Elasticity of an object I was studying elasticity, and I thought of certain questions that I couldn't find an answer to. I'll will try to explain my queries


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*If in the elastic limit, the strain on say a rectangular block of wood is proportional to the stress, then when this stress is removed will the object oscillate between getting contracted and extended? I think it should because the restoring force should have a parabolic potential curve for atleast small strain although I'm not sure if that's case.

*If the bulk modulus of an (hypothetical) material happens to be negative then by the volume strain-stress relation the material should expand on applying pressure, and my professor said that such a material would be at unstable equilibrium and will continue to expand on it's own once disturbed, in such a case how would the potential curve for the forces between atoms looks like? 

*In the above case why couldn't the object just expand and then go back to it's form when deforming forces are removed? 
Edit: changed hyperbolic to parabolic.
 A: Objects like wood do indeed oscillate when a load is suddenly applied, or removed. You can demonstrate this by striking a block of wood with a hammer and listening to the result. 
An object that responds by contracting when squeezed would indeed be unstable. A good example of this is a submarine. When it sinks to greater depth, the pressure on its hull increases which causes it to contract slightly. This reduces its displacement volume, which reduces the buoyant force on it, which causes it to sink faster, which increases the pressure on its hull, etc., etc. until it is suddenly crushed by the pressure. 
To prevent this sort of dynamic instability requires external control because the system quickly runs out of control. The perfect example of this is the Chernobyl explosion. The RBMK-1000 reactor exhibits a fundamental instability mode, in which a perturbation in its power output will grow without bound unless positive control is exerted faster than a human could respond. The reactor operators disabled all the nested power control networks to run their experiment, and when a sudden surge occurred there was nothing stopping its growth and the plant blew up. 
A: 
1) Does the sudden release of a compressed elastic material result in the oscillation of its bulk mass?

Yes, an elastic material will oscillate after releasing a compressive force. 


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*Example: Consider a sphere of perfectly elastic material.  To isolate the oscillatory effect, place this perfect Superball in orbit around the earth, to prevent contact with walls or objects which would draw off energy.  Likewise, assume a perfect vacuum so it cannot transmit acoustic energy to molecular collisions.   

*1) Strike the Perfect Superball on opposite sides, simultaneously, with two hammers. With no modes of energy loss, the energy supplied by the compressive Work will stay inside the bulk material almost forever, (i.e., neglect energy lost to gravitational waves).

*2) The energy supplied during compression is turned into potential energy initially.  Upon release of the compressive force, a) the potential energy of compression converts into kinetic energy (the atoms accelerate under the repulsive force of the compressed orbitals). The velocity of the sphere's surface accelerates until mid-cycle (a neutral sphere) and then decelerates to zero velocity - the point of maximum excursion and internal tension (a sphere with bulges on both sides). b) At the point of maximum expansion/zero kinetic energy, the force of tension predominates and begins compressing the Perfect Superball.  c) The cycle of compression and tension repeats in an eternal oscillation, trading potential and kinetic energy.


Oscillation is due to the properties of an elastic material, which are: 


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*1) The energy of compression (Work against a repulsive force) is retained in the bulk of the material as potential energy resulting from the conservative force of intermolecular repulsion. (Note: energy is a secondary/meta-phenomenon, dependent upon the primary properties of mass and fields to exert force). Upon release of the compressive force, the potential energy stored during compression is converted to kinetic energy.  A material in a static state is composed of atoms in equilibrium (a forceless state where further compression results in repulsion, and further tension results in attraction). Compression from that state displaces atoms in the lattice, resulting in a net repulsion between the molecules. The interatomic forces of compression and tension in an elastic material are conservative. (Note: this statement is redundant, as "elastic" implies a conservative force, just as "inelastic" implies non-conservative force.) Because of the conservation of energy, and the associated conservative force field, the Work done to the material by compression or tension, is available to do Work on an external object (for conversion into kinetic or potential energy).  

*2) In a perfectly elastic material, no energy is lost to: a) deformation (i.e., energy stored permanently in shortened or lengthened bonds, or plane slippage), or b) heat (randomly directed kinetic energy), c) radiation (e.g., activated orbitals that decay, molecular vibration that radiates, etc.).  


