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Hybrid finite elements for axially loaded elasto-plastic bars
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This work reports on the formulation, implementation and validation of hybrid finite
elements for elasto-plastic axially loaded bars, under quasi-static conditions.
The equations governing the axially loaded bars are defined for both elastic and elastoplastic
ranges. Two different hardening models are considered, kinematic and isotropic.
The solution of the governing equations involves their discretization in time and space.
The discretization in time is made by expanding in Euler series the time variation of
the involved quantities and the integration in space is performed using the hybrid finite
element method.
Independent approximations of the displacement and plastic strain fields are made in
the domain of each finite element using Chebyshev polynomials. Unlike conforming
displacement finite elements, the bases are hierarchical and not linked in any way to the
nodes of the mesh.
The hybrid finite element formulation is derived by enforcing the weak form of the governing
equations using the Garlerkin method. The computational implementation of the
formulation is developed in the Matlab environment. The implementation offers considerable
flexibility for the definition of the structure and its loads, the time steps and the
finite element mesh.
To validate the implementation and assess the convergence properties of the hybrid formulation,
a problem with known analytic solution is used. The displacement and stress
solution errors are measured and their reduction rates under mesh (h-), basis (p-) and
time step (t-) refinements are computed to understand their relative effect on the quality
of the solution. A second problem with a higher complexity level is used to illustrate
the performance of the formulation when confronted to multiple loading and unloading
cycles, that lead to partial and total yielding under traction and compression regimes.
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Elasto-plastic problems Axially loaded rods Finite elements method Hybrid finite elements