Abstract
The paper deals with the Free Material Design {(text {FMD})} problem aimed at constructing the least compliant structures from an elastic material, the constitutive field of which plays the role of design variable in the form of a tensor valued measure lambda supported in the design domain. Point-wise the constitutive tensor is referred to a given anisotropy class mathscr {H}, while the integral of a cost c(lambda ) is bounded from above. The convex p-homogeneous elastic potential j is parameterized by the constitutive tensor. The work puts forward the existence result and shows that the original problem can be reduced to the Linear Constrained Problem (mathrm {LCP}) known from the theory of optimal mass distribution by G. Bouchitté and G. Buttazzo. A theorem linking solutions of {(text {FMD})} and (mathrm {LCP}) allows to effectively solve the original problem. The developed theory encompasses several optimal anisotropy design problems known in the literature as well as it unlocks new ones. By employing the derived optimality conditions we give several analytical examples of optimal designs.
Highlights
Under the term compliance of an elastic structure, the properties of which are characterized by a constitutive tensor field λ, we understand the value of the elastic energy induced by a Communicated by A
This renders C = C(λ) convex and weak-* lower semi-continuous, which shall in turn furnish the existence result: Theorem 1.1 Assuming that j satisfies (H1)–(H5) the Free Material Design problem (FMD) admits a solution λonce F is balanced
In the present subsection it will appear that a "mirror" relation may be established between the polar ρ0 and the conjugate function j∗, which will be fundamental for connecting two of minimization problems in this work: (P∗) and (3.9)
Summary
For a chosen cost function c : H → R+ by Free Material Design (FMD) we shall call the problem of finding the constitutive field being a tensor valued measure λ ∈ M( ; H ). The present work puts forward a far more general framework where (FMD) is a family of optimal design problems parameterized by: the elastic potential j, the cost function c, and the set of admissible Hooke tensors H. 2 for a fixed function u the properties of the elastic energy functional M( ; H ) λ → j λ, e(u) are established, including its upper semi-continuity in the weak-* topology proved in Proposition 2.2 This renders C = C(λ) convex and weak-* lower semi-continuous, which shall in turn furnish the existence result: Theorem 1.1 Assuming that j satisfies (H1)–(H5) the Free Material Design problem (FMD) admits a solution λonce F is balanced. The present work essentially adapts and generalizes the methods of (MOP) to provide a rigorous mathematical
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