Nano Tribology and Fracture Mechanics

Table of Contents

Chapter No Topic Page Number

01 Nano technology In Mechanical Engineering
1.1 Introduction
1.2 Scope of Nanotechnology
1.3 Fundamental Concepts
1.4 Nano materials
1.5 Mechanical Properties of Nano Materials
1.6 Importance of Nano Materials:
1.7 Nano composite
1.8 Nano-composites – basic ingredients:
1.9 Significance and advantages
1.10 Nano Composites
1.11 Selected Application of Nano materials And Nanotechnology
1.12 Role of nano-structured materials in mechanical engineering applications;
1.13 Disadvantages of nano materials
1.14 Safety of nano materials

02 Introduction to nano mechanics
2.1 Materials, Film coatings
2.2 Industrial considerations
2.3 Structures & Geometries
2.4 Mechanical behaviour of nano particles
2.5 Defects And Testing of Nano Structures;
2.6 Different failure mechanism

03 Introduction To Nano Tribology
3.1 Definition of Nano Tribology;
3.2 Need of Nano Tribology;
3.3 Understand nano tribology;
3.4 Introduction to Atomic Force Microscope (AFM),
3.5 An introduction to the Atomic Force Microscope (AFM) and FFM
3.6 The use of AFM to Study Nano Tribology;
3.7 Scratching, wear and machining; For micro scale scratching, micro
3.8 Surface potential measurements
3.9 Nano Indentation Measurements
3.10 Boundary lubrication measurements;
3.11 Identifying materials with low friction and good adhesion for nanotechnology applications:
3.12 Applications of Nano Tribolgy in current products

04 Fracture of Nano Material
4.1 Introduction-Nano Indentation;
4.2 Fracture Mechanism of Brittle Thin Films;
4.3 Crack patterns in brittle film/hard substrate
4.4 Crack patterns in brittle film/ductile substrate;
4.5 Fracture of Mono & Multi Layers of Gold Nano Particle;
4.6 Fracture of Hollow Silica Nano Particles (HSNP’s)
4.7 Fracture Mechanism of Solid Lubricant Nano Particle
4.8 Fracture Mode of Ultrasonic Treated Nickel Nano Particles

05 Advanced Nano Materials
5.1 Block Copolymer Systems-Introduction
5.2 Self - Assembly of Block Copolymers
5.3 Precursors for Novel Composite Materials –Introduction
5.4 CNT–Metal based nano particle Composites;
5.5 CNT–AuNP Composite;
5.6 Nano ceramics- Applications
5.7 Carbon Nano tubes Adsorbents for purification purpose
5.8 Nano material Imprinting Technique;
5.9 Fullerene contained Nanostructures
5.10 Combined CNT with Bio molecules: Advancements and future Challenges;

CHAPTER- 1 NANOTECHNOLOGY IN MECHANICAL ENGINEERING

1.1 Introduction

Nanotechnology is science, engineering and technology conducted at the nano scale, which is about 1 to 100 nm where nano denotes the scale range of 10-9 and nanotechnology refers the properties of atoms and molecules measuring thoroughly 0.1 to 1000 nm. Nanotechnology is highly interdisciplinary as a field, and it requires knowledge drawn from a variety of scientific and engineering areas. Nanotechnology has become an all-embracing term, which means different things to different people. Nanotechnology is interface technologies that are include many different science and applications area. Nanotechnology falls into this category and offers fundamentally new capabilities to architect a board array of the novel materials, composites and structure on a molecular scale. Here discusses on some of the applications for nanotechnology and shows a few cases of them. That is believed to have the highest probability of success in competitive industry. The nanotechnology that are economically promising for the future include those that have applications in information technology, electronics, building materials, household appliances, textiles, cosmetics, food, environmental technologies, energy technologies and medicine etc.,

Nanotechnology deals with studies of phenomena and manipulation with elements of matter at the atomic, molecular and macromolecular level (rangefrom1to100nm), where the properties of matter are significantly different from their properties at larger scales of dimensions. There are two main types of approaches to nanotechnology: the first approach is Top-down and another one is Bottom-up approach. The Top-down approach involves taking layer structures that are either reduced down size until they reach the nano-scale or deacon structured into their composite parts. The other hand the Bottom-up approach is where materials are constructed from the atomic or molecular components. Designing at the nanoscale is working in a world where physics, chemistry, electrical engineering, mechanical engineering, and even biology become unified into an integrated field. “Building blocks” for nanomaterials include carbon-based components and organics, semiconductors, metals, and metal oxides; nanomaterials are the infrastructure, or building blocks, for nanotechnology. The last decade has seen advancement in every side of nanotechnology such as: nanoparticles and powders; nanolayers and coats; electrical, optic and mechanical nanodevices; and nanostructure biological materials. Presently, nanotechnology is estimated to be influential in the next 20-30 years, in all fields of science and technology.

