  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.
