Analytical limit solutions for tenth order theory of edge-loaded semi-infinite elastic sheets

w

-('

,

,fr' r

Xin Sun Dale G. Karr

Research Scientist, Advanced Manufacturing Group Associate Professor, Depart. of Naval Arch. and Marine Eng.

B attelle Columbus Laboratories The University of Michigan

Columbus, OH 4320 1-2693 Ann Arbor, MI 48109-2145

ABSTRACT

Analytical solution for the tenth order theory of edge loaded semi-infinite elastic sheet is obtained for the limit state when the sheet thickness approaches infinity. In contrast to some previous studies, this study proves that there is no transition to plane strain solution for the tenth order theory in the limit as plate thickness approaches infinity. This is due to the discrepancy between the original assumption of the theory, which states that stress varies in a parabolic fashion in the thickness direction, and the actual plane strain stress distribution. Fourier transform and inverse Fourier transform are used in obtaining the solution and the results are compared with numerical results obtained previously.

INTRODUCTION

The tenth order theory of stretching of transversely isotropic sheets developed by Reissner (1942) and Clark and Reissner (1984) is applied to boundary value problems for prescribed edge tractions. The homogeneous and isotropic sheet is hounded by two parallel

planes z = ±c which are considered to be traction free, see Figure 1. Solutions for the stress state within the entire sheet can be obtained using Fourier transforms and inverse Fourier transforms (Karr and Choi. 1989, Sun et al., 1996).

In this study, analytical solutions (for the stresses at the center of the loadings) are obtained in the limit as the sheet thickness approaches infinity. Clark (1985) obtained solutions for sinusoidally varying edge loading; analytic solutions for this loading were obtained for the limit condition as the thickness to wave length ratio approaches infinity. Clark noted that his limit solutions for the transverse stress showed the qualitative trend for a transition to plane strain only in the middle portion of the plate. Karr and Chol (1989) obtained solutions for a uniform edge loading of finite length; limit solution as plate thickness approaching infinity was speculated to reach the plane strain value. Sun et al. (1996) obtained solutions for three cases of more general edge loading, and the solutions were compared to the results obtained using finite element method. In this study, analytical solutions for the limit condition are derived for arbitrary edge loading. It is shown that as thickness approaches infinite, the limit solution does not correspond to the solution of plane strain. In this sense, this work serves as a sequel to Sun el al. (1996) and a correction toKarr and

Choi (1989Ys limit solution as aspect ration y - O.

GOVERNING EQUATIONS

Clark and Reissner (1984) derived a tenth-order system of differential equations for approximating the three dimensional corrections to the generalized plane stress theory of stretching of elastic sheets. The stresses are assumed to vary across the plate thickness in the parabolic form which are compatible with the equilibrium equations. According to these assumptions, the stress distributions

are:

(NN,N» (R.R).JRX»

(s,s»

r

(o.a.r)

= + Z"(z),

(r r)

Z'(z),

--Z(z).

(1)

2c 2c ' Zr 2c

Where Z(z) =

(c2

/ 4)(l - z2/ r2)2. The resulting tenth-order system can be reduced to three uncoupled partial differential

equar.ion.s for three stress functions ç, Q and (Clark, 1985): =

0, A0cVc2 - Q

=0, A c4V4yi+A í = 0 (2)

From these equations we can see that if one formally put c=O, then all quantities except o

,o' r, vanish. Equation (2) reduces

to a biharrnonic function which represents the Airy Stress function of elementary plane stress. In the following sections, uniform through-thickness edge loadings P(x)are considered which are applied over an edge distance of 2a (Figure 1), and P(0) ₌

Karr and Chai (1989) applied the method of Fourier transforms to this type of problem in which the edge traction is constant. The development of the governing equations for a more general edge load P(x) is performed in a similar manner by Sun et al. (1996). The resulting system of equations are:

TC1

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ANALYTICAL LIMIT SOLUTIONS FOR TENTH ORDER THEORY OF

EDGE-LOADED SEMI-INFINETE ELASTIC SHEETS

in which, R is the Fourier transform of the edge loading P(x) The five functions A(e). B(). C(). D() and E(e) in equaions (3)-(7) are used to evaluate the Fourier transforms of the stress resultants, and the final stress components can be obtained using

inverse Fourier transforms as shown in Karr and Choi (1989) arid Sun et al. (1996).

