(c 6 ) . Therefore, the division would alter the accuracy, if we do not add additional terms that are already neglected and in general would also include summands in new (i.e., higher-order) displacement coefficients. Dividing by the common factors 3, 9/5 or 9 is also not advisable since this would disturb the symmetry of the PDE system (Tab. 1). It may be further noted, that the RHS of equation (e) in Tab. 1 is set to zero, since due to (12) and (11) the load term would be of order O(c 6 ) . Before we start with the reduction of the equations in Tab. 1, we introduce new variables
3 , ,
3
3 1 , ,
1
5 , ;
5
3 3 , ,
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1 , ,
1
(14)
3 5 ,
3
5
( ijk is the completely skew symmetric permutation tensor).
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Applying the divergence operator to equations (b), (d) and (f) on the one hand, and application of the rotary operator to the same equations on the other, with (14) leads to two, decoupled PDE systems: one formulated in 0 w, 2 w, 4 w, 1 , 3 and 5 and another one formulated in 1 , 3 and 5 depicted in Tab. 2 and 3. Tab. 2: PDE system for 0 w , 2 w , 4 w , 1 , 3 and 5 . w
1
1
c2
1 (2 )c 2
c2
(1 2 )c 2
3c 2
9 3c 2 (2 ) c 4 5
9 4 c 5
9 (1 4 ) c 4 5
3
w
5
3c 2
9 4 c 5
9c 4
(1 2 )c 2
9 3c 2 (2 ) c 4 5
9 (1 4 ) c 4 5
9c 4
9 4(2 )c 2 c 4 5
9 (3 2 ) c 4 5
1
3
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w
PT
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2
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0
9c 4
(2 )
72 4 c 5
0
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9c 4
9 (3 2 ) c 4 5
Tab. 3: PDE system for 1 , 3 and 5 .
1
3
5
3 3c 2 1 c 2 9c 4 5 3 27 3c 2 1 c 2 c2 0 5 5 1 c2
9c 4
0
0
RHS 0
(a)
0
(b)
0
(c)
5
4
(2 )
72 4 c 5
0
RHS
a P Gh 0
a 2 3c P Gh
(a ) (b) (c )
81 4 c 5
0
0
0
(d )
0
0
0
0
(e)
0
0
0
0
(f)
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3 Pseudo-reduction method The uniform-approximation argument that is proposed for the derivation of the plate equations was extended [11, 7, 13] as to apply 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, and a main differential equation system, that are entirely formulated in main variables. In addition, one has to find a set of reduction differential equations, in which all non-main variables are expressed in terms of the main variables. The original PDE system must be identically solved by inserting the reduction differential equations and omitting all terms of order O(c 2( N 1) ) , if the main variables are a solution of the main differential equation system, i.e., the reduced PDE system and the original are equivalent despite terms of the order O(c 2( N 1) ) (for more details see [6]).
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Since the PDEs are actually truncated power series in the characteristic plate parameter c 2 , multiplication or division by powers of c 2 would change the accuracy of the given equations. Therefore, every product of different powers of the characteristic parameter with the same displacement coefficient 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 c 2 and neglecting again the arising terms of order O(c 2( N 1) ) . Note that the equations generated by multiplications by powers of c 2 are formulated in terms of the original displacement coefficients of Tab. 1, whereas equations generated by divisions would in general contain terms in additional displacement coefficients.
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Before we proceed along this line, let us recall that we already used tacitly a model assumption every time we differentiated a power series of the type of equations (6) and (8). It has been assumed throughout and will be assumed in the following that (15)
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if f ( w, ) O(c k ) f ( w, ) O(c k ) ,
i.e., differentiation with respect to the dimensionless variables does not change the order of magnitude of a quantity. This assumption has to be changed when treating the terms 1 , 3
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and 5 , what was discussed in detail in [5,16]. The quantity 1 introduced in (14) is given by
1 2,1 1 1,2 .
(16)
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1
Employing the normal hypothesis of the classical Kirchhoff-Love-type theory, 1 0 w, , this term vanishes or is negligible as of higher order of magnitude, i.e.,
O(c 2 ) .
(17)
In this sense, the quantity 1 measures the deviation from the normal hypothesis and may therefore be regarded as a measure of the transverse-shear deformation. From Reissner’s theory we know that 1 is a fast decaying function and describes edge effects. It is qualitatively of the appearance [14]
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1
f e c ,e
1
2 c
.
(18)
This leads to the necessary second modeling assumption, cf. [5]: if g ( ) O(c k ) c2 g ( ) O(c k ) .
(19)
c 4 3
1 2v 4 0 c w O (c 6 ); 6 1 v
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Choosing 0 w and 1 as main variables and observing these two modeling assumptions, (15, 19), pseudo reduction of the equations of Tab. 2 is straight forward and yields the following reduction differential equations: 9 1 4v 2 0 3 31 18v v 2 4 1 3c 2 3 9c 4 5 c 4 4 w 0 w c w c 0 w 2 5 2 1 v 20 (1 v) 3 1 6v a 2 c P O(c 6 ), 8 1 v Gh 3 4 0 c 2 1 c 2 0 w c w O (c 6 ), 1 v c 4 1 c 4 0 w O (c 6 );
(20)
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18 3 1 10v v 2 4 0 1 v 2 0 c2 2 w c2 4 w c w c w 5 20 (1 v) 2 2 1 v 3 1 2v a 2 c P O (c 6 ), 8 1 v Gh 1 v 4 0 c4 2 w c w O (c 6 ). 2 1 v
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Note that multiplying (20.a) by c 2 and neglecting terms of O(c 6 ) delivers
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c 2 1 3c 4 3 c 2 0 w
1 4v 4 0 c w O(c6 ) 2 1 v
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i.e., the same result as if c 2 1 and 3c 4 3 would be replaced by (20.b) and (20.d), respectively. Thus, the reduction equations are inherently consistent. Substituting equations (20) into the equations of Tab 2, it turns out that (b-f) are satisfied identically, whereas (a) yields 3 4v 2 3 v 2 K 0 w c 0 w a3 P c P 5 1 v 2 1 v
(21)
with the plate stiffness Ghc 2 a 2 Eh3 . K 2 1 v 12(1 v 2 )
(22)
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Neglecting all terms of O(c 4 ) (note that K a 2 (22) contains the factor c 2 and P (12) is of order O(c 2 ) ) results in the classical Kirchhoff-Love plate-differential equation w
a3 P . K
(23)
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Thus the classical plate theory of Kirchhoff may be regarded as a consistent first-order approximation including boundary conditions involving Kirchhoff’s effective shear forces [7]. Multiplication of (21) with c 2 and neglection of higher-order terms leads to c 2 K 0 w c 2a3 P O(c6 ) .
