Nonlinear Analysis of Oscillatory Indentation in Elastic and Viscoelastic Solids

Determining the mechanical properties at micro- and nanometer length scales using nanoindentation or atomic force microscopy is important to many areas of science and engineering. Here we establish equations for obtaining storage and loss modulus from oscillatory indentations by performing a nonlinear analysis of conical and spherical indentation in elastic and viscoelastic solids. We show that, when the conical indenter is driven by a sinusoidal force, the square of displacement is a sinusoidal function of time, not the displacement itself, which is commonly assumed. Similar conclusions hold for spherical indentations. Well-known difficulties associated with measuring contact area and correcting thermal drift may be circumvented using the newly derived equations. These results may help improve methods of using oscillatory indentation for determining elastic and viscoelastic properties of solids.

Finite Element Solutions for Plane Strain Mode I Crack with Strain Grading Effect

In this paper, a new phenomenological theory with strain gradient effects is proposed to account for the size dependence of plastic deformation at micro- and submicro-length scales. The theory fits within the framework of general couple stress theory and three rotational degrees of freedom omega(i) are introduced in addition to the conventional three translational degrees of freedom mu(i). omega(i) is called micro-rotation and is the sum of material rotation plus the particles' relative rotation. While the new theory is used to analyze the crack tip field or the indentation problems, the stretch gradient is considered through a new hardening law. The key features of the theory are that the rotation gradient influences the material character through the interaction between the Cauchy stresses and the couple stresses; the term of stretch gradient is represented as an internal variable to increase the tangent modulus. In fact the present new strain gradient theory is the combination of the strain gradient theory proposed by Chen and Wang (Int. J. Plast., in press) and the hardening law given by Chen and Wang (Acta Mater. 48 (2000a) 3997). In this paper we focus on the finite element method to investigate material fracture for an elastic-power law hardening solid. With remotely imposed classical K fields, the full field solutions are obtained numerically. It is found that the size of the strain gradient dominance zone is characterized by the intrinsic material length l(1). Outside the strain gradient dominance zone, the computed stress field tends to be a classical plasticity field and then K field. The singularity of stresses ahead of the crack tip is higher than that of the classical field and tends to the square root singularity, which has important consequences for crack growth in materials by decohesion at the atomic scale. (C) 2002 Elsevier Science Ltd. All rights reserved.

Liquefaction and Displacement of Saturated Sand Under Vertical Vibration Loading

In order to investigate the influence of the vertical vibration loading on the liquefaction of saturated sand, one dimensional model for the saturated sand with a vertical vibration is presented based on the two phase continuous media theory. The development of the liquefaction and the liquefaction region are analyzed. It is shown that the vertical vibration loading could induce liquefaction. The rate of the liquefaction increases with the increase of the initial limit strain or initial porosity or amplitude and frequency of loading, and increases with the decrease of the permeability or initial modulus. It is shown also that there is a phase lag in the sand column. When the sand permeability distribution is non-uniform, the pore pressure and the strain will rise sharply where the permeability is the smallest, and fracture might be induced. With the development of liquefaction, the strength of the soil foundation becomes smaller and smaller. In the limiting case, landslides or debris flows could occur.

Torsional impact of transversely isotropic solid with functionally graded shear moduli and a penny-shaped crack

The torsional impact response of a penny-shaped crack in an unbounded transversely isotropic solid is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace transform and Hankel transform are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress fields are obtained. Investigated are the influence of material nonhomogeneity and orthotropy on the dynamic stress intensity factor. The peak value of the dynamic stress intensity factor can be suppressed by increasing the shear moduli's gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface.

An Energy-Based Method for Analyzing Instrumented Spherical Indentation Experiments

Using dimensional analysis and finite element calculation, we studied spherical indentation in elastic-plastic solids with work hardening. We report two previously unknown relationships between hardness, reduced modulus, indentation depth, indenter radius, and work of indentation. These relationships, together with the relationship between initial unloading stiffness and reduced modulus, provide an energy-based method for determining contact area, reduced modulus, and hardness of materials from instrumented spherical indentation measurements. This method also provides a means for calibrating the effective radius of imperfectly shaped spherical indenters. Finally, the method is applied to the analysis of instrumented spherical indentation experiments on copper, aluminum, tungsten, and fused silica.

