Zigzags

The structure is combined as herring bones from laminates or from anisotropic initial phases. These structures disperse the direction of anisotropy, and they result is either isotropic mixtures or in mixtures with lower anisotropy rate.

$$D_{\mbox{\footnotesize {zigzag}}}= L( R(\phi) D R^T(\phi), R(-\phi) D R^T(-\phi), m, n_0)$$

where $R(\phi)$ is the rotation matrix. The construction can be repeated many times leading to a fractal geometry.

Zigzag structure

Zigzags may model the natural high order laminate structures, as in the pictures .. These structures helped to solve several optimization problems The zigzag structures are known to have:

• The extremal value of bulk modulus and the whole range of values of shear modulus, depending on the angle of zigzags (actually, the zigzag construction should be repeated infinitely many times to achieve isotropy of the final composite)
• The minimum stiffness under a given shear-type load, in both two and three dimensions (the angle must be specially adjusted)
• Negative Poisson coefficient (close to its theoretical bound)