Optimized constitutive distributions visualized by lamina formulas

For optimal design most parameters may be classified in size, shape, and topology, such as simple density variables and parameters for surface description. Density and surface can be rather directly visualized. Extending the design to material design in sense of design of distributions of constitutive matrices, a practical visualization is more complicated but may be based on classical laminate analysis.

In rotational transformation of constitutive matrices, some practical quantities are often termed invariants, but the invariance relates to an unchanged reference direction. Rotating this reference direction, the practical quantities do change and this point is clarified with derived rotational transformation for these practical quantities.

The theoretical and numerical background for design of optimal anisotropic constitutive matrices are presented. Then design results are applied in a 2D visualization of optimized constitutive matrices that are distributed in a finite element model where each element has a specific reference direction.

The visualized distributions of physical quantities are stiffest material direction, material stiffest longitudinal constitutive component, level of anisotropy, absolute or relative shear stiffness, and test of orthotropy.