Date of Award

May 2019

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mechanical Engineering

Committee Member

Joshua Summers

Committee Member

Georges Fadel

Committee Member

Nicole Coutris

Committee Member

Gang Li


When the material properties of cellular materials are tested, small samples may have different apparent material properties than large samples. This difference is known as a size effect. This work examines size effects in periodic thin walled cellular materials using a discrete beam element lattice model.

Chapter 2 uses this lattice model to characterize size effects for a variety of lattice topologies and boundary conditions. It also compares those size effects to results coming from a nominally equivalent micropolar model. Micropolar elasticity extends classical elasticity to incorporate size dependent behaviors and can therefore capture size effects in a continuum model. The micropolar model used here typically under predicts the size effects seen in the lattice model by an order of magnitude. The lattice size effects are examined for patterns and the size effect patterns found can be explained by the shape of the free edges, and by the specifics of how material is distributed within the material domain. These are causes for size effects that are not captured in the micropolar model.

Chapter 3 examines two hypotheses taken from the literature. The literature has attributed stiffening size effects to the local beam bending behavior of lattice materials; there is an additional stiffness connected to the rotation of the beam nodes at lattice vertices, that causes a stiffening size effect. This work decomposes the strain energy of the beams in the lattice into energy from axial stretching and beam bending, and shows that size effects are connected to bending strain energy in certain situations.

Other literature has shown a softening size effect in stochastic foams caused by damaged or incomplete cells on the free surfaces. This work uses a continuum like strain map of the lattice model to show that these edge softening effects can appear in periodic cellular materials when the surface cells are neither damaged nor incomplete. This effect is only observed for certain lattice topologies and is quantified and connected to a global size effect. In conjunction with the beam bending size effect, this edge effect is able to explain the origin of size effects for a variety of lattice topologies and boundary conditions.

Chapter 4 examines the ability of micropolar elasticity to predict size effects in periodic cellular materials in both bending and shear. It shows that a set of micropolar material properties that predict size effects accurately in bending, is inaccurate for shear. A different set of properties does well for shear, but poorly for bending. This suggests that, for periodic cellular materials, size effects in shear and bending arise from different mechanisms.

Chapter 4 presents a novel mechanism causing size effects in periodic cellular materials in bending, called a material distribution effect. In bending, material far from the neutral axis contributes more to stiffness than material close to the neutral axis. A sample with only a few unit cells can have a relatively large amount of its material close to the neutral axis, or far away, depending on topology and choice of unit cell. A sample with many cells must have its material spread more evenly. This can cause either a stiffening or softening size effect. This work derives formulas to predict the magnitude and direction of these size effects, and shows that these formulas are able to predict size effects for a variety of different cellular materials in different types of bending boundary conditions.

These chapters provide an explanation of the origin of size effects in periodic cellular materials for a variety of boundary conditions and lattice topologies.