Elastic modulus from contact stiffness

In this relation, k. is the cantilever stiffness and ki is the tip-surface effective stiffness given by dFIBS. From the relations (1.1 to 1.3), it can be seen that k depends on the static contact force, the tip geometry and the surface elastic modulus. Knowing the cantilever stiffness, the static contact force and the tip geometry and dimensions, it is thus theoretically possible to determine the surface elastic modulus from the dynamic response. 

A typical load-displacement curve is shown in Fig. 2. The loading portion of the curve results from both plastic and elastic deformation response of the contact, while the unloading portion of the curve is related to the elastic recovery of the contact. If the indenter geometry and materials properties are known, the modulus can be obtained by fitting the unloading curve to determine the contact stiffness at maximum load (i4, 17). In this case,... 

Ultralow load indentation, also known as nanoindentation, is a widely used tool for measuring the mechanical properties of thin fdms and small volumes of material. The principle is to pushing in a hard material tip called the indenter into the analyzed sample and to measure the curve load-penetration. A modified commercial nanoindenter (Nano indenter XP - MTS) was used to characterize coated materials. The device allows to measure the contact stiffness with superimposing a harmonic oscillation (small amplitude of 3 nm, constant frequency of 32 Hz) to the continuous penetration of the indenter into the sample. This specificity allows one to continually measure the elastic modulus and hardness according to the penetration depth. Loubet et al. demonstrated that reduced Young modulus and hardness for a Berkovich indenter with a dynamic measurement method could be deduced from the following equations [11] ... 

Yuya et al. [243] extracted the elastic modulus of single electrospun PAN nanofibre dynamically through the natural frequencies of a pair of AFM microcantilevers linked by a nanofibre segment (Fig. 4.24b). The theory of this technique is based on the dynamic relationship between the fibre stiffness (i.e. spring constant) and the resonance frequencies of cantilever vibration mode. On the other hand, Liu et al. [244] used atomic force acoustic microscopy (AFAM) based on ultrasonic frequency oscillations to excite an AFM cantilever when the tip was in contact with a sample. A different approach based on a model of the resonant frequency that is dependent on the bob s free flight was employed to measure the elastic modulus of as-spun nylon 6, 6. A ball was glued to a nanofibre and suspended from a cantilever beam that was attached to a piezoelectric-actuated base [245]. 

If brought into contact, the (theoretical) distance assuming an LJ interaction potential would be equal 1.12a. Elastic deformation of the sphere and the flat surface have not been considered (infinite stiffness was assumed for the bodies). In such cases, the contact ideally is a point contact. However, if the Young s modulus (modulus of elasticity) of one of the bodies (or both) has a finite value, then the contact point becomes a contact circle with a radius a. The value of the contact radius a depends in such cases on the elastic properties of the spheres, on the Young s moduli E and E2, and on the Poisson s ratios q and v2, of the two contacting materials, respectively. The value of the contact radius a for two spheres pressed together can be calculated from the following formula ... 

For both elastic and viscoelastic materials, the response of the contact ((X a) in Figure 2) is simply related to the contact radius a(t). Starting from equation 1, assuming the cantilever to be perfectly elastic with torsional stiffness Ke, and describing the contact response by a complex shear modulus G =, ... 

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