81
ON CERTAIN INTEGRABLE NONLINEAR MEMBRANE SOLUTIONS*
BY
CHIEN-HENG WU
University of Illinois, Chicago
1. Introduction. We study the class of axially symmetrical deformations of an
elastic sheet characterized by a strain energy function satisfying certain inequalities
pertinent to finite elasticity. It is found that the system of nonlinear equations can be
reduced to quadratures if either the undeformed or the deformed middle surface is
cylindrical.
When the undeformed state is a cylindrical surface, the equation of equilibrium for
the direction tangent to the meridian curve has a first integral if the strain energy
function does not depend on position explicitly. This fact has been discovered independ-
ently by Pipkin [1] and Wu [2], While a specific problem was solved in [2], the deduction
given in [1] was completely general. The fact that the equation of equilibrium for the
direction normal to the middle surface has a first integral is known [3]. We use these
two integrals to obtain the general solution of our problem.
When the deformed state is a cylindrical surface, the equations of equilibrium reduce
to very simple forms which can be integrated even if the material is meridianly inhomo-
geneous.
2. Formulation. We consider the class of axially symmetrical problems in which
the middle surface of an elastic sheet forms a surface of revolution before and after
deformation. Let the middle surface M of the undeformed sheet be characterized by a
meridian curve C and let the middle surface m of the deformed sheet be characterized
by a meridian curve c. The thickness H of the undeformed sheet is assumed to be con-
stant.
We choose a cylindrical coordinate system (R, 9, Z), the z-axis of which coincides
with the axis of symmetry of the sheet. We assume that any point P(R, 9, Z) in M is
carried by the deformation to the point p(r, 9, z) in m. Let C have arc length S and let
c have arc length s. We introduce a set of unit vectors (a! , a2 , a3), a.i being tangent to
the curve c and pointing to the direction of increasing s, a2 being tangent to the lines of
azimuth and pointing to the direction of increasing 6. From the symmetry of the system
it follows that (a! , a2 , a3) are the principal directions of strain at P. If we denote by
Xi , X2 , X3 the principal extension ratios in these three directions, then
= ds/dS, (2.1)
X2 = r/R, (2.2)
X3 = h/H (2.3)
where h is the thickness of the deformed sheet.
* Received April 23, 1969. The research reported herein was supported by the Air Force Office of
Scientific Research under Grant AFOSR-537-67. The paper was written while the author was a visiting
member at the Courant Institute of Mathematical Sciences, New York University.
82 C. -H. WU [Vol. XXVIII, No. 1
We now assume that the elastic sheet is characterized by a strain energy function
<o = u(Xt , X2, S). It is shown in Appendix I that, for such an elastic sheet, the meridian
and azimuthal stress resultants ti and l2 are given by
ti = ~r~ (2.4)
a2
t2 = — uXa (2.5)
where the subscripts on w denote partial differentiation with respect to the indicated
argument. We require that the strain energy function oj satisfies the conditions
— — ^ ^ (^X,X, — '°x") ^ X"WX,X. > 0, — C0xax, > 0.
(2.6)
These inequalities correspond respectively to the conditions of "invertibility of force-
stretch" and "tension-extension" discussed by Truesdell and Toupin [4],
It is also shown in Appendix I that the stress resultants satisfy the equations of
equilibrium
d.(rti)/ds = t2 clr/ds. (2.7)
ti(d<t>/ds) + (sin <j>)t2/r = q (2.8)
where — qa3 is the externally applied force per unit deformed area and <£ is the tangent
angle such that
dr/ds = cos <j>, (2.9)
dz/ds = sin <f>. (2.10)
Thus we have eight equations (2.1), (2.2), (2.4), (2.5), (2.7), (2.8), (2.9) and (2.10)
for the eight unknowns Xj , X2 , t, , t2 , r, z, s and 4>. In general, the unknown h does not
enter into the analysis. If the elastic sheet is made of Mooney material (cf. Appendix I),
then the thickness h of the deformed sheet can be determined from the condition of
incompressibility
h = H\3 = H/XtX2 . (2.11)
Various numerical procedures are used to integrate the system of equations. In
Sees. 3 and 4, we show that the system can be reduced to quadratures if either the un-
deformed or the deformed middle surface is cylindrical.
3. Deformation from a tube. The middle surface M of the undeformed sheet is
assumed to be a circular cylindrical surface and the elastic sheet is homogeneous, i.e.
co does not depend explicitly on S.
Let M be generated by the meridian curve
C: R = I, Z = S, 0<<S<L<OO. (3.1)
The principal extension ratios become
= ds/dZ, (3.2)
X2 = r. (3.3)
1970] ON INTEGRABLE NONLINEAR MEMBRANE SOLUTIONS 83
The problem now is to solve the system of Eqs. (3.2), (3.3), (2.4), (2.5), (2.7), (2.8),
(2.9) and (2.10).
Eq. (2.7) can be written as
dirt^/dr = t2 (3.4)
if dr/ds does not vanish. Multiplying (3.4) by Aid\2/dr (= Xi) and using (2.4), (2.5),
and (3.3), we obtain
wx, d\2/dr = Xt du^Jdr
= dfyiOSxj/dr — d\/dr.
Hence,
w — = a (3.5)
where a is a constant of integration. This deduction was given by Pipkin [1].
If the elastic sheet is made of isotropic incompressible Mooney material (cf. Ap-
pendix I), then (3.5) reduces to
X? - 3(rX,r2 + fc(rXi)2 - 3fcX^2 - r2 - kr'2 = a.
This expression was obtained by Wu [2], using a different approach.
We consider in great detail the general solutions of the following two cases.
Case I. Pressure q is a given function of r.
