Stress-smoothing holes in an elastic plate: From the square lattice to the checkerboard

Abstract

Minimization of the hoop stresses variation was recently advanced to form a novel and efficient criterion of the stress state optimization in a statically loaded elastic domain with a finite number of free-form holes. As a radical generalization of the well-known equi-stress principle, it offers substantial analytical and numerical advantages over direct minimization of the local stress concentration factor. Here we extend the proposed criterion beyond its primary application to the shape optimization in a regular perforated structure with the checkerboard arrangement of identical traction-free holes. The averaging nature of the stresses variation allows its effective numerical implementation in a wide range of the governing parameters at a modest computational cost. The results obtained describe in depth the elastic response of the optimal structure to applied loads.

Simply and doubly periodic arrangements of the equi-stress holes in a perforated elastic plane: The single-layer potential approach

“…The inevitable flattening against the cell boundary keeps the ESS from further evolving. A similar situation was observed previously in the square checkerboard lattice [4].…”

“…The inevitable flattening against the cell boundary keeps the ESS from further evolving. A similar situation was observed previously in the square checkerboard lattice [4]. Figure 7 compares the centrally located ESS of the same volume fraction for the two lattices with dependence on the load anisotropy.…”

“…A memory- and time-efficient encoding scheme for this type of 2D shape optimization problems was proposed in a series of our previous publications (see, for example, [2, 4] and references therein). With this in mind, our focus is narrowed to the smooth closed curve γ which can be conformally mapped onto a unit circle ϕ by an analytical function ω ( τ ) with a relatively small number M of first non-zero Laurent terms [11]…”

“…Finally, after some algebra over equations (3), (4) and (10) and subsequent integration along each λ j , j = 1, N, one arrives at the traction-free conditions in either of the two Cartesian displacements…”

A generalization of the equi-stress principle in optimizing the mechanical performance of two-dimensional grained composites

“…For a closed contour we require t n = t 0 . In essence, (1.2) indirectly generalizes the V criterion in the more simple case of minimizing the hoop stresses along traction-free holes [7,8], although the bi-material structure is technically harder to tackle. For this reason, a detailed analysis is performed here to justify the applicability of the same three-component numerical optimizer as before.…”