LIMIT SOLUTIONS FOR PLATE THICKNESS

c

+

To simplify algebra, we divide both sides of equation (6) by , anddefine as in Clark (1985): (cy)2 =

(c)2

+ 21/2

2 2 2 2

22

2 2

(cß) = (ce) + , (cß) =(ce) +p ,M = (2121)(2 - V

(1-69v

/7O)p 1(3(1 +y)))

2/21 - 1(1 - v)/21.

M 2/21 +

1(1 - v).[i/ 21. Denoting sign()

= /jJ=

J/,equations (3)-(7) become:

2

-

v/(3(1 +v)) lc2

B() -

iy(4c2/21)C(ç)+ [1+

(c)2M ]D(ç)+ [1

+

(c)2

1E()

= 0,

22

₁₍₃₍₁

_{+ v))B()}

_{(2c2 /21)2C()}

+ic2çß

M D()

+

ic2ç ßE()

= 0,

sign() A(e) - B()

+

(2v/l5)c {cß D() 4E()]=

o,

cß D() cE() - icC(c)

= 0.

Now let iC()

= G() and break D(), E(e).ß, ,M. into real and imaginary parts: $ = ¡3,. +iß1 .. =

ßr - j$.

in which, Mr = 2/21, M1 =

(1v)/21, cßr

=

(b+b2

+90/4)12, cß1

=

3I(4cßr)

(13)

(c)2

fact that is

of D() (Karr and

where b= +3. In doing so, we have taken the advantage of the E(e) the complex conjugate

r r i L

In equations (14)-(18), the notation showing the dependency of A,B,C,D and E on has been dropped. Equation (14) leads to Dr =

(RIa0 -

A)/(4vc22 /15). Substitution of Dr back into (17) gives

D1

=[(sign()_ßr/)A_B+(ßr/)(R/aO))/(4VC2ß./15).

22

A () +

(2v/15)c

[D () + E()J = RIa0,

(3)

-

v/(3(1

v))llc2 B() +

+ (2/21)(2 - - 2(1 -69v2 /70)/(63(1 +

v))c22p2]D()

(4)

22

2 2 1 2

21

2 21

-

v

/(3(1 + v))B() (2c /21)

C()

icÇj(c)

+p 12(2 - v)121-2(1 - 69v /70)/(63(1

+ v))IL JD()

2

2r

2

21

+ icV(c)

+ P [2(2 -

v)/21

2(1 -69v /70)/(63(1 +v))p

j

E(e)= 0 (5)

2112

2 -

L2

-

2 I

+(pic) D(ç)+q; +(/c) E(ç)j=O

(6) 2 ₊

(/c)2

D()

2 +

(Ic)2E(ç)

- iC()

= 0 (7)

Choi, 1989). The real quantity equations for equation (8)-(12) are the following:

A+(4v/15)c22D =R/a

(14) r Q

_v/(3(1+v))Jc2B_y(2/2l)G+2[l+ (C)2M]D-2(C)2MD.0,

(15)

22

2 2 2

-

/(3(l+v))B +(2c /21)ç G+2r

[(ß M

ßM)D (ß M+ßM )D]=0,

, (16)

sign() A

B+(/15)(c2)2[ß

Dr

ßD]=O,

r

i i r L (17)

cG+2cß D 2cßD =0.

(18)

i

i

Also from equation (17) and (18) we have: G = (B - sign()A)/(2v c - /15)

i

2

Substituting Dr D and G into equations (15) or (16), dividing both sides of the equations by (c while letting (ce)

*

and using the fact that c2ß =3- OEsign(c)/4,we derive B in the following form: B =

(RIa0)sign()l0(1 v2)/(10 3v2)

As (c)2

, from equation (13) we have: ß,./ =sigri(). Substituting this and the expression of B into D yields:

35v R

'110(10-3v)

In the expression of Dr, since the numerator should be a finite number, as (c)2

*

Dr should vanish.

Having B, G, D Dr for (c)2 -

, we calculate the stiess components at the point (0,0,0) for this limiting case.