(24)
3 8 3v 2 K 0 w a3 1 c P. 10 1 v
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With (24) and (15), the second term of the left-hand side of (21) may be replaced by 3 4v 2 3 c a P leading finally to our first main differential equation 5 1 v (25)
c 4 3 c 2 1 O(c 6 ),
(26)
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c 4 1 0 O(c 6 ).
c2 (5 9 )(5 27 ) 1 O(c 6 ), 15
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3c 2 3 9c 4 5
1
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Since the differential equation system in Tab 3 is homogeneous, we have some freedom to reduce the equations. For an arbitrary chosen constant we obtain
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With it, (c) of Tab 3 is satisfied and both (a) and (b) deliver the same main differential equation 3 c 4 1 0 O(c 6 ). 5 27
AC
c 2 1
In earlier papers [4, 13, 15], we did not use this freedom but employed 3 c 2 1 c 2 1 0 O(c 6 ). 2
(27) 1 and obtained 9
(28)
The proper choice of will be given shortly. Before we can calculate the stress resultants, we need, in addition, the corresponding reduction differential equations derived from the original differential system (Tab. 1). With (20), (26) and (27) we find
ACCEPTED MANUSCRIPT
9 5
3c 2 3 9c 4 5 c 4 4 w, c 2 2 w
1
18 2 4 c w , 5
2 2 0 2 v(5 27 ) 2 3 8 3v ac 2 1 1 c w, c 3 , P, O(c 6 ), 5 1 v 10 1 v Gh 1 v (29)
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3 2 0 3 v 4 c 2 1 c 2 1 c w, c 3 1 , O(c 6 ), 2 1 v 1 v 1 2v 0 v c 4 3 c 4 w, 3 1 , O(c 6 ). 6 1 v 1 v
Rendering also these reduction equations consistent, we proceed similar as in the paragraph following equation (20). We multiply (29.a) by c 2 and replace the ensuing quantities by (29.b) and (29.c) leading to
AN US
2 5 (5 27 ) 1 . 5 54
It may be mentioned that the corresponding differential equation system for anisotropic plates does not allow for a free choice of an arbitrary constant since the equation system is inhomogeneous. This will be dealt with in a forthcoming paper [16].
M
Summing up this chapter, the pseudo-reduction method leads to two main differential equations
PT
ED
3 8 3v 2 K 0 w a 3 1 c P 10 1 v 6 c 2 1 c 2 1 0 5
AC
CE
and to the eight reduction differential equations
(30)
ACCEPTED MANUSCRIPT 9 18 2 4 c w , 5 5 2 2 0 v 4 3 8 3v ac 2 1 c w, c 3 1 , P, , 1 v 10 1 v Gh 1 v 3 2 0 3 v 4 c 2 1 c 2 1 c w, c 3 1 , , 2 1 v 1 v 1 2v 0 v c 4 3 c 4 w, 3 1 , , 6 1 v 1 v 18 3 1 10v v 2 4 0 1 v 2 0 c2 2 w c2 4 w c w c w 5 20 (1 v) 2 2 1 v 3 1 2v a 2 c P O (c 6 ), 8 1 v Gh 1 v 4 0 c4 2 w c w O (c 6 ); 2 1 v 25 1 3c 2 3 9c 4 5 c 2 1 O (c 6 ) 36 5 c 4 3 c 2 1 O (c 6 ) 54 4 c O (c 6 )
3c 2 3 9c 4 5 c 4 4 w, c 2 2 w
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1
AN US
(31)
M
With that, all necessary equations are provided to calculate the stress resultants.
mij a
h 2a
PT
ED
4 Stress resultants The stress resultants can either be calculated by the constitutive relations (8) in explicit form or by the usual definition
1
ij
d ,
(32)
CE
h 2a
AC
where (5), (4) and (3) have to be inserted. Due to the special plate-displacement ansatz (3), only plate-stress resultants occur whereas the corresponding disc-stress resultants vanish. We obtain the following relations [7]
M : 1m Ghac 2 1 , 1 , ( 1 , 2 2 w)
1
9 c 2 3 , 3 , ( 3 , 4 4 w) O(c 6 ), 5 9 3 M : 3m Gha3 c 4 1 , 1 , ( 1 , 2 2 w) O(c6 ), 5 2 1 6 m O(c ), 2;
1
9 m33 Ghac 2 2(2 ) 2 w 1 , c 2 4(2 ) 4 w 3 , O(c 6 ) 5
(33)
ACCEPTED MANUSCRIPT 9 m33 Gha3 c 4 2(2 ) 2 w 1 , O(c 6 ), 5 2 1 m33 O(c6 ), 2;
3
(34)
9 Q : 0 m 3 Gh ( o w, 1 ) c 2 ( 2 w, 3 3 ) c 4 ( 4 w, 5 5 ) O(c 6 ), 5 9 2 Q : 2 m 3 Gha 2c 2 ( 0 w, 1 ) c 2 ( 2 w, 3 3 ) O(c 6 ), 5 9 4 Q : 4 m 3 Gha 4 c 4 0 w, 1 O(c 6 ), 5 2 6 m 3 O(c ), 2.
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0
(35)
The equations of equilibrium for a second-order consistent theory written in stress resultants are given by
(36)
ED
M
AN US
10 Q , P, a 11 M , 0Q 0, a 12 Q , 2 1m33 3a 2 c 2 P, a 13 M , 3 2Q 0, a 14 Q , 4 3 m33 9a 4 c 4 P O (c 6 ). a
PT
On inserting (33) - (35) into (36), the governing PDE system of Tab. 1 is recovered, i.e., the equations of (36) and Tab. 1 are equivalent. The infinite system of equilibrium equations
CE
1 m j , a 1 a
1
m3 j c Pj ,
AC
which contain the disc equations as well, is equivalent to the exact problem of the threedimensional elasticity, cf. [5, 14]. Now we insert the reduction differential equations (31) into the relations for the stress resultants (33) - (35) in order to express them exclusively by the main variables w and to obtain [15] K 1 3 8 3v 2 0 Q 2 0 w, v 3 1 , c aP, O(c 6 ), a 2 10 1 v M
1
K 3 8v 2 c (1 v) 0 w, v 0 w, 1 a 10 1 v 3 6 v 2 2 v c 2 3 1 , 3 1 , c a P O (c 6 ); , 5 5 1 v
(37)
ACCEPTED MANUSCRIPT 6 m33 c 2 a 2 P O(c 6 ), 5 3 m33 O(c 6 ), 3 3 2 Q Kc 2 0 w, O(c 6 ) c 2 a 2 0Q O (c 6 ), 5 5 4 6 Q O(c ),
1
3
(38)
9 9 M Ka c 2 (1 v) 0 w, v 0 w, O(c 6 ) c 2 a 2 1M O (c 6 ). 5 5
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Thus it turns out that within the second-order approximation, the higher-order moments and shear forces are expressible by their classical counterparts, whereas the non-classical stress resultants m33 , that neither have a resulting moment nor a resulting shear force, are either directly determined by the external load or to be neglected.
