Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

4, 39, 2001

MICROPOLAR PLATES SUBJECT TO A NORMAL

POLYHARMONIC LOADING

Grzegorz Jemielita

Institute of Structural Mechanics, Warsaw University of Technology

e-mail: gjemiel@il.pw.edu.pl

Following thepreviouslypublished considerations thepresentpaperaims
at determination of a displacement vector and infinitesimal rotation vec-
tor describing the bending of theGrioli-Toupinplate subject to a normal
polyharmonic loading.The presented biharmonic representation reduces
the problem of equilibrium of such a plate to a non-homogeneous bihar-
monic equation involving a function of plate deflection. A semi-inverse
method in an explicit form has been obtained together with relation-
ships for force and moment stresses. Formulas for determination of the
functions gi and fi of the variable ξ and coefficients Ai in a recurrent
form have been given as well.

Key words: Grioli-Toupin material, micropolar plates, polyharmonic
loading

1. Introduction

Exact distributions of displacements and streses in a plate subject to lo-
adings normal to constraining planes were found within the symmetric the-
ory of elasticity by making use of various methods described by Jemielita
(1991). The problemwas reduced to the determination of a solution to a non-
homogeneous biharmonic equation.
The displacement and stress distribution corresponding to the non-

homogeneous equation and satisfying the equations of the symmetric theory
of elasticity was found in the case of constant loadings, see Garbedian (1925),
Love (1927), Sokołowski (1958) and Lekhnitskǐı (1963), the loadings being
harmonic functions, see Gutman (1940), Stevenson (1942), Hata (1953) and
Negoro (1954), polyharmonic ones (Jemielita, 1993), for constant mass for-
ces (Gutman, 1941), and biharmonic (Dougall, 1904). Numerous methods of



960 G.Jemielita

asymptotic expansions, such as power series, Legendre’s polynomials infinite
differential operators, Birkhoff’smethod, and semi-inversemethod, see Jemie-
lita (1991, 1993), were used for that purpose.
A generalized plane stress state (GPSS) in a plate made of the Grioli-

Toupin material was defined by Jemielita (1992b). The bending problem in
such a plate was reduced to a single biharmonic equation involving a function
of deflection. In the paper by Jemielita (1992a) non-homogeneous flexure pro-
blems were considered in a plate under its own weight and normal uniform
loading acting on its faces.
Now, we investigate a plate under the normal loading q(xα) acting on the

faces antisymmetrical with respect to the middle plane. We assume that the
function q(xα) satisfies the equation

∇
2n+2q(xα)= 0 (1.1)

where n is an arbitrary natural number or zero.
The summation convention is adopted. Latin indices have the range 1, 2, 3,

Greek indices - take the values 1, 2 only.Commadenote partial differentiation.
We introduce also the non-dimensional variable ξ

ξ =
2z
h

(1.2)

where z is the coordinate normal to themiddleplane of the plate.Aderivative
with respect to this variable we denote as d(·)/dξ =(·)′.
The constitutive equations of an isotropic, homogeneous and centrosym-

metricmediumare assumed in the following form (Nowacki, 1970; Sokołowski,
1972)

σij = µ
(
uj,i+ui,j +

2ν
1−2ν

uk,kδij
)
−
1
2
ǫkjiµ

lk
,l

(1.3)

µij = γ(ϕi,j +ϕj,i)+ε(ϕi,j −ϕj,i)= 4µl
2(ϕi,j +ηϕj,i)

where
ϕi =

1
2
ǫi
jkuk,j l

2 =
γ +ε
4µ

η =
γ −ε

γ +ε
(1.4)

and
σji – stress tensor
µij – couple-stress tensor
ǫkji – Levi-Civitá’s permutation symbol
δij – Kronecker’s delta
ui – components of the displacement vector



Micropolar plates subject to... 961

ϕi – components of the vector of infinitesimal rotations
µ – Lamé’s constant
ν – Poisson’s ratio
γ,ε,l,η – material constants of the Grioli-Toupin material.
Neglecting the body forces one can write the equations of equilibrium in

the following form

σji,j =0 ǫ
ijkσjk+µ

ji
,j =0 (1.5)

while the equilibrium equations expressed in terms of displacements can be
cast as follows

∇̃
2ui+

1
1−2ν

uk,ki+ l
2
∇̃
2(uk,ki−∇̃

2ui)= 0 (1.6)

where ∇̃2 represents the Laplace operator in R3

∇̃
2 =∇2+

∂2

∂z2
=∇2+

4
h2

∂2

∂ξ2
(1.7)

