This journal is © The Royal Society of Chemistry 2022 Soft Matter, 2022, 18, 2979–2991 | 2979 Cite this: Soft Matter, 2022, 18, 2979 Interaction of a defect with the reference curvature of an elastic surface Manish Singh, Animesh Pandey and Anurag Gupta * The morphological response of two-dimensional curved elastic sheets to an isolated defect (dislocation/ disclination) is investigated within the framework of Fo ¨ ppl–von Ka ´ rma ´ n shallow shell theory. The reference surface, obtained as a shell configuration in the absence of defect and external forces, accordingly has a finite non-zero curvature. The interaction of the defect with the curvature of the reference surface is emphasized through the problem of defect driven buckling of an elastic sheet. Detailed bifurcation diagrams, including the post-buckling deformation behaviour, are plotted for several combinations of defect types, reference curvatures, and boundary conditions. A pitchfork bifurcation is obtained when the reference surface is flat irrespective of the defect type and boundary condition. For curved reference surfaces there are some cases where the pitchfork bifurcation persists and others where it does not. The varied response demonstrates the rich interaction of the defects with the curvature of the reference surface. 1 Introduction The mechanical behaviour of a two-dimensional (2D) elastic surface, in response to an isolated defect (dislocation/disclination), is fundamentally different from that of a three-dimensional (3D) elastic body. Whereas a dislocation (or a disclination) in a 3D elastic body necessarily causes a far field stress distribution, 1 the elastic surface can explore the third (out-of-plane) dimen- sion so as to keep the stress fields localized around the defect point. 2 Consequently, disclinations in elastic surfaces are fea- sible (energetically) while disclinations in 3D elastic solids are not. The response of an elastic surface to a defect is therefore predominantly geometric, characterized in terms of kine- matical quantities such as transverse deflection and Gaussian curvature. This understanding has led to significant literature in the recent past where defect induced shape transformations have been used to explain novel physical characteristics of 2D crystalline and amorphous surfaces. 3–8 When posed as a boundary value problem, the isolated defect problem has been well studied assuming the elastic sheet to be a Fo ¨ppl–von Ka ´rma ´n plate with a flat reference surface and free boundary condition. 2,9–11 The reference surface refers to the configuration achieved in the absence of defects and external forces. In accordance with the physical observations, depending on a dimensionless parameter, which is a function of the plate size, elastic constants, and defect strength, the plate may choose to respond (to the defect, in the absence of external forces) either through pure stretching, thereby remaining flat, or by buckling out of the plane. The buckled plate due to a positive disclination, for instance, acquires a conical shape whereas it acquires a saddle shape due to a negative disclination. 2 The buckled solution in fact appears as a symmetric pitchfork bifurcation regardless of the defect type (dislocation or disclination) and boundary conditions, as long as the reference surface remains flat; this is further elaborated in a paragraph below, see also Fig. 1. Our interest is in studying the deformation behaviour of a 2D elastic sheet with an isolated defect whose reference surface has a finite non-zero curvature. We restrict ourselves to elastic Fig. 1 The global bifurcation diagram for a positive disclination in an elastic sheet with a flat reference surface and free boundary. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, 208016, India. E-mail: ag@iitk.ac.in Received 26th January 2022, Accepted 14th March 2022 DOI: 10.1039/d2sm00126h rsc.li/soft-matter-journal Soft Matter PAPER Published on 15 March 2022. Downloaded by Indian Institute of Technology Kanpur on 7/28/2022 7:59:49 PM. View Article Online View Journal | View Issue