AN US
On substitution of (38) it is seen that the equilibrium equations (36.c-e) are satisfied identically whereas substitution of (37) into (36.a and b) leads to
a3 3 8 3v 2 c P 1 K 10 1 v
10 Q , P a
0 w
11 M , 0Q a
6 c 1 c 2 1 , 0. 5 2
(39)
ED
M
Equation (39.a) is identical to our main differential equation (30.a). By integration of (39.b) and setting the integration constants to zero, the second main differential equation (30.b) is recovered.
PT
The variational principle also delivers boundary conditions [15] for the prescribed stress resultants M and Q or the displacement quantities and w along the boundary of the plate’s midplane as
CE
M n 1M n
AC
Q n 0Q n
or
3 8v
6 v
1 c 2 0 w, 3 c 2 1 , , 10 1 v 5 1 v
(40) or
3 v 2 0 w 1 c w . 10 1 v
The quantities w and may be interpreted as energetic means over the plate thickness. It is
handsome to replace the displacements 0 w and 1 by their means. We introduce therefore
ACCEPTED MANUSCRIPT 3 v 2 0 c 2 wK c 2 1 c w O (c 6 ) 10 1 v c 4 w K c 4 0 w O (c 6 ) 3 8v 2 0 6 v c 2 K c 2 1 c w, 3 c 2 1 , O(c 6 ) 5 1 v 10 1 v 2 K 21 6 c c O (c )
(41)
K c 4 , c 4 1 , O(c 6 ).
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The factor c 2 is applied in (41) because only the terms c 2 0 w, c 2 1 and c 2 1 and their derivatives appear in the differential equations (30, 31) and in the equations for the stress resultants (37, 38). If the factor c 2 in (41) had been omitted, further terms of order O(c 4 ) would have been to be added in the relations.
K a
12 K c 2 (1 v) w, vw,K 1 5(1 v) 3 v c 2 3 , K 3 ,K , 5 6 v 2 2 c a P O(c 6 ), 5 1 v
(42)
M
M K
AN US
For the moments we obtain
ED
and two equivalent variants for the transverse-shear forces 6 v 2 12 1 c 2 w, K v 3 ,K c aP, O (c 6 ), 1 2 5(1 v) 5 1 v K 1 6 2v 2 QK 2 w, K v 3 ,K c aP, O(c 6 ). a 2 5 1 v K a2
(43)
CE
PT
QK
The equilibrium equations now become with (43.b)
AC
1 K Q , P a
wK
a3 6 2 v 2 c P , 1 K 5 1 v
(44)
and with (42) and (43.a) (see the comment after equation (39))
1 M K, QK a
6 c 2 1 c 2 K 0 , 5
and the boundary conditions are
(45)
ACCEPTED MANUSCRIPT M n M K n Q n QK n
or * K ,
(46)
or w wK .
With (41-46) we obtain a most compact representation of the consistent second-order plate theory.
u a 1 3 3 5 5 O(c 7 ) , u3 a 0 w 2 w 2 4 w 4 6 w 6 O(c8 ) .
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5 Displacements We return to our power-series expansion of the displacements (3)
(47)
u a 1 0 1 2 c2 1 4 c4
AN US
Although no integration over the plate thickness has been performed yet, we use the symbol h h , i.e. 2 3c2 . Thus 2 is of the order of c2 O(cn ), since the range of is 2a 2a and a misunderstanding of the notation is not apprehended. In the following, we will consider as many terms in the series expansion as it turns out to be necessary during the following calculation. Guided by the reduction equations (31), the individual displacement coefficients may be represented in power series of c2 themselves, leading to the ansatz
M
3 3 0 3 2 c2 5 5 0 O(c7 ) ,
u3 a 0w0 0w2 c2 0 w4 c4 0 w6 c6
(48)
w w c w c 2
0
2
2 2
2
4 4
ED
2
4 4 w0 4 w2 c2 6 6 w0 O(c8 ) ,
CE
PT
where the new double indexed quantities are terms depending merely on E, v, wK , K and derivatives of the latter. First, we introduce the definition of wK . With (41.a) and (41.b) we have
AC
w0c2 0w0c2 0w2c4 O(c6 ) wK c2
3 4 v c wK O(c6 ). 10 1 v
Thus we can identify w0 wK , 3 v 0 2 w wK . 10 1 v 0
(49)
Before preceding further we have to transfer the reduction equations (31) to wK and K with the result
ACCEPTED MANUSCRIPT
0
9 18 w, 1 3c2 3 9c4 5 c4 4w, c2 2w c2 4w , 5 5 v 2v 2 12 c2 w,K 3 ,K c4 w,K O (c6 ), 2 1 v 1 v 5 (1 v)
c2 1 c4 1
3 10 v K 3 v c2 w,K c4 w, 3 ,K O(c6 ), 2 1 v 10 1 v 4 K 6 c w, O(c ),
(50)
1 2 v K 1 v c4 w, 3 ,K O(c6 ), 6 1 v 6 1 v 18 v 3 1 1 c2 2 w c2 4 w c2 wK c4 wK O (c6 ), 2 5 5 (1 v) 2 1 v 1 v c4 2 w c4 wK O (c6 ). 2 1 v
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c4 3
AN US
It is easy to prove, that on insertion of (50) into the PDE system depicted in Tab. 1, the main differential equations (44) and (45) follow from (a) and (b), whereas (c) - (f) are satisfied identically. From (50), the following ansatz coefficients are identified
(50.d): (50.c): (50.b):
M
0 w,K ,
1
3 10 v K 3 v w, 3 ,K , 10 1 v 2 1 v 3 9 4v 9 1 4 w,K 0w,4 4w,0 3 3 2 9 5 0 . 2 5 (1 v) 5
2
1
CE
(50.a):
w0
ED
(50.e):
1 v wK , 2 1 v 3 1 18 2 2 w wK 4w0 , 2 5 (1 v) 5 1 2v K 1 v 3 0 w, 3 ,K , 6 1 v 6 1 v 2
PT
(50.f):
AC
Replacing the identified quantities (49) and (51) in (48) yields
(51)
ACCEPTED MANUSCRIPT 3 10 v K 3 v u a w,K c2 w, 3 ,K 2 1 v 10 1 v 9 3 9 4v c4 w,K 0w,4 4w,0 3 3 2 9 5 0 2 5 5 (1 v) 1 2 v K 1 v 2 w, 3 , c2 3 2 4 5 0 O(c6 ) , 6 1 v 6 1 v
(52)
4 4 w0 c2 4 w2 6 6 w0 O (c8 ) .