1.2 Scope of Nanotechnology in Mechanical Engineering

The nanotechnology in mechanical engineering and manufacturing is immensely useful to the field. Nanotechnology can be used to increasing the life of the components and automobile parts. A many number of materials can be enhanced by the use of nanotechnology. Nanomaterials exhibit unique physical and chemical properties and impart enhancements to engineered materials. There including better magnetic properties, improved mechanical activity and increased optical properties. Developments are being made to improve the properties of the materials and to find alternative precursors that can give desirable properties on the materials.

1.3 Fundamental Concepts in Nano Technology

Nanotechnology involve the ability to see and to control individual atoms and molecules, everything on earth is made up of atoms the food we eat the cloths we wear the building and houses we live in, and our own bodies. Here are a few illustrative examples,

- There are 25,400,000 nano meter an inch.
- A sheet of news paper is about 100,000 nano meter thick.

The microscopes needed to see things at the nano scale were invented relatively recently. Although modern nano science and nanotechnology are quite new, nano scale materials were used for centuries. Today’s scientists and engineers are finding a wide variety of ways to deliberately make materials at the nano science to take advantage of their enhanced properties. such as higher strength, lighter weight, increased control of light spectrum and greater chemical reactivity than their large-scale counter parts.

1.4 Nano materials

Nanoscale materials are defined as a set of substances where at least one dimension is less than approximately 100 nanometers. A nanometer is one millionth of a millimetre approximately 100,000 times smaller than the diameter of a human hair. Nanomaterials are of interest because at this scale unique optical, magnetic, electrical, and other properties emerge. These emergent properties have the potential for great impacts in electronics, medicine, and other fields. Nanomaterials are resources designed at the molecular (nanometre) level to take advantage of their small size and novel properties which are generally not seen in their conventional, bulk counterparts. The two main reasons why materials at the nano scale can have different properties are increased relative surface area and new quantum effects. To understand the convective effect of nano material brazillian crystal shown in fig.1.1 and lotus effect are interpreted which is as Shown in fig.1.2

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Fig.1.1 Brazilian Crystal Fig.1.2 Lotus effect

1.5 Mechanical Properties of Nano Materials:

Nano materials have the structural features in between of those of atoms and the bulk while most micro structured materials have similar properties to the corresponding bulk materials, the properties of materials with nano meter dimensions are significantly different from those of atoms and bulks materials. This is mainly due to the nano meter size of the materials which render them: (i) large fraction of surface atoms; (ii) high surface energy; (iii) spatial confinement; (iv) reduced imperfections, which do not exist in the corresponding bulk materials. Due to their small dimensions, nano materials have extremely large surface area to volume ratio, which makes a large to be the surface or interfacial atoms, resulting in more “surface” dependent material properties. Nanoparticles are of great scientific interest as they are effectively a bridge between bulk materials and atomic or molecular structures. A bulk material should have constant physical properties regardless of its size, but at the Nano-scale this is often not the case. Size-dependent properties are observed such as quantum confinement in semiconductor particles, surface Plasmon resonance in some metal particles and super Para magnetism in magnetic materials. Nanoparticles exhibit a number of special properties relative to bulk material. Copper nanoparticles smaller than 50 nm are considered super hard materials that do not exhibit the same malleability and ductility as bulk copper. The often very high surface area to volume ratio of nanoparticles provides a tremendous driving force for diffusion, especially at elevated temperatures. Sintering is possible at lower temperatures and over shorter durations than for larger particles. This theoretically does not affect the density of the final product, though flow difficulties and the tendency of nanoparticles to agglomerate do complicate matters. They also offer good physical & mechanical properties in Nano structure and lightweight. Porosity is an important property for Nano materials.