Substituting D Dr into equation (A.9) of Sun et al. (1996), we have:

210v R

T(c,0) =

(10-3v2) C

Taking inverse Fourier transform on both sides of equation (19) leads to:

210v 1 °

T(x,0)=

2 J Re d

(10-3v )c2X_

Since R is the Fourier transform of edge loading P, using the property of Fourier transforms (Sneddon, 1951), we have:

- I Red=P(x)

_2r

_{_}

- 210v

a0

, Substitution of T(0,0) into equation (1) yields:

Therefore equation (20) gives: T(0,0)=

(10-3v2)

C

a (0.0.0)

z

4(10-3v2)

2'

2

i

2v

From equation (AA) of Sun et al. (1996) we have:

2sign()B --().L D

+

E) - (A +(c)[D

+EI).

N0 15 15

Substitutions of equation (8) and the values of B, Dr D into the above equation lead to: i(0)=

2cR,so N(0,0)

=

2ca.

_

(°)

2

2

vc IçiB 2

From equation (A.4) of Sun et al. (1996) we have: r

(1 (cß) M}D

+

(1 (cß) M}E

+ + C

N0

3(1v)

21

10v2 Substitutions of equation (9) and the values of Dr D into the above equation lead to: R(,0)

2 2cR, so

10-3v 10v2

R(0,0)

2

2c(a )

Substitution of N(0,0) and R(0,O) into equation (1) gives:

10-3v

0 o(O,0.0) 10v2 a(O,O,±c) 10v2

=1

2ad

--1+2

2

10-3v

a0 10-3v J

For Poisson's ratio y =0.3, equations (21) and (22) give:

a (0,0,0)_t

_a(O,0.0)

_c(0,0,±c)

- 0.80935;

-

1.0924; 0.815 (23)

a0

a0

a0

Note that at the center of the applied load the ratio of the resulting stresses to the applied stress is independent of the distribution of the applied stress. The limit solutions presented in equation (23) correspond very well to the numerical results obtained by Sun et aI. (1996) for three loading conditions in the limit state as aspect ratio y

+

0, which represents infinite sheet thickness.

Furthermore, for infinite sheet thickness c, the elementary plane strain condition is not achieved, since for plane strain -l.O;ando(O,O,O)/ü = -06 forv =0.3.

Substituting N(0.0) and R(0,0) into equation (1)yields the distribution of o along the line (0,0,z) when c .-4 °°:

--1--

3-

(24)

OO 10-3v

From equation (24) we can see that o(O,O. z) is parabolic in z as prescribed in equation (1), andit is also a function of y

-DISCUSSIONS

The analytical solutions for

(°°°) and o,

(0,0,0) obtained in equations (21) and (22) for vanishing aspect ratio are different from the plane strain solutions. The underlying assumptions of the tenth order theory involve aprescribed parabolic distribution of the transverse stress which vanishes at z = ±c . These assumptions preclude obtaining the plane strain condition for which it is assumed that there is no functional dependency with respect to the z coordinate.

Using linear elastic stress-strain relations andstrain-displacement relations, we also find the displacement component in the

z-direction u (0,0.

z) when c + o:

uz(o.o.z)E

[

10v3 105v

-

[

35v 10v3 21v

-1 2v+

2 2 2 2 2

L

10-3v

4(10-3v ) L2(10-3' )

10-3v

4(10-3v

where E is the Young's modulus and i = z fc - Note that the displacement involves contractionwithin the central portion of the sheet and extension in the outer portion of the sheet. This is again in contrast toplane strain conditions in which the transverse displacement is assumed to be zero throughout.

REFERENCES

CLARK, R-A.: REISSNER, E.: A tenth-order theory of stretching of transversely isotropic sheets. ZAMP 35(1984), 883-889. CLARK. R.A.: Three-dimensional corrections for a plane stress problem. mt. J. Solids Structures 21(1985)1, 3-10.

KARR. DG.; CHOI, S.K.: Three-dimensionalelasticity solutions for edge loaded semi-infinite sheets. Zangew.Math.Mech. 69(1989) 10, 329-337.

REISSNER, E.: On the calculation of three-dimensional corrections for thetwo-dimensional theory of plane stress. In: Proc. 15th Semi-Annual Ea.srern Photoelosticity conference1942, pp. 23-3 1.

SNEDDON, I.N.: Fourier transforms, 2 edition, McGraw Hill Book Co., Inc., New York, 1951.

SUN. X.. KARR. D.G. and HAN, C.: Acombined analytical and numericalapproach for the solution of an edge loaded semi-infinite elastic sheet, Proc. ASME Computers in Engineering Conference, August 18-22, 1996, Irvine, California.

Fig. I Semi-in.firtire sheet withcompressive edge loading.