Material elasticity determines scaling behaviour of cracking dynamics in porous materials: a precursor to crack percolation

Abstract The crack statistics of a 3-dimensional disordered porous material subjected to constant axial compressive loading have been investigated with regard to the elastic properties of the system. While cracking is a complex dynamic that involves material intrinsic properties like grain shape and size distribution, elastic properties of grain and cementing materials, and extrinsic properties of loading, in this work, the focus has been to check the dependence on the elastic properties of the bonding material. Three-dimensional disordered systems were constructed from spherical balls of varying average radius with a size distribution chosen randomly from a log-normal distribution. The growth of micro-cracks with increasing compressive strain was monitored till the limit of the percolation crack. The two parameters varied were the bond stiffness constant and the strength of the bonding material. Two distinct regimes of cracking rates were observed across a critical strain ϵ knee that manifested as a knee in the cumulative crack-strain plot. The average coordination number the the particles was also monitored with increasing strain. The critical strain ϵ knee and the strain at the percolation point ϵ perc showed a power law dependence. Individual micro-cracks were observed to grow sharply to a maximum value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> <mml:mi>a</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msubsup> </mml:math> , after which the number of new micro-cracks decreased, showing a long tail. The maximum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> <mml:mi>a</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msubsup> </mml:math> was found to occur at ϵ knee and corresponded to ≈66% of the average coordination number of bonds connected to a particle. Thus, rapid growth of 33% micro-cracks pre- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> <mml:mi>a</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msubsup> </mml:math> was indicative of brittle failure, followed by ductile cracking behavior of the system. Finally, we show that there exists a robust power law relation between ϵ knee and ϵ perc . The crack statistics were found to be independent of the average particle size and elastic properties of the material. As ϵ knee can be determined from acoustic signals associated with micro-cracks, our proposed relation can act as a warning towards critical strain resulting in crack percolation.