2. Biharmonic representation of the non-homogeneous problem

Let us investigate a plate of the thickness h subject to a normal loading
acting on the faces x3 = z =±h/2 and satisfying equation (1.1).
The solution to the differential equations (1.6) with the boundary condi-

tions on the faces

σ3α
(
xβ,±

h

2

)
=0 σ33

(
xβ,±

h

2

)
=±

q

2
µ3i
(
xβ,±

h

2

)
=0

(2.1)
will be sought with the help of the semi-inverse method.
Let us present the components of the displacement vector ui and the

components of the infinitesimal rotation vector ϕi in the form (Jemielita,
1992b)

uα(x
β,ξ)=−

h

2
ξw,α(x

β)+h3f(ξ)∇2w,α(x
β)+

1
µ

n∑

i=0

h2i+1fi(ξ)∇
2iq,α

(2.2)

u3(x
β,ξ)= w(xβ)+h2g(ξ)∇2w(xβ)+

1
µ

n∑

i=0

h2i+1gi(ξ)∇
2iq



962 G.Jemielita

ϕα(x
γ,ξ)= ǫα

β
[
w,β +h

2t(ξ)∇2w,β +
1
2µ

n∑

i=0

h2i+1
(
gi(ξ)−2f

′

i(ξ)
)
∇
2iq,β
]

(2.3)

ϕ3 =0

where fi(ξ) and gi(ξ) are the unknown functions. They satisfy the conditions

fi(ξ)=−fi(−ξ) gi(ξ)= gi(−ξ) (2.4)

The functions s(ξ), f(ξ) and t(ξ) are defined by (see Jemielita, 1992b)

f(ξ) = −
2−ν
48(1−ν)

ξ(C2− ξ
2)−k2

sinh(k̂ξ)

sinh k̂

g(ξ) = −
1

24(1−ν)

[
6
(
1−

ν

2
ξ2
)
− (2−ν)C2

]

(2.5)

t(ξ) =
1
2
(g−2f ′)=

=
1

24(1−ν)

[
(2−ν)C2−3−3(1−ν)ξ

2+12k(1−ν)
cosh(k̂ξ)

sinh k̂

]

where
k =
1
h

k̂ =
1
2k

The constant C2 determines the physical meaning of w(xα). In the theories
of flexural plates the following functions representing the plate deflection are
used:
— deflection of the plate faces ŵ(xα)

ŵ(xα)
def
= u3

(
xα,±

h

2

)
(2.6)

— deflection of the mid-plane
♦
w (xα)

♦
w (xα)

def
= u3(x

α,0) (2.7)

— simple average
∗

w (xα)

∗

w (xα)
def
=
1
h

h/2∫

−h/2

u3(x
α,z) dz (2.8)



Micropolar plates subject to... 963

—weighted average
◦

w (xα)

◦

w (xα)
def
=
3
2h

h/2∫

−h/2

(
1−4

z2

h2

)
u3(x

α,z) dz (2.9)

Then using the second equation of (2.2) and definitions (2.6)-(2.9) we ob-
tain the following values of the constant C2
— for ŵ(xα)

C2 =3 gi(±1)= 0 (2.10)

— for
♦
w (xα)

C2 =
6
2−ν

gi(0)= 0 (2.11)

— for
∗

w (xα)

C2 =
6−ν
2−ν

1∫

−1

gi(ξ) dξ =0 (2.12)

— for
◦

w (xα)

C2 =
3(10−ν)
5(2−ν)

1∫

−1

(1− ξ2)gi(ξ) dξ =0 (2.13)

It can be clearly seen that, depending on the definition of deflection, the for-
mulae for displacements and stresses assume different forms.We assume that
the function w(xα) satisfies the equation

D∇
4w =

n∑

i=0

h2iAi∇
2iq (2.14)

where D is the rigidity of the micropolar plate determined by

D= Dθ θ =1+24(1−ν)k2 D =
µh3

6(1−ν)
(2.15)

and Ai are some unknown constants.



964 G.Jemielita

UsingEqs (1.3), (2.2) and (2.3) we arrive at the following formulae for the
stresses and couple stresses

σαβ =−
2µ
1−ν

{h
2
ξ
(
(1−ν)w,αβ +ν∇

2wδαβ
)
+

+
h3

48

[
(2−ν)ξ(C2− ξ

2)+48(1−ν)k2
sinh(k̂ξ)

sinh k̂

]
∇
2w,αβ

}
+

+
2

1−2ν

n∑

i=0

h2i
{
h2(1−2ν)fi∇

2iq,αβ +ν
[
fi−1+2g

′

i+
6(1−ν)

θ
s
]
∇
2iqδαβ

}

σα3 =−
µh2

4(1−ν)

[
1− ξ2+8(1−ν)k

cosh(k̂ξ)

sinh k̂

]
∇
2w,α+

+
n∑

i=0

h2i+1
{
gi+2f

′

i −k
2
[
gi−1−2f

′

i−1+4(g
′′

i −2f
′′′

i )+
12(1−ν)

