Is it possible to mechanically isomerize an sp3 hybridized carbon center? Imagine I have an sp3 hybridized carbon attached to four separate polyethylene chains.  By pulling on the polyethylene chains in some manner, is it possible for me to mechanically isomerize the chiral center prior to breaking any carbon-carbon bonds?  
Perhaps a more realistic scenario would involve shining a laser on a compound similar to 2,2-dimethylpropane (i.e. a fully saturated carbon compound where four carbons are attached to an sp3 hybridized carbon), but with asymmetric functionalizations on the bonded carbons to allow for detection of isomerization.  Just as in the previous scenario, can mechanical isomerization occur prior to breaking any carbon-carbon bonds?

Quoting from Georg's answer: "In "practice" there is a big problem, You would need two Laplacian demons (with two hands each) to do the experiment..."
It's not clear to me that this is an impossible experiment.  For example, you could attach one polyethylene chain to a surface and then try to make a covalent bond between the other relevant chain and, say, an AFM tip.  If this is successful, one would then proceed in a manner similar to that reported in "How Strong is a Covalent Bond?" by Michel Grandbois et. al. (http://www.sciencemag.org/content/283/5408/1727.short), which estimates a ~4.0 nanonewton rupture force for a C-C bond (See Fig. 4) and directly measures the rupture of a single Si-C bond to be ~2.0 +/- 0.3 nanonewtons.  
In the case where one wants to look for mechanical isomerization of the sp3-hybridized carbon, one would look for an earlier sub-4.0 nN peak in the force vs. extension curve (or whatever critical force is experimentally measured for breaking a C-C bond), perhaps allow relaxation at some critical force to complete the isomerization process, and then compare the observed displacement with a geometrical model.
Sounds ridiculously hard, but I see no fundamental reason it couldn't be done.
 A: The exact answer is a problem of computational chemistry, 
maybe somebody has caculated the energy of such a center 
in a planar configuration, but I don't know. 
In "practice" there is a big problem, You would need two Laplacian demons 
(with two hands each) to do the experiment :=)
Generally, on the level of such a molecule, "mechanic" is a unrealistic simplification. 
Nature on this level is not electric as opposed to mechanic or what ever.
So forcing the four ligands in a plane (the activated complex of thiś isomerisation) 
will lead to rupture of one of the bonds, either forming a radical pair or 
an ion pair, this depends on the milieu (gas phase, polar liquid, unpolar etc) 
The possible (imaginable) configurations of the four ligands make a energy hypersurface
and system will go over the lowest pass  to a more stable configuration. 
Why can one be so shure that it impossible? Think of reason for that "hybridisation"!
Why is the cabon not bound to three neighbours at 90 degrees like PH3 (nearly) by three bonds
via a pure p-orbital and the forth to a pure s-orbital? Reason is simply space. 
There is not enough room around a carbon , contrary to atoms in higher periods like phosphor or sulfur.  In this sense the hybridisation is forced by "mechanics". 
Think of Gillespie-Nyholm rules and retropolate "down" for the crowded space 
around a C-Atom.
