Finite element modeling of nanoindentation on an elastic-plastic microsphere

The understanding of the mechanical indentation on a curved specimen (e.g., microspheres and microfibers) is of paramount importance in the characterization of curved micro-structured materials, but there has been no reliable theoretical method to evaluate the mechanical behavior of nanoindentation on a microsphere. This article reports a computational study on the instrumented nanoindentation of elastic-plastic microsphere materials via finite element simulation. The finite element analyses indicate that all loading curves are parabolic curves and the loading curve for different materials can be calculated from one single indentation. The results demonstrate that the Oliver-Pharr formula is unsuitable for calculating the elastic modulus of nanoindentation involving cured surfaces. The surface of the test specimen of a microsphere requires prepolishing to achieve accurate results of indentation on a micro-spherical material. This study provides new insight into the establishment of nanoindentation models that can effectively be used to simulate the mechanical behavior of a microsphere.


Introduction
Instrumented nanoindentation is perhaps the most commonly adopted and used technique in the characterization of mechanical behavior of microplastics, thin films, coatings, powders, small crystals and materials at small scales. One of the great advantages of the technique is that many mechanical properties of materials can be determined by the analyses of indentation load-displacement data alone, thereby avoiding the need to measure the area of hardness impression by imaging and facilitating property measurement at the sub-micron scale [1,2]. In a nanoindentation test, an axisymmetric diamond indenter with a geometry known to high precision (usually a Berkovich tip, which has threesided pyramid geometry with the same area-to-depth ratio as the four-sided Vickers pyramid used commonly in microhardness testing) is pushed into the surface of test specimen with an increased force or displacement. As the force or displacement reaches a user-defined maximum value (sometimes moving just a few hundreds of atom into solid surface, indentation depth 20 or 30 nanometers), the load is then withdrawn. While loading-unloading is in progress, force-displacement curves are recorded via a Nanoindenter® instrument. The unloading curve is used to extract the mechanical properties (including elastic modulus and hardness) of test specimen via an analytical method, such as the Oliver-Pharr method [3,4]. For Oliver-Pharr method, the test specimen is assumed to be a flat surface with linear isotropic elastic-perfectly plastic material properties. A permanent hardness impression is formed during loading and unloading. When the indenter is unloaded, the elastic strains are recovered. Thus, instrumented nanoindentation has elastic and plastic deformation during loading, but only elastic deformation during unloading.
Cheng and Cheng derived several scaling relationships for conical indentation in elastic-plastic solids with work hardening using dimensional analysis and finite element calculations [5]. They pointed out that some properties such as the elastic modulus are size independent [6]. The measured values from macroscopic experiments are consistent with that predicted from first-principles quantum mechanics calculations. Oliver and Pharr reviewed the mechanics governing elastic-plastic indentation as they pertain to load and depth-sensing indentation testing of monolithic materials [7]. The measurement of contact stiffness by dynamic techniques allows for continuous measurement of properties as a function of depth. Stiffness is measured continuously during the loading of the indenter by imposing a small dynamic oscillation on the force (or displacement) signal and measuring the amplitude and phase of the corresponding displacement (or force) signal by means of a frequencyspecific amplifier [7]. The elastic and plastic properties of materials by employing instrumented sharp (geometrically self-similar indenters like Vickers, Pyramids, Berkovich or Cones) indentation may be computed from a single loading-displacement curve through a general theoretical framework proposed by Giannakopoulos and Suresh [8]. Their procedure can be used to accurately predict the indentation response from a given set of elastic-plastic properties (forward algorithms), and to extract elasticplastic properties from a given set of indentation data (reverse algorithms) [9]. Pileup (or sink-in) leads to contact areas that are greater than (or less than) the cross-sectional area of the indenter at a given depth. These effects lead to errors in the absolute measurement of mechanical properties by nanoindentation. The measured indentation modulus and hardness would be too high in the case of pileup and too low in the case of sink-in without accounting for the difference between the actual contact area and the cross-sectional area of the indenter [10]. Saha and Nix examined the effects of substrate on determining the mechanical properties of thin films by nanoindentation [11]. Compared to hardness, the nanoindentation measurement of the elastic modulus of thin films is more strongly affected by substrate. True contact area and true hardness of film can be determined from the measured contact stiffness, irrespective of the effects of pileup or sink-in around the indenter.
Much research has been done on the indentation problem of a half-space by a rigid indenter [12][13][14].
However, not all small scale structures are flat. Examples of small scale microplastics and fibers (typical diameter 10-20um) require material characterization. The material properties are not affected by the geometry of the test specimen, but the Oliver-Pharr procedure to obtain material properties will vary according to the geometry of the testing specimen. There has been no reliable theoretical method to evaluate the mechanical behavior of nanoindentation on a curved specimen. It is necessary to conduct reliable numerical simulations to evaluate the mechanical behavior of nanoindentation on a microsphere. The numerical simulations are usually carried out via the finite element method [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].
Phadikar and Karlsson investigated the possibility of extending instrumented indentation to non-flat surfaces [21][22][27][28][29]. In this study, finite element method had been used to systematically investigate the mechanical behavior of nanoindentation on a microsphere.

