Dimension reduction for elastoplastic rods and homogenization of elastoplastic lattices
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Date
2021
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Abstract
We derive effective equations for periodic lattices of linearly elastoplastic rods in
the limit of both infinitesimal periodicity and infinitesimal relative width of the
rods. For this derivation we use the method of evolutionary Γ-convergence for
quadratic rate-independent systems.
As a first step towards this goal we derive effective equations for a single
rod. After introducing appropriate scalings, the main difficulty lies in the proof
of Γ-convergence for the stored energy. For the study of periodic lattices we
then introduce the notion of periodic graph frameworks, discuss infinitesimal
rigidity properties of such frameworks and define a notion of two-scale
convergence. The stored energy of a lattice of rods ist just the sum of the
energies of the individual rods, coupled by boundary conditions at the nodes.
For this energy we again prove Γ-convergence. In the presence of volume loads
we observe qualitatively different behaviour depending on the relative rate of
convergence of the periodicity parameter and the thickness parameter.
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Keywords
Partial differential equations, Elastoplasticity, Dimension reduction, Homogenization