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3 v u3 a wK c2 wK c4 0w4 c6 0w6 10 1 v 1 v 18 3 1 2 wK c2 wK 4w0 c4 2w4 2 5 5 (1 v) 2 1 v
AN US
It turns out that not all displacement coefficients are fixed individually by the reduction differential equations. We will use this freedom to satisfy the traction-boundary conditions along the plate faces and the local equilibrium conditions in the following. To this extent, we calculate the stresses. For isotropic material, Hooke’s law leads to
1 E 1 v u , u3,3 u , u , a 1 v 2 1 2v
E 1 v2
ED
1 E (1 v)u3,3 vu , a (1 v)(1 2v)
PT
33
M
1 K K 2 2 K K (1 v) w, vw v 9c 3 , 3 , , 12 2 3 2 1 v 6 11v v2 v 1 v 4 0 K K c w 24 w (1 v)(10 v)w, 10 1 v 1 2v 1 2v 1 v v(1 v) 4 0 2 (2 v) w,K wK 24 w O(c4 ) , 6 1 2v 1 2v
E 3 2 2 K 24 0 c 4 10v v w 24(1 v) w (1 v )(1 2v) 10 1 2 v(2 v)wK 24(1 v)2 4w0 6 9 3 v(9 v) c4 wK v(1 v) 0w4 v (1 v ) 4w0 5 5 1 v
AC
CE
2
2(1 v)2 2 w4 3v(1 v) 3 2 , 9v(1 v) 5 0 ,
2 2 c2 4 1 v 4w2 v(1 v ) 3 2 ,
4 6 1 v 6 w0 v(1 v ) 5 0 , O (c6 ) ,
(53)
ACCEPTED MANUSCRIPT 1 E u3, u ,3 a 2(1 v)
3
E 2 3 K 3 c w, v 3 ,K 2 1 v 2 4 1 1 2 w,K v 3 ,K 2 4 9 3 9 3 9 4v c4 w,K (1 v) 4w,0 (1 v) 3 2 (1 v) 5 0 10 2 2 10 1 v 9 3 3 1 2 c2 w,K (1 v) 4w,0 (1 v) 3 2 5 2 10 1 v 5 1 4 1 v 4 w,0 (1 v) 5 0 O(c6 ) . 2 2
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AN US
It may be mentioned that on insertion of the stresses (53) into the formula for the stress K K resultants (32), the moments M (42) and Q (43) are recovered (with the use of (24) and (41)) and the relations for the non-classical (higher-order) stress resultants m33 , Q and
M
M (38) are confirmed. The so-far undetermined displacement coefficients do not contribute anything to the stress resultants of a second-order approximation. They may be chosen arbitrarily (e.g., set to zero) without changing the main PDEs (44) and (45) from which wK and K are determined uniquely when the boundary conditions (46) are taken into account.
h 2a
E 0 O ( c4 ) . 2 1 v
PT
3
ED
Therefore, the following determination of the additional displacement coefficients is a complete a posteriori procedure! It may be mentioned further that the shear stresses 3 satisfy already the traction boundary conditions of a first-order approximation
CE
In order to satisfy the traction boundary conditions for the normal stresses 33 within a firstorder approximation, merely 4w0 has to be determined
AC
33
h 2a
1 1 1 v P O(c4 ) 4w0 wK 2 24 1 v
(54)
leading to
33
E 1 v2
1 2 2 K 4 c 9 2 w O(c ) . c 6
Thus with (49), (51) and (54) (and setting 0w4 , 2w4 , 4w2 , 6w0 , 32 and 5 0 to zero), a consistent second-order plate theory may be established which, however, satisfies the traction boundary conditions only up to the order of O(c4 ) .
ACCEPTED MANUSCRIPT
Note that with (54), the inplane stresses (53.a) are completely determined. The traction-boundary conditions within a second-order approximation deliver
33
h 2a
h 2a
0 O ( c6 )
P O ( c6 ) 2
2
w4
2
3
4
1 3 2v w,K 6 5 0 , 2 5 (1 v)
w0
1 1 v wK , 24 1 v
(55)
3 32 7v v3 1 v wK 0w4 6 4w2 27 6w0 . 3 80 (1 v) 2 1 v
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3
Finally, we adopt the local homogeneous equilibrium conditions and find
4
E 1 (1 v)(1 2v) 1 1 5 0 0 O(c6 ) 6w0 wK , , 2 2 1 v 1440 (1 v) 12 1 v
AN US
3, 33,3
1 17 18v v2 2v3 3 1 5 0 wK , ; 3 160 (1 v) 4 1 v E 1 3v 1 v 0 O(c4 ) 5 0 w,K 3 ,K . 2 1 v 120 1 v 120 1 v
w2
, 3 ,3
(56)
ED
M
Two coefficients, which have no influence on the stress distribution, 0 w4 and 0 w6 , remain undetermined. They may be kept for cosmetical reasons or for comparison with results of other authors, which we, however, are not aware of. In summing up, the series expansion for the displacements read
2 1 1 4 2 36(25 13 v ) 6 15 12 v v (3 v )(1 v ) w,K 2 2 4 120 (1 v) c c
CE
PT
1 c2 2 K 4 0 4 u a w,K 9(10 v ) 5(2 v ) w, c w, 30 1 v c2
1 v 2 2 1 v 4 2 4 c 9 2 3 ,K c 9 6 2 4 3 ,K O (c6 ) , 6 1 v c 120 1 v c c
AC
1 vc2 2 K u3 a wK 3 5 2 w 10 1 v c 2 4 1 1 2 2 c4 0w4 18 5 v 5 1 v wK 2 2 4 120 (1 v) c c 2 2 1 v 1 1 25 13v c6 0 w6 0w4 54 2 2 1 v c 720 (1 v)2 1 v c2 4 6 2 K 8 9 10 v v (2 v )(1 v ) w O ( c ) . c4 c6
(57)
ACCEPTED MANUSCRIPT This displacement ansatz delivers the stresses E (1 v)w,K vwK 1 v2 1 5v c2 9(10 v)w,K 9v wK 30 1 v 2 5 2 (2 v)w,K vwK c 2 1 c2 v 9 2 3 ,K 3 ,K , O (c4 ) , 12 c E 1 2 2 33 c 9 wK 2 2 1 v 6 c
3
1 4 25 13v 2 5 3v 4 c 9 6 2 4 wK O (c6 ) , 60 1 v c 1 v c
E 1 2 2 K 3 2 c w, 1 v2 2 c
(58)
AN US
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1 4 25 13v 2 c 3 5 2 w,K 30 1 v c
1 2 2 c v 3 60 K c2 3 5 2 K , 120 c
M
h and the local 2a three-dimensional equilibrium conditions within a consistent second-order approximation. As long as wK and K are determined from (44)-(46) the stress distribution within the plate follows from (58).
PT
ED
which satisfy the traction-boundary conditions along the plate faces
CE
We hardly believe that neither (57) nor (58) could be estimated, even not by an educated guess, a priori.
AC
6 Comparison with other theories The equations (42-46) will be used to compare the proposed theory with those of other authors. 6.1 Schneider’s approach In the paper of Schneider et al. [5], the same uniform-approximation technique in combination with the pseudo-reduction method was applied for anisotropic plates but using - instead of monomials - suitably scaled Legendre polynomials as basis for the Fourier-series expansion in thickness direction. Further details may be found in [14]. The coefficients of the two series expansions are thus differently defined. As has been shown in [5, 14], the list of corresponding quantities is given by
ACCEPTED MANUSCRIPT 1 v 2 0 1 v 2 K c 2 wS c 2 1 c w c 2 1 c w 5 1 v 2 1 v 4 S 40 4 K c w c wc w
(59)
c c c , 2
S
21
2
K
c S c 4 1 c 4 K . 4
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Note that wS and S are geometric means, whereas wK and K are energetic means. The terms c2 wS (59. a) and c2 wK (41.a), therefore, differ from each other. However, the geometric mean and the energetic mean c2 S and c2 K , respectively, coincide within a consistent second-order approximation, cf. (59.c).