Theoretical Background
The analysis of Sneddon for the indentation of an elastic half space by a flat, cylindrical punch leads to a simple relation between P and h of the form [31] where P is the indenter load, h is the displacement of the indenter relative to the initial undeformed surface, a is the radius of the cylinder, G is shear modulus, and  is Poisson's ratio of testing specimen. Noting that the contact area A is equal to 2 a  (i.e., the projected area or cross sectional area of elastic contact) and that shear modulus is equal to where / S dP dh  is the initial stiffness of the unloading curve, defined as the slope of the upper portion of the unloading curve during the initial stages of unloading (also called contact stiffness), and E is the Young's modulus of testing specimen. For the Berkovich and Vickers pyramids, the equivalent cone angle is 70.296°, and the area-to-depth relationship, also known as the area function, is given by where A is the cross-sectional area of the indenter at a distance c h (contact depth) back from its tip.
Known the contact depth, and the shape of the indenter determined through the "area function", the contact area is then determined. If the contact stiffness and contact area were known, Equation 2 and can be used to measure the elastic modulus of a material. Taking one complete cycle of loading and unloading data, three quantities are measured: one is the maximum load, another is the maximum displacement max h (the maximum displacement of the indenter relative to the initial undeformed surface), and the third is the unloading stiffness.
Effects of non-rigid indenters on the load-displacement behavior can be effectively accounted for by defining an effective elastic modulus through the equation where i E and i  are the Young's modulus and Poisson's ratio of the indenter. If the indenter is a rigid body (i.e., i E   ), for any axisymmetric indenter, the effective elastic modulus eff E can be derived as If the indenter is a conical indenter, then The normal definition of hardness H is where max P is the peak indentation load.
The Young's modulus and hardness extracted from the Oliver-Pharr method are dependent on the initial stiffness of the unloading curve and the projected area of the indentation at the contact depth c h .
To correct Oliver-Pharr's solution accounting for the radial displacements, Hay used finite element method to calibrate Equation 2 and included a "correction factor" [32]. The correction factor is dependent on the half-included angle of indenter and Poisson's ratio of a material where  is the correction factor, and OP E is the Young's modulus extracted according to Oliver-Pharr

Finite element Model
To reduce testing, finite element analysis is used to calculate the load-displacement curves of nanoindentation during the loading and unloading, and the unloading curve can be used to determine the Young's modulus of a material using the Oliver-Pharr method (reverse algorithms). Finite element analyses of nanoindentation tests were carried out on an elastic-plastic microsphere. Figure 1a shows the finite element model of a microsphere with 11.5µm radius and the mesh generated using ABAQUS, in which two dimensional CAX4R (continuum, axisymmetric, quadrilateral four-node reduced integration) and CAX3 elements were used in the mesh discretization of the microsphere.       Figure 4 correspond to the "correction factor" extracted from the Oliver-Pharr method. The correction factor is not a constant with the increase of normalized maximum indentation depth even for the same / y E  . This means that the calculated elastic modulus of a microsphere using the Oliver-Pharr method according to simulated unloading curve is dependent on the indentation depth. As a consequence, suggesting that formula 2 is unsuitable for calculating the elastic modulus of nanoindentation involving cured surfaces. The surface of the test specimen of a microsphere requires prepolishing to achieve accurate results of indentation on a microsphere material. Figure 4b shows that as long as the ratio of Young's modulus of the microsphere over the initial yield

Conclusion
A computational study was undertaken to simulate the instrumented nanoindentation of elasticplastic microsphere materials. The ratio of Young's modulus of the microsphere over the initial yield stress of the microsphere, / y E  , was systematically varied from 10 to 1000 to cover the most mechanical properties of materials encountered in engineering. Finite element simulation results indicate that the load is proportional to the square of the indenter displacement during loading as The calculated elastic modulus of a microsphere using Oliver-Pharr formula according to simulated unloading curve was found to be dependent on the indentation depth, suggesting that this formula is unsuitable for calculating the elastic modulus of nanoindentation involving cured surfaces. The surface of the test specimen of a microsphere requires prepolishing to achieve accurate results of indentation on a micro-spherical material.