Determination of displacement fields at the sub-nanometric scale

Macroscopic behavior of materials depends on interactions of atoms and molecules at nanometer/sub-nanometer scale. Experimental mechanics (EM) can be used for assessing relationships between the macro world and the atomic realm. Theoretical models developed at nanometric and sub-nanometric scales may be verified using EM techniques with the final goal of deriving comprehensive but manageable models. Recently, the authors have carried out studies on EM determination of displacements and their derivatives at the macro and microscopic scales. Here, these techniques were applied to the analysis of high-resolution transmission electron microscopy patterns of a crystalline array containing dislocations. Utilizing atomic positions as carriers of information and comparing undeformed and deformed configurations of observed area, displacements and their derivatives, as well as stresses, have been obtained in the Eulerian description of deformed crystal. Two approaches are introduced. The first establishes an analogy between the basic crystalline structure and a 120° strain gage rosette. The other relies on the fact that, if displacement information along three directions is available, it is possible to reconstruct the displacement field; all necessary equations are provided in the paper. Remarkably, the validity of the Cauchy-Born conjecture is proven to be correct within the range of observed deformations.