θ
tAi
]}
∇
2iq,α

(2.16)

σ3α =−
µh2

4(1−ν)
(1− ξ2)∇2w,α+

+
n∑

i=0

h2i+1
{
gi+2f

′

i +k
2
[
gi−1−2f

′

i−1+4(g
′′

i −2f
′′′

i )+
12(1−ν)

θ
tAi
]}
∇
2iq,α

σ33 =
2

1−2ν

n∑

i=0

h2i
[
2(1−ν)g′i+ν

(
fi−1+

6(1−ν)
θ

sAi
)]
∇
2iq

µαβ =4µh
2k2
[
ǫβ
γ(w,γα+h

2t∇2w,γα)+ηǫα
γ(w,γβ +h

2t∇2w,γβ)
]
+

+2h2k2
[
ǫβ
γ
n∑

i=0

h2i+1(gi−2f
′

i)∇
2iq,αγ +ηǫα

γ
n∑

i=0

h2i+1(gi−2f
′

i)∇
2iq,βγ

]

µ3α =−2µk
2h3ǫα

β
[
µ
(
ξ−
sinh(k̂ξ)

sinh k̂

)
∇
2w,β −2

n∑

i=0

h2i−1(g′i−2f
′′

i )∇
2q,β
]

µα3 = ηµ3α µ33 =0

3. Functions fi, gi and coefficients Ai

Substituting (2.2) and (2.3) into Eqs (1.6), we find nonzero solutions to
this set when the function w(xα) satisfies Eq. (2.14) and the functions fi(ξ)



Micropolar plates subject to... 965

and gi(ξ) satisfy the following simple ordinary differential equations

4k2fIVi −f
′′

i = F(ξ)

g′′i =−
1−2ν
8(1−ν)

gi−1−
1

4(1−ν)
f ′i−1−

(3.1)

−
3
4θ

[
(1−2ν)g+2f ′− (1−2ν)k

cosh(k̂ξ)

cosh k̂

]
Ai+

+
1−2ν
8(1−ν)

k2
[
gi−2−2f

′

i−2+4(g
′′

i−1−2f
′′′

i−1)+
12(1−ν)

θ
tAi−1

]

where i =0,1, ...,n and

F(ξ)=
1

2(1−2ν)

[
(1−ν)fi−1+g

′

i+
6(1−ν)2

θ
fAi
]
+

+
k2

2

[
g′i−1+4g

′′′

i −2f
′′

i−1−
3(1−ν)

θ

(
ξ−
sinh(k̂ξ)

sinh k̂

)
Ai
]

gi(ξ)= fi(ξ)= 0 for i < 0

Boundary conditions (2.1) can be rewritten as follows

gi(1)+2f
′

i(1)+k
2
[
gi−1(1)−2f

′

i−1(1)+4
(
g′′i (1)−2f

′′′

i (1)
)
+

+
12(1−ν)

θ
t(1)Ai

]
=0

(3.2)

2(1−ν)g′i(1)+ν
[
fi−1(1)+

12(1−ν)
θ

f(1)Ai
]
=
1−2ν
4

δ0i

g′i(1)−2f
′′

i−1(1)= 0 i =0,1, ...,n

Adefinite integral of the third equation of the first of the equilibrium ones
(1.5) within the limits 0, 1 combined with boundary conditions given by Eqs
(2.1)2 and (3.2)3, yields the following formula for Ai

Ai = δ0i+
1∫

0

gi−1 dξ +2fi−1(1)−

(3.3)

− k2
[ 1∫

0

gi−2 dξ −2fi−2(1)+
12(1−ν)

θ
Ai−1

1∫

0

t dξ
]



966 G.Jemielita

The solution to system (3.1) can be written in the following form

gi = −
1−2ν
8(1−ν)

ξ∫

0

ξ∫

0

gi−1 dξdξ−
1

4(1−ν)

ξ∫

0

fi−1 dξ −

−
3
4

[
(1−2ν)

ξ∫

0

ξ∫

0

g dξdξ+2

ξ∫

0

f dξ−4(1−2ν)k3
sinh(k̂ξ)

sinh k̂

]Ai
θ
+

+ k2
1−2ν
8(1−ν)

[ ξ∫

0

ξ∫

0

gi−1 dξdξ−2

ξ∫

0

fi−2 dξ +4(gi−1−2f
′

i−1)+

+
12(1−ν)

θ
Ai−1

ξ∫

0

ξ∫

0

t dξdξ
]
+Ci

(3.4)

fi = B1ik̂ξ+B2i sinh(k̂ξ)−

ξ∫

0

[
k̂(ξ −ψ)− sinh

(
k̂(ξ −ψ)

)]
F(ψ)dψ

The constants B1i, B2i can be determined from boundary conditions (3.2)1,3,
while the coefficients Ci result fromconditions (2.10)-(2.13), dependingon the
considered deflection. The obtained recurrent formulae (3.3) and (3.4) allow
one to determine all the sought-after functions fi, gi and coefficients Ai.
Up till now, equations (2.2), (2-3), and (3.4) have not been reported in

the literature in such a general form. These equations allow for an accurate
representation of displacement vector (2.2) for a variety of deflections w(xα)
leading to a solution to non-homogeneous equation (2.14). The exact values
of the coefficients Ai, appearing in this equation can be obtained explicitly
from Eq. (3.3).

