Heterogeneous Deformation Modes in Elastic Foams
Solid foams are light materials which may be employed in numerous applications. In packaging, for example, polymeric solid foams are very effective at shielding fragile products from the jolts associated with transportation and handling. In the aeronautical industry, foams are the most commonly used materials in the cores of structural panels. Elastic foams are widely used in car seats, where they are ideally suited to provide comfort to the occupant. (However, much research is need to elucidate how the seat interacts with the occupant in the event of a crash.) Closer to everyday life, many foodstuffs happen to be solid foams, too, and most research into the behavior of brittle solid foams has been aimed at characterizing the ``crunchiness'' of snacks and crackers. Solid foams also occur in the form of biological materials, among which cancellous bone has elicited perhaps the most interest.
Our interest is in elastic foams. Because their microstructures are composed of slender cells, elastic foams are highly nonlinear materials. Under service conditions they are usually in the large-strain range. They display hysteresis, path dependence and configurational transitions -a vast phenomenology.
Heterogeneous Modes: Experimental Observation
Subject an elastic foam to a uniform compressive stress: the spatial distribution of stretches will be spatially heterogeneous. We performed this experiment, and measured the displacements on the surface of the foam specimen using the Digital Image Correlation technique. The figure shows the results in the form of a contour plot of the vertical component of the displacement on the surface. The contours are loci of equal displacement. The compression is in the vertical direction.The darker regions, where the successive contours are closer to each other, are regions of relatively large stretch. The lighter regions are regions of relatively low stretch. The regions of high and low stretch form alternating, roughly horizontal strata. The deformation mode is spatially heterogeneous. Our aim is to understand these heterogeneous deformation modes.
Heterogeneous Modes: Their Anatomy
The figure shows a plot of the displacement distribution along one vertical line in the contour plot. In this plot the slope is a measure of the local stretch. The stratified nature of the stretch distribution is apparent. We denote the high- and low-stretch strata by H and L, respectively. The figure reveals that the distribution of local values of stretch is bimodal: the high-stretch strata are characterized by a single value of stretch, and the same is true of the low-stretch strata. We have measured similar displacement distributions when the foam was subjected to different values of overall applied stretch, and verified that both characteristic stretches remain invariant. This suggests that the characteristic stretches may represent material properties somehow related to the microstructure of the foam.
The Micromechanics of Cell Collapse
The figure shows a sequence of micrographs of a polyether polyurethane foam being compressed in the vertical direction. The sequence progresses from left to right and top to bottom. The microstructure of the foam is composed of open cells. In the conventional theory of elastic foams, these cells are assumed to buckle in the sense of Euler (bifurcation point). However, the micrographs indicate that the cells do not buckle in the sense of Euler; instead, they collapse by snap-through buckling (limit point).We have used the Kármán theory of rods to
model a single cell undergoing snap-through buckling. After homogenizing
the behavior of a single cell, we have computed strain energy functions,
and used these functions to study the energetics of compressed elastic
foams. As we shall see, the result is a straightforward characterization
of the heterogeneous deformation modes observed in experiments.
Energetics
When the density of a foam is relatively low, the attendant strain energy function is nonconvex. Then, we can use conventional tools of nonconvex analysis to conclude that the total energy of a foam specimen is minimized when the distribution of local stretch is bimodal, i.e., when the local stretch is everywhere equal to either of two characteristic values of stretch. The characteristic stretches correspond to specific configurational phases of the microstructure, which we have denoted by H and L. The overall applied stretch is accommodated by mixing appropriate volume fractions of H and L. As the applied overall stretch diminishes during a test, the volume fraction of H increases from 0 to 1; stretching occurs in the form of a phase transformation between the two configurational phases of the microstructure. The phase transformation takes place at a constant value of stress -the Maxwell stress. As we shall see in the following, these theoretical predictions are beared out by the experimental evidence. The characteristic stretches and the Maxwell stress can be determined using a simple graphical construction, as shown in the figure. Because they do not depend of the applied overall stretch, they can be construed as material properties.
Nucleation and Growth
The figure shows plots of the vertical displacement along a vertical line for a series of values of applied overall stretch. We measured the displacements using the digital image correlation technique. The stretch distribution is initially homogenous. Later, four narrow strata of the high-stretch phase nucleate and subsequently grow in thickness. Eventually, a fifth stratum nucleates, which in turn grows in thickness. Because the characteristic stretches are invariant, it is clear that a change in the overall stretch cannot be accommodated by changes in the local stretches. Instead, when the applied stretch changes, the volume fraction occupied by the high-stretch strata increases (or diminishes) at the expense of the volume fraction occupied by the low-stretch strata. This is precisely what the figure shows. The experimental results are therefore in agreement with the theoretical prediction of a phase tranformation.
Mechanical Response: Measurements and Predictions
The figure show the experimentally measured stress-overall stretch curves (dashed gray lines) for six elastic foams of different densities, and theoretical predictions (black lines). For each density the stress-stretch curve derives from a single, smooth strain energy function. When the density is lower than the critical density, the strain energy functions are nonconvex, the stress-stretch curve displays a constant-stress plateau with the values of the Maxwell stress, and the deformation modes are heterogeneous, composed of alternating strata of the two configurational phases of the microstructure. When the density is higher than the critical density, the strain energy functions are convex, the stress increases monotonically, and the deformation modes are homogeneous.Gioia, G., Wang, Y., and Cuitiño, A.M.
The Energetics of Heterogeneous Deformation in Open-Cell Solid Foams
Proc. Royal Soc. London A 457, 1079-1096, 2001.
[PDF 1.1 Mb]
Wang, Y., Gioia, G., and Cuitiño, A.M.
The Deformation Habits of Compressed Open-Cell Solid Foams
J. Engrg. Mater. Technol. 122, 376-380, 2000.
[PDF 180 kb]