Solution space of axisymmetric capsules enclosed by elastic membranes

Two principal methodologies for computing equilibrium shapes of axisymmetric capsules enclosed by elastic membranes are discussed. First, a variational formulation based on a bending energy functional defined in terms of the principal curvatures is considered. Constrained minimization results in a nonlinear, second-order differential equation for the meridional curvature with respect to the slope angle or arc length along the membrane contour in a meridional plane. Numerical solutions of the relevant boundary-value problem are presented over a broad range of conditions to illustrate the rich structure of the solution space. Second, the problem is formulated from the viewpoint of shell theory in terms of stress resultants and bending moments using force and torque equilibrium balances. Assuming isotropic tensions and linear constitutive equations relating the bending moments to the principal curvatures, the problem is reduced to solving a nonlinear third-order differential equation, which is distinct from that obtained using the variational method. Numerical solutions uncover a variety of physically realizable capsule shapes, including those resembling the biconcave, disk-like shape of healthy red blood cells.