Jute is very weak. Try breaking a strand of it.
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How weak is jute really?
Of plant fibers, flax and hemp are said to be the strongest fibers, yet if you take a narrow strand of flax, you can easily break it. So that's not really a good test. A similar "strand" of osage would break even much faster (see below), yet no one would dare to say that osage wood makes poor bow backs.
It is said that jute is inferior to flax and hemp (e.g., TBB2: strings), but how does that relate to jute as a backing material compared to other fibers, and to other woods (unbacked)?
Suppose you'd elongate a plant/wood fiber by 1% (about the maximum stretch a normal self bow would ever encounter on its back) how much force would you need?
Let's use Osage as the bench mark of strength, the most prized bow wood on this forum, and give it a strength score of 10, so we can compare how other fibers or woods relate to that for a same strand/wood diameter.
Osage: 10
Yew: 8.0
Black locust: 12.1
Lemonwood: 13.5
Ipe: 19.0
Bulletwood: 19.8
strong Flax: 60.1
weak flax: 43.0
Strong hemp: 51.5
Weak hemp: 25.8
Strong jute: 43.0
weak jute: 17.2
high grade Sisal: 18.9
Though not all woods can take 1% elongation, all these plant fibers can. Many will even stretch to 2 or sometimes even 3% (which is still way below what sinew can take, but that's another story).
So in other words, even rather low quality jute fibers will be as strong in tension as the strongest wood (bamboo not considered).
the reason for this is that the actual tensile strength of plant and wood fibers is determined entirely by cellulose. The higher the cellulose content, the stronger the fiber or wood. Flax has the highest cellulose content of any plant fiber (65-85%), Hemp is a bit lower (60-77%), jute even lower (45-63%), just like sisal (50-64%). But that's still more than most woods, which have at most 40-50% cellulose content.
So if decently applied, any OK plant fiber backing will be stronger and more elastic than a wood backing, even fibers generally rated as poor like jute and sisal.
Of all these fibers, it seems that sisal can stretch the furthest in an elastic way (see table 11 in chapter below, making it the most sinew-like).
Source of the plant fiber strength data (and pdf of article available on request):
Lilholt H. and Lawther J.M. Natural organic fibres. In: Comprehensive composite materials, editors: Kelly A. and Zweben C. Vol. 1. New York: Pergamon Press; 2000.
Source of the wood strength data: The wood handbook: wood as an engineering material;
www.wood-database.com