Date of Award

Fall 12-1-2020

Level of Access

Campus-Only Thesis

Degree Name

Master of Science in Mechanical Engineering (MSME)


Mechanical Engineering


Senthil S. Vel

Second Committee Member

Zhihe Jin

Third Committee Member

Robert J. Lad

Additional Committee Members

Yingchao Yang


A two-dimensional (2D) material is a material composed of one layer of atoms, or a few layers of tightly bonded atomic planes. Examples include Molybdenum disulfide (MoS2), Black Phosphorus (BP), and Rhenium disulfide (ReS2). The application areas of these materials include flexible electronics, nano-medicine and multi-logic devices. In order to effectively use this class of materials in industrial applications, especially with the emergence of the strain engineering field, it is important to understand their nonlinear elastic behavior. The nonlinear elastic behavior of 2D materials of hexagonal and cubic symmetries has been probed in the literature. However, more general material symmetry classes, such as orthorhombic and tricilinc materials, have not been considered.

We employ the thermodynamically rigorous hyperelastic constitutive model proposed by Murnaghan (1937), wherein the strain energy density is expanded as a partial Maclaurin series in terms of the Green-St. Venant strains. The constants in the strain energy expansion are the elastic coefficients for the 2D material. The strain energy density can be expanded as a polynomial of any order. However, as the order of the polynomial increases, the number of elastic coefficients to be determined increases. For example, if a fifth order polynomial expansion is assumed for a triclinic material like ReS2, whose crystal model is shown in the figure to the right, there are $52$ non-vanishing linearly independent elastic coefficients that need to be determined. Therefore, a systematic, general and rigorous approach is needed to determine the elastic coefficients of 2D materials of any symmetry and for polynomial expansions of any order.

In this thesis we propose and validate a general methodology to determine the elastic coefficients for 2D materials of arbitrary symmetries. We use a ray based approach to sample the strain energy density in strain space and curve fit for the elastic coefficients. The ray based approach is independent of the methodology used to calculate the strain energies. In this thesis, we use a first principles calculation technique based on density functional theory for the strain energy calculations due to the assumption used in its formulation, i.e., slowly varying electron densities, which translates into the thermodynamic equilibrium assumption for the equivalent continua. The density functional theory calculations are preformed using open-source software package Quantum-Espresso (QE).

The ray based methodology is validated by comparing the nonlinear elastic coefficients for graphene with the published literature. Subsequently, the nonlinear elastic coefficients for hexagonal MoS2 and orthorhombic Black Phosphorus are obtained and validated. The effect of the order of the constitutive polynomial on the accuracy of the predicted strain energy density and stresses is evaluated. Furthermore, the effects of the number of rays and number of sampling points per ray are assessed. Preliminary results are presented for ReS2. In addition, the theory is linearized for each material to obtain the engineering properties, such as the directional variation of the Young's modulus, Poisson's ratio and shear modulus, which are important for linear elastic studies. We have also developed a modular software package to automatically perform QE simulations and generate elastic coefficients.

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