Nonlinear boundary value problems for shallow membrane caps, II

Suppose a shallow membrane cap, with an undeformed shape described in cylindrical coordinates by z = C(1−rγ) (where 0⩽r⩽1 and γ>1), is subjected to a uniform vertical pressure P. If the resulting deformed shape is radially symmetric, then under certain assumptions, the radial stress Sr satisfies the ordinary differential equation r2Sr″ + 3rSr′ = λ2r2y−22 + βvr2Sr − r28Sr2, for 0<r⩽1 and either the boundary condition Sr(1) = S>0 (if the boundary stress S is specified) or Sr′(1)+(1 − v)Sr(1) = ⌜ (if the boundary displacement ⌜ is specified). Here v(0⩽v<0.5) is the Poisson ratio, and λ and β are positive constants depending on the pressure P, the thickness of the membrane, and Youngʹs modulus. We show that if γ > 1, a radially symmetric solution Sr(r), positive for 0<r⩽1, exists, and if S⩽1(4βv) or ⌜(1−v)⩽1(4βv), the solution is unique. In the case γ⩽43, if λ is fairly large, it may happen that Sr(r) → 0 as r → 0. In all other cases, Sr(r) has a positive limit as r → 0. Rather detailed information on the behavior of solutions Sr(r) is provided. Conditions are obtained which guarantee monotonicity of Sr. In any case, Sr has at most one critical point and is monotone in some neighborhood of r = 0. A computational algorithm, making use of the qualitative behavior of Sr, is discussed and some numerical results are included.