What is Geometric frustration and why do we study it?
Let me begin by discussing what leads to an interest in studying geometric frustration or incompatibility.
Keep a piece of string on the top of a table. Consider two cases,
Stretch on both ends of the string, now cut the string from the middle. What happens? The string moves away from the point where it is cut.
Do not apply any force. What happens? The string does not move on cutting!
If you do not apply the force on the string, on cutting, the string would not have moved. If you are convinced of this, you will agree with me that it was the external force that caused the string to move when it was cut. People who have some familiarity with physics at late school or entry college level, can understand it as the external force leading to an internal tension in the string. The internal tension leads to the string moving when it is cut. Let me make a leap and say that the string will move on cutting at a point if and only if it has some internal tension at that point!
We note that the external force we have applied is the source of the internal tension in the string.
Let us do another experiment. Take a piece of carrot and cut it from the top but not all the way, notice what happens.
The two cut parts of the carrot move apart.
If you agree with me on what I said earlier, then you would agree that the carrot has some internal tension. But you have not applied any external force!! Why do the half cut pieces of carrot move apart? What is the source of this internal tension?
The answer to this question is 'geometric frustration' or incompatibility.
Elastic solids usually have a characteristic shape or reference configuration, that they relax to, in the absence of applied forces. However these reference configurations are fixed by local processes like plasticity which are heterogeneous in nature. In other words, different regions of a solid body have their own locally prescribed reference configurations. However these configurations might not fit with each other. We refer to such an incompatibility in the reference configuration of the elastic solid as 'Geometric Frustration.'
Geometric Frustration acts as a source of non trivial internal stress in an elastic solid even in the absence of external forces. Following are some research topics of interest in relation to Geometric frustration:
Mathematical characterisation and classification of Geometric frustration
What are the sources of Geometric frustration?
Response of an elastic solid to the presence of Geometric frustration
Relation of Geometric frustration to processes like plasticity, activity, biological growth