Min Zhou

The virial stress is the most commonly used definition of stress in discrete particle systems. This quantity includes two parts. The first part depends on the mass and velocity (or, in some versions, the fluctuation part of the velocity) of atomic particles, reflecting an assertion that mass transfer causes mechanical stress to be applied on stationary spatial surfaces external to an atomic‐particle system. The second part depends on interatomic forces and atomic positions, providing a continuum measure for the internal mechanical interactions between particles. Historic derivations of the virial stress include generalization from the virial theorem of Clausius (1870) for gas pressure and solution of the spatial equation of balance of momentum. The virial stress is stress‐like a measure for momentum change in space. This paper shows that, contrary to the generally accepted view, the virial stress is not a measure for mechanical force between material points and cannot be regarded as a measure for mechanical stress in any sense. The lack of physical significance is both at the individual atom level in a time‐resolved sense and at the system level in a statistical sense. It is demonstrated that the interatomic force term alone is a valid stress measure and can be identified with the Cauchy stress. The proof in this paper consists of two parts. First, for the simple conditions of rigid translation, uniform tension and tension with thermal oscillations, the virial stress yields clearly erroneous interpretations of stress. Second, the conceptual flaw in the generalization from the virial theorem for gas pressure to stress and the confusion over spatial and material equations of balance of momentum in theoretical derivations of the virial stress that led to its erroneous acceptance as the Cauchy stress are pointed out. Interpretation of the virial stress as a measure for mechanical force violates balance of momentum and is inconsistent with the basic definition of stress. The versions of the virial‐stress formula that involve total particle velocity and the thermal fluctuation part of the velocity are demonstrated to be measures of spatial momentum flow relative to, respectively, a fixed reference frame and a moving frame with a velocity equal to the part of particle velocity not included in the virial formula. To further illustrate the irrelevance of mass transfer to the evaluation of stress, an equivalent continuum (EC) for dynamically deforming atomistic particle systems is defined. The equivalence of the continuum to discrete atomic systems includes (i) preservation of linear and angular momenta, (ii) conservation of internal, external and inertial work rates, and (iii) conservation of mass. This equivalence allows fields of work‐ and momentum‐preserving Cauchy stress, surface traction, body force and deformation to be determined. The resulting stress field depends only on interatomic forces, providing an independent proof that as a measure for internal material interaction stress is independent of kinetic energy or mass transfer.

Molecular dynamics simulations are performed to characterize the response of zinc oxide (ZnO) nanobelts to tensile loading. The ultimate tensile strength (UTS) and Young's modulus are obtained as functions of size and growth orientation. Nanobelts in three growth orientations are generated by assembling the unit wurtzite cell along the [0001], , and crystalline axes. Following the geometric construction, dynamic relaxation is carried out to yield free-standing nanobelts at 300 K. Two distinct configurations are observed in the [0001] and orientations. When the lateral dimensions are above 10 Å, nanobelts with rectangular cross-sections are seen. Below this critical size, tubular structures involving two concentric shells similar to double-walled carbon nanotubes are obtained. Quasi-static deformations of belts with and orientations consist of three stages, including initial elastic stretching, wurtzite-ZnO to graphitic-ZnO structural transformation, and cleavage fracture. On the other hand, [0001] belts do not undergo any structural transformation and fail through cleavage along (0001) planes. Calculations show that the UTS and Young's modulus of the belts are size dependent and are higher than the corresponding values for bulk ZnO. Specifically, as the lateral dimensions increase from 10 to 40 Å, decreases between 38–76% and 24–63% are observed for the UTS and Young's modulus, respectively. This effect is attributed to the size-dependent compressive stress induced by tensile surface stress in the nanobelts. and nanobelts with multi-walled tubular structures are seen to have higher values of elastic moduli (∼340 GPa) and UTS (∼36 GPa) compared to their wurtzite counterparts, echoing a similar trend in multi-walled carbon nanotubes.