Analysis of indentation loading curves obtained using conical indenters

Using dimensional analysis and finite-element calculations we determine the functional form of indentation loading curves for a rigid conical indenter indenting into elastic-perfectly plastic solids. The new results are compared with the existing theories of indentation using conical indenters, including the slip-line theory for rigid-plastic solids, Sneddon's result for elastic solids, and Johnson's model for elastic-perfectly plastic solids. In the limit of small ratio of yield strength (Y) to Young's modulus (E), both the new results and Johnson's model approach that predicted by slip-line theory for rigid-plastic solids. In the limit of large Y/E, the new results agree with that for elastic solids. For a wide range of Y/E, some difference is found between Johnson's model-and the present result. This study also demonstrates the possibilities and limitations of using indentation loading curves to extract fundamental mechanical properties of solids.

Resistance of cast polyurethane elastomers to solid particle erosion

The rates of erosive wear have been measured for a series of eight polyester-based one-component castable polyurethane elastomers, with widely varying mechanical properties. Erosion tests were conducted with airborne silica sand, 120μm in particle size, at an impact velocity of 50 ms-1 and impact angles of 30° and 90°. For these materials, which all showed similar values of rebound resilience, the erosion rate increased with increasing hardness, tensile modulus and tensile strength. These findings are at variance with those expected for wear by abrasion, perhaps because of differences in the strain rate or strain levels imposed on the elastomer during erosion and abrasion.

Dynamic stress intensity factor of a functionally graded material under antiplane shear loading

The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.

Amorphous Coatings Deposited on Aluminum Alloy by Plasma Electrolytic Oxidation

Amorphous [Al-Si-O] coatings were deposited on aluminum alloy by plasma electrolytic oxidation (PEO). The process parameters, composition, micrograph, and mechanical property of PEO amorphous coatings were investigated. It is found that the growth rate of PEO coatings reaches 4.44 mu m/min if the current density is 0.9 mA/mm(2). XRD results show that the PEO coatings are amorphous in the current density range of 0.3-0.9 mA/mm(2). EDS results show that the coatings are composed of O, Si and At elements. SEM results show that the coatings are porous. Nano indentation results show that the hardness of the coatings is about 3 - 4 times of that of the substrate, while the elastic modulus is about the same with the substrate. Furthermore, a formation mechanism of amorphous PEO coatings was proposed.

Substrate Effects on the Micro/Nanomechanical Properties of TiN Coatings

Nanoindentation and nanoscratch tests were performed for titanium nitride (TiN) coatings on different tool steel substrates to investigate the indentation/scratch induced deformation behavior of the coatings and the adhesion of the coating–substrate interfaces and their tribological property. In this work, TiN coatings with a thickness of about 500 nm were grown on GT35, 9Cr18 and 40CrNiMo steels using vacuum magnetic-filtering arc plasma deposition. In the nanoindentation tests, the hardness and modulus curves for TiN/GT35 reduced the slowest around the film thickness 500 nm with the increase of indentation depth, followed by TiN/9Cr18 and TiN/40CrNiMo. Improving adhesion properties of coating and substrate can decrease the differences of internal stress field. The scratch tests showed that the scratch response was controlled by plastic deformation in the substrate. The substrate plays an important role in determining the mechanical properties and wear resistance of such coatings. TiN/GT35 exhibited the best load-carrying capacity and scratch/wear resistance. As a consequence, GT35 is the best substrate for TiN coatings of the substrate materials tested.