1 12 7v 2 K wS a 3 1 c P, 5 1 v 6 c 2 1 c 2 S 0, 5 and the stress resultants are given as
1 12 7v 2 S 1 S c aP, , w, v 3 , 2 5 1 v K 12 v 2 1 c (1 v) w,S vw,S a 5(1 v) 3 6 v 2 2 v c 2 ( 3 S , 3 S , ), c a P . 5 5 1 v
(60)
M
K a2
(61)
PT
ED
QS S M
AN US
When the load resultants are also defined correspondingly, the differential equations for the special case of isotropic materials follow to be
CE
The equilibrium equations
1 S Q , P. a 1 S M , QS 0 a
AC
(62)
lead to (60). Use is made again (compare (24)) of the relation c2 K wS c2 a3 P O(c6 ) . Just by insertion of (59) into (60) and (61), a complete equivalence is established with the corresponding equations (25), (37) and (40) or with (44)-(46). Although the boundary condition written in wS 1 v 2 S w 1 c w 5 1 v
,
(63)
ACCEPTED MANUSCRIPT may have not such a concise appearance as that written in wK , cf. (46. b), both conditions are equivalent by insertion of (59.a). 6.2 Reissner’s approach
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Although Mindlin [17] investigated the influence of rotatory inertia and introduced a shear correction factor of 2 12 , his approach has much in common with Reissner’s to the extent that the theory is frequently referred to as Reissner-Mindlin-plate theory. Reissner’s approach (cf. [18, 19] and later papers) belongs to the class of classical approaches relying on a set of a priori assumptions. In the following we transpose Reissner’s notation to dimensionless coordinates and dimensionless displacement quantities in order to compare his theory with ours. First, he introduces an energetic mean of the transverse displacement [7] 3 v 2 0 ! K c w w 10 1 v
(64)
AN US
wRs 0 w
and an energetically motivated shear correction factor Rs
5 6
(65)
2 5 0.82 are quite similar) and arrives 0.83 and 12 6 finally at the differential equation for the transverse displacement as
M
(note that the numerical values of
a3 1 2v 2 w 1 c P K Rs 1 v
ED
Rs
(66)
PT
a3 6 2 v 2 1 c P. K 5 1 v
CE
Equation (66) coincides with (44), and Reissner’s engineering intuition is remarkable in choosing a set of partly self-contradicting a priori assumptions to arrive at a consistent theory.
AC
In order to introduce stress resultants, Reissner splits the displacements and the transverse shear forces into homogeneous and particular parts by
wRs wHRs wPRs
QRs QRsH QRsP
(67)
and introduces a shear-force potential by
K 1 wHRs, 3 , 2 a a K 6 2v 2 2 wPRs, c aP, . a 5 1 v
QRsH Rs P
Q
(68)
ACCEPTED MANUSCRIPT Combining (68. a and b) yields QRs
K 1 6 2v 2 w,Rs 3 , c aP, . 2 a a 5 1 v
(69)
The moments are defined by
K Rs (1 v) w,Rs vw, a 6 6 v 2 2 ac 2 QRs, QRs, c a P a . 5 5 1 v
Rs M
This leads with (69) after some rearrangements to Rs M
K 12 Rs c 2 (1 v) w,Rs vw, 1 a 5(1 v)
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(70)
(71)
AN US
6 6 v 2 2 c 2 ( 3 , 3 , ), c a P O(c 6 ). 5 5 1 v
M
Note that in deriving (71) use was made of c4 P O(c6 ) and c2 K wRs c2 a3 P O(c6 ) . The 1 1 Rs Rs equilibrium equation QRs, P delivers (66) and M , Q 0 leads to Reissner’s a a second equation 6 5
(72)
ED
c 2 0.
K1 v K . a 2
(73)
CE
PT
On comparing (69, 71) with (43.b, 42), respectively, a one-to-one relation is established between and as
AC
Thus it turns out that Reissner’s theory fully coincides with the theory proposed here up to terms of order O(c 6 ) . 6.3 Marguerre/Hencky’s approach As in the classical Kirchhoff theory, Marguerre [20] and Hencky [21] assume that straight fibres vertical to the plate midplane remain straight and do not change length, but release the normal hypothesis, i.e., 0 w and 1 are independent variables. Referring back to Reissner [18], the transverse displacement wM is understood as energetic mean. Marguerre uses, in contrary to Hencky, the shear correction factor 5 6 . The main difference to Reissner’s theory is the assumption of plane-stress conditions in the plate plane. Thus the stress resultants are immediately given by
ACCEPTED MANUSCRIPT K 1 v M , M, v M, , a 2 5 Eh QM GhS M w,M with GhS . 12 1 v M M
(74)
With the abbreviation
M 3M,
(75)
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and the equilibrium equations (36.a and b), Marguerre obtains, finally, two PDEs
12 K wM a3 1 c2 P , 5(1 v) 6 M c2 M 0 5
12
6 v
AN US
and the “reduction equation”
(76)
12
aP
M 1 c2 w,M c23 ,M c2 , . 5 1 v 5(1 v) GhS 5(1 v)
M
The stress resultants follow in turn as
K 12 1 c2 (1 v) w,M vw,M a 5(1 v) 3 v c2 3 ,M 3 ,M , O(c6 ), 5 K 1 12 QM 2 w,M v 3 ,M c2 aP, O (c6 ). a 2 5(1 v)
(77)
PT
ED
M M
CE
Hencky’s equations are obtained by simply setting the factor 6 5 equal to 1, where ever it appears.
AC
Clearly, wM and M may be identified with wK and K , respectively. The mere difference between (76.a), (77) and (42)-(45) consists in the missing factor 5 v 2 2 caP 6 1 v
in (76.a), (77.a) and (77.b), which is attributed to the plane stress assumption. The advantage of a straight-forward derivation of the theory is counterbalanced by a resulting non-consistent approach. 6.4 Zhilin’s approach
ACCEPTED MANUSCRIPT A quite different approach is proposed by Zhilin [22]. From the beginning, he introduces geometrical averages of the transverse displacement w and the rotation , such that
c 2 wZ c 2 wS ,
(78)
c 2 Z c 2 S c 2 K .
Z
5 . 6v
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In order to omit additional algebraic manipulations, it appears to be most appropriate to compare Zhilin’s theory with Schneider’s. A shear correction factor Z is introduced as (79)
Zhilin’s plate differential equations are given finally as
AN US
2 1 v 2 K wZ a 3 1 c P 1 v z 2 1 12 7v 2 a3 1 c P, 5 1 v 6v 2 c F 0. 1 5
(80)
M
Equation (80.a) coincides with (60.a). The quantity F will be discussed and identified in the following.
