Department of Physics, University of Massachusetts, Amherst, MA 01003;
Department of Physics, University of Massachusetts, Amherst, MA 01003.
Proc Natl Acad Sci U S A. 2019 Jan 29;116(5):1483-1488. doi: 10.1073/pnas.1815507116. Epub 2018 Dec 27.
The complex morphologies exhibited by spatially confined thin objects have long challenged human efforts to understand and manipulate them, from the representation of patterns in draped fabric in Renaissance art to current-day efforts to engineer flexible sensors that conform to the human body. We introduce a theoretical principle, broadly generalizing Euler's -a core concept of continuum mechanics that invokes the energetic preference of bending over straining a thin solid object and that has been widely applied to classical and modern studies of beams and rods. We define a class of geometrically incompatible confinement problems, whereby the topography imposed on a thin solid body is incompatible with its intrinsic ("target") metric and, as a consequence of Gauss' , induces strain. By focusing on a prototypical example of a sheet attached to a spherical substrate, numerical simulations and analytical study demonstrate that the mechanics is governed by a principle, which we call the "Gauss-Euler " This emergent rule states that-despite the unavoidable strain in such an incompatible confinement-the ratio between the energies stored in straining and bending the solid may be arbitrarily small. The Gauss-Euler underlies a theoretical framework that greatly simplifies the daunting task of solving the highly nonlinear equations that describe thin solids at mechanical equilibrium. This development thus opens possibilities for attacking a broad class of phenomena governed by the coupling of geometry and mechanics.
空间受限的薄物体所表现出的复杂形态长期以来一直挑战着人类对其进行理解和操控的能力，从文艺复兴时期艺术中对悬垂织物图案的表现到当今对能够顺应人体的柔性传感器的工程设计。我们引入了一个理论原理，该原理广泛推广了欧几里得的 - 这是连续介质力学的核心概念，它调用了弯曲比拉伸薄固体物体的能量偏好，并且已经广泛应用于梁和杆的经典和现代研究。我们定义了一类几何不兼容的约束问题，其中施加在薄固体上的地形与它的固有（“目标”）度量不兼容，并且由于高斯的作用，会产生应变。通过专注于附着在球形基底上的薄片的原型示例，数值模拟和分析研究表明，力学由一个我们称之为“高斯 - 欧拉”的原理控制。这个新出现的规则表明 - 尽管在这种不兼容的约束中不可避免地存在应变 - 但固体中应变和弯曲储存的能量之间的比值可以任意小。高斯 - 欧拉原理为解决描述机械平衡的薄固体的高度非线性方程这一艰巨任务提供了一个简化的理论框架。因此，这一发展为解决由几何和力学耦合控制的广泛现象提供了可能性。
