Many biological systems are coated by thin films for protection, selective absorption, or transmembrane transport. A typical example is the mucous membrane covering the airways, the esophagus, and the intestine. Biological surfaces typically display a distinct mechanical behavior from the bulk; in particular, they may grow at different rates. Growth, morphological instabilities, and buckling of biological surfaces have been studied intensely by approximating the surface as a layer of finite thickness; however, growth has never been attributed to the surface itself. Here, we establish a theory of continua with boundary energies and growing surfaces of zero thickness in which the surface is equipped with its own potential energy and is allowed to grow independently of the bulk. In complete analogy to the kinematic equations, the balance equations, and the constitutive equations of a growing solid body, we derive the governing equations for a growing surface. We illustrate their spatial discretization using the finite element method, and discuss their consistent algorithmic linearization. To demonstrate the conceptual differences between volume and surface growth, we simulate the constrained growth of the inner layer of a cylindrical tube. Our novel approach toward continua with growing surfaces is capable of predicting extreme growth of the inner cylindrical surface, which more than doubles its initial area. The underlying algorithmic framework is robust and stable; it allows to predict morphological changes due to surface growth during the onset of buckling and beyond. The modeling of surface growth has immediate biomedical applications in the diagnosis and treatment of asthma, gastritis, obstructive sleep apnoea, and tumor invasion. Beyond biomedical applications, the scientific understanding of growth-induced morphological instabilities and surface wrinkling has important implications in material sciences, manufacturing, and microfabrication, with applications in soft lithography, metrology, and flexible electronics.

许多生物系统都涂有薄膜以进行保护、选择性吸收或跨膜运输。一个典型的例子是覆盖气道、食道和肠道的粘膜。生物表面通常表现出与整体不同的机械行为；特别是，它们可能以不同的速度增长。通过将表面近似为有限厚度的层，已经深入研究了生物表面的生长、形态不稳定性和屈曲；然而，增长从未归因于表面本身。在这里，我们建立了具有边界能和零厚度生长表面的连续体理论，其中表面配备有自己的势能，并允许独立于体积生长。完全类似于运动学方程，平衡方程，和生长实体的本构方程，我们推导出生长表面的控制方程。我们使用有限元方法说明了它们的空间离散化，并讨论了它们的一致算法线性化。为了演示体积和表面生长之间的概念差异，我们模拟了圆柱管内层的受约束生长。我们对具有生长表面的连续体的新方法能够预测内圆柱表面的极端生长，使其初始面积增加一倍以上。底层算法框架健壮稳定；它允许预测在屈曲开始期间及以后由于表面生长而导致的形态变化。表面生长建模在诊断和治疗哮喘、胃炎、阻塞性睡眠呼吸暂停和肿瘤侵袭。除了生物医学应用之外，对生长引起的形态不稳定性和表面起皱的科学理解在材料科学、制造和微制造中具有重要意义，在软光刻、计量学和柔性电子学中的应用。