Dynamic Garden Hoses (Magic Hose) - 2

Back to Magic Hoses.

Most advertizements of Magic Hoses (X-Hose, Pocket Hose or what ever with this structure and expanding mechanism) feature:

1) Automatically expandable in the axial direction (emphasized) or extensible, stretchable and automatically contractible (retractable) but expandable in the radial direction as well though not very much emphasized.

2) Light weight

3) Flexible

4) Kink Free

All these are due to that they use a rubber or rubber like tube as a inner tube or they utilize the very good elasticity of rubber and the other rubber nature.

So we think about Elasticity. We already saw some basic equations of Force - Stress - Strain - Displacement and even Elasticity in the previous posts.

Elasticity (physics) - wiki (6th-July-2015)

"  
Linear elasticity

Main article: Linear elasticity

As noted above, for small deformations, most elastic materials such as springs exhibit linear elasticity and can be described by a linear relation between the stress and strain. This relationship is known as Hooke's law. A geometry-dependent version of the idea^[4] was first formulated by Robert Hooke in 1675 as a Latin anagram, "ceiiinosssttuv". He published the answer in 1678: "Ut tensio, sic vis" meaning "As the extension, so the force",^[5][6][7] a linear relationship commonly referred to as Hooke's law. This law can be stated as a relationship between force F and displacement x,

where k is a constant known as the rate or spring constant. It can also be stated as a relationship between stress σ and strain :

where E Is known as the elastic modulus or Young's modulus.
Although the general proportionality constant between stress and strain in three dimensions is a 4th order tensor, systems that exhibit symmetry, such as a one-dimensional rod, can often be reduced to applications of Hooke's law.


"

From the equation

     -------------  (1)

to the equation

          -------------  (2)

is a big jump of generalization as well as understanding more generally by using Stress - Strain and the relationship between them plus complicated <tensor> (σ: Stress and ɛ: Strain are expressed as a tensor). But the key idea is the same and simple - linear change proportionality with Constant E, which depends on mainly the material property and supposing the same E everywhere.

(1) shows F (Force) relates with the constant <k> (which depends on mainly the material property of the object under study) and the displacement <x>. <-> sign is not strict and depends on which direction is <+> or <->. In terms of unit <k> should be N(Newton) / m(meter). It means "Force changes the length of the object with a ratio of <-k>. If we take F as the vertical axis and <x> as the horizontal axis we can draw a declining straight line with the rate <k>.

<F>

I
I    \
I     \
I      \
I       \                      F = - kx
I        \
I         \
I          \
I_________________  <x>
 


The more force the less <x>. The less force the more <x>. Oops ! a mistake ? 

We must stat from the equilibrium where F = 0, x = 0 (no Force, no displacement)


                 <F>
           
     \              I             
       \            I           
         \          I         
           \        I                         F = - kx
             \      I     
               \    I   
                 \  I  
_________ \I________________  <x>
                    I\
                    I  \
                    I    \
                    I      \
                    I        \
                    I          \
                    I            \

or the same thing

                <-F>

                    I              /
                    I            /
                    I          /
                    I        /                   - F = kx
                    I      /
                    I    /
                    I  /
__________I/_________________  <x>
                   /I
                 /  I
               /    I
             /      I
           /        I
         /          I
        /           I



Now equation (2)



Where 

E: Elastic modulus
σ: Stress
ɛ: Strain

plus <tensor>. We must add one more concept - Poisson ratio. Generally Garden Hoses can be theoretically or rather mostly mathematically analyzed by using the equations below and most of which we have seen in the previous posts but  just as given without explanation. Garden Hoses belong to one group of cylinder shape objects, to which the same equations can apply. So these equations are generalized.




Concept of stress tensor






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