ED
The transverse-shear forces and the bending and twisting moments are defined in the following way
K 1 v 2 2 z z z c a P . (1 v) , , v , a 2 1 v
(81)
CE
z M
PT
Qz Gh z w,z z ,
AC
For z , Zhilin applies a Helmholtz decomposition in the plane involving the two (scalar) displacement potentials and F as
z , 3 F, .
(82)
Thus
z , , 3 z , F . From (82) with (20.c), (59.b) and (80.b) it follows immediately that
(83)
ACCEPTED MANUSCRIPT c 4 wS c 4 O(c 6 ), 5 c 2 S c 2 F F O(c 6 ). 6v
(84)
From further equations of Zhilin’s derivation it follows that
(85)
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2 6v 2 z c 2 c 2 1 c w O(c 6 ), 5 1 v 2 2 6 v 2 z 1 12 7v c a 1 c w P O(c 6 ). 5 1 v Gh z 5 1 v
As a consequence the stress resultants read after elimination of , and higher-order terms as
K 1 12 7v 2 w,Z Gh Z 3 F, c aP O (c 6 ), 2 a 5 1 v K 2 6v 2 1 c (1 v) w,Z vw,Z a 5 1 v K1 (1 v)( 3 F, 3 F, ), a 2 v 2 2 c a P O (c 6 ). 1 v
(86)
M
Z M
AN US
QZ
ED
Due to (78.a), the following conclusion may be drawn from a comparison between (61) and (86)
QZ QS O(c 6 ), M
(87)
CE
c 4 ,S
v c2 M K w,S a 3 P O(c 6 ), 5 a 5 1 v 2 c F, O(c 6 ). 6 v S
PT
Z
AC
Due to (60.b), (84.b) and (87.c) do not contradict each other. Obviously, it is not possible to obtain (87.c) from (84.b) by differentiating (84.b) twice and multiplying by c2 , because our second modeling assumption (19) must be observed. Instead an equation similar to (60.b) has 1 Z to be invoked. Firstly, we observe that the equilibrium conditions Q , P and a 1 Z M , QZ 0 lead to the equations (80). Secondly, whereas the transverse shear resultants a of both approaches coincide, the moments exhibit a difference in the O(c 4 ) -terms. This additional term vanishes identically when substituted into the moment equilibrium condition S and may thus be interpreted as an added equilibrium group to M . It is expected that the S
Z
difference between M and M is rather insignificant. The resulting difference in the stress distribution over the thickness of the plate might be compensated by a proper choice of the
ACCEPTED MANUSCRIPT free displacement-ansatz coefficients (cf. chapter 5). Despite of this deviation, Zhilin’s and the consistent second-order approach lead to the same differential equations and may be regarded as equivalent within the order of O(c 6 ) . Again, the engineering intuition which guided Zhilin is admirable. 6.5 Ambartsumyan’s approach
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Ambartsumyan [23] uses a displacement ansatz, which is reversely constructed from a priori assumptions concerning the distribution of shear stresses, resulting in an intrinsically overdetermination of the system of the equations of the linear theory of elasticity. This is compensated by a semi-inversion of Hooke’s law, which assumes 33 to be given. In fact, since Hooke’s law for the stresses 13 and 23 is decoupled from the other stresses (this holds even for monotropic materials), these two stresses can be computed in accordance with the semi-inversion without the knowledge of 33 . This determines 33 afterwards by the corresponding equilibrium condition, which in turn determines all other stresses by use of the semi-inverted Hooke’s law.
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By converting Ambartsumyan’s notation to ours (which is by-far not an easy task and will be dealt with in a forthcoming paper) and introducing isotropy, the equations [23], (II. 3.3-5) or (II. 5.5-7), are rewritten as
, aP,
K 6 c2 A w (1 v) , , , a2 5 1 v
(88)
M
h3 h3 and have been replaced by 1 and 2 , respectively. The 12 12 quantities and , cf. (II. 2.18, 19), have the dimensions N m4 ).
ED
(The ansatz coefficients
PT
Similar to (14), we introduce
, ,
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3 ,
(89)
AC
and obtain from (88) aP,
K 6 c2 A w (2 v), (1 v) 3 , . , a2 5 1 v
(90)
Applying the divergence operator to (90.b) yields in combination with (90.a)
K 6 2v 2 w A c aP . 2 a 5 1 v
Obviously, is of the order O(c 2 ) and thus
(91)
ACCEPTED MANUSCRIPT c2 ac2 P O(c6 ).
(92)
Invoking our first modeling assumption (15) the term c 2 may be replaced by ac 2 P and (91) becomes 6 2v 2 K w A a3 1 c P. 5 1 v
(93)
We return to (90.b). With (92), we have
K 6 2v 2 6 w,A c aP, c 2 3 , . 2 a 5 1 v 5
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If we identify w A with wK or wRs (which is by no means justified by Ambartsumyan’s displacement ansatz nor does he mention that w A is to be understood as an energetic mean of the transverse displacement), (93) coincides with (44).
(94)
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Now, after differentiation with respect to and multiplication by c 2 , it turns out that the second term of the right-hand side of (94) is of order O(c 6 ) and has to be neglected, whereas the third term has due to our second modeling assumption (19) to be retained leading to K 2 A 6 4 c w, c 3 , O(c 6 ). a2 5
(95)
M
c 2 ,
QA ,
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The stress resultants can be calculated by insertion of equations [23], either (II. 2.38-42) or (II. 5.11-14) as
CE
PT
K (1 v) w,A vw,A a 6 1 c2a (1 v)( , , ) v , . 5 1 v
A M
Using (94), (95) and (92), we find
AC
K 6 2v 2 6 w,A c aP, c 2 3 , , 2 a 5 1 v 5 K 12 1 c 2 (1 v) w,A vw, a 5(1 v)
QA
A M
(96)
6 v 2 2 36 c a P ac 4 3 , 3 , , . 5 1 v 25
By comparison of (96.a and b) with (43.b) and (42), respectively, the quantity is identified uniquely as
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c2
5 v Ghc 2 K . 6 1 v
The equilibrium conditions
(97)
1 A 1 A A Q , P and M , Q 0 in combination with (96) yield a a
6 2v 2 K w A a 3 1 c P 5 1 v 6 c 2 1 c 2 0. 5
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(98)
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With w A wK and (97), the differential equations (98.a and b) and (44), (45) coincide and the stress resultants (96.a and b) and (43.b and 42) are identical, respectively. Thus it turns out that Ambartsumyan’s theory could be interpreted as a consistent second-order plate theory, provided that w A is identified as energetic mean of the transverse displacement w , which was not introduced by Ambartsumyan. In [23], Ambartsumyan treats also several assumptions and simplifications. One of them is to neglect the normal stress 33 in thickness direction ( A1 A2 ( A3 ) 0, cf. [23], chapter II.5). The resulting equations coincide exactly with those of Marguerre, (76) and (77) and are, therefore, not dealt with in detail.
