Multi-material topology optimization of adhesive backing layers via J-integral and strain energy minimizations

Strong adhesives rely on reduced stress concentrations, often obtained via specific geometry or composition of materials. In many examples in nature and engineering prototypes, the adhesive performance relies on structural rigidity being placed in specific locations. A few design principles have been formulated, based on parametric optimization, while a general design tool is still missing. We propose to use topology optimization to achieve optimal stiffness distribution in a multi-material adhesive backing layer, reducing stress concentration at specified locations. The method involves the minimization of a linear combination of J-integral and strain energy. While the J-integral minimization is aimed at reducing stress concentration, we observe that the combination of these two objectives ultimately provides the best results. We analyze three cases in plane strain conditions, namely (i) double-edged crack and (ii) center crack in tension and (iii) edge crack under shear. Each case evidences a different optimal topology with (i) and (ii) providing similar results. The optimal topology allocates stiffness in regions that are far away from the crack tip, intuitively, but the allocation of softer materials over stiffer ones can be non-trivial. To test our solutions, we plot the contact stress distribution across the interface. In all observed cases, we eliminate the stress singularity at the crack tip. Stress concentrations might arise in locations far away from the crack tip, but the final results are independent of crack size. Our method ultimately provides optimal, flaw tolerant, adhesives where the crack location is known.