Simulations of Indentation at Continuum and Atomic levels

TR Number



Journal Title

Journal ISSN

Volume Title


Virginia Tech


The main goal of this work is to determine values of elastic constants of orthotropic, transversely isotropic and cubic materials through indentation tests on thin layers bonded to rigid substrates. Accordingly, we first use the Stroh formalism to provide an analytical solution for generalized plane strain deformations of a linear elastic anisotropic layer bonded to a rigid substrate, and indented by a rigid cylindrical indenter. The mixed boundary-value problem is challenging since the deformed indented surface of the layer contacting the rigid cylinder is unknown a priori, and is to be determined as a part of the solution of the problem. For a rigid parabolic prismatic indenter contacting either an isotropic layer or an orthotropic layer, the computed solution is found to compare well with solutions available in the literature. Parametric studies have been conducted to delimit the length and the thickness of the layer for which the derived relation between the axial load and the indentation depth is valid.

We then derive an expression relating the axial load, the indentation depth, and the elastic constants of an orthotropic material. This relation is specialized to a cubic material (e.g., an FCC single crystal). By using results of three virtual (i.e., numerical) indentation tests on the same specimen oriented differently, we compute values of the elastic moduli, and show that they agree well with their expected values. The technique can be extended to other anisotropic materials.

We review the literature on relations between deformations at the atomic level and stresses and strains defined at the continuum level. These are then used to compare stress and strain distributions in mechanical tests performed on atomic systems and their equivalent continuum structures. Whereas averaged stresses and strains defined in terms of the overall deformations of the atomic system match well with those derived from the continuum description of the body, their local spatial distributions differ.



continuum equivalence, nanoindentation, analytical solution, FCC, load-displacement relation, atomic simulation, Cylindrical contact