Mechanics of Solids: Equilibrium for Rigid vs Deformable Bodies

[heman]

In a lecture of Mechanics of solids ,the instructor said,

For an rigid body to be in equilibrium it should have no tendency to either displace or rotate(that's okay!)but for deformable bodies each & every subsystem should be in equilibrium!
I wonder how it is that body can be in equilibrium but its subspaces are not!

 

[sniffer]

when the centre of mass of the system remains static. subsystem mass are not necessarily static.

 

[heman]

Obviously i think the basic constituent particles are not at rest but what would be the definition of subsystem here!it is at which level!

 

[faust9]

I wonder how it is that body can be in equilibrium but its subspaces are not!

How did you make this leap? Sure, within a crystal structure or rather at a molecular level there is motion, but at a macroscopic level for a body to be in equilibrium the entire body as well as the differential bodies must also be in equilibrium. What that means is ME's disregard molecular motion and deal with macroscopic concepts for equilibrium in that when you look at beam bending you integrate along the length of the beam as if each differential section behaved as a solid with the same characteristics as the whole body---unless there is a/are known point(s) of change is composition or geometry.

your fist statement:

but for deformable bodies each & every subsystem should be in equilibrium!

means that if you have a column beam (those things that support the roof of the parthanon) or a cantalevered beam(a diving board) or any other beam/truss/whatever setup the beam(s) will tend toward a state of lowest potential energy. What that means is if you do not over load the beams they will bend or buckle, thus deforming, in proportion to the forces applied such that after some amount of time with constant forces applied the structure will no longer translate or rotate. The entire structure will be in equilibrium as will the differential areas or cross sections of the beams themselves.

Even if you do overload the beams they will still tend toward the lowest potential energy; however, the lowest potential energy for an overloaded beam may be at the bottom of a heap of rubble in an extreme case or the lowest PE may be the beam experiences a little plastic deformation. Anyway...

If you looked at the smallest cross section of a beam in equilibrium(assuming macroscopic properties) then that smallest cross section wil be stationary just like the entire body.

Hope this helped because the above concept is the basis for some pretty important concepts in mechanics like Castigliono's(sp?) theorms. These theorms assume macroscopic properties(molecular motion is inconsequential thus neglected---or rather factored in as a simple temperature term) in order to easily use this minimal potential energy concept. Differential cross sections are assumed to be in equilibrium if the entire beam is in equilibrium.

If you want to look at molecular motion then you're getting a little more into the materials aspect of engineering and away from the mechanics aspect. Even with molecular motion, at that level the motion will still be in equilibrium if the entire beam is in equilibrium.

 

[heman]

but at a macroscopic level for a body to be in equilibrium the entire body as well as the differential bodies must also be in equilibrium. .

Well doesn't there exist even a single case when the body is in equilibrium but its subspaces are not!
I have heard that it exists in case of Deformable bodies,and since only few initial lectures are done we haven't reached till cantilevers etc.etc.!

your fist statement:


means that if you have a column beam (those things that support the roof of the parthanon) or a cantalevered beam(a diving board) or any other beam/truss/whatever setup the beam(s) will tend toward a state of lowest potential energy. .

Agree!

What that means is if you do not over load the beams they will bend or buckle,.

Sorry,i can't realize this!
DO You mean body with proper loading will bend or buckle to achieve min. energy!
Or rather shouldn't the body move into state of min. potential energy irrespective of the lesser/over loading!

thus deforming, in proportion to the forces applied such that after some amount of time with constant forces applied the structure will no longer translate or rotate. The entire structure will be in equilibrium as will the differential areas or cross sections of the beams themselves.

Even if you do overload the beams they will still tend toward the lowest potential energy; however, the lowest potential energy for an overloaded beam may be at the bottom of a heap of rubble in an extreme case or the lowest PE may be the beam experiences a little plastic deformation. Anyway...

If you looked at the smallest cross section of a beam in equilibrium(assuming macroscopic properties) then that smallest cross section wil be stationary just like the entire body.

Hope this helped because the above concept is the basis for some pretty important concepts in mechanics like Castigliono's(sp?) theorms. These theorms assume macroscopic properties(molecular motion is inconsequential thus neglected---or rather factored in as a simple temperature term) in order to easily use this minimal potential energy concept. Differential cross sections are assumed to be in equilibrium if the entire beam is in equilibrium.

If you want to look at molecular motion then you're getting a little more into the materials aspect of engineering and away from the mechanics aspect. Even with molecular motion, at that level the motion will still be in equilibrium if the entire beam is in equilibrium.