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Size effects in cellular solids

(2007) Tekoğlu, Cihan

The mechanical properties of metal foams (and other cellular solids) depend on the properties of the metal that they are made from, on their relative density, and on the cell topology (i.e., cell size, cell shape, open or closed cell morphology, etc.). The cell size of commercially available metal foams is about 1 to 10 mm. This is on the order of the smallest structural length of specimens in many applications. In such cases, the individual response to a load differs significantly from one cell to another, and the fundamental assumption of the classical continuum theory that the (physical, chemical, mechanical, etc.) properties of a material are uniformly distributed throughout its volume fails. Another situation where the classical continuum theory loses its accuracy is when the characteristic wavelength of loading is comparable to the cell size. An important technological consequence of this is the occurrence of size effects. The term "size effect" designates the effect of the macroscopic (sample) size, relative to the cell size, on the mechanical behaviour. In the last decade evidence of this appeared in a number of experimental studies (see section 1.4). To theoretically account for size effects, one may take the cellular morphology into account by discretely modelling each cell wall and/or cell face. This allows for an accurate representation of the microstructural deformation mechanisms, the bending and stretching of cell walls and faces. Such a microstructural model can predict how the overall (macroscopic) response is related to the microstructural parameters. In view of size effects, the most important feature is that it incorporates, in a physically sound manner, the material length scale in the problem, i.e. the cell size. However, such a discrete model can become computationally expensive for complex (random) microstructures, especially in three dimensions. Another approach is to use a generalized continuum theory in which many microstructural details are averaged out, but in which a "characteristic" length scale is retained. The goal of this thesis is twofold:
1) To explore the physical mechanisms that are responsible for the size-dependent elastic behaviour of cellular solids by using a discrete microstructural model.
2) To assess the capability of generalized continuum theories to capture size effects through a careful comparison with the discrete simulations.
The first chapter thesis introduces the experimental evidence for the size effects, and gives an historical overview of the generalized continuum theories used to model these size effects. In chapter 2, we use two-dimensional beam networks to mimic real (three-dimensional) foams, which allow us to account for the discreteness of their microstructure. We perform simple shear, uniaxial compression, and pure bending tests on a large variety of samples, and calculate the change in the macroscopic mechanical properties corresponding to a change in size. We close the chapter with a summary of the size effects that we observed in our calculations and discuss the possible mechanisms behind these size effects.
Chapter 3 uses the micropolar theory to capture the size effects observed in chapter 2. We fit the elastic constants of the micropolar continuum theory by comparing the analytical solution of the simple shear problem with the discrete analyses, in terms of the best agreement in the macroscopic shear stiffness of the samples. We develop a strain mapping procedure and evaluate the performance of the fitted micropolar constants in predicting the local deformation fields, the microrotations and shear strains. Finally, we solve the pure bending problem analytically for the micropolar theory and close the chapter with a discussion on the limitations of the Cosserat-type theories.
In chapter 4, we propose a generalized continuum theory (strain divergence theory), which associates energy to the divergence of strain. We derive the equilibrium equations and the boundary conditions for the strain divergence continuum, and develop a finite element implementation of the theory. We solve the simple shear and the pure bending problems analytically and compare the solutions with the discrete calculations, as well as with the analytical solutions for the couple stress theory.
Chapter 5 explores the strain concentration problem around a cylindrical hole in a field of uniaxial tension. First, we perform discrete calculations on samples with different hole sizes and show the effect of the hole size on the strain distribution near the hole. Then we compare the discrete analyses with the analytical solutions for the classical, couple stress and strain divergence theories.
Finally, in Chapter 6, we summarize the size effects that we observed in the mechanical behaviour of the two dimensional cellular solids, and we compare the different generalized continuum theories with respect to their ability in capturing size effects.