# Worst case boundary loads for bolted or joined composites

Composite materials and metallic alloys often fail near structural features where stress can concentrate. Examples include neighborhoods surrounding welds, lap joints or bolt holes where composite or metallic structures are fastened or joined, see for example (Tong and Soutis, 2003) and (Esaklul, 2002). Large boundary loads can increase the overall energy near structural features and initiate failure. These aspects provide motivation for a better understanding of energy concentration inside structures. State of the art applies the Saint-Venant principle (Maremonti and Russo, 1994), (von Mises, 1942), (Saint-Venant, 1885) to characterize the rate of decay of the magnitude of the stress or strain away from the boundary and study its effect on interior subdomains.

Unfortunately the Saint-Venant principle can not predict the worst case boundary load that would cause a composite elastic structure to fail. With this in mind we pursue a refined analysis and address the problem from an energy based perspective. Here we will consider traction loads on the boundary $\partial\omega^\ast$ of the structural domain $\omega^\ast$.

The worst case load is defined to be the one that delivers the largest portion of a given input energy to a prescribed interior domain of interest. Together P. Sinz and M. Stuebner (Lipton, Sinz, and Stuebner, 2016) we have successfully characterized the worst case load for composite structures as the solution to a special eigenvalue problem. It is shown to correspond to the largest singular value of the restriction operator $R$ acting on space of solutions to the equation of elasticity with zero right hand side over $\omega^*$. Here $R$ is the restriction of the solution over the structural domain $\omega^*$ to the domain of interest $\omega$. We then develop the computational method for computing the worst case boundary loads and associated energy contained inside a prescribed subdomain through the numerical solution of the eigenvalue problem. We apply this computational method to bound the worst case load associated with an ensemble of random boundary loads given by a second order random process. Several examples are carried out on heterogeneous structures to illustrate the method. We note that this is the same type of spectral problem identified as delivering the optimal local bases with MS-GFEM for scalar and elastic problems, see (Babuska and Lipton, 2011) and (Babuska, Huang, and Lipton, 2014).

The work discussed here has appeared in R. Lipton, R. Sinz, and M. Stuebner. Uncertain loading and quantifying maximum energy concentration within composite structures. J. Comput. Phys., 325:38-52, 2016.
This work is supported by NSF Grant DMS-1211066

## References

1. I. Babuska, X. Huang, and R. Lipton. Machine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method. Mathematical Modeling and Numerical Analysis (M2AN), 48 Number 2 (2014) 493-515.
2. I. Babuska and R. Lipton. Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems. Multiscale Modeling and Simulation, SIAM 9 (2011) 373-406.
3. Khlefa Alarbe Esaklul. Handbook of Case Histories in Failure Analysis. ASM The Materials Information Society, Materials Park OH, 2002.
4. R. Lipton, R. Sinz, and M. Stuebner. Uncertain loading and quantifying maximum energy concentration within composite structures. J. Comput. Phys., 325:38-52, 2016.
5. P. Maremonti and R. Russo. On the von-mises-sternberg version of st venants principle in plane linear elastostatics. Arch. Rational Mech Anal., 128:207-221, 1994.
6. A.J.C.B. Saint-Venant. Memoire sur la torsion des prismes. Mem. Divers Savants, 14:233-560, 1885.
7. L. Tong and C. Soutis. Recent Advances in Structural Joints and Repairs for Composite Mate- rials. Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
8. R. von Mises. On saint-venant's principle. Bull. AMS, 51:555-562, 1945.