Oscillation in a bulk elastic material occurs when:


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*a) Potential Energy is stored in an elastic material by compressive Work by a force acting over a distance; 

*b) The compressive force is released.  The potential energy stored in the interatomic bonds is converted into kinetic energy (Force acting on a mass produces acceleration); 

*c) At the neutral/flat/zero-displacement/original surface, the potential energy is zero, and the kinetic energy is at its maximum.  There is no restraining force, and the inertial force of the bulk material propels the molecules of bulk material into tension;  

*d) The tension of the interatomic bonds act against the kinetic energy of the molecules, slowing them down until they stop at the point of full conversion from kinetic energy to the potential energy of tension; 

*e) The unopposed tension at the point of maximum outward bulging of the elastic material accelerates the molecules inward and returns that potential energy into kinetic energy. 


Thus, the energy increment supplied by compression will alternately convert from a) the potential energy of compression, b) the kinetic energy of expansion, c) the potential energy of tension, and then return to d) the kinetic energy of compression.  The energy cycles back and forth between potential and kinetic energy.  In a system with losses, the oscillation will continue until the energy losses to other modes have completely dissipated the initial energy input.


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*An elastic material will oscillate forever if it is isolated from all possible pathways of energy loss, such as heat, sound, radiation, permanent deformation, etc.

*A material is partially elastic if some of the energy input to these various energy losses as the bulk material cycles between compression and tension. A partially elastic system will exhibit damped oscillation, losing some displacement amplitude with each cycle. A piece of wood struck with a hammer is an example of a system that loses energy to permanent deformation upon its initial compression. The sound made by a hammer strike on wood is evidence of the displacement of air molecules that fill the compression indentation and then participate in the oscillatory cycles of contraction and expansion of the partially elastic wood molecules.  The sound of the hammer strike continues until the energy input is dissipated as acoustic and thermal energy.  The oscillation period is short, indicating the energy loss rate is large, resulting in rapid decay. 

*Above a critical value of damping (viscosity) the system will not oscillate, it will only return to its original configuration.  An example includes the spongy rubber used for stress balls, which upon compression-release only returns to its original shape without oscillation.

*As noted, a perfectly elastic material will cycle energy between compressive potential energy, kinetic energy, tension potential energy, kinetic energy, oscillating endlessly.



2) Would a hypothetical/imaginary elastic material with a reversed bulk modulus expand forever if an impulse of compressive pressure caused expansion? 

(Note: since imaginary substances are unregulated by reality, we cannot refer to the real world, and its real physical laws to explain its behaviors.  We must instead decide what laws the imaginary substance follows, and then describe its behavior based on that ruleset.)


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*Since this is an imaginary substance, the question is "How does the substance store the energy of compression or tension?"

*If we assume this imaginary material stores the energy of deformation as kinetic energy, then any small energy input will convert into an expansion which will never stop. A larger energy input will cause a more rapid expansion.  This is not an explosion, but rather like a balloon that continues to inflate once initiated. 

*Consider for comparison, the case of an impulse applied to a normal/real elastic material. An impulse supplies an increment of energy to the molecules of the material. A normal/real material absorbs kinetic energy, converting it to potential energy, which then oscillates between kinetic energy and potential energy.

*If we assume this imaginary material does not convert the Work of compression into normal potential energy, and we assume the conservation of energy, then the externally applied Work must convert into kinetic energy.  Once the material begins to expand (the motion of mass expanding is kinetic energy), it will continue to expand forever (unless a tension force is applied).  



3) Why will the material expand forever after the removal of the compressive force?



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*This system is, of course, unrealistic, but it illustrates that if we honor the principle of energy conservation, that the energy input must be converted into another form. In this case, the compressive force produces expansion without an increase in potential energy, thus kinetic energy is the only energy compartment available for energy storage.

*If we identify the character of this negative bulk modulus material as storing its energy after compression as kinetic energy in the direction of expansion, then the expansion will continue forever.  

*In the opposite case, if this material is put in tension (e.g., stretched) it will begin to contract, and continue to contract to its limit.  



What does the stress-strain curve look like, (i.e., stress on the x-axis, and strain on the y-axis)?



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*Any small stress (whether tension or compression) will produce the maximum expansion or contraction of the system. A greater force will result in a more rapid expansion or contraction.

*1) Any external compressive stress will initiate a maximum expansion. 

*2) Any externally applied tension will initiate a maximum contraction.  

*The stress vs strain curve for compression will be a series of vertical lines tracing upward to infinity. These vertical compression lines will be to the left of the origin.  That is, every value of compressive stress will produce a maximum displacement of expansion.

*The stress vs strain curve for tension will be a series of vertical lines tracing downward. These vertical compression lines will be to the right of the origin.  That is, every value of tension stress will produce a maximum displacement of contraction.