1.6 Importance of Nano Materials:

These materials have created a high interest in recent years by virtue of their unusual mechanical, electrical, optical and magnetic properties. Some examples are given below:

- Nanophase ceramics are of particular interest because they are more ductile at elevated temperatures as compared to the coarse-grained ceramics.
- Nanostructured semiconductors are known to show various non-linear optical properties. Semiconductor Q-particles also show quantum confinement effects which may lead to special properties, like the luminescence in silicon powders and silicon germanium quantum dots as infrared optoelectronic devices. Nanostructured semiconductors are used as window layers in solar cells.
- Nanosized metallic powders have been used for the production of gas tight materials, dense parts and porous coatings. Cold welding properties combined with the ductility make them suitable for metal-metal bonding especially in the electronic industry.

1.7 Nano composite:

Nano composites can be made with a variety of enhanced physical, thermal and other unique properties. They have properties that are superior to conventional micro scale composites and can be synthesized using simple and inexpensive techniques. Materials are needed to meet a wide range of energy efficient applications with light weight, high mechanical strength, unique color, electrical properties and high reliability in extreme environments. The term nano composite encompasses a wide range of materials right from three dimensional metal matrix composites, two dimensional lamellar composites and nano-wires of single dimension to zero-dimensional core-shells all representing many variations of nano-mixed & layered materials. Though various composite materials like fiberglass and reinforced plastics are now in wide use for numerous applications, there has been continued demand for novel composites with desirable properties for many other applications.

1.8 Nano-composites – basic ingredients:

There has been a great deal of interest in polymer nano composites over the last few years. There are different types of commercially available nano-particles that can be incorporated into the polymer matrix to form polymer nano composites. Polymer nano composites consist of a polymeric material (e.g., thermoplastics, thermosets or elastomers) with reinforcement of nano-particles. Polymeric nanocomposites can be broadly classified as

- Nanoclay-reinforced composites
- Carbon nanotube-reinforced composites
- Nanofibre-reinforced composites, and
- Inorganic particle-reinforced composites.

1.9 Significance and advantages

Nano Composites are termed as evolutionary the class of materials. It has very high impact in developing a new generation of composites with enhanced functionality and a wide range of applications. The data on processing, characterization and applications helps researchers in understanding and utilizing the special chemical and material principles underlying these cutting-edge polymer nano composites. Although Nano composites are realizing many key applications in numerous industrial fields, a number of key technical and economic barriers exist to widespread commercialization. These include impact performance, the complex formulation relationships and routes to achieving and measuring nanofiller dispersion and exfoliation in the polymer matrix. Investment in state-of-the-art equipment and the enlargement of core research team’s is another bottleneck to bring out innovative technologies on nano composites.

- Improve the macroscopic properties of products
- Nano composites differ from conventional composite materials due to the exceptionally high surface to volume ratio of the reinforcing phase and/or its exceptionally high aspect ratio
- Improves electrical and thermal conductivity by adding particles like carbon nano tubes.
- Other kinds of nano particulates may result in enhanced optical properties, dielectric properties, heat resistance or mechanical properties such as stiffness, strength and resistance to wear and damage.
- Nano composites can greatly enhance the properties of materials as it leads to the formation of nano scale aluminide secondary phases in aluminium alloys, thereby increasing their strength and corrosion resistance.

