Stress-Free Morphing

We study the morphing of soft materials within the framework of non-linear elasticity with large distortions: a distortion field induces a target metric, and the configuration which is effectively realized by a body is the one that minimizes the distance, measured through the elastic energy, between the target metric and the actual one.

Deformations due to distortions, in contrast to those generated by forces, can have a peculiar feature: they can be stress-free; if this is the case, the distortions field is called compatible.

We maintain that the morphing through compatible distortions is a key strategy exploited by many soft biological materials, which can exhibit very large shape-change in response to distortions controlled by chemicals or by temperature changes, while keeping their stress state almost null.

Thus, the study of compatible distortions, and of the related shape-changes, is quite important. Here, we show how the notions of metric tensor and of Riemannian curvature can be used to assess the compatibility of a distortion field.