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Titanium is the best material for a mountain bike to it's ability to absorb impact energy as well as impressive corrosion resistance.

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Carbon Fibre is best suited to Road Bikes for it's amazing specific stiffness and strength. A cycle will not bleed power into frame deflection and the lightweight allows fast acceleration

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This article shows carbon fibre can absorb impact energy and is biologically inert as well as un reactive to environmental stressors such as oxygen.

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Material Advantages and most suited Bicycle Applications: Carbon Fibre

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Material Advantages and most suited Bicycle Applications: Carbon Fibre

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https://www.livestrong.com/article/271286-titanium-vs-carbon-bikes/
The Summary of the article provides information that states carbon Fibre is a highly strong material but very costly to produce due to an intensive, labour intensive production process. The carbon fibre can be brittle in some loading scenarios which would not be ideal in mountain bikes. However, carbon fibre is very light and could be used in Road bikes which are not exposed to high impact forces and do not require high energy absorption characteristics.

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This website describes the most commonly used materials in bike manufacture.
In summary, they are:
Aluminium Alloy
Stainless Steel
Titanium
Carbon Fibre
Magnesium

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Material Advantages and most suited Bicycle Applications: Titanium

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Material Advantages and most suited Bicycle Applications: Titanium

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Titanium has the best strength to weight ratio of the metals. Is it highly corrosion resistance. Ideal for mountain bikes due to high toughness. Denser than Aluminium but less volume required for reduced weight. high impact toughness so the expected crashes and falls will not harm the bike.

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research filtered for currency (one week) on advanced google search shows there are various types of bicycle with different purposes.

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research filtered for currency (one week) on advanced google search shows there are various types of bicycle with different purposes.

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These are:
Road Bikes
Racing Bikes
Mountain Bikes
Folding bikes
Electrically Assisted Bikes

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Material Advantages and most suited Bicycle Applications: Aluminium Alloy

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Material Advantages and most suited Bicycle Applications: Aluminium Alloy

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Aluminium Alloy: Highly available and relatively cheap to obtain. Can be treated with a variety of heat treatments for increased toughness. can be alloy with many different elements for improved properties. Good Strength to weight ratio and corrosion resistance

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https://www.redbull.com/gb-en/buying-a-mountain-bike-everything-you-need-to-know
Article showing how mountain bikes must be able to absorb high quantities of absorbed energy through the frame and springs.
material TOUGHNESS is a high priority. Titanium is ideal but high cost.

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A 2017 and CURRENT article showing how Road bikes benefit from light and rigid frames as flexible frames bleed out input energy.
https://www.forbes.com/sites/kevinmurnane/2017/01/11/new-materials-have-advantages-over-carbon-fiber-for-bike-frames/#43dae18a10d6