4. Plate made of the Hooke material

Stresses and displacements occurring in plate made of the Hooke material
can be obtained from the limit passage as l → 0. Calculating the limiting
values of Eqs (2.2), (2.5), (2.16), (3.3) and (3.4) as k → 0 we arrive at
the formulae for displacements, stress and coefficients Ai given by Jemielita
(1993).



Micropolar plates subject to... 967

References

1. Dougall J., 1904, An Analytical Theory of the Equilibrium of an Isotropic
Elastic Plate,Trans. R. Soc. Edinburgh, 41, Part 1, 8, 129-228

2. Garabedian C.A., 1925, Solution du probléme de la plaque rectangulaire
épaisse encastrée ou posée, portant une charge uniformément répartie ou con-
centrée en son centre,C. R. Acad. Sci., 180, 257-259, (Errata p.1191)

3. Gutman S.G., 1940, Raschet tolstykh uprugikh plit pod nepreryvno raspre-
delennym davleniem, Izv. Nauch. Issl. Inst. Gidrotekh., 28, 212-238

4. Gutman S.G., 1941, Raschet tolstykh uprugikh plit pod děıstviem sobstven-
nogo vesa, Izv. Nauch. Issl. Inst. Gidrotekh., 29, 153-156

5. Hata K.-I., 1953, On the Thick Plate Problem I, Mem. Fac. Eng. Hokkaido
Univ., 9, 3, 428-477

6. Jemielita G., 1991, Meandry teorii płyt, Prace Naukowe Politechniki War-
szawskiej. Budownictwo, 117,Warszawa

7. Jemielita G., 1992a, Bending of a Cosserat Plate Under its OwnWeight and
Normal Uniform Loading, J. Theor. Appl. Mech., 30, 2, 369-377

8. Jemielita G., 1992b, Biharmonic Representation in the Analysis of Plates
Made of the Grioli-ToupinMaterial, J. Theor. Appl. Mech., 30, 1, 9-99

9. Jemielita G., 1993, Ścisłe równania teorii płyt i tarcz, Prace Naukowe Poli-
techniki Warszawskiej. Budownictwo, 24, Warszawa

10. Lekhnitskij S.G., 1963,Ploskoenapryazhennoe sostoyanie i izgib neodnorod-
nǒı transversal’no-izotropnǒı plity, Izv. AN SSSR OTN 1, 61-67

11. Love A.E.H., 1927, A Treatise on the Mathematical Theory of Elasticity, 4
edit., Oxford

12. Negoro S., 1954, On a Method of Solving Elastic Problems of Plates, Proc.
4th Japan Nat. Congr. Appl. Mech., 153-156

13. Nowacki W., 1970,Teoria sprężystości, PWNWarszawa

14. Sokołowski M., 1958, The Bending of Transversally Non-Homogeneous Pla-
tes ofModerate Thickness,Arch. Mech. Stos., 10, 3, 315-328;Bull. Acad. Pol.
Sci. Sér. Sci. Tech., 6, 4, 181-186

15. Sokołowski M., 1972,O teorii naprężeń momentowych w ośrodkach ze zwią-
zanymi obrotami, PWNWarszawa

16. Stevenson A.C., 1942, On the Equilibrium of Plates,Phil. Mag., Ser. 7, 33,
224, 639-661



968 G.Jemielita

Poliharmoniczne obciążenia płyt mikropolarnych

Streszczenie

Wniniejszej pracy (korzystając zwcześniej opublikowanych rozważań)wyznaczo-
no przedstawienie wektora przemieszczenia i infinitezymalnego obrotu, opisujące zgi-
nanie płytywykonanej zmateriałuGrioli-Toupinawywołaneobciążeniemnormalnym
poliharmonicznym.Przedstawiona reprezentacja biharmoniczna sprowadza zagadnie-
nie równowagi takiej płyty do rozwiązania niejednorodnego równania biharmonicz-
nego na funkcję przedstawiającą ugięcie płyty. Metodą półodwrotną uzyskano wzory
na naprężenia siłowe i momentowe w postaci jawnej. Podano też, w postaci rekuren-
cyjnej, wzory na poszukiwane funkcje gi, fi zmiennej ξ oraz współczynników Ai.

Manuscript received November 22, 2000; accepted for print February 5, 2001