M
6.6 Vekua’s approach
PT
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Hierarchical shell theories have been established by Vekua [24, 25]. The approximation order of this theory is defined by the maximum order of polynomials (Legendre polynomials there) which are taken into account, i.e., the first-order theory considers a linear distribution of the displacement quantities with respect to the plate thickness, the second-order theory allows for a quadratic distribution, etc. Specializing the shell equations to plates by neglecting all contributions due to the curvature tensor, we obtain for Vekua’s first-order theory 1 2v . (1 v)2
(99)
CE
K wV 1 a3 P
AC
This is not astonishing since a linear displacement approximation leads to the consistent Kirchhoff equation only, if additionally the plane-stress assumption and the normal hypothesis would be invoked [7]. The quadratic displacement ansatz then leads to the consistent first-order result K wV 2 a3 P.
(100)
The same results (99) and (100) have been obtained by Preusser [26] by using a linear and a quadratic displacement ansatz, respectively. From the cubic displacement ansatz onwards, her higher-order theories coincide - after pseudo reduction - with (25), cf. [7]. Higher-order theories are not further elaborated in Vekua’s book [25]. It would be interesting to compare a fifth-order Vekua theory with the consistent second-order approach. It is
ACCEPTED MANUSCRIPT conjectured that both theories coincide, if additional terms of order O(c 6 ) are neglected within Vekua’s approach (pseudo-reduction approach). 6.7 Reddy’s third-order shear theory
Ry M
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In order to improve Reissner’s plate theory (which is called in Reddy’s nomenclature a firstorder shear theory) Reddy and coworkers [27, 28] propose a third-order shear-deformable plate theory. By introducing an ansatz for the displacements involving cubic polynomials in thickness direction with the three unknowns 0 w and 1 , and assuming further plane-stress conditions in the middle plane of the plate, he defines stress resultants as h 2
x dx 3
3
M ,
h 2
PRy
h 2
x dx 3 3
3
M ,
3
h 2
QRy
h 2
dx 3
Q ,
3
RRy
h 2
2 3 3
x dx3
2
Q ,
ED
h 2
(101)
M
h 2
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and establishes the following equations of equilibrium
PT
1 Ry 4 Ry 4 Ry Q 2 R 2 2 P , P, a h 3 h a ,
(102)
CE
1 Ry 4 Ry Ry 4 Ry M 2 P Q 2 R 0. a 3h h ,
AC
Insertion of (101) into (102) leads to
5c 2 4c 2 15 a 1 0 w 1 1 , P 8 Gh 28(1 v) 7(1 v) 4c 2 17 1 v 1 0 w, 1 c 2 1 , , 0. 1 14 1 v 7(1 v)
(103)
We did not introduce specific symbols since 0 w and 1 coincide in our’s and Reddy’s notation provided that dimensionless coordinates and displacements (1) - (3) are used. Solving for 0 w delivers
ACCEPTED MANUSCRIPT 1 17 K 1 c 2 0 w a3 1 c 2 P. 35(1 v) 7(1 v)
(104)
Equation (104) is identical to (7.4.24) in combination with (7.4.23) in [28] despite of terms of the order O(c 6 ) . By use of (24) we arrive finally at
12 K 0 w a3 1 c 2 P. 5(1 v)
(105)
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Application of the operator rot to (103.b) yields with 1 3 1 , 17 c 2 1 c 2 1 0. 14
(106)
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It may be mentioned that the factor 17c 2 14 in (106) is equal to 17h2 168a 2 which has already been reported by Lewinski [29]. Equation (105) coincides with Marguerre’s (non-consistent) plate theory [20],cf.(76.a), which is derived, however, in a much more straight-forward manner. On comparing (106) with (30.b), it is observed that the factors and
6 1.20 5
M
17 1.21 14
(107)
ED
are quite similar. Anyway, it turns out that Reddy’s third-order shear theory is not a consistent second-order plate theory and does not improve the plate theory of Reissner.
1 (36.d) 0. 9c 2 a 2
CE
(36.b)
PT
The origin of the discrepancy may be explained by the following reasoning. The second equilibrium equation (102.b) may be regarded simply as a combination of (108)
AC
Dividing an equation by c 2 , however, is only justified, if higher-order terms are taken into account additionally (see the discussion in Section 3). Therefore, terms of order c 2 are missing in (106) and cause the discrepancy with (30.b). The same argument applies to (102.a) since it may be interpreted as
36.a
1 1 (36.d), . 9c 2 a 2 a
(109)
Since plane-stress conditions and a constant displacement ansatz for w are assumed, a higher-order load term is not present on the right-hand side of (103.a). By replacing P in (103.a) - in view of (20.f) and (21) - by
ACCEPTED MANUSCRIPT v 2 P 1 c P, 1 v
(110)
and by choosing 9 10 , equations (105) and (25) would coincide. The deviation between (106) and (30.b), however, would remain. 7 Concluding remarks
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By expanding the displacements in thickness directions into an abstract Fourier series with respect to a suitable basis and integrating over the plate thickness, the densities of both the elastic strain energy W and the potential of external forces V per unit of midplane area of the plate appear as power series in the plate parameter c2 h2 (12a 2 ) . By truncating the series expansion in such a way that all terms appearing in W V up to the order c 2 n , n N , are retained and all terms of the order c 2 m , m N are omitted, an Nth -order consistent plate theory is established, where the governing field equations and the associated boundary conditions follow from a variational principle, which delivers 3( N 1) partial differential equations (PDEs) for 3( N 1) unknown displacement coefficients “living” on the twodimensional midplane of the plate. Involving the pseudo-reduction method, the system of PDEs may be reduced to a system of 3( N 1) PDEs of higher (differential) order in main unknowns. In this way, a hierarchy of consistent Nth -order plate theories is established.
M
It is shown that the rather classical Kirchhoff-Love plate theory is a consistent first-order ( N 1) approximation resulting in one ( 1) fourth-order PDE formulated in the transverse displacement w .
PT
ED
The consistent second-order ( N 2) approximation results in a system of two ( 2) PDEs. For isotropic materials, we obtain one PDE of fourth order in the transverse displacement w and one PDE of second order for the transverse-shear-deformation measure , which describes fast decaying edge effects of the plate. In order to treat this quantity properly, a special modeling assumption had to be invoked.