Since X2 = r, (3.5) can be solved to yield
Xi = X*(r, a). (3.6)
This is guaranteed by the fact that the implicit function condition for (3.5) is satisfied
by the assumptions (2.6) imposed on u. Then Eqs. (2.4), (2.5) yield
ti = t*(r, a) = o)Xl(X? , r)/r, (3.7)
<2 = t%(r, a) = wXj(Xf , r)/Xf . (3.8)
It is noted that for a given problem, i.e. for a specific value of a, the values of Xj, tl and t2
at a point p(r, 9, z) of m depend only on r but not on z.
Multiplying (2.8) by r(dr/ds) and using (2.7), we obtain
dr
Tds
d , , . . dr
Integration j'ields
rti sin <t> =
J q(r)r dr + b (3.9)
where b is a constant of integration. Eq. (3.9) shows that b is proportional to the total
(axial) force applied to a circular cross-section of the sheet. If rh ^ 0, then (3.9) implies
<j> = <£*(r, a, b) = Sin
Q" qr dr + bj^rttj* (3.10)
S4 C. -H. WU [Vol. XXVIII, No. 1
From equation (3.2)
ds/dZ = (ds/dr)(dr/dZ) = X, (3.11)
and hence
dr/dZ = Xi cos <fi. (3.12)
This equation can be integrated to yield
Z
where r0 = r|z=0-Itis also convenient to set
s = 0, z = 0 at Z = 0. (3.14)
It follows from (2.9), (2.10) and (3.14) that
C' fir
s = s*(r,a,b,r0) = (3-15)
z = z*(r, a, b, r0) = f tan </>*(rj) drt . (3.16)
J T 0
It should be mentioned that the integrals appearing in equations (3.14), (3.15) and
(3.16) are improper when cos 4>* = 0. However, a simple calculation given in Appendix II
shows that these integrals do exist.
The system of Eqs. (3.2), (3.3), (2.4), (2.5), (2.7-2.10) is of fifth order and has the
general solution given by (3.6), (3.7), (3.8), (3.10), (3.13), (3.15) and (3.16). Since we
have conveniently set up the coordinates in such a way that (3.14) is satisfied, the
general solution contains only three constants a, b and r0. These constants must then be
chosen to satisfy the set of transcendental equations corresponding to a given set of
boundary conditions. The number of solutions of these transcendental equations will
be considered in subsequent work.
We now consider the solution of the initial value problem with the initial conditions:
s = 0, 2 = 0, r = r0 , <j> = 4>0 , tx = t0 at Z = 0. (3.17)
Substituting the third and the fifth conditions of (3.17) into (2.4), we find
Xl|z-0 = Xo(fo l to)
since cox^./Mz-o ^ 0- Then (3.5) yields
a = a0(r0, t0) = (co — XjtoxJIx.-x,, ; x,-r„ • (3.18)
Thus, the solution of the initial value problem is
Xi = X?(r, r0 , t0) = X*(r, a0(r0 , <„)), (3.19)
t, = t°(r, r0 , t0) = «Xl(x; , r)/r, (3.20)
t2 = t°(r, r„ , <„) = «x.(x; , r)/x; , (3.21)
4> = <#>°(r, r0 , t0 , <t>0) = sin-1 ^ dr, + r0«0 sin [rt\ , (3.22)
1970] ON INTEGRABLE NONLINEAR MEMBRANE SOLUTIONS 85
r dr
Z = Z°(r, r0, to, to) = I xo(ri) , (3.23)
r dr
s = s°(r, r0 , , *„) = J^ cos ^T) - (3.24)
z = a°(r, r0 , t0 , <t>o) = ^ tan <£°(r,) . (3.25)
The problem of a circular cylindrical sheet stretched in the direction of its axis by
pulling the end circles apart while maintianing them undeformed has been considered
by Stoker [5], The equations were solved numerically by a finite difference method.
This step can be eliminated by using the present results. To illustrate the procedure,
let the radius of the cylindrical sheet be unity and let the length be 2L. The sheet is
stretched to a length 21 > 2L. We choose a cylindrical coordinate system (It, 6, Z),
the Z-axis of which coincides with the axis of symmetry of the cylindrical sheet; the
plane Z = 0 contains one of the end circles. Because of the symmetry of the deformation,
we consider only one half of the sheet, i.e. 0 < Z < L.
The problem now is to solve the system of Eqs. (3.2), (3.3), (2.4), (2.5), (2.7-2.10)
with 5 = 0 subject to the boundary conditions:
s = 0, z = 0, r = 1 at Z = 0 (3.26)
z = I, <i> = tt/2 at Z = L (3.27)
The solution is given by f*(r, a, b, r0) where / is a generic symbol and the constants
a, b, rQ are the solutions of the transcendental equations:
r0 = 1,
Z*(r1,a,b,ro)=L> (3.28)
, a, b) = t/2,
z*{r1 , a, b, ra) = I.
In the above equations n = r\z-L is the deformed radius at the "throat".
Alternatively, we may wish to consider the initial value problem with the initial
conditions:
s = 0, z = 0, r — 1, t = t0, 4> — <f>o at Z — 0.
The solution is given by f(r, r0 , t0 , 4>a) where / is again a generic symbol and r0 = 1.
The constants t0 and 4>0 have to be adjusted in such a way that both conditions (3.27)
at Z = L are satisfied.
Case II. The middle surface m of the deformed sheet is given.
Let m be generated by a meridian curve c which is a portion of a given curve paramet-
rized by
r = p(o-), z = f(ff), r(0) = 0 (3.29)
where p and f are continuous functions of the arc length <r. The functions p and f may
have discontinuous derivatives.
The undeformed circular cylindrical sheet (3.1) is applied onto the surface of revolu-