Some of the mechanical properties are:

1.Hardness test

2.Study of Elastic Modulus

3.Friction

In the case whereby the physical properties are referring to the nanoparticles intrinsic properties, you are then required to analyze the nanoparticles on a substrate via atomic force microscopy (AFM) under quantitative nano-mechanical (QNM) or nano-indentation mode.

To address the question on the available mechanical properties to be studied, properties such as hardness, Young's modulus, ultimate strength and yield strength can be obtained.

Refer to the article attached, it can give you basic review about the mechanical properties of nanoparticles. Further a short answer note is presented below.

Mechanical properties of solids depend on the microstructure, i.e. the chemical composition, the arrangement of the atoms (the atomic structure) and the size of a solid in one, two or three dimensions. The most well-known example of the correlation between the atomic structure and the properties of a bulk material is the variation in the hardness of carbon when it transforms from diamond to graphite. The important aspects related to structure are: atomic defects, dislocations and strains, grain boundaries and interfaces, porosity, connectivity and percolation and short range order.

Characterizing the mechanical properties of individual nanotubes/nanowires/nanobelt [called one-dimensional (1-D) nanostructure] is a challenge to many existing testing and measuring techniques because of the following constrains. First, the size (diameter and length) is rather small, prohibiting the applications of the well-established testing techniques. Tensile and creep testing require that the size of the sample be sufficiently large to be clamped rigidly by the sample holder without sliding. This is impossible for 1-D nanomaterials using conventional means. Secondly, the small size of the nanostructure makes their manipulation rather difficult, and specialized techniques are needed for picking up and installing individual nanostructure. Therefore new methods and methodologies must be developed to quantify the properties of individual nanostructure.  A number of methods have been developed for mechanical testing of nanowires (NWs) including resonance in scanning or transmission electron microscopes (SEM/TEM), bending or contact resonance using atomic force microscopy (AFM), uniaxial tension in SEM or TEM, and nanoindentation. In particular, in situ SEM/TEM tensile testing of NWs enabled by microelectromechanical systems has attracted a lot of recent attention. However each technique has its own merits and demerits in invoking the mechanical parameters of nanowires.

Mechanical properties of carbon nanotube

The carbon nanotube (CNT) is a rolled-up sheet of graphene and has three types depending upon the rolling direction such as armchair, zigzag and chiral. The bond between carbons is similar to that of graphite and the mechanical property is closely related to the bond nature between the carbon atoms. The electronic structure of carbon is 1s2 2s2 2p2 and when carbon atoms combine to form graphite, sp2 hybridization will occurs. In this process, one s-orbital and two p-orbitals combine to form three hybrid sp2 -orbitals at 120° to each other within a plane. This in-plane bond is referred to as a σ-bond (sigma–bond). This is a strong covalent bond that binds the atoms in the plane, and results in the high stiffness and high strength of a CNT. The remaining p-orbital is perpendicular to the plane of the σ-bonds. It contributes mainly to the interlayer interaction and is called the Π-bond (pi–bond). These out-of planes, delocalized Π-bonds interact with the Π-bonds on the neighbouring layer. This interlayer interaction of atom pairs on neighbouring layers is much weaker than a sigma bond. Also Unlike bulk materials, the density of the defects in nanotubes is presumably less and therefore the strength is presumablysignificantly higher at the nanoscale.