AC
CE
Once a solution of the boundary value problem given by the two main PDEs of the secondorder approximation is obtained only a subset of the coefficients of the series expansion for the displacements is determined uniquely by the reduction equations gained by the pseudoreduction procedure. The remaining coefficients are determined a posteriori from the traction boundary conditions along the plate faces at h 2a and from the local (threedimensional) equilibrium conditions. In this way, the stress distribution within the plate continuum can be calculated up to the same order of accuracy, i.e., beside terms of order O (c 6 ) . The PDE system, the stress resultants and the boundary conditions have been compared with those of other authors. Within a consistent second-order approximation, Reissner’s plate theory coincides completely with our approach. A consistent improvement of Reissner’s plate theory is only possible within a uniform third-order approximation, i.e., if all terms of the order O(c 6 ) would have been retained. This implies a series expansion of the displacements including seventh-order polynomials in (cf. eq. (3)). Other theories treated in the paper
ACCEPTED MANUSCRIPT show more or less pronounced differences. The origin of the discrepancies are discussed in some detail. For anisotropic plates, the PDEs for w and are coupled and may be given in the following schematic form [4, 5, 14, 16]
c D
c 4 D6 w c 4 D 4
4
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a 1 c 2 D 2 P, h 4 2 4 c D 4 w c D0 c D2 0. 2
The symbolic operator Dn is given by n
Dn bi i 0
n , 1i 2n i
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where the coefficients bi , which are different in the operators with and without tilde, depend on the elastic constants and are given explicitly in [5, 14, 16].
M
The first-order approximation leads after pseudo reduction to a Kirchhoff-type PDE for anisotropic plates attributed to Huber [30]. The equations of the second-order approximation coincide with those of Ambartsumyan [23], provided that the transvers displacement w A is (in contrary to his original work) interpreted as an energetic mean and under the condition that none of his suggested additional simplifications are introduced. Details will be given in a forthcoming paper.
ED
The uniform-approximation technique in combination with the pseudo-reduction technique has been used to derive material conservation laws within a consistent second-order plate theory [15]. It has also been applied to shells [11].
CE
PT
Recently, the same technique was applied to beams. Although merely one-dimensional, beam theory is more advanced than plate theory because the series expansion has to be performed with respect to two directions. Thus two parameters have to be considered, a height and a width parameter, which results in a more complex pseudo-reduction problem. First results of this investigation are given in [8].
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Plates with variable stiffness distributions in thickness direction like functionally graded plates and plates with variable thickness are the subject of ongoing research. Acknowledgment The research work of R.K. has been partly funded by a Distinguished Visiting Fellowship of the Royal Academy of Engineering. The generous support is greatly acknowledged. References [1] Szabo, I. (1987) Geschichte der mechanischen Prinzipien, Birkhäuser, Basel [2] Altenbach, J., Altenbach, H., Eremeyev, V. A. (2010) On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Archive of Applied Mechanics 80, 73-92 [3] Naghdi, P. M. (1972) The theory of plates and shells. In: Flügge, S. (Ed.), Handbuch der Physik, Springer, Berlin VI, A2, 425-646
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[18]
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Kienzler, R., Schneider P. (2012) Consistent theories of isotropic and anisotropic plates. Journal of Theoretical and Applied Mechanics 50, 755-768 Schneider, P., Kienzler, R., Böhm, M. (2014) Modeling of consistent second-order plate theories for anisotropic materials. Zeitschrift für Angewandte Mathematik und Mechanik 94, 21-42 Schneider, P., Kienzler, R. (2011) An algorithm for the automatization of pseudo reductions of PDE systems arising from the uniform-approximation technique. In: Altenbach, H., Eremeyev, V. A. (Eds.) Shell-like structures; non-classical theories and applications. Advanced Structured Materials 15, Springer, Berlin, 377-390 Kienzler, R. (2002) On consistent plate theories. Archive of Applied Mechanics 72, 229-247 Schneider, P., Kienzler, R. (2015) On exact rod/beam/shaft theories and the coupling among them due to arbitrary material anisotropies. International Journal of Solids and Structures 56-57, 265-279 Koiter, W. T., Simmonds, J. G. (1973) Foundations of shell theories, In: Becker, E., Mikhailov, G. K. (Eds.) Theoretical and Applied Mechanics. Proceedings of the 13th International Congress of Theoretical and Applied Mechanics, 150-176 Lo, K. H., Christensen, R. M., Wu, E. M. (1977) A higher-order theory of plate deformation, Part I: Homogeneous plates. Journal of Applied Mechanics 44, 663-668 Kienzler, R. (1982) Eine Erweiterung der klassischen Schalentheorie; der Einfluss von Dickenverzerrungen und Querschnittsverwölbungen. Ingenieur-Archiv 52, 311-322; Extended version in: PhD-Thesis, Technische Hochschule Darmstadt, Darmstadt (1980) Krätzig, W. B. (1989) On the structure of consistent plate theories, In: Koiter, W. T., Mikhailov, G. K. (Eds.) Proceedings of the 3rd IUTAM Symposium on Shell Theory. North-Holland, Amsterdam, 353-368 Kienzler, R. (2004) On consistent second-order plate theories. In: Kienzler, R., Altenbach, H., Ott, I. (Eds.) Theories of plates and shells: critical review and new applications. Springer, Berlin, 85-96 Schneider, P. (2010) Eine konsistente Plattentheorie zweiter Ordnung für monotropes Material. Diploma thesis, University of Bremen, Bremen Bose, D. K., Kienzler, R. (2006) On material conservation laws for a consistent plate theory. Archive of Applied Mechanics 75, 606-617 Schneider, P., Kienzler, R. (2016) A Reissner-type plate theory for monoclinic material derived by extending the uniform approximation technique by orthogonal tensor decompostions of the nth-order gradient. Submitted Mindlin, R. D. (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 18, 31-38 Reissner, E. (1944) On the theory of bending of elastic plates. Journal of Mathematical Physics 23, 184-191 Reissner, E. (1945) On the effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 12, 69-77 Marguerre, K., Wörnle, H. T. (1975) Elastische Platten. Mannheim, BI Wissenschaftsverlag Hencky, H. (1947) Über die Berücksichtigung der Schubverzerrungen in ebenen Platten. Ingenieur Archiv 16 (1947) Zhilin, P. (1992) On the Poisson and Kirchhoff plate theories from the point of view of the modern plate theory (In Russian). Izvesta Akademii Nauk Rossii, Mekhanica Averdogotela, Moscow 3, 48-64 Ambartsumyan, S. A. (1970) Theory of anisotropic plates. Progress in Material Science Series II, Technonic, Stamford, Conn.
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Vekua, I. N. (1955) On one method of calculating prismatic shells (In Russia). Trudy Tbilis. Mat. Inst. 21, 191-259 Vekua, I. N. (1982) Shell theory: General methods of construction. Pitman, Boston et al. Preusser, G. (1984) Eine systematische Herleitung verbesserter Plattengleichungen. Archive of Applied Mechanics 54, 51-61; Extended version in: PhD-Thesis, Technische Hochschule Darmstadt, Darmstadt (1982) Reddy, J. N. (1984) A refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures 22, 881-896 Wang, C. M., Reddy, J. N., Lee, K. H. (2000) Shear deformable beams and plates. Elsevier, Amsterdam et al. Lewinski, T. (1987) On refined plate models based on kinematical assumptions. Ingenieur-Archiv 57, 133-146 Huber, M. T. (1929) Probleme der Statik technisch wichtiger orthotroper Platten. Gebetner and Woff, Warsaw
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[24]