Abstract
We discuss compliance minimization from the broad perspective of Free Material Design (FMD) with Young’s moduli in place of Hooke’s tensor. By this, we attempt to optimize these elastic moduli which tightly correspond to principal components of stress at each point in the design space. The optimization task under study takes a form of the following constrained variational problem: In a given design space, fix two variables for which the total stored energy functional is minimal: (i) design variable - the directional Young modulus field restricted by an isoperimetric condition, and (ii) state variable - the statically admissible stress field. The functional is parametrically dependent on the Poisson ratios; these moduli are not optimized in the proposed approach. Similarly to FMD, our problem also reduces to minimizing a functional of linear growth in stresses. A characteristic feature is that the functional to be minimized is Michell-like, i.e. it takes a form of an integral of absolute values of principal components of stress tensor if the Poisson ratios are set to zero.
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Acknowledgement
The paper was prepared within the Research Grant no 2013/11/B/ST8/04436 financed by the National Science Centre (Poland), entitled: Topology optimization of engineering structures. An approach synthesizing the methods of: free material design, composite design and Michell-like trusses.
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Dzierżanowski, G., Lewiński, T. (2018). Young’s Modulus Control in Material and Topology Optimization. In: Schumacher, A., Vietor, T., Fiebig, S., Bletzinger, KU., Maute, K. (eds) Advances in Structural and Multidisciplinary Optimization. WCSMO 2017. Springer, Cham. https://doi.org/10.1007/978-3-319-67988-4_103
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