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Principle of virtual work and conservation of linear momentum

Submitted by bruno-page on

I've been reading Prof.Bower's text on solid mechanics:

http://solidmechanics.org/text/Chapter2_4/Chapter2_4.htm#Sect2_4_1

 

Can anyone plz help me to understand how principle of virtual work is another form of conservation of linear momentum?

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Computational Mechanics Forum
arash_yavari

To be precise, "principle of virtual work" is not really "another form of conservation of linear momentum". If you restrict yourself to rigid translations, principle of virtual work gives you conservation of linear momentum.



It is true that one can start with the local form of balance of linear momentum and contracting it with an arbitrary vector field and integrating over the body one can recover the principle of virtual work. But the problem is what is obtained is "assumed" to be valid even for situations for which differential equations do not necessarily make sense.



Regards,

Arash

Mon, 02/15/2010 - 17:48 Permalink
bruno-page

I do remember in  one of the posts regarding principle of virtual work (Virtual forces *Real Displacement = virtual stresses *Real strains) is a weak  form of conservation of linear momentum.

Can you explain and justify this statment?

From basic physics the conservation of linear momentum is - momentum of bodies before impact = moemntum after impact----In ther words, this (conservation of linear momentum)can be got from Newtons second law-> F = m(dv/dt) if there is no resultant force then there is no change in momentum that is momentum is conserved.

Can you explain how theprinciple of virtual work is weak form of conservation of linear momentum based on above in italic?

Mon, 02/15/2010 - 21:32 Permalink
bruno-page

I do remember in  one of the posts regarding principle of virtual work (Virtual forces *Real Displacement = virtual stresses *Real strains) you had mentioned that the principle of virtual work is a weak  form of conservation of linear momentum.

Can you explain and justify this statment?

From basic physics the conservation of linear momentum is - momentum of bodies before impact = moemntum after impact----In ther words, this (conservation of linear momentum)can be got from Newtons second law-> F = m(dv/dt) if there is no resultant force then there is no change in momentum that is momentum is conserved.

Can you explain how theprinciple of virtual work is weak form of conservation of linear momentum based on above in italic?

Mon, 02/15/2010 - 21:34 Permalink
arash_yavari

Dear Julie:



Let's not worry about the "weak form" part. That just means you can deal with discontinuities too. As I mentioned earlier, looking at principle of virtual work and restricting ourselves to rigid displacements (a constant vector field), (integral form of) balance of linear momentum is implied. Principle of virtual work is not just a "weak form" of balance of linear momentum; balance of angular momentum can be derived from it too. It can be shown that principle of virtual work can be derived using energy balance and it's covariance. I'm not sure if I said priniple of virtual work is equivalent to balance of linear momentum?



Regards,

Arash

Wed, 02/17/2010 - 20:57 Permalink
bruno-page

sorry for the late reply Arash,

you has actually said;

http://imechanica.org/node/5806

Now, in the paragraph;

Principle of virtual work in elasticity states that the work done by all the real "forces" (inertial, body forces, and tractions) in any virtual deformation of the body is zero. This is nothing but a weak form of balance of linear momentum and is widely used, for example, in finite element formulations.

Can you justify when you sa- weak form of conservatio of linear momentum?Then, in the paragraph;

Work of real forces acting on virtual displacements is equal to the corresponding stored energy. In the proof you would start with your real forces and corresponding real displacements and then impose the (compatible) virtual displacements. Balance of energy then will give you everything.

Can you justify when you say-' Balance of energy will give you everything'?

Sat, 02/20/2010 - 02:31 Permalink