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Simultaneous design of optimal shape and local cubic material characteristics

Czubacki R., Lewiński T., Ganghoffer J. F.

Engineering Transactions

2017

Vol. 65, iss. 1

11--17

This paper deals with the minimum compliance problem of the femur bone made of a nonhomogeneous elastic material with cubic symmetry. The elastic moduli as well as the trajectories of anisotropy directions are design variables. The isoperimetric condition determines the value of the cost of the design expressed as the integral of the trace of the Hooke’s tensor. The optimum design is found for a selected design domain and a single load case. The optimal cubic material characteristics are reflected by the properties of the underlying microstructure. Admissible microstructures are reconstructed, thus delivering a deeper insight into the optimum design. The obtained microstructures are second- rank laminates composed of an isotropic material and voids. To eliminate the degeneracy of the design at least three load cases should be considered.

Connecting Euler and Lagrange systems as nonlocally related systems of dynamical nonlinear elasticity

Bluman G., Ganghoffer J. F.

Archives of Mechanics

2011

Vol. 63, nr 4

363-382

Nonlocally related systems for the Euler and Lagrange systems of twodimensional dynamical nonlinear elasticity are constructed. Using the continuity equation, i.e., conservation of mass of the Euler system to represent the density and Eulerian velocity components as the curl of a potential vector, one obtains the Euler potential system that is nonlocally related to the Euler system. It is shown that the Euler potential system also serves as a potential system of the Lagrange system. As a consequence, a direct connection is established between the Euler and Lagrange systems within a tree of nonlocally related systems. This extends the known situation for one-dimensional dynamical nonlinear elasticity to two spatial dimensions.