1.10 Nano Composites for Sustainable, Performance Enhancing Mechanical Members.

Due to the high corrosive nature of metals used in oil and gas pipelines, and failure rate of conventional materials the importance of nano composites cannot be over-emphasised, this is because nano composites are materials with a nano scale structure that improve the macroscopic properties of products. In mechanical terms, nano composites differ from conventional composite materials due to the exceptionally high surface to volume ratio of the reinforcing phase and/or its exceptionally high aspect ratio. The reinforcing material can be made up of particles (e.g. minerals), sheets (e.g. exfoliated clay stacks) or fibres (e.g. carbon nano tubes or electrospun fibres). The area of the interface between the matrix and reinforcement phase(s) is typically an order of magnitude greater than conventional composite materials. The large amount of reinforcement surface area means that a relatively small amount of nanoscale reinforcement can have an observable effect on the macro scale properties of the composite. For example, adding carbon nano tubes improves the electrical and thermal conductivity. Other kinds of nanoparticulates may result in enhanced optical properties, dielectric properties, heat resistance or mechanical properties such as stiffness, strength and resistance to wear and damage. In general, the nano reinforcement is dispersed into the matrix during processing. The percentage by weight (called mass fraction) of the nano particulates introduced can remain very low (on the order of 0.5% to 5%) due to the low filler percolation threshold, especially for the most commonly used non-spherical, high aspect ratio fillers (e.g. nano meter-thin platelets, such as clays, or nanometer-diameter cylinders, such as carbon nanotubes). Nano composites can greatly enhance the properties of materials as it leads to the formation of nanoscale aluminide secondary phases in aluminium alloys, thereby increasing their strength and corrosion resistance. Magnetic multilayered materials are one of the most important aspects of nanocomposites, as they have led to significant advances in storage media. Nanocomposites are becoming very popular today due to the enormous benefits being derived from it, this explains its acceptability and why leading manufacturing companies are spending millions of dollars on its research and development. The objectives of the project is to conduct a thorough research on the topic and then look for better approach to manufacture better and more durable nanocomposite products for oil and gas pipelines, that will serve as better alternatives to metal and other conventional materials.

1.11 Selected Application of Nano materials And Nanotechnology:

Nanotechnology and Nanomaterials having wide range of applications in the field of energy sectors, It is evident that nanomaterials split their conventional counterparts because of their superior chemical, physical, and mechanical properties and of their exceptional formability.

1.11.1 Heavy industry’s:

An inevitable use of nanotechnology will be in heavy industries.

Aerospace:

Lighter and stronger materials will be of immense use to aircraft manufactures, leading to increased performance, spacecraft will also benefit where weight is a major factor. Nanotechnology would help to reduce the size of equipment and there by decrease of fuel-consumption required to get it airplane.

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Fig. 1.3 Nano Aero Space

Catalysis:

The application of potential nanoparticles in catalysis ranges from fuel cell to catalytic converters and photo catalytic devices. The synthesis provides novel “nanomaterials” and in the long run, superior processes such as “self-assembly” will enable energy time preserving strategies. Platinum nanoparticles are now being considered in the next generation of automotive catalytic converters because the very high surface area of nanoparticles could reduce the amount of platinum required.

Automobile industry:

The present-days automobile vehicle has more inner components parts in the system. Those parts are more hard wearing and more heat-resistant. The auto engine wastes loft of fuel and to create a population because of incomplete gas combustion. Now nanotechnology and nanomaterials are likely to play a significant role in sparkplugs. Since nanomaterials are strongest, harder and resist wear and erosion, they are currently being considered for the use in sparkplug.

Fuel Tanks:

The ability of nanoclay incorporation to reduce solvent transmission through polymers such as polyamides has been demonstrated. Available data reveals significant reductions in fuel transmission through polyamide–6/66 polymers by incorporation of nanoclay filler. As a result, considerable interest is now being seen in these materials as both fuel tank and fuel line components for cars. Of further interest for this type of application, the reduced fuel transmission characteristics are accompanied by significant material cost reductions.

Coatings:

Nanocoating refers to the act of covering a material with a layer on the nanometer scale or to cover a nanoscaled entity. Nanocoating forms a nanocomposite that comprises a combination of two or more different substances of nanometer size, thereby producing a material that generally has enhanced or specific targeted properties due to the combined properties and/or structuring effects of the components.

Steel coatings:

The nanotechnology in steel material its help to improve the physical properties of steel, fatigue or the structural failure of steel is due to cyclic loading. Steel cables can be strength the using carbon nanotubes are stronger cables are reduce the costs of the constructions.

Food coatings:

A nanocomposite coating process could improve food packaging by placing anti-microbial agents directly on the surface of the coated film. They can also improve the mechanical and heat-resistance properties and lower the oxygen transmission rate. Research is being performed to apply nanotechnology to the detection of chemical and biological substances for sensing’s in foods.

1.11.2 Glass coatings and other areas

Glass also makes use of nanotechnology. TiO2 nanoparticles are used to coat glazing it has sterilizing and anti-fouling properties. TiO2is hydrophilic which can attract rain drops that wash off the dirt particles. Glass is using a clear instrument layer sandwiched between glass panels formed of silica nanoparticles which turns into a rigid and opaque fire shield when heated.

