Mechanical deformations induced by expansion within an elastic material which is spirally-wound in layers with a thin inextensible reinforcing material are considered. The motivation is to understand behaviour of spirally-wound batteries where both the active material and the metal current collectors expand due to changes in lithiation and/or temperature. This paper considers a spiral made from a single reinforcing layer with a matrix layer of linear elastic material, whose properties may vary through the layer. The layers undergo prescribed isotropic expansion, where the matrix expansion may depend on the macroscopic radial coordinate. Asymptotic homogenisation, exploiting the small scale of the layer thickness relative to the large scale of the overall spiral structure, reveals the bulk of the spiral has an unexpected simple behaviour while there are boundary layers in a surface region near the inner and outer windings. There are further finer-structure boundary layers at the very beginning and very end of the spiral. In all these regions analytical solutions are found providing simple expressions for the deformations and in particular the tension in the inextensible layer. Comparisons are shown between these expressions and detailed finite-element solutions of the problem. These reduced-order models provide a simple way of accounting for stresses induced by expansion of the spiral structure.

考虑了由弹性材料内的膨胀引起的机械变形，该弹性材料是用薄的不可延伸的增强材料螺旋缠绕在层中的。动机是了解螺旋缠绕电池的行为，其中活性材料和金属集电器都会因锂化和/或温度的变化而膨胀。本文考虑由单个增强层和线性弹性材料基体层制成的螺旋，其特性可能随层而变化。这些层经历规定的各向同性膨胀，其中矩阵膨胀可能取决于宏观径向坐标。渐近均匀化，利用层厚度相对于整体螺旋结构的大尺度的小尺度，揭示了螺旋的大部分具有意想不到的简单行为，而在靠近内部和外部绕组的表面区域中存在边界层。在螺旋的最开始和最末端还有更精细结构的边界层。在所有这些区域中，解析解提供了变形的简单表达式，特别是不可延伸层中的张力。这些表达式与问题的详细有限元解之间进行了比较。这些降阶模型提供了一种简单的方法来解释螺旋结构膨胀引起的应力。在所有这些区域中，解析解提供了变形的简单表达式，特别是不可延伸层中的张力。这些表达式与问题的详细有限元解之间进行了比较。这些降阶模型提供了一种简单的方法来解释螺旋结构膨胀引起的应力。在所有这些区域中，解析解提供了变形的简单表达式，特别是不可延伸层中的张力。这些表达式与问题的详细有限元解之间进行了比较。这些降阶模型提供了一种简单的方法来解释螺旋结构膨胀引起的应力。