The Kirsch problem in second strain gradient elasticity

The Kirsch problem, namely the problem of an infinite plane with a circular hole under uniaxial tension, is one of the cornerstone problems in elasticity. The Kirsch problem is rooted in classical elasticity theory, which cannot explain the size effects. We investigate the Kirsch problem for the first time in the framework of Mindlin's second strain gradient elasticity theory. By presenting the fundamental equations in polar coordinates, we use the stress function to derive the closed-form solution of the Kirsch problem. Then, we compare the solution to its counterparts associated with first strain gradient elasticity and classical elasticity. Our results indicate that the strain gradient effects can increase stiffness and reduce the stress concentration around the hole. Under the second strain gradient elasticity theory, a larger circular hole radius results in higher stress concentration, exhibiting a significant size effect. Furthermore, we establish a mixed finite-element method for second strain gradient elasticity. The results of computational simulations are in close agreement with the solution proposed in this contribution. This work extends the Kirsch problem to second strain gradient elasticity providing a benchmark solution highlighting the importance of higher gradient effects in predicting the elastic behaviour of materials at smaller scales.