This paper presents a novel conic programming approach for modeling the wrinkling of isotropic hyperelastic membranes subject to nonlinear loading and finite strains, employing convex potentials derived from tension field theory. The incompressible neo-Hookean strain energy is recast as a minimization problem over a set of cones, with a semidefinite constraint on the deformed surface metric. To address pneumatic membranes, a linearized volumetric potential is introduced, reestablishing convexity for follower forces, and Boyle's law is expressed as a minimization problem over the exponential cone.

The primal–dual interior point method solves the total potential energy minimization problem, with convergence verified using the analytical solution of a sheared planar membrane. The proposed model, applied to examples of increasing complexity, reveals intriguing membrane behaviors rarely discussed in existing literature. The method is shown to be robust, even when handling highly wrinkled membranes, which are challenging to address using direct nonlinear equilibrium analysis or nonlinear interior point methods.

Implementation using high-level automatic code generation tools (FEniCS) results in concise, extensible code. The findings point to several avenues for future research, such as exploring complex material models, mesh refinement, dynamics, and parametric design. Additionally, the linearization process implies potential applicability of the methods to nonlinear problems beyond membrane mechanics.

本文提出了一种新颖的圆锥规划方法，利用从张力场理论导出的凸势来模拟受非线性载荷和有限应变影响的各向同性超弹性膜的起皱。不可压缩的新胡克应变能被改写为一组锥体上的最小化问题，并对变形表面度量具有半定约束。为了解决气动膜的问题，引入了线性化体积势，重新建立了从动力的凸性，并将波义耳定律表示为指数锥上的最小化问题。

原对偶内点法解决了总势能最小化问题，并使用剪切平面膜的解析解验证了收敛性。所提出的模型应用于复杂性不断增加的例子，揭示了现有文献中很少讨论的有趣的膜行为。该方法被证明是稳健的，即使在处理高度皱纹的膜时也是如此，而使用直接非线性平衡分析或非线性内点方法很难解决这一问题。

使用高级自动代码生成工具 (FEniCS) 的实现会产生简洁、可扩展的代码。研究结果指出了未来研究的几种途径，例如探索复杂材料模型、网格细化、动力学和参数化设计。此外，线性化过程意味着该方法对膜力学之外的非线性问题的潜在适用性。