1.11.3 Fire retardation:

Polymers containing a few weight per cent of nanoparticle clays have greatly improved fire resistance as reported by Gilman. The thermal properties of the PNC are improved, melting and dripping are delayed, and rate of burning is greatly reduced (by more than half). The presence of flake-like clay nanoparticles reduces the diffusion of polymer decomposition volatiles (the fuel) to the burning surface and reduces diffusion of air into the polymer.

1.11.4 Corrosion protection:

Corrosion protection of metals and alloys is normally achieved by surface coatings which must resist both mechanical damage (scratching, impact, abrasion) and chemical attack (salts, acids and bases, solvents). It should also not be damaged (cracked) by having a coefficient of thermal expansion greatly different from the metal to be protected. PNCs have improved scratch and abrasion resistance, due to their higher hardness combined with improved elastic.

1.11.5 Sports:

Nanotechnology may also play a role in sports such as soccer, football, and baseball. Materials for new athletic shoes may be made in order to make the shoe lighter (and the athlete faster).Baseball bats already on the market are made with carbon nanotubes which reinforce the resin, which is said to improve its performance by making it lighter.

1.12 Role of nano-structured materials in mechanical engineering applications;

1.12.1 Application in automobile industry

1. Spark plug

Since Nano materials are stronger, harder, and resist wear and erosion, they are currently being considered for use in spark plugs. Nano electrodes would make spark plugs long lasting and fue lefficient. The rail plug made by Nano material creates powerful sparks that burn fuel better.

2. Engine coatings

Automobiles waste huge amounts of energy by way of heat loss from the engine, especially from the diesel ones. Engineers are currently looking at coating engine cylinders with Nano crystalline ceramics, like zirconia and alumina, to help preserve heat efficiency and increase fuel combustion. Engineers are currently looking at coating engine cylinders with Nano crystalline ceramics, like zirconia and alumina, to help preserve heat efficiency and increase fuel combustion.

3. Engine

Today's car engines are only 25 per cent efficient; meaning only a quarter of the energy stored in fuel is actually converted to useful work. Fuel cells - devices that work by harnessing the chemical attraction between oxygen and hydrogen to produce electricity - are, by contrast, 50 per cent efficient. Further, because they use oxygen – which is taken from the air, and hydrogen - the most abundant element in the universe, they have the potential to produce clean and cheap energy. The only by-products are heat and water.

4. Braking system, Body panel

Use of aluminium nanotube composite in the braking system results effective braking performance. It will reduce brake system weight while increasing acceleration. In addition to being lighter, Nano composites are significantly more resistant to wear and tear in case of body panels.

5. Chassis

Chassis are structural support of a vehicle. If we can reduce its weight it will become more fuel efficient. So we can use Nano size steel instead of Aluminium. It will offer lower weight, dimensional accuracy, corrosion resistance and aesthetics.

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Fig. 1.4 Automobile chassis

1.12.2 Application in the field of construction

Steel has been widely available material and has a major role in the construction industry. The theoretical strength of steel is 27.30 GPa (in <111> direction). There are two ways of achieving high strength in steels. One method is by reducing the size of a crystal to such an extent that it is free of any defects. Second method is by introducing a very large density of defects in a metal sample that act as an obstacle to the motion of dislocations. The Nano size steel produce stronger steel cables which can be used in bridge construction. The strengthening arises due to the presence of Nano scale cementite/ferrite lamellar structure. The high carbon steel wire is an important engineering material used for reinforcing automobile tires, galvanized wires etc.

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Fig.1.5 Nano steel structured building

1. Steel

Steel is a widely available material that has a major role in the construction industry. The use of nanotechnology in steel helps to improve the physical properties of steel. Fatigue, or the structural failure of steel, is due to cyclic loading. Current steel designs are based on the reduction in the allowable stress, service life or regular inspection regime. This has a significant impact on the life-cycle costs of structures and limits the effective use of resources. Advancements in this technology through the use of nanoparticles would lead to increased safety, less need for regular inspection, and more efficient materials free from fatigue issues for construction. Steel cables can be strengthened using carbon nanotubes. Stronger cables reduce the costs and period of construction, especially in suspension bridges, as the cables are run from end to end of the span. The use of vanadium and molybdenum nanoparticles improves the delayed fracture problems associated with high strength bolts. This reduces the effects of hydrogen embrittlement and improves steel micro-structure by reducing the effects of the inter-granular cementite phase.

