Mechanics of moving defects in growing sheets: 3-d, small deformation theory

Amit Acharya,Shankar C Venkataramani

doi:10.1186/s41313-020-00018-w

Amit Acharya, Shankar C Venkataramani

Open Access

https://doi.org/10.1186/s41313-020-00018-w

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Journal: Materials Theory	Publication Date: Apr 16, 2020
Citations: 5	License type: open-access

Affiliation: Carnegie Mellon University, University of Arizona

Abstract

Growth and other dynamical processes in soft materials can create novel types of mesoscopic defects including discontinuities for the second and higher derivatives of the deformation, and terminating defects for these discontinuities. These higher-order defects move “easily", and can thus confer a great degree of flexibility to the material. We develop a general continuum mechanical framework from which we can derive the dynamics of higher order defects in a thermodynamically consistent manner. We illustrate our framework by obtaining the explicit dynamical equations for the next higher order defects in an elastic body beyond dislocations, phase boundaries, and disclinations, namely, surfaces of inflection and branch lines.

Highlights

As Margaret Wertheim writes in her delightful essay “Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe" (Wertheim 2016) – We have built a world of largely straight lines – the houses we live in, the skyscrapers we work in and the streets we drive on our daily commutes
In “Kinematics” section and “Thermodynamics” section we develop a continuum mechanical theory that encompasses the mechanics of such folds in connected domains within a setting that allows for deformations with less smoothness
Our work aims to provides a concrete, tangible, and compelling justification - that the precise treatment of defects in the deformation and its higher order gradients is the raison d’être for higher order theory in continuum mechanics

Summary

Introduction

Hyperbolic sheets abound in nature (see Fig. 1). As Margaret Wertheim writes in her delightful essay “Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe" (Wertheim 2016) – We have built a world of largely straight lines – the houses we live in, the skyscrapers we work in and the streets we drive on our daily commutes. This is first done by explicitly constructing a continuously differentiable deformation of a non- connected domain whose second derivative has a prescribed, constant jump across a planar surface in the body.

Results

Conclusion