We measured the elastic properties and intrinsic breaking strength of free-standing monolayer graphene membranes by nanoindentation in an atomic force microscope. The force-displacement behavior is interpreted within a framework of nonlinear elastic stress-strain response, and yields second- and third-order elastic stiffnesses of 340 newtons per meter (N m(-1)) and -690 Nm(-1), respectively. The breaking strength is 42 N m(-1) and represents the intrinsic strength of a defect-free sheet. These quantities correspond to a Young's modulus of E = 1.0 terapascals, third-order elastic stiffness of D = -2.0 terapascals, and intrinsic strength of sigma(int) = 130 gigapascals for bulk graphite. These experiments establish graphene as the strongest material ever measured, and show that atomically perfect nanoscale materials can be mechanically tested to deformations well beyond the linear regime.

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Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene

Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

Ultrasonic, pressure-volume, and shock wave measurements of NaCl are reviewed, with the purpose of representing as much as possible within a single theoretical framework, A large portion of the data is consistent, to within reasonable uncertainties, with the Eulerian formulation of finite strain, in the BE2 form. This contains three parameters, of which two, K0 and K0′, are obtainable from single-crystal ultrasonic measurements, while the third, K0″, may be found with the aid of shock wave data. The combination of the values of Spetzler et al. (1972a) with the isotherm of Fritz et al. (1971) gives an equation of state consistent with the pressure-volume measurements to 300 kbar, to within a kilobar or two. There remain small but possibly significant discrepancies with the high-precision measurements of linear change at low pressures. Various two-parameter isotherms are also examined and found to fail either at low or at high compressions: While other three-parameter forms may be fitted to the data, they do not afford a satisfactory general treatment of wave propagation. Finite strain theory is applied to project the effective elastic coefficients of single-crystal NaCl to 270 kbar. The anisotropy index, 2B44(B11–B12), falls from 0.7 at P = 0 to 0.12 at 270 kbar, and the usual methods of averaging to find the properties of a quasi-isotropic aggregate diverge increasingly as pressure increases. The methods of Kroner and of Peresada give fair agreement with recent measurements of velocities in NaCl aggregates to 270 kbar.

Finite strain isotherm and velocities for single-crystal and polycrystalline NaCl at high pressures and 300°K

The nondestructive assessment of the damage that occurs in components during service plays a key role for condition monitoring and residual life estimation of in-service components/structures. Ultrasound has been widely utilized for this; however most of these conventional methods using ultrasonic characteristics in the linear elastic region are only sensitive to gross defects but much less sensitive to micro-damage. Recently, the nonlinear ultrasonic technique, which uses nonlinear ultrasonic behavior such as higher-harmonic generation, subharmonic generation, nonlinear resonance, or mixed frequency response, has been studied as a positive method for overcoming this limitation. In this paper, overall progress in this technique is reviewed with the brief introduction of basic principle in the application of each nonlinear ultrasonic phenomenon.

Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: A review

General thermodynamic definitions of the higher order elastic coefficients of thermoelastic media are presented in tensor and engineering notation. They are natural generalizations of the customary definitions of second-order coefficients, they retain the usual conventions relating tensor and engineering stresses and strains, and they simplify thermodynamic calculations.

Thermodynamic Definition of Higher Order Elastic Coefficients

Third-order elastic constants can be determined from the velocity of small amplitude sound waves in statically stressed media. For this purpose exact expressions are derived for the sound velocity and for a natural velocity and their stress derivatives, evaluated at zero stress, in terms of second- and third-order elastic constants. The formulas apply to arbitrary crystal symmetry and to arbitrary stress systems depending on a single scalar variable. Special formulas for hydrostatic pressure and uniaxial stress are listed for the cubic point groups $O$, ${O}_{h}$, ${T}_{d}$, and for isotropic materials. Attention is given to the proper variation of propagation direction with static stress in order to maintain propagation normal to a given crystal face as in ultrasonic experiments, and to the proper separation of isothermal and isentropic coefficients in the results. The simplest and most convenient from of the results employs the natural velocity (natural unstressed length at the same temperature divided by the transit time), which is computed directly from experimental data without correcting the path length for the effect of stress.

R. N. Thurston

Papers

Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media