Microstructure and mechanical properties at TiCp/Ni-alloy interfaces in laser-synthesized coatings

Titanium carbide particle (TiCp) reinforced Ni alloy composite coatings were synthesized by laser cladding using a cw 3 kW CO2 laser. Two kinds of coatings were present in terms of TiCp origins, i.e. undissolved and in situ reacted TiCp, respectively. The former came from the TiCp pre-coated on the sample, whereas the latter from in situ reaction between titanium and graphite in the molten pool during laser irradiation. Conventional and high-resolution transmission electron microscope observations showed the epitaxial growth of TiC, the precipitation of CrB, and the chemical reaction between Ti and B elements around phase interfaces of undissolved TiCp. The hardness, H, and elastic modulus, E, were measured by nanoindentation of the matrix near the TiCp interface. For undissolved TiCp, the loading curve revealed pop-in phenomena caused by the plastic deformation of the crack formation or debounding of TiCp from the matrix. As for in situ generated TiCp, no pop-in mark appears. On the other hand, in situ reacted TiCp led to much higher hardness and modulus than that in the case of undissolved TiCp. The coating reinforced by in situ generated TiCp displayed the highest impact wear resistance at both low and high impact conditions, as compared with coatings with undissolved TiCp and without TiCp. The impact wear resistance of the coating reinforced by undissolved TiCp increases at a low impact work but decreases at a high impact work, as compared with the single Ni alloy coating. The degree of wear for the composite coating depends primarily on the debonding removal of TiCp.

Effect of inclusions and holes on the stiffness and strength of honeycombs

A finite element study has been performed on the effects of holes and rigid inclusions on the elastic modulus and yield strength of regular honeycombs under biaxial loading. The focus is on honeycombs that have already been weakened by a small degree of geometrical imperfection, such as a random distribution of fractured cell walls, as these imperfect honeycombs resemble commercially available metallic foams. Hashin-Shtrikman lower and upper bounds and self-consistent estimates of elastic moduli are derived to provide reference solutions to the finite element calculations. It is found that the strength of an imperfect honeycomb is relatively insensitive to the presence of holes and inclusions, consistent with recent experimental observations on commercial aluminum alloy foams.

Crack Tip Field and J-Integral with Strain Gradient Effect

The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani. The theory retains the essential structure of the incremental version of the conventional J_2 deformation theory No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required. The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus. The strain gradient measures are included into the tangent modulus as internal parameters. Therefore the boundary value problem is the same as that in the conventional theory Two typical crack Problems are studied: (a) the crack tip field under the small scale yielding condition induced by a linear elastic mode-I K-field and (b) the complete field for a compact tension specimen. The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it. Consequently, the J-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases.

Capacitive nanoelectromechanical switch based on suspended carbon nanotube array

We present the fabrication and high frequency characterization of a capacitive nanoelectromechanical system (NEMS) switch using a dense array of horizontally aligned single-wall carbon nanotubes (CNTs). The nanotubes are directly grown onto metal layers with prepatterned catalysts with horizontal alignment in the gas flow direction. Subsequent wetting-induced compaction by isopropanol increases the nanotube density by one order of magnitude. The actuation voltage of 6 V is low for a NEMS device, and corresponds to CNT arrays with an equivalent Young's modulus of 4.5-8.5 GPa, and resistivity of under 0.0077 Ω·cm. The high frequency characterization shows an isolation of -10 dB at 5 GHz. © 2010 American Institute of Physics.

What is indentation hardness?

Using dimensional analysis and finite element calculations, we derive simple scaling relationships for loading and unloading curve, contact depth, and hardness. The relationship between hardness and the basic mechanical properties of solids, such as Young's modulus, initial yield strength, and work-hardening exponent, is then obtained. The conditions for 'piling-up' and 'sinking-in' of surface profiles during indentation are determined. A method for estimating contact depth from initial unloading slope is examined. The work done during indentation is also studied. A relationship between the ratio of hardness to elastic modulus and the ratio of irreversible work to total work is discovered. This relationship offers a new method for obtaining hardness and elastic modulus. Finally, a scaling theory for indentation in power-law creep solids using self-similar indenters is developed. A connection between creep and 'indentation size effect' is established.