2. Glass

Research is being carried out on the application of nanotechnology to glass, another important material in construction. Titanium dioxide (TiO2) nanoparticles are used to coat glazing since it has sterilizing and anti-fouling properties. The particles catalyse powerful reactions that break down organic pollutants, volatile organic compounds and bacterial membranes. TiO2 is hydrophilic (attraction to water), which can attract rain drops that then wash off the dirt particles. Thus the introduction of nanotechnology in the Glass industry incorporates the self-cleaning property of glass.

1.12.3 Effect of nano materials on bio-social environment

1. Effect of nanoparticles on health and environment: Nanoparticles may also enter the body if building water supplies are filtered through commercially available nano filters. Airborne and waterborne nanoparticles enter from building ventilation and wastewater systems.
2. Effect of nanoparticles on societal issues: As sensors become commonplace, a loss of privacy and autonomy may result from users interacting with increasingly intelligent building components.

1.13 Disadvantages of nano materials

- The nanostructured steels exhibit inadequate ductility.
- Instability of the particles - Retaining the active metal nanoparticles is highly
Challenging, as the kinetics associated with Nano materials is rapid.
- Nanostructured materials require non-traditional processing. It will lead to high cost.
- Impurity - Because nanoparticles are highly reactive, they inherently interact with impurities as well.
- The workers who engaged in the production can cause health hazard.
- Biologically harmful – Nano materials are usually considered harmful as they become transparent to the cell-dermis. Toxicity of Nano materials also appears predominant owing to their high surface area and enhanced surface activity.
- Fine metal particles act as strong explosives owing to their high surface area coming In direct contact with oxygen and due to this exothermic combustion can cause Explosion.
- Recycling and disposal - There are no hardand- fast safe disposal policies evolved for Nano materials. Issues of their toxicity are still under question

1.14 Safety of nano materials;

Nanoparticles behave differently than other similarly sized particles. It is therefore necessary to develop specialized approaches to testing and monitoring their effects on human health and on the environment. When materials are made as nanoparticles, their surface area to volume ratio will be more. The greater specific surface area (surface area per unit weight) may lead to increased rate of absorption through the skin, lungs, or digestive tract and may cause unwanted effects to the lungs as well as other organs. However, the particles must be absorbed in sufficient quantities in order to pose health risks. The result of a study in 2008 showed that iron oxide nanoparticles caused little DNA damage and were non-toxic. Zinc oxide nanoparticles were slightly worse. Titanium dioxide caused only DNA damage. Carbon nanotubes caused DNA damage at low levels. Copper oxide was found to be the worst offender, and was the only nanomaterial identifie by the researchers as a clear health risk. Though Nano materials are not confirmed as a health risk to workers who produce them, NIOSH recommends that exposure precautions and personal protective equipment be used to protect workers until the risks of nanomaterial manufacture are better understood.

CHAPTER-2 INTRODUCTION TO NANO MECHANICS

2.1 Materials, Film coatings, Industrial considerations, Structures & geometries

2.1.1 Materials;

Metals;

Nano indentation is the fastest way to measure the Young’s modulus and Vickers hardness of metals. Testing is completely automated, because the residual hardness impression doesn’t have to be imaged. With Nanomechanics’ equipment and standardized test methods, the entire process, from test method to final report, is radically faster and easier than what most people expect from a micro hardness tester.

Ceramics;

Nanoindentation is the simplest way to measure the Young’s modulus and Vickers hardness of ceramics. Nanoindentation can be done on an as-manufactured part, thus eliminating the need to produce dog-bone specimens for tensile testing. Nanoindentation is also the best way to measure high-temperature properties, because the test requires only a small volume of material which can be heated more quickly and uniformly. Fracture toughness can also be estimated through nanoindentation testing by taking advantage of the brittle nature of ceramics and cracking that occurs during contact.

Polymers;

Dynamic nanoindentation gives exactly the same information as Dynamic Mechanical Analysis (DMA), but testing is faster, easier, and can be done on thin films and other small volumes that cannot be tested by DMA. The engineers at Nanomechanics, Inc. pioneered the technique of dynamic indentation, which oscillates the indenter over a range of frequencies to measure storage modulus, which characterizes the elasticity of the material, and loss modulus, which characterizes the damping of the material.

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