A singular problem in incompressible nonlinear elastostatics (Q2764897)

The work deals with the numerics of problems in the incompressible nonlinear theory of elasticity, with particular interest in plane problems involving corners. These corners require special attention. An example of this is provided by a compressed bonded block problem corresponding to the compression of an incompressible elastic block of rectangular cross-section and infinite transverse length between two opposing bonded rigid surfaces, with the two remaining lateral faces left traction-free. To examine this problem and particularly the behavior of the solution at a corner where a bonded end is adjacent to a free lateral side, the authors use a finite element method. The latter is based on a reduced and selective integration technique with penalization. This method converges everywhere except in a small neighborhood of the said corner. Then an apriori inequality concerning the angle of shear is employed to show that the numerical computations in the neighborhood of this corner are inaccurate. A more refined study is needed. A way out is offered by a conjecture concerning the local shape of the deformed free lateral surface at the corner. This is a smart paper illustrated by efficient figures including a sketch of the distribution of principal stresses of Cauchy stress tensor.