SUMMARYIn this paper, we present a homogenization approach that can be used in the geometrically nonlinear regime for stress-driven and strain-driven homogenization and even a combination of both. Special attention is paid to the straightforward implementation in combination with the finite-element method. The formulation follows directly from the principle of virtual work, the periodic boundary conditions, and the Hill-Mandel principle of macro-homogeneity. The periodic boundary conditions are implemented using the Lagrange multiplier method to link macroscopic strain to the boundary displacements of the computational model of a representative volume element. We include the macroscopic strain as a set of additional degrees of freedom in the formulation. Via the Lagrange multipliers, the macroscopic stress naturally arises as the associated 'forces' that are conjugate to the macroscopic strain 'displacements'. In contrast to most homogenization schemes, the second Piola-Kirchhoff stress and Green-Lagrange strain have been chosen for the macroscopic stress and strain measures in this formulation. The usage of other stress and strain measures such as the first Piola-Kirchhoff stress and the deformation gradient is discussed in the Appendix.
The cylindrical orthotropy, inherent timedependency response, and variation between and within samples make the stiffness characterisation of wood more challenging than most other structural materials. The purpose of the present study is to compare static loading with resonant ultrasound spectroscopy (RUS) and to investigate how to combine the advantages of each of these two methods to improve the estimation of the full set of elastic parameters of a unique sample. The behavior of wood as an orthotropic mechanical material was quantified by elastic engineering parameters, i.e. Poisson's ratios and Young's and shear moduli. Recent and waterlogged archaeological oak impregnated with polyethylene glycol (PEG) from the Vasa warship built in 1628 was in focus. The experimental results were compared, and the difference between RUS and static loading was studied. This study contributes additional information on the influence of PEG and degradation on the elastic engineering parameters of wood. Finally, the shear moduli and Poisson's ratios were experimentally determined for Vasa archaeological oak for the first time.
This contribution presents a novel and versatile approach to geometrically nonlinear topology optimization by combining the level-set method with the element connectivity parameterization method or ECP. The combined advantages of both methods open up the possibility to treat a wide range of optimization problems involving complex physical and/or geometrical nonlinearities in a general and elegant manner. The level-set method features shape optimization on a fixed mesh, leading to intrinsically black-andwhite designs. This approach allows a clear description of location and orientation of the interface, whereas topological changes can still be handled easily. A popular concept used in conventional level-set methods is to map the levelset function to volume-fraction design variables for every element of a finite element mesh. The resulting element density variables are then used to scale the Young's modulus in each element using the Ersatz material approach. In this work we employ a modified material interpolation method, in which the element density variables, based on a per-element integration of a regularized Heaviside operator applied to the level-set function, are used as element
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