What are the associated expression of the crack extension force and its physical origin?
To understand the origin of this question, please refer to the Q&A : Why are brittle materials stronger in compression than in tension?
Take into account the boundary conditions and loads distribution which determine stress-strain state of material. The brittle auxetics, having negative Poisson's ratio, seemed to be sensitive to surface tractions (for example, friction stress) arising during compression.
When you uniaxially compress a plate which contains a circular crack in the middle, there occur radial and circumferential stresses around the crack. Some of these stresses are tension stresses. Due to these tensile stresses, the failure occurs. When the crack is ellipsoidal, these tensile stresses may be larger. If you wanna see which stresses occur round the crack, please look at the fracture mechanics book of Perez.
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