A linear poroelastic analysis of equilibrium asymptotic fields around stationary sharp V-notches in polymer gels

Abstract

In this paper, an asymptotic solution is obtained for mode I stress and solvent concentration fields around stationary planar sharp V-notches in polymer gels at equilibrium state via poroelasticity theory. To this end, the mechanical equilibrium equations are first solved by employing a proper Airy stress function and then the solvent concentration field is obtained accordingly. It is shown that the stress field is similar to its corresponding linear elasticity solution with both real and complex eigenvalues. Furthermore, it is found that the solvent concentration field, around the notch tip, possesses the same singularity as the stress field and exhibits cosine variations with respect to the angular coordinate. The asymptotic solution is given for both plane stress and plane strain conditions. Next, a numerical study was performed on single edge notched (SEN) specimens with notch opening angles of 30° and 60° to explore the accuracy of the obtained asymptotic solution. It is shown that there is a very good match between the numerical results and the predictions of the asymptotic solution which confirms the validity of the present solution. The comparative study indicates that the first two or three terms of the asymptotic solution calibrated with the finite element over deterministic (FEOD) method is enough to accurately capture the stress and concentration fields at radial distances of about r/a = 0.2 from the notch tip. It is further shown that thicker SEN specimens absorb more solvent molecules per volume around their notch tips for similar far field applied strains and hence have more propensity to brittle fracture.